i
c
Austen
Lamacraft, 2010
Atomic Condensates
Collective Phenomena in Ultracold Gases
Austen Lamacraft
Contents
List of illustrations
List of tables
Part I
vi
viii
Fundamentals
1
1
1.1
Quantum statistics
Quantum indistinguishability: bosons and fermions
1.2
1.3
Example: particles on a ring
Strong interactions and a surprise
2
2.1
Ideal quantum gases
Occupation numbers in the ideal gas
2.2
Ideal gases of bosons and fermions
1.1.1
2.1.1
2.2.1
2.2.2
2.2.3
2.2.4
3
3.1
3.2
One particle operators
Correlation functions
Two particle operators and interactions
Number and phase as conjugate variables
A double slit experiment with condensates
A quantum measurement perspective
The experimental system
Atomic properties
4.1.1
4.1.2
4.1.3
4.2
Density of states
The Fermi gas
The Bose gas
Bose gas in an harmonic trap
Interference of condensates
3.3.1
3.3.2
3.3.3
4
4.1
Canonical approach
Second Quantization
Creation and annihilation operators
Representation of operators
3.2.1
3.2.2
3.2.3
3.3
From two to many
3
3
5
7
9
11
11
11
14
14
16
18
20
23
23
28
28
32
35
36
37
38
42
45
46
46
46
47
50
Boson or Fermion?
The Zeeman effect
Excited states and polarizabilty
Trapping and imaging
iii
Contents
iv
4.2.1
4.2.2
4.2.3
4.3
Magnetic traps
Optical traps
Imaging
Interactions
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
Effective interaction between like species
Interaction between species
Three body recombination
The Feshbach resonance
Dipolar interactions
Part II
Condensates of bosons and fermions
51
51
52
52
53
56
57
57
60
63
5
5.1
5.2
5.3
5.4
Static properties of Bose condensates
The Gross–Pitaevskii approximation
Condensates in trap potentials
Rotational states: A simple example
A single vortex
65
65
67
67
69
6
6.1
6.2
6.3
6.4
6.5
Dynamics of condensates
Time-dependent Gross–Pitaevskii theory
Bogoliubov–de Gennes equations
Interlude: structure factors and sum rules
Hydrodynamics of condensates
Hydrodynamic description
70
70
71
74
75
76
7
7.1
7.2
Bose condensation and superfluidity
BEC and off-diagonal long-range order
Superfluidity defined
7.3
Experimental status
78
78
79
80
82
84
8
8.1
8.2
8.3
8.4
8.5
Bogoliubov theory
Pair approximation and ground state energy
Interlude: Bogoliubov transformations and squeezed states
Structure of the ground state and excitations
Generalization to the trapped case
Application: condensate fluctuations
86
86
89
89
92
92
9
9.1
9.2
Fermi superfluids
Fermionic condensates
The BCS theory
7.2.1
7.2.2
9.2.1
9.2.2
9.2.3
Non-classical rotational intertia
Metastability of superflow and vortices
The pairing hypothesis
The BCS-BEC crossover
Quasiparticle excitations
9.3
BCS theory from the Bogoliubov transformation
9.4
The effect of ‘magnetization’
9.3.1
Effect of Temperature
93
93
95
95
100
103
104
105
106
Contents
9.4.1
9.4.2
Sarma state
Magnetization in the BCS-BEC crossover
Part III
Low-dimensional systems
v
108
109
113
10
10.1
10.2
10.3
Quantum Hydrodynamics
Hamiltonian and commutation relations
Mode expansion
Correlation functions
115
115
116
117
11
11.1
Quantum Fluids in One Dimension
Bose fluids in one dimension: the Tonks gas
119
119
Part IV
123
Part V
Appendix A
Notes
Atoms in optical lattices
Multicomponent gases and atomic magnetism
Symbology
125
127
128
Illustrations
1.1
1.2
2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
4.1
4.2
4.3
4.4
5.1
7.1
7.2
7.3
8.1
9.1
Particles distinguished by their trajectories in classical mechanics
Nodal surfaces xi = xj for three fermions. Because of the periodic boundary
conditions, the three dimensional space of particle coordinates is divided into
two regions, corresponding to the even (123, 231, 312) and odd (132, 321,
213) permutations.
The Fermi function
The Bose distribution
Counting states in k-space
Illustration of the various factors in the integrand of Equation (2.14) for the
Fermi case at finite temperature
From [1]
Single particle density matrix for the Fermi gas.
Density correlation function for the Fermi gas
From Ref. [4]
From Ref. [4]
Beam splitter configuration with two condensates
Magnetic field dependence of atomic states of an atom with J = 1/2, I = 3/2
Absorption image of around 7 × 105 atoms just above, at, and below the
Bose- Einstein condensation temperature. The gas is allowed to expand for
6 ms. Reproduced from [4]
Sketch of the interatomic potentials for two alkali atoms with valence electrons in singlet and triplet states
A Feshbach resonance is caused by hybridization of a closed channel bound
state with the open channel.
From [3]
Thought experiment used to define the superfluid fraction. An annulus of
fluid is rotated slowly
E(L/N ~), showing metastable minima.
Vortices in an atomic gas of 23 Na. Reproduced from Ref. [? ].
Momentum transfer per particle for a BEC (open circles) and expanded cloud
(closed circles), for a given value of the wavevector q = q1 − q2 in Bragg
spectroscopy. Note how the peak is suppressed and shifted to higher energies
in the BEC. Broadening is due to the inhomogeneity of the condensate.
Reproduced from Ref. [? ]
Schematic phase diagram of a system of two species of fermions (equal in
number). The dashed line represents a naive expectation, without accounting
for the Cooper phenomenon
vi
3
8
13
14
15
18
21
31
34
41
42
43
47
53
54
58
68
80
83
84
91
94
Illustrations
9.2
9.3
9.4
9.5
9.6
9.7
9.8
11.1
Anderson spin configurations and the associated distribution functions for
the free fermi gas (top) and the BCS state (bottom).
Variation of gap and chemical
potential. Also plotted is the result for the
p
gap on the BEC side 16/(kF as )/π. Note the exponential dependence of
the gap on the BCS side, consistent with the analytic result Eq. (9.28)
Tc in the mean-field theory (solid line), compared with the result of a treatment that smoothly interpolates to TBEC (dashed). Reproduced from Ref. [1]
Schematic plots of ground state energy versus ∆ for different h.
The tricritical point in the (T, h) plane. Reproduced from Ref. [3]
Magnetization versus field at zero (black line) and finite temperature (red
line), and the resulting region of phase separation (PS) in the (T, m) plane.
a) Schematic phase diagram of the zero temperature BCS-BEC crossover
with magnetization. The red dot marks a tricritical point that separates
the trivial second-order phase transition when the number of pairs N↓ /2
goes to zero from the phase separated region. Point A is where the normal
state becomes fully polarized, and point B is where the free energy of the
superfluid state with gap ∆s (kF aS ) is equal to that of the normal state in
magnetic field h = ∆s . b) The same, but at finite temperature. Now the
second tricritical point on the BCS side is visible.
The curve shows the function e(γ) obtained in Ref. [? ] for the δ-function
Bose gas in 1D, in terms of which the energy is E = N n2 ~2 e(γ)/2m.
vii
98
102
107
109
110
110
111
120
Tables
A.1
Notation used in the text
127
viii
Part I
Fundamentals
1
1
Quantum statistics
1.1 Quantum indistinguishability: bosons and fermions
The phenomena that we will discuss involve the quantum behavior of large numbers of identical particles. Thus it is important to realize at the outset that
classical and quantum mechanics differ in a very fundamental way in their treatment of particles that cannot be distinguished by any measurement of their mass,
electric charge, etc.. This is because in classical mechanics any labeling of the particles that we may imagine at one instant — by noting the positions of each, for
example — can in principle be maintained at an subsequent times by following
the particles’ trajectory. In quantum theory, however, the notion of trajectory
has no place. Thus even if each particle had started in its own special box, there
is no trajectory stretching backward in time that ties the present day position of
the particle to that box (Figure 1.1).
Let’s now focus on a pair of identical particles. The above discussion suggests
that quantum mechanics should make identical predictions about the system
irrespective of which particle we decide to label 1 and which we label 2. Since the
probabilities predicted by quantum mechanics are related to the modulus squared
Figure 1.1 Particles distinguished by their trajectories in classical
mechanics
3
Quantum statistics
4
of the wavefunction, this tells us1
|Ψ(y, x)|2 = |Ψ(x, y)|2
(1.1)
More precisely, Ψ(x, y) and Ψ(y, x) must correspond to the same quantum state,
meaning that they can differ only by a (constant) phase
Ψ(y, x) = eiθ Ψ(x, y)
What can we say about θ? Take another two positions r01 , and r02 , then
Ψ(y0 , x0 ) = eiθ Ψ(x0 , y0 ),
but there’s nothing to stop us taking x = y0 , and y = x0 , in which case we can
combine these two expressions to give
Ψ(x, y) = e2iθ Ψ(x, y)
that is eiθ = ±1 and the wavefunction for two particles is either symmetric or
antisymmetric.
A more formal approach is to consider the exchange operator P12 that exchanges the two arguments
P12 Ψ(x, y) = Ψ(y, x)
which is evidently a linear operator that furthermore squares to give the iden2
tity P12
= 1. The eigenvalues of P12 are therefore ±1, with the corresponding
eigenstates being symmetric or antisymmetric respectively.
Exercise 1.1.1 Confirm that a state satsifies Equation (1.1) only if it is an
eigenstate of P12 , and is therefore symmetric or antisymmetric.
The exchange operator commutes with the Hamiltonian of a pair of identical
particles. As an example, consider the Hamiltonian for a pair of electrons
H=−
e2
~2 2
∇x + ∇2y +
2me
|x − y|
(1.2)
evidently it doesn’t matter whether we apply P12 before or after H. Thus [H, P12 ] =
0 and basic ideas of quantum mechanics tell us that
1. Eigenstates of the Hamiltonian have definite exchange symmetry2 .
2. The symmetry of the wavefunction is a constant of the motion.
1
2
We ignore spin for now.
In a real two-electron system overall antisymmetry is guaranteed by the appropriate choice of spin
wavefunction.
1.1 Quantum indistinguishability: bosons and fermions
5
1.1.1 From two to many
What changes when we consider systems of N identical particles? The wavefunction is now a function of the N coordinates Ψ(r1 , r2 , . . . , rN ), and we can
consider an exchange operator Pij that swaps the ith and j th arguments. The
same physical reasoning as before singles out eigenstates of the {Pij }. Note that
[P12 , P23 ] 6= 0, and thus one might worry whether it is mathematically possible
to have a simultaneous eigenstate of all the Pij . We see straightaway that a totally symmetric state with all eigenvalues equal to +1 is allowed. What about a
totally antisymmetric state with all eigenvalues −1? This too is possible for the
following reason. Any given permutation may be written in many different ways
as a product of exchanges (see the next exercise for an example). For a given permutation, however, these different possibilities either involve only even numbers
of exchanges, or only odd numbers, a fact we’ll prove below.
Exercise 1.1.2 By showing that P12 P23 P12 = P13 = P23 P12 P23 prove that a
simultaneous eigenstate of all the Pij must have all eigenvalues +1 or all
−1, that is, be totally symmetric or totally antisymmetric.
All of this shows that any given species of quantum particle will fall into one of
two fundamental classes: symmetric bosons and antisymmetric fermions, named
for Bose and Fermi respectively (the whimsical terminology is Dirac’s). The distinction works equally well for composite particles, providing we ignore the internal degrees of freedom and discuss only the center of mass coordinate. All
matter in the universe is made up of fermions: electrons, quarks, etc., but you
can easily convince yourself that an even number of fermions make a composite
boson (e.g. a 4 He atom with two electrons, two neutrons and two protons) and
an odd number make a composite fermion (3 He has one fewer neutron, which in
turn is made up of 3 quarks).3
If we dealt with distinguishable particles, the wavefunction of a pair of particles
in states ϕ1 and ϕ2 would be
Ψ(r1 , r2 ) = ϕ1 (r1 )ϕ2 (r2 ).
(1.3)
Accounting for indistinguishability, we have either
1
Ψ(r1 , r2 ) = √ [ϕ1 (r1 )ϕ2 (r2 ) ± ϕ2 (r1 )ϕ1 (r2 )].
2
(1.4)
with the upper sign for bosons and the lower for fermions. Note in particular
that when ϕ1 = ϕ2 the fermion wavefunction vanishes. This illustrates the Pauli
exclusion principle, that no two identical fermions can be in the same quantum
state. There is no such restriction for bosons.
The Hamiltonian of a system of N identical noninteracting particles is a sum
of N identical single particle Hamiltonians, that is, with each term acting on a
3
I am being a little glib here: ‘The solution with antisymmetrical eigenfunctions, though, is probably
the correct one for gas molecules, since it is known to be the correct one for electrons in an atom,
and one would expect molecules to resemble electrons more closely than light-quanta.’ [1]
Quantum statistics
6
different particle coordinate
H=
N X
i=1
~2 2
−
∇ + V (ri )
2m i
(1.5)
where m is the particle mass, and V (ri ) is a potential experienced by the particles.
Let’s denote the eigenstates of the single particle Hamiltonian by {ϕα (r)}, and
the corresponding eigenenergies by {Eα }, where α is a shorthand for whatever
quantum numbers are used to label the states. A set of labels {αi } i = 1, 2, . . . N
tells us the state of each of the particles. P
Thus we can write an eigenstate of N
N
distinguishable particles with energy E = i=1 Eαi as4
|Ψα1 α2 ···αN i = ϕα1 (r1 )ϕα2 (r1 ) · · · ϕαN (r1 )
(1.6)
A general state will be expressed as a superposition of such states, of course. As
we’ve just discussed, however, we should really be dealing with a totally symmetric or totally antisymmetric wavefunction, depending on whether our identical
particles are bosons or fermions. To write these down we introduce the operators
of symmetrization and antisymmetrization
1 X
1 X
S=
P,
A=
sgn(P )P
(1.7)
N! P
N! P
The sums are over all N ! permutations of N objects, P denotes the corresponding
permutation operator, and sgn(P ) is the signature of the permutation, equal to
+1 for permutations involving an even number of exchanges, and −1 for an odd
number. This allows us to write the totally symmetric and totally antisymmetric
versions of Equation (1.6) as
s
N!
S ϕα1 (r1 )ϕα2 (r1 ) · · · ϕαN (r1 )
α Nα !
A
√
Ψα α ···α = N !A ϕα1 (r1 )ϕα2 (r1 ) · · · ϕαN (r1 )
1 2
N
S
Ψα α ···α =
1 2
N
Q
(1.8)
The normalization factor in the boson case involves the occupation numbers {Nα }
giving the number of particles in state α. In the fermion case each Nα is either
0 or 1 so the prefactor simplifies. Since the order of the α indices is irrelevant in
the boson case, and amounts only to a sign in the fermion case, states based on
a given set of single particle states are more efficiently labeled by the occupation
numbers. In terms of these numbers the total energy is
E=
N
X
i=1
4
Eαi =
X
Nα Eα
(1.9)
α
We will frequently switch between the wavefunction (ϕ(x)) and bra-ket notation (|ϕi). In the latter
notation the product wavefunction in Equation (1.3) is written |ϕ1 i |ϕ1 i
1.2 Example: particles on a ring
7
Exercise 1.1.3 Find a. the totally antisymmetric state of three fermions in
states ϕ1 , ϕ2ga , and ϕ3 and b. the totally symmetric state of three bosons,
with two in state ϕ1 and one in state ϕ2 .
Exercise 1.1.4 Verify that the normalization factors in Equation (1.8) are correct.
A more formal way of putting things is as follows. We first consider the space
spanned by states of the form Equation (1.6). Then we introduce the operators
S and A, noting that S 2 = S and A2 = A. In other words, there’s no point
symmetrizing or antisymmetrizing more than once (we say that the operators
are idempotent). Any eigenvalue of one of these operators is therefore either one
or zero. The states with S = 1 are the symmetric states, and those with A = 1
are antisymmetric. You can easily convince yourself that if a state has one of S
or A equal to one, the other is zero. This defines symmetric and antisymmetric
subspaces, consisting of the admissible boson and fermion wavefunctions.
Note that the fermion wavefunction takes the form of a determinant (usually
called a Slater determinant)
ϕα1 (r1 ) ϕα1 (r2 ) · · · ϕα1 (rN ) ϕα2 (r1 )
A
1
·
·
·
·
·
·
·
·
·
Ψα α ···α = √ (1.10)
1 2
N
···
···
· · · N! · · ·
ϕαN (r1 )
···
· · · ϕαN (rN )
The vanishing of a determinant when two rows or two columns are identical means
that the wavefunction is zero if two particle coordinates coincide (ri = rj ), or if
two particles occupy the same state (αi = αj )
1.2 Example: particles on a ring
Let’s consider perhaps the simplest many particle system one can think of: noninteracting particles on a ring, or equivalently in one dimension with periodic
boundary conditions. If the ring has circumference L, the single particle eigenstates are
1
(1.11)
ϕn (x) = √ e2πinx/L , n = 0, ±1, ±2, . . .
L
2
2
h n
with energies En = 2mL
2 . Let’s find the N particle ground state. For bosons every
particle is in the state ϕ0 with zero energy: N0 = N . Thus (ignoring normalization)
ΨS (x1 , x2 , . . . xN ) = 1
That was easy! The fermion case is harder. Since the occupation of each level is
at most one, the lowest energy is obtained by filling each level with one particle,
starting at the bottom. If we have an odd number of particles, this means filling
the levels with n = −(N −1)/2, −(N −3)/2, . . . , −1, 0, 1 . . . (N −1)/2 (for an even
number of particles we have to decide whether to put the last particle at n =
Quantum statistics
8
1.0
x2
0.5
0.0
1.0
x3
0.5
0.0
0.0
0.5
x1
1.0
Figure 1.2 Nodal surfaces xi = xj for three fermions. Because of the
periodic boundary conditions, the three dimensional space of particle
coordinates is divided into two regions, corresponding to the even (123, 231,
312) and odd (132, 321, 213) permutations.
±N/2). Introducing the complex variables
in Equation (1.10) becomes
−(N −1)/2 −(N −1)/2
z1
−(N −3)/2 z2
z
···
1
···
·
··
z (N −1)/2
·
··
1
zi = e2πixi /L , the Slater determinant
−(N −1)/2 · · · zN
···
···
···
···
(N −1)/2
· · · zN
(1.12)
Let’s evaluate this complicated looking expression in a simple case. With three
particles we have
−1 −1 −1 z1
z2
z3 z1 z2 z3 z1 z2 z3
1
1
1 =
−
+
−
+
−
(1.13)
z1
z2
z3 z2 z1 z1 z3 z3 z2
r
r r
r r
r z3
z1
z1
z2
z2
z3
=
−
−
−
(1.14)
z1
z3
z2
z1
z3
z2
π[x1 − x2 ]
π[x3 − x1 ]
π[x2 − x3 ]
∝ sin
sin
sin
(1.15)
L
L
L
The vanishing of the wavefunction when xi = xj (see Figure 1.2) is consistent
with the Pauli principle. You should check that additionally it is periodic and
totally antisymmetric.
1.3 Strong interactions and a surprise
9
Exercise 1.2.1 Show that this generalizes for any (odd) N so that Equation (1.12) is proportional to
N
Y
π[xi − xj ]
sin
(1.16)
L
i<j
You will need the vandermonde determinant
1
1
···
1 N
z1
z2
· · · · · · Y
=
(zi − zj )
z12
z22
· · · · · · i<j
N −1 · · · z1N −1 z2N −1 · · · zN
(1.17)
which can be proved in a variety of ways. Note that proving directly that
Equation (1.16) is an eigenstate of the Hamiltonian is not easy!
Exercise 1.2.2 Check carefully that the above result is periodic and totally
antisymmetric. Note that explicitly finding a totally antisymmetric function of N variables is tantamount to proving the statement of Section 1.1.1
that a given permutation can be written in terms of only even numbers of
exchanges, or only odd numbers, not both.
Let’s take the opportunity to introduce some terminology. The wavevector of
the last fermion added is called the Fermi wavevector and denoted kF . In this case
. The corresponding momentum pF = ~kF is the Fermi momentum;
kF = (N −1)π
L
2 2
kF
the corresponding energy EF = ~2m
the Fermi energy, and so on.
1.3 Strong interactions and a surprise
The ground state of the non-interacting Bose gas turned out to be extremely dull.
To make things interesting, we have to add interactions between the particles.
Let’s then consider the Hamiltonian
H=
N
X
i=1
N
−
X
~2 ∂ 2
+
g
δ(xi − xj )
2m ∂x2i
i<j
(1.18)
The second term describes an interaction between pairs of particles that is nonzero only when the coordinates of a pair coincide. We usually speak of a δfunction interaction. Note that the strength of the interaction g has units of
(Energy) × (Length). The sum in the interaction term in Equation (1.18) ensures
that we count each pair of particles once.
It is a standard problem in quantum mechanics to work out the boundary
conditions on the wavefunction for a δ-function potential. The present case only
amounts to a small modification of the argument, leading to the result
xj + mg ∂i Ψxj − = 2 Ψxi =xj
(1.19)
~
that is, the jump in the derivative with respect to the position of one particle
10
Quantum statistics
when it coincides with another is determined by the wavefunction at the point
where they coincide.
In general this is a hard but not impossible problem to solve. We will return to
the general case later, but for now we are going to note a remarkable simplification
that occurs in the limit of infinitely strong repulsion. If the jump in the derivative
is going to be finite, it is necessary for Ψxi =xj → 0 as g → ∞5 . Thus we seek a
function that a. solves the Schrödinger equation whenever the particle positions
are non-coincident and b. goes to zero for coincident positions. By a stroke of
luck, we have such a function, Equation (1.16). The only problem is that it is
antisymmetric rather than symmetric, but that is easily fixed
N Y
π[x
−
x
]
i
j
sin
(1.20)
L
i<j
Taking the modulus has no effect on wavefunction’s validity as a solution to the
Schrödinger equation when no coordinates are coincident.
Exercise 1.3.1 By considering the expectation value of the energy, show that
taking the modulus has no effect on the kinetic energy, and explain why the
potential energy of the gas is zero in the g → ∞ limit. Note that a gas of
fermions is completely oblivious to the presence of the δ-function interaction
as a consequence of the Pauli principle.
This strongly interacting 1D gas is known as a Tonks-Girardeau gas [2]. Many
of its properties can be understood in terms of the free fermion gas, as we’ll see
in more detail later. The emergence of simplicity from (presumed) complexity is
one of the recurring pleasures of many body physics.
References
[1] Dirac, PAM. 1926. On the theory of quantum mechanics. Proceedings of the Royal Society of
London. Series A, Containing Papers of a Mathematical and Physical Character, 661–677.
[2] Girardeau, M. 1960. Relationship between Systems of Impenetrable Bosons and Fermions
in One Dimension. Journal of Mathematical Physics, 1(6), 516–523.
5
What happens as g → −∞? Think about the case of two particles.
2
Ideal quantum gases
2.1 Occupation numbers in the ideal gas
If we restrict ourselves to an ideal gas, the total energy is just the sum of the
kinetic energy of each particle. Usually these will be the ‘particle in a box’ energy
levels familiar from our earlier calculation of the partition function. We can write
this energy in terms of the occupation numbers of each energy level
X
Etot =
Nα Eα .
α
That is, Nα is the number of particles in the αth energy level, of energy Eα . In
the past we might have instead thought of Etot as the energy of particle 1 plus
the energy of particle 2, and so on. The reason for switching to the occupation
number picture is that these numbers uniquely specify the quantum state because
the particles are indistinguishable. That is, it doesn’t make any sense to ask at
which energy particle number 1 sits.
If we now consider a system in thermal equilibrium, the total internal energy
can be written
X
U=
hNα i Eα .
α
where the average is to be taken over the Boltzmann distribution.
2.1.1 Canonical approach
We first evaluate the average occupation numbers in the canonical ensemble.
Fermions
The average occupation of an energy level (call it 1) is
hN1 i =
1
1
1 X X
· · · N1 e−β(N1 E1 +N2 E2 +··· ) ,
Z(N ) N =0 N =0
1
subject to N1 + N2 + · · · = N.
2
(2.1)
Here β = kB1T , where T is the temperature and kB is Boltzmann’s constant.
Z(N ) is the normalization factor or partition function. Its has the form of the
same nested sum, but without a factor N1 .
11
Ideal quantum gases
12
Now we focus on the first sum. When N1 = 0, the summand vanishes: the only
contribution comes from N1 = 1, so we can write
hN1 i =
1
X
1
e−βE1
· · · e−β(N2 E2 +··· ) ,
Z(N )
N =0
subject to N2 + · · · = N − 1.
2
Now the clever part. Rewrite this sum as
hN1 i =
1
1
X
X
1
e−βE1
· · · (1−N1 )e−β(N1 E1 +N2 E2 +··· ) ,
Z(N )
N =0 N =0
1
subject to N1 +N2 +· · · = N −1.
2
P1
We’ve put the sum N1 =0 back in, but with a factor (1 − N1 that ensures only
the N1 = 0 term contributes. Why do this? Because the right hand side now looks
like the average of 1 − N1 , but with one fewer particle. More precisely, we have
hN1 iN =
Z(N − 1) −βE1
e
h1 − N1 iN −1
Z(N )
(2.2)
where we’ve now added subscripts to remind us that the averages on the left and
right hand sides are different. Now we make an assumption: that in a macroscopic system they in fact coincide: hN1 iN = hN1 iN −1 . This allows us to solve
Equation (2.2) for hN1 i
1
hN1 iF = β(E1 −µ)
e
+1
Where the subscript emphasizes that we are dealing with Fermions, and we have
Z(N )
identified e−βµ = Z(N
. Taking logs, you can see why
−1)
∂F Z(N )
= F (N ) − F (N − 1) =
Z(N − 1)
∂N V,T
Here F is the Helmholtz free energy and the relation µ = ∂F
is from classical
∂µ V,T
thermodynamics . Obviously there is nothing special about the state we have
chosen, so the same result holds for a state of any energy: hNα i = f (Eα ), where
the function
1
f () = β(−µ)
(2.3)
e
+1
µ = −kB T ln
is called the Fermi–Dirac distribution or just Fermi function (see Figure 2.1).
Note that it is always less than one, as it must be, and only achieves that value
asymptotically as the energy goes to far below the chemical potential (on a scale
set by kB T ). For energy equal to the chemical potential, it is exactly 1/2.
How do we find the chemical potential if N is fixed? It is chosen so that the
equation
X
hNα i = N
(2.4)
α
is satisfied.
2.1 Occupation numbers in the ideal gas
f HΕL
1.0
13
0.8
0.6
0.4
0.2
-4
-2
2
4
Β HΕ - ΜL
Figure 2.1 The Fermi function
Bosons
Now let’s try the same thing with bosons, the expression Equation (2.1) is altered
slightly
hN1 i =
∞
∞
1 X X
· · · N1 e−β(N1 E1 +N2 E2 +··· ) ,
Z(N ) N =0 N =0
1
subject to N1 + N2 + · · · = N.
2
(2.5)
Now the sums go all the way to ∞ (of course, none of the them actually do,
because of the constraint of fixed particle number. Again, the term with N1 = 0
doesn’t contribute, and we can write
hN1 i =
∞ X
∞
X
1
e−βE1
· · · (1+N1 )e−β(N1 E1 +N2 E2 +··· ) ,
Z(N )
N =0 N =0
1
subject to N1 +N2 +· · · = N −1.
2
Thus the N1 = 0 term is now providing the contribution that the N1 = 1 did,
and so on. In analogy to Equation (2.2) we have
hN1 iN =
Z(N − 1) −βE1
e
h1 + N1 iN −1
Z(N )
(note the plus sign!) and so
1
eβ(E1 −µ) − 1
The Bose–Einstein distribution, or Bose function, has the form
hN1 iB =
b() =
1
eβ(−µ)
−1
(2.6)
This diverges as approached µ from above (see Figure 2.2), suggesting that this
analysis holds as long as µ is less than the lowest energy level of the gas.
Note that both distribution functions have the same behavior at high energies
hNα iB/F → e−β(E−µ)
Ideal quantum gases
14
bHΕL
1.5
1.0
0.5
1
2
3
4
5
Β HΕ - ΜL
Figure 2.2 The Bose distribution
Exercise 2.1.1 Recover these results using the grand canonical formuation, in
which particle number can vary and the distribution of occupation numbers
is e−β(N1 E1 +N2 E2 +···−µN ) /Z, where the grand partition function is
XX
Z=
· · · e−β(N1 E1 +N2 E2 +···−µN ) ,
N = N1 + N2 + · · · .
(2.7)
N1
N2
and sums go from 0 to ∞ in the case of bosons, from 0 to 1 in the case of
fermions.
Exercise 2.1.2 Using the method of this section, or of the previous exercise,
to show that the variance in the occupancy is
2
Var Nα ≡ Nα2 − hNα i = hNα i [1 ± hNα i]
(2.8)
with the plus sign for bosons, and minus sign for fermions.
2.2 Ideal gases of bosons and fermions
2.2.1 Density of states
We can therefore express thermal averages of the energy and particle number as
X
hU i =
Eα hNα iB/F
(2.9)
α
hN i =
X
hNα iB/F
(2.10)
α
where the average occupation number hNα iB/F is given by the Bose or Fermi
function. The next step is to evaluate these expressions for an appropriate set of
single particle energy levels {Eα }. For particles in a box, we have
E=
~2 k 2
,
2m
k=
π
(nx , ny , nz ),
L
nx,y,z = 1, 2, . . .
(2.11)
2.2 Ideal gases of bosons and fermions
15
Figure 2.3 Counting states in k-space
that lie in a thin spherical shell.
corresponding to a cubic lattice of points in k-space with lattice spacing Lπ occupying the positive octant. The density of states per unit interval of k is then
(Figure 2.3)1
3
1
L
× 4πk 2 dk
dN =
π
8
The more useful quantity is the density of states per unit interval of energy,
which we’ll denote D(E)
D(E) =
dN m
dN dk
=
dk dE
dk ~2 k
or
L3
D(E) = 3 2
~ π
r
√
m3 E
=C E
2
(2.12)
q
√
3
3
where for convenience we have defined C = ~L3 π2 m2 . The crucial part is the E
dependence on energy. Sometimes you will see the density of states defined per
unit volume, in which case the L3 factor is missing.
Sums over wavevectors appear constantly in many-body physics. The above
1
Though dk is to be taken to zero, this approximation assumes dk system is implicit.
2π
,
L
so the limit of an infinite
Ideal quantum gases
16
manipulations quickly become second nature, to the extend that the notations
Z
dk
1 X
(2.13)
, and
V k
(2π)3
with V = L3 the volume of the system, are often used interchangeably.
Exercise 2.2.1 How does the density of states depend on energy in one and
two dimensions?
Exercise 2.2.2 Convince yourself that the result Equation (2.12) is unchanged
if you were to chose periodic rather than hard wall boundary conditions
Using the function D(E), we can write the sums in Equation (2.9) as integrals
Z ∞
ED(E)
dE β(E−µ)
hU iB/F =
e
∓1
Z0 ∞
D(E)
dE β(E−µ)
hN iB/F =
(2.14)
e
∓1
0
note that because of the L3 in D(E), both quantities are proportional to the volume of the system, as we would expect. Although we can evaluate these integrals
analytically, it is preferable to develop a physical understanding of what is going
on. Thus we discuss the case of Bosons and Fermions separately.
2.2.2 The Fermi gas
Let’s consider first the T = 0 limit. We expect the system to be in its ground
state, which, based on our earlier discussion, consists of a Fermi sea of singly
occupied states forming a sphere in k-space centered at the origin. How is this
reflected in the Fermi function? From Figure 2.1 you can see that scale over which
f () changes from 0 to 1 with increasing energy is of order kB T , so
f (E) =
1
eβ(E−µ)
+1
−−−→ Θ(µ − E)
T →0
where Θ(x) is the step function
(
1, if x ≥ 0
Θ(x) =
0, if x < 0
Thus at zero temperature, µ is just EF , the Fermi energy corresponding to the
energy of the last fermion added.
The two integrals in Equation (2.14) are then simple
Z µ
Z µ
2
U=
ED(E) = C
E 3/2 = Cµ5/2
(2.15)
5
0
0
Z µ
Z µ
2
N=
D(E) = C
E 1/2 = Cµ3/2
(2.16)
3
0
0
2.2 Ideal gases of bosons and fermions
17
From which we conclude
3
U = N EF
5
(2.17)
This makes sense. The energy of a fermion is on the scale of the Fermi energy,
and there are N fermions. Of course, only the fermions at the Fermi surface have
energy EF , which explains why the factor in Equation (2.17) is less than one.
Exercise 2.2.3 Consider the cases of dimension two and one. Can you say,
without doing any calculations, how the numerical factor in the relation
analogous to Equation (2.17) will compare with 12 ?
Writing the factor C explicitly, we see that the density n =
1
n= 2
6π
2
2mEF
~2
3/2
=
N
L3
is
kF3
6π 2
(2.18)
2/3
n
Inverting this to give EF ∝ ~ 2m
makes it clear that parametrically EF is the
kinetic energy of a particle with de Broglie wavelength equal to the interparticle
separation n−1/3 .
As an example, let’s consider the element Lithium. In the solid state, Lithium is
a simple metal with one conduction electron per atom. The density of the electron
gas formed from these electrons is of order 1029 m−3 . This gives a Fermi energy of
order 1 eV, or Fermi temperature TF (defined by EF = kB TF ) of around 104 K.
As far as the electrons are concerned, a ordinary metal at room temperature is
close to absolute zero!
6
Now let’s compare with the case of the fermionic isotope Li popular in atomic
physics. A typical density is of order 1020 m−3 , and the atoms are some 104 times
heavier than the electron. This gives a Fermi temperature that differs by a factor
(10−9 )2/3 × 10−4 = 10−10 from that of the electrons in solid Lithium, that is,
around 1 µK!
Now let’s consider what happens at T 6= 0. If the temperature remains small
compared to µ (a good approximation for a room temperature metal!) the only
difference from the zero temperature result must arise from the smearing of the
step in the Fermi function over an energy range of order kB T (Figure 2.4). Let’s
suppose we are interested in a physical quantity that involves an integrand consisting of an arbitrary function φ(E) times the Fermi function.
Exercise 2.2.4 Show that
Z E
Z
φ(E)f (E)dE =
I=
0
0
µ
φ(E)dE +
dφ π2
(kB T )2
+ ...
6
dE E=µ
(2.19)
Sommerfeld’s formula, as Equation (2.19) is known, shows that as an expansion in powers of kB T , the first effect of finite temperature is determined
solely by the derivative of the function φ(E) at the Fermi surface.
Exercise 2.2.5 By applying this result to the energy and number integrals,
18
Ideal quantum gases
Figure 2.4 Illustration of the various factors in the integrand of
Equation (2.14) for the Fermi case at finite temperature
show that at low temperatures the heat capacity (at constant volume) CV =
∂U
of a Fermi gas is
∂T
π2 2
k D(EF )T
3 B
Hint: don’t forget to include the change in µ at finite temperature!
CV =
(2.20)
2.2.3 The Bose gas
Now we turn to the Bose gas. We know already that the picture at zero temperature is radically different from the Fermi case: all N particles sit in the lowest
energy state: N0 = N . For particles in a box (see Equation (2.11)) the ground
state then has energy
3N π 2 ~2
E = N E0 =
2mL2
How is this state of affairs reflected in the behavior of the average occupation
hNα iB =
1
eβ(Eα −µ) − 1
as β → 0? In order to get hN0 i = N , hN1 i = hN2 i = . . . = 0 we require that µ
approach E0 from below as β → ∞. To be precise
µ → E0 −
kB T
, as
N
T →0
(2.21)
2.2 Ideal gases of bosons and fermions
19
As we make the system size larger keeping the density n = LN3 fixed, both terms
go to zero, so µ → 0. Now let’s instead start at high temperatures and imagine
cooling the system. As we’ve already said, the chemical potential should be fixed
by solving Equation (2.14) for N . µ is negative for the Bose distribution to make
sense, but one can see that, as the temperature falls, µ must increase towards
zero to compensate for the faster decay of the distribution. Things come to a
head when µ hits zero, at a temperature determined by the equation
Z ∞
D(E)
dE βE
N=
e −1
0
√
Z ∞ √
Z ∞
E
x
3/2
= C(kB T )
(2.22)
=C
β(E−µ)
x
e
−1
e −1
0
0
The final dimensionless integral is
√
√
Z ∞ √
x
π
π
=
ζ(3/2) ∼
2.6124
x
e −1
2
2
0
P∞
where ζ(s) = n=1 n1s is the Riemann ζ-function. Beneath the temperature defined by Equation (2.22) it is no longer possible to satisfy the number integral.
Precisely, this temperature is
2/3
N
2π
~2 2/3
2
=
n
(2.23)
kB TB = √
(ζ(3/2))2/3 m
πζ(3/2) C
and is referred to variously as Bose temperature, Bose-Einstein temperature, or
critical temperature. Note that it is parametrically equal to the Fermi temperature. EF represents only an overall temperature scale at which quantum statistics
become important, however. In contrast, something much more dramatic happens
at TB . The expressions in Equation (2.14) upon which we have been basing our
analysis seem to have failed. What went wrong? The answer is that in turning the
sums in Equation (2.9) into integrals, we assumed that every interval of energy
E → E + dE contains a number of particles that is, in the limit, proportional to
dE. We already know, however, that at zero temperature this fails for the interval
that contains E0 , because it includes all of the particles! We should separate the
contribution of the first energy level from the rest of the sum, which can be then
safely be turned into an integral
X
N=
Nα
α
Z
= N0 +
∞
dE
0
D(E)
eβE − 1
(2.24)
We say that the ground state is macroscopically occupied, meaning that N0 is
proportional to the number of particles in the system. In this limit Equation (2.21)
is replaced by
kB T
µ → E0 −
, as N → ∞
(2.25)
N0
20
Ideal quantum gases
Exercise 2.2.6 Convince yourself that the number of particles in the first excited state is not macroscopically large
Now the integral that appears in Equation (2.24) is the same as that appearing
in Equation (2.22), so we solve for N0 in terms of TB
"
3/2 #
T
(2.26)
N0 = N 1 −
TB
A result that approaches N as T → 0, as it should. Note that this doesn’t mean
that N0 is equal to zero for T > TB , just that it is not macroscopically large.
The appearance of a finite fraction of the atoms in the ground state beneath a
critical temperature was dubbed ‘condensation’ by Einstein in analogy with the
liquification of a gas (though the processes are quite different), and is universally
known as Bose-Einstein condensation today.
2.2.4 Bose gas in an harmonic trap
The experimental realization of quantum gases with which we will be concerned
involves trapping and cooling a relatively small number ∼ 105 − 107 of atoms.
The resulting gas is confined not by the hard wall potential responsible for the
particle-in-a-box energy levels in Equation (2.11) that were the input to our
density of states calculation, but by a potential that can usually be approximated
as harmonic
1
(2.27)
V (x, y, x) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )
2
Here we are allowing for an arbitrary anisotropy in the potential. The corresponding energy levels are
1
1
1
Enx ,ny ,nz = nx +
~ωx + ny +
~ωy + nz +
~ωz
(2.28)
2
2
2
Ignoring the zero point energy in each of the oscillators, the states corresponding
to a given energy E lie on a plane on in (nx , ny , nz ) space that touches the three
axes at E/~ωx,y,x . The volume between this plane and the planes defined by
x = 0, y = 0, and z = 0
E3
6~ωx ωy ωz
but this is just the number of states with energy less than E, so the density of
states is just the derivative
D(E) =
E2
2~3 ωx ωy ωz
(2.29)
2.2 Ideal gases of bosons and fermions
Exercise 2.2.7 Show that in this case the condensate fraction obeys
"
3 #
T
N0 = N 1 −
TB
21
(2.30)
and identify TB .
Exercise 2.2.8 By thinking about whether the number equation can be satisfied at all temperatures, say whether Bose condensation occurs in the following situations a. a two dimensional gas in a box with hard walls, b. a gas
trapped in a two dimensional harmonic oscillator trap.
Exercise 2.2.9 Show that p = 32 u for an ideal gas of bosons or fermions at any
temperature. You might find it useful to use the formula
∂Φ p=−
∂V µ,T
where Φ = −kB T ln Z is the grand potential, defined in terms of the grand
partition function that we met earlier.
Figure 2.5 From [1]
A classic on the ideal Bose gas, that includes a discussion of condensate
fluctuations, is [2]
22
Ideal quantum gases
References
[1] Jin, D. 2002. A Fermi gas of atoms. Physics World, 27–31.
[2] Ziff, R.M., Uhlenbeck, G.E., and Kac, M. 1977. The ideal Bose-Einstein gas, revisited.
Physics Reports, 32, 169–248.
3
Second Quantization
The formalism of totally symmetric or antisymmetric wavefunctions developed
in Chapter 1 is not a convenient one when we come to discuss the quantum mechanics of a truly macroscopic number of particles. Only in certain special cases,
such as the Slater determinant for fermions in one dimension that we discussed
in Section 1.2, can a compact expression for the wavefunction be obtained for
an arbitrary number of particles. Even then, evaluating observables typically involves integrals over every particle coordinate: an arduous task. In this chapter
we will develop a formalism that is well suited to the general case, in which
indistinguishability is built in from the outset rather than imposed upon the
wavefunction. This formalism is called second quantization
3.1 Creation and annihilation operators
We have already gleaned the essential idea behind second quantization in Chap- E
S/A
ter 1. Instead of working with totally symmetric or antisymmetric states Ψα1 α2 ···αN ,
we’d rather label states purely by the occupation numbers that describe how particles populate the single particle states. The second quantization formalism is
based upon creation and annihilation operators that add and remove particles
from the one particles states, that is, change the occupation numbers Nα 1 . We
generally want the number of particles to be conserved, so observables of interest
are typically products of equal number of creation and annihilation operators
whose overall effect is to redistribute particles among the single particle states.
The operators are defined by their effect on the states in Equation 1.6. A first
guess at defining a creation operator that puts a particle in state |ϕα i would be
√
√
(3.1)
cα : |Ψα1 α2 ···αN i → N + 1 |ϕα i |Ψα1 α2 ···αN i = N + 1 |Ψαα1 α2 ···αN i
√
(the origin of the N + 1 will become clear shortly). This operator acts on a state
with N particles and produces one with N + 1 particles2 . Since any state can be
written as a linear combination of the states {|Ψα1 α2 ···αN i}, the action of cα can
be extended to any state by linearity. Equation 3.1 has an obvious shortcoming,
1
It is often convenient, but by no means necessary, to chose the single particle states {ϕα } to be
eigenstates of the single particle Hamiltonian as we did in Chatper 1
2
z
}|
{
Formally, the space of distinguishable N particle states is HN = H1 ⊗ · · · H1 . The creation operator
acts in the space C ⊕ H1 ⊕ H2 ⊕ · · ·
n times
23
24
Second Quantization
however, in that it does not preserve the symmetry of the wavefunction. This is
easily remedied by applying the symmetrization operator that we introduced in
Equation 1.7 after adding a particle (we discuss the boson case first), so a better
definition is
√
cα : ΨSα1 α2 ···αN → N + 1S |ϕα i ΨSα1 α2 ···αN , for bosons.
(3.2)
Exercise 3.1.1 Bearing in mind the normalization factors in Equation 1.8,
show that this implies
p
(3.3)
cα : ΨSα1 α2 ···αN → Nα + 1 ΨSαα1 α2 ···αN .
we label
If
totally symmetric states by their occupation numbers, so that
ΨSα α ···α = |N0 , N1 . . .i, this equation may be written
1 2
N
p
cα |N0 , N1 , . . . Nα , . . .i → Nα + 1 |N0 , N1 , . . . Nα + 1, . . .i .
(3.4)
By considering matrix elements hN0 , N1 , . . . | cα | N00 , N10 , . . .i we can conclude
p
c†α |N0 , N1 , . . . Nα , . . .i → Nα |N0 , N1 , . . . Nα − 1, . . .i .
(3.5)
In other words, the conjugate of the creation operator is a destruction operator,
that removes one particle from a given state. From Equation 3.4 and Equation 3.5
come the fundamental relationships
†
cα , cβ = δαβ
cα , cβ = c†α , c†β = 0.
(3.6)
For reasons that will become clear shortly we normally write things in terms of
the annihilation operator aα ≡ c†α , in which case the above takes the form
aα , a†β = δαβ
aα , aβ = a†α , a†β = 0
(3.7)
The combination Nα ≡ a†α aα is called the number operator for state α for obvious
reasons
Nα |N0 , N1 , . . . Nα , . . .i = Nα |N0 , N1 , . . . Nα , . . .i .
(3.8)
From Equation 3.1 it follows that
aα , Nα = aα
†
aα , Nα = −a†α .
You can think of the first of these
count’, for example.
It follows that a normalized state
be written as
(3.9)
as ‘count then destroy minus destroy then
S
Ψα
1 α2 ...αN
of the many boson system may
N0 † N1
a†0
a
√1
· · · |0, 0, . . .i
|N0 , N1 , . . .i = √
N0 !
N1 !
(3.10)
3.1 Creation and annihilation operators
25
The state with no particles |0, 0, . . .i is known as the vacuum state. We will often
denote it by |VACi for brevity3 . In terms of a wavefunction, you can think of it
as equal to 1, so that Equation 3.2 works out. Alternatively, you can take the
relations in Equation 3.1 as fundamental, in which case |VACi has the defining
property
aα |VACi = 0,
for all α
Now we move on the slightly trickier matter of fermions. Equation 3.2 suggests
that the creation operator should be defined by
√
N + 1A |ϕα i ΨA
for fermions.
(3.11)
cα : ΨA
α1 α2 ···αN ,
α1 α2 ···αN →
For the fermion annihilation operators aα = c†α the result corresponding to Equation 3.1 is
aα , a†β = δαβ
aα , aβ = a†α , a†β = 0
(3.12)
where {A, B} = AB + BA denotes the anticommutator.
Exercise 3.1.2 Prove this.
Of course, you could calculate the commutator, but it proves to be complicated
(and uninteresting). The number operators, together with Equation 3.9 (with
2
commutators!), work as in the boson case. Equation 3.1 implies that a†α = 0,
so we can’t add two particles to the same single particle state, consistent with
2
the Pauli principle. Similarly, since Nα = 0 or 1, aα = 0, meaning that we
can’t remove two particles from the same state.
4
The state ΨA
α1 α2 ···αN has the representation
N0 † N1
|N0 , N1 , . . .i = a†0
a1
· · · |0, 0, . . .i ,
(3.13)
which is the same as Equation 3.10, since Nα = 0 or 1 only are allowed.
Suppose we want to move to a different basis of single particle states {|ϕ̃α i},
corresponding to a unitary transformation
X
|ϕ̃α i =
hϕβ | ϕ̃α i |ϕβ i .
(3.14)
β
From Equations 3.4
and 3.11,
the one particle states with the wavefunctions
{ϕα (r)} are just a†α |VACi . So we see that the above basis transformation
gives a new set of creation operators
X
hϕβ | ϕ̃α i a†β .
(3.15)
a˜†α ≡
β
3
4
Not to be confused with |0i, which will denote the ground state of our many body system
Note that this equation defines the overall sign of the state |N0 , N1 , . . .i. Two states ΨA
α1 α2 ···αN and
0 } a permutation of {α }, may
ΨA
with
the
same
occupation
numbers,
that
is,
with
{α
n
0
0
0
n
α α ···α
1
2
N
differ by a sign depending upon the signature of the permutation.
Second Quantization
26
Often we will work in the basis of position eigenstates {|ri}. In this case the matrix
elements of the unitary transformation are hϕβ | ri = ϕ∗β (r), just the complex
conjugate of the wavefunction. Denoting the corresponding creation operator as
ψ † (r), Equation 3.15 becomes
X
ψ † (r) ≡
ϕ∗β (r)a†β .
(3.16)
β
Now we can see why we chose to work with the annihilation operator rather than
the creation operator. The conjugate of Equation 3.16 is
X
ψ(r) ≡
ϕβ (r)aβ ,
(3.17)
β
and involves the wavefunctions ϕβ (r), rather than their conjugates. The relations
satisfied by these operators are easily found from the corresponding relations in
Equations 3.4 and 3.11, together with the completeness relation
X
ϕ∗α (r)ϕα (r0 ) = δ(r − r0 ).
(3.18)
α
We find
ψ(r)ψ † (r0 ) ∓ ψ † (r0 )ψ(r) = δ(r − r0 )
ψ(r)ψ(r0 ) ∓ ψ(r0 )ψ(r) = ψ † (r)ψ † (r0 ) ∓ ψ † (r0 )ψ † (r) = 0
(3.19)
with the upper sign for bosons, and the lower for fermions. Just as the position
representation is very one convenient one for quantum states, the position basis
creation and annihilation operators provide a convenient basis for many of the
many body operators we will encounter.
As an example, let our original basis be the eigenbasis of the free particle
p2
with periodic boundary conditions
Hamiltonian H = 2m
eik·r
nx ny nz
,
,
, nx,y,z integer,
(3.20)
|ki = √ , k = 2π
Lx Ly Lz
V
with V = Lx Ly Lz . The matrix elements of the transformation
√ between this orig−ik·r
inal basis and the position basis {|ri} are hk | ri = e
/ V , so we have
1 X −ik·r †
ψ † (r) ≡ √
e
ak ,
(3.21)
V k
and similarly
1 X ik·r
ψ(r) ≡ √
e ak .
V k
(3.22)
n
Eo
S/A
Taking a breath at this point, we can see that the cumbersome basis set Ψα1α2 ···αN
has been hidden away behind a much more compact algebra of operators that generates it. Once we figure out how to write physical observables in terms of these
3.1 Creation and annihilation operators
27
operators, we can – if we wish – purge our formalism completely of wavefunctions
with N arguments!
Exercise 3.1.3 Consider the Hamiltonian
H = a† a + b† b + ∆ a† b† + ba
This Hamiltonian is slightly unusual, as it doesn’t conserve the number of
particles.
1.
Let’s first consider the case of bosons. Then all operators commute except
[a, a† ] = [b, b† ] = 1
Define new operators
α = a cosh κ − b† sinh κ
β = b cosh κ − a† sinh κ
for some κ to be determined. Show that α, α† , and β, β † satisfy the
same commutation relations.
2. Show that κ can be chosen so that, when written in terms of α and β,
there are no ‘anomalous’ terms in H (i.e. no terms αβ or α† β † ). In this
way find the eigenvalues of the Hamiltonian.
3. Now repeat the problem for fermions. The first thing you will need to
figure out is what kind of transformation preserves the anticommutation
relations.
4. It’s natural to expect that, since the algebraic relations are preserved
by this transformation, it may be written as a unitary transformation
on the operators
α = U aU †
β = U bU † ,
U † = U −1
Show that U = exp κ a† b† − ba does the job for both fermions and
bosons.
Exercise 3.1.4 Consider the Hamiltonian
X
H = a† a +
ξp b†p bp + a† aγp b†p + bp
p
involving a fermion a and bosons bp .
Transform the Hamiltonian
H → U iHU † using the unitary transforh
P
mation U = exp a† a p γξpp b†k − bk . You should find that you have
eliminated the coupling of the fermion to the bosons.
2. Try to give a simple physical interpretation for what you have done.
1.
28
Second Quantization
3.2 Representation of operators
We now turn to the matter of representing operators of the many particle system
in terms of creation and annihilation operators.
3.2.1 One particle operators
A one particle operator consists of a sum of terms, one for each particle, with
each term acting solely on that particle’s coordinate5 . In the terminology that
we introduced in Chapter 1, a one particle operator is a sum of single particle
operators, one for each particle. By assumption each term is the same, consistent
with the indistinguishability of the particles. We’ve already met one important
example, namely the Hamiltonian for identical noninteracting particles in Equation 1.5. In general, the action of an operator A on the single particle states may
written in terms of the matrix elements hϕα | A | ϕβ i
X
|ϕα i hϕα | A | ϕβ i .
(3.23)
A |ϕβ i =
α
In words, the action of A is to take the particle from state |ϕβ i to a superposition
of states with amplitudes given by the matrix elements Aαβ ≡ hϕα | A | ϕβ i. Now
it’s not hard to see that this action can be replicated on the one particle states
a†α |VACi with
X
 ≡
Aαβ a†α aβ
(3.24)
α,β
(for the moment we’ll use hats for the second quantized operator, but later we’ll
drop this distinction). To see this, first note that Equation (3.9) can be generalized
to
aα , a†β aγ = δαβ aγ
† † aα , aβ aγ = −δαγ a†β .
(3.25)
Using the second of these relations, together with the fact that  |VACi = 0
h
i
 a†β |VACi = Â, a†β + a†β  |VACi
X
=
Aαβ a†α |VACi ,
(3.26)
α
which is precisely Equation (3.23).
Exercise 3.2.1 Show that  acts in the desired way on N particle states,
namely
!
N
E X
E
X
S/A
S/A
 Ψβ1 β2 ···βN =
Aαβ Ψβ1 β2 ···αi ···βN
(3.27)
i=1
5
αi
More formally, each term acts on one factor in the product H1 ⊗ · · · H1 of single particle Hilbert
spaces
3.2 Representation of operators
29
NoticePthat  looks formally like the expectation value of A in a single particle
state α aα |ϕα i. The difference, of course, is that the aα in  are operators, so
that the order is important, while those in the preceding expression are amplitudes. The replacement of amplitudes, or wavefunctions, by operators is the origin
of the rather clumsy name ‘second quantization’, which is traditionally introduced
with the caveat that what we are doing is not in any way ‘more quantum’ than
before.
To repeat the above prescription for emphasis: A one particle operator  has a
second quantized representation formally identical to the expectation value of its
single particle counterpart A.
This probably all looks a bit abstract, so let’s turn to a one particle operator that we have already met, namely the noninteracting Hamiltonian in Equation (1.5). According to the above prescription, this should have the second quantized form
X
hϕα | H | ϕβ i a†α aβ ,
(3.28)
Ĥ ≡
α,β
2
~
∇2i + V (ri ). This takes on a
where H is the single particle Hamiltonian H = − 2m
very simple form if the basis {|ϕα i} is just the eigenbasis of this Hamiltonian, in
which case hϕα | H | ϕβ i = Eα δαβ and
X
X
Ĥ ≡
Eα a†α aα =
Eα Nα .
(3.29)
α
α
Evidently this
is correct:
E the eigenstates of this operator are just the N particle
S/A
basis states Ψα1α2 ···αN , and eigenvalues coincide with Equation (1.9).
Alternatively, we can look at things in the position basis. By recalling how the
expectation value of the Hamiltonian looks in this basis, we come up with
Z
~2 †
2
†
Ĥ = dr −
ψ (r)∇ ψ(r) + V (r)ψ (r)ψ(r)
2m
Z
~2
†
†
= dr
∇ψ (r) · ∇ψ(r) + V (r)ψ (r)ψ(r) ,
(3.30)
2m
where in the second line we have integrated by parts, assuming that boundary
terms at infinity vanish. The equality of (3.30) and (3.29) may be seen by using
Equation (3.17).
As a second example, consider the particle density. This is not something that
one encounters very often in few particle quantum mechanics, but is obviously an
observable of interest in a extended system of many particles. The single particle
operator for the density at x is
ρ(x) ≡ δ(x − r).
(3.31)
This may look like a rather strange definition, but its expectation value on a
single particle state ϕ(r) is just ρ(x) = |ϕ(x)|2 , which is just the probability to
Second Quantization
30
find the particle at x. Following our prescription, the second quantized form of
the operator is then
ρ̂(x) ≡ ψ † (x)ψ(x).
(3.32)
As a check, integrating over position should give the total number of particles
Z
X
X
Nα ,
(3.33)
a†α aa =
N̂ = dx ψ † (x)ψ(x) =
α
α
as it does! Another useful thing to know is the expectation value of the density
on a basis state |N0 , N1 . . .i
X
2
hN0 , N1 . . . | ρ̂(r) | N0 , N1 . . .i =
Nα |ϕα (r)| .
(3.34)
α
which is most easily proved by substituting the representation (3.17). This seems
like a very reasonable generalization of the single particle result: the density is
given by sum of the probability densities in each of the constituent single particle
state, weighted by the occupancy of the state. Note that the symmetry of the
states played no role here.
As a final example of a one particle operator, the particle current has the second
quantized form
~ †
ĵ(r) = −i
ψ (r) ∇ψ(r) − ∇ψ † (r) ψ(r) .
(3.35)
2m
Often we consider the Fourier components of the density or current
Z
X †
ρ̂q ≡ dr ρ̂(r)e−iq·r =
ak−q/2 ak+q/2
k
Z
ĵq ≡
dr ĵ(r)e−iq·r
X k †
ak−q/2 ak+q/2 .
=
m
k
The q = 0 modes are just the total particle number and
momentum, respectively.
(3.36)
1
m
times the total
Exercise 3.2.2 The operator for the density of spin of a system of spin-1/2
fermions is
1X †
S(r) =
ψ (r)σ ss0 ψ(r),
2 s,s0 s
where σ = (σx , σy , σx ) are the Pauli matrices and ψ † (r), ψ(r) satisfy
{ψs (r), ψs†0 (r0 )} = δss0 δ(r − r0 ),
1. Find the commutation relations [Si (r), Sj (r0 )].
2. For the Hamiltonian
Z
~2 X
d3 r ∇ψs† ∇ψs
H=
2m s
3.2 Representation of operators
31
Figure 3.1 Single particle density matrix for the Fermi gas.
find the form of the Heisenberg equation of motion ∂t S(r, t) = i[H, S(r, t)]/~.
Interpret your result.
Exercise 3.2.3 By considering the expectation value of the double commutator
h0 | [[H, ρq ], ρ−q ] | 0i, prove the f-sum rule
X
n
N ~2 q2
(En − E0 ) | n ρ†q 0 |2 =
2m
where N is the number of particles, |0i is the ground state of the N particle
system, and the sum is over all excited states hn|. ρq is the Fourier transform
of the density operator defined in Equation (3.36).
We can also define a quantity, whose usefulness will become apparent as we go
on, called the single particle density matrix
g(r, r0 ) ≡ ψ † (r)ψ(r)
(3.37)
Notice that g(r, r) = hρ̂(r)i. A slight generalization of the above calculation gives
for the state |N0 , N1 . . .i
X
g(r, r0 ) =
Nα ϕ∗α (r)ϕα (r0 ).
(3.38)
α
Let’s evaluate this for the ground state of noninteracting Fermi gas. Recall that
in this case Nk = 1 for |k| < kF , and 0 otherwise. Thus we have
Z
1 X ik·(r0 −r)
dk ik·(r0 −r)
g(r, r0 ) =
e
=
e
3
V
|k|<kF (2π)
|k|<kF
kF3 sin (kF |r0 − r|) cos (kF |r0 − r|)
= 2
−
(3.39)
2π
(kF |r0 − r|)3
(kF |r0 − r|)2
3
kF
(see Figure 3.1). Note that g(r, r) = 6π
2 = n (using Equation (2.18)), as it should.
Also, g(r, r0 ) → 0 as |r−r0 | → ∞. Let’s compare this behavior with that of a Bose
Second Quantization
32
gas below the condensation temperature. If we evaluate Equation (3.38) using the
appropriate (average) occupation numbers, we will have to single out the single
particle ground state for special treatment as in Chapter 2 before converting the
sum to an integral
Z
0
eik·(r −r)
dk
N0
0
+
.
(3.40)
g(r, r ) =
V
(2π)3 eβ~2 k2 /2m − 1
At large distances the second term falls off like 1/r, showing that in this case
g(r, r0 ) tends to a constant value at large separations, given by the density of
particles in the zero momentum state. This property, characteristic of Bose condensation, is sufficiently important to have a technical term: off diagonal long
range order [9]. We will return to it later.
3.2.2 Correlation functions
0
The function g(r, r ) appears as an ingredient in many calculations. As an example, let’s consider the density-density correlation function. This is defined as6
Cρ (r, r0 ) ≡ h: ρ(r) ρ(r0 ) :i .
(3.41)
Sandwiching operators between colons denotes the operation of normal ordering,
meaning that we write the constituent creation and annihilation operators so that
all annihilation operators stand to the right of all creation operators. In the case
of fermions we include the signature of the permutation (the sign that tells us
whether the permutation corresponding to our rearrangement is even or odd).
Thus
aα a†β = 1 ± a†β aα , while : aα a†β := ±a†β aα ,
with the upper sign for bosons, and the lower for fermions.
In the present case you can easily convince yourself that
hρ(r) ρ(r0 )i = h: ρ(r) ρ(r0 ) :i + δ(r − r0 ) hρ(r)i .
(3.42)
Before plowing on, let’s first motivate the above definition. Imagine dividing space
into a fine lattice of cubic sites, such that the probability of finding two particles
inside one cube can be neglected. This means that the cube should have linear
dimension much smaller than the interparticle spacing. We denote by Ni the
number of particles in cube i, centered at ri . Of course
Z
Ni =
dr ρ(r).
(3.43)
cube i
Since Ni is only 0 or 1 by assumption, the probability of cube i being occupied is
P (cube i occupied) = hNi i ,
6
We are now going to drop the hats for second quantized operators: the distinction is no longer
important
3.2 Representation of operators
33
that is, the mean occupancy. Likewise the probability of having cubes i and j
occupied is
Z
Z
dr0 hρ(r) ρ(r0 )i .
dr
P (i and j occupied) = hNi Nj i =
cube i
cube j
This tells us that hρ(r) ρ(r0 )i is equal to the joint probability density of finding
a pair of particles at r and r0 , but this interpretation only works for r 6= r0 , since
for i = j
hNi Nj i = Ni2 = hNi i ,
which shows that hρ(r) ρ(r0 )i always has an additional contribution δ(r−r0 ) hρ(r)i,
determined by the mean density. Equation (3.42) shows that normal ordering
automatically removes this piece. Why? Imagine removing particles at r and r0
by applying the product of annihilation operator ψ(r)ψ(r0 ) to the original state,
call it |Ψi. This has the effect of projecting onto that part of the wavefunction
where two particles are localized at these positions. The probability (density)
of finding two particles at these locations is then the squared modulus of the
resulting state, that is
kψ(r)ψ(r0 ) |Ψi k2 = ψ † (r0 )ψ † (r)ψ(r)ψ(r0 ) ,
(3.44)
which is just the normal ordered form h: ρ(r) ρ(r0 ) :i!
Exercise 3.2.4 Prove the formula
†
eλa
a
=: e(e
λ
−1)a† a
:
With the physical meaning of Cρ (r, r0 ) established, let’s proceed to the calculation. As in the derivation of Equation (3.34), we substitute the representation (3.17) to give
X
hρ(r) ρ(r0 )i =
ϕ∗α (r)ϕβ (r)ϕ∗γ (r0 )ϕδ (r0 ) a†α aβ a†γ a†δ .
(3.45)
α,β,γ,δ
If we are considering the expectation in a state of the form |N0 , N1 , . . .i, we
can see that an annihilation operator for a given single particle state must be
accompanied by a creation operator for the same state. There are therefore two
possibilities
α = β, γ = δ, or
α = δ, β = γ,
which give rise to two groups of terms. The first contains the average a†α aα a†γ aγ =
Nα Nγ , while the second involves
†
aα aγ a†γ aα = Nα (1 ± Nγ ).
Second Quantization
34
Figure 3.2 Density correlation function for the Fermi gas
Here we have used
a†α aα = Nα
aγ a†γ = 1 ± a†γ aγ = 1 ± Nγ .
(3.46)
Overall we have7
hρ(r) ρ(r0 )i =
X
ϕ∗α (r)ϕα (r)ϕ∗γ (r0 )ϕγ (r0 )Nα Nγ
α,γ
+
X
ϕ∗α (r)ϕγ (r)ϕ∗γ (r0 )ϕα (r0 )Nα (1 ± Nγ ).
(3.47)
α,γ
The result that expectation values in a state |N0 , N1 , . . .i can be computed by
pairing the indices of creation and annihilation operators in all possible ways
and using Equation (3.46) is known as Wick’s theorem. Using the completeness
relation Equation (3.18) the above result can be written8
hρ(r) ρ(r0 )i = δ(r − r0 ) hρ(r)i + hρ(r)i hρ(r0 )i ± g(r, r0 )g(r0 , r).
(3.48)
or
Cρ (r, r0 ) = hρ(r)i hρ(r0 )i ± g(r, r0 )g(r0 , r).
(3.49)
For the fermion case, we see that the correlation function vanishes as the separation |r − r0 | → 0, because g(r, r) = hρ(r)i (Figure 3.2). This is, of course, another
manifestation of the exclusion principle: it is not possible for two fermions to sit
on top of each other. The scale of the ‘hole’ in the correlation function is of course
set by the mean interparticle separation, which is to say the Fermi wavelength.
7
8
The attentive reader of legend will have already spotted that this expression does not handle the
case α = β = γ = δ correctly in the case of bosons, and convinced herself that in the limit of a large
system this does not matter (consider the case of properly normalized plane waves if you are not
sure).
Of course, one can work directly with the normal ordered version in which case the δ-function term
is absent.
3.2 Representation of operators
35
For bosons the situation is very different. Suppose we have a non-condensed
Bose gas, so that g(r, r0 ) → 0 as |r − r0 | → ∞ (see the discussion after Equation (3.40)). Then the value of the correlation function as |r − r0 | → 0 is twice
the value at |r − r0 | → ∞. This characteristic behavior is often termed bunching:
a pair of bosons is more likely to be found at two nearby points than at two
distant points. Note also that the bunching effect is absent in the Bose gas at
zero temperature.
Exercise 3.2.5 Re-derive Equation (3.48) by starting from the many-body
wavefunction of the ground state (that is the totally symmetric or antiS/A
symmetric function Ψα1 α2 ···αN (r1 , r2 , . . . rN )). Remember that in the first
PN
quantized representation, the density operator is ρ(x) = i δ(x − ri ).
3.2.3 Two particle operators and interactions
A two particle operator consists of a sum of terms acting pairwise on the particles.
The most important example of such an operator is that describing the potential
energy of interaction between pairs of particles, one version of which appeared
already in Equation (1.18)
Hint =
N
X
U (ri − rj ).
(3.50)
i<j
Here U (r − r0 ) is the potential energy of a pair of particles at positions r and r0 ,
and the sum ensures that we count each pair of particles once. Recalling
PN the form
of the density operator in the first quantized representation: ρ(x) = i δ(x−ri ),
we see that
Z
? 1
Hint =
drdr0 ρ(r)U (r − r0 )ρ(r0 )
(3.51)
2
almost replicates Equation (3.50). This expression is familiar from electrostatics,
where it represents the electric potential energy of a continuous charge distribution, with U (r − r0 ) the Coulomb potential. The factor of 12 ensures that we only
sum over each pair once. The problem with Equation (3.51), however, is that
it includes the terms with i = j that are excluded from Equation (3.50), that
is, it allows a particle to interact with itself! From Exercise 3.2.5 you will recall
that it is precisely these terms that are responsible for generating the δ-function
contribution to Equation (3.48). Here, as before, the remedy is to normal order
the operators
Z
Z
1
1
0
0
0
Hint =
drdr : ρ(r)U (r−r )ρ(r ) :=
drdr0 ψ † (r)ψ † (r0 )U (r−r0 )ψ(r0 )ψ(r).
2
2
(3.52)
After doing so, it is clear that at least two particles, rather than one, are required for the interaction energy to be non-vanishing. The removal of these ‘selfinteraction’ terms was of particular importance historically. The divergent ‘self-
Second Quantization
36
energy’ of the electron, due to the singular nature of the Coulomb interaction,
was an entrenched difficulty of the classical theory. Jordan and Klein [5], who
pioneered the modern form of second quantization for bosons9 , saw the existence
of the above simple prescription in quantum mechanics as a particular benefit.
We have taken the more usual modern approach (in nonrelativistic physics) of
assuming interactions between pairs only at the outset.
We pause to present the final form of the second quantized Hamiltonian, including an external potential and pairwise interactions
Z
H=
~2
†
†
dr
∇ψ (r) · ∇ψ(r) + V (r)ψ (r)ψ(r)
2m
Z
+ dr1 dr2 ψ † (r)ψ † (r0 )U (r − r0 )ψ(r0 )ψ(r).
(3.53)
Understanding the properties of this Hamiltonian, and its extensions that include
different spin states and particle species, is the central problem of many body
physics.
More generally, a two particle operator has the second quantized representation
X
Aαβγδ a†α a†β aγ aδ .
(3.54)
αβγδ
for some coefficients Aαβγδ . The generalization to three particle operators, four
particle operators, and so on should be obvious.
3.3 Interference of condensates
Let’s consider a gas of N noninteracting bosons occupying the lowest energy level
of some potential well. If the ground state wavefunction is ϕ0 (r), the N -body
wavefunction for such a state is
Ψ(r1 , r2 , . . . , rN ) =
N
Y
ϕ0 (ri ),
(3.55)
i
which we can write in second quantized notation as
N
1
a†0 |VACi ,
|Ψi = √
N!
(3.56)
where a†0 creates a particle in the state ϕ0 (r). Imagine that we took another well,
also filled with N bosons, and placed it alongside the first. If we switched off the
potentials at some instant, the particles will fly out, with particles orginating in
the two wells overlapping. What should we expect to see?
9
The formalism was subsequently extended to fermions by Jordan and Wigner [6]
3.3 Interference of condensates
37
3.3.1 Number and phase as conjugate variables
We are going to need two kinds of states to discuss the problem. The first, which
we will call Fock states, are the kind of states we have been using as basis states
up to now, with a definite number of particles, for example
N
1
a†0 |VACi .
|N iF ≡ √
N!
(3.57)
In this state the single particle density matrix takes the form
gF (r, r0 ) = N ψ † (r)ψ(r0 ) N F = N ϕ∗0 (r)ϕ0 (r).
The second type of states are the coherent states
2
|αiC ≡ e−|α|
/2
exp(αa†0 ) |VACi .
This is a superposition of Fock states, that is, a superposition of states with
definite particle number.
Exercise 3.3.1 1.
Show that these states satisfy
a |αi = α |αi
2. Show that the above normalization is correct and find the overlap hα | βi.
3. Show that the normalized state can also be written exp αa† − α∗ a
4. Find the probability distribution of the number of quanta.
5. Consider the density correlations in this state (don’t forget to normal
order, with all the annihilation operators to the right of the creation
operators). What changes compared to the case we considered before
(Fock states)? Do bosons ‘bunch’ (i.e. prefer to arrive together) in this
case?
While this might be a reasonable state for photons, which can be created and
destroyed, it certainly is never the state of an isolated system of atoms. The
coherent state has, however, the nice property that
† m
(3.58)
α (a0 ) (a0 )n α C = (α∗ )m αn .
In other words, a0 acquires an expectation value α, which can be complex. The
property (3.58) gives us the density matrix
GC (r, r0 ) = α ψ † (r)ψ(r0 ) α C = ϕ∗0 (r)ϕ(r0 )|α|2 ,
which coincides with the Fock state case after the identification N ↔ |α|2 . The
difference is that now ψ(r) itself has an expectation value
α ψ(r) α = ϕ0 (r)α,
whereas this is zero in the Fock state, simply because ψ(r) changes the number
of particles in the system.
38
Second Quantization
Without worrying about normalization, the Fock states may be written in
terms of the coherent states as
Z 2π
dθ −iθN iθ (3.59)
e C,
|N iF ∝
e
2π
0
i.e. coherent states and Fock states are conjugate in more or less the same was
as momentum and position states in ordinary single particle quantum mechanics.
Since the number is fixed for a system of isolated particles, the phase is completely
unknown.
The situation is quite different, however, when we consider that atoms may
occupy two possible states ϕ0 (r) and ϕ1 (r), not necessarily orthogonal. The state
N
a†0 + αa†1 |VACi
(3.60)
is clearly a physically sensible state, and quite distinct from the product of two
Fock states
|n, N − ni ≡ (a†0 )n (a†1 )N −n |VACi .
We do, however, have a relation just like Equation (3.59)
Z
N
dθ −iθn †
e
a0 + |α|eiθ a†1 |VACi .
|n, N − ni ∝
2π
(3.61)
(3.62)
We see that Equation (3.60) describes a state in which atoms in states 0 and
1 have a definite relative phase, while Equation (3.61) can be thought of as a
state in which that phase fluctuates. As we will now show, this is not just a
mathematical nicety.
3.3.2 A double slit experiment with condensates
With this background, let us consider the case where the states ϕL and ϕR represent the ground states of two spatially separated potential wells . One can prepare
a system in a state like Equation (3.60) of definite relative phase or in the Fock
state Equation (3.61), depending on whether a gas of bosons is divided in two
below or above the Bose condensation temperature.
Suppose that the two states are initially sufficiently separate that they can be
thought of as orthogonal. Then the state of definite relative phase θ is
"r
#N
r
1
N̄
N̄
L −iθ/2 †
R iθ/2 †
N̄L , N̄R ≡ √
e
aL +
e aR |VACi ,
(3.63)
θ
N
N
N!
where N̄L,R are the expectation values of particle number in each state N =
N̄L + N̄R . We allow the system to evolve for some time T , so that the two ‘clouds’
begin to overlap (typically achieved by allowing free expansion i.e. turning off
the confining potentials). Ignoring interactions between the particles, the manyparticle state is just (3.63) with the wavefunctions ϕL,R evolving freely. We compute the subsequent expectation value of the density using the second quantized
3.3 Interference of condensates
39
representation
ρ(r) = ψ † (r)ψ(r),
ψ(r) = ϕL (r)aL + ϕR (r)aR + · · ·
where the dots denote the other states in some complete orthogonal set that
includes ϕL (r) and ϕR (r): we can ignore them because they are empty. A simple
calculation gives
≡ρint(br,T )
}|
{
zp
iθ ∗
2
2
hρ(r, T )iθ = N̄L |ϕL (r, T )| + N̄R |ϕR (r, T )| + 2 N̄L N̄R Re e ϕL (r, T )ϕR (r, T ) .
(3.64)
If the clouds begin to overlap, the last term in Equation (3.64) comes into play. Its
origin is in quantum interference between the two coherent subsystems, showing
that the relative phase has a real physical effect.
As an illustration, consider the evolution of two Gaussian wavepackets with
width R0 at T = 0, separated by a distance d R0
"
#
2
(r ± d/2) (1 + i~T /mR02 ))
1
exp −
,
(3.65)
ϕL,R (r, T ) =
3/2
2RT2
(πRT )
with
RT2
=
R02
+
~t
mR0
2
.
Equation (3.65) illustrates a very important point about the expansion of a gas.
After a long period of expansion, the final density distribution is a reflection of
the initial momentum distribution. This is simply because faster moving atoms
fly further, so after time T an atom with velocity v will be at position r = vT
from the center of the trap, provided that this distance is large compared to R0 ,
the initial radius of the gas. The T → ∞ limit of Equation (3.65) gives
" 2 #
mR
[r
±
d/2]
0
,
(3.66)
|ϕL,R (r, T → ∞)|2 ∝ exp −
~T
reflecting a Gaussian initial momentum distribution of width ~/R0 . Imaging the
density distribution after expansion is one of the most commonly used experimental techniques in ultracold physics, and yields information about the momentum
distribution n(p) ≡ a†p ap before expansion.
The final term of Equation (3.64) is then
~r · d
ρint (r, T ) = A(r, T ) cos θ +
T
mR02 RT2
p
2
2 N̄L N̄R
r + d2 /4
A(r, T ) =
exp −
(3.67)
RT3
RT2
The interference term therefore consists of regularly spaced fringes, with a separation at long times of 2π~T /md.
Second Quantization
40
Now we consider doing the same thing with two condensates of fixed particle
number, which bear no phase relation to one another. The system is described
by the Fock state
|NL , NR iF ≡ √
1
(a† )NL (a†R )NR |VACi .
NL !NR ! L
Computing the density in the same way yields
hρ(r, T )iF = NL |ϕL (r, T )|2 + NR |ϕR (r, T )|2 ,
(3.68)
which differs from the previous result by the absence of the interference term.
This is consistent with the principle expressed in Equation (3.62), that we can
think of the Fock state as a kind of ‘phase-averaged’ state.
This is not the end of the story, however. When we look at an absorption
image of the gas, we are not looking at an expectation value of ρ(r) but rather
the measured value of some observable(s) ρ(r). The expectation value just tells
us the result we would expect to get if we repeated the same experiment many
times and averaged the result. We get more information by thinking about the
correlation function of the density at two different points.
Exercise 3.3.2 Show that
hρ(r)ρ(r0 )iF = hρ(r)iF hρ(r0 )iF + NL (NR + 1)ϕ∗L (r)ϕ∗R (r0 )ϕL (r0 )ϕR (r)
+ NR (NL + 1)ϕ∗R (r)ϕ∗L (r0 )ϕR (r0 )ϕL (r).
(3.69)
We see that the second line contains interference fringes, with the same spacing
as before. The correlation function gives the relative probability of finding an
atom at r0 if there is one at r. It seems that in each measurement of the density,
fringes are present but with a phase that varies between measurements, even if
the samples are identically prepared10 . Indeed one can show that, for NL , NR 1
Z 2π
dθ
0
hρ(r)ρ(r )iF =
hρ(r)iθ hρ(r0 )iθ ,
(3.70)
2π
0
The rather surprising implication is that predictions for measured quantities for
a system in a Fock state are the same as in a relative phase state, but with a
subsequent averaging over the phase.
Exercise 3.3.3 Prove this by showing that the density matrix
Z 2π
dθ ρ=
N̄L , N̄R θ N̄R , N̄L θ
2π
0
coincides with that of a mixture of Fock states with binomial distribution of
10
We note that Equation (3.69) is not real, so that strictly h[ρ(r), ρ(r0 )]i 6= 0 and the density measured
at two points does not correspond to two commuting variables. This is, however, only an O(N )
effect, compared to the O(N 2 ) magnitude of the correlator. In the limit of a large number of
particles, simultaneous measurement of ρ(r) at different points is possible.
ity of the solar corona. 1. Boundary shearing of an
s. Res. 101, 13445–13460 (1996).
Reconnection-driven current filamentation in solar
ma, T. Nonlinear growth of Rayleigh–Taylor
. J. 358, L57–L61 (1990).
e spontaneous fast reconnection mechanism in three
servation and detailed photometry of a great Ha two-
ER spectral observations of post-flare supra-arcade
K. Downflow motions associated with impulsive
y 23 solar flare. Astrophys. J. 605, L77–L80 (2004).
eiss, A. Asai and D. H. Brooks for comments. Use
supported by the Japan–UK Cooperation Science
and N. O. Weiss) and a Grant-in-Aid for the 21st
sality in Physics’ from MEXT, Japan. The
Earth Simulator.
bution. We performed the experiment with a Bose gas initially in
density distribution after sudden switch-off of the trapping potenthe Mott insulator regime , where repulsive interactions pin the
tial and a fixed period of free expansion (the ‘time of flight’).
atomic density to exactly an integer number of atoms per lattice site,
typically between one and three. In this Mott insulator phase, the
Resonant absorption of a probe laser24 yields the two-dimensional
average density distribution after expansion is simply given by the
column density of theAcknowledgements
cloud, that
is,thankthe
density
integrated
The authors
N. O. Weiss,
A. Asai and D.profile
H. Brooks for comments.
Use
of TRACE data is acknowledged. This work was supported by the Japan–UK Cooperation Science
along the probe line of
sight,
illustrated
It should
be
Program
of the JSPSas
(Principal
investigators K.S. andin
N. O. Fig.
Weiss) and1a.
a Grant-in-Aid
for the 21st
Century COE ‘Centre for Diversity and Universality in Physics’ from MEXT, Japan. The
computation
performed of
on theflight
Earth Simulator.
noted that the densitynumerical
after
thewastime
reflects the in-trap
momentum distribution
rather
than
the
density
Competing interests
statement
The authors
declare in-trap
that they have no competing
financial distribution. We performedinterests.
the experiment with a Bose gas initially in
Correspondence
8–10and requests for materials should be addressed to H.I.
, where repulsive interactions pin the
the Mott insulator regime
(isobe@kwasan.kyoto-u.ac.jp).
atomic density to exactly an integer number of atoms per lattice site,
typically between one and three. In this Mott insulator phase, the
..............................................................
average density distribution
after 3.3
expansion
is simply given
the
Interference
of by
condensates
41
ribbon flare. Sol. Phys 125, 321–332 (1990).
24. Innes, D. E., McKenzie, D. E. & Wang, T. SUMER spectral observations of post-flare supra-arcade
inflows. Sol. Phys. 217, 247–265 (2003).
25. Asai, A., Yokoyama, T., Shimojo, M. & Shibata, K. Downflow motions associated with impulsive
nonthermal emissions observed in the 2002 July 23 solar flare. Astrophys. J. 605, L77–L80 (2004).
Spatial quantum noise interferometry
in expanding ultracold atom clouds
Simon Fölling, Fabrice Gerbier, Artur Widera, Olaf Mandel,
Tatjana Gericke & Immanuel Bloch
Institut für Physik, Johannes Gutenberg-Universität, Staudingerweg 7,
D-55099 Mainz, Germany
clare that they have no competing financial
.............................................................................................................................................................................
In a pioneering experiment1, Hanbury Brown and Twiss (HBT)
demonstrated that noise correlations could be used to probe the
properties of a (bosonic) particle source through quantum
statistics; the effect relies on quantum interference between
possible detection paths for two indistinguishable particles.
HBT correlations—together with their fermionic counterparts2–4 —find numerous applications, ranging from quantum
optics5 to nuclear and elementary particle physics6. Spatial
HBT interferometry has been suggested7 as a means to probe
hidden order in strongly correlated phases of ultracold atoms.
Here we report such a measurement on the Mott insulator8–10
phase of a rubidium Bose gas as it is released from an optical
lattice trap. We show that strong periodic quantum correlations
exist between density fluctuations in the expanding atom cloud.
These spatial correlations reflect the underlying ordering in the
lattice, and find a natural interpretation in terms of a multiplewave HBT interference effect. The method should provide a
useful tool for identifying complex quantum phases of ultracold
bosonic and fermionic atoms11–15.
Although quantum noise correlation analysis is now a basic tool
in various areas of physics, applications to the field of cold atoms
have been scarce. Most of these concentrate on photon correlation
techniques from quantum optics5,16. It was not until 1996 that
bunching of cold (but non-degenerate) bosonic atom clouds could
be directly measured17, followed by the observation of reduced
inelastic losses due to a modification of local few-body correlations
by quantum degeneracy18–20.
In our experiment, we directly measure the spatial correlation
function of the
fluctuations in a freely expanding atomic
Ldensity R
uld be addressed to H.I.
...............................
oise interferometry
cold atom clouds
Figure 1 Illustration of the atom detection scheme and the origin of quantum correlations.
a, The cloud of atoms is imaged to a detector plane and sampled by the pixels of a CCD
camera. Two pixels P1 and P2 are highlighted, each of which registers the atoms in a
column along its line of sight. Depending on their spatial separation d, their signals show
correlated quantum fluctuations, as illustrated in b. b, When two atoms initially trapped at
lattice sites i and j (separated by the lattice spacing a lat) are released and detected
independently at P1 and P2, the two indistinguishable quantum mechanical paths,
illustrated as solid and dashed lines, interfere constructively for bosons (or destructively
for fermions). c, The resulting joint detection probability (correlation amplitude) of
simultaneously finding an atom at each detector is modulated sinusoidally as a function of
d (black curve). The multiple wave generalization to a regular array of six sources with the
same spacing is shown in green. a.u., arbitrary units.
Figure 3.3 From Ref. [4]
ur Widera, Olaf Mandel,
-Universität, Staudingerweg 7,
atoms into states 0, 1. At large N this distribution becomes sharply peaked
at occupations N , N ]
.......................................................................................
anbury Brown and Twiss (HBT)
tions could be used to probe the
ticle source through quantum
quantum interference between
wo indistinguishable particles.
with their fermionic countercations, ranging from quantum
ntary particle physics6. Spatial
suggested7 as a means to probe
lated phases of ultracold atoms.
ement on the Mott insulator8–10
as it is released from an optical
g periodic quantum correlations
ns in the expanding atom cloud.
t the underlying ordering in the
pretation in terms of a multipleThe method should provide a
lex quantum phases of ultracold
15
.
elation analysis is now a basic tool
cations to the field of cold atoms
oncentrate on photon correlation
cs5,16. It was not until 1996 that
erate) bosonic atom clouds could
d by the observation of reduced
ion of local few-body correlations
y measure the spatial correlation
ons in a freely expanding atomic
ature.com/nature
8–10
!
!
NATURE | VOL 434 | 24 MARCH 2005 | www.nature.com/nature
!
!
© !""# Nature Publishing Group!
481
The interference of two independent condensates was observed in 1997 in
Ref. [2]. The related question of whether two independent light sources give rise
to interference was discussed much earlier in Ref. [7]11 .
The occurrence of interference fringes in a correlation function does not depend
upon Bose condensation, although the phenomenon is very striking in this case.
As another example, let’s consider the problem of noise correlations in time-offlight images of an expanded gas. As we saw above, the density distribution after
expansion reflects the initial momentum distribution. A crucial realization [1] was
that this observation extends not just to the average of the momentum distribution n(p) ≡ a†p ap , but also to its correlation functions. Thus even images from
a single experiment that seem to show only small fluctuations in density superimposed
on a smooth background can reveal information when the correlation
Figure 1 Illustration of the atom detection scheme and the origin of quantum correlations.
function
computed
(Figure
3.4)
a, The cloud of is
atoms
is imaged to a detector
plane and
sampled by the pixels of a CCD
camera.
Two pixels
P1 and
P2 are highlighted,
each of whichprepared
registers the atoms
a optical lattice in a Mott state,
In Ref.
[4],
atoms
were initially
in inan
column along its line of sight. Depending on their spatial separation d, their signals show
meaning
that
each
site
in
the
lattice
had
a
fixed
number of atoms (we will study
correlated quantum fluctuations, as illustrated in b. b, When two atoms initially trapped at
these
extensively
Chapter
??).andThe
lattice sitesstates
i and j (separated
by the lattice in
spacing
a lat) are released
detectedwavefunction of such a state may
independently
at P1 and P2, the two indistinguishable quantum mechanical paths,
then
be written
illustrated as solid and dashed lines, interfere constructively for bosons (or destructively
Y †amplitude) of
for fermions). c, The resulting joint detection probability (correlation
ai |VACi
, of
(3.71)
simultaneously finding an atom at each detector is modulated sinusoidally
as a function
i of six sources with the
d (black curve). The multiple wave generalization to a regular array
same spacing is shown in green. a.u., arbitrary units.
where a† creates a particle localized at site 481
ri in the lattice, with (let’s say)
i !
© !""# Nature Publishing Group
11
This article opens by recalling Dirac’s comment from the introduction to his Principles of Quantum
Mechanics [3]: ‘Each photon then interferes only with itself. Interference between two different
photons never occurs’. Since Dirac invented second quantization (though the term itself is Jordan’s)
to describe many photon states we give him a pass.
!
#$%&'!()*!$(&+!%,!*('-).!/+0+&1(&'!%&!)-+(2!*0#)(#$!*+0#2#)(%&!d3!)-+(2!*('&#$*!*-%4!5%22+$#)+1!
67#&)78!,$75)7#)(%&*3!#*!($$7*)2#)+1!(&!b.!b3!9-+&!)4%!#)%8*!(&()(#$$:!)2#00+1!#)!$#))(5+!*()+*!(!
#&1!;!<*+0#2#)+1!=:!)-+!$#))(5+!*0#5(&'!a$#)>!#2+!2+$+#*+1!#&1!1+)+5)+1!(&1+0+&1+&)$:!#)!?@!#&1!
?A3!)-+!)4%!(&1(*)(&'7(*-#=$+!67#&)78!8+5-#&(5#$!0#)-*3!($$7*)2#)+1!#*!*%$(1!#&1!1#*-+1!
$(&+*3!(&)+2,+2+!5%&*)275)(B+$:!,%2!=%*%&*!<%2!1+*)275)(B+$:!,%2!,+28(%&*>.!c3!C-+!2+*7$)(&'!
;%(&)!1+)+5)(%&!02%=#=($():!<5%22+$#)(%&!#80$()71+>!%,!*(87$)#&+%7*$:!,(&1(&'!#&!#)%8!#)!+#5-!
1+)+5)%2!(*!8%17$#)+1!*(&7*%(1#$$:!#*!#!,7&5)(%&!%,!d!<=$#5D!572B+>.!C-+!87$)(0$+!4#B+!
'+&+2#$(E#)(%&!)%!#!2+'7$#2!#22#:!%,!*(F!*%725+*!4()-!)-+!*#8+!*0#5(&'!(*!*-%4&!(&!'2++&.!#.7.3!
Second Quantization
#2=()2#2:!7&()*.!
42
Gaussian
Figure 2!G(&'$+!*-%)!#=*%20)(%&!(8#'+!(&5$71(&'!67#&)78!,$75)7#)(%&*!#&1!)-+!#**%5(#)+1!
*0#)(#$!5%22+$#)(%&!,7&5)(%&.!a3!C4%H1(8+&*(%&#$!5%$78&!1+&*():!1(*)2(=7)(%&!%,!#!I%))!
Figure 3.4 From Ref. [4]
(&*7$#)(&'!#)%8(5!5$%71!5%&)#(&(&'!J!@KL!#)%8*3!2+$+#*+1!,2%8!#!)-2++H1(8+&*(%&#$!%0)(5#$!
$#))(5+!0%)+&)(#$!4()-!#!$#))(5+!1+0)-!%,!LKE2.!C-+!4-()+!=#2*!(&1(5#)+!)-+!2+5(02%5#$!$#))(5+!
*5#$+!l!1+,(&+1!(&!+67#)(%&!<A>.!b3!M%2(E%&)#$!*+5)(%&!<=$#5D!$(&+>!)-2%7'-!)-+!5+&)2+!%,!)-+!
wavefunction
(8#'+!(&!a3!#&1!'#7**(#&!,()!<2+1!$(&+>!)%!)-+!#B+2#'+!%B+2!NO!(&1+0+&1+&)!(8#'+*3!+#5-!%&+!
"
#
2
*(8($#2!)%!a.!c3!G0#)(#$!&%(*+!5%22+$#)(%&!,7&5)(%&!%=)#(&+1!=:!#&#$:*(&'!)-+!*#8+!*+)!%,!
1
(r − ri )
ϕi (r) =
exp −
(8#'+*3!4-(5-!*-%4*!#!2+'7$#2!0#))+2&!2+B+#$(&'!)-+!$#))(5+!%21+2!%,!)-+!0#2)(5$+*!(&!)-+!)2#0.!
3/2
2R2
(πR)
d3!M%2(E%&)#$!02%,($+!)-2%7'-!)-+!5+&)2+!%,!)-+!0#))+2&3!5%&)#(&(&'!)-+!0+#D*!*+0#2#)+1!=:!
(&)+'+2!87$)(0$+*!%,!l.!C-+!4(1)-!%,!)-+!(&1(B(17#$!0+#D*!(*!1+)+28(&+1!=:!)-+!%0)(5#$!
2+*%$7)(%&!%,!%72!(8#'(&'!*:*)+8.!
Exercise 3.3.4 Show that the correlation function of the occupancies of the
momentum states is
X
"!!
h: n(p)n(p0 ) :i =
ϕ̃∗i (p)ϕ̃i (p)ϕ̃∗j (p0 )ϕ̃j (p0 ) ± ϕ̃∗i (p)ϕ̃j (p)ϕ̃∗j (p0 )ϕ̃i (p0 ).
i,j
(3.72)
where ϕ̃i (p) is the Fourier transform of the spatial wavefunction. Evaluate
the Fourier transform and explain the structure of the noise correlations in
Figure 3.4.
3.3.3 A quantum measurement perspective
Turning this observation around suggests that after we have observed fringes,
the system is in a state of definite relative phase. It is useful to consider how
this happens, but in the simpler situation depicted in Figure 3.5. Atoms from
two condensates are released to pass through a beam splitter. Assuming that
there is some way to determine when k atoms have passed the splitter, without
measuring whether they came from condensate 0 or 1, we can ask for the resulting
probability that k+ arrive at the counter, with the other k− ≡ k − k+ passing
along the other arm of the beam splitter. The simplicity of this situation lies in
the fact that there are only two possible final positions for the atoms. Assuming
that these final states |±i are created by √12 [a0 ± a1 ]a†± (i.e. the beam splitter
3.3 Interference of condensates
43
Figure 3.5 Beam splitter configuration with two condensates
is 50 : 50 with no relative phase shift), the observation of k+ counts leaves the
condensate, initially in the Fock state |N/2, N/2i, in the state
Z π
dθ
k
k
[cos(θ/2)] + [sin(θ/2)] − |(N − k)/2, (N − k)/2iθ ,
[a0 +a1 ]k+ [a0 −a1 ]k− |N/2, N/2i ∝
2π
−π
(3.73)
where we have used the relationship between the Fock and phase states to resolve
the resulting state into phase states. We see that the effect of the measurement
is to create uncertainty in the relative number of atoms remaining in the two
condensates. From our discussion of the conjugate relationship between particle
number and phase, this means that the relative phase can become more precisely
known. At large k± , we can write the integral in Equation (3.73) in the stationary
phase approximation as
Z π
i
2
dθ h −k(θ−θ0 )2
k
(3.74)
e
+ (−) − e−k(θ+θ0 ) |(N − k)/2, (N − k)/2iθ ,
−π 2π
where θ0 is the solution in [0, π] of k+ = k cos2 (θ0 /2). This corresponds to the
value of the phase that we infer (up to a sign) from a measurement of the fraction
√
k+ /k. Equation (3.74) shows that the residual uncertainty in the phase is 1/ 2k.
The measurement of the phase is shot noise-limited : the fraction k+ /k estimates
the magnitude of a wavefunction of definite phase, but the discreteness of the
individual measurements provides a limit to how accurately we can know it. An
even more fundamental point is that a necessary condition for the phase to be a
sharp observable is that we are dealing with a large number of particles in the
first place.
An experiment that very closely resembles this situation has recently been
performed. In Ref. [8], the beam-splitter consists of the crossed laser set-up used
to perform Bragg spectroscopy, as described in Section 6.1. The input states
Second Quantization
44
correspond to two trapped condensates, as in Figure 3.5, and the two output
states correspond to momentum zero and p respectively. The number of atoms
kp ending up in p can be monitored through the number of photons transferred
from one beam to another. The rate of this process is given by Equation (6.10)
2π V02
[S(p, ω) − S(−p, −ω)] .
(3.75)
~ 4
The relation to our previous discussion is that we take k → ∞ with k+ /k → 0
such that the expected number of counts k+ is Γτ , with τ the time for which
the Bragg pulse is applied. The analogue of k+ = k cos2 (ϕ/2) is then the following formula relating the structure factor to the relative phase of two spatially
separated condensates:
Γ(p, ω) =
Exercise 3.3.5 Show that the structure factor of two condensates of
definite phase is.... (impulse approximation). Point out that it
creates a grating in momentum space
References
[1] Altman, E., Demler, E., and Lukin, M.D. 2004. Probing many-body states of ultracold
atoms via noise correlations. Physical Review A, 70(1), 13603.
[2] Andrews, MR, Townsend, CG, Miesner, H.J., Durfee, DS, Kurn, DM, and Ketterle, W. 1997.
Observation of Interference Between Two Bose Condensates. Science, 275(5300), 637–641.
[3] Dirac, P.A.M. 1981. The principles of quantum mechanics. Oxford University Press, USA.
[4] Fölling, S., Gerbier, F., Widera, A., Mandel, O., Gericke, T., and Bloch, I. 2005. Spatial
quantum noise interferometry in expanding ultracold atom clouds. Nature, 434(7032), 481–
484.
[5] Jordan, P., and Klein, O. 1927. Zum Mehrk
”orperproblem der Quantentheorie. Zeitschrift f
”ur Physik A Hadrons and Nuclei, 45(11), 751–765.
[6] Jordan, P., and Wigner, E. 1928.
”uber das paulische
”aquivalenzverbot. Zeitschrift f
”ur Physik A Hadrons and Nuclei, 47(9), 631–651.
[7] Magyar, G., and Mandel, L. 1963. Interference fringes produced by superposition of two
independent Maser light beams. Nature, 198(4877), 255–256.
[8] Saba, M., Pasquini, TA, Sanner, C., Shin, Y., Ketterle, W., and Pritchard, DE. 2005. Light
Scattering to Determine the Relative Phase of Two Bose-Einstein Condensates.
[9] Yang, CN. 1962. Concept of off-diagonal long-range order and the quantum phases of liquid
He and of superconductors. Reviews of Modern Physics, 34(4), 694–704.
4
The experimental system
We are going to try and give a lightning introduction to the most important
aspects of the atomic gases that are actually realized in experiments. The goal is
to provide justifications for the kinds of models that we will be considering in the
following lectures, and to gain some feel for when experimental reality is likely to
intrude!
We will be concerned with the properties of gases of neutral alkali atoms. The
number of atoms in typical experiments range from 104 to 107 : often N → ∞ will
be a good enough approximation. The atoms are confined in a trapping potential
of magnetic or optical origin, with peak densities at the centre of the trap ranging
from 1013 cm−3 to 1015 cm−3 .
As we just discussed in Chapter 2, the observation of quantum phenomena
like Bose-Einstein condensation requires a phase-space density of order one, or
nλ3dB ∼ 1. The above densities then correspond to temperatures
T ∼
~2 n2/3
∼ 100nK − few µK
mk
p
At these temperatures the atoms move at speeds of ∼ kT /m ∼ 1 cm s−1 ,
which should be compared with around 500 m s−1 for molecules in this room,
and ∼ 106 m s−1 for electrons in a metal at zero temperature. Achieving the
regime nλdB ∼ 1, through sufficient cooling is the principle experimental advance
that gave birth to this new field of physics.
It should be noted that such low densities of atoms1 are in fact a necessity.
We are dealing with systems whose equilibrium state is a solid (that is, a lump
of Sodium, Rubidium, etc.). The first stage in the formation of a solid would be
the combination of pairs of atoms into diatomic molecules, but this process is
hardly possible without the involvement of a third atom to carry away the excess
energy. The rate per atom of such three-body processes is 10−29 − 10−30 cm6 s−1 ,
leading to a lifetime of several seconds to several minutes2 . These relatively long
timescales suggest that working with equilibrium concepts may be a useful first
approximation.
1
2
Compare 1019 cm−3 for the number density of air molecules at ground level, and ∼ 1022 cm−3 for
atomic densities in liquids and solids.
This three-body rate is reduced by the appearance of Bose-Einstein condensation, further enhancing
the sample lifetime.
45
The experimental system
46
4.1 Atomic properties
4.1.1 Boson or Fermion?
Since the alkali elements have odd atomic number Z, we readily see that alkali
atoms with odd mass number are bosons, and those with even mass number are
fermions.
Alkali atoms have a single valence electron in an nS state, so have electronic
spin J = S = 1/2. Thus bosonic and fermionic alkalis have half integer and integer nuclear spin respectively. We list the following experimental ‘star players’
Bosons
87
Rb
23
Na
7
Li
Fermions
6
Li
40
K
Nuclear spin, I
3/2
3/2
3/2
1
4
4.1.2 The Zeeman effect
The Zeeman effect plays a crucial role in the trapping of atoms. Assuming that
we deal with atoms in a state of zero orbital angular momentum, the effect of
magnetic field is described by the Hamiltonian
HZ = AI · J + gµB B · J,
(4.1)
where I and J are the nuclear and electronic angular momenta respectively, and
the first term originates from the hyperfine interaction. We can ignore the Zeeman
effect of the nuclear spin, as the nuclear magneton is approximately me /mp ∼
1/2000 of the Bohr magneton.
We will consider this case I = 3/2, exemplified by 87 Rb, 23 Na, and 7 Li. Solving the Hamiltonian (4.1) gives energy levels labeled by the conserved quantum
numbers F (F = I + J) and mF , the component parallel to the field. In the
future, when we speak of an atomic species, we will normally mean one of these
hyperfine-Zeeman states. The zero-field splitting between the F = 2 and 1 states
is 2A, and we can define a crossover scale Bbf ≡ |A|/µB beyond which the complexities of the hyperfine coupling become unimportant. The full field dependence
of the energy levels is3
3
We shunt all the energy levels up by A/4 for convenience
4.1 Atomic properties
47
Figure 4.1 Magnetic field dependence of atomic states of an atom with
J = 1/2, I = 3/2
mF
2
1
0
−1
−2
F
2
2, 1
2, 1
2, 1
2
E(B)
A (1 + B/Bhf )
2
±A[1 + B/Bhf + (B/Bhf ) ]1/2
2 1/2
±A[1 + (B/Bhf ) ]
2
±A[1 − B/Bhf + (B/Bhf ) ]1/2
A (1 − B/Bhf )
Exercise 4.1.1 Prove this.
These are plotted in Figure 4.1. Many experiments are done in the regime
B Bhf , so a linear expansion of the above energies suffices.
1
(4.2)
E(B) = ± A + µB mF B ,
2
with plus (minus) for the upper (lower) multiplet. Since magnetic traps have a
local minimum in the field, it is the low field seekers with positive gradient that
can be trapped. In the present case these are F = 2, mF = 2, 1, 0, and F = 1,
mF = −1.
We will see that in general collisions between atoms can convert low field seekers
to high field seekers that are then lost from the trap. Two states are however of
special experimental importance in being immune to this process. They are the
doubly polarized state with F = I + 1/2, mF = F and the maximally stretched
state with F = I − 1/2, mF = −F .
4.1.3 Excited states and polarizabilty
A second approach to trapping atoms is to use the potential that they feel in the
presence of the electric fields created by a laser. The field polarizes the atoms,
giving them an electric dipole moment that in turn interacts with the field.
The experimental system
48
Let us start by considering the effect of a static electric field. Second-order
perturbation theory gives us an expression for the polarizability, defined through
the quadratic energy shift in the presence of an electric field ∆E = −αE 2 /2
α=2
X | hn | d · ε̂ | 0i |2
n
En − E0
,
d ≡ −e
X
rj .
(4.3)
j
We leave in the unit direction vector of the electric field ε̂ in order to avoid
the tensorial structure of α. It is convenient to write this result in terms of the
dimensionless oscillator strengths
fn0 =
2me (En − E0 )
| hn | d · ε̂ | 0i |2 .
e2 ~2
These satisfy the f-sum rule (or Thomas-Reiche-Kuhn sum rule)
X
fn0 = Z
(4.4)
n
Exercise 4.1.2 Prove this by considering [[H, di ] , di ] (no sum).
We have
α=
e2 X fn0
,
2
me n ωn0
(4.5)
with ωn0 = (En − E0 ) /~.
The response to external fields is mostly determined by the fundamental transition of the valence electron nS → nP (a doublet due to spin-orbit coupling). The
wavelength of this transition is in the range 500 − 700 nm. We can use these facts
to estimate the polarizability. First we assume that the valence electron states
are well-approximated by those of a single electron moving in a coulomb potential due to the nucleus and core electrons. Thus their oscillator strengths satisfy
Equation (4.4) with Z = 1. Next we neglect all but the nS → nP transition,
giving
1
.
α∼
(∆E)2
The result should be understood in atomic units with α measured in units of a30 ,
and energies in e2 /a0 ∼ 27.2 eV.
Can do this next bit much more nicely using Floquet...
Polarizability in the presence of an oscillating field requires a slightly different
framework, since energy is no longer conserved. Instead, consider writing the
formal solution to the time-dependent Schrödinger equation
−i~
∂
|Ψi = H(t) |Ψi ,
∂t
(4.6)
4.1 Atomic properties
49
in terms of the time evolution operator from initial time ti to final time tf
Z
i tf
U (tf , ti ) ≡ T exp −
H(t) dt ,
(4.7)
~ ti
where T denotes the operation of time ordering. If H(t) is periodic with period
n
T , then U (ti + nT, ti ) = (U (ti + T, ti )) for integer n. Since U (ti + T, ti ) is a unitary operator, it has a complete set of orthonormal eigenstates with unimodular
eigenvalues. Of course, we are free to take the initial time ti anywhere in the
cycle: if we switch to another point t0i the corresponding evolution operator is
related by unitary transformation
U (t0i + T, t0i ) = U (t0i + T, ti + T )U (ti + T, ti )U (ti , t0i ) = U † (ti , t0i )U (ti + T, ti )U (ti , t0i ),
and thus has the same eigenvalues. We write the eigenvalues as eiEα T /~ , in terms
of pseudoenergies {Eα }. If we denote the eigenstates of U (ti + T, ti ) by {|αiti }
then the eigenstates of U (t0i + T, t0i ) are {U (t0i , ti ) |αiti }.
The case of oscillating fields can be treated by considering the ground state to
ground state amplitude in second order perturbation theory in the field E(t) =
Eω e−iωt + E−ω eiωt , with E−ω = Eω∗ 4
Z t
1
dt1 dt2 h0 | T d(t1 ) · Eω d(t2 ) · Eω∗ | 0i e−iω(t1 −t2 ) + {ω → −ω}
h0 | 0it = 1 − 2
2~ 0
−i(ωn0 +ω)t
1
i
i X
t−
e
− 1 | h0 | d · Eω | ni |2
=1+ 2
~ n ωn0 + ω
ωn0 + ω
+ {ω → −ω}.
(4.8)
After time averaging the imaginary linear in t term can be thought of as a shift in
the energy of the ground state, leading to a phase factor e−i∆Et/~ ∼ 1 − i∆Et/~,
equal to
1 X | h0 | d · Eω | ni |2
∆E = −
+ {ω → −ω}.
~ n
ωn0 + ω
This gives us the (real part of the) dynamical polarizability through ∆E =
−α0 (ω) hE 2 (t)i /2
α0 (ω) =
X 2 (En − E0 ) | hn | d · ε̂ | 0i |2
2
n
2
=
2
(En − E0 ) − (~ω)
e X fn0
,
2
me n ωn0
− ω2
(4.9)
which generalizes the static results Equation (??) and Equation (4.5). Note that
in contrast to the static case, where an attractive force is always felt towards
4
Field-theoretic types may prefer to think of the self-energy of the atom at second order in the
‘dressing’ electric field
50
The experimental system
regions of high field intensity, the dynamical polarizability can be of either sign.
In particular, when the polarizability is dominated by a single transition we have
α0 (ω) ∼
| hn | d · ε̂ | 0i |2
,
~ (ωn0 − ω)
(4.10)
and the sign from positive to negative as we go from ω < ωn0 (red detuning) to
ω > ωn0 (blue detuning).
Finally, inclusion of a finite excited state lifetime Γe tends to broaden this
behaviour. The effect can be included by putting an imaginary part in the denominator of Equation (4.10) to obtain the complex polarizability
α(ω) ∼
| hn | d · ε̂ | 0i |2
.
~ (ωn0 − ω − iΓe )
(4.11)
In particular the imaginary part of this expression can be thought of as giving
the rate of transitions out of the ground state (an ‘imaginary part to the energy’
Im ∆E = −α00 (ω) hE(t)2 i)
1
2
Γg = − Im ∆E = α00 (ω) E(t)2
~
~
(4.12)
Excited state lifetimes are of order 10 ns.
Exercise 4.1.3 Consider a harmonically confined electron. The equation of motion for the dipole moment d = −er is
d̈ + ω02 d =
e2
E
me
Show that the expression for the classical polarizability is
α(ω) =
1
e2
me ω02 − ω 2
This correpsonds to Equation (4.5) with one oscillator strength equal to one
at ω0 . Verify that this is case in a quantum mechanical calculation.
4.2 Trapping and imaging
In the previous section we have discussed the two important features of the alkali
elements that make them such a versatile experimental system. The spin of the
valence electron allows magnetic trapping, while the wavelength of the nS → nP
transition means that lasers can be used for trapping and cooling. Cooling is
discussed in some detail in citeppethick2002. Here we discuss briefly trapping
and imaging, which are probably more important for the theorist seeking to
understand experimental papers.
4.2 Trapping and imaging
51
4.2.1 Magnetic traps
In free space B cannot attain a maximum, so magnetic traps work by creating a
field minimum in which the low field seeking states are confined. Most produce
an axially symmetric magnetic field of the form
1
1
(4.13)
|B(r, z, φ)| = B0 + αr2 + βz 2 .
2
2
Provided that the atoms move slowly, we can make an adiabatic approximation
and assume that the atoms remain in the instantaneous hyperfine-Zeeman state
appropriate to their current position, even though the direction of B(r) may
change. This is fine as long as |B(r)| does not become too small, which in practice means various strategies are required to plug such ‘holes’ where unwanted
transitions between species can occur. Details of the various types of magnetic
traps can be found in [9].
The relationship between (4.13) and the potential experienced by the atom
is then very simple in the linear regime described by Equation (4.2). There is
one subtlety that makes itself known only occasionally. The effective quantum
Hamiltonian of an atom of a particular species in the adiabatic approximation
does not simply involve a conservative potential due to the magnetic field, but
in general includes a gauge potential whose origin is the Berry phase accumulated by a varying direction of B(r). This potential is expressed in terms of the
instantaneous hyperfine-Zeeman state |α(r)i as
Aα (r) = i hα(r) | ∇ | α(r)i
(4.14)
Exercise 4.2.1 The magnetic field of a quadrupole trap is
B = Bz ẑ + B 0 (xx̂ − yŷ)
If Bz B 0 , we can think of the hyperfine states as being eigenstates of
mF , the component in the z-direction. Inversion of Bz would carry atoms
adiabatically from mF to −mF , as the direction of B rotates through π. The
rotation axis, however, depends on where they are in the trap, being about
an axis parallel to yx̂+xŷ. Show that the effect of this rotation is to multiply
the wavefunction by the position-dependent phase factor e−2imF φ , where φ
is the azimuthal angle. This technique has been used to create vortices in
Bose-Einstein condensates [6].
4.2.2 Optical traps
Optical traps work by focusing a laser to create a field maximum. If the laser
is red-detuned the result is a potential minimum for the atoms, as discussed in
Section 9.4. This type of trap is useful when we are interested in interactions that
depend on species or Feshbach resonances, which are tuned by magnetic field. In
such circumstances we don’t want the Zeeman energy confusing things.
52
The experimental system
In this context, it is useful to consider if the optical potential really is independent of species. Any species-dependence requires spin-orbit coupling to be taken
into account in the excited state. In the nP state spin-orbit causing a splitting into
nP1/2 and nP3/2 corresponding to total electron angular momentum of 1/2 and
3/2 respectively. Since these states contain Kramers doublets due to time-reversal
symmetry, the dipole matrix elements between these states and the ground state
are independent of the initial spin state of the electron for linearly polarized light.
Thus any effect will arise from the hyperfine splitting, which alters the detuning
from a particular transition. If the detuning is much larger than this splitting,
as it usually is, the effect is negligible. For circularly polarized light the matrix
elements can differ, and the greatest effect is achieved by tuning between nP1/2
and nP3/2 .
More details in [1]
Optical traps are usually made using far-detuned light with Γe /(ωn0 − ω) ∼
10−7 − 10−6 . Although the trapping potential is inversely proportional to the
detuning, the ground state lifetime Equation (4.12) goes like the inverse square.
Note that the absorption of even one photon would be a disaster for a sample,
heating it far from degeneracy.
4.2.3 Imaging
As we saw, the scale of the dependence of optical properties on frequency is
set by the excited state lifetime Γ and is typically a few MHz. This is much
less than the zero field hyperfine splitting on the GHz scale. Thus distinguishing
optically between different F values presents no problem. For the sublevels within
a multiplet, the dependence of transitions upon polarization can be exploited to
further enhance resolution. Usually it is possible to image the individual species
in a gas separately.
The images that one most frequently sees in experimental papers are simple
absorption images, in which light is absorbed by the gas, creating real transitions
and heating the gas, see Figure 4.2. Such measurements are therefore of a one-shot
character in that they destroy the sample.
The second kind of imaging is dispersive (phase-contrast) imaging that relies
on diffraction. Many images may be taken with this technique without much
heating. The creation of such images can be regarded as instantaneous.
4.3 Interactions
For the most part, interactions between alkali atoms in the parameter ranges of
experimental interest are highly amenable to theoretical analysis. The effective
range of the potential is always small compared to other length scales, and normally we are also in the dilute gas limit na3s 1 where as is the s- wave scattering
length. These happy circumstances go some way to explaining the popularity of
these systems with theorists.
4.3 Interactions
53
Figure 4.2 Absorption image of around 7 × 105 atoms just above, at, and
below the Bose- Einstein condensation temperature. The gas is allowed to
expand for 6 ms. Reproduced from [4]
4.3.1 Effective interaction between like species
Working within the standard Born-Oppenheimer approximation we consider the
interatomic potential describing the interaction between two alkali atoms. In general this potential is strongly dependent on the spin state of the valence electrons.
The singlet state has a far deeper minimum, of the order ∼ 5000K, than the triplet
state. This is because the two electrons in the singlet state can share an orbital
and form a covalent bond. By contrast, the minimum in the triplet potential
results from an interplay between the −C6 /r6 van der Waals attraction at large
distances, and a hardcore repulsion at short distances, see Figure 4.3.
Two atoms in the same hyperfine state clearly have a symmetric electron spin
wavefunction, so the triplet potential is the appropriate one5 . The van der Waals
1/4
potential defines a length r0 ≡ (2mr C6 /~2 ) , where mr is the reduced mass.
This scale, the typical extent of the last bound state in the potential, is of order
50 Angstroms, much smaller than the de Broglie wavelength. This allows us to
ignore all but s-wave scattering.
On general grounds, we expect all features in the scattering amplitude to be
at the high energy scale ∼ ~2 /2mr r02 . Then the low energy scattering of interest
is described simply by the s-wave scattering length, defined through the form of
the wavefunction in the relative displacement of two atoms
ψ(r) = const.
sin [k (r − as )]
.
r
(4.15)
The scattering length will prove to be a basic parameter of all theories of the
alkali gases. For all but the heaviest atoms theoretical calculation of scattering
5
In the next section we will consider interactions between different species, which in general will
involve both singlet and triplet potentials. As explained at the start of this chapter, however, the
strongly bound molecular states are slow to form so still have little effect, and singlet and triplet
scattering lengths can be roughly equal.
54
The experimental system
Figure 4.3 Sketch of the interatomic potentials for two alkali atoms with
valence electrons in singlet and triplet states
lengths is extremely difficult. They can, however, be reliably measured using photoassociative spectroscopy, see [9]. It is of the same order as the scale r0 introduced
above.
as normally enters into our theoretical considerations through the notion of the
pseudopotential. The idea is that provided that all other scales in the problem
are much larger than as — in particular this requires the smallness of the gas
parameter na3s — the expectation value of the interaction energy has the form6
Z
1 4πas ~2 X Y
drk δ̃(rij )|Ψ({r})|2 .
(4.16)
hHint i =
2 m i6=j
k
δ̃ (rij ) denotes a delta-function smeared on a scale much larger than as but much
smaller than all other scales. The integral in Equation (4.16) is just the (spatially
averaged) probability density for two particles to approach each other, and the
interaction energy is just proportional to this quantity summed over all pairs. It
is natural to expect that a pairwise form holds in the dilute limit and the quantity
summed over in then the energy of a pair in vacuo. To be absolutely clear about
the origin of Equation (4.16), I present an argument leading to it in more detail7 .
• At low densities interaction energy (meaning the expectation value of the interaction hamiltonian) should have a pairwise form.
6
7
This is a slight abuse of notation: the origin of this effective potential is both kinetic and potential
in general: see below
Alternative versions may be found in [7] and [8]. The interesting feature of Leggett’s argument is
that it seems to apply even when na3s 1 as long as nr03 1, e.g. near a Feshbach resonance.
4.3 Interactions
55
• Wavefunctions we will consider correspond to energy per particle Etyp much less
than ~2 /2ma2s . It’s reasonable that such wavefunctions satisfy the ‘boundary
condition’
Ψ({r}) ∼ A(1 − as /rij ),
p
at ~/ 2mEtyp rij as .
• In the case of a single pair, such a wavefunction has energy
4πas ~2 2
|A| ,
m
(4.17)
which is found by solving the Schrodinger equation in a spherical box of size
R as , then finding the normalization constant for this wavefunction. (don’t
forget the reduced mass!). Equation (4.17) is in fact the O(as /R3 ) part of the
energy, with the leading term being π 2 ~2 /4mR2 . Since it represents the leading
as dependence, however, it is reasonable to call this the ‘interaction energy’.
Note, however, that the energy can range from being all kinetic for a hard
sphere potential, to all potential for a very ‘soft’ potential. In any case, it
arises from a scale ∼ as and should be affected by long distance changes in the
wavefunction only through A.
• Finally, |A|2 corresponds to the value of the integral in Equation (4.16).
Several other ways of writing the pseudopotential are in common usage. One
is to dispense with the slightly cumbersome form of Equation (4.16) and write
the interaction Hamiltonian as a delta-function potential.
U (r) =
4πas ~2
δ(r),
m
(4.18)
dr φ† (r)φ† (r)φ(r)φ(r).
(4.19)
or in second quantized notation
Hint
1 4πas ~2
=
2 m
Z
Obviously this requires careful handling in light of the above. Another way of
approaching these difficulties is to ask how we can define a δ-function potential U0 δ(r) in three dimensions. The scattering amplitude in general satisfies the
integral equation
Z
1
dq U (k0 , q)F (k, q)
0
0
F (k, k ) = −U (k, k ) −
(4.20)
3
2
(2π) (q) − (k) − i0
where (k) = k 2 /2m. The δ-function potential can then be taken to be the limit
of the (non-translationally invariant!) separable potential
U (k, k0 ) = U0 g(k)g(k0 )
where g(k) is equal to unity at small k, but falls to zero at some cut-off scale. Then
the integral equation is solved trivially for the low energy scattering amplitude
The experimental system
56
F (k, k0 ) → −4π~2 as /m
1
m
−
=
2
4πas ~
U0
Z
dq g(k)
.
3
(2π) 2k
which is compatible with Equation (4.18), apart from the (divergent) second
term. This is just the momentum space version of our real space difficulties.
Finally, Equation (4.18) is sometimes written
U (r) =
4πas ~2
δ(r)∂r r.
m
This serves to remove the 1/r piece from the boundary condition Ψ({r}) ∼ A(1 −
as /rij ) when the expectation value is taken.
All of these complexities aside, by far the most important thing we will do
with the pseudopotential is use it to find the interaction energy for trial wavefunctions that have no additional correlations built in between particles (the
‘Gross-Pitaevskii’ approximation). Then the summand in Equation (4.16) is n/N ,
and we find the energy per particle
2πnas ~2
.
m
The remarkable thing is that, although we used the diluteness condition to derive
it, the pseudopotential can be used to compute the next order in the ground state
1/2
energy of a system of bosons as an expansion in (na3s )
1/2
2πnas ~2 E/N =
1 + α na3s
+ ... ,
(4.21)
m
√
where α = 128/15 π.
E/N =
4.3.2 Interaction between species
The natural generalization of the pseudopotential Equation (4.19) to several
species is
Z
1 X 4πaαβγδ ~2
dr φ†α (r)φ†β (r)φγ (r)φδ (r).
Hint =
(4.22)
2 αβγδ
m
Bose (Fermi) statistics allows us to take aαβγδ = (−)aβαγδ = (−)aαβδγ 8 . Where
B Bhf and the hyperfine splitting is large enough to rule out scattering to
other values of F , rotational invariance simplifies things considerably [3]. For the
case of the F = 1 multiplet, we have
aαβγδ =
8
1
[a1 δαγ δβδ + a2 Fαγ · Fβδ ± {α ↔ β}] ,
2
The definition of these quantities is again from the scattering state, where the incoming and
outgoing waves are now in the internal states |γi1 |δi2 ± |δi1 |γi2 and |αi1 |βi2 ± |βi1 |αi2
respectively
(4.23)
4.3 Interactions
57
where F is the total spin operator within the multiplet. Even when scattering
between multiplets is possible, the total angular momentum and projection mF =
mF 1 + mF 2 of the two particles is conserved.
We are now in a position to explain the importance of the low-field seeking
doubly polarized state with F = I + 1/2, mF = F and the maximally stretched
state with F = I − 1/2, mF = −F , introduced in Section 4.1.2. Atoms in the
doubly polarized state can only scatter into the same state, as no other states
have larger mF . Two atoms in the maximally stretched state could scatter so
that one ends up with mF = −F − 1, but these states lie in the F = I + 1/2
multiplet. Since we are concerned with positive splittings A on the scale 100 mK
- 1 K, collisions might not be expected to depopulate the maximally stretched
state. Transitions to other states within the F = I − 1/2 multiplet do however
occur due to the magnetic dipole interactions, see Section 4.3.5, but these are
typically much less frequent.
4.3.3 Three body recombination
4.3.4 The Feshbach resonance
If the centre-of-mass energy of two scattering atoms is close to the energy of a
bound state, the scattering amplitude can be strongly modified. This phenomenon
is called a Feshbach resonance. In recent years its exploitation by experimentalists
has made the strength of the interaction between atoms a continuously tunable
experimental parameter, something unthinkable in conventional condensed matter systems.
The idea is illustrated in Figure 4.4. The curves that we drew in Figure 4.3
are really just representative of the many that we could draw for the interatomic
potentials corresponding to different hyperfine states of the two atoms. As we
just explained, the relatively large hyperfine splitting makes it impossible for
either of the atoms to scatter at low energy into a higher multiplet (the ‘closed
channel’ of the diagram). Their wavefunctions will, however, in general hybridize
with bound or nearly bound states in these closed channels due the presence of
exchange interactions.
The simplest model for this kind of scattering is the two-channel model, that
accounts for one nearby bound state in one closed channel
X
X q
g X
+ ε0 b†q bq + √
bq a†1,q+p a†−1,−p + h.c (4.24)
H=
p a†s,p as,p +
2
V p,q
p,s
q
Here bq annhilates a molecule (bound state of two atoms in the closed channel),
and as,p annihilates an atom in the open channel. The energy of the closed channel
bound state is ε0 . We have introduced the two species s = ±1 so that we can
discuss either bosons or fermions: the atoms are distinguishable in either case 9 .
9
The same model can be applied to bosons of one species – the result for the scattering amplitude
below should be doubled – while fermions of the same species have no s-wave scattering. This
The experimental system
58
Figure 4.4 A Feshbach resonance is caused by hybridization of a closed
channel bound state with the open channel.
Exercise 4.3.1 Find the scattering amplitude for two atoms in the model
Equation (4.24).
Solution For two atoms the wavefunction in the centre-of-mass frame has
the form
"
#
X
†
†
†
|ψi = βb0 +
αp a1,p a−1,−p |VACi .
p
Substituting into Equation (4.24) gives the equations
g
2p αp + √ β = Eαp
V
g X
ε0 β + √
αp = Eβ.
V p
Eliminating β
2p αp +
X
g2
αp0 = Eαp .
V (E − ε0 ) p0
We look for a scattering state of the form
αp (E) = δp,p0 +
4π~2
f (E)
,
m 2p − E − i0
exemplifies the general relation for the scattering amplitude of identical bosons or fermions
f (θ) ± f (π − θ) in the s-wave case, when f (θ) = const
(4.25)
4.3 Interactions
59
where p0 is the wavevector of the incoming wave with 2p0 = E. Substituting
into Equation (4.25) gives
Z
g2
m
dp
f (E)
f (E) +
+
=0
3
(E − ε0 ) 4π~2
(2π) 2p − E − i0
(c.f. Equation (4.20)). In order to tame the singular behaviour of the integral,
we need to shift the detuning parameter ε0 by the infinite constant
Z
dp 1
2
,
(4.26)
ε0 → ε0 + g
3
(2π) 2p
to give
Z
g2
m
dp
1
1
f (E) +
= 0.
+ f (E)
−
3
(E − ε0 ) 4π~2
2p − E − i0 2p
(2π)
Taking real and imaginary parts now yields
#
"
√
g2
m
m3/2 E
Re f (E) +
−
Im f (E) = 0
(E − ε0 ) 4π~2
4π
√
g2
m3/2 E
Im f (E) +
Re f (E) = 0.
(E − ε0 ) 4π
The final result for the scattering amplitude is
~γ
1
√
f (E) = − √
m E − ε0 + iγ E
(4.27)
with γ = g 2 m3/2 /4π.
The pole in f (E) is lies at real energies for ε0 < γ 2 /4, passing through zero when
ε0 = 0. When the pole lies at negative values of energy, its position corresponds
to the bound state energy (modified by coupling). When the pole is no longer at
negative energy we refer to a virtual state 10 . The scattering length f (0) = −a is
~γ 1
a = −√
,
m ε0
and displays the divergence characteristic of a Feshbach resonance as we pass from
positive to negative detuning, signaling the occurrence of a bound state11 . In a
sense the model we introduced can now be discarded. The form of the scattering
amplitude Equation (4.27) is in fact the most general one allowed at low energies
[5]: the model was just a convenient physical realization where this two-parameter
asymptotic description turns out to be exact. The γ parameter characterizes the
width of the resonance. If we were only interested in energies very low compared
10
11
Although the pole is at real positive energies up to for 0 < ε0 < γ 2 /4, there is no singularity in the
scattering amplitude as the pole is not on the physical sheet, see [5].
A background scattering length is normally added to this to include the effect of non-resonant
scattering.
60
The experimental system
to γ 2 , a single parameter description in terms of the scattering length only would
suffice.
Feshbach resonances have been found in a variety of alkali atoms. Those in the
fermions 6 Li and 40 K have been exploited to great effect recently to probe the
BCS-BEC crossover that we will discuss later. The consensus view is that these
resonances are broad in the above sense, so that a single parameter description
is possible. Tuning through the resonance is achieved by varying an applied field,
as the atom and molecule generally have different magnetic moments.
The divergence of the gas parameter na3s implied by the approach to a Feshbach resonance suggests that sample lifetime will be dramatically reduced, as
three-body collisions leading to the formation of diatomic molecules become more
frequent. One should bear in mind, however, that such processes are a function
of statistics. In a fermionic system a Feshbach resonance for scattering between
two species can occur in the s-wave channel, but of any three particles scattering
in this way, two will be of the same species. The formation of a molecule of size r0
is then suppressed by some power of r0 q, for q a typical wavevector. This power
turns out to be about 3.33, so that even when as > 3000 Angstroms, the molecule
lifetime can be > 100 ms.
4.3.5 Dipolar interactions
Finally, there is a dipole-dipole interaction between the valence electron spins of
two atoms
2
µ0 (2µB )
[S1 · S2 − 3 (S1 · r̂) (S2 · r̂)] .
(4.28)
Umd =
4πr3
This interaction only conserves total angular momentum (orbital plus spin), so
can lead to a decay from the doubly polarized or maximally stretched states. It is
possible to show, however, that the rate for such processes is slow and in general
does not limit the lifetime in experiments on alkali atoms [9].
Since the creation of a Bose-Einstein condensate of Chromium atoms in 2005
[2], the magnetic dipole interaction has returned to prominence. Chromium has a
dipole moment of 6µB , six times larger than the alkalis, so the dipole interaction
is 36 times stronger. Recent theoretical work has focussed on understanding the
consequent properties of the condensate.
References
[1] Deutsch, Ivan H., and Jessen, Poul S. 1998. Quantum-state control in optical lattices. Phys.
Rev. A, 57(3), 1972–1986.
[2] Griesmaier, A., Werner, J., Hensler, S., Stuhler, J., and Pfau, T. 2005. Bose-Einstein condensation of chromium. Physical review letters, 94(16), 160401.
[3] Ho, T.L. 1998. Spinor Bose Condensates in Optical Traps. Physical Review Letters, 81(4),
742–745.
[4] Ketterle, W. 2002. Nobel lecture: When atoms behave as waves: Bose-Einstein condensation
and the atom laser. Reviews of Modern Physics, 74(4), 1131–1151.
4.3 Interactions
61
[5] Landau, L. D., and Lifshitz, E. M. 1977. Quantum Mechanics. Butterworth-Heinemann.
[6] Leanhardt, AE, G
”orlitz, A., Chikkatur, AP, Kielpinski, D., Shin, Y., Pritchard, DE, and Ketterle, W. 2002.
Imprinting vortices in a Bose-Einstein condensate using topological phases. Physical Review
Letters, 89(19), 190403.
[7] Leggett, A.J. 2001. Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Reviews of Modern Physics, 73(2), 307–356.
[8] Lieb, Eliot H. 1965. The Bose Fluid. In: Brittin, W. E. (ed), Lectures in Theoretical Physics,
vol. VII C. University of Colorado Press.
[9] Pethick, C., and Smith, H. 2002. Bose-Einstein condensation in dilute gases. Cambridge
Univ Pr.
Part II
Condensates of bosons and fermions
63
5
Static properties of Bose condensates
In this chapter we explore the properties of Bose condensates using a very simple
and versatile variational wavefunction proposed by Gross and Pitaevskii [1, 2]
that is capable of describing the effect upon the ground state of (sufficiently
weak) repulsive interactions between the particles. Both of these early works
were concerned with the structure of quantized vortex lines in the condensate.
Undoubtedly this is because this is an obviously important situation where the
nonuniformity of the condensate plays a vital role. In the case of trapped atomic
gases, the trapping potential provides another source of inhomogeneity. We will
explore both of these applications in this chapter.
5.1 The Gross–Pitaevskii approximation
The ground state of a noninteracting Bose gas has the form
Ψ(r1 , r2 , . . . , rN ) =
N
Y
ϕ0 (ri ),
(5.1)
i
where ϕ0 (r) is the (normalized) ground state wavefunction for a single particle.
The idea behind the Gross–Pitaevskii approach is to describe an interacting Bose
gas using a variational wavefunction of the same (totally symmetric) product
form. That is, the total energy of the system is written as a functional of ϕ0 (r),
after which the function ϕ0 (r) is varied to find the lowest possible energy. Solving
the variational problem therefore calls for the calculus of variations, rather than
simple multivariable calculus as in the case of a discrete number of variational
parameters.
The Hamiltonian of the N particle system is
X
N X
~2 2
∇ + V (ri ) +
U0 δ(ri − rj ),
(5.2)
H=
−
2m i
i<j
i=1
2
where we have introduced the interaction parameter U0 ≡ 4π~
as for brevity. The
m
expectation value of the energy in this state is
2
Z
Z
~
1
2
2
hHi = N dr
|∇ϕ0 | + V (r)|ϕ0 (r)| + N (N − 1)U0 dr|ϕ0 (r)|4 . (5.3)
2m
2
For large N , we can neglect the difference between N and N + 1. Now we want
65
66
Static properties of Bose condensates
to extremize this functional with respect to ϕ0 (r), keeping ϕ0 (r) normalized1 . To
do this we introduce a Lagrange multiplier µN (the factor of N will become clear
shortly) and extremize the functional
Z
hHi − µN dr|ϕ0 (r)|2 .
To find the extremum requires a standard application of the calculus of variations,
yielding the equation2
~2 2
2
(5.4)
∇ − µ + V (r) + N U0 |ϕ0 (r)| ϕ0 (r) = 0.
−
2m
It’s convenient
to deal with the stray factor of N in Equation (5.4) by defining
√
ϕ(r) ≡ N ϕ0 (r). ϕ(r) is known as the condensate wavefunction or order parameter. Later we will give a more general definition that does not depend on
the above variational approximation. We have thus obtained the Gross–Pitaevskii
equation
~2 2
2
∇ − µ + V (r) + U0 |ϕ(r)| ϕ(r) = 0.
(5.5)
−
2m
µ. This is done by demanding that
RIt remains2 to fix the Lagrange multiplier
R
dr |ϕ(r)| = N . Since hHi − µ dr|ϕ(r)|2 = hHi − µN was extremized under
general variations, including those that changed N , we must have
∂ hHi
(5.6)
∂N
so that µ is identified with the chemical potential.
A fundamental effect of the nonlinearity of the GP equation is that there exists
a length scale set by the typical value of |ϕ(r)|2 ∼ n and the interaction strength
−1/2
2mnU0
−1/2
= (8πnas )
.
(5.7)
ξ≡
2
~
µ=
This healing length determines the scale over which ϕ(r) is disturbed by the
introduction of a localized potential of scale ξ. It is a fundamental length scale
in the system.
Exercise 5.1.1 Show that near a hard wall, where the condensate wavefunction
goes to zero, ϕ(r) is given by
x
ϕ(x) = ϕ∞ tanh √
2ξ
√
where x is the distance from the wall, and ϕ∞ = n∞ is fixed by the density
of the condensate far from the wall.
1
2
hΨ | H | Ψi
Alternatively, one could extremize hHi = hΨ | Ψi , but this is not so convenient
A standard question raised by those applying the usual formalism to complex functions for the first
time is: how can I treat the variations due to δϕ0 (r) and δϕ∗0 (r) as independent? The answer is that
these quantities are related by complex conjugation, so if one vanishes, so does the other.
5.2 Condensates in trap potentials
67
Note that in the dilute limit when na3s 1, ξ the interparticle separation.
The fact that Ψ(r) varies on such long scales compared to the distance between
particles is another physical justification for the present mean-field approach3 . A
typical value of ξ may be around 4000
√ Angstoms.
In a uniform system with ϕ(r) = n, the GP energy density hHi /V = n2 U0 /2
provides us with a formula for the sound velocity via the hydrodynamic relation
c2s =
n ∂ 2 (E/V)
nU0
=
2
m ∂n
m
(5.8)
√
Note that mcs = ~/ 2ξ.
With the ansatz Eq. (5.1) for the wavefunction, we can obtain various observables without difficulty. The particle density is just
ρ(r) = g(r, r) = |ϕ(r)|2 .
While, using Equation (3.35), the current density is
~
[ϕ∗ (r) (∇ϕ(r)) − (∇ϕ∗ (r)) ϕ(r)]
(5.9)
2m
Alternatively, one may write this is terms of the velocity field of the gas, using
j = nv, and the decomposition of ϕ(r) into magnitude and phase
p
ϕ(r) = n(r)eiθ(r) .
(5.10)
j(r) = −i
Then Equation (5.9) defines the superfluid velocity
~
∇θ.
(5.11)
m
The name is to distinguish this contribution from that arising from thermal excitations. For the moment we are considering zero temperature, so this is all there
is. Finally, the total z-component of angular momentum is
Z
Lz = −i~ dr|ϕ(r)|2 ∂φ ϕ(r)
(5.12)
vs ≡
where φ is the angular coordinate in cylindrical coordinates.
5.2 Condensates in trap potentials
5.3 Rotational states: A simple example
Consider a Bose gas on a one-dimensional ring of radius R. The single particle
eigenstates are
1
ϕ` (θ) = √
ei`θ .
(5.13)
2πR
3
In real systems ξ n−1/3 is actually a severe exaggeration, as their ratio is only ∼ (na3s )−1/6 , and
the gas parameter is maybe 10−4 . Recall from Section 4.3.1, however, that the expansion parameter
justifying the present approximation is (nas )1/2 , which is still small.
neously (<100 !s) extinguishing the toroidal trap. The
persistent flow after 10 s, as seen in Fig. 2. This mis
initial hole in the BEC [see Fig. 1(b)] is filled in during
ment is due to a relative drift (0:4 !m=s) betwee
expansion [23]. In Fig. 1(d), we show the corresponding
position of the plug beam and the center of the
TOF image of a circulating BEC after transfer of @ of
magnetic trap, which drifts due to thermal cycling d
OAM. The hole is clearly visible, providing a simple way
each realization of the experiment. We substantially
to detect circulation [22,24].
pensate for this drift by applying a linear voltage ra
In order to investigate persistent flow in a condensate,
the external coils to generate a linear temporal shift
we measured the decay of the circulation as a function of
location of the TOP trap center, but the residua
Static properties
of Bose
condensates
eventually
(after 10 s) leads to a loss of flow. Th
time in the toroidal68trap. For this experiment,
the plug
FIG. 1 (color). (a) Toroidal trap from the combined potentials of the TOP trap and Gaussian plug beam. (b) In situ image of a B
Figure
5.1 from
Fromthe
[3] toroidal trap. (d) TOF image of a circulating BEC, re
the toroidal trap. (c) TOF image of a noncirculating BEC
released
after transfer of @ of OAM.
Since the states have constant magnitude, these functions also solve the Gross–
Pitaevskii equation (after normalizing260401-2
to N instead of 1), as the non-linear term
is a constant, giving
µ=
~2 `2
+ U0 n.
2mR2
(5.14)
Now of course this situation is somewhat trivial, as angular momentum is a good
quantum number. Equation (5.12) tells us
Lz = N ~`,
that is, every particle has ` quanta of angular momentum. The following problem
explores what happens when this symmetry is broken.
Exercise 5.3.1 We are going to consider the above one-dimensional ring again,
but will restrict ourselves to the two lowest angular momentum states ` =
0, 1. We now wish to include, however, the effect of a small deviation from
cylindrical symmetry due to a potential V (θ), whose effect is to mix these
two states.
1.
If a†0 and a†1 create atoms in the ` = 0, 1 states, show that when restricted
to these states the Hamiltonian is
Hrot = ~ωc a†1 a1 − a†0 a0 + V0 a†0 a1 + h.c.
U0 † †
a0 a0 a0 a0 + a†1 a†1 a1 a1 + 4a†1 a†0 a0 a1
(5.15)
+
2V
and identify ωc and V0 .
If we introduce the Gross–Pitaevskii wavefunction
h
iN
χ
χ
cos eiϕ/2 a†0 + sin e−iϕ/2 a†1 |VACi ,
2
2
show that:
2. The order parameter has a node for χ = π/2. If V0 is due to a localized
potential, this node will coincide with the position of that potential.
5.4 A single vortex
3.
69
The GP variational energy is (up to a constant, and ignoring terms lower
order in N )
E(χ)/N = −~ωc cos χ + V0 sin χ +
nU0
sin2 χ,
2
while the angular momentum is
1
(1 − cos χ)
2
4. A metastable minimum exists for 2U0 > V0 (assuming U0 and V0 are both
much less than ~ωc ). That is, for small enough deviations from perfect
symmetry, metastable configurations are possible, and have their origin
in the repulsive interactions. The point χ = π/2 that corresponds to an
order parameter with a node is then a maximum of the energy.
5. Repeating the argument for a pair of angular momentum states `, ` + 1,
show that even when V0 goes to zero, metastable configurations are
2
only possible
pwhen nU0 >√` ~ωc . This corresponds to a critical velocity
of `/mR = 2nU0 /m = 2cs , and coincides (parametrically, at least),
with the famous Landau criterion.
6. Finally, consider what would happen for Fock states of the form
N0 † N1
1
√
a†0
a1
|textV ACi .
(5.16)
N0 !N1 !
L(χ)/N ~ =
This situation — macroscopic occupations of more than one single particle state — is sometimes called a fragmented condensate.
5.4 A single vortex
References
[1] Gross, E.P. 1961. Structure of a quantized vortex in boson systems. Il Nuovo Cimento
(1955-1965), 20(3), 454–477.
[2] Pitaevskii, LP. 1961. Vortex lines in an imperfect Bose gas. Sov. Phys. JETP, 13(2),
451–454.
[3] Ryu, C., Andersen, M. F., Cladé, P., Natarajan, Vasant, Helmerson, K., and Phillips, W. D.
2007. Observation of Persistent Flow of a Bose-Einstein Condensate in a Toroidal Trap.
Phys. Rev. Lett., 99(26), 260401.
6
Dynamics of condensates
6.1 Time-dependent Gross–Pitaevskii theory
For time dependent problems it is tempting to immediately write down
~2 2
∂ϕ(r, t)
−
∇ + Uext (r) + U0 |ϕ(r, t)|2 ϕ(r, t) = i~
.
(6.1)
2m
∂t
R
Note that this equation conserves the normalization N = dr|ϕ(r)|2 – all particles remain in the condensate. Eq. (6.1) may be derived by generalizing the
ansatz Eq. (5.1)
Y
Ψ({ri }, t) =
ϕ0 (ri , t).
(6.2)
i
Substitution into the time-dependent Schrödinger equation yields
i~
X ∂ϕ0 (ri , t) Y
i
"
X
i
−
∂t
#
ϕ0 (rj ) =
j6=i
X
Y
~2 2
∇i + Uext (ri ) + U0
δ(rk − ri ) ϕ0 (ri , t)
ϕ0 (rj )
2m
k6=i
j6=i
(6.3)
P
In order to get a closed equation for ϕ0 (r) we can replace the j6=i δ(rj − ri ) with
the expectation value of the density N |ϕ0 (ri )|2 evaluated with Eq. (6.2). With
this replacement, Eq. (6.3) is satisfied if Eq. (6.1) is, as long as we normalize ϕ(r)
to N .
The simplicity of this derivation is of course deceptive. The sleight of hand
comes at the last stage. This is somewhat clearer if we pass to an orthogonal basis
of single-particle states of which ϕ0 (r) (at some reference time) is a member. We
write the boson field operator in terms of these states
X
ψ(r) =
ϕn (r)aα .
α
Then the interaction Hamiltonian has the form
U0 X
Hint =
Mαβγδ a†α a†β aγ aδ ,
2 αβγδ
70
(6.4)
6.2 Bogoliubov–de Gennes equations
71
where the matrix elements Mαβγδ are
Z
Mαβγδ = dr ϕ∗α (r)ϕ∗β (r)ϕγ (r)ϕδ (r).
Now applied to the ansatz Eq. (6.2), we can see that
R it is the term in Eq. (6.4)
with α = β = γ = δ = 0 that gives 21 N (N − 1) U0 dr|ϕ0 (r)|4 times the original
wavefunction, and is therefore just this that is kept in the GP approximation.
One might worry that we are throwing away all sorts of complexity at this stage,
but in fact the only neglected terms correspond to α, β 6= γ = δ = 0. You should
satisfy yourself that these have the form
Y
Sϕα (r1 )ϕβ (r2 )
ϕ0 (rj ),
(6.5)
j6=1,2
where S denotes the operation of symmetrizaton. The inclusion of such effects is
thus quite tractable, but it will have to wait until the next (Bogoliubov) stage
of approximation. In the equilibrium state, their effect is to give the quantum
depletion of the condensate
fraction, leading to N0 < N , even at zero temperature.
p
The effect is of order na3s , so it is reasonable that the present approximation is
justified when this is small [? ].
6.2 Bogoliubov–de Gennes equations
The Gross–Pitaevskii theory therefore gives a very straightforward and appealing route to the computation of observables in time-dependent situations. A great
deal of intuition may be obtained from examining the dynamics of small deviations δϕ(r, t) of ϕ(r, t) from some reference solution ϕ0 (r, t), satisfying
∂ϕ(r, t)
~2 2
∇ + Uext (r) δϕ(r, t)+2U0 |ϕ0 (r, t)|2 δϕ(r, t)+U0 ϕ20 (r, t)δϕ∗ (r, t) = i~
.
−
2m
∂t
Note that the nonlinear term couples δϕ(r) and δϕ∗ (r). The presence of the chemical potential in Eq. (5.5) means that the solution of Eq. (6.1) corresponding to to
a solution of the time-independent problem is ϕ0 (r, t) = ϕ0 (r)e−iµt/~ . Introducing
the harmonic solution
δϕ(r) = e−iµt/~ u(r)e−iωt + v ∗ (r)eiωt ,
one obtains easily the Bogoliubov–de Gennes equations (BdG equations)
~2 2
2
~ωu(r) = −
∇ + Uext (r) − µ + 2U0 |ϕ0 (r, t)| u(r) + U0 ϕ20 (r)v(r)
2m
~2 2
2
−~ωv(r) = −
∇ + Uext (r) − µ + 2U0 |ϕ0 (r, t)| v(r) + U0 ϕ∗2
(6.6)
0 (r)u(r).
2m
In free space µ = nU0 , and plane wave solutions of Eq. (6.6) have the dispersion
relation ~ω(k) = E(k)
1/2
E(k) ≡ (k) (k) + 2mc2s
,
(6.7)
72
Dynamics of condensates
where we use the expression Eq. (5.8) for the speed of sound found earlier. This
is the famous Bogoliubov spectrum that we will encounter again shortly. At k ~/mcs (kξ 1) it has the linear form E(k) = cs k, crossing over to the free
particle spectrum at higher momentum.
Application: Bragg spectroscopy
We now apply this formalism to a description of the very useful experimental
technique known as Bragg spectroscopy. A schematic illustration of the set-up is
shown in Figure 6.2, involving a pair of lasers with different (angular) frequencies
ω1,2 and wavevectors q1,2 . There are two ways to describe the resulting perturbation experienced by the gas. The first is to consider the potential V (r) = −αE 2 /2
experienced by the atoms, arising from the electric field intensity. The superposition of the two beams will lead to an oscillatory cross term, taking the form of
a traveling wave
Vosc (r, t) = V0 cos (Q · r − Ωt) ,
where Q ≡ k1 − k2 and Ω ≡ ω1 − ω2 . A second, quantum mechanical viewpoint
harkens back to the microscopic expression for the AC polarizablity in Equation
(4.9), which originates from a second order process: the virtual absorption and
then re-emission of a photon. The cross term therefore corresponds to the transfer
of a photon from one beam to the other. Though the atom is returned to its
electronic ground state, the difference in wavevector and frequency of the beams
means that momentum and energy are imparted to the gas.
To probe the collective behavior
1
. This allows us to enter a regime of ω, q that probes the collective behaviour
of the system ~q/m ∼ cs ∼ 1 cm s−1 , ~ω ∼ h × 1kHz. The BdG equations can
be used to compute the resulting density response
δn(r, t) = |ϕ0 (r, t) + δϕ(r, t)|2 − |ϕ0 (r, t)|2 ∼ ϕ∗0 (r, t)δϕ(r, t) + ϕ0 (r, t)δϕ∗ (r, t),
(6.8)
1
2 /2m on absorption of an optical photon is
To put some figures to it, the typical recoil energy ~2 kop
on the kHz scale
6.2 Bogoliubov–de Gennes equations
73
giving
δn(q, ω) = −nV0
E(q)2
(q)
V0
≡ D(q, ω).
2
2
− ~ (ω + i0)
2
(6.9)
As usual, the imaginary part of this response function describes the absorption of
energy from the perturbing field. In more quantum mechanical terms, quanta are
created when their energy E(q) and momentum matches the change in energy
and momentum of photons scattering from one beam to the other. In terms of
the many body eigenstates {|ni} and eigenenergies {En }, the golden rule gives
the rate for this process as
2π V02 X δ (~ω − En ) | n eiq·ri 0 |2 + δ (~ω + En ) | n e−iq·ri 0 |2 ,
Γ(q, ω) =
~ 4 n,i
(6.10)
so that the rate of energy absorption is ~|ω|Γ(q, ω). Note that in this situation
energy can be absorbed for positive or negative ω as photons are scattered from
the more energetic beam to the less energetic one. Comparing with the response
function Eq. (6.9)
X
1
(Vol) Im D(q, ω) = −
δ (~ω − En ) | n eiq·ri 0 |2 − δ (~ω + En ) | n e−iq·ri 0 |2 ,
π
n,i
(6.11)
which implies that all of the absorption comes from a single transition with2
X N (q)
| n eiq·ri 0 |2 =
.
(6.12)
E(q)
i
P
P
Noting that i eiq·ri is just a Fourier component ρq of the density ρ(r) = i δ(r−
ri ), we can integrate over positive ω in Eq. (6.11) to obtain the line strength of
the resonance
Z ∞
1
− (Vol)
dω Im D(q, ω) = h0 | ρq ρ−q | 0i
π
0
P
where we used the completeness relation to get rid of
α . This FluctuationDissipation relation relates the response function that we found to the density
fluctuations in the ground state of our system. Our explicit form Eq. (6.9) implies
h0 | ρq ρ−q | 0i =
N (q)
N ~q
→
N,
E(q)
2mcs
p ~/ξ.
(6.13)
The only problem is that our original ground state ansatz Eq. (??) disagrees with
this result. Because there are no correlations between particles in that state, its
density fluctuations are normal, even at low p
hGP | ρq ρ−q | GP i = N,
while the result Eq. (6.13) only reaches this value at large p3 .
2
c.f. Problem ??. The only difference is the colossal ratio (∼ 1011 ) of energy scales in the two
contexts!
Dynamics of condensates
74
Nevertheless, we will see that Eq. (6.13) is correct. The time-dependent GP
approximation, though adequate for obtaining the dynamics of expectation values
of such one-body quantities such as the density, fails to describe the quantum
fluctuations implied by this dynamics through general relations like Eq. (6.11).
The remedy to this situation is the Bogoliubov approximation.
Before closing, we should mention the effect of finite temperature. As long as
the condensate fraction is close to unity, meaning that both quantum depletion
(see next section) and thermal excitation out of the condensate are small, the
GP approximation and be justified in both its static and time-dependent forms.
Obviously, the relation just explored between response and fluctuations has to be
modified when the initial state of system may be different from the ground state,
as we assumed in Eq. (6.10).
6.3 Interlude: structure factors and sum rules
Let’s take a moment to relate the analysis of the previous section to some more
general formalism. The expression on the right hand side of Eq. (6.11) is related
to the dynamical structure factor S(q, ω), defined as
X Eα − E0
| hn | ρq | 0i |2 .
(6.14)
S(q, ω) =
δ ω−
~
n
Using completeness of the eigenstates, we then have
Z ∞
dω S(q, ω) ≡ S(q) = h0 | ρq ρ−q | 0i ,
0
where S(q) is called the static structure factor. S(q, ω) satisfies the following two
sum rules (see, for instance, Ref. [? ])
Z ∞
N ~2 q 2
~ωS(q, ω) =
2m
0
Z ∞
N
S(q, ω)
lim
=
,
(6.15)
q→0 0
~ω
2mc2s
known as the f-sum and compressibility sum rules, respectively. They are the
analogues of Eq. (4.4) and Eq. (??).
Exercise 6.3.1 Convince yourself that the above form of the f-sum rule coincides with that in Exercise 3.2.3.
Exercise 6.3.2 Show that S(q, ω) is the Fourier transform of
hρ(r, t)ρ(r0 , t0 )i ,
3
(6.16)
We should point out that the finiteness of density fluctuations at q → 0 is not inconsistent with the
value exactly at q = 0 being zero, as it will be for a wavefunction describing a fixed number of
particles.
6.4 Hydrodynamics of condensates
75
and that the static structure factor S(q) is the Fourier transform of the
same, but with respect to r only, with t = t0
Exercise 6.3.3 The thermodynamic compressibility is defined as4
κ≡−
1 ∂V
.
V ∂p
At zero temperature, the inverse compressibility (or bulk modulus), can be
written as
∂ 2(E/V )
κ−1 = n2
∂n2
In the present case Eq. (6.12) implies
N (q)
E(q)
,
S(q, ω) =
δ ω−
E(q)
~
(6.17)
and you should check that this satisfies both sum rules.
Exercise 6.3.4 Use the sum rules Eq. (6.15) and the Cauchy-Schwartz inequality | hA | Bi | ≤ |A||B|, interpreting the integrals as inner products, to derive
Onsager’s inequality
N ~q
S(q) ≤
(6.18)
2mcs
Note that Eq. (6.13) shows that in the present case the inequality is saturated.
This is the case (as should be fairly obvious from the proof) whenever S(q, ω)
consists of a single mode.
6.4 Hydrodynamics of condensates
Galilean invariance
Let us go back to the time-dependent Gross–Pitaevskii theory describing the
dynamics of a condensate
∂ϕ(r, t)
~2 2
2
∇ + Uext (r) + U0 |ϕ(r, t)| ϕ(r, t) = i~
,
(6.19)
−
2m
∂t
and ask what happens when we pass to a reference frame moving with relative
velocity −v to the original frame. One can verify directly that if ϕ(r, t) is a
solution of Eq. (6.19) then
i
1
2
ϕ(r − vt, t) exp
mv · r − mv t ,
(6.20)
~
2
is also a solution. Eq. (6.20) gives the transformation property of the condensate
wavefunction under Galilean transformations. Upon making the decomposition
4
Strictly one should specify what is kept constant as we compress the system (usually temperature or
entropy), but this distinction is not important at zero temperature where both vanish
Dynamics of condensates
76
√
ϕ = neiϕ , we see that this implies a transformation law for the phase of the
condensate
1
1
2
ϕ→ϕ+
mv · r − mv t .
(6.21)
~
2
Recalling the identifications5
~
∇ϕ
µ = −~ϕ̇,
m
we have the transformation laws for the superfluid velocity and chemical potential
1
vs → vs + v
µ → µ + mv2 ,
(6.22)
2
which seems sensible. The general character of these transformation laws leads us
to suppose that they are more general that the equation of motion Eq. (6.19) that
we started from. Indeed, we could have made exactly the same arguments for the
equation of motion of the Bose field ψ(r, t), valid for arbitrary density-density
interactions.
vs =
6.5 Hydrodynamic description
We can make the hydrodynamic character of the time-dependent GP equation
more explicit. If we rewrite ∂|ϕ|2 /∂t as the difference of Eq. (6.19) and its complex
conjugate we get the continuity equation
∂n
+ ∇ · [nvs ] = 0,
(6.23)
∂t
while ϕ∗ ϕ̇ − ϕ̇∗ ϕ, which involves the sum of the equation and its conjugate, yields
√
1
~2
√ ∇2 n + U0 n + Uext = 0.
~θ̇ + mvs2 −
2
2m n
(6.24)
If n varies sufficiently slow in space – in practice this means slower than the
−1/2
√
0
healing length ξ ≡ 2mnU
– we can drop the ∇2 n term. Taking the gradient
2
~
of the resulting equation gives
1
2
mv̇s + ∇
mvs + µ(n) + Uext = 0,
(6.25)
2
where we wrote µ(n) = nU0 . For the static case we have
µ(n(r)) + Uext (r) = µ0 ,
(6.26)
which defines the Thomas–Fermi approximation, widely used to compute density
profiles of trapped gases.
The equation Eq. (6.25) coincides with the Euler equation for the flow of a
non-viscous fluid, usually written as
ρ [∂t + v · ∇] v + ∇P = 0,
5
in the lab frame
6.5 Hydrodynamic description
77
where we used ρ = mn, the mass density, and v ∧ ∇ ∧ v = ∇ (v2 ) /2 − (v · ∇) v.
The pressure gradient is introduced through ∇P = n∇µ, which follows from the
Gibbs-Duhem relation. One should not forget, however, that the velocity field is
in the present case irrotational. Note that in the one-dimensional case that we
will consider in Chapter ?? there is no vorticity, so this difference disappears.
It is illuminating to consider the linearization of the equations Eq. (6.23) and
Eq. (6.25). In many ways the analysis is clearer than the linearization of the
TDGP theory considered in Section 6.1. Assuming that n(r, t) = n0 + δn(r, t),
and that vs is of the same order as δn, one can easily derive the wave equation
¨ − c2 ∇2 δn = 0,
δn
s
with c2s = n0 U0 /m giving the sound velocity that we found before. Obviously this
description is not Galilean invariant. The requirement that vs be small
√ amounts
to a choice of reference frame. What happens if you include the ∇ n term in
Eq. (6.24)?
As in the previous section, Eq. (6.23) and Eq. (6.25) are far more general than
the weak interaction limit of the TDGP theory, and even apply to situations in
low-dimension where there is no condensate. For instance, Eq. (6.25) could have
been obtained by taking the time and spatial derivatives of the transformation
law Eq. (6.21).
References
7
Bose condensation and superfluidity
In this chapter, we will give the concepts of Bose-Einstein condensation (BEC)
and superfluidity a more general formulation, not restricted to the approximations
used in the previous chapters. We’ll also take a look at the experimental status
of these distinct phenomena.
7.1 BEC and off-diagonal long-range order
BEC, according to Einstein’s original idea, means that a finite fraction of the
total number of particles in our system occupy one single-particle state below
some critical temperature. For the usual case of periodic boundary conditions
and translational invariance, this is the zero momentum state, with energy zero.
The distribution n(k) of the number of particles in each momentum state is then
n(k) = Nc δk + · · · ,
(7.1)
where Nc is O(N ), and f ≡ Nc /N is the condensate fraction. For non-interacting
bosons f = 0 at the condensation temperature and f = 1 at T = 0. For a uniform
3D system we have
"
3/2 #
T
.
f (T ) = N 1 −
Tc
This basic idea can be elaborated in a number of ways. For the case of atomic
gases, held in a trapping potential, it is necessary to have a definition that does
not depend upon translational invariance. The most commonly used one is based
on the behavior of the single particle density matrix. We defined this quantity
in Eq. (3.37) for a single quantum state (usually the ground state). Now we
generalize that definition to a statistical mixture of states
X g(r, r0 ) ≡
pn n ψ † (r)ψ(r0 ) n
(7.2)
n
≡ hhψ † (r)ψ(r0 )ii.
(7.3)
The double angular bracket notation on the second line is often used to convey
this ‘double’ average (quantum and statistical) 1 . Since the distribution n(k) =
hhφ†k φk ii, we identify this as the Fourier transform of ρ1 (r, r0 ) ≡ g1 (|r − r0 |)
78
7.2 Superfluidity defined
79
in a translationally invariant, isotropic system. The presence of a δ-function in
Eq. (7.1) shows that
Nc
g(r) →
,
|r| → ∞,
V
The non-vanishing of the right hand side is referred to as off-diagonal long range
order, or ODLRO2 . More generally, this property implies that in the spectral
resolution of the density matrix
X
nα χ∗i (r)χi (r0 , )
ρ1 (r, r0 ) =
i
there is at least one eigenvalue of order N . This may be seen by using a trial
eigenfunction χ0 (r) = 1/V 1/2 . The useful thing about this last criterion is that
it is very general, and doesn’t require translational invariance. For trapped gases
it is therefore useful to define BEC as the presence of such an eigenvalue. The
corresponding eigenfunction χ0 (r) is called the condensate wavefunction (though
it is the solution of no Hamiltonian).
As long as BEC is simple, meaning that there is only one thermodynamically
large eigenvalue (the multicomponent case will be discussed√in Section ??), the
order parameter of the BEC can be introduced as Ψ(r) = N0 χ0 (r), where N0
and χ0 (r) are the eigenvalue and eigenfunction respectively. Writing χ0 (r) as
|χ0 (r) = |χ(r)|eiϕ(r) we define the superfluid velocity
vs (r) =
~
∇ϕ(r),
m
(7.4)
We will justify this choice microscopically when we discuss the Gross-Pitaevskii
equation. Although it looks just like the corresponding formula from elementary
quantum mechanics, it refers to a macroscopic quantity, one whose quantum
fluctuations are much smaller than its average. It immediately follows that the
superfluid velocity is irrotational : ∇∧vs (r) = 0, and its circulation (line integral)
satisfies the quantization condition
I
nh
,
n ∈ Z.
(7.5)
vs (r) · dl =
m
Finally, note that none of the definitions we made here refer exclusively to
equilibrium or even time- independent quantities – all can be considered to be
functions of time without any conceptual difficulty.
7.2 Superfluidity defined
1
2
The first-quantized version of Eq. (7.2) is
R
P
ρ1 (r, r0 ) ≡ N n pn dr2 · · · drN Ψ∗n (r, r2 , . . . , rN )Ψn (r0 , r2 , . . . , rN ), for a statistical mixture of
orthogonal states Ψn occupied with probabilities pn
If all the particles were in a finite momentum state, the RHS would tend to a plane wave, for
instance
80
Bose condensation and superfluidity
Figure 7.1 Thought experiment used to define the superfluid fraction. An
annulus of fluid is rotated slowly
Superfluidity is not really a single phenomenon but rather a complex of related
phenomena, see Ref. [? ] for a very complete discussion. Here we focus on two of
the conceptually simplest properties of those systems usually deemed superfluid.
7.2.1 Non-classical rotational intertia
Suppose we have a container in the form of a cylindrical annulus containing
some ‘matter’ consisting of N particles of mass m (Fig. 7.1). If we rotate this
at some angular frequency ω, we expect that the free energy of this system has
ω-dependence of the form
1
F (ω) = F0 + Iω 2
2
where I = N mR2 is the moment of inertia, and we neglect the mass of the container and the order d/R effects of finite container thickness. The phenomenon of
non-classical rotational inertia (NCRI) corresponds to an additional ω-dependent
contribution ∆F (ω), which at small ω has the form
1
∆F (ω) = − (ρs /ρ) Iω 2 ,
2
defining the superfluid density ρs or superfluid fraction ρs /ρ. In other words, the
equilibrium state of the system is one in which a fraction of the mass is not
rotating with the container3 .
3
To be careful that we are talking about the equilibrium state, one could imagine tuning the system
into the superfluid state while the container is in motion, thus excluding the possibility that the
system merely takes a very long time to catch up with the rotating walls.
7.2 Superfluidity defined
81
It may be surprising that we are proposing to characterize a situation that
seems intrinsically dynamical using equilibrium concepts. It is easy to show,
however, that conditions of constant ω are rather special. Consider the timedependent Hamiltonian
H(t) =
N
N
X
1 X
p2i
+ Ucont (r0i (t)) +
V (|ri − rj |, )
2m
2
i,j=1
i=1
where Ucont (r) is the potential due to the container, and
r0i (t) = (xi cos ωt + yi sin ωt, yi cos ωt − xi sin ωt, zi )
is the position of the ith particle in the rotating frame. The time evolution of
wavefunctions in the rotating frame is governed by the time-independent Hamiltonian4
Hrot = H(0) − ω · L,
(7.6)
where L is the operator of total orbital angular momentum
X
L=
r i ∧ pi .
i
Thus a formal procedure to calculate the superfluid fraction would involve obtaining the equilibrium density matrix in the rotating frame using Eq. (7.6), then
using it to calculate the average of H(0) − T S to give the free energy in the lab
frame. As a trivial example, a rigid body has H = L2 /2I, so that Hrot is minimized for L = Iω, giving hHi = Iω 2 /2, as expected. Taking quantum mechanics
into account means that L is quantized in units of ~, so that the non-classical
part of the energy in this example is of order ~2 /2I. Applied to our annulus, we
see that for our defintion of NCRI to make sense we must have N mR2 ω/~ → ∞.
At the same time we require, for reasons that will become clear, mR2 ω/~ → 0.
With these two conditions and d/R → 0 met we are free to take both the thermodynamic limit and ω → 0.
Thus we see that the superfluid density is defined through a response of an
equilibrium system to an infinitesimal perturbation. Specializing now to zero
temperature, we see that, since hHrot iω = −ω hLiω /2 for small ω (by perturbation
theory in ω, for example)5
1 2
2
ρs /ρ = lim
hHrot iω − hHrot i0 + Iω .
(7.7)
ω→0 Iω 2
2
We now note that the quantity in square brackets is the expectation value of the
Hamiltonian
N
2
X
[pi − mω ∧ ri ]
Hω =
+ ...
2m
i=1
4
5
Note that this procedure would break down e.g. for the case of the magnetic dipole interaction if the
magnetic field does not rotate with the container. We assume such effects to be negligible.
More generally, ∂ hHrot iω /∂ω = − hLiω is a consequence of the Hellman-Feynmann theorem.
82
Bose condensation and superfluidity
The ‘vector potential’6 mω ∧ ri can be removed by a gauge transformation which
returns us to the original Hamiltonian H(0) but changes the boundary conditions
from periodic to ‘twisted’
Ψ(θ1 , . . . , θi + 2π, . . . , θN , {rj , zj }) = eiϕ Ψ(θ1 , . . . , θi , . . . , θN , {rj , zj }),
with ϕ = 2πR2 mω/~. The definition Eq. (7.7) is thus seen to be equivalent to
4π 2 I ∂ 2 E0 (ϕ)
,
ϕ→0 N 2 ~2
∂ϕ2
ρs /ρ = lim
(7.8)
(the strange notation is to remind us that ϕ 1/N by the earlier discussion of
the thermodynamic limit) where E0 (ϕ) is the ground state energy . The definition in terms of ‘rigidity’ to twisted boundary conditions is very appealing, and
constitutes a pleasing abstraction of the original thought experiment.
Exercise 7.2.1 Satisfy yourself that a noninteracting Bose gas at zero temperature has ρs /ρ = 1. What about a non-interacting Fermi gas?
Exercise 7.2.2 (for enthusiasts, but related) The definition Eq. (7.8) seems to
coincide with the ‘Drude weight’ in Kohn’s theory of the insulating state.
In that theory, a metal has a finite Drude weight. But a metal is not a
superconductor (charged superfluid). What’s going on?
7.2.2 Metastability of superflow and vortices
A characteristic of superfluidity, which is probably more familiar than the resistance to rotation at low angular velocity just discussed, is the property of persistent circulation. This refers to the ability of a superfluid to keep rotating after its
container has stopped, without slowing due to friction. The state with no rotation is evidently lowest in energy, so the implication is that there are metastable
configurations of the fluid with finite angular velocity. Such configurations can
exist up to some critical angular velocity ωc (or velocity vc ).
Using ∂ hHrot iω /∂ω = − hLiω , we can formally introduce the Legendre transform
E(L) = hHrot iω(L) + Lω(L),
where ω(L) is the function inverse to hLiω , assuming it exists. From Eq. (7.6),
it’s clear that E(L) = hH(0)i when the container is rotating at angular velocity
ω(L). It seems reasonable that in order to be metastable when the rotation stops,
E(L) must have regions of negative curvature E 00 (L) < 0, see Fig. 7.2. But since
E 00 (L) = ω 0 (L), and ω(L) is the inverse of some function with hLi0 = 0, this
can’t be true, and the assumption that the inverse exists was wrong. Barring the
unrealistic scenario that hLiω decreases over some region where ω increases, we
conclude: metastability implies jumps in hLiω 7 .
6
This formulation makes the analogy between the superfluid response and Meissner effect in a
superconductor clear.
7.2 Superfluidity defined
83
2.5
2
1.5
1
0.5
1
2
3
4
5
Figure 7.2 E(L/N ~), showing metastable minima.
Unlike the definition of the superfluid fraction, there is no general formalism
that tells us whether metastable superflow is possible, so this is about as far as
we can get without discussing a concrete physical system. If we are dealing with
a Bose-Einstein condensate, with a macroscopic number of atoms in the same
state, things are immediately a lot clearer. Since the atoms behave as one, their
quantized angular momentum n~ becomes a quantized macroscopic quantity N n~
(if we ignore all deviations from axial symmetry and the effect of interactions).
This explains the resistance of the condensate to rotation at small ω discussed
earlier: the n = 0 and 1 states have hHrot i = 0 and N ~2 /2mR2 −ωN ~ respectively,
so n = 1 is not favored until ω > ω1 ≡ ~/2mR2 . It’s possible to argue that the
inclusion of interactions only quantitatively changes this conclusion.
Metastability, on the other hand, implies that if the angular momentum is
made to deviate from N n~, there is a energy barrier (Fig. 7.2). We will see that
the origin of this barrier is the (repulsive) interaction between particles, which
penalizes the order parameter Ψ(r) going to zero in some part of the annulus. The
resulting metastable configurations satisfy the quantization condition Eq. (7.5),
where the phase of the order parameter winds through 2π n times, corresponding
to the angular momentum quantum number just discussed. Note that the mere
existence of the order parameter is almost enough to explain persistent flow,
and requires only some reasonable assumptions about its stable configurations
together with the definition of the superfluid velocity.
When we introduced the order parameter, we stressed that we required BEC to
7
In the interests of full disclosure, I have to say that I don’t really like this argument. The problem is
that it sneaks in the physically attractive idea that E(L) tells us about the possible rotating states
at ω = 0, even though it is just a formal transform introduced at finite ω. Probably the only honest
thing one can say is that metastability is associated with a first order transition as ω is changed,
and that the jump in hLiω is an associated discontinuity
84
Bose condensation and superfluidity
Figure 7.3 Vortices in an atomic gas of
23
Na. Reproduced from Ref. [? ].
be simple (one component). We will see later that in multicomponent condensates
the issue of metastability is considerably more complicated.
If the asymmetry of the container is too large we might not get any metastable
configurations (see Problem ?? in the next chapter). This is the situation for so
called ‘weak links’ or Josephson junctions, which does not stop such situations
displaying superfluidity in the sense of the previous section.
What happens if we don’t have an annular container but an (approximately)
cylindrical one? If Ψ(r) is finite everywhere, then n = 0 in the quantization
condition. n 6= 0 requires, by Stokes’ theorem, that the irrotationality condition
∇ ∧ vs (r) breaks down somewhere inside any surface bounded by the contour we
integrate around. For such configurations to have finite energy, Ψ(r) must vanish
at this point. The resulting line defect is a called a vortex. The simplest vortex
configuration, for a vortex along the r = 0 line in cylindrical coordinates
n~ 1
.
(7.9)
mr
The metastable states resulting from halting a rotation with Iω & N ~ in this
simply connected geometry are generically (multi-)vortex configurations.
vsφ (φ, r, z) =
7.3 Experimental status
The experimental situation regarding the demonstration of the phenomena of
BEC and superfluidity in atomic gases is in some ways the reverse of that in
the study of other quantum fluids like liquid Helium. There, the belief that BEC
is the cause of the observed superfluid behaviour was part of the theoretical
explanation of superfluidity, not an independently verified experimental fact. In
contrast, BEC in atomic gases was observed in 1995, but the first experiments
confirming superfluidity had to wait until 1999.
As we mentioned in the previous chapter, absorption images of the trapped
gas are one of the most common experimental probes. If the gas is allowed to
expand freely for some time T , the resulting density profile corresponds to the
distribution n(k) in momentum space introduced earlier (as long as vT the
trapped cloud). A central peak in absorption in such images (corresponding to
7.3 Experimental status
85
the logarithm of n(k) column integrated along the line of sight) therefore provides
a direct measurement of condensation (see Fig. 4.2).
The other major experimental confirmation of BEC relates to the observation
of certain interference phenomena that we will discuss in Section ??. As for
superfluidity, experiments in which a laser was used to ‘stir’ the gas revealed
dramatic arrays of vortices in subsequent imaging, which speak for themselves,
see e.g. Fig. 7.3.
In order that the vorticity matches, in a coarse-grained fashion, that of a rigid
body ∇ ∧ vs (r) = 2ωẑ, we must have a area density of quantized vortices nv =
2mω/h. In the trapped case, this is only an approximate statement.
Note that in a trap, while the quantization condition Eq. (7.5) remains exact,
we expect that L(ω) is not in general quantized, since the angular momentum
density in general depends on the magnitude of the order parameter (see next
chapter). The jumps in L(ω) that result from metastability should still be present,
of course.
8
Bogoliubov theory
8.1 Pair approximation and ground state energy
With the shortcomings of the GP approximation now manifest, we now turn to
their resolution. We have already seen the germ of the idea when we discussed
the derivation of the time-dependent GP equation. We saw there that the action
of the interaction Hamiltonian on the GP ground state generates pairwise occupation of single particle orbitals orthogonal to χ0 . For now we will stick with
the translationally
invariant case (and periodic boundary conditions), so that
√
χ0 = 1/ V and the single particle orbitals are plane waves. We can thus imagine
an improved ground state ansatz of the form
|pairi ≡
X
c{nk }
{nk }
Y
Λnkk |GP i,
(8.1)
k
where Λk = a†k a†−k a0 a0 creates a (+k, −k) pair out of the ground state and we
sum overall all assignments {nk } of the number of such pairs to each k. Provided
the mean number of particles N0 in the zero momentum state remains close to
N , the state that results from this one after an application of the Hamiltonian
is approximately of the same form (with different coefficients, in general) if the
interaction is weak, as the c’s are in general small in the interaction. Putting it
another way, any order of perturbation theory in the interaction, starting from
the GP state, is clearly going to be dominated by repeated excitation of pairs out
of this state1 .
The crucial thing about the use of variational states of the form Eq. (8.1) is
that
hpair|H|pairi = hpair|Hpair |pairi,
1
(8.2)
It’s necessary to point out that the method of this section should be understood as applying strictly
to the case of weak interatomic potentials U (r), where the Born approximation
`
´R
as ∼ m/4π~2
dr U (r) is valid. This is not the same as the diluteness condition na3s 1, as
strong potentials can give small scattering lengths. The derivation of Eq. (4.21), say, with as the
true scattering length, is considerably more complicated, see Ref. [? ].
86
8.1 Pair approximation and ground state energy
87
where the Hpair is
Hpair
0
U0
U0 X †
= Hkin +
α α + α0† αp + 2np n0
N (N − 1) +
2V
2V p p 0
+
0
U0 X
np nq + αp† αq .
2V p6=q
(8.3)
Here αp† = a†p a†−p creates a (p, −p) pair, np = a†p ap is the occupation number of
P0
an orbital, and
indicates that the zero momentum state is to be excluded. Note
that it is probably not justified to keep the terms on the second line, however.
The last term changes a (q, −q) pair to a (p, −p) pair, when we have ignored
the possibility of creating a triple with p + q + r = 0. It turns out that these
contributions can be dropped for weak interactions (though strictly doing so
destroys the bounding property of Hpair , see below).
The pair Hamiltonian (without the second line) is usually obtained directly
from the original
Hamiltonian, by similarly arguing that BEC means that a0 is
√
of ”order N ”. It’s important to realize that these two points of view – starting
from a variational state of the form Eq. (8.1) or jumping straight to the pair
Hamiltonian Eq. (8.3) – are equivalent in terms of the physical intuition they
embody. Furthermore, the strength of having a reduced Hamiltonian is that one
can obtain excited states as well as the ground states, imagining that they have a
similar form. Mathematically, the nice thing about the relation Eq. (8.2) is that
we know we are going to end up with an upper bound for the ground state energy.
Can we solve Hpair ? If we were able to treat a0 and n0 as O(N ) c-numbers
(and neglect the last term), then we would be left with a quadratic Hamiltonian,
and the answer would clearly be yes. The one glitch, however, is that ha0 i = 0
on any state with a fixed number of particles. The usual approach is to consider
states that are superpositions of different particle number, introducing a chemical
potential to fix the average particle number to N . The resulting state, after
projection, should be a good approximation to the true ground state of Hpair at
large N√
, since the fluctuations in particle number in the unprojected state are
only ∼ N .
−1/2
An alternative approach is to introduce the operator bp = a†0 (n0 + 1)
ap
†
and its conjugate. These satisfy the canonical relations bp , bp = 1 for p 6= 0. An
1/2
operator like αp† α0 is then (n0 (n0 + 1)) b†p b†−p . Since bp conserves total particle
number c-number substitutions in the resulting Hamiltonian are free from the
above problem. Dropping the second line of Eq. (8.3), we have, following the
replacement of n0 with hn0 i = N (assuming that the depletion N − N0 can be
neglected)
Hpair =
0
X
U0
U0 N X † †
N (N − 1) +
(p)b†p bp +
b b + bp b−p + 2np ,
2V
2V p p −p
p
(8.4)
which is the same as Bogoliubov’s non-conserving Hamiltonian. It is diagonalized
Bogoliubov theory
88
by the transformation
2
βp = bp cosh κp − b†−p sinh κp
nU0
tanh 2κp =
.
(p) + nU0
The transformed Hamiltonian is
0
X
1
1
[E(p) − (q) − nU0 ] + E(p)βp† βp ,
H = nU0 N +
2
2
p
(8.5)
(8.6)
where E(p) is the Bogoliubov dispersion relation introduced in Eq. (6.7). The
ground state corresponds to
βp |0i = 0,
(8.7)
and the ground state energy
0
X1
1
E0 = nU0 N +
[E(p) − (q) − nU0 ] .
2
2
p
The integral is divergent in the ultraviolet, but this can be fixed by writing
"
#
0
X
1
1 X U0
(nU0 )2
1
E(p) − (q) − nU0 +
.
E0 = nU0 N 1 −
+
2
V p 2(p)
2
2(p)
p
In this form, the term we have added and subtracted is recognized as the next
order in the Born approximation for the scattering length as = a0 + a1 + · · · . The
second term can now be evaluated to give
1/2
1
4π~2
1
4π~2
128
E0 = nN
(a0 + a1 ) + nN
a0 √ na30
.
2
m
2
m
15 π
(8.8)
Where
a0 = (m/4π~2 )U0 ,
a1 = −(m/4π~2 )
U02 X 1
.
V p 2(p)
This closely resembles the result quoted in Eq. (4.21) for the first two terms of
1/2
the ground state energy of a system of bosons as an expansion in (na3s ) . The
Bogoliubov approximation, as just described, is not able to reproduce that result
in all its glory, for reasons we’ll explain below.
Exercise 8.1.1 What do you expect to happen if the interactions between particles are attractive, so that U0 < 0? Think about the compressibility and
the Bogoliubov spectrum.
2
Note that the transformation will not exist at low p if µ is negative, meaning that the Hamiltonian
has no ground state. The stems from the the neglect of the second line of Eq. (8.3). While systems
with negative scattering length are believed to be thermodynamically unstable, their Hamiltonians
should still make sense!
8.2 Interlude: Bogoliubov transformations and squeezed states
89
8.2 Interlude: Bogoliubov transformations and squeezed states
8.3 Structure of the ground state and excitations
We now turn our attention to the nature of the ground state of Eq. (8.4). It is not
difficult to see that the ‘Bogoliubov vacuum’ conditions Eq. (8.7) are satisfied by
|Bi ≡
0
Y
N
1
a†0 |0i,
|N i0 = √
N!
† †
e(cp /2)bp b−p |N i0
p
(8.9)
if cp = tanh κp (we have not normalized). The factor of 1/2 arises from accounting
for p and −p contributions when βp is applied.
A slightly different wavefunction, that coincides with Eq. (8.9)
for N 1, is
√
†
†
more easily given physical interpretation. Using bp = ap a0 / N , one can show
that in this limit
"
#N/2
0
X
1
† †
† †
|B; N i = √
a0 a0 +
cp ap a−p
|0i
(8.10)
N!
p
(assuming N even) is an equivalent choice. This is almost, but not quite, the
number projection of the non-conserving Bogoliubov ground state
|B; ζi ≡
0
Y
†
†
eζ(cp /2)ap a−p |ζi
|ζi = e
√
N ζa†0
|0i,
|ζ| = 1.
(8.11)
p
I emphasize that for N 1 all three of Eq. (8.9-8.11) give the same result
(Eq. (8.8)) when the energy is evaluated. Eq. (8.10) has the pleasant property
of being a member of the class Eq. (8.1) in which the amplitudes for multi-pair
states exactly factorize into those for the constituent single-pair states. In first
quantized language we have
Y
|B, N i = S
ϕ(ri − rj ),
(8.12)
i<j
(c.f. Eq. (6.5) with
ϕ(r) =
X
cp eip·r
p
cp ∼ −nU0 /2(p) at large p, reproducing the scattering wavefunction at lowest
order. Since it doesn’t work to all orders, there is clearly no way that even the
leading order term in (na3s )1/2 in E0 can come out right, as it depends on the
scattering wavefunction being written in terms of the true scattering length. It
is possible to obtain the result Eq. (4.21), however, by using a variational wavefunction of the form Eq. (8.10), but retaining the full pair Hamiltonian Hpair [? ?
]. Furthermore, in the thermodynamic limit, the exact solution of Hpair is belived
to be of this form.
The excitations described by Eq. (8.6) have the same spectrum that we found
before. Indeed, it should be clear that cosh κp and sinh κp are determined by the
Bogoliubov theory
90
same Bogoliubov-de Gennes equations that determined u(r) and v(r) (compare
the quadratic form in Eq. (8.4) with the matrix structure of Eq. (6.6)). In a sense
the Bogoliubov theory can be thought of as a quantization of those oscillations.
This allows the Bogoliubov approximation to be extended to the inhomogeneous
case quite straightforwardly (see e.g. Ref. [? ] for a brief discussion), though in
light of the small depletion discussed below this normally isn’t necessary.
An intuitive picture of the excitations can be obtained by considering the action
of the density ρq on the ground state. Keeping only the parts of the operator that
move particles in or out of the condensate gives
√ √ ρq |Bi ∼ N a†q a0 + a†0 a−q |Bi = N b†q + b−q |Bi
√
= N [cosh κp − sinh κp ] βp† |Bi, (8.13)
(c.f. Eq. (6.8)) so that an excitation coincides with a density fluctuation. This
allows us to immediately obtain the structure factor
2
S(q) = hB|ρq ρ−q |Bi = N [cosh κp − sinh κp ] =
N (p)
Np
→
,
E(p)
2mcs
q ~/ξ,
(8.14)
matching the result Eq. (6.13) and showing that the structure factor saturates
the inequality Eq. (6.18). We therefore see that all of the weight in S(q, ω) lies
in the Bogoliubov modes
S(q, ω) = S(q)δ(ω − E(q)/~).
The result Eq. (8.14) can be interpreted as resulting from interference between
terms with different number of excited pairs at q, −q. That is, both
|N − 2ni0 |niq |ni−q ,
and
|N − 2n − 2i0 |n + 1iq |n + 1i−q
terms of Eq. (8.9) contribute to the |N − 2n − 1i0 |n + 1iq |ni−q component when
ρq is applied, with amplitudes cosh κq and sinh κq respectively.
I want to emphasize that response functions in the Bogoliubov approximation,
calculated via the dynamical structure factor, coincide with those obtained in
the time-dependent Gross-Pitaevskii theory. Thus experiments on scattering (see
Fig.??) that are consistent with these results do not really bear on the validity of
the Bogoliubov theory. The qualitatively new feature of the Bogoliubov ground
state is that the zero momentum state is depleted. We find the momentum distribution
mcs
|cp |2
→
,
p ξ −1 .
n(p) =
2
1 − |cp |
2p
The radial density distribution 4πp2 n(p) is peaked around ~/ξ. Summing over p
gives the fraction of atoms not in the condensate
1 X
8 p 3
n(p) = √
nas ,
(8.15)
N p
3 π
where we used the Born approximation for the scattering length as =
4π~2 U0
.
m
8.3 Structure of the ground state and excitations
91
Figure 8.1 Momentum transfer per particle for a BEC (open circles) and
expanded cloud (closed circles), for a given value of the wavevector
q = q1 − q2 in Bragg spectroscopy. Note how the peak is suppressed and
shifted to higher energies in the BEC. Broadening is due to the
inhomogeneity of the condensate. Reproduced from Ref. [? ]
Under typical experimental conditions the depletion does not much exceed 0.01,
which justifies the use of the GP approximation. In the presence of an optical
lattice, however, we shall see that the condensate can be depleted to zero, causing
a quantum phase transition out of the superfluid state.
What is beyond the Bogoliubov approximation? Just two brief comments
• The finite temperature depletion of the condensate, and the self-consistent
effect it has on the Bogoliubov approximation, can be included in a relatively
straightforward way. This is known as the Popov approximation.
• Since the Bogoliubov excitations are of course not exact eigenstates, an improved calculation will lead to their interaction. A notable process is Beliaev
damping, corresponding to the decay of one Bogoliubov excitation into two.
This process comes from terms like a†p a†p0 ap+p0 a0 that we discarded in making
the pair ansatz. Such processes are however not significant at low momenta.
Exercise 8.3.1 Consider a system of two types of bosons, denoted by ψ± , of
equal density. Suppose that the intra-species interaction is the same for
the two species U+ = U− , and that additionally there is an inter -species
interaction
Z
Hinter = U+−
1.
† †
dr ψ+
ψ− ψ− ψ+
Find the Bogoliubov spectra, and discuss how their behavior depends
on the sign of U+−
2. Can you interpret what is going on in the first part?
92
Bogoliubov theory
Exercise 8.3.2 How does the superfluid fraction discussed in Section 7.2.1
change in the Bogoliubov approximation?
8.4 Generalization to the trapped case
8.5 Application: condensate fluctuations
9
Fermi superfluids
9.1 Fermionic condensates
In Chapter 7, we saw that superfluidity was a natural consequence of BoseEinstein condensation. The existence of the condensate order parameter and the
definition of the superfluid velocity in terms of it make superfluidity almost inevitable. Thus it is natural to ask whether superfluidity is a phenomenon which
is entirely confined to systems of bosons. It is worth bearing in mind, however,
that so far we have been treating composite objects – alkali atoms – as indivisible
interacting quantum particles. But an atom is made of electrons, protons, and
neutrons (if we stop at this level of description), which are all fermions. So the
answer to our question is clearly negative.
To make the discussion simpler, consider a gas of two different types of fermions
with attractive interactions between the atoms of different types. Since the two
types are distinguishable, there is s- wave scattering, with an associated scattering
length as , as we have before. If the potential between the two types of fermion
is made sufficiently attractive, a bound state may form. The resulting molecule
can be thought of as a boson, in the sense that if we swap the positions of the
two atoms with the corresponding coordinates of another two atoms bound in
another molecule, the wavefunction is unchanged. If we are concerned only with
energy scales low compared to the binding energy of the molecule, so that the
wavefunction of the relative coordinate can be almost always be taken to be the
bound state wavefunction (except when we are considering collisions between
molecules), we can forget about the internal structure altogether, and the system
can bose condense. Putting it more formally, the system displays ODLRO, but
in the two-particle density matrix. By analogy with our discussion in Section 7.1,
and using the label s =↑, ↓ to denote the two types of fermions
ρ2 (r1 , r2 ; r01 , r02 ) ≡ hhψ↑† (r1 )ψ↓† (r2 )ψ↑ (r01 )ψ↓ (r02 )ii
Nc ∗
→
F (r1 − r2 )F (r01 − r02 ),
|r1 − r01 |, |r2 − r02 | → ∞,(9.1)
V
where F (r) is the wavefunction of the molecule, and Nc , as before, is the number
of particles (this time diatomic molecules) in the condensate. Thus we might
conjecture that a superfluid can result only if we have a bound state, and then only
if we are at temperatures below the binding energy (and of course, the degeneracy
temperature), see Fig. 9.1. The surprising thing is that this expectation is not
93
94
Fermi superfluids
Figure 9.1 Schematic phase diagram of a system of two species of fermions
(equal in number). The dashed line represents a naive expectation, without
accounting for the Cooper phenomenon
borne out. The long-range order described by Eq. (9.1) can be present even when
the interaction between two isolated particles is not strong enough to create a
bound state, as a consequence of the restrictions imposed by the Pauli principle
in a many-fermion sysetem. This is known as the Cooper phenomenon after Leon
Cooper, who later became the C in the famous BCS theory of superconductivity,
in which this effect plays a prominent role.
As a historical note, the idea that superconductivity (the analog of superfluidity
in a charged system) in metals is caused by the condensation of electron pairs
existed before BCS, and in this context a bound fermion pair is sometimes called
a Schafrorth pair. The problem for these early authors was that they could not
figure out any way that two electrons could bind.
Returning to our original discussion, we can start out with weakly attractive
interactions and describe the resulting condensate using the BCS theory, but
then what happens if we continue to increase the strength of the attraction between fermions? A two-particle bound state eventually will become possible, and
the picture we outlined above, of the condensation of tightly bound molecules,
will apply. The suggestion is that, at least as far as the ground state goes, the
evolution from the BCS to the BEC descriptions is smooth. The experimental
realization of this BCS-BEC crossover in 2004 was perhaps the greatest triumph
since the creation of bosonic condensates in 1995 (see Ref. [2] for a non-technical
introduction to the 40 K experiments at JILA). Understanding this phenomenon
more closely will be one of main goals in this chapter
9.2 The BCS theory
95
9.2 The BCS theory
9.2.1 The pairing hypothesis
We start from the obvious Hamiltonian describing two species of Fermions of
equal mass, which we will denote with a label s =↑, ↓, deferring a discussion of
the actual experimental system until later.
X
U0 X †
ap+q↑ a†−p↓ a−p0 ↓ ap0 +q↑ .
(9.2)
H=
ξp a†p,s ap,s +
V
p,s
p,p0 ,q
We take the interaction between the two species to be attractive U0 < 0. Starting
from the ground state of the non-interacting problem
Y † †
ap↑ ap↓ |0i
(9.3)
|F Si =
|p|<pF
describing a fermi sea with all states below the Fermi energy filled, we note that
the application of the interaction terms generates terms of the form
a†p+q↑ a†−p↓ a−p0 ↓ ap0 +q↑ |F Si.
|p|, |p + q| > pF , |p0 |, |p0 + q| < pF .
Note the difference from the Bose case: because the state p = 0 plays no special
role – like every other state below the Fermi surface it is occupied with one fermion
of each species – we do not just create pair excitations with zero centre of mass
momentum q = 0. Nevertheless, the BCS theory starts from the assumption
that such finite momentum pairs do not contribute significantly to the ground
state, and makes pair ansatz of essentially the same form that was used in the
Bogoliubov theory1
Y n
X
(9.4)
Λpp0 |0i,
|pairi ≡
c{nPp }
P
p
nP
p =N/2
p
where Λk = a†p↑ a†−p↓ creates a (+p, −p) pair and the numbers np,p0 are either 0
or 1. Note that in writing Eq. (9.4) the number of each species is assumed to be
exactly N/2 – the more general case will be discussed later. As in the Bogoliubov
case, such a form allows us to work with a pair Hamiltonian2
X
U0 X † †
a a a 0 a 0 .
(9.5)
Hpair =
ξp a†p,s ap,s +
V p,p0 p↑ −p↓ −p ↓ p ↑
p,s
That is, just Eq. (9.2) with all but the q = 0 part discarded. Now, can we
solve Eq. (9.5)? It is illuminating to introduce the operators b†p = a†p↑ a†−p↓ and
its conjugate, that create and destroy a (+p, −p) pair. Because our pair ansatz
Eq. (9.4) only includes amplitudes for a given (+p, −p) pair of momenta having
1
2
In the book Ref. [4], Schreiffer explains that the BCS wavefunction was directly motivated by such
treatments.
One difference from the Bogoliubov case is that hpair|H|pairi = hpair|Hpair |pairi + EHartree , where
the Hartree energy is EHartree /V = U0 n↑ n↓ , and ns is the expectation value of the density of
species s. Since the overall Hamiltonian commutes with the total number of each component, our
variational calculations are not affected by dropping this term.
Fermi superfluids
96
either none or two fermions, the pair Hamiltonian can be written in terms of the
pair operators b†p , bp as
Hpair = 2
X
ξp b†p bp +
p
0
U0 X †
b b 0.
V p,p0 p p
(9.6)
This may now look like a quadratic problem, but the pair operators bp , while
commuting with those at another momentum
[bp , bp0 ] = [b†p , b†p0 ] = [b†p , bp0 ] = 0
p 6= p0 ,
(9.7)
obey the hardcore constraint
(b†p )2 = 0,
(9.8)
†
which is a result of b being composed of a pair of fermions obeying the exclusion
principle. Nevertheless, given that a pair that ‘hops’ into a level at p could come
from any other level p0 , it seems reasonable to try, as a variational state, one in
which the amplitudes for the occupancy of each level are uncorrelated
"
#N/2
X
|N i ≡
cp b†p
|0i,
(9.9)
p
corresponding to Eq. (9.4) but with a set of coefficients c{nPp } that factorizes.
Finding the variational energy of Eq. (9.9) is still a tricky problem. For instance,
what is the expectation value of the kinetic energy?
X
X
K.E = 2
ξp hb†p bp i ≡ 2
ξp hnPp i,
(9.10)
p
p
Finding the average number of pairs hnPp i in Eq. (9.9) is however not obvious.
We can make it so by following the route taken by BCS, and considering instead
the normalized wavefunction
Y
|BCSi =
vp b†p + up |0i
|up |2 + |vp |2 = 1.
(9.11)
p
This is a superposition of states with different total number of particles. One can
see quite easily, however, that the projection onto a fixed number N of particles
corresponds exactly to Eq. (9.9) if cp = vp /up . Since Eq. (9.11) is a product of
factors corresponding to each momentum separately, hnPp i is easily found to be
vp2 . The total variational energy of this state is
X
U0 X ∗
up vp up0 vp∗ 0 .
(9.12)
hHiBCS = 2
ξp vp2 +
V
p
p,p0
What about our use of a non-conserving wavefunction? The expectation value
of any operator that itself conserves the number of particles can evidently be
written
X
hOiBCS =
PN hOiN ,
N
9.2 The BCS theory
97
where h· · · iN denotes the expectation with respect to the N -particle projection
Eq. (9.9). The probabilities PN are
X Y
PN =
nPp vp2 + 1 − nPp u2p ,
P
p
nP
p =N/2
which is strongly peaked around hN i = 2
that is O(N ). Thus at large N
P
p
vp2 = 2
P
p hnp i,
P
hOiBCS → hOihN i .
with a variance
(9.13)
In the thermodynamic limit, we might as well work with the non-conserving form
Eq. (9.11).
There is an interesting alternative interpretation of the pair Hamiltonian Eq. (9.6).
Acting within the pair subspace, the three operators bp , b†p , and b†p bp −1/2 behave
as spin-1/2 operators (Anderson spins)
†
bp , bp = 2 b†p bp − 1/2
†
bp , b†p bp − 1/2 = −b†p ,
(9.14)
meaning that Eq. (9.6) can be written as a spin chain
X
U0 X + −
Sp Sp0 .
Hpair = 2
ξp Spz +
V
0
p
p,p
(9.15)
If we parameterize (up , vp ) as (cos(θ/2)eiϕ/2 , sin(θ/2)e−iϕ/2 ) then the variational
energy Eq. (9.12) has the form (except for a constant)
X
U0 X
hHiBCS = −
ξp cos θp +
sin θp sin θp0 cos (ϕp − ϕp0 ) .
(9.16)
4V p,p0
p
The interpretation of Eq. (9.16) is the following. The first term tends to align
the spins with the z- axis in the - direction for ξp < 0 and in the + direction
for ξp > 0. On the other hand, the second term, originating from the potential
energy between the constituents of a pair, wants the spins to lie in the x-y plane3 .
It remains to actually minimize the energy Eq. (9.12) to determine the u’s
and v’s, or equivalently, the configurations of the spins. For U0 > 0 (repulsive
interactions), the spins all point in the ±z direction, forming a ‘domain wall’
where ξp changes sign at the fermi surface, see Fig. 9.2. The relationship between
the spin picture and the average number of pairs is
hnPp i = vp2 = [1 − cos θp ] /2,
so we see that this corresponds simply to a sharp fermi step. For U0 < 0, the
system can lower its energy by taking sin θp 6= 0. The lowering of the interaction
energy more than compensates the increase in kinetic energy that comes from
smearing the step, see Eq. (9.10). Clearly all of the angles ϕp , describing the
3
Of course the spins are really quantum mechanical spin-1/2 operators. The direction corresponds to
the direction of hSi in the BCS state
Fermi superfluids
98
Figure 9.2 Anderson spin configurations and the associated distribution
functions for the free fermi gas (top) and the BCS state (bottom).
angle in the x-y plane, should be equal. Taking the extremum of Eq. (9.16) with
respect to the angles θp gives the condition
ξp sin θp − |∆| cos θp = 0,
where it is convenient to introduce the gap parameter
U0 X iϕ
U0 X
∆=−
e sin θp = −
up vp∗ .
2V p
V p
(9.17)
Thus we have
cos θp =
ξp
,
Ep
sin θp =
|∆|
,
Ep
Ep =
p
ξ(p)2 + |∆|2 .
(9.18)
The meaning of these solutions is very simple. They correspond to the alignment
of the spin vector with the direction of the effective ‘magnetic field’
(Re ∆.Im ∆, ξp )
To be self-consistent, the solution must further satisfy
U0 X ∆
∆=−
.
2V p Ep
(9.19)
(9.20)
Eq. (9.20) is a fundamental relation of the BCS theory. It is clear that for U0 > 0
there are no non-trivial solutions (∆ = 0 always), while for any U0 < 0 there
is always a solution at finite ∆ (One can also show that it corresponds to a
9.2 The BCS theory
99
minimum of energy. In the repulsive case, the solution at ∆ = 0, which always
exists, corresponds to a maximum). Passing to the continuum limit we have
Z
dp ∆
U0
(9.21)
∆=−
3
2
(2π) Ep
This integral is divergent in the ultraviolet. We turn to the question of how to
regularize it in the next section. More significant, however, is the dependence of
the right hand side on ∆ for small ∆. This is
∼−
U0
ν(µ)∆ ln Λ/∆,
2
where Λ is the UV cut-off (we will shortly identify it with the Fermi energy). This
shows that no matter how small the attraction U0 < 0, there will always be a
solution of Eq. (9.21) with finite ∆. This is the essence of the Cooper phenomenon.
It should be compared with the situation in which there are not a macroscopically
large number of particles present. In that case µ = 0, so that ξp = (p) > 0.
Then the right hand side of Eq. (9.21) has no divergence at low energies because
the density of states ν(E) vanishes4 . The finite value of ∆ is thus seen to be a
consequence of the fermi sea.
What are the implications for the order of the system? In the BCS state
hBCS|a†p↑ a†−p↓ |BCSi = u∗p vp .
Obviously this is a consequence of the non-conserving form of the wavefunction.
But based on our previous argument we have
hN |a†p↑ a†−p↓ ap0 ↑ a−p0 ↓ |N i = u∗p vp up0 vp∗ 0 ,
which is equivalent to our statment of ODLRO, Eq. (9.1), with
Fp = up vp∗ .
Thus we have demonstrated the presence of a fermi condensate for an attractive
interaction, no matter how weak5 . To be more honest, what we have really shown
is that the BCS state always has a lower energy than the free fermi problem.
Since this certainly is the ground state at U0 = 0, and the BCS is smoothly
connected to it, it seems clear that the BCS state can be trusted at least for
small coupling. At larger coupling some other state of matter could win out, but
given our argument that we should eventually arrive in a BEC state of molecules,
which also has ODLRO of the form Eq. (9.1), this seems unlikely.
4
5
As should be clear, this picture in fact depends on dimensionality. In d ≤ 2, the form of the density
of states at low p means that a bound state will always form. It is only in three dimensions that a
critical strength of U0 is required. In contrast, our discussion of the Cooper phenomena for µ > 0 is
independent of dimensionality.
The statement that pairing always occurs is strictly true only for exactly equal numbers of the two
species, as we will see in Section 9.4 below
100
Fermi superfluids
Exercise 9.2.1 As well as the average occupancy of a given momentum state
we can consider the correlations between the occupancy of different p states
Css0 (p, p0 ) ≡ hnp,s np0 ,s0 i − hnp,s ihnp0 ,s0 i
(9.22)
Show that for the BCS state
C↑↓ (p1 , p2 ) = δp1 ,−p2 u2p1 vp2 1 = δp1 ,−p2
C↑↑ (p1 , p2 ) = δp1 ,p2
|∆|2
4Ep2 1
|∆|2
4Ep2 1
Interpret these two expressions.
9.2.2 The BCS-BEC crossover
Taking our argument one step further, we can argue that not only is the order
specified by Eq. (9.1) the same whether two fermions can form a bound state or
not, but the wavefunction in the extreme molecular limit is of precisely the same
form as the BCS wavefunction. This can be seen from the first quantized form of
the number conserving wavefunction Eq. (9.9)
N/2
|N i = A
Y
ϕ(ri↑ − rj↓ ),
(9.23)
i<j
where ϕp = vp /up . At weak coupling the extent of the ‘pair wavefunction’ ϕ(r) is
large compared to the separation between pairs. In this limit the antisymmetrization operation, required by the exclusion principle, plays a dominant role, as we
have seen. When the pair wavefunction ϕ(r) has a much smaller extent than
the typical separation between pairs, we can expect that the anitsymmetrization
operation in Eq. (9.23) is not too important, as two fermions of the same type
rarely overlap. In this limit, any given momentum state has a low average occupancy, and the hardcore constraint Eq. (9.8) does not play a significant role. Then
Eq. (9.6) can really be thought of as a Hamiltonian for isolated pairs, with the
corresponding binding energy (see also the discussion at the end of the previous
section). The resulting wavefunction is then essentially a Gross-Pitaevskii state
of molecules, which was the picture that led us to suggest Eq. (9.1) in the first
place. This suggests that we can use Eq. (9.11) as a variational wavefunction all
the way through the BCS-BEC crossover.
First we have to address the issue of regularizing Eq. (9.21). Suppose we had
worked with a finite-range interaction with fourier transform U0 (p). It is easy to
see that the only way that the equations of the previous section are modified is
through the gap parameter ∆ becoming momentum dependent
X
1 X
U0 (p − p0 ) sin θp0 = −
U0 (p − p0 )up0 vp∗ 0 ,
(9.24)
∆p = −
2V p0
p0
9.2 The BCS theory
101
so that the self-consistent equation is now
∆p = −
1 X
∆p0
U0 (p − p0 )
.
2V p0
Ep0
(9.25)
The solution of Eq. (9.25) is in general very difficult, but it can be greatly simplified if we assume that the range of the potential U0 (r) is the shortest length
scale in the problem. This is of course the case for the systems in which we are
interested. With this assumption we can write Eq. (9.25) as
U0 (p) X ∆0
∆0
∆p = −
−
V
2Ep0
2p0
p0
∆p0
1X
U (p − p0 )
.
(9.26)
−
V p0
2p0
We have added a subtracted a term to the right hand side. The first term of the
resulting expression converges on a scale set by the momentum corresponding to
the larger of ∆p or µ, which by assumption is much less than the scale on which
U0 (p) varies, and justifies replacing U0 (p − p0 ) with U0 (p), and ∆p0 with ∆0 . We
now make the following ansatz for ∆p
∆p = −
m
∆0 F (0, p),
4π~2 as
in terms of the scattering amplitude F (p, p0 ) and scattering length as corresponding to the potential U0 (r). From the integral equation Eq. (4.20) satisfied by the
scattering amplitude we see immediately that
m
∆0
1 X ∆0
−
−
.
(9.27)
∆0 =
4π~2 as
V p 2Ep0
2p0
In the weak-coupling limit, the gap ∆ (we drop the subscript 0 from now on, as
we never need to discuss the high momentum behaviour of ∆p again) is expected
to be much smaller than the Fermi energy, and the chemical potential is just
equal to the Fermi energy EF = p2F /2m. The integral in Eq. (9.27) can then be
done explicitly to give the gap
π
8
(9.28)
∆BCS = 2 EF exp −
e
2|kF as |
Outside of the weak-coupling limit, we have to account for a change in the chemical potential, in order to keep a fixed density. This is apparent from the equation
X
X
X
ξp
P
2
N =2
hnp i = 2
vp =
1−
(9.29)
Ep
p
p
p
The two equations Eq. (9.27) and Eq. (9.29) are conveniently cast in the dimen-
Fermi superfluids
102
1.5
1
0.5
-2
-1
1
2
-0.5
-1
-1.5
Figure 9.3 Variation of gap and chemical
potential. Also plotted is the
p
result for the gap on the BEC side 16/(kF as )/π. Note the exponential
dependence of the gap on the BCS side, consistent with the analytic result
Eq. (9.28)
sionless form
π
=
2kF as
Z
∞
"
x2
#
dx 1 − p
(x2 − µ)2 + ∆20
#
"
Z ∞
2
2
x
−
µ
=
x2 1 − p
3
(x2 − µ)2 + ∆20
0
0
(9.30)
where µ and ∆0 are measured in units of EF = p2F /2m, and the unit of length
2
is p−1
F . In these units the total density of particles of both types is 1/3π . The
behavior of the gap and chemical potential is shown in Fig. 9.3. Recall from
Section 4.3.4 that the point 1/kF as = 0, where the scattering length diverges,
corresponds to the formation of a bound state. This is an interesting part of
the phase diagram (sometimes called the unitary point), because here (if the
temperature is zero) there is only one energy scale (the Fermi energy) and only
one length scale (the fermi wavelength). All quantities such as ∆ and µ are simply
some universal fraction of the Fermi energy. In particular the equation of state
of the system is
3
E/V = α EF n ∝ n5/3 .
(9.31)
5
The numerical factors are to emphasize the resemblance of the unitary gas to the
free fermi gas, where α = 1. The mean field theory above gives α = 0.59, while
a recent Monte Carlo calculation found α = 0.44 ± 0.01 [? ]. There is of course
no reason to believe the quantitative predictions of the mean-field theory in the
region where interactions are so strong.
Exercise 9.2.2 Show that the in the BCS limit (1/kF a large and negative,
∆ EF , and µ = EF ), the variational enenergy Eq. (9.12) of the BCS state
9.2 The BCS theory
103
relative to the free fermi gas is
1
EBCS (∆) − EBCS (0) = − ν(EF )|∆|2
(9.32)
2
F
where ν(EF ) = mp
is the fermi surface density of states. Note that this
2π 2
arises from the sum of a large increase in the kinetic energy, and a large
decrease in the pairing energy.
Exercise 9.2.3 When µ becomes large and negative for 1/kF a > 0, the angles
θp in Eq. (9.16) are all close to 0, since cos θp = ξp /Ep . Expand Eq. (9.16)
in small deviations from θp = 0 and interpret the result.
Exercise 9.2.4 In Section 4.3.4 we introduced the simplest two-channel model
that displays a Feshbach resonance, Eq. (4.24)
X q
X
g X
bq a†↑,q+p a†↓,−p +h.c. (9.33)
p a†s,p as,p +
+ ε0 b†q bq + √
H=
2
V
q
p,q
p,s
Construct a mean-field theory to describe condensation in this model by
replacing the operator b†0 creating a bosonic molecular state a zero momentum with a c-number b∗0 . Show that the self- consistent equation Eq. (9.27)
is replaced by
ε0
1X 1
1
=
−
,
(9.34)
g2
V p 2Ep 2p
√
and ∆ = gb0 = g nb V, where nb is the density of b molecules. [Hint: you
will need to shift the detuning ε0 by an infinite constant as in Eq. (4.26) of
Problem 4.3.1]
Show that the number equation is then
1X
ξp
n = nb +
1−
.
(9.35)
V p
Ep
Argue that in the limit of a broad resonance, where the parameter γ =
g 2 m3/2 /4π → ∞, the mean-field is equivalent to that considered previously.
Note that in this limit the occupancy of the molecular state goes to zero.
This demonstrates in the many-body context that broad resonances are
adequately described by the single channel model.
9.2.3 Quasiparticle excitations
Like the Bogoliubov theory, this BCS theory also lets us discuss excitations out
of the ground state. We didn’t solve the BCS hamiltonian by a Bogoliubov transformation, as is often done, but we can introduce the Bogoliubov-type excitations
after the fact. Recalling the BCS state
Y
(9.36)
|BCSi =
vp a†p↑ a†−p↓ + up |0i,
p
104
Fermi superfluids
it’s easy to see that the operators
αp↑ = up ap↑ − vp a†−p↓
αp↓ = up ap↓ + vp a†−p↑ ,
(9.37)
satisfy the canonical fermion anticommutation relations and annihilate the BCS
state αp,s |BCSi = 0. Consider the state
Y
†
vp a†p↑ a†−p↓ + up |0i,
|p, si = αp,s
|BCSi = a†p,s
p0 6=p
corresponding to the momentum state p certainly containing one particle with
(pseudo-)spin s, and the (−p, −s) state certainly being empty. The result is an
eigenstate of momentum and spin, but is it an energy eigenstate, and thus a
sharply defined excitation? Note that if we chose s =↑ so that the (p, ↑) state
is certainly occupied it means that a†p↑ a†−p↓ |p, ↑i = 0, so that the corresponding
term no longer appears in the interaction term when it is applied to this state. The
level is said to be ‘blocked’. Thus it certainly is an eigenstate of the pair problem,
if |BCSi is. What is its energy? We have to take into account the kinetic energy
as well as the loss of attractive interaction energy, see Eq. (9.16)
(p,s) occupied (−p,−s) empty
z }| {
Es (p) = ξp [ 1 − hnPp i
z }| {
−hnPp i
‘blocking0
z }| {
] + ∆ sin θp = Ep
Note that these quasiparticle excitations always have a gap ∆s given by
(
∆,
µ > 0,
∆s = min Ep = √ 2
2
p
∆ + µ , µ < 0.
(9.38)
(9.39)
As µ turns from positive to negative as we pass from BCS to BEC (see Fig. 9.3),
the density of states of the quasiparticle excitations turns from having a square
−1/2
root singularity ν() ∼ ( − ∆s )
to vanishing like a square root ν() ∼
1/2
( − ∆s ) .
9.3 BCS theory from the Bogoliubov transformation
Given the similarities between the BCS theory and the Bogoliubov theory of the
Bose gas, it is natural to re-cast the former to look more like the latter. To this
end, we seek to reduce the full Hamiltonian to a quadratic one. Allowing for an
external potential independent of spin, we have
Z
X ~2
†
†
H = dr
∇ψs (r) · ∇ψs (r) + [V (r) − µ] ψs (r)ψs (r)
2m
s=↑,↓
Z
+U0 dr ψ↑† (r)ψ↓† (r)ψ↓ (r)ψ↑ (r).
(9.40)
Now for A and B are any bilinears in the field operators, let us write
AB → hAiB + AhBi − hAihBi.
9.3 BCS theory from the Bogoliubov transformation
105
The idea is to replace A and B by their average values, with the last term preventing overcounting. This prescription yields the quadratic Hamiltonian
Z
X
∇ψs† (r) · ∇ψs (r) + [V (r) − µ + U0 ρ−s (r)] ψs† (r)ψs (r)
H = dr
s=↑,↓
Z
dr ∆∗ (r)ψ↓ (r)ψ↑ (r) + ∆(r)ψ↑† (r)ψ↓† (r)
Z
1
2
− dr
|∆(r)| + U0 ρ↑ (r)ρ↓ (r)
U0
+
(9.41)
(Here −s denotes the spin state opposite to s) where
∆(r) = U0 hψ↓ (r)ψ↑ (r)i
(9.42)
and ρs (r) is the density of particles of spin s. In common with the Bogoliubov
Hamiltonian, the above contains anomalous terms. We can write the first two
lines of Equation (9.41) in a compact matrix from by introducing the Nambu
spinor notation
ψ↑ (r)
Ψ(r) =
(9.43)
ψ↓† (r)
ψ↑ (r) =
X
u∗α (r)aα,↑ + vα (r)a†α,↓
(9.44)
−vα∗ (r)aα,↑ + uα (r)a†α,↓
(9.45)
α
ψ↓† (r) =
X
α
9.3.1 Effect of Temperature
When we discussed the quasiparticle excitations, we didn’t account for the effect
that they have on the self-consistent equation. This is fine when there are few
excitations in the system, but when many levels are blocked, we have to take
this effect into account, leading to a reduction in the gap parameter. The obvious
example of this effect is a system at finite temperature, where the quasiparticles,
being fermionic, have the fermi-dirac distribution function
ns (p) =
1
,
eβEs (p) + 1
β = 1/kB T,
(9.46)
where we allow the quasiparticle energies, and hence distributions, to differ for
the two species. Since the occupancy of a given state can be zero or one only
ns (p) is also the probability for that state to be occupied. Thus the probability of
a (p, s; −p, −s) state being blocked is ns (p) [1 − n−s (−p)] + n−s (−p) [1 − ns (p)].
What if we have both a (p, s) and a (−p, −s) quasiparticle present?
Y
†
†
αp↑
α−p↓
|BCSi = up a†p↑ a†−p↓ − vp
vp a†p↑ a†−p↓ + up |0i,
(9.47)
p0 6=p
106
Fermi superfluids
which is orthogonal to the BCS state. In the Anderson spin language this corresponds to a single flipped spin, and has energy given by twice the ‘magnetic field’
each pair experiences 2E(p), see Eq. (9.19). The spin interpretation allows us to
see easily that this state contributes −up vp to the self consistent equation, and
occurs with probability n↑ (p)n↓ (−p). The overall result is thus
two qp
no qp
}|
{ z
}|
{
U0 X ∆ z
∆=
([1 − n↑ (p)][1 − n↓ (−p)] − n↑ (p)n↓ (−p))
V p 2Ep0
U0 X ∆
=
[1 − n↑ (p) − n↓ (−p)].
V p 2Ep0
(9.48)
For the case Es (p) = E(p), and in the BCS limit where µ ∼ EF , ∆ EF , this
is conveniently presented in the form
Z
1
E
∆
√
=−
dE tanh
,
2
U0 ν(EF )
2T
E − ∆2
∆
where ν(EF ) is the fermi surface density of states, and we have left out the
upper cut-off. This expression needs to be regularized as in the zero temperature
case considered before. The resulting ∆(T ) varies from ∆(0) = ∆BCS given by
Eq. (9.28), to zero at
γ
kB Tc = ∆BCS .
(9.49)
π
The key point is that the variation is smooth, suggesting a second order transition
to a normal (non- superfluid) state above Tc . It looks like we have gone some way
to verifying the schematic phase diagram that we sketched in Fig. 9.1, with a small
but finite transition temperature on the ‘weak’ side (now identified with 1/kF a <
0). Unfortunately there is a problem. If we repeat the calculation throughout the
crossover, the temperature at which a non-zero ∆ develops continues to increase
indefinitely on the BEC side [1]
3/2
kB Tc ∼ Eb /2 [ln Eb /EF ]
, 1/kF a 0
(9.50)
where Eb = ~2 /ma2 is the binding energy of a pair. This temperature should
be thought of as a dissociation temperature below which pairs can form (the
logarithmic factor is entropic in origin). The temperature at which these pairs
condense will however be lower, tending to the ideal bose TBEC in the 1/kF a → ∞
limit. A more sophisticated treatment is required to find a smooth interpolating
Tc (1/kF a), see Fig. 9.4.
9.4 The effect of ‘magnetization’
We turn now to the effect of having differing densities of the two species of
fermions. This problem has a long history in the theory of superconductivity,
where it corresponds to the magnetization of the system of electron spins in
an external field. There, the Meissner effect and other complications originating
9.4 The effect of ‘magnetization’
107
Figure 9.4 Tc in the mean-field theory (solid line), compared with the
result of a treatment that smoothly interpolates to TBEC (dashed).
Reproduced from Ref. [1]
from the orbital effect of the magnetic field on the electrons often dominate the
behaviour of the system. A neutral gas is free of such complications, and moreover
achieving finite ‘magnetization’ is straightforward
To study this situation we introduce a conjugate variable h that couples to
the imbalance in number between the two species, just as the chemical potential
couples to the total number6
Hµ,h = H − µ [N↑ + N↓ ] − h [N↑ − N↓ ] .
The resulting free energy F (µ, h) gives the thermodynamic relations
−
∂F
= N ≡ N↑ + N↓ ,
∂µ
−
∂F
= M ≡ N↑ − N↓ ,
∂h
(9.51)
but in fact it is possible to write down the complete set of equations that we need
without computing F (µ, h). Since the quasiparticles carry spin, it is natural that
the energy Es (p) is
Es (p) = Ep − sh,
(9.52)
and this energy appears in the equilibrium quasiparticle distribution function
6
In studies of the BCS-BEC crossover in 6 Li, a Feshbach resonance between the
|F, mF i = |1/2, ±1/2i states has been used, whereas for 40 K, the two states are |9/2, −9/2i, and
|9/2, −7/2i. Thus the gases really are magnetized at finite species imbalance. We stress, though,
that the magnetic field h is simply a conjugate variable. The real magnetic field is varied to tune
through the resonance, see Section 4.3.4.
Fermi superfluids
108
Eq. (9.46). Thus we have
1X
1
m
1
=
−
[1 − n↑ (p) − n↓ (−p)] −
4π~2 as
V p 2Ep0
2p
X
N=
2vp2 + u2p − vp2 [n↑ (p) + n↓ (−p)]
p
M=
X
n↑ (p) − n↓ (p).
(9.53)
p
This is a complete set of equations to study the problem of finite M , although
their analysis may be numerically quite difficult. The interpretation of the equation for N , which we have not seen before, is actually straightforward. With no
quasiparticles we get a contribution 2vp2 , the occupancy of the paired state as
before. With one quasiparticle we get a contribution of one (one state of the pair
(p, −p) occupied), and with two a contribution of u2p , see Eq. (9.47).
9.4.1 Sarma state
If h is positive, only ↑ quasiparticles may exist, and then only if h > ∆s . In the
BCS limit at T = 0 the analysis is simple enough
√ to be done analytically. If h > ∆
quasiparticle states are occupied for |ξp | > h2 − ∆2 . By taking the difference
between the self-consistent equation at h = 0 (solved by ∆BCS ) and finite h, we
get the condition
√
h + h2 − ∆2 ∆BCS
√
,
(9.54)
0 = ln
h − h2 − ∆ 2 ∆
with solution
∆Sarma
=
∆BCS
r
2h
− 1.
∆BCS
(9.55)
Thus for ∆BCS /2 < h < ∆BCS there are three solutions of the self-consistent
equation: ∆ = 0 (always a solution, though we have often divided through by ∆),
∆BCS , and ∆Sarma . It is useful to think about what this means for the variational
energy of the system as function of ∆, see Fig. 9.5, and one quikly concludes that
the appearance of a new solution at h = ∆BCS /2 correpsonds to the maximum
at ∆ = 0 turning to a minimum, with a maximum moving away to finite ∆Sarma .
Note that for ∆ > h, the potential coincides with its zero field value. Thus we
can use the result Eq. (9.32) for the condensation energy to determine when the
∆ = 0 minimum becomes the global minimum of the potential, assuming this
occurs for h < ∆BCS . The free energy density of the magnetized normal system
is
E(0, h)/V = −ν(EF )h2 ,
h EF
9.4 The effect of ‘magnetization’
109
Figure 9.5 Schematic plots of ground state energy versus ∆ for different h.
√
so that for h > hc = ∆BCS / 2, the system makes a first order transition to a
normal fermi gas, with magnetization density
∂E(0, h)/V mN = −
= 2ν(EF )hc
∂h
hc
We have already argued in Section 9.3.1 that the transition to the superfluid
state at finite temperature is second order at h = 0. Together with the result just
found at zero temperature, we are led to the conclusion that at some point on
the superfluid-normal phase boundary in the (T, h) plane, the transition changes
from being first order to second order: a tricritical point, see Fig. 9.6
Since we are interested in an experimental situation where m is fixed and not
h, it is important to realize that values of m in the region 0 < m < mN cannot
be accommodated within a single phase, see Fig, 9.7. At finite temperature, the
superfluid state can be magnetized, as thermally excited quasiparticles will be
principally of ↑ type, but beneath the tricritical point there will still be a jump
from some mS to some larger mN at a critical hc . The result is phase separation,
leading to two coexisting but spatially separated regions, one of superfluid with
m = mS , the other a magnetized normal fluid. It is straightforward to show that
the fraction p of superfluid is
mN − m
,
mS < m < mN
p=
mN − mS
9.4.2 Magnetization in the BCS-BEC crossover
Now we discuss in a completely qualitative way what happens as we pass through
the BCS-BEC crossover [? ]. The main difference from the analysis of the BCS
limit is that it is possible to have h > ∆s , so that there is a finite concentration
of quasiparticles in the superfluid, before the normal state becomes favoured.
Physically, we expect that on the far BEC side we can have a coexisting mixture
110
Fermi superfluids
Figure 9.6 The tricritical point in the (T, h) plane. Reproduced from
Ref. [3]
Figure 9.7 Magnetization versus field at zero (black line) and finite
temperature (red line), and the resulting region of phase separation (PS) in
the (T, m) plane.
of N↓ /2 bosonic molecules with the remaining M unpaired ‘majority’ fermions,
see Fig. 9.8. At finite temperature, the phase diagram will in general have two
tricritical points, one on each side of the crossover, which meet near the unitary
point at some temperature of order of the Fermi energy.
9.4 The effect of ‘magnetization’
111
Figure 9.8 a) Schematic phase diagram of the zero temperature BCS-BEC
crossover with magnetization. The red dot marks a tricritical point that
separates the trivial second-order phase transition when the number of pairs
N↓ /2 goes to zero from the phase separated region. Point A is where the
normal state becomes fully polarized, and point B is where the free energy of
the superfluid state with gap ∆s (kF aS ) is equal to that of the normal state
in magnetic field h = ∆s . b) The same, but at finite temperature. Now the
second tricritical point on the BCS side is visible.
References
[1] 1993. Crossover from BCS to Bose superconductivity: Transition temperature and timedependent Ginzburg-Landau theory. Phys. Rev. Lett., 71, 3202.
[2] Greiner, Markus, Regal, Cindy A, and Jin, Deborah S. 2005. Fermi Condensates.
[3] Sarma, G. 1963. J. Phys. Chem. Solids, 24, 1029.
[4] Schreiffer, J. R. 1964. Theory of Superconductivity. Perseus Books, Reading, MA.
Part III
Low-dimensional systems
113
10
Quantum Hydrodynamics
In Section 6.4 we gave a hydrodynamic interpretation to the dynamics described
by the Gross–Pitaevskii equation. That discussion centered on the classical equations of motion obeyed by the density and phase of the condensate. Now we show
how, without recourse to microscopic theory, this description can be quantized
to reveal some generic features of interacting condensates.
10.1 Hamiltonian and commutation relations
Our starting point is the hydrodynamic equations (continuity and Euler) that we
found previously
∂n
+ ∇ · [nvs ] = 0,
∂t
1
2
mv̇s + ∇
mvs + µ(n) = 0,
2
√
√
∇2 n n/ξ 2 .
(10.1)
(we have dropped the external potential). This set of equations can be obtained
from the Hamiltonian
Z
1
H = dr
vρv + V (ρ) ,
(10.2)
2
together with the commutation relation
[ρ(r), ϕ(r0 )] = iδ (r − r0 )
(10.3)
where v = ∇ϕ (I am setting ~ = m = 1 from now on, so that the mass density
ρ equals the number density n), and µ = dV /dρ. The Hamiltonian Eq. (10.2) is
just what one would expect for a fluid, only with an irrotational velocity v = ∇ϕ.
Let’s check the continuity equation
Z
i
ρ̇ = i [H, ρ] =
dr vρ [v, ρ] + [v, ρ] ρv
2
1
= − ∇ · (vρ + ρv) = −∇ · ρv,
(10.4)
2
and you should also check that the Euler equation is reproduced. The commutation relation Eq. (10.3) plays a fundamental role in the following development,
115
Quantum Hydrodynamics
116
and follows from the fundamental quantum commutator. It can be obtained from
the microscopic expressions for the density and current
X
ρ(r) =
δ(r − ri )
i
1X
pi δ (r − ri ) + δ (r − ri ) pi
j(r) =
2 i
pi = −i∇i ,
(10.5)
from which one readily obtains
[j(r), ρ(r0 )] = −iρ(r)∇r δ(r − r0 ).
If we use j = (ρv + vρ) /2, then the commutation relation for ϕ(r) follows immediately.
10.2 Mode expansion
It is convenient to re-write the commutation relation in terms of the Fourier
modes
of the fields. Introducing the canonical boson operators βk , βk† , with βk , βk† 0 =
δk,k0 , the commutation relations are consistent with
√ X −κk
ρ(r) = ρ0 + ρ0
e βk eik·r + h.c.
k
r
ϕ(r) =
1 X
−ieκk βk eik·r + h.c.,
4ρ0 k
(10.6)
where κk is for now a free parameter that will be fixed by some dynamical input
(i.e. the Hamiltonian). We now write the Hamiltonian Eq. (10.2) as
Z
1
1
(10.7)
H = dr ρ0 v2 + U0 (ρ − ρ0 )2 .
2
2
At this point we have approximated the full Hamiltonian by a quadratic one.
This assumes that v and ρ − ρ0 are small, and is equivalent to the linearization of
the equations of motion discussed at the end of Section 6.4. In a narrow sense we
could think of the interaction term as the usual quadratic approximation from
the Gross-Pitaevskii theory, where U0 is a microscopic interaction parameter (or
pseudopotential, at the next level of sophistication). The Hamiltonian Eq. (10.7)
is more general, however, and if the scale of variation of ρ is much larger than the
interparticle separation, U0 can be thought of as V 00 (ρ0 ), with V (ρ) the potential
energy density of the fluid. In general this has a more complicated relationship to
microscopic parameters. We will see how this works when we discuss the Tonks
gas.
Substituting the mode expansion Eq. (10.6) into the Hamiltonian Eq. (10.7)
yields a quadratic form in the operators {βk , βk† }, which in general contains β−k βk
terms and their complex conjugates. A judicious choice of κk sets such terms to
10.3 Correlation functions
117
zero and yields (aside from an infinite constant)
X
cs |k|βk† βk ,
H=
1/d
kρ0
e−2κk =
~|k|
,
2mcs
c2s =
nU0
m
(10.8)
Of course, this is recognizable as nothing more than the Bogoliubov transformation in another guise (we have restored the units of mass for familiarity’s sake).
In line with the above discussion, we have we have included only low wavevectors
in Eq. (10.8).
10.3 Correlation functions
To see the power of this approach, let’s consider the density matrix of the bosons.
Using
the density-phase representation, we write the boson operator as b(r) =
p
ρ(r)eiϕ(r) . Assuming that the phase-phase correlations are sufficiently small, we
can expand the exponents to obtain for the density matrix
1
0 2
0
†
0
(10.9)
ρ (r, r ) = hb (r)b(r )i ∼ ρ0 1 − h(ϕ(r) − ϕ(r )) i , |r − r0 | → ∞
2
It turns out that in d > 1 this expansion is a reasonable thing to do, because
1
mcs
2
h(ϕ(r) − ϕ(r0 )) i → −cd
2
ρ0 |r − r0 |d−1
|r − r0 | → ∞,
(the constant cd depends on dimension) giving for the momentum distribution
mcs
,
|p| → 0.
n(p) = no δp,0 +
2|p|
We see that the effect of interactions is to give rise to a ground state in which some
particles have been removed from the zero-momentum state. The effect is called
the quantum depletion of the condensate, and is more commonly discussed in
the context of Bogoliubov’s theory of the weakly interacting gas. The advantage
of that approach is that the full wavevector dependence of quantities can be
calculated, not just the k ξ −1 asymptotes. In this way one can show that the
total depletion is
1 X
8 p 3
n(p) = √
nas ,
(10.10)
N p
3 π
2
where we used the Born approximation for the scattering length as = 4π~m U0 .
Under typical experimental conditions the depletion does not much exceed 0.01,
which justifies the use of the GP approximation.
The nice thing about the present discussion, however, is that it is not restricted
to weak interactions. As we’ll see in the next chapter, it applies even when the
condensate is totally depleted.
118
Quantum Hydrodynamics
References
11
Quantum Fluids in One Dimension
11.1 Bose fluids in one dimension: the Tonks gas
So far, we have worked in the limit of weak interactions, where energy scales such
as the chemical potential are simply proportional to U0 (µ = nU0 ). What happens
as interactions become strong? There is no general answer to this very difficult
question, but one situation where progress is often possible is in one dimension 1 .
In fact, the case of δ-function interactions that we have been considering can
be exactly solved in one dimension to yield the wavefunction for the ground and
excited states. We will briefly discuss the character of this solution. Firstly, notice
that we can form the dimensionless parameter
mU0
,
~2 n
because the density n has the units of inverse length in one dimension, while the
strength of the δ-function has units [energy] × [length]. Our earlier result for the
energy density at small U0 goes through as before and can be written
γ≡
E/Ω1 = U0 n2 /2 = ~2 n3 γ/2m.
In general then, the energy density will be of the form E/Ω1 = n3 ~2 e(γ)/2m,
where the function e(γ) is shown in Fig. 11.1. Notice that e(γ → ∞) = π 2 /3.
How can we understand this result? Surprisingly, the wavefunction has a simple
form in this limit, which can be obtained by noticing that for infinitely repulsive
interactions the many-body wavefunction has to vanish whenever the coordinates
of two particles coincide. When none coincide, the wavefunction satisfies the free
Schrödinger equation because of the δ-function nature of the interaction. There
is an obvious class of wavefunctions that satisfy both of these properties, namely
the Slater determinants. The drawback that these function are completely antisymmetric, rather than symmetric as dictated by bose statistics, is readily solved
by taking the modulus. Thus any eigenstate of the γ → ∞ problem can be written
ΨB (x1 , · · · , xN ) = |ΨF (x1 , · · · , xN )|,
(11.1)
where ΨF (x1 , · · · , xN ) is an eigenstate of a system of non-interacting fermions.
Such a one-dimensional system of impenetrable bosons is known as a TonksGiradeau gas. Let’s check this idea by calculating the ground state energy. If the
1
In a sense, the interactions are always strong in one dimension, see Problem ??.
119
Quantum Fluids in One Dimension
120
Figure 11.1 The curve shows the function e(γ) obtained in Ref. [? ] for the
δ-function Bose gas in 1D, in terms of which the energy is
E = N n2 ~2 e(γ)/2m.
fermi gas has fermi wavevector kF , the total energy density is
Z kF
dk ~2 k 2
~2 kF3
n3 ~2 π 2
E/Ω1 =
=
=
,
6πm
2m 3
−kF 2π 2m
using n = kF /π for the density. The chemical potential is ∂E/∂N = ~2 kF2 /2m ≡
EF and the hydrodynamic speed of sound is
n ∂ 2 (E/Ω)
~kF
,
cs =
≡ vF .
(11.2)
2
m ∂n
m
Another very useful feature of the fermion mapping is that it allows us to calculate
any observable that depends on the local density operator ρ(r) = ψ † (r)ψ(r), as
the modulus in Eq. (11.1) doesn’t interfere with such a calculation.
c2s =
Exercise 11.1.1 Using the harmonic Hamiltonian Eq. (10.7)
1.
Show
h0|ρq ρ−q |0i =
2.
N ~|q|
,
2mcs
|q| → 0,
(11.3)
consistent with Onsager’s inequality.
Show that in 1D the behaviour of the phase correlation function is such
that the full exponential has to be retained in Eq. (10.9) to give
n(p) ∝
1
1− η1
|p|
,
|p| → 0,
η≡
2π~n
mcs
11.1 Bose fluids in one dimension: the Tonks gas
121
In particular, this result tells us that there is no condensate in a onedimensional system.
Exercise 11.1.2 Let’s compute some properties of the Tonks gas using the
fermionic mapping Eq. (11.1).
1. First show that the dynamical structure factor defined in Eq. (6.14) is
(
qpF
qpF
~q 2 ~q 2 Nm
<
ω
<
−
+
m
2m
m
2m S(q, ω) = 2~qpF
(11.4)
0
otherwise
2.
Check that this is consistent with the sum rules Eq. (6.15) as well as
the inequality Eq. (6.18), with cs = vF ≡ pF /m. What is the value of
η, and the resulting n(p), for the Tonks gas? Note that deriving this
result for the momentum distribution is hard to do starting from the
wavefunction. In particular, it does not coincide with the momentum
distribution of a fermi gas.
References
Part IV
Atoms in optical lattices
123
Part V
Multicomponent gases and atomic
magnetism
125
Appendix A
Symbology
I’ve tried to stick to the following notation
Notation
|ni
En
Eα
ϕα (r)
Nα
Nα
S/A
Ψα1 α2 ···αN
|N0 , N1 . . .i
|VACi
ρ(r)
φi (r)
χρ
Cρ
g(r, r0 )
Meaning
Energy eigenstate of a many-body system
Eigenergy of a many body system
Single particle eigenenergies
Single particle state (usually eigenstate of single particle Hamiltonian)
Occupation number of single particle state α
Number operator for single particle state α
Symmetrized / antisymmetrized N particle state formed from single particle
states {ϕαi }
The same, but labeled by occupation number
The vacuum state (no particles)
Density operator
Eigenfunction of the single particle density matrix
Density response function
Density correlator
Single particle density matrix
Table A.1 Notation used in the text
127
Notes
128