Polarization gradient cooling (PG cooling) is a technique in laser cooling of atoms. It was proposed to explain the experimental observation of cooling below the doppler limit.[1] Shortly after the theory was introduced experiments were performed that verified the theoretical predictions.[2] While Doppler cooling allows atoms to be cooled to hundreds of microkelvin, PG cooling allows atoms to be cooled to a few microkelvin or less.[3] [4]
The superposition of two counterpropagating beams of light with orthogonal polarizations creates a gradient where the polarization varies in space. The gradient depends on which type of polarization is used. Orthogonal linear polarizations (the lin⊥lin configuration) results in the polarization varying between linear and circular polarization in the range of half a wavelength. However, if orthogonal circular polarizations (the σ+σ− configuration) are used, the result is a linear polarization that rotates along the axis of propagation. Both configurations can be used for cooling and yield similar results, however, the physical mechanisms involved are very different. For the lin⊥lin case, the polarization gradient causes periodic light shifts in Zeeman sublevels of the atomic ground state that allows for a Sisyphus effect to occur. 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. Both configurations achieve sub-Doppler cooling and instead reach the recoil limit. While the limit of PG cooling is lower than that of Doppler cooling, the capture range of PG cooling is lower and thus an atomic gas must be pre-cooled before PG cooling.
When laser cooling of atoms was first proposed in 1975, the only cooling mechanism considered was Doppler cooling.[5] As such the limit on the temperature was predicted to be the Doppler limit:[6]
kBT=
| \hbar\Gamma | |
| 2 |
Here kb is the Boltzmann 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.[7] However, in 1988 experiments began to report temperatures below the Doppler limit.[1] 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
\vec{E1}=
| ikz | |
| E | |
| 0e |
\hat{x}
\vec{E2}=
| -ikz | |
| E | |
| 0e |
\hat{y}
k=
|
\vec{E1}
\vec{E2}
\vec{E}tot=
| E0 | \left(\cos(kz) | |
| \sqrt2 |
| \hat{x | |
| +\hat{y}}{\sqrt2} |
-i\sin(kz)
| -\hat{x | |
| +\hat{y}}{\sqrt2}\right) |
Introducing a new pair of coordinates
\hat{x}'=
|
\hat{y}'=
|
\vec{E}tot=
| E0 | |
| \sqrt2 |
\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
z=0
\hat{x}'
z=
|
z=
|
\hat{y}'
z=
|
z=
|
\hat{x}'
Consider an atom interacting with the field detuned below the transition from atomic states
Fg=
|
Fe=
|
\hbar{}\omega{}field<Eeg
| |g,m | |||||
|
| |e,m | |||||
|
| |g,m | |||||
|
| |e,m | |||||
|
\sigma-
| |g,m | |||||
|
| |e,m | |||||
|
\sigma+
| |g,m | |||||
|
| |e,m | |||||
|
|
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
\tau
kv ≈ \tau{}-1
For the case of counterpropagating waves with orthogonal circular polarizations the resulting polarization is linear everywhere, but rotates about
\hat{z}
-kz
F=1
Consider two EM waves detuned from an atomic transition
Fg=1 → Fe=2
\vec{E1}=
| ikz | |
| E | |
| 0e |
|
\vec{E2}=
| -ikz | |
| E | |
| 0e |
|
\vec{Etot
As previously stated, the polarization of the total field is linear, but rotated around
\hat{z}
-kz
\hat{y}
Consider an atom moving along z with some velocity v. The atom sees the polarization rotating with a frequency of
kv
\hat{H}rot=kvFz
Here we see the inertial term looks like a magnetic field along
\hat{z}
Choosing the polarization in the rotating frame to be fixed along
\hat{y}
\hat{F}y
Fg=1
|g,mf=0\rangley
|g,mf=\pm{1}\rangley
|g,mf=0\rangley
|g,mf=\pm{1}\rangley
\langle\hat{F}y\rangle=0
\langle\hat{F}z\rangle
\langle\hat{F}z\rangle=
| 40\hbar{ | |
| kv}{17\Delta |
| '} | |
| 0 |
Where
| ' | |
| \Delta | |
| 0 |
mF=0
| ' | |
| \Delta | |
| 0 |
|g,mf=-1\rangle
|g,mf=1\rangle
|g,mf=-1\rangle
\sigma-
\sigma+
|g,mf=1>
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
|
\Gamma
|
| ' | |
| \Delta | |
| 0 |
\ll\Gamma
Both methods of PG cooling surpass the Doppler limit and instead are limited by the one-photon recoil limit:
kTrecoil=
| \hbar{ | |
| 2k |
2}{2M}
Where M is the atomic mass.
For a given detuning
\delta
\Omega
\Omega\ll|\delta|
\delta\gg\Gamma
kT=\alpha{}
| \hbar{ | |
| \Omega |
2}{|\delta|}
Where
\alpha
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
| 2S | |
| |6 | |
| 1/2 |
F=4\rangle
| 2P | |
| |6 | |
| 3/2 |
F'=5\rangle
| 2S | |
| |6 | |
| 1/2 |
F=3\rangle
| 2P | |
| |6 | |
| 3/2 |
F'=4\rangle
| 2S | |
| |6 | |
| 1/2 |
F=3\rangle
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.[9]
An important theoretical result is that in the regime where PG cooling functions, the temperature only depends on the ratio of
\Omega2
|\gamma|
\mu
\mu