Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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Quantum Degenerate Gases 76.1 Elements of Quantum Field Theory 1109

k ≡{p,α},wherep stands for quantum numbers of the

center of the mass and α for the quantum numbers of the

internal state, it is expedient to deﬁne a quantum ﬁeld

for each species α as

(x) =



pα

(x)a

pα

. (76.13)

Mechanisms that cause transitions between the internal

states couple the ﬁelds

(x) for different α.

States of Bosons. To complete the transformation from

wave function quantum mechanics to second quantiza-

tion, the state of the system must be speciﬁed in second

quantization. For instance, take the Hamiltonian

H and

the particle number operator

N. According to statistical

mechanics a system characterized by the temperature T

and chemical potential µ is in the state with the density

operator

ρ =e

−(

H−µ

N)/k

/Z, where the grand partition

function is Z = Tr e

−(

H−µ

N)/k

Bose–Einstein Condensate. The state of a boson sys-

tem of particular interest here is the BEC.Inanideal

gas condensation entails a macroscopic fraction of the

particles occupying the ground state of the c.m. motion.

Condensation is a phase transition that occurs when ei-

ther the density of the gas is increased or the temperature

is lowered. In a homogeneous ideal Bose gas the gov-

erning parameter is the phase space density ζ deﬁned

ζ =



2π



3/2

n , (76.14)

where m is the mass of the condensing atoms, T is

the temperature, and n is the density of the condensing

species. For the purposes of quantum degeneracy, each

internal state of an atom behaves as a separate species.

Bose–Einstein condensation takes place when the phase

space density satisﬁes ζ =2.612. Depending on whether

density or temperature is regarded as a constant, (76.14)

may be regarded as an equation for the critical tem-

perature T

or the critical density n

for Bose–Einstein

condensation.

76.1.2 Fermions

Particles with a half-integer angular momentum obey

the Fermi–Dirac statistics. Each Fock state may then

only have the occupation numbers n

= 0, 1. The con-

ventional deﬁnition of the annihilation operator contains

a phase factor,

, n

,... ,n

∞



= n

(−1)



k−1

p=0

, n

,... ,n

−1,... ,n

∞

 ,

(76.15)

and fermion operators are governed by the anticommu-

tator

[A, B]

≡ AB+BA (76.16)

rather than the commutator. For instance,

, a



]

= 0,



, a

†



= δ



. (76.17)

Except for the use of anticommutators in lieu of com-

mutators, all formal expressions for ﬁeld operators and

one- and two-particle operators written down for bosons

in Sect. 76.1.1 remain valid as stated.

Degenerate Fermi Gas. A degenerate Fermi gas realized

in a dilute atom vapor is the fermion counterpart of

a BEC. The basic parameter of a free noninteracting

Fermi gas is the Fermi energy, the chemical potential at

zero temperature. It is given by



; k



6π



1/3

, (76.18)

where n once more is the density for the relevant fermion

species. In the limit of zero temperature the Fermi gas

makes a Fermi sea; the states below the Fermi energy are

ﬁlled with one particle each, the states above the Fermi

energy are empty. The gas begins to show substantial

deviations from the Maxwell-Boltzmann statistics and

may be regarded as degenerate when the temperature is

below the Fermi temperature, T ≤ T

= 

. Except

for a numerical factor, in terms of density and tempera-

ture the condition T ≤ T

is the same as the condition

for Bose–Einstein condensation.

76.1.3 Bosons versus Fermions

Isotopes of alkali metals with an odd mass number



Li,

Na,

Rb,

133



make Bose–Einstein

gases, while isotopes with an even mass number



Li,



make Fermi–Dirac gases. Atoms are composite

particles consisting of fermions, and how they may act

as bosons is a legitimate question. Whether a satisfactory

formal answer to this question exists may be debat-

able, but in practice atoms seem to obey the correct

statistics in processes that do not expose their individual

constituents.

Part F 76.1

1110 Part F Quantum Optics

When two bosonic atoms with integer angular mo-

menta combine into a molecule, the molecule has an

integer angular momentum and behaves as a boson. On

the other hand, two fermionic atoms also make a bosonic

molecule. Models for this latter type of system are basic-

ally ad hoc since, at this point in time, no microscopic

theory for such a reorganization of the statistics exists.

Nonetheless, empirically, diatomic molecules formed by

combining two fermionic atoms indeed appear to be

bosons.

76.2 Basic Properties of Degenerate Gases

Atoms Are Trapped. Quantum degenerate alkali va-

pors are typically prepared in an atom trap. Close to

the bottom almost every trap is a three-dimensional

harmonic oscillator potential completely characterized

by the principal-axis directions and the corresponding

trap frequencies ω

, the (angular) frequencies at which

a single atom would oscillate back and forth in the given

principal-axis direction. In the principal-axis coordinate

system the trapping potential reads

V(x) =



i=1

mω

. (76.19)

It is convenient to introduce the characteristic harmonic-

oscillator frequency scale and the corresponding

harmonic-oscillator length scale as

ω = (ω

)

1/3

,=



(76.20)

Atom–Atom Interactions. At low temperatures/ener-

gies only s-wave collisions are signiﬁcant. In the theory

of quantum degenerate gases these are frequently repre-

sented by a pseudopotential tailored to give the correct

s-wave phase shift. For two atoms the atom–atom inter-

action is

u(x

, x

) =

4π

δ(x

−x

), (76.21)

where a is the s-wave scattering length. Qualitatively, the

scattering length is positive (negative) if the interaction

is repulsive (attractive).

Model Hamiltonian. Quantum ﬁeld theory for a single-

component Bose gas usually starts with the Hamiltonian

H =



H (x), (76.22)

where the Hamiltonian density is

H (x) =

†

(x)



−

∇

+V(x)



ψ(x)

2π

†

(x)

†

(x)

ψ(x)

ψ(x). (76.23)

As in Sect. 76.1.1,

ψ(r) is the boson ﬁeld operator, the

kinetic energy −

∇

/2m and trapping potential V(x)

are one-particle operators, and atom–atom interactions

are governed by the two-body operator u of (76.21).

Analogous models can be written down for multicom-

ponent boson and fermion ﬁelds, for coupling between

atoms and molecules, and so on.

76.2.1 Bosons

Gross–Pitaevskii Equation

Mean-Field approximation. Conventionally, the next

step for bosons is to go over to the corresponding clas-

sical ﬁeld theory. The result is referred to as mean-ﬁeld

theory or semiclassical theory.

Formally, one ﬁrst writes down explicitly the Heisen-

berg equation of motion for the boson ﬁeld

ψ,

∂

∂t

ψ(x, t) =



ψ(x, t),



(76.24)

and then declares that in the equations of motion

ψ →ψ

is a classical ﬁeld not a quantum ﬁeld anymore. We

call ψ the macroscopic wave function of the conden-

sate. This approximation is precisely analogous to using

the classical instead of the quantum description for the

electric and magnetic ﬁelds of the light coming out of

a laser.

Time-Dependent Gross–Pitaevskii Equation. The

time-dependent Gross–Pitaevskii equation (GPE)is

∂

∂t

ψ(r, t) =



−

∇

+V(r)



ψ(r, t)

4π

|ψ(r, t)|

ψ(r, t). (76.25)

This equation is nonlinear, and normalization of the

macroscopic wave function ψ is important. Quantum

mechanically, the particle number operator is given

by (76.8), so that the normalization for a system with

N particles naturally reads



x |ψ(x, t)|

= N . (76.26)

Part F 76.2

Quantum Degenerate Gases 76.2 Basic Properties of Degenerate Gases 1111

Time evolution under (76.25) preserves the normaliza-

tion. Obviously, and in accordance with (76.9),

n(x) =|ψ(x)|

(76.27)

is the local density of the gas.

Time-Independent Gross–Pitaevskii Equation. So-

lutions to the time-dependent GPE of the form

ψ(x, t) = φ(x) e

−iµt/

are stationary states with no time

evolution in the physics. The analog of the energy of

a stationary state is called the chemical potential µ.

The corresponding wave function φ satisﬁes the time-

independent GPE

µφ =



−

∇



φ +

4π

|φ|

φ. (76.28)

Both GPEs are nonlinear variants of the Schrödinger

equation, and in other contexts they are often referred to

as nonlinear Schrödinger equations. The nonlinear term

approximates the interaction energy of an atom with

the other atoms in an averaged way by relying on the

local density of the atoms, hence the term mean-ﬁeld

theory.

Sign of Scattering Length. The qualitative properties

of a condensate, as per the GPE, depend on the sign

of the scattering length. For repulsive atom–atom in-

teractions or no atom–atom interactions, a ≥0, both

the time-dependent and the time-independent forms are

mathematically well behaved. Unless otherwise noted,

the scattering length is always assumed non-negative.

In the case of a negative scattering length a BEC

may, in principle, decrease its energy without a bound

by collapsing to a point. Mechanisms such as three-body

recombination or molecule formation would eventually

set in as the density increases and the collapse would

stop, but the condensate must then be presumed lost.

In the absence of an external potential, a condensate

with a negative scattering length is unconditionally un-

stable against collapse. For a bounded condensate the

increase in the kinetic energy coming with the decreas-

ing size may hold off the collapse, provided the number

of atoms in the condensate is sufﬁciently small. Simple

dimensional-analysis arguments give the condition of

stability in a harmonic trap as N|a|

 .

Behavior attributed to a collapse has been observed

Li for trapped states with a negative scattering

length. By using a Feshbach resonance it is also possible

to adjust the (apparent) scattering length (Sect. 76.5.1)

which has led to further demonstrations of collapse-like

physics.

Healing Length. Consider the time-independent

GPE (76.28) without an external potential, and scale

the various quantities as follows:

φ =

√

φ, µ =

4π

x = ξ

x; ξ =

√

8πna

. (76.29)

Here n is the density scale for the gas, and the length

scale is ξ. In terms of these new variables the time-

independent GPE reads

φ =−

∇

φ +



φ.

(76.30)

There is a solution in all of space with

µ = 1,

φ = 1. If

for some reason, such as at an edge of the sample, the

condensate wave function must vanish, the length scale

over which the wave function grows back to one (in the

scaled units) is of the order of unity. In fact, (76.30)has

the solution

φ(

x) =tanh



√



in the half-space

z ≥ 0.

The quantity ξ is the minimum length scale over which

a condensate wave function can build up to the density n.

It is called the healing length.

Thomas–Fermi Approximation. Without atom–atom

interactions, the ground state of the trapping potential

V(x) would be the lowest-energy (lowest µ) solution

to (76.28). However, experience has shown that even

modest repulsive atom–atom interactions (a > 0) spread

out the macroscopic wave function of the condensate

a great deal. With increasing size comes decreasing

kinetic energy, according to the Heisenberg uncertainty

principle. This suggests the Thomas–Fermi approxima-

tion, in which the kinetic energy term in (76.28)issimply

ignored. The density of the gas is then easily solved to

n(x) =









µ −V(x)



4 π

,µ>V(x)

0, otherwise,

(76.31)

an inverted image of the trapping potential. The normal-

ization (76.26) can be used to ﬁnd the relation between

chemical potential and particle number, and all of the

unknown quantities may, in principle, be found.

For a harmonic potential the Thomas–Fermi approx-

imation can be worked out explicitly with the results

µ =



15Na





2/5

, R = 



15Na





1/5

n(0) =

8π





15Na





2/5

. (76.32)

Part F 76.2

1112 Part F Quantum Optics

The quantity R represents the size of the condensate. In

particular, it equals the radius of the spherical condensate

if the trap is isotropic with ω

= ω

. Finally, n(0)

is the central, maximum, density of the atoms.

The relevant dimensionless parameter is Na/,

which can easily be much larger than unity in the ex-

periments. When the Thomas–Fermi approximation is

accurate, the chemical potential exceeds the typical level

spacing of the harmonic-oscillator trap, and the conden-

sate is larger than the ground-state wave function of the

harmonic oscillator would be. Ordinarily, the condensate

is also much larger than the healing length.

Small Excitations in a BEC

Linearizing the GPE. The time-dependent GPE is

nonlinear, but may be linearized around a stationary

solution. Consider the special case without a trapping

potential, V(x) ≡0. The stationary solutions are plane

waves,

φ(x) =

√

n e

i p·x

. (76.33)

This corresponds to a ﬂow of the gas at the velocity

v =

p/m and with a momentum p per atom. The

chemical potential of such a mode is

= 



, (76.34)

where



,

, c =

√

8πna

mξ

(76.35)

are the dispersion relation of free atoms, a peculiar ana-

log of the rest energy, and the speed of sound in the

BEC.

The ansatz for small deviations from the stationary

solution is written as

ψ(x, t) =

√

n e

i( p·x−



1 +u e

i(q·x−νt)

∗

−i(q·x−ν

∗



(76.36)

where q and ν are the momentum and energy associ-

ated with the excitation relative to the momentum and

energy of the original ﬂow. The GPE mixes the ﬁeld ψ

and its complex conjugate ψ

∗

,sothattwosmallam-

plitudes u and v are needed for the excitations. The

ansatz (76.36) is a solution to the time-dependent GPE

to the lowest nontrivial order in u and v if these am-

plitudes and the frequency of the excitation satisfy the

eigenvalue equation





+

+q ·v 



+

−q ·v





= ν



−v



(76.37)

The remaining problem is that the eigenvalue equa-

tion has two solutions for each q, which gives twice

as many small-excitation modes as there are degrees of

freedom. The extra modes are the penalty one pays for

the linearization of the GPE. The criterion |u|

−|v|

> 0

picks out the correct small-excitation modes. The cor-

responding dispersion relation for the excitations is

ν(q) = q·v +





(

+2

). (76.38)

For a stationary BEC with v = 0 and in the limit q → 0,

(76.38)givesν  cq. This conﬁrms the identiﬁcation

of c as the speed of sound.

In the BEC experiments the condensates are trapped,

but in principle the same analysis of small-excitation

modes may be carried out both numerically and in a myr-

iad of analytical approximations. The generic result is

that the trap frequencies lend their frequency scale to

small excitations. At low enough temperatures, excita-

tion frequencies calculated in this way agree well with

the experiments.

Within the mean-ﬁeld approximation small excita-

tions may be analyzed similarly in all boson systems, for

instance, in a multi-component Bose–Einstein conden-

sate or a joint atom–molecule condensate. The evolution

frequencies may be complex, which signals a dynam-

ical instability of the stationary conﬁguration; there are

small-excitation modes that grow exponentially. The in-

stability of a free gas with a negative scattering length,

which is apparent in (76.38)for

< 0, is a simple

example.

Bogoliubov Theory. Bogoliubov theory is the many-

body quantum version of the analysis of small

excitations. The idea is to treat the condensate mode ψ

containing n

atoms, separately in the ﬁeld operator

ψ =

√

+δ

ψ, (76.39)

expand the Hamiltonian in the lowest nontrivial (second)

order in the remnant quantum ﬁeld δ

ψ, and diagonalize.

The result is small-excitation modes with the annihi-

lation operators A

,wherek stands for the appropriate

quantum numbers. It turns out that the core mathematics

of Bogoliubov theory is the same as the mathemat-

ics of small excitations, but two features are added.

Part F 76.2

Quantum Degenerate Gases 76.2 Basic Properties of Degenerate Gases 1113

First, Bogoliubov theory explicitly shows that the co-

efﬁcients u and v in the analog of (76.37) need to satisfy

|u|

−|v|

= 1 to ensure boson commutators for the

operators A

. Second, with quantum ﬂuctuations, atom–

atom interactions force atoms out of the condensate even

at zero temperature. In a homogeneous (untrapped) con-

densate, in the limit na

 1, at T = 0, the fraction of

noncondensate atoms is

N −n



(76.40)

When the gas parameter na

is much smaller than

unity, at low enough temperatures most of the atoms are

in the condensate. Mean-ﬁeld theory and the GPE are

expected to apply, and empirically, they do.

Numerical Methods for GPE

Mathematical Properties of the GPE. Let us momen-

tarily assume that by separation of variables, or by some

ﬁat, the problem of solving the time-independent GPE

has been rendered one-dimensional. The Schrödinger

equation is linear and any constant multiple of a solution

is also a solution. One parameter, e.g., the logarith-

mic derivate of the wave function at a given point in

space, determines a stationary state completely. This

does not hold for the corresponding GPE, for which the

values of the wave function and its derivative can be

speciﬁed independently at (almost) every ﬁxed point in

space. As a result of the added ﬂexibility, and unlike

the Schrödinger equation, the GPE has bounded solu-

tions for continuous ranges of the values of the chemical

potential µ.

However, the time-independent GPE (76.28)comes

with the added normalization condition (76.26). Nor-

malization quantizes the values of µ for the bound states.

In practice one might, for instance, ﬁnd a solution that

satisﬁes the boundary conditions for a given µ with the

shooting method, then adjust the value of µ until nor-

malization holds. Techniques used in the ﬁrst numerical

analyses of the time-independent GPE in the context of

atom vapor condensates were variations of this theme.

Such schemes are not feasible in spatial dimensions

greater than one.

In general there is one solution to the time-

independent GPE that can be chosen to be positive

everywhere, the ground state with the lowest chemical

potential. Excited steady states exist, but only a few,

such as the ﬂowing states of (76.33) and vortices dis-

cussed in Sect. 76.4.1, have obvious physical meanings.

As the GPE is nonlinear, excited states are not the same

as small excitations.

Split-Step Fourier Method. The superposition principle

does not hold for the solutions of the time-dependent

GPE, and the excited states are usually not orthogonal

to one another in any useful sense. Methods based on

eigenstate expansions for solving the time-dependent

GPE are cumbersome at best. Instead, one often sim-

ply integrates the GPE as a partial differential equation

in time. A number of different methods are used, but

here we only discuss an elementary split-step Fourier

method [76.7]. This is an exceedingly popular algorithm

for parabolic equations, easy to implement, and with

minor modiﬁcations also solves the time-independent

GPE in any number of dimensions.

Thus, consider integration of (76.25) forward in time

over a step from t to t +∆t. For this purpose assume ﬁrst

that |ψ|

in the GPE were a constant equal to its value

at time t, then the evolution over the time step ∆t would

be given by

ψ(x, t +∆t) =exp



−i∆t



−

∇

+U(x)



× ψ(x, t),

(76.41)

where U(x) is a given function of position. In the al-

gorithm the exponential is ﬁrst split approximately, for

instance, as

exp



−i ∆t



−

∇

+U(x)



 exp



∆t

∇



exp



−i ∆tU(x)



×exp



∆t

∇



≡

T .

(76.42)

The exponential of the kinetic-energy operator is diago-

nal in the Fourier representation. Consequently, carrying

out the Fourier transform F and its inverse with the aid

of the Fast Fourier Transformation gives the split-step

algorithm

ψ(t +∆t) = F

−1

TF ψ(t)

(76.43)

with obvious efﬁcient implementations. The inaccurate

constant |ψ|

may be improved upon in a corrector

step in which the average of the initial wave func-

tion ψ(t) and the ψ(t +∆t) obtained in the ﬁrst pass

is used as |ψ|

,andstep(76.43) is taken again. This

split-step algorithm preserves the normalization of the

macroscopic wave function, and features a high-order

approximation to the exponential operator of the kinetic

energy.

Part F 76.2

1114 Part F Quantum Optics

Integration in Imaginary Time. The split-step algo-

rithm also provides a global method to ﬁnd the ground

state. To this end the time-dependent GPE is integrated

in imaginary time, i. e., replacing ∆t →−i∆t, starting

from a more or less random initial wave function and

normalizing after every step. If the GPE were linear, this

procedure would emphasize the lowest-energy compon-

ent of the wave function until it is the only one that

remains to within a prescribed accuracy. It is not clear

that the same should apply to the nonlinear GPE,but

often this is the case.

In nonlinear problems, split-operator methods, in

spite of their seeming simplicity, often exhibit spurious

behavior. Successful applications of these techniques

require skill and experience in the art of numerical

methods.

Local-Density Approximation

While an experimental BEC is usually trapped, it is often

much easier to study the theory for a formally inﬁnite ho-

mogeneous condensate. As long as the phenomena under

investigation involve length scales much smaller than the

size of the condensate and time scales much shorter than

the inverse trap frequencies, trapping cannot affect the

behavior of the gas locally. Under such conditions one

may analyze the gas at each position x as if it were ho-

mogeneous, and at the end of the calculations average

over the density distribution. The unit-normalized dis-

tribution of the density of the gas used in the averaging

P() =





 −n(x)



n(x) d



n(x) d

(76.44)

For instance,

P() =



−

5/2

H()H(n

−) (76.45)

holds for the Thomas–Fermi approximation with the

maximum density n

≡ n(0). The Heaviside step func-

tions H restrict the density to the correct range

0 ≤  ≤ n

. As an example, the average density in the

Thomas–Fermi model is



P() d =

. (76.46)

76.2.2 Meaning of Macroscopic

Wave Function

Here the macroscopic wave function ψ has been intro-

duced by replacing a boson ﬁeld theory with a classical

ﬁeld theory.

The intuitive interpretation is that, for interacting

particles, the atoms condense not to the ground state

of the conﬁning potential, but to the one-body state

whose wave function is the macroscopic wave func-

tion. This notion may be criticized on various grounds,

but in practice it makes a useful picture.

A precise formal meaning of the macroscopic wave

function, and of Bose–Einstein condensation for inter-

acting systems, is found by considering the one-particle

density matrix

ρ(x, x



) =



†

(x)

ψ(x



)



, (76.47)

which is sufﬁcient to determine the expectation value

of any one-particle operator. This is the position repre-

sentation of a positive Hermitian operator with the trace

equal to particle number (or its expectation value) N.In

this way, with an orthonormal set of functions {ψ

(x)}

and nonnegative eigenvalues n

, such that



= N , (76.48)

an expansion of the form

ρ(x, x



) =



(x)ψ



) (76.49)

exists. The system is a BEC if at least one eigen-

value n

is of the order of the number of particles and

does not formally go to zero in the thermodynamic

limit (if the limit exists and is sensible). The usual

case is that only one eigenstate, call it k = 0, has such

a large eigenvalue. The macroscopic wave function is

the corresponding eigenfunction ψ ≡ ψ

,andn

gives

the number of condensate atoms. If there is more than

one macroscopic eigenvalue, the condensate is called

fragmented.

Another interpretation of the macroscopic wave

function comes from statistical mechanics. In a continu-

ous (second-order) phase transition typically a symme-

try of the system is spontaneously broken. For example,

below the Curie temperature a single-domain ferro-

magnet magnetizes in some speciﬁc direction, and the

state has a lower symmetry than the rotationally in-

variant Hamiltonian of an isotropic ferromagnet. Any

quantity that appears in a continuous phase transition

and characterizes the breaking of the symmetry may

be called an order parameter. The macroscopic wave

function can be viewed as the order parameter asso-

ciated with spontaneous breaking of the global phase

or “gauge” symmetry of quantum mechanics. Specific-

ally, in quantum mechanics the state of the system is

Part F 76.2

Quantum Degenerate Gases 76.3 Experimental 1115

unchanged if the wave function is multiplied by an ar-

bitrary complex phase factor e

iϕ

. But to write the wave

function as ψ(x) already implies a preferred phase, and

likewise even if the wave function is adorned with ran-

dom but, for any given condensate, ﬁxed phase, as in

iϕ

ψ(x).

For one condensate the random phase is inconse-

quential. Suppose, however, that two BECs with the

wave functions e

iϕ

(x) and e

iϕ

(x) are combined.

If the macroscopic wave functions behave as wave

functions should, the combination of two condensates

displays the density n(x) =|e

iϕ

(x) + e

iϕ

(x)|

There should be an interference pattern between the

condensates. Two BECs indeed produce an interference

pattern when they are combined, although the random-

ness or the absence thereof of the phases ϕ

1,2

is difﬁcult

to verify experimentally.

From the quantum optics viewpoint, a condensate is

a given number of atoms in a given one-particle state,

a number state, and cannot possess any phase at all. This

seems to contradict the observations of an interference

pattern. The resolution is that the process of measure-

ment in itself produces a phase difference between the

condensates even if there initially is none.

76.2.3 Fermions

Static Fermi Gas

Thomas–Fermi Approximation. Consider an ideal

single-species Fermi gas of trapped atoms. The origi-

nal Thomas–Fermi approximation (see Chapt. 20)was

formulated for fermions, namely, electrons, and in the

present case it is modiﬁed as follows. For the atom

density n(x) at position x, at a low temperature, the

corresponding local internal chemical potential is ap-

proximated according to (76.18)as



(x) =



6π

n(x)



2/3

(76.50)

Given the trapping potential V(x), the density of the gas

adjusts in such a way that the sum of the external poten-

tial energy and the local internal chemical potential, the

Fermi energy, is a constant across the gas,

V(x) +



6π

n(x)



2/3

= µ,

(76.51)

the global chemical potential. One may solve the density

for a given chemical potential as

n(x) =











√

2 m



µ −V(x)



3 π

2 3

, V(x)<µ;

0, otherwise.

(76.52)

Finally, the integral of the density over all space should

equal the atom number, which gives an equation to

determine the chemical potential µ.

For a harmonic trap this program can be carried out

in an exact, analytical manner with the result that

µ = 6

1/3

ω, R = 2

2/3

1/3

1/6

,

n(0) =

√

3π



. (76.53)

The quantities

ω, ,andR have the same meaning as

in the BEC case. The Thomas–Fermi approximation for

fermions should be applicable whenever N  1.

In a one-component Fermi gas at low tempera-

ture atom–atom interactions are typically negligible for

a multitude of reasons. There is no s-wave scattering,

and the presence of the Fermi sea tends to suppress re-

pulsive interactions. However, in the case of attractive

interactions between two species, the Fermi sea may be

thermodynamically unstable; the energy may be low-

ered by pairing fermions into Cooper pairs. This is the

mechanism behind the BCS theory of superconductiv-

ity [76.8].

Excitations in a Fermi Gas

If the interactions do not render a fermion system into

a superﬂuid, see Sect. 76.5.1, the elementary excitations

of a degenerate Fermi gas with short-range interactions

are basically atom–hole pairs. What happens in the con-

trary case for trapped and strongly interacting atoms is

presently an active area of research.

76.3 Experimental

76.3.1 Preparing a BEC

In a trapped gas the density n(x) is self-determined from

the atom number N, and the condition for a BEC in an

ideal gas is most readily expressed in terms of the total

number of atoms as

= 0.94

ωN

1/3

. (76.54)

Part F 76.3

1116 Part F Quantum Optics

In practice, at the bottom of the trap the conditions on

temperature and density for a BEC are similar to the con-

ditions for a BEC in a free gas. In the thermodynamic

limit, such that

ω → 0, N →∞with

ωN

1/3

held con-

stant, below the critical temperature T

the fraction of

condensate atoms behaves as a function of temperature T

= 1 −





. (76.55)

The experimental realizations of alkali vapor con-

densates are based on techniques of laser cooling and

trapping of atoms. The following discussion relies heav-

ily on material from Chapt. 75.

A BEC in a dilute atomic gas is usually prepared

using a two-stage process. First, a magneto-optical trap

is used to capture a sample of cold atoms and to cool it to

a temperature of the order of a few tens of microkelvin.

The atoms are then transferred to a magnetic trap for

evaporative cooling that leads to condensation.

A magnetic trap is based on a combination of two

ideas. First, if an atom that starts out with its magnetic

moment antiparallel to the magnetic ﬁeld moves slowly

enough in a position dependent magnetic ﬁeld, its mag-

netic moment remains adiabatically locked antiparallel

to the magnetic ﬁeld. The energy of the atom is then

a minimum where the magnetic ﬁeld is a minimum.

Second, the absolute value of the magnetic ﬁeld may

have a minimum in free space. The minimum is then

a trap for atoms whose magnetic moments are suitably

oriented. The downside is that only atoms in the right

magnetic (Zeeman) states are trapped. While the atoms

cool down, they accumulate at the center of the trap.

The center should not be a zero of the magnetic ﬁeld,

because at zero ﬁeld an atom would lose the lock be-

tween the directions of the magnetic moment and the

magnetic ﬁeld necessary for trapping.

A time orbiting potential (TOP) trap starts with the

same kind of magnetic ﬁeld that is used in a magneto-

optical trap. A time-dependent magnetic ﬁeld is then

added in such a way that the zero of the magnetic ﬁeld

orbits around the center of the trap. If the frequency at

which the zero orbits is high enough so that the atoms

cannot follow, they see an effective potential with a min-

imum at the center of the trap and do not sample the

zero. Alternatively, it is possible to wind a coil in such

a way that it makes a magnetic ﬁeld whose absolute

value has a minimum that is not zero. In this type of

a Ioffe–Pritchard trap the winding of the wire resembles

the seams on a US baseball.

The basic idea of evaporative cooling is that the most

energetic atoms escape from the trap, then the remaining

atoms thermalize to a lower temperature. Some atoms

are lost in the process, but with the decreasing tem-

perature the density at the trap center nonetheless tends

to increase and the phase space density increases even

more due to the cooling.

The cooling is usually forced by an rf drive. The tran-

sition frequency between the Zeeman states depends on

the magnetic ﬁeld, and increases toward the edges of

the trap. Atoms are removed where the rf frequency is

on resonance and drives transitions to untrapped Zee-

man states. Thus, while the atoms cool and concentrate

at the center, the radio frequency is swept down in such

a way that the “rf knife” removing the atoms slides

in from the edge of the trap. At some radio frequency

a condensate abruptly emerges. The temperature can be

further lowered by continuing evaporative cooling, al-

beit at the expense of loss of atoms. As a rule of thumb,

an atom needs to experience a hundred collisions before

condensation occurs, and a typical time needed to pre-

pare a condensate is a few seconds. In a good vacuum

a condensate may live for tens of seconds.

It is also possible to condense atoms trapped in a far-

off resonant optical trap based on the dipole forces of

light, instead of the magnetic trap [76.9]. For tuning

below the resonance, atoms are strong-ﬁeld seekers.

A focused laser beam is a three-dimensional trap for

atoms, as is an arrangement with two crossed beams

focused to the same spot. Furthermore, with extreme

off-resonant light from a CO

or a Nd-YAG laser, ab-

sorption of photons and the associated photon recoil

kicks and heating may be negligible.

An optical trap may also be added after a BEC is

prepared in a magnetic trap. The advantage is that an op-

tical trap will hold the atoms regardless of their magnetic

state, so that multicomponent “spinor” condensates may

be studied. Moreover, while an adiabatic change of the

strength of a trap cannot change the phase space dens-

ity, the phase space density may be altered by changing

the shape of the trap by adding a tight optical trap to

the bottom of a much wider magnetic trap. Reversible

condensation inside an added optical subtrap based on

such an increase in the phase space density has been

demonstrated.

Methods to condense atoms that might be suited

for future technological applications are being pursued.

For instance, by lithographic techniques it is possible

to put conducting wires on a substrate to make an

atom chip. With currents ﬂowing, the wires produce

magnetic ﬁelds that guide the atoms. Condensation

Part F 76.3

Quantum Degenerate Gases 76.4 BEC Superﬂuid 1117

in such a conﬁguration has been reported [76.10].

Two-dimensional condensation in what is known as

a gravito-optical surface trap has also been achieved

experimentally [76.11].

There is an analogy between a condensate and

a beam of light from a laser that we rely on extensively.

However, the analogy is only partial. By dropping a con-

densate under gravity one makes a pulsed atom laser,

and by coupling a trapped Zeeman state to an untrapped

state by rf excitation it is possible to make a conden-

sate leak slowly out of the trap. Nonetheless, at this

time a method to produce a continuous beam of conden-

sate atoms, a continuous-wave atom laser, is yet to be

demonstrated.

76.3.2 Preparing a Degenerate Fermi Gas

A single-species, very-low temperature Fermi gas is an

uninteresting system, as the Fermi–Dirac statistics for-

bids s-wave interactions between the atoms and the gas

is nearly ideal. In experiments the gas usually has two

species, different states of the same atom. The interac-

tions between the species are comparable in strength to

the interactions between bosonic atoms.

Evaporative cooling works in a two-species gas, and

can be used to prepare a degenerate Fermi gas either

in a magnetic trap [76.12] or in an optical trap [76.13].

Second, one can use a gas of bosons, and indeed a BEC,

as a refrigerator [76.14]. At this writing the lowest tem-

peratures are of the order of 0.1 T

. Reaching lower

temperatures is complicated by various factors, such as

the very low heat capacity of a BEC and collisions be-

coming inefﬁcient at low temperature because the inert

Fermi sea reduces the available phase space. Nonethe-

less, lower temperatures seem to be mainly a matter of

advances in technology.

76.3.3 Monitoring Degenerate Gases

Orders of Magnitude. As a rule of thumb, the trapping

frequencies in a magnetic trap are

ω ≈ 2π × 10 Hz, while

the frequencies in an optical trap may reach into the kHz

regime. A typical oscillator length is  ≈1 µm. A usual

number of atoms is N ≈ 10

. Scattering lengths are of

the order of a ≈10 nm. The size of a degenerate gas

is in the neighborhood of R ≈0.1 mm, the maximum

density is about n

≈10

−3

,andtheBEC transition

temperature and the Fermi temperature are of the order

of T

≈ T

≈ 1 µK. However, much lower temperatures

are readily reached in a BEC.

Phase Contrast Imaging. It is possible to monitor con-

densate features substantially larger than the wavelength

of the light used in the measurements nondestructively,

in situ, by using phase contrast imaging. In this method

the light is detuned far off resonance so that absorptions

with the accompanying photon recoil kicks on the atoms

are rare, but the phase of the light nonetheless changes

upon propagation through the sample. The phase change

may be detected by interfering the transmitted light with

the original light, with the phase of the latter suitably

shifted.

Time-of-Flight Imaging. Usually, though, the obser-

vation of a degenerate gas at the end of an experiment is

by time-of-ﬂight imaging. The trap is suddenly removed,

whereupon the gas expands freely. After the atom cloud

has grown to a size large enough compared to the wave-

length of the resonant light used to monitor the gas, an

absorption image of the cloud is taken. This gives the

projection of the density of the gas onto a plane per-

pendicular to the direction of propagation of the light.

Except for the effects of atom–atom interactions, after

a sufﬁciently long time of free ﬂight the density reﬂects

the initial momentum distribution of the atoms. Time-

of-ﬂight images bear the signs of both condensation in

a Bose gas and quantum degeneracy in a Fermi gas. Non-

trivially, other features of interest such as vortex cores

are also preserved and can be detected after the free ex-

pansion. The downside is that the time-of-ﬂight method

is destructive. After each snapshot the sample will have

to be prepared again.

76.4 BEC Superﬂuid

76.4.1 Vortices

Flow Velocity in a Superﬂuid. By manipulating the

Heisenberg equations of motion for a Bose ﬁeld under

the Hamiltonian (76.22) it is easy to derive the equation

of continuity for the atoms,

∂

∂t

n+∇·

J = 0 ,

(76.56)

n =

†

ψ,

J = i



ψ∇ψ

†

−ψ

†

∇ψ



, (76.57)

Part F 76.4

1118 Part F Quantum Optics

which identiﬁes

n and

J as the operators for atom density

and atom current density. The corresponding mean-ﬁeld

quantities are obtained when again the boson ﬁelds are

replaced with the corresponding classical ﬁelds. Writing

the classical ﬁeld in terms of the density n(x, t) and phase

ϕ(x, t) in the form

ψ =

√

n e

iϕ

, (76.58)

the local ﬂow velocity v = j/n becomes

v =

∇ϕ.

(76.59)

The velocity ﬁeld is irrotational.

Quantization of Circulation. Integration of the ﬂow

velocity around an arbitrary loop gives



dl·v =

∆ϕ = p

2π

p = 0, ±1, ±2,... ,

(76.60)

since the change of the phase ∆ϕ around a closed loop

must be an integer multiple of 2π. Equation (76.60)

expresses the quantization of circulation in a superﬂuid.

A medium described by a macroscopic wave function,

such as a BEC, cannot sustain arbitrary ﬂow velocities.

Vortices. As an example, in bulk rotation at an angu-

lar velocity Ω the line integral around a loop at the

distance r from the axis of rotation would be 2πr

Ω,

which is not permitted for an arbitrary r. Instead, upon

an attempt to make a BEC rotate, the angular velocity

will be carried by vortex lines. These are lines through

the condensate, entering and exiting at the surface, such

that each vortex carries one quantum of circulation. At

the core of a vortex the ﬂow velocity should be inﬁnite

to sustain a ﬁnite circulation, which is physically impos-

sible. Nature solves this problem by making the vortex

core normal (not BEC), so that the macroscopic wave

function does not apply. The diameter of the vortex core

is of the order of the healing length ξ,givenby(76.29).

When the trapping potential on the atoms is rotated,

it is convenient to study the physics in the co-rotating

frame. Given a frame rotating at the angular velocity Ω

and the angular momentum operator L = x× p per par-

ticle, transformation to the rotating frame adds the

one-particle term

=−Ω · L (76.61)

to the Hamiltonian. Any particular conﬁguration of vor-

tices is a thermodynamically stable equilibrium if it is

the minimum of energy in the co-rotating frame. For

a trapped condensate, at zero rotation velocity the state

without vortices is the energy minimum, and increas-

ing the rotation speed makes states with an increasing

number of vortices the stable conﬁguration. However,

a vortex conﬁguration may be metastable and live for

a long time even if it is not the minimum of energy.

Conversely, even the energy-minimum conﬁguration of

vortices must ﬁrst be nucleated. Since the circulation

can only have quantized values, it cannot change in

a continuous process. It takes a zero condensate density

somewhere to create or destroy a vortex.

These alternatives provide a large number of experi-

mental scenarios involving rotation of the trap or stirring

of the condensate, condensation of a rotating normal gas

by taking it across the transition temperature, and so

forth. For instance, when a trap containing a BEC is ro-

tated, vortices are generated at the surface where they

start their lives as dynamical instabilities. The vortices

then drift in and form a regular vortex array [76.15].

When the rotation is halted, the vortices drift out to the

surface and disappear.

76.4.2 Superﬂuidity

A BEC also has the remarkable property that it may sus-

tain persistent currents that are completely immune to

viscosity. The qualitative reason may be seen from the

dispersion relation of small excitations (76.38). As long

as the ﬂow speed |v| is less than the speed of sound c,all

excitation energies are positive, so that the ﬂowing state

is the state of lowest energy and is thermodynamically

stable. On the other hand, when the ﬂow velocity ex-

ceeds the speed of sound, the system has excitations that

lower the energy, ν(q)<0forsomeq. The ﬂowing state

is then not a minimum of energy. The ﬂow is not ther-

modynamically stable, and it decays when it interacts

with an environment by sending off small excitations.

The speed of sound gives the Landau critical velocity

for superﬂuidity.

The critical velocity c tends to zero when the

atom–atom interactions vanish with a → 0. While the

condensate wave function may be written down whether

the atoms interact or not, superﬂuidity and persistent

ﬂows rely on the interactions. The same applies to vor-

tices, as in the limit of a noninteracting gas the healing

length and the radius of the vortex core tend to inﬁnity.

The conventional picture is that superﬂuid ﬂow in

an inhomogeneous medium is unstable if the local ﬂow

velocity exceeds the local (density dependent) speed of

sound. In practice, in liquid He experiments and nu-

merical simulations of dilute condensates the current

Part F 76.4