Polarization gradient cooling
When laser cooling of atoms was first proposed in 1975, the only cooling mechanism considered was Doppler cooling. As such the limit on the temperature was predicted to be the Doppler limit:
formula_1
Here kb is the Boltzman constant, T is the temperature of the atoms, and Γ is the inverse of the excited state's radiative lifetime. Early experiments seemed to be in agreement with this limit. However, in 1988 experiments began to report temperatures below the Doppler limit. These observations would take the theory of PG cooling to explain. There are two different configurations that form polarization gradients: lin⊥lin and σ+σ−. Both configurations provide cooling, however, the type of polarization gradient and the physical mechanism for cooling are different between the two. In the lin⊥lin configuration cooling is achieved via a Sisyphus effect. Consider two counterpropagating electromagnetic waves with equal amplitude and orthogonal linear polarizations formula_2 and formula_3, where k is the wavenumber formula_4. The superposition of formula_5 and formula_6 is given as:
formula_7
Introducing a new pair of coordinates formula_8 and formula_9 the field can be written as:
formula_10
The polarization of the total field changes with z. For example: we see that at formula_11 the field is linearly polarized along formula_12, at formula_13 the field has left circular polarization, at formula_14 the field is linearly polarized along formula_15, at formula_16 the field has right circular polarization, and at formula_17 the field is again linearly polarized along formula_12.
Consider an atom interacting with the field detuned below the transition from atomic states formula_19 and formula_20 (formula_21). The variation of the polarization along z results in a variation in the light shifts of the atomic Zeeman sublevels with z. The Clebsch-Gordan coefficient connecting the formula_22 state to the formula_23 state is 3 times larger than connecting the formula_24 state to the formula_25 state. Thus for formula_26 polarization the light shift is three times larger for the formula_22 state than for the formula_28 state. The situation is reversed for formula_29 polarization, with the light shift being three times larger for the formula_24 state than the formula_25 state. When the polarization is linear, there is no difference in the light shifts between the two states. Thus the energies of the states will oscillate in z with period formula_32. As an atom moves along z, it will be optically pumped to the state with the largest negative light shift. However, the optical pumping process takes some finite time formula_33. For field wavenumber k and atomic velocity v such that formula_34, the atom will travel mostly uphill as it moves along z before being pumped back down to the lowest state. In this velocity range, the atom travels more uphill than downhill and gradually loses kinetic energy, lowering its temperature. This is called the Sisyphus effect after the mythological Greek character. Note that this initial condition for velocity requires the atom to be cooled already, for example through Doppler cooling. For the case of counterpropagating waves with orthogonal circular polarizations the resulting polarization is linear everywhere, but rotates about formula_35 at an angle formula_36. As a result, there is no Sisyphus effect. The rotating polarization instead leads to motion-induced population imbalances in the Zeeman levels that cause imbalances in radiation pressure leading to a damping of the atomic motion. These population imbalances are only present for states with formula_37 or higher. Consider two EM waves detuned from an atomic transition formula_38 with equal amplitudes: formula_39 and formula_40. The superposition of these two waves is:
formula_41
As previously stated, the polarization of the total field is linear, but rotated around formula_35 by an angle formula_36 with respect to formula_44.
Consider an atomic moving along z with some velocity v. The atom sees the polarization rotating with a frequency of formula_45. In the rotating frame, the polarization is fixed, however, there is an inertial field due to the frame rotating. This inertial term appears in the Hamiltonian as follows. formula_46
Here we see the inertial term looks like a magnetic field along formula_35 with an amplitude such that the Larmor precession frequency is equal to rotation frequency in the lab frame. For small v, this term in Hamiltonian can be treated using perturbation theory. Choosing the polarization in the rotating frame to be fixed along formula_44, the unperturbed atomic eigenstates are the eigenstates of formula_49. The rotating term in the Hamiltonian causes perturbations in the atomic eigenstates such that the Zeeman sublevels become contaminated by each other. For formula_50 the formula_51 is light shifted more than the formula_52 states. Thus the steady state population of the formula_51 is higher than that of the other states. The populations are equal for the formula_52 states. Thus states are balanced with formula_55. However, when we change basis we see that populations are not balanced in the z-basis and there is a non-zero value of formula_56 proportional to the atom's velocity:
formula_57
Where formula_58 is the light shift for the formula_59 state. There is a motion induced population imbalance in the Zeeman sublevels in the z basis. For red detuned light, formula_58 is negative, and thus there will be a higher population in the formula_61 state when the atom is moving to the right (positive velocity) and a higher population in the formula_62 state when the atom is moving to the left (negative velocity). From the Clebsch-Gordan coefficients, we see that the formula_61 state has a six times greater probability of absorbing a formula_26 photon moving to the left than a formula_29 photon moving to the right. The opposite is true for the formula_66 state. When the atom moves to the right it is more likely to absorb a photon moving to the left and likewise when the atom moves to the left it is more likely to absorb a photon moving to the right. Thus there is an unbalanced radiation pressure when the atom moves which dampens the motion of the atom, lowering its velocity and therefore its temperature. Note the similarity to Doppler cooling in the unbalanced radiation pressures due to the atomic motion. The unbalanced pressure in PG cooling is not due to a Doppler shift but an induced population imbalance. Doppler cooling depends on the parameter formula_67 where formula_68 is the scattering rate, whereas PG cooling depends on formula_69. At low intensity formula_70 and thus PG cooling works at lower atomic velocities (temperatures) than Doppler Cooling. Both methods of PG cooling surpass the Doppler limit and instead are limited by the one-photon recoil limit:
formula_71
Where M is the atomic mass. For a given detuning formula_72 and Rabi frequency formula_73, dependent on the light intensity, both configurations display a similar scaling at low intensity (formula_74) and large detuning (formula_75):
formula_76
Where formula_77 is a dimensionless constant dependent on the configuration and atomic species. See ref for a full derivation of these results. PG cooling is typically performed using a 3D optical setup with three pairs of perpendicular laser beams with an atomic ensemble in the center. Each beam is prepared with an orthogonal polarization to its counterpropagating beam. The laser frequency detuned from a selected transition between the ground and excited states of the atom. Since the cooling processes rely on multiple transitions between care must be taken such that the atomic does not fall out of these two states. This is done by using a second, "repumping", laser to pump any atoms that fall out back into the ground state of the transition. For example: in cesium cooling experiments, the cooling laser is typically chosen to be detuned from the formula_78 to formula_79 transition and a repumping laser tuned to the formula_80 to formula_81 transition is also used to prevent the Cs atoms from being pumped into the formula_80 state. The atoms must be cooled before the PG cooling, this can be done using the same setup via Doppler cooling. If the atoms are precooled with Doppler cooling, the laser intensity must be lowered and the detuning increased for PG cooling to be achieved. The atomic temperature can be measured using the time of flight (ToF) technique. In this technique, the laser beams are suddenly turned off and the atomic ensemble is allowed to expand. After a set time delay t, a probe beam is turned on to image the ensemble and obtain the spatial extent of the ensemble at time t. By imaging the ensemble at several time delays, the rate of expansion is found. By measuring the rate of expansion of the ensemble the velocity distribution is measured and from this, the temperature is inferred. An important theoretical result is that in the regime where PG cooling functions, the temperature only depends on the ratio of formula_83 to formula_84 and that the cooling approaches the recoil limit. These predictions were confirmed experimentally in 1990 when W.D. Phillips et al. observed such scaling in their cesium atoms as well as a temperature of 2.5formula_85K, 12 times the recoil temperature of 0.198formula_85K for the D2 line of cesium used in the experiment.
Theory ; The σ+σ− Configuration
However, if orthogonal circular polarizations (the σ+σ− configuration) are used, the result is a linear polarization that rotates along the axis of propagation. In the σ+-σ− configuration, the rotating polarization creates a motion-induced population imbalance in the Zeeman sublevels of the atomic ground state resulting in an imbalance in the radiation pressure that opposes the motion of the atom.