This set of lectures is supposed to provide a short introduction to the condensed matter physics of atomic gases. Over the past ten years, the field of cold atomic gases has emerged as one of the most exciting in physics. These gases offer a beautifully clean system in which to study quantum many-body physics, free from many of the material complications common to other forms of matter. These lectures will cover those phenomena that may (covetously) be said to belong to condensed matter physics. Of course, as condensed matter physicists, we have no particular right to dictate what is interesting in a branch of what is essentially atomic physics. Nevertheless, there are many elements of the subject that fit into a traditional condensed matter context. An incomplete but illustrative list of invaluable concepts includes states of matter, equilibrium phase diagrams, single particle vs. collective behavior, the role of dimensionality, and long-range order.
Probably the biggest omission from these lectures is that in discussing exclusively gases of bosons I will be ignoring the huge progress that has taken place in the last few years in creating degenerate fermion systems and observing their condensation. The reason for this is that other lectures at this school will focus on the theory of fermi systems, of which
the atomic gases are a very ideal realization. The article Ref. [1] provides a non-technical
introduction to the
40
K experiments at JILA that were groundbreaking in this regard.
We will be concerned with the properties of gases of neutral alkali atoms. The number of atoms in typical experiments range from 10
4 to 10
7
: 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 10
13 cm
− 3 to 10
15 cm
− 3 .
As we’ll discuss in more detail in the next chapter, the observation of quantum phenomena like Bose-Einstein condensation requires a phase-space density of order one, or q nλ 3 dB
∼ 1 , where λ dB
≡
2 π
~
2 mkT is the thermal de-Broglie wavelength. The above densities then correspond to temperatures
T ∼
~
2 n
2 / 3 mk
∼ 100 nK − few µK
1

At these temperatures the atoms move at speeds of ∼ compared with around 500 m s
− 1 p kT /m ∼ for molecules in this room, and
1 cm s
∼ 10
6
− 1
, which should be m s
− 1 for electrons in a metal at zero temperature. Achieving the regime nλ
3 dB
∼ 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 atoms 1
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 cm
6 s
− 1
, leading to a lifetime of several seconds to several minutes
2
. These
relatively long timescales suggest that working with equilibrium concepts may be a useful first approximation.
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 Nuclear spin, I
87
Rb 3/2
23 Na
7 Li
3/2
3/2
Fermions
6 Li
40
K
1
4
The hyperfine coupling between electronic and nuclear spin splits the ground state manifold into two multiplets with total spin F = I ± 1 / 2 . The Zeeman splitting of these multiplets
by a magnetic field (see Fig. 1.1) forms the basis of magnetic trapping.
1 Compare 10 19 cm
− 3 cm
− 3 for atomic densities in liquids and solids.
2
This three-body rate is reduced by the appearance of Bose-Einstein condensation, further enhancing the sample lifetime.
for the number density of air molecules at ground level, and ∼ 10 22
2

Figure 1.1: Magnetic field dependence of atomic states of an atom with J = 1 / 2 , I = 3 / 2 , e.g.
87
Rb. 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 and F = 1 , m
F
= − 1
F = 2 , m
F
= 2 , 1 , 0 ,
3

In 1923 Bose gave a derivation of Planck’s radiation law that treated light quanta as indistinguishable particles to yield the equilibrium distribution that now bears his name n ( E ) =
1 e E/k
B
T − 1
.
(2.1)
The following year, in applying this distribution to massive bosonic particles, Einstein made the following observation. Because the number of particles is conserved, we introduce a chemical potential µ and chose it such that the total number is fixed
N =
X e ( k
1 k
− µ ) /k
B
T − 1
, k
=
~
2 k
2
2 m
(2.2)
Recalling P k have for d = 3
−−−−→
Ω d
→∞
Ω d
R d d k
(2 π ) d in d dimensions, where Ω d is the d -dimensional volume, we n ≡
=
Z
N
Ω
3
=
Z d 3 k 1
(2 π )
3 e ( k
− µ ) /k
B
T − 1 k
2 dk
2 π 2
1 e (
~
2 k 2 / 2 m − µ ) /k
B
T
=
1
~
3 k
B
2
T m
π
3 / 2
Li
3 / 2
( µ/k
B
T ) (2.3)
( Li n
( z ) = P
∞ k =1 z k
/k n is the polylogarithm). Note that ( k
B
T m )
1 / 2 is the typical magnitude of a particle’s momentum. As the density of particles increases, or the temperature falls,
µ n p increases to zero at some critical value of ‘phase space density’
∼ ( k
B
T m )
− 3 / 2
), corresponding to n × n p
∼
~
− 3 (where k
B
T c
= α
~
2 m n
2 / 3
, α ≡ 2 π/ [ ζ (3 / 2)]
2 / 3
Clearly µ
has to be negative for the distribution Eq. (2.2) to make sense, so what happens?
4

Figure 2.1: Absorption image of around 7 × 10
5 atoms just above, at, and below the Bose-
Einstein condensation temperature, reproduced from Ref. [2]. The gas is allowed to expand
for 6 ms. In such measurements light is absorbed by the gas, creating real transitions and heating the gas. Such measurements are therefore of a one-shot character in that they destroy the sample. A second kind of imaging is dispersive (phase-contrast) imaging that relies on diffraction. Many images may be taken with this second technique without much heating.
Let’s think about zero temperature first. Since the particles are bosons, the ground state consists of every particle sitting in the lowest energy state ( k = 0 , if we think of a box with periodic boundary conditions). But such a singular distribution was excluded by the above replacement of the sum by the integral. Supposing that at T < T c we have a finite fraction f ( T ) of particles sitting in this state. Then the chemical potential may stay equal to zero and
Eq. (2.3) gives
n = n
" f ( T ) +
T
T c
3 / 2 #
, so f ( T ) =
"
1 −
T
T c
3 / 2 #
.
(2.4)
The unusual, highly quantum degenerate state emerging below T c
Bose-Einstein condensate (BEC) came to be known as a
Problem 1 (Ideal Bose gas in a harmonic trap) Show that in a harmonic trap of energy
~
ω ho the corresponding relation is f ho
( T ) = N
"
1 −
T
T ho
BEC
3
#
.
with k
B
T ho
BEC
2 .
40411 . . .
)
=
~
ω ho
N
ζ (3)
1 / 3
(you will need the integral R
∞
0 dx x
2
/ ( e x − 1) = 2 ζ (3) ∼
5

As should be clear from the above, we are dealing with an intrinsically quantum phenomenon, occurring when the phase space density approaches
~
− 3 .
Bose condensation, 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 ) = N c
δ k , 0
+ · · · , (2.5) where N c is O ( N ) , and f ≡ N c
/N is the condensate fraction . For non-interacting bosons
Eq. (2.4) tells us that
f = 0 at the condensation temperature T c and f = 1 at T = 0 . 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 behaviour of the one-body density matrix
ρ
1
( r , r
0
) ≡ hh φ
†
( r ) φ ( r
0
) ii .
(2.6) hh· · · ii is the average over the many-body density matrix of the system
1
. Since the distri-
bution n ( k ) = hh φ
† k
φ k ii , we identify this as the Fourier transform of ρ
1
( r , r
0
) ≡ g
1
( | r − r
0 | ) in a translationally invariant, isotropic system. The presence of a δ
-function in Eq. (2.5) shows
that g ( r ) →
N c
,
Ω
| r | → ∞ ,
The non-vanishing of the right hand side is referred to as off-diagonal long range order , or
ODLRO 2 . More generally, this property implies that in the spectral resolution of the density
matrix
ρ
1
( r , r
0
) =
X n
α
χ
∗ i
( r ) χ i
( r
0
, ) i there is at least one eigenvalue of order N . This may be seen by using a trial eigenfunction
χ
0
( r ) = 1 / Ω
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 χ
0
( 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 order parameter of the BEC can be introduced as Ψ( r ) = N
0
χ
0
( r ) , where N
0 and
χ
0
( r ) are the eigenvalue and eigenfunction respectively. Writing χ
0
( r ) as | χ
0
( r ) = | χ ( r ) | e iϕ ( r ) we define the superfluid velocity (we’ll discuss superfluidity shortly – for now it’s just a definition) v s
( r ) =
~ m
∇ ϕ ( r ) , (2.7)
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
1
The first-quantized version of Eq. (2.6) is
for a statistical mixture of orthogonal states
ρ
Ψ
1 n
( r , r
0
) ≡ N P n p n
R d r
2
· · · d r
N occupied with probabilities p n
Ψ
∗ n
( r , r
2
, . . . , r
N
)Ψ n
( r
0
, r
2
, . . . , r
N
) ,
2 If all the particles were in a finite momentum state, the RHS would tend to a plane wave, for instance
6

refers to a macroscopic quantity , one whose quantum fluctuations are much smaller than its average. It immediately follows that the superfluid velocity is irrotational : ∇ ∧ v s
( r ) = 0 , and its circulation (line integral) satisfies the quantization condition
I v s
( r ) · d l = nh
, m n ∈
Z
.
(2.8)
The existence of such a macroscopic irrotational velocity is the key to the superfluid properties of BECs.
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.
2.3.1
Time-independent Gross-Pitaevskii theory
We now see how these ideas work by extending out discussion of Bose gases in two directions to include the non-ideal case where there are interactions between particles, and the possibility of an external potential resulting in an inhomogeneous condensate. We start at zero temperature, where a system of N -particles will be described by a wavefunction
Ψ( r
1
, . . . , r
N
) . In the case of the uniform ideal Bose gas every particle sits in the zero momentum state. The Gross-Pitaevskii approximation consists of a variational wavefunction that slightly generalizes this
Ψ( { r i
} ) =
Y
χ
0
( r i
) i
(2.9)
Such a wavefunction of course displays BEC with the density matrix having an eigenfunction
χ
0
( r ) with eigenvalue N . We now use this wavefunction to find the expectation value of the many-body Hamiltonian
H
N
=
N
X ~
2
−
2 m
∇
2 i i
+ U ext
( r i
) +
X
U
0
δ ( r i
− r j
) , i<j where we have introduced a short-range two-body interaction of the form U
0
δ ( r − r
0
) expectation value is
3
. The
h H
N i = N
Z d r
~
2
2 m
|∇ χ
0
| 2
+ U ext
( r ) | χ
0
( r ) | 2
+
1
2
N ( N − 1) U
0
Z d r | χ
0
( r ) | 4
, (2.10)
For large N , we can neglect the difference between N and N + 1 . Minimizing with respect to
χ
0
( r ) , and introducing a Lagrange multiplier to maintain the normalization of χ
0
( r ) gives the equation
~
2
−
2 m
∇
2
− µ + U ext
( r ) + N U
0
| χ
0
( r ) |
2
χ
0
( r ) = 0 .
3 The true interactions between atoms are in general the sum of a short-range hardcore repulsion and an attractive van der Waals potential that goes like ∝ 1 /r
6 at large distances, but see the next footnote.
7

The multiplier µ = ∂ h H i /∂N , so is identified with the chemical potential. Rewriting in terms of the order parameter Ψ( r ) gives the Gross-Pitaevskii
(GP) equation 4
~
2
−
2 m
∇
2
− µ + U ext
( r ) + U
0
| Ψ( r ) |
2
Ψ( r ) = 0 .
(2.11)
A fundamental effect of the nonlinearity of the GP equation is that there exists a length scale set by the typical value of | Ψ( r ) | 2 ∼ n and the interaction strength
ξ ≡
2 mnU
0
~
2
− 1 / 2
= (8 πna s
)
− 1 / 2
.
(2.12)
This healing length determines the scale over which Ψ( r ) is disturbed by the introduction of a localized potential of scale ξ . It is a fundamental length scale in the system. Note that in the dilute limit when the gas parameter na
3 s
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 approach
5
. A typical value of
ξ may be around 4000 Angstoms.
In a uniform system with Ψ( r ) =
√ n , the GP energy density h H i / Ω = n
2
U
0
/ 2 provides us with a formula for the sound velocity via the hydrodynamic relation c
2 s
= n ∂
2
( E/ Ω)
= nU
0
(2.13) m ∂n 2 m
Note that mc s
= (
~
/ξ ) /
√
2 .
With the ansatz Eq. (2.9) for the wavefunction, we can obtain various observables without
difficulty. The particle density is just
ρ ( r ) = ρ
1
( r , r ) = | Ψ( r ) |
2
.
Note that this refers to the number density, not the mass density as in the previous chapter.
The current density is j ( r ) =
− i
~
2 m
∇ r
− ∇
0 r
ρ
1
( r , r
0
) | r
0 → r
=
~ m
| Ψ( r ) |
2
∇ ϕ ( r ) .
Dividing one by the other yields the superfluid velocity defined in Eq. (2.7), though that rela-
tion is in fact the more general one.
4
The Gross-Pitaevskii description is in fact rather more general than this derivation suggests. We built no correlations into the wavefunction to accommodate the interaction, so it is the bare interaction parameter U
0 that appears. If the separation between particles ( ∼ n
− 1 / 3 ) is sufficiently small compared to the range of the interaction, however, one can imagine adjusting the ( r i
, r j
) dependence of the wavefunction to the exact twobody wavefunction whenever r i and r j approach each other (ignoring the smaller probability of three particles
approaching each other). The effect of this adjustment turns out (see Ref. [3] for a detailed discussion) to
amount to replacing the microscopic U
0
in Eq. (2.11) by the
pseudopotential U
0
= 4 π
~
2 a s
/m , depending on the s-wave scattering length a s
(note that this is the relation between microscopic parameters and the scattering length within the Born approximation). In this way the GP description applies even to the case of a hard sphere potential, where the potential energy is strictly zero, and the only the kinetic energy is affected by the interactions.
5
In real systems ξ n
− 1 / 3 is actually a severe exaggeration, as their ratio is only ∼ ( na 3 s
)
− 1 / 6 , and the gas parameter is maybe 10
− 4 . We will see, however, that the expansion parameter justifying the present approximation is ( na s
)
1 / 2
, which is still small.
8

Figure 2.2: Thought experiment used to define the superfluid fraction. An annulus of fluid is rotated slowly.
With this formalism in place, we can now discuss one of the most dramatic consequences of
Bose-Einstein condensation: the phenomenon of superfluidity.
The colloquial idea of a superfluid is that it can flow without resistance. A more precise description is that it is rigid to certain kinds of perturbation: those that want to create vorticity
( ∇ ∧ v ). The usual thought experiment used to define a superfluid goes as follows. Suppose we have a container in the form of a cylindrical annulus containing some ‘matter’ consisting of N particles of mass m
(Fig. 2.2). If we rotate this at some angular frequency
ω , we expect that the free energy of this system has ω -dependence of the form
F ( ω ) = F
0
+
1
2
Iω
2 where I = N mR 2 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
∆ F ( ω ) = −
1
2
( ρ s
/ρ ) Iω
2
, defining the superfluid density ρ s or superfluid fraction ρ s
/ρ . In other words, the equilibrium
state of the system is one in which a fraction of the mass is not rotating with the container
6
.
Now what does this have to do with Bose condensation? For a completely smooth annulus that is rotationally symmetric and infinite in length, it’s clear that the possible solutions of
6
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.
9

the GP equation have the form
Ψ n
( φ, r, z ) = Ψ
0
( r ) e inφ where Ψ
0
( r ) is some real function, nonzero everywhere, and n = 0 , ± 1 , ± 2 , etc. Using
Eq. (2.7), this corresponds to a superfluid velocity
v s
=
~ n m R so that the circulation is quantized as
I v s
( r ) · d l = nh
.
m
When the annulus is rotated, then, it is clear that only slight distortions of Ψ n
( φ, r, z ) can result, even if is not completely smooth. In particular, the fluid cannot begin to move at the (small) ω of the container, as that would generally violate the quantization of circulation.
Thus within the GP description, the superfluid fraction is unity. Of course, for sufficiently large ω , the fluid could jump to a state with n = 0 . Given the fact the critical value ω c
~
/ 2 mR
2 ∼ 10
− 14 sec
− 1 (with R = 10
− 1 m
≡
) is hardly fast, one may wonder what this has to do with the robust phenomena that are actually observed. The answer is that what is usually called superfluidity is the related metastability of the moving fluid. Configurations with finite (and sufficiently small) n cannot be distorted into configurations with different n without passing through a configuration where Ψ vanishes. If the interactions between particles are repulsive, this necessarily constitutes an (extensive) energy barrier. Thus it is very hard for superflow to decay when the container is halted: the dissipative degrees of freedom present in a ordinary fluid just don’t exist. This feature of condensates (which can be eliminated by a sufficiently rough container, our reason for not introducing it as the fundamental characteristic
of superfluids) is explored in Problem 2.
Problem 2 (Metastable superflow and repulsive interactions) [See Ref.
[3]
Section
VI.D.2] Using the GP approximation, we can give a more informed discussion of the way in which repulsive interactions allow the existence of metastable rotational states. Again we consider a cylindrical annulus, and the two lowest angular momentum states n = 0 , 1 .
We now wish to include, however, the effect of a small deviation from cylindrical symmetry, whose effect is to mix these two states. If a
†
0 and model version of the rotating frame Hamiltonian H a
†
1 rot create atoms in the
7
` = 0 , 1 states, a that includes the kinetic energy, the asymmetry effect, and interactions is
H rot
= −
~
( ω − ω
1
) h a
†
1 a
1
− a
†
0 a
0 i
− V
0 h a
†
0 a
1
+ h .
c .
i
+
U
0
2Ω h a
†
0 a
†
0 a
0 a
0
+ a
†
1 a
†
1 a
1 a
1
+ 4 a
†
1 a
†
0 a
0 a
1 i
(2.14)
7
The rotating frame Hamiltonian is in general H rot finite angular momentum.
= H − ω · L , so that at ω = 0 the ground state can have
10

where ω
1
=
~
/ 2 mR
2 is the critical angular velocity at which the energy. If we introduce the GP wavefunction
` = 1 state has the lower h cos
χ
2 e iϕ/ 2 a
†
0
+ sin
χ
2 e
− iϕ/ 2 a
†
1 i
N
| 0 i , show that
• The order parameter has a node for χ = π/ 2 . If V
0 node will coincide with the position of that potential.
is due to a localized potential, this
• The GP variational energy is (up to a constant, and ignoring terms lower order in N )
E ( χ ) /N =
~
( ω − ω
1
) cos χ − V
0 sin χ + nU
0
2 sin
2
χ, while the angular momentum is
L ( χ ) /N
~
=
1
2
(1 − cos χ )
• A metastable minimum exists for 2 U
0
> V
0
(assuming U
0 and V
0 are both much less than
~
ω
1
). 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.
• Repeating the argument with a state of angular momentum n greater than one, show that even when V
0 nU
0
> ` 2
~
ω
1
. This corresponds to a critical velocity of `
~
/mR = p
2 nU and coincides (parametrically, at least), with the famous Landau criterion.
0
/m =
√
2 c s
,
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 = 0 requires, by
Stokes’ theorem, that the irrotationality condition ∇ ∧ v s
( r ) breaks down somewhere inside any surface bounded by the contour we integrate around. For such configurations to have finite energy, Ψ( r ) mush 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 v
φ s
( φ, r, z ) = n
~ m
1 r
.
(2.15)
The metastable states with ω > ω c
)vortex configurations.
in this simply connected geometry are generically (multi-
For a very complete discussion of the relationships between the various phenomena that
come under the umbrella of superfluidity, see Ref. [4].
11

Figure 2.3: Vortices in an atomic gas of
23
Na. Reproduced from Ref. [5].
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 the logarithm of n ( k ) column integrated along the line of sight)
therefore provides a direct measurement of condensation (see Fig. 2.1).
The other major experimental confirmation of BEC relates to the observation of certain
interference phenomena that we will not have time to discuss (but see Ref. [3, 14, 15]). 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 Fig. 2.3.
In order that the vorticity matches, in a coarse-grained fashion, that of a rigid body ∇ ∧ v s
( r ) = 2 ω ˆ , we must have a area density of quantized vortices case, this is only an approximate statement.
n v
= 2 mω/h . In the trapped
Note that in a trap, while the quantization condition Eq. (2.8) 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.
12

For time dependent problems it is tempting to immediately write down
~
2
−
2 m
∇
2
+ U ext
( r ) + U
0
| Ψ( r , t ) |
2
Ψ( r , t ) = i
~
∂ Ψ( r , t )
∂t
.
(3.1)
Note that this equation conserves the normalization N = R d r | Ψ( r ) | 2
– all particles remain in
the condensate. Eq. (3.1) may be derived by generalizing the ansatz Eq. (2.9)
Ψ( { r i
} , t ) =
Y
χ
0
( r i
, t ) .
i
Substitution into the time-dependent Schr ¨odinger equation yields
(3.2)
X

~
2

−
2 m
∇
2 i i

+ U ext
( r i
) + U
0
X
δ ( r k
− r i
)

χ o
( r i
, t )
Y
χ
0
( r j
) = i
~
X
∂χ
0
( r i
, t )
∂t k = i j = i i
Y
χ
0
( r j
) j = i
(3.3)
In order to get a closed equation for χ
0
( r ) we can replace the pectation value of the density N | χ
0
( r i
) | 2
P j = i
δ ( r j
− r i
) with the ex-
evaluated with Eq. (3.2). With this replacement,
Eq. (3.3) is satisfied if Eq. (3.1) is, as long as we normalize
Ψ( 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 χ
0
( r ) (at some reference time) is a member. We write the boson field operator in terms of these states
φ ( r ) =
X
χ n
( r ) a
α
.
α
Then the interaction Hamiltonian has the form
H int
=
U
0
2
X
M
αβγδ a
†
α a
†
β a
γ a
δ
,
αβγδ
(3.4)
13

where the matrix elements M
αβγδ are
M
αβγδ
=
Z d r χ
∗
α
( r ) χ
∗
β
( r ) χ
γ
( r ) χ
δ
( r ) .
Now applied to the ansatz Eq. (3.2), we can see that it is the term in Eq. (3.4) with
α =
β = γ = δ = 0 that gives
1
2
N ( N − 1) U
0
R d r | χ
0
( 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 = γ = δ = 0 . You should satisfy yourself that these have the form
S χ
α
( r
1
) χ
β
( r
2
)
Y j =1 , 2
χ
0
( r j
) , (3.5) where S denotes the operation of symmetrizaton. The inclusion of such effects is thus quite tractable, but it will have to wait unitl the next (Bogoliubov) stage of approximation. In the equilibrium state, their effect is to give the quantum depletion of the condensate fraction, leading to N
0
< N , even at zero temperature. The effect is of order p na 3 s
, so it is reasonable
that the present approximation is justified when this is small [6].
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 δ Ψ( r , t ) of Ψ( r , t ) from some reference solution Ψ
0
( r , t ) , satisfying
~
2
−
2 m
∇
2
+ U ext
( r ) δ Ψ( r , t ) + 2 U
0
| Ψ
0
( r , t ) |
2
δ Ψ( r , t ) + U
0
Ψ
2
0
( r , t ) δ Ψ
∗
( r , t ) = i
~
∂ Ψ( r , t )
.
∂t
Note that the nonlinear term couples δ Ψ( r ) and δ Ψ
∗
( r ) . The presence of the chemical po-
tential in Eq. (2.11) means that the solution of Eq. (3.1) corresponding to to a solution of the
time-independent problem is Ψ
0
( r , t ) = Ψ
0
( r ) e
− iµt/
~ . Introducing the harmonic solution
δ Ψ( r ) = e
− iµt/
~ u ( r ) e
− iωt
+ v
∗
( r ) e iωt
, one obtains easily the Bogoliubov-de Gennes equations (BdG equations)
~
2
~
ωu ( r ) = −
2 m
∇
2
+ U ext
( r ) − µ + 2 U
0
| Ψ
0
( r , t ) |
2
~
2
−
~
ωv ( r ) = −
2 m
∇ 2
+ U ext
( r ) − µ + 2 U
0
| Ψ
0
( r , t ) | 2 u ( r ) + U
0
Ψ
2
0
( r ) v ( r ) v ( r ) + U
0
Ψ
∗ 2
0
( r ) u ( r ) .
(3.6)
In free space µ = nU
0
, and plane wave solutions of Eq. (3.6) have the dispersion relation
~
ω ( k ) = E ( k )
E ( k ) ≡ ( k ) ( k ) + 2 mc
2 s
1 / 2
, (3.7)
where we use the expression Eq. (2.13) for the speed of sound found earlier. This is the
famous Bogoliubov spectrum that we will encounter again shortly. At k
~
/mc s
( kξ 1 ) it has the linear form E ( k ) = c s k , crossing over to the free particle spectrum at higher momentum.
14

Problem 3 (Density response of a condensate) Let’s consider the effect of a weak timedependent perturbing potential U ext
( r , t ) . Consider the plane wave perturbation
U ext
( r , t ) = V
0 cos ( q · r − ωt )
This describes the effect of a pair of laser beams ( Bragg spectroscopy ) with different wavevectors q = q
1
− q
2 and frequency difference
tuning from the fundamental transition
1
ω , generally much smaller than the de-
. This allows us to enter a regime of
ω , q that probes the collective behaviour of the system
~ q /m ∼ c s
∼ 1 cm s
− 1 ,
~
ω ∼ h × 1 kHz.
•
Use the BdG equations Eq. (3.6) to compute the resulting density response
δn ( r , t ) = | Ψ
0
( r , t ) + δ Ψ( r , t ) |
2
− | Ψ
0
( r , t ) |
2
∼ Ψ
∗
0
( r , t ) δ Ψ( r , t ) + Ψ
0
( r , t ) δ Ψ
∗
( r , t ) , (3.8) to give
δn ( q , ω ) = − nV
0
E ( q ) 2
( q )
−
~
2 ( ω + i 0) 2
≡
V
0
D ( q , ω ) .
2
(3.9)
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. The golden rule gives the rate for this process as
Γ( q , ω ) =
2 π
~
V
0
2
4
X
α,i
δ (
~
ω − E
α
) |h α | e i q · r i | 0 i|
2
+ δ (
~
ω + E
α
) |h α | e
− i q · r i | 0 i|
2
, (3.10) 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.
•
By comparing with the imaginary part of the response function Eq. (3.9) deduce the
spectral representation
1
π
(Vol) Im D ( q , ω ) = −
X
α,i
δ (
~
ω − E
α
) |h α | e i q · r i | 0 i|
2
− δ (
~
ω + E
α
) |h α | e
− i q · r i | 0 i|
2
,
(3.11) which implies that all of the absorption comes from a single transition with
X
|h α | e i q · r i | 0 i|
2
=
N ( q )
.
E ( q ) i
(3.12)
Now note that P i e i q · r i is just a Fourier component ρ q of the density
In this way find the integrated line strength of the resonance
ρ ( r ) = P i
δ ( r − r i
) .
−
1
π
(Vol)
Z
∞
0 dω Im D ( q , ω ) = h 0 | ρ q
ρ
− q
| 0 i
This Fluctuation-Dissipation relation relates the response function that we found to the density fluctuations in the ground state of our system.
1
To put some figures to it, the typical recoil energy
~
2 k
2 op
/ 2 m on absorption of an optical photon is on the kHz scale
15

Figure 3.1: Momentum transfer per particle for a BEC (open circles) and expanded cloud
(closed circles), for a given value of the wavevector q = q
1
− q
2 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. [7].
•
Show that our explicit form Eq. (3.9) implies
h 0 | ρ q
ρ
− q
| 0 i =
N ( q )
E ( q )
→
N
~ q
2 mc s
N, p
~
/ξ.
(3.13)
The only problem is that our original Gross-Pitaevskii ground state ansatz disagrees with this result. Because there are no correlations between particles in that state, its density fluctuations are normal , even at low p h GP | ρ q
ρ
− q
| GP i = N,
while the result Eq. (3.13) only reaches this value at large p
2
. The resolution of this
problem lies in the quantization of the condensate oscillations, as we’ll see below.
Problem 4 (Sum rules and Onsager’s inequality) Let’s relate the analysis of the previous
problem to some more general formalism. The expression on the right hand side of Eq. (3.11)
is related to the dynamical structure factor S ( q , ω ) , defined as
S ( q , ω ) ≡
X
δ ω −
E
α
− E
0
α
~
|h α | ρ q
| 0 i|
2
.
(3.14)
Using completeness of the eigenstates, we then have
Z
∞ dω S ( q , ω ) ≡ S ( q ) = h 0 | ρ q
ρ
− q
| 0 i ,
0
2
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.
16

where S ( q , ω ) is called the static structure factor.
S ( q , ω ) satisfies the following two sum
rules (see, for instance, Ref. [8])
Z
∞
~
ωS ( q , ω ) =
0 lim q → 0
Z
∞
S ( q , ω )
0
~
ω
=
N
~
2 q
2
2 m
N
2 mc 2 s
, (3.15) known as the f-sum and compressibility sum rules, respectively.
In the present case
Eq. (3.12) implies
S ( q , ω ) =
N ( q )
δ ω −
E ( q )
E ( q )
~
, (3.16) and you should check that this satisfies both sum rules.
•
Use the sum rules Eq. (3.15) and the Cauchy-Schwartz inequality
|h A | B i| ≤ | A || B | , interpreting the integrals as inner products, to derive Onsager’s inequality
S ( q ) ≤
N
~ q
2 mc s
(3.17)
Note that Eq. (3.13) 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.
3.2.1
Galilean invariance
Let us go back to the time-dependent Gross-Pitaevskii theory describing the dynamics of a condensate
~
2
−
2 m
∇
2
+ U ext
( r ) + U
0
| Ψ( r , t ) |
2
Ψ( r , t ) = i
~
∂ Ψ( r , t )
∂t
, (3.18) 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 Ψ( r , t )
is a solution of Eq. (3.18) then
Ψ( r − v t, t ) exp i
~ m v · r −
1
2 m v
2 t , (3.19) tion under Galilean transformations. Upon making the decomposition Ψ = that this implies a transformation law for the phase of the condensate
√ ne iϕ
, we see
1 ϕ → ϕ +
~ m v · r −
1
2 m v
2 t .
(3.20)
Recalling the identifications
3
v s
=
~ m
∇ ϕ µ = −
~
˙
3 in the lab frame
17

we have the transformation laws for the superfluid velocity and chemical potential v s
→ v s
+ v µ → µ +
1
2 m v
2
, (3.21) which seems sensible. The general character of these transformation laws leads us to sup-
pose that they are more general that the equation of motion Eq. (3.18) that we started from.
Indeed, we could have made exactly the same arguments for the equation of motion of the
Bose field φ ( r , t ) , valid for arbitrary density-density interactions.
3.2.2
Hydrodynamic description
We can make the hydrodynamic character of the time-dependent GP equation more explicit.
If we rewrite ∂ | Ψ | 2
/∂t
as the difference of Eq. (3.18) and its complex conjugate we get the
continuity equation
∂n
+ ∇ · [ n v s
] = 0 , (3.22)
∂t while Ψ
∗ ˙ − Ψ
∗
Ψ , which involves the sum of the equation and its conjugate, yields
~
˙ +
1
2 m v
2 s
−
~
2
2 m
√ n
∇ 2
√ n + U
0 n + U ext
= 0 .
(3.23)
If n varies sufficiently slow in space – in practice this means slower than the healing length
√
ξ ≡
2 mnU
0
~
2
− 1 / 2
– we can drop the ∇ 2 n term. Taking the gradient of the resulting equation gives m ˙ s
+ ∇
1
2 m v
2 s
+ µ ( n ) + U ext
= 0 , (3.24) where we wrote µ ( n ) = nU
0
. For the static case we have
µ ( n ( r )) + U ext
( r ) = µ
0
, (3.25) which defines the Thomas-Fermi approximation , widely used to compute density profiles of trapped gases.
The equation Eq. (3.24) coincides with the Euler equation for the flow of a non-viscous
fluid, usually written as
ρ [ ∂ t
+ v · ∇ ] v + ∇ P = 0 , where we used ρ = mn , the mass density, and v ∧∇∧ v = ∇ v
2
/ 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 4 there is no vorticity, so
this difference disappears.
It is illuminating to consider the linearization of the equations Eq. (3.22) and Eq. (3.24).
In many ways the analysis is clearer than the linearization of the TDGP theory considered
in Section 3.1. Assuming that
n ( r , t ) = n
0
+ δn ( r , t ) , and that v s is of the same order as δn , one can easily derive the wave equation
¨ − c
2 s
∇
2
δn = 0 ,
18

with c
2 s
= n
0
U
0
/m giving the sound velocity that we found before. Obviously this description is not Galilean invariant. The requirement that frame. What happens if you include the ∇ v s be small amounts to a choice of reference n
term in Eq. (3.23)?
As in the previous subsection, Eq. (3.22) and Eq. (3.24) 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. (3.24) could have been obtained by taking
the time and spatial derivatives of the transformation law Eq. (3.20).
So far, our hydrodynamic description has been purely classical. Now we show how, without recourse to microscopic theory, this description can be quantized to reveal some generic features of interacting condensates.
3.3.1
Hamiltonian and commutation relations
Our starting point is the hydrodynamic equations (continuity and Euler) that we found previously
∂n
+ ∇ · [ n v s
] = 0 ,
∂t m ˙ s
+ ∇
1
2 m v
2 s
+ µ ( n ) = 0 , ∇ 2
√ n
√ n/ξ
2
.
(3.26)
(we have dropped the external potential). This set of equations can be obtained from the
Hamiltonian
H =
Z d r
1
2 v ρ v + V ( ρ ) , (3.27) together with the commutation relation
ρ ( r ) , ϕ ( r
0
) = iδ r − r
0
(3.28) 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. (3.27) is just what one would expect
for a fluid, only with an irrotational velocity v = ∇ ϕ . Let’s check the continuity equation i Z
ρ ˙ = i [ H, ρ ] = d r v ρ [ v , ρ ] + [ v , ρ ] ρ v
2
= −
1
2
∇ · ( v ρ + ρ v ) = −∇ · ρ v , (3.29) and you should also check that the Euler equation is reproduced. The commutation relation
Eq. (3.28) plays a fundamental role in the following development, and follows from the fun-
damental quantum commutator. It can be obtained from the microscopic expressions for the density and current
ρ j
(
( r r
)
)
=
=
X
δ ( r − r i
)
1 i
2
X p i
δ ( r − r i
) + δ ( r − r i
) p i i p i
= − i ∇ i
, (3.30)
19

from which one readily obtains j ( r ) , ρ ( r
0
) = − iρ ( r ) ∇ r
δ ( r − r
0
) .
If we use j = ( ρ v + v ρ ) / 2 , then the commutation relation for ϕ ( r ) follows immediately.
3.3.2
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 β k
, β
† k
, with h
β k
, β
† k
0 i
= δ k , k
0
, the commutation relations are consistent with
ρ ( r ) = ρ
0
+
√
ρ
0
X e
− κ k β k e i k · r k
+ h .
c .
r
1 ϕ ( r ) =
4 ρ
0
X
− ie
κ k β k e i k · r k
+ h .
c ., (3.31) where κ k is for now a free parameter that will be fixed by some dynamical input (i.e. the
Hamiltonian). We now write the Hamiltonian Eq. (3.27) as
H =
Z d r
1
2
ρ
0 v
2
+
1
2
U
0
( ρ − ρ
0
)
2
.
(3.32)
At this point we have approximated the full Hamiltonian by a quadratic one. This assumes that v and ρ − ρ
0 are small, and is equivalent to the linearization of the equations of motion
discussed at the end of Section 3.2.2. In a narrow sense we could think of the interaction
term as the usual quadratic approximation from the Gross-Pitaevskii theory, where U
0 is a microscopic interaction parameter (or pseudopotential, at the next level of sophistication).
The Hamiltonian Eq. (3.32) is more general, however, and if the scale of variation of
ρ is much larger than the interparticle separation, U
0 can be thought of as V
00
( ρ
0
) , 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. (3.31) into the Hamiltonian Eq. (3.32) yields a
quadratic form in the operators { β k
, β
† k
} , which in general contains β
− k
β k terms and their complex conjugates. A judicious choice of κ k sets such terms to zero and yields (aside from an infinite constant)
H =
X e
− 2 κ k = k ρ
1 /d
0
~
| k |
2 mc s
, c s
| k | β
† k
β k
, c
2 s
= nU
0 m
(3.33)
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. (3.33).
20

3.3.3
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 ) e iϕ ( r )
. Assuming that the phase-phase correlations are sufficiently small, we can expand the exponents to obtain for the density matrix
ρ r , r
0
= h b
†
( r ) b ( r
0
) i ∼ ρ
0
1 −
1
2 h ϕ ( r ) − ϕ ( r
0
)
2 i , | r − r
0
| → ∞
It turns out that in d > 1 this expansion is a reasonable thing to do, because
1
2 h ϕ ( r ) − ϕ ( r
0
)
2 i → − c d mc s
ρ
0
| r − r 0 | d − 1
| r − r
0
| → ∞ ,
(the constant c d depends on dimension) giving for the momentum distribution n ( p ) = n o
δ p , 0
+ mc s
,
2 | p |
| p | → 0 .
(3.34)
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
N
X n ( p ) = p
3
√
π p na 3 s
, (3.35) where we used the Born approximation for the scattering length a s
=
4 π ~
2 m
U
0 . 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.
21

The realm of strong interactions is interesting in its own right, providing a challenge to manybody theorists. But strongly correlated lattice systems also offer the promise of providing extremely clean realizations of the lattice models of traditional condensed matter physics, which may in turn lead us to new insights into real solids.
A natural question is: how generic is Bose-Einstein condensation for a system of bose particles? Can anything else happen as we move to zero temperature? In this chapter we describe two situations in which the quantum depletion examined previously can be total , leading to the destruction of the condensate.
So far, we have worked in the limit of weak interactions, where energy scales such as the chemical potential are simply proportional to U
0
( µ = nU
0
). 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
γ ≡ mU
0
~
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
U
0 goes through as before and can be written
E/ Ω
1
= U
0 n
2
/ 2 =
~
2 n
3
γ/ 2 m.
In general then, the energy density will be of the form E/ Ω
1 function e ( γ )
is shown in Fig. 4.1. Notice that
e ( γ → ∞ ) = π
2
/ 3
= n
3
~
2 e ( γ ) / 2 m , where the
. How can we understand this result? Surprisingly, the wavefunction has a simple form in this limit, which can be obtained
1
In a sense, the interactions are always strong in one dimension, see Problem 5.
22

Figure 4.1: The curve shows the function e ( γ )
obtained in Ref. [9] for the
δ -function Bose gas in 1D, in terms of which the energy is E = N n 2
~
2 e ( γ ) / 2 m .
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 Schr ¨odinger 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
Ψ
B
( x
1
, · · · , x
N
) = | Ψ
F
( x
1
, · · · , x
N
) | , (4.1) where Ψ
F
( x
1
, · · · , x
N
) is an eigenstate of a system of non-interacting fermions. Such a onedimensional system of impenetrable bosons is known as a Tonks-Giradeau gas . Let’s check this idea by calculating the ground state energy. If the fermi gas has fermi wavevector k
F
, the total energy density is
E/ Ω
1
=
Z k
F
− k
F dk
~
2 k
2
2 π 2 m
=
~
2 k
3
F
6 πm
= n
3
~
2
2 m
π
2
,
3 using n = k
F
/π for the density. The chemical potential is ∂E/∂N = hydrodynamic speed of sound is
~
2 k
2
F
/ 2 m ≡ E
F and the c
2 s
= n m
∂ 2 ( E/ Ω)
,
∂n 2 c s
=
~ k m
F
≡ v
F
.
(4.2)
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 ) = b
†
( r ) b ( r ) , as the modulus in
Eq. (4.1) doesn’t interfere with such a calculation.
23

Problem 5 (Momentum distribution in a 1D Bose fluid) Using the harmonic Hamiltonian
Eq. (3.32)
• Show h 0 | ρ q
ρ
− q
| 0 i =
N
~
| q |
,
2 mc s consistent with Onsager’s inequality.
| q | → 0 , (4.3)
• Show that in 1D the behaviour of the phase correlation function is such that the full
exponential has to be retained in Eq. (3.34) to give
n ( p ) ∝
1
| p |
1 −
1
η
, | p | → 0 , η ≡
2 π
~ n mc s
In particular, this result tells us that there is no condensate in a one-dimensional system.
Problem 6 (Structure factor and momentum distribution for the Tonks gas) Let’s com-
pute some properties of the Tonks gas using the fermionic mapping Eq. (4.1).
•
First show that the dynamical structure factor defined in Eq. (3.14) is
S ( q, ω ) =
(
N m
2 ~ qp
F
0 qp
F m
− ~ q
2
2 m otherwise
< ω < qp
F m
+ ~ q
2
2 m
(4.4)
•
Check that this is consistent with the sum rules Eq. (3.15) as well as the inequality
Eq. (3.17), with
c s
= v
F
≡ p
F
/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.
4.2.1
Optical lattices
Another way in which the rather weak interactions between atoms in a dilute gas can be made strong is by quenching the kinetic energy by confining them to an optical lattice . Such optical potentials are created by the interaction between the oscillating electric field of a laser and the electric dipole moment it induces in an atom. Thus the strength of resulting potential is proportional to the square of the field
V opt .
= − α ( ω ) | E
ω
( r ) | 2
, where α ( ω ) is the polarizability of the atom. In particular, when the polarizability is dominated by a single atomic transition of angular frequency ω n 0 we have
α
0
( ω ) ∼
|h n | d · ˆ | 0 i| 2
,
~
( ω n 0
− ω )
(4.5)
24

Figure 4.2: Counterpropagating lasers form a) a two-dimensional and b) a three-dimensional
lattice. Reproduced from the recent review Ref. [10].
( ˆ is the direction of E ) and the sign changes from positive (attractive potential) to negative
(repulsive) as we go from ω < ω n 0
( red detuning ) to ω > ω n 0
( blue detuning ).
By superimposing counterpropagating lasers one can form an optical standing wave with period λ/ 2 that generates a periodic potential. In this way one can create a one-, two- or
three-dimensional lattice, see Fig. 4.2. The quantum states of a particle propagating in a
periodic particle are the Bloch waves , characterized by (pseudo-)momentum p and band index n
Ψ n
( p ) = e i p · r ϕ n
( r ) , ϕ ( r + a i
) = ϕ ( r ) where { a i
} are the lattice vectors. If we are concerned only with low energies in the lowest band, the following tight-binding model is an adequate description of the kinetic and lattice parts of the Hamiltonian
H tb
= − t
X b
† i b j
.
h ij i
(4.6)
(where h ij i denotes nearest neightbours – we are thinking of the one-dimensional case).
This can be diagonalized to give the dispersion relation ε ( k ) = − 2 t cos k with bandwidth
4 t . With the experimental parameters such as lattice period and depth in the hands of the experimentalist, t can be made small so that we enter the regime of small effective mass and strong interactions.
25

4.2.2
The Bose-Hubbard model
Adding the simplest on-site interactions to the tight-binding Hamiltonian Eq. (4.6) gives the
Bose Hubbard model
H
BH
= − t
X b
† i b j h ij i
+ ( U/ 2)
X n i
( n i
− 1) .
i
(4.7)
Given the short-range character of interatomic interactions this is in fact a very good description of the result of confining bosonic atoms to an optical lattice. One could discuss the behaviour of this model starting from small U/t using the Bogoliubov theory (if we had covered it). This would tell us that, as in the case of bosons in free space, the condensate fraction is depleted with increasing interaction strength: in this case the relevant parameter is U/t .
A simpler approach is to start in the limit of strong interactions with U/t → ∞ . In this case
we can neglect the hopping term in Eq. (4.7), so that the Hamiltonian becomes a sum of on-
site Hamiltonians. After including the chemical potential, the free energy at zero temperature n is minimized by a state Q i b
† i
| 0 i , with integer n particles on each site and n = [ µ/U + 1 / 2] , (4.8) where the square brackets [ · · · ] denote the nearest integer. What happens when t/U = 0 ?
Let’s fix µ at a value corresponding to n bosons per site, so that µ/U = n − 1 / 2 + α , with − 1 / 2 < α < 1 / 2 . Then there is an energy (1 / 2 ∓ α ) U to add or remove a particle (add a hole).
An added particle (or hole) can hop freely, giving a contribution to the energy of order − t
(recall the dispersion relation of the model Eq. (4.6)).
Thus if t .
min ((1 / 2 + α ) U, (1 / 2 − α ) U ) , we expect that no extra particles will be added or removed from the ground state in some finite region of t/U = 0 , provided we stay away from the degeneracy points where α = ± 1 / 2 . For larger values of t , particles enter the ground state and (presumably) condense to form a BEC.
We can make this argument more precise by noting that the minimum hopping energy of a particle or hole is − 2 td in d -dimensions. Thus we expect the asymptotic phase boundaries t c
( µ ) ∼
µ
2 d
(1 + O ( t/U )) − ∞ < µ < U/ 2 with corrections that are higher order in t/U . This result is readily generalized to the states with larger numbers of particles where the hopping energy to add a particle when there are k particles per site already, or a hole when there are k + 1 , is − 2 td ( k + 1) , leading to t c
( µ ) ∼
( µ
2 d
(1 +
| µ − kU |
2 d ( k +1)
O ( t/U
(1 + O
))
( t/U ))
−∞
( k −
< µ < U/
1 / 2)
2
U < µ < ( k + 1 / 2) U, k ≥ 1 .
(4.9)
In this way we arrive at the schematic phase diagram in Fig. 4.3. The states with fixed number
that prevail at low hopping are characterized by vanishing compressibility − ∂ h n i /∂µ and in condensed matter physics are known as Mott insulators , owing their incompressibility to interparticle interactions. The transition between this state and the condensate is sometimes called the superfluid-insulator transition . Its observation in a gas of
87
Rb in 2002 [12] is what
really made the community wake up to the potential of cold gases for doing fundamental condensed matter physics.
26

Figure 4.3: Schematic phase diagram of the Bose Hubbard model, including the estimate
Eq. (4.9). The regions marked
ρ = 0 , 1 , 2 , etc. are the Mott phases with different fillings.
Reproduced from Ref. [11].
Problem 7 (More phases of the Bose-Hubbard model) In the lectures we discussed the
phase diagram of the Bose Hubbard model Eq. (4.7) as a function of chemical potential and
U/t .
• Qualitatively, what happens when we add an additional term describing repulsion between neighbouring sites?
H
0
BH
= − t
X b
† i b j h ij i
+ ( U/ 2)
X n i
( n i
− 1) + V i
X n i n j
, h ij i
(4.10)
Hint: as in the lectures, start by considering the case of zero hopping. See Ref. [13] for
more details.
27

There are now a couple of very readable books on the subject of BEC in atomic
gases [14, 15], as well as the concise but wonderfully clear review Ref. [3]. All are now
a few years old, so don’t include the latest progress on fermions and lattice physics, for
which Ref. [1] and Ref. [10] are good non-technical introductions. Some more detailed
notes, including fermions, are available at http://www-thphys.physics.ox.ac.uk/user/
AustenLamacraft/ .
28

[1] M. Greiner, C. A. Regal, and D. S. Jin (2005), URL http://arxiv.org/abs/cond-mat/
0502539 .
[2] W. Ketterle, Rev. Mod. Phys.
74 , 1131 (2002).
[3] A. J. Leggett, Rev. Mod. Phys.
73 , 307 (2001).
[4] A. J. Leggett, in Bose-Einstein Condensation: from Atomic Physics to Quantum Fluids,
Proceedings of the 13th Physics Summer School , edited by C. M. Savage and M. Das
(World Scientific, Singapore, 2000).
[5] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Science 292 , 476 (2001).
[6] Y. Castin and R. Dum, Phys. Rev. A 57 , 3008 (1998).
[7] D. Stamper-Kurn and W. Ketterle, in Coherent Atomic Matter Waves, Proceedings of the
Les Houches Summer School, Course LXXII, 1999 , edited by R. Kaiser, C. Westbrook, and F. David (Springer, New York, 2001), http://arxiv.org/abs/cond-mat/0005001 .
[8] P. Nozieres and D. Pines, The Theory of Quantum Liquids (Perseus Books, Cambridge,
1966).
[9] E. H. Lieb and W. Liniger, Phys. Rev.
130 , 1605 (1963).
[10] I. Bloch, Nature Physics 1 , 23 (2005).
[11] R. Fernandez, J. Froehlich, and D. Ueltschi, Mott transition in lattice boson models (2005), URL http://www.citebase.org/abstract?id=oai:arXiv.org:math-ph/
0509060 .
[12] M. Greiner, O. Mandel, T. Esslinger, T. W. H ¨ansch, and I. Bloch, Nature 415 , 39 (2002).
[13] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 1999).
[14] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (CUP, 2002).
[15] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press,
2003).
29