Laser Cooling and Optical Molasses

An atom in free space feels no force from a distant laser unless it absorbs light, and each absorption is also a momentum transfer. A photon of wavevector $k$ carries momentum $ℏ k$ ; absorbing it recoils the atom by $ℏ k$ . The atom then returns to the ground state, and if it does so by spontaneous emission the recoil kicks point in random directions and average to zero over many cycles. The directed absorption recoils accumulate; the undirected emission recoils cancel. A resonant beam therefore exerts a steady force in its own direction, and the rate of that force is set by how fast the atom can cycle absorption and emission.

This is the mechanism behind every technique in modern cold-atom physics. This lesson derives the scattering force from the two-level saturation of the excited state, shows how the Doppler shift converts a pair of counter-propagating beams into a velocity-dependent friction, computes the equilibrium temperature at which spontaneous-emission heating balances that friction, and adds a magnetic gradient to turn the friction into a trap.

The scattering force

Model the atom as a two-level system: ground state $∣ g ⟩$ , excited state $∣ e ⟩$ separated by $ℏ ω_{0}$ , with the excited state decaying at rate $Γ$ (inverse of the natural lifetime $τ = 1/Γ$ ). A laser of frequency $ω$ drives the transition. The steady-state excited-state population, from the optical Bloch equations for a two-level atom in a classical field,¹ is

ρ_{ee} = \frac{1}{2} \frac{s _{0}}{1 + s _{0} + ( 2 δ /Γ ) ^{2}}, δ \equiv ω - ω_{0},

where $δ$ is the detuning of the laser from resonance and

s_{0} \equiv \frac{I}{I _{sat}}, I_{sat} = \frac{π h c Γ}{3 λ ^{3}},

is the on-resonance saturation parameter, the ratio of the intensity $I$ to the saturation intensity $I_{sat}$ . For the rubidium-87 $D_{2}$ line ( $λ = 780 nm$ , $Γ/2 π = 6.07 MHz$ ), $I_{sat} = 1.67 mW c m^{- 2}$ . As $s_{0} \to \infty$ the population saturates at $ρ_{ee} \to \frac{1}{2}$ : the atom spends at most half its time in the excited state, because stimulated emission returns population to the ground state just as fast as absorption removes it.

Each atom in the excited state decays at rate $Γ$ , so the scattering rate (photons scattered per second) is

R_{sc} = Γ ρ_{ee} = \frac{Γ}{2} \frac{s _{0}}{1 + s _{0} + ( 2 δ /Γ ) ^{2}} .

Every scattered photon is absorbed from the beam (momentum $+ ℏ k$ ) and re-emitted in a random direction (momentum averaging to zero), so the mean force is the momentum per photon times the scattering rate:

F_{sc} = ℏ k R_{sc} = ℏ k \frac{Γ}{2} \frac{s _{0}}{1 + s _{0} + ( 2 δ /Γ ) ^{2}} .

The saturated force is enormous on an atomic scale. For rubidium, $F_{max} = ℏ k Γ/2$ produces an acceleration $a_{max} = ℏ k Γ/2 m \approx 1.1 \times 1 0^{5} m s^{- 2}$ , about $1 0^{4}$ times gravity. A thermal rubidium atom leaving an oven at $300 m s^{- 1}$ can be brought to rest in under a millisecond over a distance of tens of centimetres. This is the basis of the Zeeman slower that loads a trap from an atomic beam.²

Scattering rate versus detuning is a Lorentzian of half-width (Gamma/2) sqrt(1+s0); increasing intensity raises the peak toward Gamma/2 and power-broadens the line.

The Doppler cooling mechanism

A single beam only pushes; it cannot slow a moving atom below zero velocity. The trick is to use two counter-propagating beams and to detune them below resonance ( $δ < 0$ , red detuning). An atom moving with velocity $v$ toward one beam sees that beam Doppler-shifted up in frequency by $k v$ , and the opposing beam shifted down. The beam the atom moves against is therefore shifted closer to resonance and scatters more strongly; the beam the atom moves with is shifted farther away and scatters less. The imbalance produces a net force opposing the motion.

Let the two beams propagate along $\pm \overset{x}{^}$ , each of intensity $I$ giving the same $s_{0}$ . In the atom's rest frame the beam it moves against has effective detuning $δ - k v$ (moving toward a beam raises its frequency, moving red-detuned laser closer to resonance when $δ < 0$ requires the sign care below) and the co-moving beam has $δ + k v$ . The net force along $\overset{x}{^}$ is the difference of two scattering forces:

F (v) = ℏ k \frac{Γ}{2} \frac{s _{0}}{1 + s _{0} + ( \frac{2 ( δ - k v )}{Γ} ) ^{2}} - \frac{s _{0}}{1 + s _{0} + ( \frac{2 ( δ + k v )}{Γ} ) ^{2}} .

The first term (from the $- \overset{x}{^}$ beam, pushing in $+ \overset{x}{^}$ ) and the second (from the $+ \overset{x}{^}$ beam, pushing in $- \overset{x}{^}$ ) enter with opposite signs. For $δ < 0$ and $v > 0$ the co-moving-against beam is nearer resonance, the bracket is negative, and the force opposes $v$ : the motion is damped.

An atom moving right sees the left-going beam blue-shifted toward resonance (stronger push) and the right-going beam red-shifted away (weaker push); the imbalance opposes the velocity.

The friction coefficient

Near $v = 0$ the force is linear in velocity. Expand $F (v)$ to first order. With $x_{0} \equiv 2 δ /Γ$ and treating $2 k v /Γ$ as small,

F (v) \approx \frac{8ℏ k ^{2} δ s _{0}}{Γ ( 1 + s _{0} + x _{0}^{2} ) ^{2}} v \equiv - α v, α = - \frac{8ℏ k ^{2} s _{0} ( δ /Γ )}{( 1 + s _{0} + ( 2 δ /Γ ) ^{2} ) ^{2}} .

For red detuning $δ < 0$ the coefficient $α > 0$ , so $F = - α v$ is a friction force: the medium of light acts like a viscous fluid. This configuration is optical molasses. A displaced or moving atom decelerates on a timescale $m / α$ , which for typical parameters is tens of microseconds.

Net force in a two-beam molasses versus velocity is the difference of two Lorentzians; near v=0 it is linear with negative slope (friction), and it captures atoms within the capture range set by the detuning.

The linear range extends only to velocities where $k v \sim Γ$ ; beyond that the Doppler shift outruns the linewidth and the force falls off. The capture velocity $v_{c} \sim Γ/ k$ is a few metres per second for rubidium, so molasses cools atoms already slowed to that range but does not stop a thermal beam by itself.

Momentum diffusion and the Doppler limit

Friction alone would cool to absolute zero. It does not, because the same photon scattering that provides the friction also delivers random momentum kicks. Two sources of randomness heat the atom:

Absorption fluctuations. The number of photons absorbed in a time interval fluctuates; each absorption is a discrete kick $ℏ k$ .
Spontaneous-emission recoil. Each emitted photon leaves in a random direction, so the emission recoils execute a random walk in momentum with step $ℏ k$ .

Both contribute a momentum diffusion: $⟨ p^{2} ⟩$ grows linearly in time, $d ⟨ p^{2} ⟩ / d t = 2 D$ , with diffusion constant $D$ of order $(ℏ k)^{2} R_{sc}$ summed over the beams. The friction removes energy at rate $α v^{2}$ (averaged, $α ⟨ v^{2} ⟩$ ), while diffusion adds it at rate $D / m$ . In steady state the two balance:

\frac{d}{d t} ⟨ \frac{1}{2} m v^{2} ⟩ = - \frac{α}{m} ⟨ p^{2} ⟩_{cool} + \frac{D}{m} = 0 ⟹ \frac{1}{2} k_{B} T = \frac{D}{2 α} per degree of freedom.

Carrying the low-intensity limit ( $s_{0} ≪ 1$ ) through with the explicit $D$ and $α$ gives the equilibrium temperature³

k_{B} T = \frac{ℏΓ}{4} \frac{1 + ( 2 δ /Γ ) ^{2}}{2∣ δ ∣/Γ} .

Minimising over detuning, $d T / d δ = 0$ gives $2∣ δ ∣/Γ = 1$ , i.e. $δ = - Γ/2$ , and the minimum temperature is the Doppler cooling limit:

k_{B} T_{D} = \frac{ℏΓ}{2}, T_{D} = \frac{ℏΓ}{2 k _{B}} .

Numerically $T_{D}$ depends only on the linewidth. For the alkali cooling transitions:

Atom	Transition	$Γ/2 π$	$T_{D} = ℏΓ/2 k_{B}$
Sodium	$D_{2}$ , 589 nm	$9.79 MHz$	$235 μ K$
Rubidium-87	$D_{2}$ , 780 nm	$6.07 MHz$	$146 μ K$
Cesium	$D_{2}$ , 852 nm	$5.22 MHz$	$125 μ K$

The corresponding root-mean-square speed for rubidium at $146 μ K$ is about $12 cm s^{- 1}$ , five orders of magnitude below the thermal speed at room temperature. When the first molasses experiments measured temperatures below $T_{D}$ , the two-level model had to be extended; that sub-Doppler physics is the subject of the next lesson.

Cooling by friction (removing energy as alpha times v-squared) balances heating by photon-recoil diffusion (adding energy at rate D over m); their steady state fixes the temperature, minimized at detuning minus half-Gamma.

The magneto-optical trap

Molasses damps velocity but exerts no restoring force on position: an atom diffuses out of the beam overlap region over time. A spatial confinement is added by making the detuning depend on position through the Zeeman effect. Superpose a quadrupole magnetic field from a pair of coils carrying opposite currents (anti-Helmholtz), which vanishes at the centre and grows linearly, $B (x) = B^{'} x$ . The Zeeman shift of the excited-state magnetic sublevels then grows linearly with displacement, and by choosing circular polarizations for the two beams the atom is always pushed back toward the centre.

Consider one dimension with a $J = 0 \to J = 1$ transition. The excited state splits into $m_{J} = - 1, 0, + 1$ sublevels with energies shifting as $\pm μ_{B} g B^{'} x$ . The beam from $+ x$ carries $σ^{-}$ polarization (driving $Δ m = - 1$ ) and the beam from $- x$ carries $σ^{+}$ ( $Δ m = + 1$ ). At a displacement $x > 0$ the $m_{J} = - 1$ sublevel is Zeeman-shifted toward resonance with the red-detuned $σ^{-}$ beam, which pushes the atom back toward the centre; at $x < 0$ the mirror-image geometry pushes the other way. The force acquires a position-dependent term:

F = - α v - κ x, κ = \frac{μ _{B} g B ^{'}}{ℏ k} α,

the equation of a damped harmonic oscillator. The friction $α$ cools; the spring constant $κ$ confines. This is the magneto-optical trap (MOT), the standard first stage of nearly every cold-atom experiment.

A magneto-optical trap: three orthogonal pairs of counter-propagating circularly polarized beams overlap a quadrupole field from anti-Helmholtz coils, giving both velocity damping and a position-dependent restoring force.

A MOT loaded from a slowed beam or from background vapour collects $1 0^{7}$ to $1 0^{10}$ atoms in a fraction of a second, at temperatures near the Doppler limit and densities of $1 0^{10}$ – $1 0^{11} c m^{- 3}$ . Its density is capped by two effects: re-absorption of scattered photons produces an outward radiation-pressure repulsion between atoms, and light-assisted collisions eject atom pairs. Reaching quantum degeneracy therefore requires the trap to be turned off and the cooling continued by mechanisms that do not scatter photons, developed in the following lessons.

One-dimensional MOT level scheme: the excited sublevels split linearly with position, so at positive x the m=minus-one sublevel Zeeman-tunes toward the red-detuned sigma-minus beam and the resulting scattering pushes the atom back.

The MOT combines the two ingredients built in this lesson: the velocity dependence of the scattering force (friction, from the Doppler shift) and a position dependence (confinement, from the Zeeman shift). The Doppler limit sets the temperature of the loaded cloud, and the recoil kicks that impose that limit are the same kicks that, uncontrolled, prevent Doppler cooling from reaching the ground state of motion.

Foot, Atomic Physics, §9.1 and §7.3–7.6, derives $ρ_{ee}$ and the scattering rate from the optical Bloch equations; the saturation intensity $I_{sat} = π h c Γ/3 λ^{3}$ is the standard two-level result. See also Metcalf & van der Straten, Laser Cooling and Trapping, Ch. 2–3. ↩
Metcalf & van der Straten, Laser Cooling and Trapping, Ch. 6 — Deceleration of an Atomic Beam; the Zeeman slower uses a spatially varying magnetic field to hold the Doppler-shifted atom on resonance as it decelerates. Rubidium-87 $D_{2}$ data: $Γ/2 π = 6.065 MHz$ , $λ = 780.24 nm$ (D. A. Steck, Rubidium 87 D Line Data, https://steck.us/alkalidata/). ↩
Foot, Atomic Physics, §9.3, gives the diffusion–friction balance $k_{B} T = D / α$ and the minimum $k_{B} T_{D} = ℏΓ/2$ at $δ = - Γ/2$ ; the same result is in Metcalf & van der Straten, Ch. 7 §7.1. Linewidths: NIST Atomic Spectra Database, https://www.nist.gov/pml/atomic-spectra-database. ↩