Laser cooling and trapping techniques rely on selectively exciting transitions between atomic substates by controlling the polarization, propagation direction, and frequency of the laser light.

For simplicity consider a F = 0 to F = 1 transition.  Assume that the ground state has total angular momentum F = 0 and therefore no magnetic substructure.  The F = 1 excited state has a magnetic moment m and three magnetically degenerate substates characterized by the quantum numbers m_f = 0, ±1.  Assume the resonance frequency for the transition is w₀.  In the absence of an external magnetic field each of the degenerate substates emits photons with frequency w₀ with the same probability G.  The FWHM of the Lorentzian-shaped emission line is G.  In the presence of an applied magnetic B along the z-direction the degeneracy of the excited state is removed.  The m_f = 1 state is shifted up in energy by DE_n,l,j,f,1 = g_Fm_BB, and the m_f = -1 is shifted down in energy by DE_n,l,j,f,-1 = -g_Fm_BB, relative to the m_f = 0 state.

Assume that a laser beam is propagating through an atomic vapor, and that the wave vector k of the laser light is pointing into the -z direction.  Then conservation of angular momentum dictates that right circular polarized light induces Dm_f = 1 transitions and left-circular polarized light induces Dm_f = -1 transitions between the ground state and the exited states.

First consider laser cooling in one dimension with a fixed frequency laser in the absence of an external magnetic field.  A counter-propagating laser beam with wave vector k illuminates an atom traveling with velocity v.  Let v point into the z and k into the -z direction.  The lab frequency of the laser is w_L.  In the absence of an external magnetic field the F = 0 to F = 1 atomic transition is a pure two-level transition and the polarization of the laser light can be ignored.  The probability of absorption as a function of the laser frequency peaks at w_L = w₀ - Dw, where Dw = w₀v/c = kv.  The Doppler effect shifts the laser frequency by Dw towards higher frequencies in the atom’s rest frame.

When the atom absorbs a photon, it suffers a momentum loss hk and slows down.  After a time t ~ 1/G the atom spontaneously emits a photon and returns to the ground state.  If the momentum of the emitted photon is hk’, then the change in the atoms momentum is -hk’.  The net change in the momentum of the atom due to a single absorption - emission event is Dp = h(k-k’).  Since |k| ~ |k’| , Dp can range from 0 to ~2hk.  After N absorption - emission events, however, the momentum change due to absorption is Nhk while the momentum change due to spontaneous emission averages to zero, because the spontaneous emission has an isotropic angular distribution.  The total momentum change of the atom is therefore Dp = Nhk in the direction of propagation of the laser.  The atom’s translational kinetic energy KE = p²/2m decreases and it cools.  Laser cooling involves converting the atom’s translational kinetic energy into optical energy carried away by spontaneously emitted photons.

The acceleration of the atom depends on the number of photons per second absorbed and then re-emitted via  spontaneous emission.  The maximum value of the acceleration is a_max = hkG/(2m), which requires that the saturation parameter s = I/I_s >>1.   Here I is the laser intensity and I_s is called the saturation intensity.  It is given by

I_s = phwG/(3l²).

If the atom has an initial velocity v, and the laser frequency is adjusted such that w_L = w₀ (1- v/c), then, as the atom cools and its speed decreases, the laser light will no longer be resonant with the transition.  We expect that further cooling can be accomplished as long as w₀Dv/c < ~G, or Dv < ~G/k.  Power broadening the atomic transition by using ultra-high intensity lasers can overcome this problem.

But further cooling can also be accomplished by maintaining resonance between the laser frequency and the atomic transition frequency.  Two common ways to maintain resonance are sweeping the atomic transition frequency with an inhomogeneous magnetic field (Zeeman tuned cooling) and sweeping the laser frequency (chirped cooling).   Both methods produce high-flux slow atomic beams having a narrow range of final velocities.  Zeeman tuned cooling, however, can produce cold atoms continuously, while the number of slow atoms produce by chirped cooling is limited by the frequency sweep duty cycle.

In the presence of an external field B pointing in the z-direction the m_f = 1 state is shifted up in energy by DE_n,l,j,f,1 = g_Fm_BB, and the m_f = -1 is shifted down in energy by DE_n,l,j,f,-1 = -g_Fm_BB, relative to the m_f = 0 state.  Right circular polarized light (s⁺) propagating into the negative z-direction induces Dm_f = 1 transitions and left-circular polarized light (s ^-) propagating into the negative z-direction induces Dm_f = -1 transitions.  Two possible configurations for slowing down the atoms using Zeeman tuned cooling are shown in the figures below.

For s⁺ cooling, the s⁺ polarized laser is tuned to be resonant with (or slightly above) the atomic transition frequency in the absence of a magnetic field.  The atoms enter the magnetic field at z = 0, and exit at z = L.  The magnitude of the magnetic field decreases from z = 0 to z = L, B(0) > B(L) = 0.  At z = 0 atoms with velocity v₀ in the z-direction are in resonance with the laser, if v₀ is given by the relationship

Dw_D = w₀v₀/c = g_Fm_BB/h.

The Dm_f = 1 transition frequency has increased by g_Fm_BB/h because of the presence of the external field, and the laser frequency has been shifted up by Dw_D = w₀v₀/c because of the Doppler effect.

As the atoms move through the field region they slow down.  The Doppler shift decreases, but the external field also decreases and thus the Zeeman shift decreases.  By adjusting B(z) so that the Doppler shift always matches the Zeeman shift, resonance between the laser frequency and the Dm_f = 1 transition frequency can be maintained.  Atoms with speeds v_i < v₀ are slowed down starting at a location z_i > 0.  The matching of the Doppler shift and the Zeeman shift therefore does not have to be perfect.   If atoms starting with speed v₀ at z = 0 are cooled to speed v_i before they reach z_i, then they will just drift to position z_i before they will be cooled further.  As long as the magnitude of the field gradient stays below a maximum value, cooling can be maintained.

All atoms exit the field region at z = L with the same final velocity v_f.  Both the "capture velocity" v₀ and the "cooling velocity" v_f are determined by the laser frequency and the change in the magnetic field magnitude along z.

For s ^- cooling, the s ^- polarized laser is tuned to be resonant with the Doppler-shifted atomic transition frequency for atoms moving with velocity v₀. The atoms enter the magnetic field at z = 0, and exit at z = L.  The magnitude of the magnetic field increase from z = 0 to z = L, B(0) = 0 < B(L).  At z = 0 atoms with velocity v₀ in the z-direction are in resonance with the laser with frequency w₀ - Dw_D(0), if v₀ is given by the relationship

Dw_D(0) = w₀v₀/c.

The Dm_f = -1 transition frequency decreases by Dw' = g_Fm_BB/h in the presence of an external field.  At position z the resonance frequency is w₀ - Dw_D(z) - Dw'(z).  For atoms which have slowed down Dw_D(z) has decreased.  Resonance can be maintained if Dw_D(0) = Dw_D(z) + Dw'(z).  The Doppler shift decreases, but the Zeeman shift increases as a function of z.  By adjusting B(z) so that the decrease in the Doppler shift matches the increase in the Zeeman shift, resonance between the laser frequency and the frequency of the Dm_f = -1 transition can be maintained.  Again, as long as the magnitude of the field gradient stays below a maximum value, cooling can be maintained.

Zeeman-shift-induced changes to the atomic transition frequency must occur more slowly as a function of position z than cooling-induced changes in the Doppler shift.  When the acceleration a is equal to a_max, then the maximum allowable magnetic field gradient is given by

(dB/dz)_max = h²k²G²/(2mvm).

We need (dB/dz) < (dB/dz)_max. Magnetic field gradients that violate this inequality simply tune the atomic transition frequency out of resonance with the laser frequency and thus stop the slowing process.

The assumption that the cooling transition is a F = 0 to F = 1 transition is not a limiting assumption.  Laser cooling in an inhomogeneous magnetic field also works for atoms with more general values for the angular momenta.

To create optical molasses in one-dimension two lasers tuned slightly below the resonant frequency of an atomic transition in the absence of a magnetic field counter-propagate along the z-axis.

Assume an atom is moving with velocity v in the z-direction.  In the atom’s rest-frame, the frequency of the laser propagating in the -z direction is shifted up in frequency, bringing it closer to resonance with the transition frequency w₀.  The frequency of the laser propagating in the +z direction is shifted down in frequency in the rest frame of the atom, driving it farther from resonance with the transition frequency w₀.

The atom therefore absorbs a larger number of photons from the laser traveling in the -z direction than from the laser traveling in the +z direction and it begins to slow down, because contributions to the atom’s momentum due to spontaneously emitted photons again average to zero.  As the atom cools, the difference in the absorption rates decreases.  For a stationary atom the absorption rates for the two lasers are equal.  Thus the atom is cooled towards and maintained at v ~ 0.  A plot of the average force (in the z-direction) on the atom as a function of the scaled velocity v’ = 2kv/G is shown in the figure below.

The Doppler cooling limit of optical molasses can be derived by considering the balance between the rate at which atoms lose and gain kinetic energy while interacting with the cooling lasers.  This yields T_D = hG/(2k_B) for the minimum temperature that can be achieved via Doppler cooling.

Optical molasses in one-dimension can easily be generalized to three dimensions.  We need three pairs of mutually perpendicular, counter-propagating lasers. To first order T_D does not change.

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