The lin⊥lin Configuration

In the lin⊥lin configuration cooling is achieved via a Sisyphus effect. Consider two counterpropagating electromagnetic waves with equal amplitude and orthogonal linear polarizations ${\displaystyle {\vec {E_{1}}}=E_{0}e^{ikz}{\hat {x}}}$ and ${\displaystyle {\vec {E_{2}}}=E_{0}e^{-ikz}{\hat {y}}}$, where k is the wavenumber ${\displaystyle k=\textstyle {\frac {2\pi }{\lambda }}}$. The superposition of ${\displaystyle {\vec {E_{1}}}}$ and ${\displaystyle {\vec {E_{2}}}}$ is given as:

${\displaystyle {\vec {E}}_{tot}={\frac {E_{0}}{\sqrt {2}}}\left(\cos(kz){\frac {{\hat {x}}+{\hat {y}}}{\sqrt {2}}}-i\sin(kz){\frac {-{\hat {x}}+{\hat {y}}}{\sqrt {2}}}\right)}$

Introducing a new pair of coordinates ${\displaystyle {\hat {x}}'=\textstyle {\frac {{\hat {x}}+{\hat {y}}}{\sqrt {2}}}}$ and ${\displaystyle {\hat {y}}'=\textstyle {\frac {-{\hat {x}}+{\hat {y}}}{\sqrt {2}}}}$ the field can be written as:

${\displaystyle {\vec {E}}_{tot}={\frac {E_{0}}{\sqrt {2}}}\left(\cos(kz){\hat {x}}'-i\sin(kz){\hat {y}}'\right)}$

The polarization of the total field changes with z. For example: we see that at ${\displaystyle z=0}$ the field is linearly polarized along ${\displaystyle {\hat {x}}'}$, at ${\displaystyle z=\textstyle {\frac {\lambda }{8}}}$ the field has left circular polarization, at ${\displaystyle z=\textstyle {\frac {\lambda }{4}}}$ the field is linearly polarized along ${\displaystyle {\hat {y}}'}$, at ${\displaystyle z=\textstyle {\frac {3\lambda }{8}}}$ the field has right circular polarization, and at ${\displaystyle z=\textstyle {\frac {\lambda }{2}}}$ the field is again linearly polarized along ${\displaystyle {\hat {x}}'}$.

The Sisyphus effect. The ground state light shifts cause the Zeeman sublevels to oscillate with a period of ${\displaystyle \textstyle {\frac {\lambda }{2}}}$. An atom moves uphill to the right, converting its kinetic energy into potential energy. Near the top of the hill, the atom is excited into an F=3/2 state and decays back into the lowest energy state at the bottom of the hill resulting in an irreversible loss of kinetic energy. As the atom moves through the polarization gradient, this process happens many times.

Consider an atom interacting with the field detuned below the transition from atomic states ${\displaystyle F_{g}=\textstyle {\frac {1}{2}}}$ and ${\displaystyle F_{e}=\textstyle {\frac {3}{2}}}$ (${\displaystyle \hbar {}\omega {}_{field}<E_{eg}}$). 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 ${\displaystyle |g,m_{F}=-\textstyle {\frac {1}{2}}\rangle }$ state to the ${\displaystyle |e,m_{F}=-\textstyle {\frac {3}{2}}\rangle }$ state is 3 times larger than connecting the ${\displaystyle |g,m_{F}=\textstyle {\frac {1}{2}}\rangle }$ state to the ${\displaystyle |e,m_{F}=-\textstyle {\frac {1}{2}}\rangle }$ state. Thus for ${\displaystyle \sigma ^{-}}$ polarization the light shift is three times larger for the ${\displaystyle |g,m_{F}=-\textstyle {\frac {1}{2}}\rangle }$ state than for the ${\displaystyle |e,m_{F}=\textstyle {\frac {1}{2}}\rangle }$ state. The situation is reversed for ${\displaystyle \sigma ^{+}}$ polarization, with the light shift being three times larger for the ${\displaystyle |g,m_{F}=\textstyle {\frac {1}{2}}\rangle }$ state than the ${\displaystyle |e,m_{F}=-\textstyle {\frac {1}{2}}\rangle }$ 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 ${\displaystyle \textstyle {\frac {\lambda }{2}}}$.

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 ${\displaystyle \tau }$. For field wavenumber k and atomic velocity v such that ${\displaystyle kv\approx \tau {}^{-1}}$, 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.

The σ⁺σ⁻ Configuration

For the case of counterpropagating waves with orthogonal circular polarizations the resulting polarization is linear everywhere, but rotates about ${\displaystyle {\hat {z}}}$ at an angle ${\displaystyle -kz}$. 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 ${\displaystyle F=1}$ or higher.

Consider two EM waves detuned from an atomic transition ${\displaystyle F_{g}=1\rightarrow F_{e}=2}$ with equal amplitudes: ${\displaystyle {\vec {E_{1}}}=E_{0}e^{ikz}\textstyle {\frac {-{\hat {x}}-i{\hat {y}}}{\sqrt {2}}}}$ and ${\displaystyle {\vec {E_{2}}}=E_{0}e^{-ikz}\textstyle {\frac {{\hat {x}}-i{\hat {y}}}{\sqrt {2}}}}$. The superposition of these two waves is:

${\displaystyle {\vec {E_{tot}}}={\sqrt {2}}E_{0}(\sin(kz){\hat {x}}+\cos(kz){\hat {y}})}$

As previously stated, the polarization of the total field is linear, but rotated around ${\displaystyle {\hat {z}}}$ by an angle ${\displaystyle -kz}$ with respect to ${\displaystyle {\hat {y}}}$.

Consider an atomic moving along z with some velocity v. The atom sees the polarization rotating with a frequency of ${\displaystyle kv}$. 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.

${\displaystyle {\hat {H}}_{rot}=kvF_{z}}$

Here we see the inertial term looks like a magnetic field along ${\displaystyle {\hat {z}}}$ 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 ${\displaystyle {\hat {y}}}$, the unperturbed atomic eigenstates are the eigenstates of ${\displaystyle {\hat {F}}_{y}}$. The rotating term in the Hamiltonian causes perturbations in the atomic eigenstates such that the Zeeman sublevels become contaminated by each other. For ${\displaystyle F_{g}=1}$ the ${\displaystyle |g,m_{f}=0\rangle _{y}}$ is light shifted more than the ${\displaystyle |g,m_{f}=\pm {1}\rangle _{y}}$ states. Thus the steady state population of the ${\displaystyle |g,m_{f}=0\rangle _{y}}$ is higher than that of the other states. The populations are equal for the ${\displaystyle |g,m_{f}=\pm {1}\rangle _{y}}$ states. Thus states are balanced with ${\displaystyle \langle {\hat {F}}_{y}\rangle =0}$. However, when we change basis we see that populations are not balanced in the z-basis and there is a non-zero value of ${\displaystyle \langle {\hat {F}}_{z}\rangle }$ proportional to the atom's velocity:^[8]

${\displaystyle \langle {\hat {F}}_{z}\rangle ={\frac {40\hbar {}kv}{17\Delta _{0}^{'}}}}$

Clebsch Gordan coefficients for the ${\displaystyle F_{g}=1}$ to ${\displaystyle F_{e}=2}$ transition

Where ${\displaystyle \Delta _{0}^{'}}$ is the light shift for the ${\displaystyle m_{F}=0}$ state. There is a motion induced population imbalance in the Zeeman sublevels in the z basis. For red detuned light, ${\displaystyle \Delta _{0}^{'}}$ is negative, and thus there will be a higher population in the ${\displaystyle |g,m_{f}=-1\rangle }$ state when the atom is moving to the right (positive velocity) and a higher population in the ${\displaystyle |g,m_{f}=1\rangle }$ state when the atom is moving to the left (negative velocity). From the Clebsch-Gordan coefficients, we see that the ${\displaystyle |g,m_{f}=-1\rangle }$ state has a six times greater probability of absorbing a ${\displaystyle \sigma ^{-}}$ photon moving to the left than a ${\displaystyle \sigma ^{+}}$ photon moving to the right. The opposite is true for the ${\displaystyle |g,m_{f}=1>}$ 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 ${\displaystyle \textstyle {\frac {kv}{\Gamma }}}$ where ${\displaystyle \Gamma }$ is the scattering rate, whereas PG cooling depends on ${\displaystyle \textstyle {\frac {kv}{\Delta _{0}^{'}}}}$. At low intensity ${\displaystyle \Delta _{0}^{'}\ll \Gamma }$ and thus PG cooling works at lower atomic velocities (temperatures) than Doppler Cooling.

Limits and Scaling

Both methods of PG cooling surpass the Doppler limit and instead are limited by the one-photon recoil limit:

${\displaystyle kT_{recoil}={\frac {\hbar {}^{2}k^{2}}{2M}}}$

Where M is the atomic mass.

For a given detuning ${\displaystyle \delta }$ and Rabi frequency ${\displaystyle \Omega }$, dependent on the light intensity, both configurations display a similar scaling at low intensity (${\displaystyle \Omega \ll |\delta |}$) and large detuning (${\displaystyle \delta \gg \Gamma }$):

${\displaystyle kT=\alpha {}{\frac {\hbar {}\Omega ^{2}}{|\delta |}}}$

Where ${\displaystyle \alpha }$ is a dimensionless constant dependent on the configuration and atomic species. See ref ^[8] for a full derivation of these results.