Magneto-Optical Traps

A magneto-optical trap, or MOT, combines red-detuned optical molasses with a magnetic-field gradient and polarization-selective transitions. The Doppler shift makes the force oppose velocity. The Zeeman shift makes the force oppose displacement. Near the trap center,

$$\mathbf F \simeq - \boldsymbol{\kappa}\mathbf r - \boldsymbol{\alpha}\mathbf v.$$

The first term confines; the second damps. Neither the light alone nor the quadrupole magnetic field alone generally provides the standard MOT force. The trap exists because displacement changes which counterpropagating beam is closer to resonance.

A MOT is an open, driven, dissipative system. It continually scatters photons, diffuses momentum, optically pumps internal states, loads atoms from an external source, and loses them through collisions. Its steady state is not simply thermal equilibrium in a conservative potential.

Required background. Doppler cooling provides the velocity-dependent scattering rate; radiation pressure relates that rate to momentum transfer; atomic Zeeman shifts determine which beam becomes resonant after displacement.

One-dimensional model

Conventions

Take two beams along $\pm x$, both with wave-number magnitude $k$, and a closed $J=0\to J'=1$ transition as the simplest model. The ground state is unshifted; the two circular components address oppositely shifted excited sublevels. Use atom-minus-laser detuning:

$$\Delta = \omega_0-\omega_L.$$

Red detuning means $\Delta>0$. Let the local magnetic field be

$$B_x(x) = B'x.$$

The two beam helicities address transitions with opposite Zeeman shifts. Label the polarizations so that, for positive effective magnetic moment $\mu_{\mathrm{eff}}$,

$$\beta = \frac{\mu_{\mathrm{eff}}B'}{\hbar} > 0.$$

For velocity $v$ along $+x$, define the effective detunings

$$\begin{aligned} \Delta_+(x,v) &= \Delta + kv + \beta x, \\ \Delta_-(x,v) &= \Delta - kv - \beta x. \end{aligned}$$

The subscripts identify beam momentum $\pm\hbar k$. These equations already encode the chosen helicities. Reversing either the field gradient or both beam helicities reverses the sign of the position term.

Why the displaced atom is pushed inward

Set $v=0$ and place the atom at $x>0$. Then

$$\Delta_+ = \Delta+\beta x, \qquad \Delta_- = \Delta-\beta x.$$

For red detuning and small displacement, the $-x$ beam is closer to resonance:

$$|\Delta_-| < |\Delta_+|.$$

It therefore scatters more strongly and supplies momentum $-\hbar k$, toward the origin. At $x<0$, the roles reverse. The polarization assignment is correct only if this sign test succeeds.

At $x>0$, the Zeeman shift moves the inward-propagating beam closer to resonance. Panel (b) uses the exact weak-beam force at $\Delta=\Gamma/2$, with $u=\beta x/\Gamma$ and $F_0=\hbar k\Gamma s/2$. The dashed tangent is $F/F_0=-2u$. Panel (c) solves the loading equation for the same $\mathcal L$ and $\gamma$, with $A=\beta_2/V_2$ and $A\mathcal L/\gamma^2=0$ or $2$. Both curves have the same initial loading slope; two-body loss reduces the asymptotic population.

Restoring-force derivation

Weak-saturation scattering rate

Let $\Gamma$ be the excited-population decay rate. If the sum of the beams’ detuned excitation parameters is small, use

$$R(\delta) = \frac{\Gamma s}{2} \frac{1}{ 1+(2\delta/\Gamma)^2 }.$$

The net force is

$$F(x,v) = \hbar k \left[ R(\Delta_+) - R(\Delta_-) \right].$$

This compact expression contains both trapping and damping. It remains a weak-excitation transition model: real MOTs require ground-manifold optical pumping, repumping, beam saturation, and a position-dependent quantization axis.

Linear expansion

Define

$$q = kv+\beta x.$$

Near $x=v=0$,

$$R(\Delta\pm q) \simeq R(\Delta) \pm qR'(\Delta).$$

Therefore

$$\begin{aligned} F(x,v) &\simeq 2\hbar kR'(\Delta) \left( kv+\beta x \right) \\ &= -\alpha v-\kappa x, \end{aligned}$$

with

$$\alpha = -2\hbar k^2R'(\Delta),$$

and

$$\kappa = -2\hbar k\beta R'(\Delta).$$

For

$$D = 1+\left(\frac{2\Delta}{\Gamma}\right)^2,$$

the derivative is

$$R'(\Delta) = - \frac{ 4s(\Delta/\Gamma) }{ D^2 }.$$

Hence

$$\alpha = 8\hbar k^2s \frac{ \Delta/\Gamma }{ D^2 },$$

and

$$\kappa = 8\hbar k\beta s \frac{ \Delta/\Gamma }{ D^2 }.$$

For $\Delta>0$ and $\beta>0$, both coefficients are positive. A useful consistency relation is

$$\frac{\kappa}{\alpha} = \frac{\beta}{k} = \frac{ \mu_{\mathrm{eff}}B' }{ \hbar k }.$$

The same spectral slope sets damping and confinement; the field gradient converts displacement into an effective Doppler shift.

Saturation and shared excited-state population

At higher intensity, one often replaces the denominator by

$$1+s_{\mathrm{tot}} + \left( \frac{2\delta}{\Gamma} \right)^2.$$

This is only a representative correction. Counterpropagating beams share the same excited-state population, while multilevel Clebsch–Gordan coefficients and optical pumping make the saturation state dependent. A quantitative MOT model generally solves position- and velocity-dependent optical Bloch or rate equations.

Three-dimensional geometry

Quadrupole field

An anti-Helmholtz coil pair creates a field zero near the trap center. If $B'>0$ denotes the axial gradient magnitude, an ideal local field is

$$\mathbf B(\mathbf r) = B' \left( \frac{x}{2}\hat{\mathbf x} + \frac{y}{2}\hat{\mathbf y} - z\hat{\mathbf z} \right).$$

The factors are required by

$$\boldsymbol{\nabla}\cdot\mathbf B = 0.$$

Here $\partial_zB_z=-B'$; the signed axial gradient is negative. The radial gradient is half the axial magnitude. Consequently, equal beam parameters do not imply equal spring constants in all directions.

Six-beam configuration

The conventional three-dimensional MOT uses one counterpropagating beam pair along each Cartesian axis. Relative helicities are chosen so that the beam propagating toward the field zero addresses the locally favored transition.

The labels $\sigma^+$ and $\sigma^-$ are defined relative to a quantization axis, not merely by a laboratory drawing of circular polarization. Because the quadrupole field changes direction through the cloud, a correct sign argument must track:

1. beam propagation direction;
2. local magnetic-field direction;
3. the addressed $\Delta m_F=\pm1$ transition;
4. the transition’s effective Zeeman shift.

Near the exact field zero, the quantization axis is not well defined. The simple independent-axis picture is then qualitative; full multilevel dynamics determines optical pumping and dark-state structure.

Alternative optical layouts

Mirror MOTs, pyramidal MOTs, and grating MOTs generate the required beam directions with reflections or diffraction. Two-dimensional MOTs cool and collimate an atomic beam while allowing longitudinal flux. Narrow-line MOTs use a small $\Gamma$, so gravity, recoil, and field curvature become more prominent. Molecular and blue-detuned MOTs can require type-II transitions, polarization or magnetic remixing, and force mechanisms beyond the simple two-level type-I model.

The local test remains the same: calculate the actual transition shifts and show that displacement increases inward scattering.

Linear trap dynamics

Damped oscillator

Along one principal axis,

$$m\ddot x + \alpha\dot x + \kappa x = 0.$$

Define

$$\omega_0 = \sqrt{\frac{\kappa}{m}}, \qquad \Gamma_m = \frac{\alpha}{2m}.$$

The characteristic exponents are

$$r_\pm = -\Gamma_m \pm \sqrt{ \Gamma_m^2-\omega_0^2 }.$$

The motion is underdamped when $\Gamma_m<\omega_0$, critically damped when they are equal, and overdamped when $\Gamma_m>\omega_0$. Many broad-line MOTs are strongly damped, so a displaced cloud may return without visible oscillation.

The measured relaxation rates need not equal this two-level prediction. Sub-Doppler forces, beam imbalance, multiple scattering, changing cloud size, and internal-state delays alter them.

Local effective potential

The position part of the linear force can be represented by

$$U_{\mathrm{eff}}(x) = \frac{1}{2}\kappa x^2.$$

This is a local effective potential, useful for cloud-size and sag estimates. The full MOT force is dissipative, velocity dependent, saturating, and generally not derivable from a global scalar potential.

Cloud size

If one principal direction is approximately harmonic and the motional distribution is thermal,

$$n(x) \propto \exp \left( - \frac{\kappa x^2}{2k_{\mathrm B}T} \right).$$

The rms width is

$$\sigma_x = \sqrt{ \frac{k_{\mathrm B}T}{\kappa} }.$$

This relation can infer $\kappa$ from an independently measured temperature and cloud width. Anisotropy alone does not invalidate a per-axis relation: independent harmonic directions can have different effective temperatures. Collective expansion, spatially varying diffusion, or nonharmonic forces require a more detailed model. In fact, stationary second-moment equations for linear damping and constant diffusion already imply $\kappa\langle x^2\rangle=m\langle v^2\rangle$, even without assuming a Gaussian shape beforehand.

Gravity and constant forces

Along vertical coordinate $z$, add gravity:

$$F_z = -\kappa_z z-\alpha_z v_z-mg.$$

The equilibrium sag is

$$z_g = -\frac{mg}{\kappa_z}.$$

For broad-line alkali MOTs this may be small. In narrow-line MOTs, the maximum optical force and spring constant can be much smaller, so gravity substantially shifts the cloud and changes which part of the Zeeman profile is sampled.

A beam imbalance or stray radiation pressure similarly adds a constant force $F_0$ and shifts the center by

$$x_0 = \frac{F_0}{\kappa}.$$

A uniform magnetic bias shifts the quadrupole zero. Center position versus bias field is therefore a useful alignment diagnostic.

Capture

Force limit

For one monochromatic beam driving an isolated closed transition in its steady state (no extra dephasing), the maximum over detuning is

$$F_{\max} \simeq \frac{\hbar k\Gamma}{2} \frac{s}{1+s}.$$

The corresponding maximum acceleration is

$$a_{\max} = \frac{F_{\max}}{m}.$$

If a particle experiences that acceleration over distance $L$, an optimistic kinematic upper bound is

$$v_c \lesssim \sqrt{2a_{\max}L}.$$

Actual capture velocities are smaller because the force is resonant only over part of the trajectory, Gaussian beams have finite diameter, optical pumping leaks population, and atoms can leave transversely.

Spectral capture window

Substantial scattering requires at least one beam to satisfy roughly

$$\left| \Delta \pm(kv+\beta x) \right| \lesssim \frac{\Gamma}{2} \sqrt{1+s_{\mathrm{tot}}}.$$

The two signs are correlated as in $\Delta_\pm$; they are not independent choices. This linewidth estimate is not a sharp boundary, since the Lorentzian has tails. The gradient helps bring displaced atoms into resonance, but an excessively large gradient can make the resonant shell too narrow and reduce the capture volume. Detuning, intensity, gradient, and beam diameter must be optimized together.

Loading sources

Common loading architectures include:

• direct capture from a low-pressure vapor;
• a slowed atomic beam;
• a two-dimensional MOT feeding a science chamber;
• buffer-gas or cryogenic molecular beams followed by laser slowing;
• recapture from another optical or magnetic stage.

The relevant source metric is phase-space flux into the MOT’s capture acceptance, not total particle flux alone.

Loading and loss dynamics

General rate equation

Let $\mathcal L$ be the loading rate, $\gamma$ the one-body loss rate, and $\beta_2$ the two-body atom-loss coefficient, with units of volume per time. This is distinct from the frequency gradient $\beta$ used above. The number of atoms lost per collision is included in $\beta_2$, so no additional factor of two is implied. Then

$$\frac{dN}{dt} = \mathcal L - \gamma N - \beta_2 \int n(\mathbf r,t)^2\,d^3r.$$

One-body loss includes collisions with background gas and source particles. The quadratic term includes light-assisted cold collisions and other density-dependent processes.

For a Gaussian density profile (with widths fixed when solving the number equation),

$$n(\mathbf r) = n_0 \exp \left[ - \sum_i \frac{x_i^2}{2\sigma_i^2} \right],$$

and

$$\int n^2\,d^3r = \frac{N^2}{V_2}, \qquad V_2 = 8\pi^{3/2} \sigma_x\sigma_y\sigma_z.$$

The number equation becomes

$$\dot N = \mathcal L - \gamma N - \frac{\beta_2}{V_2}N^2.$$

One-body limit

When two-body loss is negligible and $N(0)=0$,

$$N(t) = \frac{\mathcal L}{\gamma} \left( 1-e^{-\gamma t} \right).$$

Thus

$$N_{\mathrm{ss}} = \frac{\mathcal L}{\gamma}, \qquad \tau_{\mathrm{load}} = \frac{1}{\gamma}.$$

Changing vapor pressure often raises both $\mathcal L$ and $\gamma$, so the largest loading rate need not maximize steady-state number or lifetime.

Two-body loss

For constant $V_2$, define

$$A = \frac{\beta_2}{V_2}.$$

The steady state solves

$$AN_{\mathrm{ss}}^2 + \gamma N_{\mathrm{ss}} - \mathcal L = 0,$$

giving

$$N_{\mathrm{ss}} = \frac{ 2\mathcal L }{ \gamma + \sqrt{ \gamma^2+4A\mathcal L } }.$$

For $N(0)=0$, define $\lambda=\sqrt{\gamma^2+4A\mathcal L}$. The exact constant-volume loading curve is

$$N(t)=N_{\rm ss}\frac{1-e^{-\lambda t}} {1+[(\lambda-\gamma)/(\lambda+\gamma)]e^{-\lambda t}}.$$

It is not a single exponential. Its initial slope is $\mathcal L$, while small deviations from the steady state relax at $\lambda=\gamma+2AN_{\rm ss}$.

At high density, $V_2$ may grow with $N$ because rescattered photons expand the cloud. A constant-volume quadratic fit can then misidentify the loss coefficient.

Separating collision coefficients

Measure decay after switching off loading and independently image the cloud volume. Vary density to separate one- and two-body contributions. In a constant-peak-density cloud, volume grows with atom number and even genuine two-body collisions can produce a loss term proportional to $N$ (Exercise 1). A fitted exponential lifetime alone does not identify a collision mechanism.

Density limits

Light-assisted collisions

Near-resonant light couples colliding atom pairs to excited molecular potentials. The released kinetic energy or radiative escape can eject one or both particles. The resulting $\beta_2$ depends on detuning, intensity, hyperfine state, molecular potentials, and trap depth.

Radiation trapping and multiple scattering

A spontaneously emitted photon can be reabsorbed by another atom. Repeated scattering produces an effective repulsion and additional momentum diffusion. At sufficiently large $N$, the cloud can enter a constant-density regime in which its radius grows while peak density changes little.

This is collective radiative transport, not a modification of the single-atom spring constant alone.

Dark spontaneous-force traps

A dark SPOT reduces the repump intensity in the cloud center, shelving most atoms in a hyperfine ground state that interacts weakly with the cooling light. The outer bright shell continues to confine and load. Lower central excited-state fraction suppresses rescattering and light-assisted loss, allowing higher density.

The method does not make the MOT conservative. It spatially separates the bright capture region from a darker storage region.

Temperature and phase-space density

The magnetic gradient provides confinement but does not by itself determine temperature. Momentum diffusion and velocity-dependent forces still set the motional distribution. Depending on species, transition type, field, and polarization:

• a two-level Doppler estimate may be adequate;
• polarization-gradient cooling may lower temperature;
• the quadrupole field may disrupt sub-Doppler coherences;
• narrow-line recoil and gravity may dominate;
• multiple scattering may heat or broaden the cloud;
• anisotropic beams may produce different temperatures by axis.

For this reason, many experiments switch off the field gradient and use a short optical-molasses stage after MOT loading.

For a dilute thermal Gaussian cloud, peak phase-space density is

$$\mathcal D_0 = n_0 \left( \frac{ 2\pi\hbar^2 }{ mk_{\mathrm B}T } \right)^{3/2}.$$

A MOT is normally a high-flux precooling stage, not the final route to quantum degeneracy. Near-resonant scattering and density-dependent loss usually limit broad-line MOT phase-space density well before degeneracy. This is a limitation of that operating regime, not a universal ban on laser cooling degenerate gases.

Operating sequence

A common broad-line alkali sequence is:

1. Load. Use large beams, substantial intensity, moderate red detuning, and a gradient chosen for capture volume.
2. Compress. Increase gradient, change detuning, and often reduce repump power to raise density.
3. Cool. Turn off or reduce the field and use optical molasses or another sub-Doppler stage.
4. State prepare. Optically pump into the desired hyperfine and Zeeman state.
5. Transfer. Load a conservative optical or magnetic trap.

The order and timing matter. Increasing gradient may compress position while heating momentum; reducing repump may raise density while slowing loading; leaving near-resonant light on during transfer may increase light-assisted loss.

Diagnostics

Fluorescence

For an optically thin cloud with background counts subtracted, if each atom scatters at total rate $R_{\mathrm{sc}}$ and the collection efficiency is $\eta$, the detected photon rate is

$$C = \eta N R_{\mathrm{sc}}.$$

Converting fluorescence to $N$ requires a model or calibration of $R_{\mathrm{sc}}$, including polarization, multilevel populations, detuning, and intensity. With appreciable reabsorption, a photon can scatter several times before escaping: the emitted flux needs a radiative-transport model, not a sum of independently detected atomic events.

Absorption imaging

Calibrated optical depth gives a spatial column density. Saturation, detuning, optical pumping during the probe, finite resolution, and multiple scattering must be controlled. Combining absorption images with time-of-flight yields number, size, and temperature.

Mechanical response

Displace the cloud with a bias field or beam imbalance, release the perturbation, and fit its return. The response constrains $\alpha$ and $\kappa$. Modulating the gradient or intensity can locate mechanical resonances, but parametric heating and delayed internal dynamics can shift the apparent frequency.

Repumping and dark states

Off-resonant excitation can move an alkali atom to the other hyperfine ground manifold. Without repumping it becomes dark to the cooling cycle. Repump intensity therefore changes fluorescence, loading, and collision rates together. A dimmer cloud need not contain fewer atoms.

For subsequent coherent experiments, optical dipole traps or optical tweezers can retain atoms after the near-resonant cooling light and quadrupole field are removed.

Exercises

1. Two-body collisions with exponential number decay

A Gaussian cloud expands as atoms are added so that its peak density $n_0$ stays fixed. Express $\int n^2d^3r$ in terms of $n_0$ and $N$, then solve the loading equation. Can an exponential curve rule out two-body loss?

Solution

Integrating the Gaussian gives $N=(2\pi)^{3/2}n_0\sigma_x\sigma_y\sigma_z$ and $\int n^2d^3r=\pi^{3/2}n_0^2\sigma_x\sigma_y\sigma_z=n_0N/2^{3/2}$. Thus

$$\dot N=\mathcal L-\gamma_{\rm eff}N,\qquad \gamma_{\rm eff}=\gamma+\frac{\beta_2n_0}{2^{3/2}}.$$

For an initially empty trap, $N(t)=\mathcal L(1-e^{-\gamma_{\rm eff}t})/\gamma_{\rm eff}$. The number dependence of loss is linear even though individual collisions involve two atoms. Imaging the changing volume separates these possibilities.

2. Check the quadrupole field

Verify that

$$\mathbf B = B' \left( \frac{x}{2}\hat{\mathbf x} + \frac{y}{2}\hat{\mathbf y} - z\hat{\mathbf z} \right)$$

obeys Maxwell’s equation in a current-free trapping region. What is the ratio of axial to radial gradient magnitudes?

Solution

The divergence is

$$\boldsymbol{\nabla}\cdot\mathbf B = \frac{B'}{2} + \frac{B'}{2} - B' = 0.$$

All cross derivatives also vanish, so $\boldsymbol\nabla\times\mathbf B=0$ as required for a static field in the current-free region. The axial gradient magnitude is $B'$, while either radial gradient magnitude is $B'/2$. The ratio is therefore $2:1$.

3. Damping regime

A trapped atom has

$$m = 1.44\times10^{-25}\,\mathrm{kg}, \quad \alpha = 3.0\times10^{-22}\,\mathrm{kg\,s^{-1}}, \quad \kappa = 2.0\times10^{-19}\,\mathrm{N\,m^{-1}}.$$

Compute $\omega_0$, $\Gamma_m$, and determine whether the motion is under- or overdamped.

Solution

The undamped frequency is

$$\omega_0 = \sqrt{\frac{\kappa}{m}} \simeq 1.18\times10^3\,\mathrm{s^{-1}}.$$

The damping parameter is

$$\Gamma_m = \frac{\alpha}{2m} \simeq 1.04\times10^3\,\mathrm{s^{-1}}.$$

Because $\Gamma_m<\omega_0$, the oscillator is underdamped, but only weakly:

$$\sqrt{\omega_0^2-\Gamma_m^2} \simeq 5.5\times10^2\,\mathrm{s^{-1}}.$$

4. One-body loading curve

A MOT loads at $\mathcal L=2.0\times10^7\,\mathrm{s^{-1}}$ and has $\gamma=0.20\,\mathrm{s^{-1}}$. Neglect two-body loss. Find the steady atom number and the time to reach $90\%$ of it.

Solution

The steady number is

$$N_{\mathrm{ss}} = \frac{\mathcal L}{\gamma} = 1.0\times10^8.$$

Set

$$0.90 = 1-e^{-\gamma t_{90}}.$$

Then

$$t_{90} = \frac{\ln 10}{\gamma} \simeq 11.5\,\mathrm s.$$

5. Two-body-limited steady state

Let

$$\mathcal L = 1.0\times10^7\,\mathrm{s^{-1}}, \quad \gamma = 0.10\,\mathrm{s^{-1}}, \quad \frac{\beta_2}{V_2} = 1.0\times10^{-9}\,\mathrm{s^{-1}}.$$

Find $N_{\mathrm{ss}}$.

Solution

With $A=\beta_2/V_2$,

$$N_{\mathrm{ss}} = \frac{ 2\mathcal L }{ \gamma+\sqrt{\gamma^2+4A\mathcal L} }.$$

Numerically,

$$\sqrt{\gamma^2+4A\mathcal L} = \sqrt{0.01+0.04} \simeq 0.224\,\mathrm{s^{-1}}.$$

Thus

$$N_{\mathrm{ss}} \simeq \frac{2.0\times10^7}{0.324} \simeq 6.2\times10^7.$$

The one-body-only prediction would be $10^8$, so the curvature is experimentally significant.

6. Cloud width

For $T=100\,\mu\mathrm K$ and $\kappa=2.0\times10^{-19}\,\mathrm{N\,m^{-1}}$, estimate the rms cloud width along one axis.

Solution

Use

$$\sigma = \sqrt{ \frac{k_{\mathrm B}T}{\kappa} }.$$

Then

$$\sigma \simeq \sqrt{ \frac{ (1.381\times10^{-23})(1.0\times10^{-4}) }{ 2.0\times10^{-19} } } \simeq 8.3\times10^{-5}\,\mathrm m.$$

So the rms width is about $83\,\mu\mathrm m$.

7. Gravitational sag

Using the mass and spring constant from Exercise 3, calculate the vertical sag.

Solution

The magnitude is

$$|z_g| = \frac{mg}{\kappa}.$$

Substitution gives

$$|z_g| = \frac{ (1.44\times10^{-25})(9.81) }{ 2.0\times10^{-19} } \simeq 7.1\times10^{-6}\,\mathrm m.$$

The cloud sags by about $7\,\mu\mathrm m$ for these parameters.

8. Diagnose an anti-trap

At $x>0$ and $v=0$, an experiment finds that the $+x$ beam is closer to resonance than the $-x$ beam. What force results, and which changes restore trapping?

Solution

The $+x$ beam then scatters more strongly and pushes the atom farther toward $+x$. The position force is anti-restoring:

$$F \simeq +|\kappa|x.$$

Reversing the magnetic-field gradient changes the Zeeman sign. Reversing both beam helicities changes which transition each beam addresses. Either operation can restore the desired condition that, at $x>0$, the $-x$ beam is closer to resonance. Reversing both the gradient and helicities together leaves their relative sign unchanged and therefore does not fix the anti-trap.

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