Cold Atoms for Condensed Matter Theorists
Austen Lamacraft
Contents
1 Introduction
1.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The ideal Bose gas: a reminder . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The experimental system
2.1 Atomic properties . . . . . . . . . . . . . . . . . .
2.1.1 Boson or Fermion? . . . . . . . . . . . . .
2.1.2 The Zeeman effect . . . . . . . . . . . . .
2.1.3 Excited states and polarizabilty . . . . . .
2.2 Trapping and imaging . . . . . . . . . . . . . . .
2.2.1 Magnetic traps . . . . . . . . . . . . . . .
2.2.2 Optical traps . . . . . . . . . . . . . . . .
2.2.3 Imaging . . . . . . . . . . . . . . . . . . .
2.3 Interactions . . . . . . . . . . . . . . . . . . . . .
2.3.1 Effective interaction between like species
2.3.2 Interaction between species . . . . . . . .
2.3.3 The Feshbach resonance . . . . . . . . .
2.3.4 Dipolar interactions . . . . . . . . . . . . .
3 Superfluidity and Bose-Einstein condensation
3.1 BEC and off-diagonal long-range order . . . .
3.2 Superfluidity defined . . . . . . . . . . . . . .
3.2.1 Non-classical rotational intertia . . . .
3.2.2 Metastability of superflow and vortices
3.3 Experimental status . . . . . . . . . . . . . .
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4 Bose superfluids
4.1 The Gross-Pitaevskii equation . . . . . . . . . . . . .
4.1.1 Time-independent Gross-Pitaevskii theory . .
4.1.2 Time-dependent Gross-Pitaevskii theory . . .
4.2 Interlude: structure factors and sum rules . . . . . .
4.3 The Bogoliubov approximation . . . . . . . . . . . .
4.3.1 Pair approximation and ground state energy .
4.3.2 Structure of the ground state and excitations
4.4 Atom optics . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Fock states and coherent states . . . . . . .
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4.4.2 Interference of two condensates . . . . . . . . .
4.4.3 Superradiance . . . . . . . . . . . . . . . . . . .
4.5 Spinor condensates . . . . . . . . . . . . . . . . . . . .
4.5.1 General considerations for multicomponent BEC
4.5.2 The Gross-Pitaevskii description . . . . . . . . .
4.5.3 Metastability of superflow . . . . . . . . . . . . .
4.5.4 Fragmented condensates . . . . . . . . . . . . .
5 Fermi superfluids
5.1 Fermionic condensates . . . . . . . . . . . . . .
5.2 The BCS theory . . . . . . . . . . . . . . . . . .
5.2.1 The pairing hypothesis . . . . . . . . . . .
5.2.2 The BCS-BEC crossover . . . . . . . . .
5.2.3 Quasiparticle excitations . . . . . . . . . .
5.2.4 Effect of Temperature . . . . . . . . . . .
5.3 The effect of ‘magnetization’ . . . . . . . . . . . .
5.3.1 Sarma state . . . . . . . . . . . . . . . . .
5.3.2 Magnetization in the BCS-BEC crossover
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49
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6 Hydrodynamics of condensates
6.1 Galilean invariance . . . . . . . . . . . . . . . .
6.2 Hydrodynamic description . . . . . . . . . . . .
6.3 Quantum Hydrodynamics . . . . . . . . . . . .
6.3.1 Hamiltonian and commutation relations
6.3.2 Mode expansion . . . . . . . . . . . . .
6.3.3 Correlation functions . . . . . . . . . . .
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7 Strong correlations: low dimensions and lattices
7.1 Bose fluids in one dimension: the Tonks gas . .
7.2 Lattice systems . . . . . . . . . . . . . . . . . .
7.2.1 Optical lattices . . . . . . . . . . . . . .
7.2.2 The Bose-Hubbard model . . . . . . . .
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2
Chapter 1
Introduction
1.1
Preamble
This set of lectures is supposed to provide a reasonably up-to-date introduction to the condensed matter physics of atomic gases. 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. All the time we should keep an eye to the fact that
our usual (often equilibrium) thinking may not be the most relevant for the experiments that
are done
Because of the pre-eminence of superfluidity and BEC as quantum states of matter at low
temperatures, the majority of the lectures will involve aspects of these phenomena. While
this has certainly been the most headline-grabbing part of the story so far, we should bear in
mind that (perhaps more mundane sounding) phases such as the ferromagnet – still a bona
fide quantum state – are likely to have their day in the near future.
There is a regime of BEC that is a bit like linear optics: the atom laser is a coherent phenomenon which nevertheless doesn’t depend on interactions. But there is some interesting
many-body physics there (even if the bodies are non-interacting), so we will discuss it a bit,
especially as we can certainly look forward to continuing cross-fertilization between quantum
optics, atomic physics and condensed matter for the next few years . From this point of view,
the condensed matter physics of atomic gases could be called the non-linear quantum optics
of matter waves!
1.2
The ideal Bose gas: a reminder
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
.
eE/kB T − 1
(1.1)
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
3
chemical potential µ and chose it such that the total number is fixed
N=
k
Recalling
P
k
−−−−→ Ωd
Ωd →∞
R
1
X
dd k
(2π)d
e(k −µ)/kB T − 1
,
k =
~2 k2
2m
(1.2)
in d dimensions, where Ωd is the d-dimensional volume, we
have for d = 3
Z
N
d3 k
1
=
3 (k −µ)/kB T
Ω3
e
−1
(2π)
Z
k2
1
1 kB T m 3/2
=
dk 2 (~2 k2 /2m−µ)/k T = 3
Li3/2 (µ/kB T )
B
2π e
~
2π
n ≡
(1.3)
P
1/2
k
n
is the typical magnitude
(Lin (z) = ∞
k=1 z /k is the polylogarithm). Note that (kB T m)
of a particle’s momentum. As the density of particles increases, or the temperature falls,
µ increases to zero at some critical value of ‘phase space density’ n × np ∼ ~−3 (where
np ∼ (kB T m)−3/2 ), corresponding to
kB Tc = α
~2 2/3
n ,
m
α ≡ 2π/ [ζ(3/2)]2/3
Clearly µ has to be negative for the distribution Eq. (1.2) to make sense, so what happens?
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 < Tc we have a finite fraction
f (T ) of particles sitting in this state. Then the chemical potential may stay equal to zero and
Eq. (1.3) gives
"
"
3/2 #
3/2 #
T
T
n = n f (T ) +
, so f (T ) = 1 −
.
(1.4)
Tc
Tc
The unusual, highly quantum degenerate state emerging below Tc came to be known as a
Bose-Einstein condensate (BEC).
4
Chapter 2
The experimental system
In this first lecture, 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 the previous chapter, 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
~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.
T ∼
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.
2
This three-body rate is reduced by the appearance of Bose-Einstein condensation, further enhancing the
sample lifetime.
1
5
2.1
Atomic properties
2.1.1
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
2.1.2
Nuclear spin, I
3/2
3/2
3/2
1
4
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,
(2.1)
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 (2.1) 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
mF
2
1
0
−1
−2
F
2
2, 1
2, 1
2, 1
2
E(B)
A (1 + B/Bhf )
±A[1 + B/Bhf + (B/Bhf )2 ]1/2
±A[1 + (B/Bhf )2 ]1/2
±A[1 − B/Bhf + (B/Bhf )2 ]1/2
A (1 − B/Bhf )
These are plotted in Fig. 2.1. Many experiments are done in the regime B Bhf , so a
linear expansion of the above energies suffices.
1
E(B) = ± A + µB mF B ,
(2.2)
2
3
We shunt all the energy levels up by A/4 for convenience
6
Figure 2.1: Magnetic field dependence of atomic states of an atom with J = 1/2, I = 3/2.
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 .
2.1.3
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.
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 .
(2.3)
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
2me (En − E0 )
fn0 =
|hn|d · ε̂|0i|2 .
e2 ~2
These satisfy the f-sum rule (or Thomas-Reiche-Kuhn sum rule)
X
fn0 = Z
(2.4)
n
7
Problem 1 Prove this by considering
H, di , di (no sum).
We have
α=
e2 X fn0
2 ,
me n ωn0
(2.5)
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 Eq. (2.4) 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.
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
h0|0it
Z t
1
= 1− 2
dt1 dt2 h0|T d(t1 ) · Eω d(t2 ) · Eω∗ |0ie−iω(t1 −t2 ) + {ω → −ω}
2~ 0
h
i
i X
1
i
−i(ωn0 +ω)t
= 1+ 2
t−
e
− 1 |h0|d · Eω |ni|2
~ n ωn0 + ω
ωn0 + ω
+{ω → −ω}.
(2.6)
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
∆E = −
1 X |h0|d · Eω |ni|2
+ {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
(En − E0 )2 − (~ω)2
X
fn0
,
2
me n ωn0 − ω 2
n
e2
(2.7)
which generalizes the static results Eq. (2.3) and Eq. (2.5). Note that in contrast to the
static case, where an attractive force is always felt towards regions of high field intensity,
4
Field-theoretic types may prefer to think of the self-energy of the atom at second order in the ”dressing”
electric field
8
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)
(2.8)
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 Eq. (7.5) to obtain
the complex polarizability
|hn|d · ε̂|0i|2
α(ω) ∼
.
(2.9)
~ (ωn0 − ω − iΓe )
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)
2
1
Γg = − Im ∆E = α00 (ω)hE(t)2 i
~
~
(2.10)
Excited state lifetimes are of order 10 ns.
Problem 2 Consider a harmonically confined electron. The equation of motion for the dipole
moment d = −er is
e2
d̈ + ω02 d =
E
me
Show that the expression for the classical polarizability is
α(ω) =
e2
1
me ω02 − ω 2
This correpsonds to Eq. (2.5) with one oscillator strength equal to one at ω0 . Verify that this
is case in a quantum mechanical calculation.
2.2
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 Ref. [1]. Here
we discuss briefly trapping and imaging, which are probably more important for the theorist
seeking to understand experimental papers.
2.2.1
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
|B(r, z, φ)| = B0 + αr2 + βz 2 .
2
2
9
(2.11)
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 Ref. [1].
The relationship between (2.11) and the potential experienced by the atom is then very
simple in the linear regime described by Eq. (2.2). 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) = ihα(r)|∇|α(r)i
(2.12)
Problem 3 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 [2].
2.2.2
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 5.3. 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.
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 .
10
Figure 2.2: Absorption image of around 7 × 105 atoms just above, at, and below the BoseEinstein condensation temperature. The gas is allowed to expand for 6 ms. Reproduced
from Ref. [3].
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 Eq. (2.10) 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.
2.2.3
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 Fig. 2.2 . 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.
2.3
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
11
Figure 2.3: Sketch of the interatomic potentials for two alkali atoms with valence electrons in
singlet and triplet states.
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.
2.3.1
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 Fig. 2.3.
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 potential defines a
1/4
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
sin [k (r − as )]
ψ(r) = const.
.
(2.13)
r
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.
12
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 lengths is extremely difficult.
They can, however, be reliably measured using photoassociative spectroscopy, see Ref. [1].
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
hHint i =
drk δ̃(rij )|Ψ({r})|2 .
(2.14)
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 Eq. (2.14) 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 Eq. (2.14), 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.
• 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 ),
at ~/
p
2mEtyp rij as .
• In the case of a single pair, such a wavefunction has energy
4πas ~2 2
|A| ,
m
(2.15)
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!). Eq. (2.15) 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 Eq. (2.14).
6
This is a slight abuse of notation: the origin of this effective potential is both kinetic and potential in general:
see below
7
Alternative versions may be found in Ref. [4] and Ref. [5]. 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.
13
Several other ways of writing the pseudopotential are in common usage. One is to dispense with the slightly cumbersome form of Eq. (2.14) and write the interaction Hamiltonian
as a delta-function potential.
4πas ~2
U (r) =
δ(r),
(2.16)
m
or in second quantized notation
Z
1 4πas ~2
dr φ† (r)φ† (r)φ(r)φ(r).
(2.17)
Hint =
2 m
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 ) −
(2.18)
2
(2π)3 (q) − (k) − i0
where (k) = k 2 /2m. The δ-function potential can then be taken to be the limit of the (nontranslationally 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 F (k, k0 ) → −4π~2 as /m
Z
m
1
dq g(k)
=
−
.
4πas ~2
U0
(2π)3 2k
which is compatible with Eq. (2.16), apart from the (divergent) second term. This is just the
momentum space version of our real space difficulties.
Finally, Eq. (2.16) 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 Eq. (2.14) is n/N , and we find the energy per particle
E/N =
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 energy of a
1/2
system of bosons as an expansion in na3s
E/N =
1/2
2πnas ~2 1 + α na3s
+ ... ,
m
√
where α = 128/15 π.
14
(2.19)
2.3.2
Interaction between species
The natural generalization of the pseudopotential Eq. (2.17) to several species is
Z
1 X 4πaαβγδ ~2
Hint =
dr φ†α (r)φ†β (r)φγ (r)φδ (r).
2
m
(2.20)
αβγδ
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 [6]. For the case of the F = 1 multiplet, we have
aαβγδ =
1
[a1 δαγ δβδ + a2 Fαγ · Fβδ ± {α ↔ β}] ,
2
(2.21)
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 2.1.2. 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 2.3.4, but these are typically much less frequent.
2.3.3
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 Fig. 2.4. The curves that we drew in Fig. 2.3 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
H=
p a†s,p as,p +
+ ε0 b†q bq + √
bq a†1,q+p a†−1,−p + h.c
(2.22)
2
V
p,s
q
p,q
8
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
15
Figure 2.4: A Feshbach resonance is caused by hybridization of a closed channel bound
state with the open channel.
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 .
Problem 4 Find the scattering amplitude for two atoms in the model Eq. (2.22).
Solution For two atoms the wavefunction in the centre-of-mass frame has the form
#
"
X
αp a†1,p a†−1,−p |0i.
|ψi = βb†0 +
p
Substituting into Eq. (2.22) 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 ) 0
(2.23)
p
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 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
16
We look for a scattering state of the form
αp (E) = δp,p0 +
4π~2
f (E)
,
m 2p − E − i0
where p0 is the wavevector of the incoming wave with 2p0 = E. Substituting into Eq. (2.23)
gives
Z
f (E)
g2
m
dp
f (E) +
=0
+
(E − ε0 ) 4π~2
(2π)3 2p − E − i0
(c.f. Eq. (2.18)). 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
,
(2.24)
ε0 → ε0 + g 2
(2π)3 2p
to give
Z
g2
m
dp
1
1
f (E) +
+ f (E)
−
= 0.
(E − ε0 ) 4π~2
(2π)3 2p − E − i0 2p
Taking real and imaginary parts now yields
#
"
√
m3/2 E
g2
m
−
Im f (E) = 0
Re f (E) +
4π
(E − ε0 ) 4π~2
√
g2
m3/2 E
Im f (E) +
Re f (E) = 0.
(E − ε0 )
4π
The final result for the scattering amplitude is
1
~γ
√
f (E) = − √
m E − ε0 + iγ E
(2.25)
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 state10 . 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 Eq. (2.25) is in fact
the most general one allowed at low energies [7]: the model was just a convenient physical
realization where this two-parameter asymptotic description turns out to be exact. The γ
10
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 Ref. [7].
11
A background scattering length is normally added to this to include the effect of non-resonant scattering.
17
parameter characterizes the width of the resonance. If we were only interested in energies
very low compared 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.
2.3.4
Dipolar interactions
Finally, there is a dipole-dipole interaction between the valence electron spins of two atoms
Umd =
µ0 (2µB )2
[S1 · S2 − 3 (S1 · r̂) (S2 · r̂)] .
4πr3
(2.26)
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 [1].
Since the creation of a Bose-Einstein condensate of Chromium atoms last year, 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.
18
Chapter 3
Superfluidity and Bose-Einstein
condensation
In this second introductory chapter, we will introduce the concepts of Bose-Einstein condensation (BEC) and superfluidity in a general way, before we move on to consider specific
models in later chapters. We’ll also take a look at the experimental status of these distinct
phenomena.
3.1
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 + · · · ,
(3.1)
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 behaviour of the
one-body density matrix
ρ1 (r, r0 ) ≡ hhφ† (r)φ(r0 )ii.
(3.2)
hh· · · ii is the average over the many-body density matrix of the system 1 . Since the distribution n(k) = hhφ†k φk ii, we identify this as the Fourier transform of ρ1 (r, r0 ) ≡ g1 (|r − r0 |) in
R
P
The first-quantized version of Eq. (3.2) is 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
1
19
a translationally invariant, isotropic system. The presence of a δ-function in Eq. (3.1) 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
ρ1 (r, r0 ) =
nα χ∗i (r)χi (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 4.5), 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),
(3.3)
m
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
vs (r) · dl =
,
n ∈ Z.
(3.4)
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.
3.2
Superfluidity defined
Superfluidity is not really a single phenomenon but rather a complex of related phenomena,
see Ref. [8] for a very complete discussion. Here we focus on two of the conceptually
simplest properties of those systems usually deemed superfluid.
3.2.1
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. 3.1). If we rotate this at some angular frequency ω,
we expect that the free energy of this system has ω-dependence of the form
2
If all the particles were in a finite momentum state, the RHS would tend to a plane wave, for instance
20
Figure 3.1: Thought experiment used to define the superfluid fraction. An annulus of fluid is
rotated slowly
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 .
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 time-dependent Hamiltonian
N
N
X
p2i
1 X
0
H(t) =
+ Ucont (ri (t)) +
V (|ri − rj |, )
2m
2
i=1
i,j=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
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.
21
the rotating frame is governed by the time-independent Hamiltonian4
Hrot = H(0) − ω · L,
(3.5)
where L is the operator of total orbital angular momentum
X
L=
ri ∧ pi .
i
Thus a formal procedure to calculate the superfluid fraction would involve obtaining the equilibrium density matrix in the rotating frame using Eq. (3.5), 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 = −ωhLiω /2 for small ω (by perturbation theory in ω, for example)5
2
1 2
ρs /ρ = lim
hH
i
−
hH
i
+
Iω
.
(3.6)
rot ω
rot 0
ω→0 Iω 2
2
We now note that the quantity in square brackets is the expectation value of the Hamiltonian
Hω =
N
X
[pi − mω ∧ ri ]2
2m
i=1
+ ...
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. (3.6) is thus seen to be equivalent to
4π 2 I ∂ 2 E0 (ϕ)
,
ϕ→0 N 2 ~2
∂ϕ2
ρs /ρ = lim
(3.7)
(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.
4
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.
5
More generally, ∂hHrot iω /∂ω = −hLiω is a consequence of the Hellman-Feynmann theorem.
6
This formulation makes the analogy between the superfluid response and Meissner effect in a superconductor clear.
22
Problem 5 Satisfy yourself that a noninteracting Bose gas at zero temperature has ρs /ρ = 1.
What about a non-interacting Fermi gas?
Problem 6 (for enthusiasts, but related) The definition Eq. (3.7) 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?
3.2.2
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. (3.5), 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. 3.2. 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 .
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. 3.2). 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. (3.4), where the phase of the order parameter winds through
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
23
2.5
2
1.5
1
0.5
1
2
3
4
5
Figure 3.2: E(L/N ~), showing metastable minima.
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 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 7 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) 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
vsφ (φ, r, z) =
n~ 1
.
mr
(3.8)
The metastable states resulting from halting a rotation with Iω & N ~ in this simply connected
geometry are generically (multi-)vortex configurations.
3.3
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 veri-
24
Figure 3.3: Vortices in an atomic gas of 23 Na. Reproduced from Ref. [9].
fied 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. 2.2).
The other major experimental confirmation of BEC relates to the observation of certain
interference phenomena that we will discuss in Section 4.4. 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. 3.3.
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. (3.4) 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.
25
Chapter 4
Bose superfluids
In this chapter, we develop the microscopic theory of the condensed phase of a Bose gas.
4.1
4.1.1
The Gross-Pitaevskii equation
Time-independent Gross-Pitaevskii theory
The first, and most versatile, approach to the problem is to use the Gross-Pitaevskii approximation. This is a variational approach that starts from the following ansatz for the ground
state
Y
Ψ({ri }) =
χ0 (ri )
(4.1)
i
Such a wavefunction of course displays BEC with the density matrix having an eigenfunction χ0 (r) with eigenvalue N . The expectation value of the energy in this state, using the
pseudopotential Eq. (2.14), is
2
Z
Z
~
1
2
2
hHi = N dr
|∇χ0 | + Uext (r)|χ0 (r)| + N (N − 1)U0 dr|χ0 (r)|4 ,
(4.2)
2m
2
where the interaction constant is U0 = 4π~as /m. For large N , we can neglect the difference
between N and N +1. Minimizing with respect to χ0 (r), and introducing a Lagrange multiplier
to maintain the normalization of χ0 (r) gives the equation
~2 2
2
−
∇ − µ + Uext (r) + N U0 |χ0 (r)| χ0 (r) = 0.
2m
The multiplier µ = ∂hHi/∂N , so is identified with the chemical potential. Rewriting in terms
of the order parameter Ψ(r) gives the Gross-Pitaevskii equation
~2 2
2
−
∇ − µ + Uext (r) + U0 |Ψ(r)| Ψ(r) = 0.
(4.3)
2m
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
2mnU0 −1/2
ξ≡
= (8πnas )−1/2 .
(4.4)
~2
26
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 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 approach1 . A typical value of ξ may be around 4000 Angstoms.
√
In a uniform system with Ψ(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
(4.5)
√
Note that mcs = ~/ 2ξ.
With the ansatz Eq. (4.1) for the wavefunction, we can obtain various observables without
difficulty. The particle density is just
ρ(r) = ρ1 (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~
~
∇r − ∇0r ρ1 (r, r0 )|r0 →r = |Ψ(r)|2 ∇ϕ(r).
2m
m
Dividing one by the other yields the superfluid velocity defined in Eq. (3.3), though that relation is in fact the more general one. The total z-component of angular momentum is
Z
Lz = −i~ dr|Ψ(r)|2 ∂φ ϕ(r)
For the (ideal) annular container considered in the previous chapter, we would have
Ψ` (φ, r, z) = Ψ0 (r)ei`φ
Where Ψ0 (r) goes to zero on the inner and outer edges of the container. The normalization
of Ψ(r) means that Lz = N `~
Problem 7 [See Ref. [4] 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. As in Chapter 3, we consider a cylindrical annulus, and the two
lowest angular momentum states ` = 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
a†1 create atoms in the ` = 0, 1 states, a model version of the rotating frame Hamiltonian Hrot
that includes the kinetic energy, the asymmetry effect, and interactions is
h
i
Hrot = −~ (ω − ω1 ) a†1 a1 − a†0 a0
h
i
−V0 a†0 a1 + h.c.
i
U0 h † †
a0 a0 a0 a0 + a†1 a†1 a1 a1 + 4a†1 a†0 a0 a1
(4.6)
+
2V
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 2.3.1, however, that the expansion parameter justifying the
present approximation is (nas )1/2 , which is still small.
1
27
where ω1 = ~/2mR2 is the critical angular velocity at which the ` = 1 state has the lower
energy. If we introduce the GP wavefunction
h
cos
iN
χ iϕ/2 †
χ
e
a0 + sin e−iϕ/2 a†1 |0i,
2
2
show that
• 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.
• The GP variational energy is (up to a constant, and ignoring terms lower order in N )
E(χ)/N = ~ (ω − ω1 ) cos χ − V0 sin χ +
nU0
sin2 χ,
2
while the angular momentum is
L(χ)/N ~ =
1
(1 − cos χ)
2
• A metastable minimum exists for 2U0 > V0 (assuming U0 and V0 are both much less
than ~ω1 ). 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 ` greater than one, show
that even when V0 goes to zero, metastable configurations are only
when
p possible √
nU0 > `2 ~ω1 . This corresponds to a critical velocity of `~/mR = 2nU0 /m = 2cs ,
and coincides (parametrically, at least), with the famous Landau criterion.
4.1.2
Time-dependent Gross-Pitaevskii theory
For time dependent problems it is tempting to immediately write down
~2 2
∂Ψ(r, t)
2
−
∇ + Uext (r) + U0 |Ψ(r, t)| Ψ(r, t) = i~
.
(4.7)
2m
∂t
R
Note that this equation conserves the normalization N = dr|Ψ(r)|2 – all particles remain in
the condensate. Eq. (4.7) may be derived by generalizing the ansatz Eq. (4.1)
Y
Ψ({ri }, t) =
χ0 (ri , t).
(4.8)
i
Substitution into the time-dependent Schrödinger equation yields


2
X
X
Y
X ∂χ0 (ri , t) Y
− ~ ∇2i + Uext (ri ) + U0
δ(rk − ri ) χo (ri , t)
χ0 (rj ) = i~
χ0 (rj ).
2m
∂t
i
k6=i
j6=i
i
j6=i
(4.9)
28
P
In order to get a closed equation for χ0 (r) we can replace the j6=i δ(rj − ri ) with the expectation value of the density N |χ0 (ri )|2 evaluated with Eq. (4.8). With this replacement,
Eq. (4.9) is satisfied if Eq. (4.7) 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 χ0 (r) (at some reference time) is a member. We write the boson field operator
in terms of these states
X
φ(r) =
χn (r)aα .
α
Then the interaction Hamiltonian has the form
U0 X
Hint =
Mαβγδ a†α a†β aγ aδ ,
2
(4.10)
αβγδ
where the matrix elements Mαβγδ are
Z
Mαβγδ = drχ∗α (r)χ∗β (r)χγ (r)χδ (r).
Now applied to the ansatz Eq. (4.8), weRcan see that it is the term in Eq. (4.10) with α =
β = γ = δ = 0 that gives 12 N (N − 1) U0 dr|χ0 (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
Sχα (r1 )χβ (r2 )
χ0 (rj ),
(4.11)
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 unitl the next (Bogoliubov) stage of approximation. In the
equilibrium state, their effect is to give the quantum depletion of p
the condensate fraction,
leading to N0 < N , even at zero temperature. The effect is of order na3s , so it is reasonable
that the present approximation is justified when this is small [10].
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 δΨ(r, t) of Ψ(r, t) from some
reference solution Ψ0 (r, t), satisfying
~2 2
∂Ψ(r, t)
−
∇ + Uext (r) δΨ(r, t) + 2U0 |Ψ0 (r, t)|2 δΨ(r, t) + U0 Ψ20 (r, t)δΨ∗ (r, t) = i~
.
2m
∂t
Note that the nonlinear term couples δΨ(r) and δΨ∗ (r). The presence of the chemical potential in Eq. (4.3) means that the solution of Eq. (4.7) corresponding to to a solution of the
time-independent problem is Ψ0 (r, t) = Ψ0 (r)e−iµt/~ . Introducing the harmonic solution
δΨ(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 |Ψ0 (r, t)| u(r) + U0 Ψ20 (r)v(r)
2m
~2 2
2
−~ωv(r) = −
∇ + Uext (r) − µ + 2U0 |Ψ0 (r, t)| v(r) + U0 Ψ∗2
0 (r)u(r). (4.12)
2m
29
In free space µ = nU0 , and plane wave solutions of Eq. (4.12) have the dispersion relation
~ω(k) = E(k)
1/2
E(k) ≡ (k) (k) + 2mc2s
,
(4.13)
where we use the expression Eq. (4.5) 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.
A natural situation to examine is the effect of a weak time-dependent perturbing potential
Uext (r, t). Consider the plane wave perturbation
Uext (r, t) = V0 cos (q · r − ωt)
This describes the effect of a pair of laser beams (Bragg spectroscopy) with different
wavevectors q = q1 − q2 and frequency difference ω, generally much smaller than the detuning from the fundamental transition2 . 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) = |Ψ0 (r, t) + δΨ(r, t)|2 − |Ψ0 (r, t)|2 ∼ Ψ∗0 (r, t)δΨ(r, t) + Ψ0 (r, t)δΨ∗ (r, t),
(4.14)
giving
δn(q, ω) = −nV0
E(q)2
(q)
V0
≡
D(q, ω).
2
2
− ~ (ω + i0)
2
(4.15)
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) =
2π V02 X δ (~ω − Eα ) |hα|eiq·ri |0i|2 + δ (~ω + Eα ) |hα|e−iq·ri |0i|2 ,
~ 4
(4.16)
α,i
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. (4.15)
X
1
(Vol) Im D(q, ω) = −
δ (~ω − Eα ) |hα|eiq·ri |0i|2 − δ (~ω + Eα ) |hα|e−iq·ri |0i|2 ,
π
α,i
(4.17)
which implies that all of the absorption comes from a single transition with3
X
|hα|eiq·ri |0i|2 =
i
2
N (q)
.
E(q)
(4.18)
2
To put some figures to it, the typical recoil energy ~2 kop
/2m on absorption of an optical photon is on the kHz
scale
3
c.f. Problem 2. The only difference is the colossal ratio (∼ 1011 ) of energy scales in the two contexts!
30
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. (4.17) to obtain the line strength of the resonance
Z ∞
1
dω Im D(q, ω) = h0|ρq ρ−q |0i
− (Vol)
π
0
P
where we used the completeness relation to get rid of α . This Fluctuation- Dissipation
relation relates the response function that we found to the density fluctuations in the ground
state of our system. Our explicit form Eq. (4.15) implies
h0|ρq ρ−q |0i =
N (q)
N ~q
N,
→
E(q)
2mcs
p ~/ξ.
(4.19)
The only problem is that our original ground state ansatz Eq. (4.1) 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. (4.19) only reaches this value at large p4 .
Nevertheless, we will see that Eq. (4.19) 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. (4.17). 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. (4.16).
4.2
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. (4.17) is related to the dynamical
structure factor S(q, ω), defined as
X Eα − E0
S(q, ω) =
δ ω−
|hα|ρq |0i|2 .
(4.20)
~
α
Using completeness of the eigenstates, we then have
Z ∞
dω S(q, ω) ≡ S(q) = h0|ρq ρ−q |0i,
0
4
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.
31
where S(q, ω) is called the static structure factor. S(q, ω) satisfies the following two sum
rules (see, for instance, Ref. [11])
Z ∞
N ~2 q 2
~ωS(q, ω) =
2m
0
Z ∞
N
S(q, ω)
=
,
(4.21)
lim
q→0 0
~ω
2mc2s
known as the f-sum and compressibility sum rules, respectively. They are the analogues of
Eq. (2.4) and Eq. (2.7). In the present case Eq. (4.18) implies
E(q)
N (q)
δ ω−
,
(4.22)
S(q, ω) =
E(q)
~
and you should check that this satisfies both sum rules.
Problem 8 Use the sum rules Eq. (4.21) and the Cauchy-Schwartz inequality |hA|Bi| ≤
|A||B|, interpreting the integrals as inner products, to derive Onsager’s inequality
S(q) ≤
N ~q
2mcs
(4.23)
Note that Eq. (4.19) 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.
4.3
4.3.1
The Bogoliubov approximation
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 timedependent 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
X
Y n
|pairi ≡
c{nk }
Λkk |GP i,
(4.24)
{nk }
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,
32
starting from the GP state, is clearly going to be dominated by repeated excitation of pairs
out of this state5 .
The crucial thing about the use of variational states of the form Eq. (4.24) is that
hpair|H|pairi = hpair|Hpair |pairi,
(4.25)
where the Hpair is
Hpair = Hkin +
+
0
U0 X †
U0
N (N − 1) +
α α + α0† αp + 2np n0
2V
2V p p 0
0
U0 X
np nq + αp† αq .
2V
(4.26)
p6=q
Here αp† = a†p a†−p creates a (p, −p) pair, np = a†p ap is the occupation number of an orbital,
P
and 0 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. (4.24)
or jumping straight to the pair Hamiltonian Eq. (4.26) – 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. (4.25) 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 theh operator
ap and its coni bp = a0 (n0 + 1)
jugate. These satisfy the canonical relations bp , b†p = 1 for p 6= 0. An operator like αp† α0
is then (n0 (n0 + 1))1/2 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. (4.26), we have, following the replacement of n0 with hn0 i = N (assuming that the
5
It’s necessary to point out that the method of this section should be understood
as´ Rapplying strictly to the
`
case of weak interatomic potentials U (r), where the Born approximation 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. (2.19), say, with as the true scattering length, is considerably more complicated, see Ref. [5].
33
depletion N − N0 can be neglected)
Hpair
0
X
U0
U0 N X † †
†
=
N (N − 1) +
(p)bp bp +
b b + bp b−p + 2np ,
2V
2V p p −p
p
(4.27)
which is the same as Bogoliubov’s non-conserving Hamiltonian. It is diagonalized by the
transformation 6
βp = bp cosh κp − b†−p sinh κp
nU0
tanh 2κp =
.
(p) + nU0
(4.28)
The transformed Hamiltonian is
0
X1
1
H = nU0 N +
[E(p) − (q) − nU0 ] + E(p)βp† βp ,
2
2
p
(4.29)
where E(p) is the Bogoliubov dispersion relation introduced in Eq. (4.13). The ground state
corresponds to
βp |0i = 0,
(4.30)
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
1
(nU0 )2
E0 = nU0 N 1 −
+
E(p) − (q) − nU0 +
.
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
Where
a0 = (m/4π~2 )U0 ,
a1 = −(m/4π~2 )
(4.31)
U02 X 1
.
V p 2(p)
This closely resembles the result quoted in Eq. (2.19) for the first two terms of the ground
1/2
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.
6
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. (4.26). While systems with negative
scattering length are believed to be thermodynamically unstable, their Hamiltonians should still make sense!
34
4.3.2
Structure of the ground state and excitations
We now turn our attention to the nature of the ground state of Eq. (4.27). It is not difficult to
see that the ‘Bogoliubov vacuum’ conditions Eq. (4.30) are satisfied by
|Bi ≡
0
Y
1 † N
|N i0 = √
a0
|0i,
N!
† †
e(cp /2)bp b−p |N i0
p
(4.32)
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. (4.32) for N 1, is more easily
given physical interpretation. Using b†p = a†p a0 / N , one can show that in this limit
"
#N/2
0
X
1
|B; N i = √
a†0 a†0 +
cp a†p a†−p
|0i
N!
p
(4.33)
(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.
(4.34)
p
I emphasize that for N 1 all three of Eq. (4.32-4.34) give the same result (Eq. (4.31))
when the energy is evaluated. Eq. (4.33) has the pleasant property of being a member of
the class Eq. (4.24) 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 ),
(4.35)
i<j
(c.f. Eq. (4.11) 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. (2.19), however, by
using a variational wavefunction of the form Eq. (4.33), but retaining the full pair Hamiltonian
Hpair [5, 12]. Furthermore, in the thermodynamic limit, the exact solution of Hpair is belived
to be of this form.
The excitations described by Eq. (4.29) have the same spectrum that we found before.
Indeed, it should be clear that cosh κp and sinh κp are determined by the same Bogoliubov-de
Gennes equations that determined u(r) and v(r) (compare the quadratic form in Eq. (4.27)
with the matrix structure of Eq. (4.12)). 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. [4] for a brief discussion),
though in light of the small depletion discussed below this normally isn’t necessary.
35
Figure 4.1: 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. [13].
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
i
i
√ h
√ h †
ρq |Bi ∼ N a†q a0 + a†0 a−q |Bi =
N bq + b−q |Bi
√
=
N [cosh κp − sinh κp ] βp† |Bi,
(4.36)
(c.f. Eq. (4.14)) so that an excitation coincides with a density fluctuation. This allows us to
immediately obtain the structure factor
S(q) = hB|ρq ρ−q |Bi = N [cosh κp − sinh κp ]2 =
N (p)
Np
→
,
E(p)
2mcs
q ~/ξ,
(4.37)
matching the result Eq. (4.19) and showing that the structure factor saturates the inequality
Eq. (4.23). 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. (4.37) 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. (4.32) 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 GrossPitaevskii theory. Thus experiments on scattering (see Fig.4.1) that are consistent with these
36
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
n(p) =
|cp |2
mcs
→
,
1 − |cp |2
2p
p ξ −1 .
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
nas ,
(4.38)
n(p) = √
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. 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.
Problem 9 How does the superfluid fraction discussed in Section 3.2.1 change in the Bogoliubov approximation?
4.4
Atom optics
Some of the most dramatic effects of Bose-Einstein condensation are quantum interference
effects that the BEC is able to ‘amplify’ to a macroscopic level. Even more surprising, however, is the tendency of separate condensates to respond coherently even when they start
out with no coherence. We explore these features in this section.
4.4.1
Fock states and coherent states
When we discussed the Gross-Pitaevskii approximation, we used a trial wavefunction for N
atoms of the form
Y
|N i =
χ0 (ri ),
i
37
or in second quantized notation
1 † N
|N i = √
a0
|0i,
N!
where a†0 creates a particle in the state χ0 (r). In general we will call states of this form Fock
states. In some discussions of BEC, one encounters the coherent states
|αi ≡ e−|α|
2 /2
exp(αa†0 )|0i.
This is a superposition of Fock states, that is, a superposition of states with definite particle
number. While this might be a reasonable state for photons, which can be created and
destroyed, it certainly is never the state of a system of atoms. The coherent state has,
however, the nice property that
hα|(a†0 )m (a0 )n |αi = (α∗ )m αn .
(4.39)
In other words, a0 acquires an expectation value α, which can be complex. Thus these are
the states that one is forced to introduce in the traditional non-conserving formulations of the
Bogoliubov approximation, see Eq. (4.34), where a0 is treated as a c- number. The property
Eq. (4.39) gives us the density matrix
ρ1 (r, r0 ) = hα|φ† (r)φ(r0 )|αi = χ∗0 (r)χ(r0 )|α|2 ,
so that for |α|2 = N we have BEC as we defined it before. The difference is that now φ(r)
itself has an expectation value
hα|φ(r)|αi = χ0 (r)α,
which leads to the idea that the phase of α is really the phase of the condensate wavefunction. Since we introduced χ0 (r) through the spectral resolution of ρ1 (r, r0 ), its phase is
arbitrary in our formulation, Another way of putting it is through the relation
Z 2π
dϕ −iϕN iϕ
|N i ∝
e
|e i,
(4.40)
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 χ0 (r) and χ1 (r), not necessarily orthogonal, which could be different orbital
states, hyperfine states, etc.. Then the Gross-Pitaevskii state
N
a†0 + αa†1
|0i
(4.41)
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 |0i.
We do, however, have a relation just like Eq. (4.39)
Z
N
dϕ −iϕn †
|n, N − ni ∝
a0 + |α|eiϕ a†1
|0i.
e
2π
(4.42)
(4.43)
We see that Eq. (4.41) describes a state in which atoms in states 1 and 2 have a definite
relative phase, while Eq. (4.42) can be thought of as a state in which that phase fluctuates.
As we will now show, this is not just a mathematical nicety.
38
4.4.2
Interference of two condensates
With this background, let us consider the case where the states χ0 and χ1 represent two
spatially separated condensates. Schematically, this is the situation in classic interference
experiment demonstrating coherence of condensates [14]. Such a system can be prepared
in a state like Eq. (4.41) of definite relative phase or in the Fock state Eq. (4.42), depending
on whether the gas is divided in two below the condensation temperature or above it.
Suppose that the two states are initially sufficiently separate that they can be thought of
as orthogonal. Then the state of definite relative phase ϕ is
"r
#N
r
1
N̄0 −iϕ/2 †
N̄1 iϕ/2 †
|N̄0 , N̄1 , ϕi ≡ √
e
a0 +
e
a1
|0i,
(4.44)
N
N
N!
where N̄0, are the expectation values of particle number in each state N = N̄0 + N̄1 . We
allow the system to evolve for some time T , so that the two ‘clouds’ begin to overlap (typically achieved by allowing free expansion). Ignoring interactions, the many-particle state
is just Eq. (4.44) with the wavefunctons χ0,1 evolving freely. We compute the subsequent
expectation value of the density
ρ(r) = φ† (r)φ(r),
φ(r) = χ0 (r)a0 + χ1 (r)a1
p
2
2
hρ(r, T )iϕ = N̄0 |χ0 (r, T )| + N̄1 |χ1 (r, T )| + 2 N̄0 N̄1 Re eiϕ χ∗0 (r, T )χ1 (r, T ).
(4.45)
If the clouds are now overlapping, the last term in Eq. (4.45) comes into play. Its origin is in
quantum interference between the two coherent subsystems, and it depends on the relative
phase, demonstrating the real macroscopic effects of this quantity.
As an illustration, consider the evolution of two Gaussian wavepackets with width R0 at
T = 0, separated by a distance d R0
"
#
(r ± d/2)2 1 + i~t/mR02 )
1
χ0,1 (r) =
exp −
,
(4.46)
2RT2
(πRT )3/2
with
RT2
=
R02
+
~t
mR0
2
.
The final term of Eq. (4.45) is then
p
~r · d
2 N̄0 N̄1
A(r, T ) cos ϕ +
T .
ρint (r, T ) =
RT3
mR02 RT2
(4.47)
The interference term therefore consists of regularly spaced fringes, with a separation at
long times of 2π~T /md.
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
|N0 , N1 i ≡ √
1
(a†0 )N0 (a†1 )N1 |0i.
N0 !N1 !
Computing the density in the same way yields
hρ(r, T )iF = N0 |χ0 (r, T )|2 + N1 |χ1 (r, T )|2 ,
39
(4.48)
i.e. the same as before, but without the interference term. This is consistent with the principle
expressed in Eq. (4.43), 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). Unlike the situation in most condensed matter experiments, where
we suppose that a kind of averaging occurs in space or time, there is no particular reason
why the expectation value should tell us everything there is to know about such a ‘one-shot’
measurement. Consider now the correlation function of the density at two different points
hρ(r)ρ(r0 )iF
= hρ(r)iF hρ(r0 )iF
+N0 (N1 + 1)χ∗0 (r)χ∗1 (r0 )χ0 (r0 )χ1 (r) + N1 (N0 + 1)χ∗1 (r)χ∗0 (r0 )χ1 (r0 )χ0 (r).
(4.49)
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 prepared7 . Indeed, it
is clear that, for N0 , N1 1
Z 2π
dϕ
hρ(r)ρ(r0 )iF =
hρ(r)iϕ hρ(r0 )iϕ ,
(4.50)
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.
Problem 10 Prove this in general by showing that the density matrix corresponding to a
statistical mixture of phase states |N̄0 , N̄1 , ϕi with random phase coincides with that of a
mixture of Fock states with binomial distribution of atoms into states 0, 1. At large N this
distribution becomes sharply peaked at occupations N0 , N1 .
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 Fig. 4.2. 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 is 50 : 50 with no relative
phase shift), the observation of k+ counts leaves the condensate, initially in the Fock state
We note that Eq. (4.49) 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. For a large number of particles, then, an absorption image can be assumed to be
an instantaneous measurement of the function ρ(r) (projected onto a plane, of course).
7
40
Figure 4.2: Beam splitter configuration with two condensates
|N/2, N/2i, in the state [15]
Z
π
dϕ
[cos(ϕ/2)]k+ [sin(ϕ/2)]k− |(N − k)/2, (N − k)/2, ϕi,
−π 2π
(4.51)
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 Eq. (4.51) in the stationary phase approximation as
Z π
i
2
dϕ h −k(ϕ−ϕ0 )2
e
+ (−)k− e−k(ϕ+ϕ0 ) |(N − k)/2, (N − k)/2, ϕi,
(4.52)
−π 2π
k+
[a0 + a1 ]
k−
[a0 − a1 ]
|N/2, N/2i ∝
where ϕ0 is the solution in [0, π] of k+ = k cos2 (ϕ0 /2). This corresponds to the value of the
phase that we infer (up to a sign) from a measurement
of the fraction k+ /k. Eq. (4.52) 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. [16], the beam-splitter consists of the crossed laser set-up used to perform Bragg
spectroscopy, as described in Section 4.1.2. The input states correspond to two trapped
condensates, as in Fig. 4.2, 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 Eq. (4.16)
Γ(p, ω) =
2π V02
[S(p, ω) − S(−p, −ω)] .
~ 4
41
(4.53)
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.
4.4.3
Superradiance
When we discussed Bragg spectroscopy in Section 4.1.2, the optical potential due to the
crossed lasers was treated classically as a ‘running wave’
Uext (r, t) = V0 cos (p · r − ωt) .
This is acceptable, because under conditions of Bragg spectroscopy, there are a large number of photons in each laser mode. There are other phenomena, however, that require the
photons to be treated more quantum mechanically. This can be done by considering the
following interaction, describing the scattering of photons (described by operators ck , c†k ) by
atoms
X
Hint =
Cklmn c†l a†n ck am δl+n−k−m .
(4.54)
k,l,m,n
Note that this expression is second order in the photon operators, and so is second order in
the electric field, so we expect a correspondence with our earlier discussion of AC Stark shift
in Section 5.3. The coupling constants Cklmn can be found by comparison with those results.
Considering a single photon mode to begin with, the number of photons with wavevector k is
nck =
0 |E|2 V
,
2~ck
whereas an electric field field of this intensity gives an AC Stark shift
∆E =
d2 E 2
,
4~∆
where d is the dipole matrix element for the transition, and ∆ is the detuning ω−ωn0 (assumed
much larger than the excited state lifetime). The l = k term of Eq. (4.54) then gives Ckkmm nck
for the shift of each atom, and we have
Ckkmm =
ckd2
.
20 V∆
For the case of crossed lasers at wavevectors k, l, the same consideration leads to
Cklm(m+k−l) =
ckd2 cos φkl
,
20 V∆
(4.55)
where φ is the angle between the axes of polarization of the two beams8 . Now, how are we
to understand a formula like Eq. (4.53) on this basis? Consider transitions out of the initial
state
|ii = |g; nck , nck−q i
8
(4.56)
It is certainly possible that l − k is on the scale of typical atomic momentum, which would lead to some
dependence on the momentum transfer. We neglect this for now, as we did implicitly in Section 5.3.
42
At lowest order in Hint , we change the momentum of one atom by q in all possible ways
through scattering of one photon between states k and k − q. The square of the matrix
element of Hint between |ii and a given final state |f i is then
|hf |Hint |ii|2 = |Ck,k−q |2 hg|ρq |f ihf |ρ†q |gi nck−q + 1 nck
P
where ρ†q = n a†n+q an is a Fourier component of the density operator. We then write down
the Golden Rule expression for the transition rate as
Γ+ =
2π X
|hf |Hint |ii|2 δ (Ef − Eg − ~ω)
~
f
=
2π X
|Ck,k−q |2 hg|ρq |f ihf |ρ†q |gi nck−q + 1 nck δ (Ef − Eg − ~ω)
~
=
2π
|Ck,k−q |2 S(q, ω) nck−q + 1 nck .
~
f
(4.57)
This is the rate at which +k is transferred. To get the total rate we must subtract the rate for
the −k transitions to give the total rate
Γ = Γ+ − Γ− =
2π
|Ck,k−q |2 S(q, ω) nck−q + 1 nck − S(−q, −ω) (nck + 1) nck−q .
~
(4.58)
In the limit of large mode occupations nc 1, the difference between nc and nc + 1 is neglected and the resulting formula, using the above expressions for C, reduces to Eq. (4.53).
The result Eq. (4.58) is much more general and includes, for example, the more familiar case
of Rayleigh scattering, in which the photon mode k − q is empty, nck−q = 0, so that only the
forward rate survives. It is via this process that light is scattered into all other modes, while
the second laser is required to preselect the momentum transfer q by creating an optical
grating. Note also that the discussion works as well for photon modes in Fock or coherent
states.
Now we have a new way of looking at this arrangement. Instead of considering the
response of the condensate to a perturbation due to the light fields, we can think instead
of the scattering of light by the condensate. An interesting case occurs when both the zero
momentum state and some finite momentum state q of the condensate are occupied. This
corresponds to the wavefunction, in the GP approximation
(a†0 )N0 (a†q )Nq |0i,
and static structure factor
S(q0 ) = N0 (Nq + 1) δq0 −q .
Ignoring all interaction effects, the dynamic structure factor is
S(q0 , ω) = N0 (Nq + 1) δq0 −q δ(ω − q ).
(4.59)
We can then use Eq. (4.58) to give the total rate for scattering of light from a mode k into an
initially empty mode k − q, leading to an increase in the number of atoms in mode q
Ṅq = Gq (Nq + 1) − Γ2,q Nq .
43
(4.60)
The gain coefficient Gq depends on N0 , nck , and geometric factors, but the crucial thing is
the ‘bosonic stimulation’ term (Nq + 1). The loss term in Eq. (4.60) represents the linewidth
of the two-photon process we are considering. The instability present for Gq > Γ2,q corresponds to the onset of superradiance: atoms begin to accumulate in the state q, and light is
emitted coherently into the mode k − q. An alternative view of this phenomenon comes from
considering scattering from a grating formed by the coherent state
"r
N0 †
a +
N 0
r
Nq †
a
N q
#N
|0i.
From our earlier discussion, this corresponds to a oscillating density hρ(r)i. Furthermore,
we know that the scattering of light by this density is going to give the same answer as
Eq. (4.59), based on the general result that the measurements on the Fock state coincide
with those for the phase state (with an average over relative phase). In more concrete terms:
though hρ(r)i has no q component in the Fock state, the density-density correlator does,
and this gives the structure factor Eq. (4.59). A detailed discussion of experiments on the
scattering of light by condensates, and the associated theory, may be found in Ref. [13].
In this section, then, we have explored the relationship between the Fock and coherent
states, and the remarkable tendency of bosons to end up in the latter, starting from the
former. We have also learnt about the reciprocal nature of atom-light scattering, where,
depending on the occupancies of the various modes, we can think about light scattering
from a grating formed by the atoms, or vice versa. Both cases have a conjugate viewpoint in
terms of bosonic stimulation of scattering by occupation of the final state.
4.5
Spinor condensates
So far our discussion of the effect of the hyperfine degree of freedom was limited to the one
and two atom level. In Chapter 2 we introduced the Hyperfine-Zeeman states of a single
atom, and briefly discussed the scattering of these states off one another. When we turn
to the many-boson problem, the existence of different atomic species leads to a number of
qualitatively new and surprising phenomena.
4.5.1
General considerations for multicomponent BEC
It is straightforward to enlarge our definition of BEC to include different species of atoms.
Our definition of the one-particle density matrix now bears indices labeling these species
ρ1 (rα, r0 α0 ) ≡ hhφ†α (r)φα0 (r0 )ii
X
=
ni χ∗i (r, α)χi (r0 , α0 ).
(4.61)
i
BEC can then be defined exactly as before, using the spectral resolution of the density
operator. We shall see that non-simple BEC, where more than one eigenvalue of order N , is
now more common.
If we do have simple BEC the condensate wavefunction and order parameter
√ are introduced exactly as in the one component case, as the eigenfunction χ0 (rα) and N0 χ0 (rα)
44
respectively. Now we meet a significant complication. χ0 (rα) is in general a non-trivial
function of α. Thus it is not immediately obvious how to generalize the decomposition into
amplitude and phase that was necessary to define the superfluid velocity. In Section 3.1 we
swept this under the carpet, effectively by assuming that the condensate wavefunction was
χ0 (rα) = χ0 (r)η̂α (r),
where η̂α (r) is some normalized vector in the hyperfine space, and then decomposing χ0 (r)
in the naive way. The problem is that this prescription is not at all well-defined, as we could
have equally well chosen a different η̂α0 (r) = eiϑ(r) η̂α (r), differing from the first by a positiondependent phase factor. The solution is to define the superfluid velocity using the ‘covariant
derivative’
D = ∇ − iη̂ † (r)∇η̂(r).
(4.62)
In this way the definition
~
Dϕ,
ϕ = Arg χ0 (r)
m
is independent of how we apportion the phase between χ0 (r) and η̂α (r). A vanishing vs
is in general inconsistent with the single-valuedness of ϕ. Thus vs is non-zero even in the
equilibrium state of the non-rotating system. Furthermore, vs is no longer irrotational, as can
be seen from the Mermin-Ho relation
vs =
∇ ∧ vs =
~
mF ijk B̂i ∇B̂j ∧ ∇B̂k ,
m
(4.63)
where B̂ is the unit vector in the direction of the magnetic field.
Problem 11 Verify Eq. (4.63) for the simplest case of the spin-1/2 spinor
iφ/2
e
cos θ/2
,
χ1/2 (θ, φ) =
e−iφ/2 sin θ/2
corresponding to the unit vector B̂ = (sin θ cos φ, sin θ sin φ, cos θ).
4.5.2
The Gross-Pitaevskii description
Now we turn to the effect of interactions, and develop the Gross-Pitaevskii approximation. It
is simplest to forget the above complexities associated with non-constant adiabatic bases,
at least to begin with. Let us therefore consider bosons with F = 1 in an optical trap, so
that all hyperfine species are present, and have equal single-particle energy. According
to the discussion of Section 2.3.2, it is clear that the Gross-Pitaevskii variational energy
corresponding to the wavefunction
Y
Ψ({ri , αi }) =
χ0 (ri αi )
(4.64)
i
is
Z
hHi =
Z
~2
1
2
2
|∇Ψ| + Uext (r)|Ψ(r)| +
dr U1 |Ψ(r)|4 + U2 [Ψ† FΨ]2 ,
dr
2m
2
45
(4.65)
where |Ψ|2 = Ψ† Ψ, U1,2 = 4π~2 a1,2 /m, and as before we have neglected the difference
between N and N − 1, although we will see that we must examine this point more carefully
later.
The following two cases immediately present themselves
• Antiferromagnetic interactions, U2 > 0. In this case the energy is minimized for hFi = 0,
the ground state energy is EA = nN U0 /2. This is the case as long as χ† ∝ (χx , χy , χz ),
for χx,y,z all real. The symmetry group of such configurations is SO(3)/U (1) i.e. rotations about the axis of Ψ are excluded. This state is sometimes called the ‘polar state’
by analogy with superfluid 3 He.
• Ferromagnetic interactions, U2 < 0. In this case we take the spin to be as big as
possible, with each particle having |hFi| = 1. Thus the ground state energy is EF =
nN (U1 + U2 )/2. The spinor with hFi = (0, 0, 1) has the form


1
1
√  −i  ,
(4.66)
2
0
and all other possible ferromagnetic configurations can be generated from the one
through appropriate SO(3) rotations. Note that because the spinor is complex, all
rotations are effective. Thus a rotation of Eq. (4.66) about the z-axis yields
 −iϕ 
e
1 
√
−ie−iϕ  .
2
0
Although the overall phase of the condensate wavefunction has no meaning, the fact
that a phase changing in space is equivalent to a rotation through differing angles at
different points can have real physical consequences, as we shall see.
4.5.3
Metastability of superflow
The internal states of the condensate have rather intriguing repercussions for the metastability of superflow. We return to our old friend the annular container to illustrate these. Suppose
that we have only two internal hyperfine states | ↑i | ↓i (F = 1/2 clearly doesn’t correspond
to a boson, but let’s not worry about that for the moment), and consider the family of order
parameter configurations
θ
θ
cos eiφ | ↑i + sin | ↓i
2
2
0 ≤ θ ≤ π.
(4.67)
(φ is the angular variable in cylindrical coordinates). This interpolates between a state with
orbital angular momentum ` = 1 and 0, just like the state considered in Problem 7 for the
single component case. The crucial difference, however, is that all members of the above
family have the same |Ψ|2 (in particular we can move from ` = 1 to 0 without passing through
a state with a node) and therefore the same interaction energy, if interactions are invariant
under rotations in the internal space. Since the kinetic energy decreases monotonically as θ
goes from 0 to π, there is no metastability.
46
Consider now the ferromagnetic state of spin-1 bosons discussed in the previous section.
We want to attempt the same trick: rotating from mF = +F = 1 to mF = −F = −1 using
a rotation axis in the x-y planes that rotates by 2π as go around the annulus, we try to undo
the winding of the phase. Thus we take the family of condensate wavefunctions


cos φ − i sin φ cos θ
1
χθ (φ) = √  − sin φ − i cos φ cos θ  .
(4.68)
2
−i sin θ
We see that
 −iφ 
e
1 
−ie−iφ  ,
χ0 (φ) = √
2
0
and furthermore that the spin rotates in the desired way


− sin φ sin θ
hF(φ)iθ =  − cos φ sin θ  .
cos θ
But what state do we end up in? Surprisingly, the answer is
 iφ 
e
1
χπ (φ) = √  −ieiφ  .
2
0
We have not got rid of the winding in the phase at all, but rather just switched its sense from
positive to negative! It is clear, then, that a winding number of two is equivalent to no winding
in this sense, and odd winding number configurations are metastable.
To summarize, when mF = 0, including in the antiferromagnetic case for F = 1, there
are metastable configurations of arbitrary winding number. For mF = 1/2, there are none,
and when mF = 1 there is just one: all even winding numbers are topologically equivalent to
zero winding number, and all odd winding numbers are equivalent to one. The general rule
that these three exemplify is: the winding number of the phase can change by 2mF without
energetic penalty. Of course, there is still the issue of the dynamics of this unwinding process
– it may take a long time – but there is no energetic barrier.
Experimentally, we are still waiting for the observation of the ferromagnetic state, but the
same physics is behind the topological vortex formation discussed in Problem 3. The result
can also be interpreted in the light of Eq. (4.63).
[A Java applet for the belt trick shown in the lecture can be found at http://www.math.
toronto.edu/∼drorbn/Students/Song/]
4.5.4
Fragmented condensates
There is a slight wrinkle to the above discussion of the antiferromagnetic case. If we assume
that all the atoms are in a single spatial orbital – the zero momentum state – but do not
constrain the spin wavefunction of this system at all, we arrive at the spin Hamiltonian
Hspin =
U1
U2 2
N (N − 1) +
S − 2N ,
2V
2V
47
(4.69)
where S is the operator of total spin. It’s clear that the ground state of this problem corresponds to the N -spin singlet state, with energy
U1
U2
N (N − 1) − N .
2V
V
This is smaller than the ground state of the Gross-Pitaevskii state that we considered earlier,
because of the (non-thermodynamic) second term. As is fairly clear, the mean number of
particles in each of the three hyperfine states of the system is N/3, with fluctuations also
of order N . We have a fragmented condensate, in which more than one eigenvalue of the
density matrix is thermodynamically large.
The consensus view of this situation is that:
• The original GP state is a better description of the ground state in a magnetic field, with
the fluctuations decreasing very rapidly
• The two states are in any case very hard to distinguish on the basis of an interference
measurement such as those discussed in Section 4.4.
48
Chapter 5
Fermi superfluids
5.1
Fermionic condensates
In Chapter 3, we saw that superfluidity was a natural consequence of Bose-Einstein 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 3.1, 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 ),
V
|r1 − r01 |, |r2 − r02 | → ∞,
(5.1)
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. 5.1. The surprising
thing is that this expectation is not borne out. The long-range order described by Eq. (5.1)
can be present even when the interaction between two isolated particles is not strong enough
49
Figure 5.1: 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
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. [17] 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
5.2
5.2.1
The BCS theory
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
50
system until later.
H=
X
ξp a†p,s ap,s +
p,s
U0 X †
ap+q↑ a†−p↓ a−p0 ↓ ap0 +q↑ .
V
0
(5.2)
p,p ,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 † †
|F Si =
ap↑ ap↓ |0i
(5.3)
|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
X
Y np
|pairi ≡
(5.4)
c{nPp }
Λp0 |0i,
P
p
p
nP
p =N/2
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. (5.4) 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
Hpair =
X
ξp a†p,s ap,s +
p,s
U0 X † †
ap↑ a−p↓ a−p0 ↓ ap0 ↑ .
V
0
(5.5)
p,p
That is, just Eq. (5.2) with all but the q = 0 part discarded. Now, can we solve Eq. (5.5)?
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. (5.4) only includes amplitudes for a
given (+p, −p) pair of momenta having 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 †
bp bp0 .
V
0
(5.6)
p,p
1
In the book Ref. [18], Schreiffer explains that the BCS wavefunction was directly motivated by such treatments.
2
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.
51
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 ,
(5.7)
obey the hardcore constraint
(b†p )2 = 0,
(5.8)
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 bp
|0i,
(5.9)
p
corresponding to Eq. (5.4) but with a set of coefficients c{nPp } that factorizes. Finding the
variational energy of Eq. (5.9) 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,
(5.10)
p
p
Finding the average number of pairs hnPp i in Eq. (5.9) is however not obvious. We can make it
so by following the route taken by BCS, and considering instead the normalized wavefunction
i
Yh
|BCSi =
vp b†p + up |0i
|up |2 + |vp |2 = 1.
(5.11)
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. (5.9) if cp = vp /up . Since Eq. (5.11) 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 ∗
hHiBCS = 2
ξp vp2 +
up vp up0 vp∗ 0 .
(5.12)
V
0
p
p,p
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
where h· · · iN denotes the expectation with respect to the N -particle projection Eq. (5.9). The
probabilities PN are
X Y
PN =
nPp vp2 + 1 − nPp u2p ,
P
p
nP
p =N/2
P
P
which is strongly peaked around hN i = 2 p vp2 = 2 p hnPp i, with a variance that is O(N ).
Thus at large N
hOiBCS → hOihN i .
(5.13)
52
In the thermodynamic limit, we might as well work with the non-conserving form Eq. (5.11).
There is an interesting alternative interpretation of the pair Hamiltonian Eq. (5.6). 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)
h
i
b†p , bp = 2 b†p bp − 1/2
h i
b†p , b†p bp − 1/2 = −b†p ,
(5.14)
meaning that Eq. (5.6) can be written as a spin chain
Hpair = 2
X
ξp Spz +
p
U0 X + −
Sp Sp0 .
V
0
(5.15)
p,p
If we parameterize (up , vp ) as (cos(θ/2)eiϕ/2 , sin(θ/2)e−iϕ/2 ) then the variational energy
Eq. (5.12) has the form (except for a constant)
hHiBCS = −
X
ξp cos θp +
p
U0 X
sin θp sin θp0 cos ϕp − ϕp0 .
4V
0
(5.16)
p,p
The interpretation of Eq. (5.16) 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. (5.12) 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. 5.2. 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. (5.10). Clearly all of the angles ϕp , describing the angle in the x-y plane, should be
equal. Taking the extremum of Eq. (5.16) 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
(5.17)
Thus we have
cos θp =
ξp
,
Ep
sin θp =
|∆|
,
Ep
Ep =
p
ξ(p)2 + |∆|2 .
(5.18)
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
53
Figure 5.2: Anderson spin configurations and the associated distribution functions for the
free fermi gas (top) and the BCS state (bottom).
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 )
(5.19)
To be self-consistent, the solution must further satisfy
∆=−
U0 X ∆
.
2V p Ep
(5.20)
Eq. (5.20) 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 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
U0
dp ∆
∆ = −
(5.21)
2
(2π)3 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. (5.21) 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. (5.21) has no divergence
54
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. (5.1), 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. (5.1), this seems unlikely.
Problem 12 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
(5.22)
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.
5.2.2
The BCS-BEC crossover
Taking our argument one step further, we can argue that not only is the order specified by
Eq. (5.1) the same whether two fermions can form a bound state or not, but the wavefunction
4
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.
5
The statement that pairing always occurs is strictly true only for exactly equal numbers of the two species,
as we will see in Section 5.3 below
55
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. (5.9)
N/2
|N i = A
Y
ϕ(ri↑ − rj↓ ),
(5.23)
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. (5.23) 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. (5.8) does not play a significant
role. Then Eq. (5.6) 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. (5.1) in the first place. This suggests that we can
use Eq. (5.11) as a variational wavefunction all the way through the BCS-BEC crossover.
First we have to address the issue of regularizing Eq. (5.21). 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
∆p = −
X
1 X
U0 (p − p0 ) sin θp0 = −
U0 (p − p0 )up0 vp∗ 0 ,
2V 0
0
p
(5.24)
p
so that the self-consistent equation is now
∆p = −
∆p0
1 X
.
U0 (p − p0 )
2V 0
Ep0
(5.25)
p
The solution of Eq. (5.25) 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. (5.25) as
U0 (p) X ∆0
∆0
∆p = −
−
V
2Ep0
2p0
0
p
∆p0
1X
−
U (p − p0 )
.
V 0
2p0
(5.26)
p
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
56
in terms of the scattering amplitude F (p, p0 ) and scattering length as corresponding to the
potential U0 (r). From the integral equation Eq. (2.18) satisfied by the scattering amplitude
we see immediately that
1 X ∆0
∆0
m
∆0 =
−
.
(5.27)
−
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. (5.27) can then be done explicitly to give the gap
8
π
(5.28)
∆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−
(5.29)
Ep
p
p
p
The two equations Eq. (5.27) and Eq. (5.29) are conveniently cast in the dimensionless form
#
Z ∞ "
π
x2
=
dx 1 − p
2kF as
(x2 − µ)2 + ∆20
0
"
#
Z ∞
2−µ
x
2
=
x2 1 − p
(5.30)
3
(x2 − µ)2 + ∆20
0
where µ and ∆0 are measured in units of EF = p2F /2m, and the unit of length is p−1
F . In
these units the total density of particles of both types is 1/3π 2 . The behavior of the gap and
chemical potential is shown in Fig. 5.3. Recall from Section 2.3.3 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 .
(5.31)
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 [19]. There is of course no reason to believe the quantitative
predictions of the mean-field theory in the region where interactions are so strong.
Problem 13 Show that the in the BCS limit (1/kF a large and negative, ∆ EF , and µ =
EF ), the variational enenergy Eq. (5.12) of the BCS state relative to the free fermi gas is
1
(5.32)
EBCS (∆) − EBCS (0) = − ν(EF )|∆|2
2
F
where ν(EF ) = mp
is the fermi surface density of states. Note that this arises from the sum
2π 2
of a large increase in the kinetic energy, and a large decrease in the pairing energy.
57
1.5
µ/EF
∆0 /EF
1
0.5
-2
-1
1
2
-0.5
-1
-1.5
1/kF a
Figure 5.3: Variation
of gap and chemical potential. Also plotted is the result for the gap on
p
the BEC side 16/(kF as )/π. Note the exponential dependence of the gap on the BCS side,
consistent with the analytic result Eq. (5.28)
Problem 14 When µ becomes large and negative for 1/kF a > 0, the angles θp in Eq. (5.16)
are all close to 0, since cos θp = ξp /Ep . Expand Eq. (5.16) in small deviations from θp = 0
and interpret the result.
Problem 15 In Section 2.3.3 we introduced the simplest two-channel model that displays a
Feshbach resonance, Eq. (2.22)
X
X q
g X
(5.33)
H=
p a†s,p as,p +
bq a†↑,q+p a†↓,−p + h.c.
+ ε0 b†q bq + √
2
V
p,s
p,q
q
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. (5.27) is replaced by
1X 1
1
ε0
=
−
,
(5.34)
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. (2.24) of Problem 4]
Show that the number equation is then
ξp
1X
n = nb +
1−
.
(5.35)
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.
58
5.2.3
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
i
Yh
|BCSi =
vp a†p↑ a†−p↓ + up |0i,
(5.36)
p
it’s easy to see that the operators
αp↑ = up ap↑ − vp a†−p↓
αp↓ = up ap↓ + vp a†−p↑ ,
(5.37)
satisfy the canonical fermion anticommutation relations and annihilate the BCS state
αp,s |BCSi = 0. Consider the state
i
Y h
†
|p, si = αp,s
|BCSi = a†p,s
vp a†p↑ a†−p↓ + up |0i,
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. (5.16)
(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 = p
2
2
p
∆ + µ , µ < 0.
(5.38)
(5.39)
As µ turns from positive to negative as we pass from BCS to BEC (see Fig. 5.3), the density
of states of the quasiparticle excitations turns from having a square root singularity ν() ∼
( − ∆s )−1/2 to vanishing like a square root ν() ∼ ( − ∆s )1/2 .
5.2.4
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
59
finite temperature, where the quasiparticles, being fermionic, have the fermi-dirac distribution
function
1
ns (p) = βE (p)
,
β = 1/kB T,
(5.40)
s
e
+1
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?
h
i Y h
i
†
†
αp↑
α−p↓
|BCSi = up a†p↑ a†−p↓ − vp
vp a†p↑ a†−p↓ + up |0i,
(5.41)
p0 6=p
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. (5.19). 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
no qp
∆ =
=
two 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
(5.42)
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. (5.28), to zero at
γ
kB Tc = ∆BCS .
(5.43)
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. 5.1, 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 [20]
kB Tc ∼ Eb /2 [ln Eb /EF ]3/2 , 1/kF a 0
(5.44)
where Eb = 1/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. 5.4.
60
Figure 5.4: Tc in the mean-field theory (solid line), compared with the result of a treatment
that smoothly interpolates to TBEC (dashed). Reproduced from Ref. [20]
5.3
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 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
(5.45)
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,
(5.46)
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 2.3.3.
61
and this energy appears in the equilibrium quasiparticle distribution function Eq. (5.40). Thus
we have
1X
1
1
m
=
[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).
(5.47)
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. (5.41).
5.3.1
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 − ∆ ∆BCS
√
0 = ln
,
(5.48)
h − h2 − ∆ ∆
with solution
r
2h
∆Sarma
=
− 1.
(5.49)
∆BCS
∆BCS
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. 5.5, 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. (5.32) 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
√
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 5.2.4 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,
62
Figure 5.5: Schematic plots of ground state energy versus ∆ for different h.
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. 5.6
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, 5.7. 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
p=
,
m S < m < mN
mN − mS
5.3.2
Magnetization in the BCS-BEC crossover
Now we discuss in a completely qualitative way what happens as we pass through the BCSBEC crossover [22]. 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 of N↓ /2 bosonic molecules with the remaining
M unpaired ‘majority’ fermions, see Fig. 5.8.
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.
63
Figure 5.6: The tricritical point in the (T, h) plane. Reproduced from Ref. [21]
Figure 5.7: 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.
64
a)
b)
Figure 5.8: 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.
65
Chapter 6
Hydrodynamics of condensates
6.1
Galilean invariance
Let us go back to the time-dependent Gross-Pitaevskii theory describing the dynamics of a
condensate
~2 2
∂Ψ(r, t)
2
−
∇ + Uext (r) + U0 |Ψ(r, t)| Ψ(r, t) = i~
,
(6.1)
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 Ψ(r, t) is a solution of Eq. (6.1) then
i
1
2
Ψ(r − vt, t) exp
mv · r − mv t ,
(6.2)
~
2
is also a solution. Eq. (6.2) gives the transformation property of the condensate wavefunction
√
under Galilean transformations. Upon making the decomposition Ψ = neiϕ , we see that
this implies a transformation law for the phase of the condensate
1
1
2
ϕ→ϕ+
mv · r − mv t .
(6.3)
~
2
Recalling the identifications1
~
∇ϕ
µ = −~ϕ̇,
m
we have the transformation laws for the superfluid velocity and chemical potential
vs =
vs → vs + v
1
µ → µ + mv2 ,
2
(6.4)
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. (6.1) 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.
1
in the lab frame
66
6.2
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. (6.1) and its complex conjugate we get the
continuity equation
∂n
+ ∇ · [nvs ] = 0,
(6.5)
∂t
while Ψ∗ Ψ̇ − Ψ̇∗ Ψ, which involves the sum of the equation and its conjugate, yields
√
~2
1
√ ∇2 n + U0 n + Uext = 0.
~ϕ̇ + mvs2 −
2
2m n
(6.6)
If n varies sufficiently slow in space – in practice this means slower than the healing length
−1/2
√
0
ξ ≡ 2mnU
– we can drop the ∇2 n term. Taking the gradient of the resulting equation
~2
gives
1
2
(6.7)
mv̇s + ∇
mvs + µ(n) + Uext = 0,
2
where we wrote µ(n) = nU0 . For the static case we have
µ(n(r)) + Uext (r) = µ0 ,
(6.8)
which defines the Thomas-Fermi approximation, widely used to compute density profiles of
trapped gases.
The equation Eq. (6.7) 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 = ∇ 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 7 there is no vorticity, so
this difference disappears.
It is illuminating to consider the linearization of the equations Eq. (6.5) and Eq. (6.7). In
many ways the analysis is clearer than the linearization of the TDGP theory considered in
Section 4.1.2. 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. (6.6)?
As in the previous section, Eq. (6.5) and Eq. (6.7) 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. (6.7) could have been obtained by taking the time
and spatial derivatives of the transformation law Eq. (6.3).
67
6.3
Quantum Hydrodynamics
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.
6.3.1
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 .
(6.9)
(we have dropped the external potential). This set of equations can be obtained from the
Hamiltonian
Z
1
H = dr
vρv + V (ρ) ,
(6.10)
2
together with the commutation relation
ρ(r), ϕ(r0 ) = iδ r − r0
(6.11)
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. (6.10) 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,
(6.12)
2
and you should also check that the Euler equation is reproduced. The commutation relation
Eq. (6.11) plays a fundamental role in the following development, 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
j(r) =
1X
pi δ (r − ri ) + δ (r − ri ) pi
2
pi = −i∇i ,
i
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.
68
(6.13)
6.3.2
Mode expansion
It is convenient to re-write the commutation relation in termshof the iFourier 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
βk eik·r + h.c.
e
ρ(r) = ρ0 + ρ0
k
r
ϕ(r) =
1 X
−ieκk βk eik·r + h.c.,
4ρ0
(6.14)
k
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. (6.10) as
Z
1
1
H = dr ρ0 v2 + U0 (ρ − ρ0 )2 .
(6.15)
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 6.2. 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. (6.15) 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. (6.14) into the Hamiltonian Eq. (6.15) 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 zero and yields (aside from
an infinite constant)
X
H=
cs |k|βk† βk ,
1/d
kρ0
e−2κk =
~|k|
,
2mcs
c2s =
nU0
m
(6.16)
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. (6.16).
6.3.3
Correlation functions
To see the power of this approach, let’s consider the density matrix ofp
the bosons. Using the
density-phase representation, we write the boson operator as b(r) = 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
(6.17)
ρ r, r = hb (r)b(r )i ∼ ρ0 1 − h ϕ(r) − ϕ(r ) i , |r − r0 | → ∞
2
69
It turns out that in d > 1 this expansion is a reasonable thing to do, because
2
1
mcs
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
n(p) = no δp,0 +
mcs
,
2|p|
|p| → 0.
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
nas ,
n(p) = √
N p
3 π
(6.18)
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.
70
Chapter 7
Strong correlations: low dimensions
and lattices
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.
7.1
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
γ≡
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
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. 7.1. 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
1
In a sense, the interactions are always strong in one dimension, see Problem 16.
71
Figure 7.1: The curve shows the function e(γ) obtained in Ref. [23] for the δ-function Bose
gas in 1D, in terms of which the energy is E = N n2 ~2 e(γ)/2m.
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 )|,
(7.1)
where ΨF (x1 , · · · , xN ) 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 kF ,
the total energy density is
Z kF
~2 kF3
dk ~2 k 2
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
c2s =
n ∂ 2 (E/Ω)
,
m ∂n2
cs =
~kF
≡ vF .
m
(7.2)
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. (7.1) doesn’t interfere with such a calculation.
72
Problem 16 (Momentum distribution in a 1D Bose fluid) Using the harmonic Hamiltonian Eq. (6.15)
• Show
h0|ρq ρ−q |0i =
N ~|q|
,
2mcs
|q| → 0,
(7.3)
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. (6.17) to give
n(p) ∝
1
1− η1
|p| → 0,
,
|p|
η≡
2π~n
mcs
In particular, this result tells us that there is no condensate in a one-dimensional system.
Problem 17 (Structure factor and momentum distribution for the Tonks gas) Let’s
compute some properties of the Tonks gas using the fermionic mapping Eq. (7.1).
• First show that the dynamical structure factor defined in Eq. (4.20) is
(
qpF
qpF
~q 2 ~q 2 Nm
−
<
ω
<
+
m
m
2m 2m S(q, ω) = 2~qpF
0
otherwise
(7.4)
• Check that this is consistent with the sum rules Eq. (4.21) as well as the inequality
Eq. (4.23), 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.
7.2
7.2.1
Lattice systems
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
Vopt. = −α(ω)|Eω (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 ωn0 we have
α0 (ω) ∼
|hn|d · ε̂|0i|2
,
~ (ωn0 − ω)
73
(7.5)
Figure 7.2: Counterpropagating lasers form a) a two-dimensional and b) a three-dimensional
lattice. Reproduced from the recent review Ref. [24].
(ε̂ is the direction of E) and the sign changes from positive (attractive potential) to negative
(repulsive) as we go from ω < ωn0 (red detuning) to ω > ωn0 (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. 7.2. The quantum states of a particle propagating in a
periodic particle are the Bloch waves, characterized by (pseudo-)momentum p and band
index n
Ψn (p) = eip·r ϕn (r),
ϕ(r + ai ) = ϕ(r)
where {ai } 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
X †
Htb = −t
bi bj .
(7.6)
hiji
(where hiji denotes nearest neightbours – we are thinking of the one-dimensional case).
This can be diagonalized to give the dispersion relation ε(k) = −2t cos k with bandwidth
4t. 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.
74
7.2.2
The Bose-Hubbard model
Adding the simplest on-site interactions to the tight-binding Hamiltonian Eq. (7.6) gives the
Bose Hubbard model
X
X †
bi bj + (U/2)
ni (ni − 1).
(7.7)
HBH = −t
i
hiji
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. (7.7), so that the Hamiltonian becomes a sum of onsite Hamiltonians. After including
the chemical potential, the free energy at zero temperature
Q † n
is minimized by a state i bi |0i, with integer n particles on each site and
n = [µ/U + 1/2],
(7.8)
where the square brackets [· · · ] denote the nearest integer. What happens when t/U 6= 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. (7.6)). 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 −2td in d-dimensions. Thus we expect the asymptotic phase boundaries
µ
tc (µ) ∼
(1 + O(t/U ))
− ∞ < µ < U/2
2d
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 −2td(k + 1), leading to
(
µ
(1 + O(t/U ))
−∞ < µ < U/2
(7.9)
tc (µ) ∼ 2d
|µ−kU |
2d(k+1) (1 + O(t/U )) (k − 1/2)U < µ < (k + 1/2)U, k ≥ 1.
In this way we arrive at the schematic phase diagram in Fig. 7.3. The states with fixed number
that prevail at low hopping are characterized by vanishing compressibility −∂hni/∂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 [26] is what
really made the community wake up to the potential of cold gases for doing fundamental
condensed matter physics.
75
Figure 7.3: Schematic phase diagram of the Bose Hubbard model, including the estimate
Eq. (7.9). The regions marked ρ = 0, 1, 2, etc. are the Mott phases with different fillings.
Reproduced from Ref. [25].
Problem 18 (More phases of the Bose-Hubbard model) In the lectures we discussed the
phase diagram of the Bose Hubbard model Eq. (7.7) as a function of chemical potential and
U/t.
• Qualitatively, what happens when we add an additional term describing repulsion between neighbouring sites?
X †
X
X
0
HBH
= −t
bi bj + (U/2)
ni (ni − 1) + V
ni nj ,
(7.10)
i
hiji
hiji
Hint: as in the lectures, start by considering the case of zero hopping. See Ref. [27] for
more details.
76
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