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

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Cooling and Trapping 75.4 Trapping and Cooling of Charged Particles 1099

zero at x =0. Suppose that the gradient of Bis chosen in

such a way that the m = 1(m =−1) magnetic substate

of the excited state has the higher (lower) energy for

x > 0, and that the atom is illuminated by σ

polarized

beams propagating in the ±x-directions, tuned below

resonance. When the atom is displaced from x =0inei-

ther direction, it is closer to resonance with the beam that

pushes it back toward x = 0. This makes the restoring

force responsible for trapping.

A magneto-optical trap can be set up also in three

dimensions. A quadrupole magnetic ﬁeld of the form

B(r) 

∂B

∂z



r=0



−



(75.48)

is produced by reversing the direction of current in one of

the two Helmholtz coils. Three orthogonal pairs of light

beams, each in the σ

–σ

−

conﬁguration, complete the

trap. The magnetic ﬁeld is sourceless. To compensate

for the ensuing signs of the ﬁeld gradients, one of the

–σ

−

corkscrews has the opposite handedness from

the other two.

The mechanism of the magneto-optical trap for the

J = 0 →1 conﬁguration is the same as the mechanism

for Doppler cooling, except that position dependent level

shifts of the excited states take the place of velocity

dependent Doppler shifts. The restoring force and the

damping coefﬁcient of Doppler cooling are closely re-

lated. For the coordinate directions u = x, y, z the relation

=−κ

u,κ

= β



∂B

∂u



(75.49)

where g

is the Landé factor for the excited state.

A magneto-optical trap may similarly be based on

the induced orientation mechanism of polarization gra-

dient cooling. In that case, the Landé factor of the

ground state, g

, should probably be used in (75.49).

This may be the true mechanism of most magneto-

optical traps, but insufﬁcient quantitative understanding

precludes ﬁrm conclusions.

Atoms in a well-aligned magneto-optical trap reside

near the zero of B, so that the magnetic ﬁeld has little

effect on polarization gradient cooling. Trapping and

cooling are achieved simultaneously.

75.3 Magnetic Trap for Atoms

The magnitude B(r) of a magnetic ﬁeld may have a min-

imum in free space, as in (75.48). A particle with

a magnetic dipole moment µ then experiences a trap-

ping potential U(r) = µB(r) if µ and B are antiparallel.

µ remains locked antiparallel to B if the ﬁeld seen by

the moving dipole satisﬁes the adiabatic condition





µB

(75.50)

(Section 73.3.4). However, if B(r) =0 at the minimum,

the adiabatic condition is violated and the dipole may

ﬂip (Majorana transition). The particle may end up in

a repulsive potential, and get expelled from the trap.

This becomes a problem at low temperatures, when the

particles accumulate near the minimum of the poten-

tial. Trap conﬁgurations are therefore designed in which

B(r) = 0 at the minimum.

Evaporative Cooling.

A magnetic trap is often combined with evaporative

cooling. The most energetic atoms from the tail of the

thermal distribution escape from the trap, whereupon the

average energy of the remaining atoms decreases. Suc-

cessful operation of evaporative cooling requires a high

enough rate of elastic collisions so that the atoms ther-

malize in a time short compared with the lifetime of the

sample. In order to sustain the rate of evaporation, the

effective depth of the trap is lowered as the atoms cool.

75.4 Trapping and Cooling of Charged Particles

Since the potential Φ(r) of a static electric ﬁeld satisﬁes

Laplace’s equation, Φ(r) cannot have an extremum in

free space. A static electric ﬁeld therefore cannot serve

as an ion trap (Earnshaw’s theorem). Paul and Penning

traps circumvent this limitation by making use of an al-

ternating electric ﬁeld and a magnetic ﬁeld, respectively.

Cooling is often employed to assist trapping.

75.4.1 Paul Trap

Trapping

Conﬁguration. Consider an ideal trap whose sur-

faces are hyperboloids of revolution; see Fig. 75.2.

The two “endcaps” and the intervening “ring”

are equipotential surfaces of the quasistatic electric

Part F 75.4

1100 Part F Quantum Optics

U–Vcos ω

⬇

Fig. 75.2 Electrode conﬁguration and voltages of an ideal

hyperboloid Paul trap

potential

Φ(x, y, z) =

(t)



−x

−y



2

, (75.51)

where 

is the distance from the center to the ring,

= 

√

2 is the distance to the endcaps, and Φ

(t) is

a voltage applied between the endcaps and the ring,

(t) =U −V cos

ωt . (75.52)

Motion of an Ion. In the ideal three-dimensional Paul

trap, Newton’s equations of motion for the coordinates

u = x, y or z may be recast as Mathieu’s equations,

dτ

+(a

−2q

cos 2τ)u = 0 , (75.53)

where τ =

ωt/2 is a dimensionless quantity proportional

to time, the parameters are

=−2a

x,y

=−

8qU



, (75.54)

=−2q

x,y

=−

4qV



, (75.55)

and m and q are the mass and charge of the particle.

Stable trapping ensues when a

and q

are such that the

motion of the ion is stable in all directions. A Paul trap

normally operates in the ﬁrst stability region of (75.53).

Stable motion may be qualitatively divided into

forced micromotion at the frequency

ω of the external

drive, and into slower secular motion of the center of the

micromotion. If U = 0, the secular motion takes place

in an effective ponderomotive potential U

, Sect. 74.2,

equal to the cycle-averaged kinetic energy in the micro-

motion. Explicitly,

(r) =

(r)



+4z





(75.56)

where E (r) is the ac ﬁeld amplitude. This is an

anisotropic harmonic oscillator potential characterized

by the oscillation frequencies

= 2ν

x,y

√

2qV

ω

. (75.57)

Quantization of C.M. Motion. The separation of micro-

motion and secular motion is excellent, and the trap

is stable, when ν

x,y,z



ω. Ignoring the micromotion,

the cm motion of the ions in the potential U

(r) may

be quantized readily. The energy of a state with n

x,y,z

quanta in the coordinate directions x, y, z is

E =



i=x,y,z





(75.58)

Variations of Paul Trap. Little practical advantage usu-

ally arises from a realization of the ideal shape. Even

a single electrode with an applied ac voltage may work as

a Paul trap. A linear trap is basically a two-dimensional

Paul trap with an added static longitudinal potential to

prevent escape of the ions from the ends of the trap.

A closed race track Paul trap is obtained by bending

a linear trap into a ring.

Cooling

Laser Cooling in One Dimension. The secular motion

of an ion may be cooled using lasers. Consider the

motion of the ion in one of the principal-axis direc-

tions x, y, z, with ν denoting the corresponding cm

frequency. In the common case where γ  ν, Doppler

cooling works basically as with a free atom. In the con-

trary case, ν  γ , cooling may be achieved by tuning

the laser to ω = ω

−ν. Resonant photoabsorption start-

ing with n cm quanta decreases the quantum number

from n to n −1, and subsequent spontaneous emission

on the average leaves the cm energy nearly untouched.

The net effect is reduction of the cm energy by

in such a Raman process. Since the oscillating ion

sees a frequency-modulated laser with sidebands, this

method of ion cooling is called sideband cooling.

For one-dimensional motion of a two-state ion in

a traveling light wave at low I, the velocity damping

Part F 75.4

Cooling and Trapping 75.4 Trapping and Cooling of Charged Particles 1101

rate is

β =

2Ω

γ∆



(∆ +ν)

+γ



(∆ −ν)

+γ



, (75.59)

and the expectation value of the cm energy is

E =

4∆



∆

+γ

+ν

+ α



(∆ −ν)

+γ



(∆ +ν)

+γ



∆

+γ



(75.60)

where α characterizes the angular distribution of sponta-

neous emission, see (75.29). The result (75.60)isuseful

when either ε

 γ or ε

 ν. The limit ν  γ is for

Doppler cooling; the temperature from (75.60) coin-

cides with (75.33)forafreeatom.Thecasewithboth

ν γ and ε

ν corresponds to sideband cooling in the

Lamb–Dicke regime, in which the cooled ion is conﬁned

to a region much smaller than λ.

In connection with sideband cooling, it is convenient

to cite the expectation number of harmonic oscillator

quanta n instead of energy or temperature; the latter

are

E =



n+



, T =





1 +n

−1



−1

(75.61)

For optimal sideband cooling, ν  γ and ∆ =ν,the

result is

n=

(1 +4α)













(75.62)

In principle, by decreasing the linewidth γ ,theionmay

be put arbitrarily close to the ground state of the cm har-

monic oscillator. Such a decrease is not practical in real

two-state systems, but is routinely achieved by using the

two-photon resonance in a three-state Λ conﬁguration

as an effective two-state system; see Sect. 73.6.2.

Laser Cooling in Three Dimensions. Either by design

or chance, no two of the ν

are precisely degenerate. If

the damping rate β and the trap frequencies ν

satisfy

 |ν

−ν

|, i = j , (75.63)

the motion of the ion in each principal axis direc-

tion of the trap is cooled independently of the other

directions. For γ  ν

x,y,z

, a single laser beam propagat-

ing approximately in the direction



√



(

)

sufﬁces to cool all components of the secular motion.

Energy in Micromotion. Possibly with the aid of com-

pensating static electric ﬁelds, one cooled ion may be

conﬁned near the zero of the trapping ac electric ﬁelds.

Then the energy in the micromotion is comparable to

the energy in the secular motion.

75.4.2 Penning Trap

Trapping

Conﬁguration. In the Penning trap, a dc voltage U is

applied between the endcaps and the ring, and a constant

magnetic ﬁeld B in the direction of the trap axis z is

added. The magnetic ﬁeld forces an ion escaping toward

theringtoturnback.

Motion of an Ion. The motion of an ion is a super-

position of three periodic components. For the same

ideal hyperboloid shape that was discussed with the Paul

trap, (75.51) and Fig. 75.2, the three components are

completely decoupled. Firstly, in the axial direction, the

ion executes oscillations at the axial frequency



2qU

m



1/2

. (75.64)

Secondly, the ion undergoes cyclotron motion in the

plane perpendicular to the trap axis. As a result of the

electric ﬁeld, the frequency of the cyclotron motion





−



1/2

(75.65)

is displaced from the cyclotron frequency ν

= qB/m

of a free ion. Thirdly, the guiding center of cyclotron

motion rotates about the trap axis at the magnetron

frequency

−



−



1/2

. (75.66)

The frequencies typically satisfy

 ν



. (75.67)

Magnetron motion has unusual properties. It takes up

the majority of the electrostatic energy in the transverse

directions, which in the absence of the magnetic ﬁeld

would lead to expulsion of the ion. Relative to a station-

ary ion at the trap center, the energy of the magnetron

motion is bounded from above by zero. The radius, as

well as velocity and kinetic energy of magnetron mo-

tion, decreases with increasing total energy. The energy

for a state with n

, n

and n

quanta in the cyclotron,

Part F 75.4

1102 Part F Quantum Optics

axial and magnetron motions is therefore

E =











−





(75.68)

Cooling

Laser Cooling. For ions in a practical Penning trap, the

frequencies ν



, ν

and ν

are <γ.Ifk is not orth-

ogonal to either the cyclotron motion or the axial motion,

Doppler cooling proceeds essentially as for a free atom.

However, energy should be added to the magnetron mo-

tion in order to reduce the magnetron radius and velocity.

The solution is to aim a ﬁnite-size laser beam off the

center of the trap in such a way that an ion experiences

a higher (lower) intensity over the part of its magnetron

orbit in which it travels in the direction of (opposite

to) the laser beam. With a proper choice of the param-

eters, the ensuing addition of energy overcomes Doppler

cooling of the magnetron motion.

Other Cooling Methods. Precision measurements are

carried out in Penning traps with objects that do not have

an internal level structure suitable for laser cooling; and

thus, other cooling methods are used.

For light particles such as electrons, characteristic

times of radiative damping of the cyclotron motion are

in the subsecond regime, and hence, so are the equi-

libration times with blackbody radiation. Cooling is

accomplished by enclosing the trap in a low-temperature

(e.g. liquid helium) environment. For protons and heav-

ier particles, the equilibration times of the cyclotron

motion with the environment are impracticably long,

and the same applies to the axial and magnetron motions

even for electrons.

A workable cooling scheme for the axial motion is

based on the charges that the oscillating particle induces

on the endcaps. The charges generate currents in an

external circuit connecting the endcaps. The endcaps

are thus coupled to a cooled resonant circuit tuned to

the axial frequency, and axial motion relaxes to thermal

equilibrium with the resonant circuit. A variant of this

resistive cooling , in which the ring is split into elec-

trically insulated segments, is used to cool the cyclotron

motion of protons and heavier ions.

Magnetron motion of an electron or proton is cooled

by sideband cooling. An electric ﬁeld with compon-

ents in both the z-direction and xy-plane, and tuned

to ω = ν

+ν

, drives transitions which may either in-

crease or decrease the number of quanta in each mode.

However, the matrix elements favor transitions with

∆n

= 1and∆n

=−1. Pumping of the axial motion

is canceled by axial cooling, while an equilibrium with

low kinetic energy ensues for the magnetron motion.

Ideally, the ratio of kinetic energies becomes

kin,m

kin,z

. (75.69)

75.4.3 Collective Effects in Ion Clouds

As soon as there is more than one ion in the trap,

Coulomb interactions between the ions profoundly

shape the physics [75.15, 16].

Ion Crystal

In the standard Paul trap radio frequency heating due

(presumably) to transfer of energy from micromotion

to secular motion limits the number of ions that can

be cooled efﬁciently by a laser. Nevertheless, at a low

temperature, the ions settle to equilibrium positions cor-

responding to a minimum of the joint trapping and

Coulomb potentials, and form a “crystal”. Depend-

ing on the trap parameters, the ions may also execute

quasiperiodic or chaotic collective motion, or move

nearly independently of one another. Changes between

crystalline and liquid forms of the ion cloud resembling

phase transitions are observed.

Strongly Coupled Plasma

Cooling of a large number of ions is possible in a Pen-

ning trap. However, magnetron motion becomes uniform

rotation of the entire cloud, and Coulomb interactions

set a lower limit on the attainable radius of the cloud.

This leads to a lower limit on the kinetic energy and

second-order Doppler shift.

In a co-rotating frame, the ions behave like a one-

component plasma on a neutralizing background. The

characteristic parameter for a one-component plasma

with charge per particle q and density n is



4πn



1/3

4π

(75.70)

essentially the ratio of the Coulomb energy between two

nearest-neighbor ions divided by the thermal kinetic en-

ergy. Γ

> 1 indicates a strongly coupled plasma; for

> 2andΓ

> 170 solid and liquid phases are ex-

pected in an inﬁnite plasma. Experiments with a Penning

trap have produced Γ

 300. Concentric shells of ions

or various more or less crystalline arrangements are seen

depending on the experimental conditions.

Part F 75.4

Cooling and Trapping 75.5 Applications of Cooling and Trapping 1103

Sympathetic Cooling

In a trap that holds two or more species of charged

particles, cooling of the motion of one species is trans-

ferred by Coulomb interactions to the other species. This

sympathetic cooling broadens the scope of ion cooling

methods.

75.5 Applications of Cooling and Trapping

Trapping and cooling offer increased interaction times

between the atoms/ions and the light. This leads

to reduced transit time broadening, and indeed to

macroscopic (> 1 s) interaction times. Laser cooling in

a magneto-optical trap routinely gives temperatures so

low that the Doppler width is below the natural linewidth

of the cooling transition. A homogeneously broadened

atomic sample is thus prepared. Cooling also enables

reduction of the second-order Doppler effect.

Various frequency measurements are the prime ben-

eﬁciary of cooling and trapping. Potential applications

range from detection of the change of natural constants

in time Chapt. 30 to such feats of technology as the

Global Positioning System.

75.5.1 Neutral Atoms

Experimental Considerations

Originally the experiments often started with a longi-

tudinal deceleration and cooling of an atomic beam by

a counterpropagating laser beam. To compensate for the

change of the Doppler shift of the atoms while they

slowed down, the position dependent magnetic ﬁeld of

a tapered solenoid shifted the transition frequency of the

atoms to keep them near resonance while they moved

down the solenoid. A magneto-optical trap then scooped

some atoms, cooled them further, and captured them.

Nowadays this Zeeman slower ist mostly supplanted

by various schemes in which a MOT directly captures

atoms from the low-velocity wing of the thermal dis-

tribution. Depths of neutral-atom traps are below 1 K.

Storage times are typically of the order of 1 s, limited at

high densities by exothermal binary collisions and at low

densities by collisions with the atoms in the background

gas.

The temperature of cooled atoms may be measured

by the time-of-ﬂight method. All cooling and trapping

ﬁelds are turned off, whereupon the atoms fall freely

under gravity. The distribution of arrival times of atoms

at a probe laser beam underneath the initial molasses is

compared with a numerical model for the disintegration

of the molasses. A ﬁt gives the temperature.

The focus is on Li, Rb and Cs, to a large extent

because the required laser frequencies can be generated

using inexpensive diode lasers. Hyperﬁne structure of

the ground state of the alkalis complicates experiments

because the atoms may end up in an inert hyperﬁne

level outside the active cooling/trapping transitions. To

counteract this, a second appropriately tuned repump

laser is added to return such atoms to circulation. A few

experiments use lanthanide atoms or metastable states

of rare-gas atoms, some isotopes of which do not have

hyperﬁne structure.

Cold Collisions

In the molecular picture of a collision involving a laser,

optical excitation takes place at the interatomic dis-

tance for which the difference between the potential

curves of the incoming and excited states equals the

energy of a laser photon [75.17]. The end products of

an inelastic collision (ﬁne or hyperﬁne structure chang-

ing collision, associative ionization, radiative escape,

etc.) are normally determined at shorter interatomic dis-

tances, when the potential curve of the excited state has

an anti-crossing with the potential curve of the product

channel. The novel feature of ultracold collisions is the

long duration due to the low velocity of the collision

partners. Spontaneous emissions and other phenomena

irrelevant at room temperature may take place during

the collision.

On the scale of typical resonance widths, at very

low temperatures the collision partners are in effect

in a single continuum state with zero energy. This fa-

cilitates photoassociation spectroscopy. Laser-induced

transitions from the initial continuum state to bound

vibrational states of the molecule are observed.

Collisions are of practical importance in that they

limit the achievable atom density in a trap: an inelastic

collision may release more kinetic energy than the trap

can contain, which results in a loss of an atom (or two

atoms) from the trap. Collision rates are actually meas-

ured by monitoring the loss rate of atoms as a function

of density. Precise energies of the molecular vibrational

states from photoassociation spectroscopy are used as

input to determine [75.18] s-wave scattering lengths

for atoms and to measure molecular parameters to an

accuracy that far exceeds the capabilities of ab-initio

calculations.

Part F 75.5

1104 Part F Quantum Optics

Frequency Standards

An atomic fountain starts with an optical molasses or

a magneto-optical trap. The laser beams are then manip-

ulated to give an upward push to the atoms. The atoms

ﬂy up against gravity for a few tens of centimeters, then

turn back and, because of the initial transverse veloci-

ties, fan out to a “fountain”. In a fountain clock [75.19],

the fountain erupts through a microwave cavity that

drives a hyperﬁne transition in the atoms. The clock

is in effect an accurate measurement of the transition

frequency. The fountain is beneﬁcial because the inter-

rogation times ≈1 s are longer and the atomic velocities

≈ 1m/s slower than in traditional beam clocks.

Bose–Einstein Condensate

At present, the most prominent basic-physics applica-

tions of cooling and trapping of atoms undoubtedly are

in studies of Bose–Einstein condensation and quantum

degenerate Fermi gases in dilute atomic vapors. This

topic is covered in Chapt. 76.

75.5.2 Trapped Particles

Experimental Considerations

Both Paul and Penning traps behave like a conserva-

tive potential, and scatter rather than conﬁne a particle

coming from the outside. One method to load a trap is

to generate the ions in situ, e.g., by letting a beam of

atoms and electrons collide inside the trap. Time depen-

dent electric potentials are another loading method. The

trapped species is injected thorough a hole in the endcap,

and the opposing endcap is raised to an electric potential

that makes the entering particles stop. The potential is

then lowered before it ejects the particles. A single elec-

tron, positron, proton, antiproton or ion may be loaded.

Typical depths of ion traps are ≈1eVor≈ 10

K. With

the aid of cooling, the storage time may be made inﬁnite

for all practical purposes.

Trap frequencies are measured by observing the res-

onance excited by added ac ﬁelds. For instance, an

electric ﬁeld near the axial cm resonance frequency may

be coupled between the ring and one endcap. A reso-

nance circuit coupled between the ring and the other

endcap is used to detect the resonance. Alternatively,

ejection of the driven ions is monitored.

For an electron in a Penning trap, the cyclotron fre-

quency is in the extreme microwave region. Detection

of the resonance is achieved indirectly. The uniform

magnetic ﬁeld is perturbed with a piece of a ferro-

magnet to make a magnetic bottle. The axial motion

and the cyclotron motion are then coupled. A resonant

microwave drive adds energy to the cyclotron motion,

which detectably alters the axial frequency.

The three trap frequencies satisfy

= ν

2

+ν

. (75.71)

This relation remains valid even if the magnetic ﬁeld

is misaligned with respect to the trap axis, and is also

insensitive to small imperfections in the cylindrical sym-

metry of the electrodes. The bare cyclotron frequency

may therefore be deduced accurately.

For an ion with a dipole-allowed resonance transi-

tion, ﬂuorescence of a single ion is readily detected.

Even absorption of a single ion may be measurable.

Various methods of ﬁnding the temperature have been

devised. At temperatures of 1 K and higher, Doppler

broadening of a dipole-allowed optical transition is ob-

servable. The size of the single-ion cloud is a measure

of temperature. Finally, motional sidebands in the ab-

sorption of a narrow transition (γ  ν), not necessarily

the same transition as the one used for cooling, may be

measured to ﬁnd n. In the Lamb–Dicke regime only

the carrier absorption at ∆ = 0 and sidebands at ∆ =±ν

are signiﬁcant, and the ratios of the peak absorptions

are

−

: α

=n

: 1 :

(

1 +n

)

(75.72)

In an ion crystal, the ions have collective vibration modes

akin to phonons, instead of the three vibration modes

along the principal axes of the trap of a single ion.

Doppler cooling and sideband cooling work for such

collective modes much like they work for the vibration

modes of a single ion.

Quantum Jumps

Ion traps make it possible to isolate an individual atomic

scale particle for studies for a practically indeﬁnite

time, which enables clean experiments on various fun-

damental aspects of quantum mechanics and quantum

electrodynamics. Quantum jumps are a case in point.

Suppose that, in addition to an optically driven two-

level system, a single ion has a third shelving state. The

ion infrequently makes a transition to the shelving state,

stays there for a long time compared with the time scale

of spontaneous emission of the active system, and then

returns to the two-level system. When the ion makes

a transition to the shelving state, ﬂuorescence from the

two-level system suddenly ceases; and the ﬂuorescence

reappears, equally abruptly, when the ion returns to the

two-level system. The jumps in light scattering are the

quantum jumps [75.20]. They are a method to detect

Part F 75.5

Cooling and Trapping References 1105

a weak transition with an enormous ampliﬁcation: a sin-

gle transition to or from the shelving state may mean the

difference between the presence or absence of billions

of ﬂuorescence photons.

g− 2 Measurements

For an electron, the cyclotron frequency ν

and the spin-

ﬂip frequency ν

are related by

gν

. (75.73)

Due to quantum electrodynamic corrections g = 2,

and so the anomaly frequency ν

= ν

−ν

∼ 10

−3

is nonzero. A magnetic ﬁeld at the anomaly fre-

quency causes a simultaneous ﬂip of the spin and

a loss or gain of one quantum of energy in the

cyclotron motion. In a magnetic bottle, the change

in the cyclotron motion causes a change in the axial

resonance frequency, which is detected. The anomaly

frequency can thus be measured accurately. Together

with a measurement of the cyclotron frequency, this

yields a measurement of the g-factor of the electron,

or positron [75.21].

Measurements of Mass Ratios

As the cyclotron frequency is inversely proportional to

the mass of the ion (or electron, positron, proton, anti-

proton, etc.), an accurate measurement of the cyclotron

frequencies of two species in the same Penning trap

amounts to an accurate measurement of the ratio of

the masses [75.22]. A sufﬁcient resolution to weigh

molecular bonds is conceivable.

Quantum System of Motional States

A vibrational mode in a trapped ion and an effective two-

state system for the internal degrees of freedom make

a realization of the Jaynes–Cummings model (discussed

in detail in Sect. 79.5.1). Moreover, sideband cooling

enables an experimenter to put this mode cleanly in its

lowest quantum state. These observations have inspired

quantum-state engineering with the objective of gen-

erating an arbitrary state of the vibrational motion of

the ion [75.23]. In many-ion crystals the collective vi-

bration modes may be used to couple and entangle the

internal degrees of freedom of two or more ions. As dis-

cussedinChapt.81, prototype quantum gates have been

demonstrated in this manner.

More generally, experiments have come to the point

when it is possible to address joint quantum states for

the internal and cm degrees of freedom almost at will.

This facilitates new cooling schemes. Time evolution

derived from a Hamiltonian can never lead to cooling;

an irreversible mechanism such as spontaneous emission

is always needed. The idea of many cooling schemes

therefore is to pump atoms optically around the quantum

states in such a way there is no pathway out of the target

state, so that the atoms eventually accumulate there.

Velocity selective coherent population trapping [75.24],

Raman sideband cooling of an ion, and Raman cooling

of an atom in an optical lattice [75.25] work in this way.

References

75.1 The Mechanical Effects of Light, J. Opt. Soc. Am. B

2(11) (1985), special issue

75.2 S. Stenholm: Rev. Mod. Phys. 58, 699 (1986)

75.3 Laser Cooling and Trapping of Atoms, J. Opt. Soc.

Am. B 6(11) (1989), special issue

75.4 C. Cohen-Tannoudji: Fundamental Systems in

Quantum Optics, ed. by J. Dalibard, J. M. Raymond,

J. Zinn-Justin (Elsevier, Amsterdam 1991) p. 1

75.5 D. J. Wineland, W. M. Itano, R. S. Van Dyck Jr.: Adv.

At.Mol.Phys.19,135(1984)

75.6 L. S. Brown, G. Gabrielse: Rev. Mod. Phys. 58,233

(1986)

75.7 Physics of Trapped Ions, J. Opt. Soc. Am. B 20(5)

(2003), special issue

75.8 Laser Cooling of Matter, J. Phys. B 36(3) (2003),

special issue

75.9 J. Javanainen: Phys. Rev. A 46, 5819 (1992)

75.10 Y. Castin, K. Mølmer: Phys. Rev. Lett. 74, 3772 (1995)

75.11 M. R. Doery, E. J. D. Vredenbregt, T. Bergeman:

Phys. Rev. A 51, 4881 (1995)

75.12 A. C. Doherty, T. W. Lynn, C. J. Hood, H. J. Kimble:

Phys. Rev. A 63, 013401 (2001)

75.13 J. Dalibard, C. Cohen-Tannoudji: J. Opt. Soc. Am. B

6, 2023 (1989)

75.14 C. Gerz, T. W. Hodapp, P. Jessen, K. M. Jones,

C. I. Westbrook, K. Mølmer: Europhys. Lett. 21, 661

(1993)

75.15 H. Walther: Adv. At. Mol. Opt. Phys. 31, 137 (1993)

75.16 J. J. Bollinger, D. J. Wineland, D. H. E. Dubin: Phys.

Plasmas 1, 1403 (1994)

75.17 P. S. Julienne, A. M. Smith, K. Burnett: Adv. At. Mol.

Opt. Phys. 30, 141 (1993)

Part F 75

1106 Part F Quantum Optics

75.18 E. G. M. van Kempen, S. J. J. M. F. Kokkelmans,

D. J. Heinzen, B. J. Verhaar: Phys. Rev. Lett. 88,

093201 (2002)

75.19 Y. Sortais, S. Bize, M. Abgrall, S. Zhang, C. Nicolas,

C. Mandache, R. Lemonde, P. Laurent, G. Santarelli,

P. Petit, A. Clairon, A. Mann, S. Chang, C. Salomon:

Phys. Scr. T95, 50 (2001)

75.20 R. Blatt, P. Zoller: Eur. J. Phys. 9, 250 (1988)

75.21 R. S. Van Dyck Jr., P. B. Schwinberg, H. G. Dehmelt:

Phys. Rev. Lett. 59, 26 (1987)

75.22 M. P. Bradley, J. V. Porto, S. Rainville, J. K. Thomp-

son, D. E. Pritchard: Phys. Rev. Lett. 83, 4510 (1999)

75.23 D. Leibfried, R. Blatt, C. Monroe, D. Wineland: Rev.

Mod. Phys. 75, 281 (2003)

75.24 J. Lawall, F. Bardou, B. Saubamea, K. Shimizu,

M. Leduc, A. Aspect, C. Cohen-Tannoudji: Phys.

Rev. Lett. 73, 1915 (1994)

75.25 S. P. Hamann, D. L. Haycock, G. Klose, P. H. Pax,

I. H. Deutsch, P. S. Jessen: Phys. Rev. Lett. 80, 4149

(1998)

Part F 75

1107

Quantum Dege

76. Quantum Degenerate Gases

The purpose of this Chapter is to summarize the

basic physics of dilute quantum degenerate gases.

Given the broad activity in the ﬁeld, many choices

have to be made regarding the topics to include

and the style of the discussion. Emphasis is placed

on AMO physics, as opposed to condensed matter

physics. One related choice is that virtually nothing

is said about temperature dependence. Inside AMO

physics the approach is in the vein of quantum

optics, as opposed to atomic/molecular structure

and collisions. For the most part, the coverage is

on elementary concepts and basic material. The

exception to this is Sect. 76.5, where a few topical

issues are addressed.

The review article [76.1]hasbecomethe

standard reference on the basic properties of

a Bose–Einstein condensate (BEC), [76.2]isits

contemporary with more of a quantum optics

slant, [76.3] concentrates on conceptual is-

sues, [76.4] makes connections between the

present theories and traditional condensed mat-

ter physics, and [76.5] is particularly explicit

about the structure and excitations of a BEC. Here,

references are usually not given for topics that

are discussed in these reviews, or where a full

discussion is easily traced from them. Otherwise,

76.1 Elements of Quantum Field Theory ........1107

76.1.1 Bosons.....................................1108

76.1.2 Fermions..................................1109

76.1.3 Bosons versus Fermions .............1109

76.2 Basic Properties of Degenerate Gases ....1110

76.2.1 Bosons.....................................1110

76.2.2 Meaning of Macroscopic Wave

Function .................................. 1114

76.2.3 Fermions.................................. 1115

76.3 Experimental ......................................1115

76.3.1 Preparing a BEC.........................1115

76.3.2 Preparing a Degenerate Fermi

Gas.......................................... 1117

76.3.3 Monitoring Degenerate Gases .....1117

76.4 BEC Superﬂuid .....................................1117

76.4.1 Vortices.................................... 1117

76.4.2 Superﬂuidity ............................1118

76.5 Current Active Topics ............................1119

76.5.1 Atom–Molecule Systems............. 1119

76.5.2 Optical Lattice with a BEC ...........1121

References .................................................. 1123

references are meant to be entries to the literature

only. Assignment of credit or priority is never

implied.

Bose–Einstein condensation in dilute alkali metal va-

pors has realized a source of atoms with properties

analogous to the properties of laser light, and more

recently, ultralow-temperature Fermi gases have come

under study. The ﬁeld of quantum degenerate gases

has become a main theme in AMO physics. Dilute-

vapor systems are weakly interacting, and subject to

a degree of experimental control not seen before in

traditional low-temperature condensed matter systems.

Ultralow-temperature gases have thereby also given

a new lease on life to investigations of superﬂuid sys-

tems in condensed matter physics. The result is a broad

interdisciplinary effort that is still expanding at the time

of writing.

76.1 Elements of Quantum Field Theory

A Bose–Einstein condensate and a degenerate Fermi gas

are both consequences of particle statistics, exchange

symmetries of the many-particle wave function. It is

possible, in principle, to deal directly with the wave

functions, but in practice analyses of many-body sys-

tems are usually carried out using the methods of second

quantization and ﬁeld theory. In ﬁrst quantization, the

particles are labeled as if each one had a unique tag

Part F 76

1108 Part F Quantum Optics

on it, and the wave function for more than one indis-

tinguishable particle must be symmetrized explicitly. In

second quantization, the question is how many particles

are in a given state without a distinction between iden-

tical particles. The exchange symmetries are then taken

care of automatically. Here we brieﬂy summarize [76.6]

elementary features of quantum ﬁeld theories for both

bosons and fermions.

76.1.1 Bosons

Particles with an integer value of the angular momen-

tum obey the Bose–Einstein statistics. The characteristic

property is that a one-particle state can accommodate an

arbitrary number of bosons.

State Space for Bosons. Speciﬁcally, ﬁrst consider one

particle whose states are completely speciﬁed by a set

of quantum numbers k. As a notational device for the

purposes of the present Chapter, all of the quantum num-

bers are assumedly mapped in a one-to-one fashion to

nonnegative integers, and correspondingly the quantum

numbers are written k =0, 1, 2,.... The quantum num-

bers written here always incorporate a description of the

state of the c.m. motion of the particle. We therefore

have an orthonormal basis of wave functions to repre-

sent any state of a particle, {u

(x)}

,wherex stands for

the c.m. coordinate.

Given the one-particle states, the postulate is that the

Fock states |n

, n

,... ,n

∞

 with n

= 0, 1, 2,... par-

ticles in the states k = 0, 1, 2,... form an orthonormal

basis for the many-body system.

Second-Quantized Operators for Bosons. The annihi-

lation operator for the state k, a

,isdeﬁnedby

, n

,... ,n

∞



√

, n

,... ,n

−1,... ,n

∞

 . (76.1)

Its Hermitian conjugate, the creation operator, behaves

†

|... ,n

,...=



+1 |... ,n

+1,....

(76.2)

It follows that

†

|... ,n

,...=n

|... ,n

,..., (76.3)

and so

= a

†

is called the number operator for the

state k. Correspondingly,

N =



†

(76.4)

is the operator for the total number of particles in the

system. The annihilation and the creation operators have

the usual boson commutators,

, a





†

, a

†



= 0,



, a

†



= δ



. (76.5)

The boson ﬁeld operator is deﬁned as

ψ(x) =



(x)a

. (76.6)

The commutator for the ﬁeld operator,



ψ(x),

†



)



= δ(x−x



), (76.7)

follows from boson commutators and the completeness

of the wave functions {u

(x)}

. The orthogonality of the

wave functions gives the expression

N =



†

(x)

ψ(x) (76.8)

for the particle number operator. The positive operator

n(x) =

†

(x)

ψ(x) (76.9)

evidently represents the density of the particles at the

position x.

The second-quantized operators introduced thus far

can be used to express all observables acting on indis-

tinguishable bosons. The most relevant here are the one-

and two-particle operators. One-particle operators, such

as the kinetic energy, act on one particle at a time, while

two-particle operators, such as atom–atom interactions,

refer to two particles. In ﬁrst quantization, these are of

the form



V(x

), O



n,n



u(x

, x



(76.10)

where the sums run over the labels of the particles. The

corresponding second-quantized operators are



†

(x)V(x)

ψ(x), (76.11)



x d



†

(x)

†



)

× u(x, x



)

ψ(x



)

ψ(x). (76.12)

When the particles have internal degrees of freedom

in addition to the c.m. motion, such as hyperﬁne and

Zeeman states, it is convenient for the present purposes

to regard particles in each internal state as a separate

species. Thus, if the quantum number breaks up into

Part F 76.1