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

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1088 Part F Quantum Optics

References

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Part F 74

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Part F 74

1091

Cooling and Tr

75. Cooling and Trapping

Interactions of light with an atomic particle

are accompanied by exchange of momentum

between the electromagnetic ﬁeld and the atom.

Narrow-band resonance radiation from tunable

lasers enhances the ensuing mechanical effects of

light to the extent that it is literally possible to

stop atoms emanating from a thermal gas, and to

trap atoms with light.

References [75.1]and[75.2]aretwoearly

sources on laser cooling and trapping of the

traditional two-state model atom. A number of

articles on cooling and trapping of atoms with

the inclusion of angular momentum degeneracy

are contained in [75.3]. While the development

based on the atom-ﬁeld dressed states is followed

sparingly in the present Chapter, an authoritative

survey of this approach is given in [75.4]. Reviews

on traps for charged particles include [75.5]

and [75.6]. These articles, as well as [75.2], also

discuss cooling of trapped particles.

Cooling and trapping of atomic particles are

now standard technologies, but development and

extension of the methods to new applications

continues. The special issues [75.7]and[75.8]

75.1 Notation .............................................1091

75.2 Control of Atomic Motion by Light .........1092

75.2.1 General Theory .........................1092

75.2.2 Two-State Atoms.......................1094

75.2.3 Multistate Atoms .......................1097

75.3 Magnetic Trap for Atoms ......................1099

75.4 Trapping and Cooling

of Charged Particles ............................. 1099

75.4.1 Paul Trap ................................. 1099

75.4.2 Penning Trap ............................ 1101

75.4.3 Collective Effects in Ion Clouds ....1102

75.5 Applications of Cooling and Trapping ....1103

75.5.1 Neutral Atoms...........................1103

75.5.2 Trapped Particles ......................1104

References .................................................. 1105

give a current snapshot. Additional references

are occasionally listed here to accentuate speciﬁc

points. These citations are to either particularly

representative papers or to the most recent articles

on the subject, and are intended as entries to the

literature. No assignment of credit or priority is

implied.

75.1 Notation

In this Chapter, the lower and upper states of an optical

transition are denoted by the respective labels g and e,for

“ground” and “excited”. The notation J

→ J

stands for

a transition in which the lower and upper levels have the

angular momentum degeneracies 2 J

+1and2J

+1.

The resonance frequency of the transition is ω

The detuning of the driving monochromatic light of

frequency ω from the atomic resonance is ∆ = ω

−ω.

Γ is the spontaneous decay rate. Spontaneous emission

is taken to be the sole mechanism of line broadening,

so that the HWHM linewidth of the transition is γ =

Γ/2. The Rabi frequency is Ω =DE/

,whereD is the

reduced dipole moment matrix element that would apply

to a transition with unit Clebsch–Gordan coefﬁcient,

and E is the electric ﬁeld amplitude of the laser. The

corresponding intensity scale is the saturation intensity

4π

cΓ

3λ

, (75.1)

deﬁned in such a way that the light intensity I satisﬁes

Ω = Γ ⇒ I = I

. (75.2)

If multiple laser beams are explicitly mentioned, laser

intensity and Rabi frequency are quoted for each of the

equally intense beams.

The momentum of a photon with the wave vector k

k. The recoil velocity

(75.3)

Part F 75

1092 Part F Quantum Optics

Table 75.1 Laser cooling parameters for the lowest S

1/2

–P

3/2

transition of hydrogen and most alkalis (the D

line). Also

shown are the nuclear spin I and the ground state hyperﬁne splitting ∆ν

hfs

. Γ is typically known to within a few per cent,

so these values of Γ , T

and I

may not all be accurate to the full displayed precision

Parameter

133

Cs Units

m 1.67 9.99 11.7 38.2 64.7 66.4 141 144 221 10

−27

λ 121.6 670.8 670.8 589.0 766.5 766.5 780.0 780.0 852.1 nm

326 9.89 8.48 2.95 1.34 1.30 0.602 0.589 0.352 cm/s

Γ 98.9 5.92 5.92 9.90 6.16 6.16 5.89 5.89 5.22 2π MHz

2390 142 142 238 148 148 141 141 125 µK

13 396 73.7 63.2 25.0 8.72 8.50 3.86 3.77 2.07 2πkHz

643 3.54 3.03 1.20 0.418 0.408 0.185 0.181 0.0992 µK

14 509 5.13 5.13 12.7 3.58 3.58 3.24 3.24 2.21 mW/cm

I 1/2 1 3/2 3/2 3/2 4 5/2 3/2 7/2

∆ν

hfs

1420 228.2 803.5 1772 461.7 1286 3036 6835 9193 2πMHz

equals the change of the velocity of an atom of mass m

when it absorbs a photon with wave number k = 2π/λ.

The kinetic energy of an atom with velocity v

and the

corresponding frequency,

R =

,ε

(75.4)

are referred to as recoil energy and recoil frequency.

Two temperatures, the Doppler limit T

and the recoil

limit T

are often cited in laser cooling. They are

, T

, (75.5)

where k

is the Boltzmann constant.

Table 75.1 lists numerical values of pertinent pa-

rameters for laser cooling and trapping using the D

line for most stable and long-lived alkali isotopes and

hydrogen.

75.2 Control of Atomic Motion by Light

75.2.1 General Theory

Hamiltonian

The mechanical effects of light may be derived from the

Hamiltonian

H =

−

d·

r), (75.6)

where

and

are the Hamiltonians for the in-

ternal degrees of freedom of the atom, center-of-mass

(cm) motion of the atom, and free electromagnetic ﬁeld.

The quantized electric ﬁeld is

r),where

r is the cm

position operator. The dipole operator

d acts on the in-

ternal degrees of freedom of the atom. Since the

d·

term couples all degrees of freedom, the possibility of in-

ﬂuencing cm motion by light immediately follows. The

inclusion of the quantized cm motion is the essential

ingredient not contained in traditional theories of light-

matter interactions. For an atom with mass m trapped in

a possibly anisotropic harmonic oscillator potential with

frequencies ν

(i = x, y, z), the cm Hamiltonian is



i=x,y,z

mν

(75.7)

where

p is the cm momentum operator. For a free atom,

= 0.

Master Equation

With the aid of Markov and Born approximations, the

vacuum modes of the electromagnetic ﬁeld may be elim-

inated as described in Sect. 78.7. This gives a master

equation for the reduced density operator

ρ that contains

the internal and cm degrees of freedom of the atom. Re-

laxation terms proportional to Γ and γ are all that is left

of the quantized ﬁelds.

Consider as an example a two-state atom in a travel-

ing wave of light with the electric ﬁeld strength

E(r, t) =

E e

i(k·r−ωt)

+c.c. . (75.8)

Part F 75.2

Cooling and Trapping 75.2 Control of Atomic Motion by Light 1093

Master equations are conveniently written using Wigner

functions to represent the cm motion. Given the internal-

state labels i and j = g or e, and the three-dimensional

variables r, p, the Wigner functions are deﬁned as

(r, p) =

(2π )



u e

iu· p/



r −

u|i|

ρ|j|r +



(75.9)

The Wigner function is one of the quantum mechan-

ical quasiprobability distributions, Sect. 78.5, with the

special property that the marginal distribution of r ob-

tained by integrating over p coincides with the correct

quantum probability distribution for position, and vice

versa with r and p interchanged. In the rotating wave

approximation, Sect. 68.3.2, the master equations are

( p) =−Γρ

( p) +i

Ω



ik·r



p−



− e

−ik·r



p−



, (75.10)

( p) = Γ



nW(

n)ρ

( p+ k

n) −

iΩ



ik·r





− e

−ik·r

( p+12 k)



, (75.11)

( p) =−(γ −i∆)

( p) −

iΩ



−ik·r

( p−

− e

ik·r





, (75.12)

( p) =−(γ +i∆)ρ

( p) +

iΩ



ik·r



p−



− e

−ik·r





. (75.13)

Here the convective derivative that describes the motion

of the atom in the absence of light is

∂

∂t

∂

∂r

−



mν

∂

∂p

; (75.14)

cf. H

in (75.7). W(

n) is the angular distribution of

spontaneous photons, and the integral runs over the unit

sphere. Representative expressions for W(

n) are

n) =

4π

8π



1 −(

e·



16π



1 +(

e·



(75.15)

These apply, respectively, for isotropic spontaneous

emission, for spontaneous emission in a ∆m = 0transi-

tion, and in ∆m =±1 transitions;

e stands for the unit

vector in the direction of the quantization axis for angu-

lar momentum. Only the pdependence has been denoted

explicitly in the Wigner functions, as the recoil effects

displayed on the right-hand sides of (75.10–75.13)take

place at a ﬁxed position r.

Semiclassical Theory

Suppose that v

v

and τ τ

,whereτ and τ

are

the time scales for light-driven changes of the internal

state and cm motion of the atom. Then the internal de-

grees of freedom may be eliminated adiabatically from

the master equations in favor of the position-momentum

distribution for the cm motion,

f(r, p, t) =



(r, p, t), (75.16)

where the sum runs over the internal states of the atom.

Technically, the recoil velocity v

is treated as an asymp-

totically small expansion parameter. The result is the

Fokker–Planck equation for the cm motion

f =−

∂

∂ p

· (F f ) +



i, j

∂

∂p

f ). (75.17)

In this semiclassical theory the cm motion of the atom

is regarded as classical. The atom moves under the op-

tical force F(r, p, t), which models the coarse-grained

ﬂow of momentum between the electromagnetic ﬁeld

and the atom. D

(r, p, t), with i, j = x, y, z,isthedif-

fusion tensor. Diffusion is an attempt to model quantum

mechanics with a classical stochastic process, including

discreteness of recoil kicks, random directions of sponta-

neous photons, and random timing of optical absorption

and emission processes.

A general prescription exists for calculating the force

and the diffusion tensor for an arbitrary atomic level

scheme and light ﬁeld [75.9]. However, the study of

diffusion amounts to an involved analysis of photon

statistics of the scattered light, and here only the force

is considered explicitly. Let

V (r) be the dipole interac-

tion operator coupling the driving ﬁeld and the internal

Part F 75.2

1094 Part F Quantum Optics

state for an atom at position r. By assumption,

V (r) has

been rendered slowly varying in time with the aid of

a suitable rotating wave approximation. To compute the

force, one takes an atom that travels along a hypothetical

trajectory unperturbed by light in such a way that at time

t it arrives at the phase space point (r, p), whereupon

the density operator of the internal degrees of freedom

. The force is then

(r, p, t) =−Tr





∂

∂r



(75.18)

Quantum Theory

When either v

 v

or τ  τ

, the full quantum the-

ory of cooling and trapping is needed. Master equations

such as (75.10–75.13) must then be solved without the

assumption that v

is small. Most practical calculations

have been numerical case studies [75.10–12]. A trun-

cated basis, e.g., of plane waves, is used to expand the cm

state. Density matrix equations are solved numerically,

either directly, or by resorting to quantum trajectory

simulations, Sect. 78.11.

Qualitative Origin of Laser Cooling

Velocity dependent dissipative forces are needed for

cooling. They arise because the evolution of the internal

state of a moving atom has a ﬁnite response time τ.The

atom conveys the memory of the ﬁeld it has sampled

over the length  = vτ on its past trajectory. If   λ,

a nonequilibrium component proportional to  is present

in the density operator of the internal state of the atom.

Further interactions with light convert this component

into a velocity dependent force of the form

F =−mβv , β ∝ Iτ.

(75.19)

If the damping constant β is positive, (75.19)de-

scribes exponential damping of the velocity on the time

scale β

−1

. In the contrary case   λ, when the atom

travels many wavelengths during the memory time, lin-

ear dependence of force on velocity breaks down. The

watershed is the critical velocity or velocity capture

range

≈

(75.20)

One-Dimensional Considerations

Most speciﬁc results cited here are one-dimensional. By

default, the propagation direction of light and the di-

rection of vector quantities other than light polarization

. The relevant components of position, velocity and

momentum are denoted by x, v,and p.

The general one-dimensional Fokker–Planck equa-

tion for a particle trapped in a harmonic oscillator

potential with a cm oscillation frequency ν is



∂

∂t

∂

∂x

−mν

∂

∂p



=−

∂

∂p

(Ff) +

∂

∂p

(Df). (75.21)

For the force (75.19) with constant β = β

and

D(z, p) = D

, the steady state of the Fokker–Planck

equation is a thermal distribution of the form

f(x, p) = K exp



−



mν



(75.22)

where K is a normalization coefﬁcient.

Since Wigner functions give correct quantum mech-

anical marginal distributions for r and p, expectation

values of kinetic and potential energy may be calculated

from the distribution function (75.22) as if it were a clas-

sical phase space density. For a free atom with ν = 0,

the temperature is directly proportional to the kinetic

energy,

T =

. (75.23)

However, for a trapped particle with ν = 0 the Fokker–

Planck equation may be valid all the way to the quantum

mechanical zero-point energy. Then temperature and en-

ergy are no longer directly proportional to one another.

For a trapped particle, the safe interpretation of (75.22)

is that the total cm energy of the particle is

E =

(75.24)

75.2.2 Two-State Atoms

A two-state or two-level atom, discussed in detail in

Sect. 68.3, stands for a closed (recycling) transition with

one lower state and one excited state. In practice, a two-

state system is often realized by driving a J → J +1

transition with circularly polarized light. This leads to

optical pumping to the states with maximal (or minimal)

component of angular momentum along the quantization

axis, say, to the transition m = J → J +1.

Two types of force are generally distinguished: light

pressure, or scattering force, or spontaneous force, and

dipole, or gradient, or induced force. However, the dis-

tinction is neither exclusive, nor exhaustive. Here the

two types of force are approached by way of examples.

Part F 75.2

Cooling and Trapping 75.2 Control of Atomic Motion by Light 1095

Traveling Waves

Light Pressure. Consider a cycle of absorption and spon-

taneous emission. In an absorption, the atom receives

a photon recoil kick in the direction of the laser beam,

while in spontaneous emission the recoil kick has a ran-

dom direction and zero average. The atom is on the

average left with a velocity change equal to v

.The

corresponding force is along k, and is given by

F = F

Ω

+∆

(v) +Ω

(75.25)

Here the maximum of light pressure force, a convenient

scale for optical forces, is

Γ, (75.26)

and

∆(v) = ∆ +kv (75.27)

is the effective detuning, which includes the Doppler

shift experienced by the moving atom.

Diffusion. For a traveling wave, the diffusion coefﬁcient

accompanying light pressure is

(1 +α)Ω



∆

(v) +γ

+Ω



−



∆

(v) −3γ



Ω



∆

(v) +γ

+Ω



, (75.28)

where

α =



nW(

n)(

(75.29)

depends on W(

n),see(75.15). Representative values are

α = 1/3 for isotropic spontaneous emission, and α =

2/5(α = 3/10) for spontaneous emission with ∆m =0

(∆m =±1) with respect to a quantization axis that is

perpendicular (parallel) to the direction

Spontaneously emitted photons cover all of the 4π

solid angle, and so do the directions of photon recoil

kicks on the atom. Absorption from a light wave travel-

ing in a particular direction leads to transverse diffusion

also in the orthogonal directions, which is not accounted

for by the one-dimensional (75.28).

Phenomenology in Multimode Fields

Doppler Cooling in Standing Waves. Next take an atom

in two counterpropagating plane waves of light. At low

intensity, Ω Γ , forces of the form (75.25)forthetwo

beams may be added when averaged over a wavelength.

For velocities well below the critical velocity

c,D

(75.30)

the wavelength-averaged force is of the form of (75.19),

F =−m

βv ,

β =

4Ω

γ∆



∆

+γ



. (75.31)

When light is tuned below the atomic resonance (“red de-

tuning” with ∆>0), exponential damping of the atomic

velocity with the time constant

−1

ensues. No matter

which way the atom moves, it is always Doppler tuned

toward resonance with the light wave that propagates

opposite to its velocity, and away from resonance with

the light wave that propagates along its velocity. Net

momentum transfer therefore opposes the motion of the

atom. This is known as Doppler cooling.

Optical Molasses. For three pairs of counterpropagating

waves in three orthogonal directions, (75.31) is valid in

all coordinate directions, and hence as a vector equa-

tion between the force F and velocity v.For∆>0an

atom experiences an isotropic viscous damping force,

as if it were moving in a thick liquid. Such a ﬁeld

conﬁguration is dubbed optical molasses. Two coun-

terpropagating beams make a one-dimensional optical

molasses.

Limit of Doppler Cooling. Under the conditions

of (75.31), the diffusion coefﬁcients for the two coun-

terpropagating beams averaged over a wavelength may

be added, and the v = 0 form sufﬁces for slow atoms.

This yields

D(v = 0)

(1 +α) Ω



∆

+γ



(75.32)

The random diffusive motion of the atom corresponds

to diffusive heating that competes with Doppler cooling.

In equilibrium, the temperature is

T =

D(v = 0)

βk

(1 +α)



∆



(75.33)

Equation (75.33) also applies to three-dimensional

Doppler cooled molasses, provided one uses α =1cor-

responding to added transverse diffusion. The minimum

temperature is reached at

∆ = γ =

(75.34)

Part F 75.2

1096 Part F Quantum Optics

For three-dimensional molasses, the Doppler limit T

of (75.5) is obtained.

For Ω>Γ, the performance of Doppler cooling de-

teriorates. Qualitatively, power broadening increases the

effective linewidth γ .

Dipole Forces. Dipole forces are the resonant analog

of ponderomotive forces discussed in Sect. 74.2.They

arise from successions of absorption and induced emis-

sion driven by photons with different momenta. Such

processes occur only if there is more than one wave

vector present in the ﬁeld, i. e., if there is an intensity

gradient. For a zero-velocity two-state atom, the gradi-

ent force in a monochromatic ﬁeld with the local total

intensity I(r) is

(r) =

∆γ

∆

+γ

[1 +2I(r)/I

]

∇I(r)

. (75.35)

The dipole force may be derived from the potential

energy

(r) =−2 ∆ ln



1 +

2γ

I(r)/I

∆

+γ



(75.36)

The atoms are strong ﬁeld seekers for ∆>0, and weak

ﬁeld seekers when ∆<0.

Optical Trap and Optical Lattice. Dipole forces are uti-

lized in the optical trap for atoms, and even molecules.

A common conﬁguration consists of a focused laser

beam tuned below resonance. The focus becomes the

trap. The detuning from resonance may be substantial;

lasers such as CO

and Nd–YAG have been used.

A standing wave of light makes a periodic array of

optical traps called an optical lattice. Optical lattices

may be set up in 1D, 2D, and 3D conﬁgurations.

Induced Diffusion. Random motion of atoms in vel-

ocity space owing to absorptions and induced emissions

of photons with different momenta leads to induced

diffusion. Contrary to diffusion in a traveling wave as

in (75.28), induced diffusion does not saturate at high

intensity. Instead, the diffusion coefﬁcient continues to

grow linearly with I. Induced diffusion is another rea-

son why the lowest Doppler cooling temperatures are

generally reached at low (I < I

) light intensities.

Sisyphus Effect. In a standing wave at high intensity

and large detuning, another kind of optical force be-

comes important that cannot be categorized either as

light pressure or gradient force.

As explained in Sect. 68.3.4, one may diagonalize

the Hamiltonian to obtain the dressed atom-ﬁeld states.

Because the light ﬁeld depends on position, so do the

energies of the dressed states and their decompositions

into plain atomic states. In Fig. 75.1 drawn for ∆<0,

the dressed state with a minimum (maximum) at the

ﬁeld nodes coincides with the bare ground state (ex-

cited state) at the nodes. At the antinodes the admixtures

of ground and excited states are evened out to some

extent.

The energy of a dressed state acts as potential energy

for the cm motion of an atom residing in that particu-

lar state. In fact, the gradient force is the force derived

from these potential energies, averaged over the occupa-

tion probabilities of the dressed states. The occupation

probability is larger for the dressed state with a larger

ground state admixture. From Fig. 75.1 one therefore

sees that the atom predominantly resides in the dressed

state that has a minimum of energy at the nodes. The

atom is a weak-ﬁeld seeker, as it should for ∆<0.

Spontaneous emission remains to be considered. It

gives rise to transitions between the dressed states. These

transitions may go both ways between the dressed states,

because the states are in general superpositions of the

bare ground state and the excited state. The rate of spon-

taneous transitions from one dressed state to another

Fig. 75.1 Qualitative origin of Sisyphus effect. The hatched

pattern represents a standing light wave. The energies of

the two dressed states are drawn as a function of position,

along with a few ﬁlled circles representing the admixture

of the ground state in each dressed state at selected ﬁeld

positions. Larger circles correspond to larger ground state

admixtures, and hence, larger equilibrium populations of

the dressed state. This ﬁgure applies for tuning of the laser

above the atomic resonance (“blue detuning”)

Part F 75.2

Cooling and Trapping 75.2 Control of Atomic Motion by Light 1097

increases (decreases) with the excited (ground) state

admixture of the initial state.

In reference to Fig. 75.1, suppose that the atom is

coming from the left in the upper dressed state. The

probability that the atom makes a transition to the lower

dressed state, as marked by the downward vertical ar-

row, is largest at the node. If this transition takes place,

near the next antinode the most probable transition is as

shown by the upward vertical arrow. In this manner the

atom spends most of its time at an uphill climb against

the potential, and is therefore slowed down. In reference

to Greek mythology, this is called the Sisyphus effect.

In the two-state model atom, cooling takes place when

the laser frequency is higher than the atomic resonance

frequency.

Exact Results for Standing Waves

Force. The force on a two-state atom in a one-

dimensional standing light wave may be expanded

analytically to ﬁrst order in velocity. With the ﬁeld phase

chosen so that the antinode is at x = 0, the force is

F(x,v)= F

(x) −mβ(x)v , (75.37)

where the gradient force is

(x) =−

k∆Ω

sin 2kx

(75.38)

and the damping coefﬁcient is

β(x) = 8∆Ω

−1

−3



1 −cos





∆

+γ

−2γ

Ω

cos

−2Ω

cos



, (75.39)

with d =∆

+γ

+2Ω

cos

kx.

Diffusion. For a standing wave, the v =0 form of the

diffusion coefﬁcient is

D(v = 0)

Ω



αγ

cos

kx +γ

sin

kx +2Ω

sin

kx cos



2γ

−

∆

Ω

sin

kx cos



∆

+5γ

+4Ω

cos



(75.40)

Semiclassical versus Quantum Theory

When γ  ε

, the r.m.s. velocity of a cooled two-state

atom is always  v

, and semiclassical theory is valid.

Under the same condition γ  ε

, the Doppler-limit

r.m.s. velocity also is less than the critical velocity v

c,D

from (75.30). Velocity expansions such as in (75.37)

and (75.40) are then justiﬁed.

In the contrary case, γ  ε

, the full quantum the-

ory of trapping and cooling must be employed. The

cooled velocity distribution is not thermal, and tem-

perature is ill-deﬁned. The lowest expectation value of

kinetic energy for a two-state atom in a linearly polar-

ized standing wave occurs at low I for ∆ = 4.4ε

,andis

equal to 0.53 R.

75.2.3 Multistate Atoms

Energy levels of atomic systems usually have angular

momentum degeneracy. In addition, the polarization of

light in general depends on position. A combination of

these aspects leads to phenomena beyond the two-state

atomic model.

Polarization Gradient Cooling

As explained in connection with (75.19), a ﬁnite mem-

ory time of the internal atomic state may lead to damping

of the cm motion. For a two-state atom, internal equi-

libration arises from spontaneous emission. The time

scale is τ

∼ Γ

−1

, and Doppler cooling ensues. How-

ever, an atom whose ground state has angular momentum

degeneracy is also subject to optical pumping. If the

polarization of light varies as a function of position, op-

tical pumping is needed to reach local equilibrium. The

pumping time scale τ

∝ I

−1

then becomes relevant for

a moving atom. The associated cooling is known as

polarization gradient cooling. Its hallmark is that, for

low I , the damping coefﬁcient β ∝ Iτ

is independent

of intensity.

Two detailed mechanisms of polarization gradi-

ent cooling have been described [75.13], although in

three-dimensional light ﬁelds they are intertwined. The

Sisyphus effect works like the Sisyphus effect for a two-

state atom, except that it relies on light shifts and optical

pumping within the ground state manifold. Induced

orientation cooling is analogous to Doppler cooling.

Velocity dependence of optical pumping in counterprop-

agating waves leads to pumping to a state for which the

force due to the wave propagating opposite to the atom

exceeds the force due to the wave propagating along

with the atom.

Lin ⊥ Lin Molasses. One-dimensional lin ⊥lin molasses

consists of two counterpropagating waves with orth-

ogonal linear polarizations. The net polarization varies

Part F 75.2

1098 Part F Quantum Optics

from linear – via elliptical – to circular over a distance

of λ/8. With the phases chosen such that light is linearly

polarized at x = 0, in the limit of nonsaturating inten-

sity, and at low velocity, the semiclassical force on an

atom with a J = 1/2 →3/2 transition is

F =−

k∆



s sin 4kx −

qvγ

−1

(1 +cos 4kx)



(75.41)

Here the saturation parameter s is

s =

Ω

∆

+γ

. (75.42)

In this conﬁguration, only the Sisyphus effect con-

tributes to cooling. Cooling takes place for ∆>0, and

the resulting temperature obtained from the position-

averaged quantities is

T =

135∆

+296γ

1080(∆

+γ

)



2IΓ

|∆|



. (75.43)

–σ

−

Molasses. One-dimensional σ

–σ

−

molasses

consists of two counterpropagating waves with opposite

circular polarizations. The net polarization is linear ev-

erywhere, but the direction of polarization rotates as the

point of observation is displaced along the propagation

axis; hence, the alternative name corkscrew molasses.

At low intensity and low velocity, the force on an atom

with a J = 1 →2 transition is

F =−

∆γ

5γ

+∆

v. (75.44)

In this conﬁguration only induced orientation cooling

contributes. Cooling again takes place for ∆>0, and

the resulting temperature is

T =

29∆

+1045γ

300



∆

+γ





2IΓ

|∆|



. (75.45)

Lin  Lin Molasses. The designation lin  lin denotes

a standing wave with the same linear polarization for

both counterpropagating beams. In one dimension there

is no polarization gradient, but three lin  lin pairs

in orthogonal directions (often with mutually orthog-

onal polarizations) make a three-dimensional optical

molasses with potential polarization gradient cooling.

Experimental Molasses. For IΓ/I

|∆| < 1and

|∆| >Γ, the temperatures (75.43)and(75.45) both

reduce to the form

T = C

Ω

|∆|

= C

2IΓ

|∆|

. (75.46)

Under these conditions the same scaling is approx-

imately observed also in three-dimensional six-beam

optical molasses operating with atoms that have a de-

generate ground state. The constant C depends on the

degeneracy of the transitions and on the polarizations of

the molasses beams. Measured values are mostly in the

range 0.25 < C < 0.5 [75.14].

Limit of Cooling. While the expressions (75.43), (75.45)

and (75.46) suggest that T goes all the way to zero as

I → 0or∆ →∞, there is a lower limit of T reached in

polarization gradient cooling. T eventually starts to rise

abruptly when |∆| is increased or I is decreased. The

empirical rule of thumb is that T ∼ 10 T

is the lowest

temperature one can expect.

Semiclassical versus Quantum Theory. According to

the semiclassical theory, T ∝ I , so the r.m.s. velocity

of cooled atoms is proportional to I

1/2

.Now,thecrit-

ical velocity of polarization gradient cooling, estimated

roughly as

c,p

≈

λγΩ

∆

+γ

, (75.47)

is proportional to I. At low enough I, v

c,p

is therefore

smaller than the velocity width of the cooled atoms.

Expansions of force and diffusion in velocity are no

longer useful, and temperature predictions of the type

(75.46) fail. This occurs, at the latest, when the r.m.s.

velocity equals a few recoil velocities.

Semiclassical theory does not lead to predictions that

grossly violate its key premise that the ensuing velocity

distribution is much broader than v

. However, reliable

theoretical limits of temperature for polarization gradi-

ent cooling may only be obtained from the full quantum

treatment.

Magneto-Optical Trap

Since spontaneous forces may be strong already at mod-

est light intensities ∼l

mathrm S

use of light pressure to

trap neutral atoms appears desirable. However, the opti-

cal Earnshaw theorem states that (in the limit of low I)

the spontaneous force on a two-state atom is sourceless.

While conﬁnement may be possible in some directions,

escape routes for atoms remain open in others. Three-

dimensional trapping of a two-state atom with light

pressure is not possible.

A Magneto-optical trap (MOT) defeats the Earnshaw

theorem by relying on angular momentum degeneracy.

Consider an atom with a J = 0 →1 transition in a mag-

netic ﬁeld Bthat depends linearly on position around the

Part F 75.2