View source for Sub-Doppler cooling

The general laser cooling methods we have seen so far may be
conceptually understood using the physics of optical pumping.
However, traditional laser cooling methods do not allow the
cooling to temperatures below a certain limit, known as the single
photon recoil limit.  Nevertheless, by generalizing laser cooling
methods beyond two-level approximations, and by considering other
degrees of freedom such as laser polarization, sub-Doppler temperatures
can be achieved, as we see below.

=== Cooling as optical pumping ===

Let us begin by generalizing what we have discussed in terms of light
forces and cooling, by saying that in some sense, all cooling
processes can be related to a process known as optical pumping.

Optical pumping can be understood through the following scenario:
::[[Image:Sub-Doppler_cooling-coolfig1.png|thumb|400px|none|]]
If level <math>|2{\rangle}</math> is a dark state, meaning that no optical transitions
occur from it, then eventually light incident on all the other
(optically accessible) transitions will pump all the atoms into that
dark state.

What does optical pumping have to do with cooling?  Consider this
scenario:
::[[Image:Sub-Doppler_cooling-coolfig2.png|thumb|408px|none|]]
Now write down the populations of the two levels,
:<math> 
\frac{n_2}{n_1} = e^{-\beta \Delta E}
</math>
and clearly as <math>n_2\rightarrow 0</math>, then the effective temperature of
the system goes to zero.

This picture can be used to understand laser cooling, by thinking of
cooling as optical pumping of motional states of the atom.  For
example, beam cooling was optical pumping into dark states of velocity.

On the other hand, laser cooling, is not just about reaching a
specific internal atomic state, but rather, external states
representing momenta of the atoms.  Typically, internal states such as
hyperfine states do not couple well to eternal degrees of motion, so
internal state cooling does not generally work well to cool the motion
of atoms.

We will now encounter optical pumping in two different scenarios.

=== Sub-Doppler cooling ===

The main new feature which allows cooling to beyond the photon recoil
temperature is the use of multiple energy levels in laser cooling.
Consider this energy level configuration, with multiple levels in the
ground state:

::[[Image:Sub-Doppler_cooling-coolfig2.png|thumb|400px|none|]]

This allows Raman resonances to be created in the ground state, which
can be extremely narrow.  The decay of the states coupled by a Raman
transition can be expressed as
:<math> 
\Gamma' = \frac{1}{\tau_p} = \frac{\Omega^2}{\delta^2} \Gamma
\,,
</math>
where <math>\delta</math> is the detuning from the virtual upper state, <math>\Gamma</math>
is the (short) lifetime of the upper state, and <math>\Omega</math> is the Rabi
frequency.

When there are long internal relaxation times, it is important that
these timescales are taken into account in the cooling process.  In
particular, such long timescales allows for large time-lags, and
recall that large time-lags gave rise to high cooling rates in our
dressed atom picture of laser cooling.  In particular, we found that
the populations of the states, under laser cooling, behave as
:<math> 
\pi(\vec{x}) = \pi_e(\vec{x} - v \tau_{relax})
\,,
</math>
so that long lifetimes lead to very strong laser cooling.

Often, the final temperature of schemes with narrow resonances lead to
:<math> 
k_B T_{\rm final} \propto \hbar\Gamma'
\,,
</math>
because of the physical intuition that narrower resonances give higher
resolution in distinguishing velocity classes of moving atoms.  Thus,
in this picture, the Doppler temperature limit is <math>T_{final}\sim
\Gamma</math>.  More precisely, for Raman cooling, it turns out to be
:<math> 
T_{final} \sim \Gamma' \frac{\Gamma}{\delta}
\,.
</math>

With ground state resonances, provided in this optical pumping
picture, features arise in the plot of force verses velocity which are
very narrow:
::[[Image:Sub-Doppler_cooling-coolfig3.png|thumb|510px|none|]]
This gives a wide feature which provides a large capture range, and a
narrow middle feature which leads to high cooling rates.  This is the
result of a specific sub-doppler cooling technique known as
polarization gradient cooling, as we see next.

=== Polarization gradient cooling ===

The following is an iconic picture of polarization gradient cooling.
If you have just a two-level atom, it doesn't care what the
polarization is.  But if you have internal levels such as in the
example above, the atoms do care about the incident polarization.
Shown here is a configuration in which the change of relative phases
of the two counter-propagating lasers lead to a gradient in which the
polarization changes from circular to linear, to opposite circular, to
opposite linear:

::[[Image:Sub-Doppler_cooling-coolfig4.png|thumb|400px|none|]]

(figure from Dalibard and Cohen-Tannoudji, {\em Laser cooling below
the Doppler limit by polarization gradients: simple theoretical
models, JOSAB, p2023})


Polarization gradient cooling is best explained using a picture of the
different Clebsch-Gordan coefficients for the transition rates (these
sample rates given for a <math>J_g=1/2 \leftrightarrow J_e=3/2</math> transition):

::[[Image:Sub-Doppler_cooling-coolfig4a.png|thumb|400px|none|]]

The strongest transitions are those on the ends.  <math>\sigma^+</math> light
moves the state to the right, and has a stronger AC Stark shift.
<math>\sigma^-</math> light does correspondingly to the transition on the left.
This results in a periodic potential in space:

::[[Image:Sub-Doppler_cooling-coolfig5.png|thumb|400px|none|]]

The steady state populations are represented by filled circles for a
<math>J_g=1/2</math> ground state in the lin <math>\perp</math> lin configuration of light
polarizations, and for negative detuning.  This results in atoms
having to climb potential hills more often than descending hills,
leading to an energy loss mechanism of this form:

::[[Image:Sub-Doppler_cooling-coolfig6.png|thumb|400px|none|]]

assuming the polarization changes between linear, <math>\sigma^-</math>, linear,
and <math>\sigma^+</math> from left to right.

This is a Sisyphus cooling mechanism, as was described
previously in the context of blue detuned optical molasses.    Here
the Sisyphus cooling is no longer between dressed states, but rather,
between hyperfine states of the atom.  Historically, it is in this
context that Sisyphus cooling became important.

=== Sub-recoil cooling ===

Sub-recoil cooling means that the residual thermal velocity of the
cooled atom is less than that due to the momentum of emitted photons.
For a long time, it was believed that laser cooling could not achieve
temperatures below this photon recoil limit.

Let us begin by motivating the discussion with a "proof" that
sub-recoil cooling is not possible.  We may calculate the change of
energy in an absorption + spontaneous emission cycle.  We first have
kinetic energy <math>m\vec{v}^2/2</math>.  After absorption, the energy is
:<math> 
\frac{1}{2}  \left(   { m\vec{v} +\hbar \vec{k}_L } \right) ^2
\,,
</math>
After emission, the energy is
:<math> 
\frac{1}{2}  \left(   { m\vec{v} +\hbar \vec{k}_L - \hbar\vec{k}_e } \right) ^2
\,,
</math>
where <math>\vec{k}_e</math> is the emitted photon's wavevector.  The energy
change is thus
:<math> 
\Delta E = 2 E_{rec} + \hbar (\vec{k}_L - \vec{k}_e) \cdot \vec{v}
\,.
</math>
The average over several cycles gives a change in mechanical energy of
the atom
:<math> 
\overline{\Delta E} = 2E_{rec} + \hbar \vec{k}_L \cdot \vec{v}
\,.
</math>
To make this as negative as possible, a counter-propagating beam should
be used (the laser beam should be anti-parallel to the velocity), giving
:<math> 
\overline{\Delta E} = 2E_{rec} - \hbar k_L v
\,,
</math>
giving that cooling is possible only if (using the fact that <math>E_{rec}
= k_L^2/m</math>) the
:<math> 
\overline{\Delta E} = 2E_{rec} - \hbar k_L v < 0
\,,
</math>
so that initial velocity must be
:<math> 
v > v_{rec} = \frac{\hbar k_L}{m}
\,.
</math>
Thus, the moment atoms reach the recoil velocity, atoms heat as much
as they cool.

How can this limit be circumvented?  Broadly speaking, spontaneous
emission is necessary for cooling.  Stimulated emission just swaps
population from one state to another, and does not change the
total system entropy.

The fact is that this limit can be circumvented if we do not treat
<math>\Delta E</math> using an average over cycles, and if we do not throw away
the graininess of the cooling process.  After all, sometimes,
atoms can emit such that their final energy is zero.  So the last
scattered photon may not be random, if things are setup
properly.  We are jostling around the atoms with random photons,
after all, so sometimes, an atom may drop to very low temperature.  As
an example, if the atoms are in a potential well which has a hole in
the center, then atoms can get trapped in the hole, and reach very low
temperatures.

This concept is used in dark state cooling.  Suppose the force
as a function of velocity looks like:

::[[Image:Sub-Doppler_cooling-coolfig7.png|thumb|400px|none|]]

This potential has an excitation that goes to zero at zero velocity.
Essentially it is the picture of velocity space optical pumping.  A
variety of methods can realize such potentials, such as Raman
excitations (Raman cooling), or velocity selective coherent population
trapping (VSCPT), or electro-magnetically-induced transparency methods.

We will not describe these methods any further, however, because in
practice, a different technique: evaporative cooling, has taken
over the field, and has become the workhorse of low temperature atom
cooling.

=== References ===

*  Important paper: J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am .B 2, 1707 (1985)  [https://cua-admin.mit.edu:8443/wiki/images/b/b7/Dali85_JOSAB.pdf Download]

[[Category:Light forces]]