Techniques for cooling to ultralow temperatures

Sub-doppler and sub-recoil cooling

We've seen two general cooling methods so far: Doppler cooling and, on trapped ions, sideband cooling.

Cooling as optical pumping

Cooling can be regarded as optical pumping. Consider this three level system:

File:Techniques for cooling to ultralow temperatures-bec1-cooling

\noindent A laser from level 2 to level 3 pumps the state to level 1, because no transitions come from level 1, which are reached by spontaneous emission from level 3. Laser cooling, however, 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.

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:

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\noindent 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

$\Gamma '={\frac {1}{\tau _{p}}}={\frac {\Omega ^{2}}{\delta ^{2}}}\Gamma \,,$

where $\delta$ is the detuning from the virtual upper state, $\Gamma$ is the (short) lifetime of the upper state, and $\Omega$ 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. Recall that we found that the populations of the states, under laser cooling, behave as

$\pi ({\vec {x}})=\pi _{e}({\vec {x}}-v\tau _{relax})\,,$

so that long lifetimes lead to very strong laser cooling.

Another way to see the usefulness of long lifetime states is to recall that the Doppler temperature limit is $T_{final}\sim \Gamma$. More precisely, for Raman cooling, it turns out to be

$T_{final}\sim \Gamma '{\frac {\Gamma }{\delta }}\,.$

Polarization gradient cooling

This 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 counterpropagating lasers lead to a gradient in which the polarization changes from circular to linear, to opposite circular, to opposite linear:

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The level diagram includes Clebsch-Gordan coefficients for the transition rates. The strongest transitions are those on the ends. $\sigma ^{+}$ light moves the state to the right, and has a stronger AC Stark shift. $\sigma ^{-}$ light does correspondingly to the transition on the left. This results in a periodic potential in space:

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This results in atoms having to climb potential hills more often than descending hills, leading to an energy loss mechanism of this form:

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This Sysiphus cooling mechanism was described previously in the context of blue detuned optical molasses. It was historically important, but is no longer widely used in practice.

Sub-recoil cooling

Let us motivate the discussion by first "proving" 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 $m{\vec {v}}^{2}/2$. After absorption, the energy is

${\frac {1}{2}}\left({{\vec {v}}+\hbar {\vec {k}}_{L}-\hbar {\vec {k}}_{e}}\right)^{2}\,,$

where ${\vec {k}}_{e}$ is the emitted photon's wavevector. The energy change is thus

$\Delta E=2E_{rec}+\hbar ({\vec {k}}_{L}-{\vec {k}}_{e})\cdot {\vec {v}}\,.$

The average over several cycles gives a change in mechanical energy of the atom

${\overline {\Delta E}}=2E_{rec}+\hbar {\vec {k}}_{L}\cdot {\vec {v}}\,.$

To make this as negative as possible, a counterpropagating beam should be used, giving

${\overline {\Delta E}}=2E_{rec}-\hbar k_{L}v\,,$

giving that cooling is possible only if the

${\overline {\Delta E}}=2E_{rec}-\hbar k_{L}v<0\,,$

so that initial velocity must be

$v>v_{rec}={\frac {\hbar k}{m}}\,.$

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 $\Delta E$ using an average over cycles, and if we do not throw away the grainess of the cooling process. After all, sometimes, atoms can emit such that their final energy is zero. We are josteling aroun 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 very low temperature.

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

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\noindent This potential has an excitation that goes to zero at zero velocity. A variety of methods can realize such potentials, such as Raman excitations, or electromagnetically-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.

Magnetic cooling

The principle of magnetic trapping is deflection by magnetic fields. This is at the heart of the Stern-Gerlach experiment in 1925. In the 1960's it was realized that magnetic confinement could be realized. It was first realized for neutrons, in 1978 by Paul. The exciting 1980's was magnetic cooling and trapping, including work by Phillips, Pritchard, and Greytak, on sodium and other atoms. Recall the Earnshaw theorem, previously discussed for the optical force, and for trapped ions. This states that trapping requires $\nabla F<0$ or $\Delta \phi >0$. For a rigid charge distribution, with positions $R+\tau _{i}$, where $R$ is the center of mass motion and $\tau _{i}$ is the relative position of charge $i$, we find that

$\phi _{tot}=\sum \phi _{i}$

must satisfy $\Delta \phi >0$. An arbitrary rigid charge distribution (with arbitrary multiple moments) cannot be kept in stable equilibrium at rest in free space in static electric fields. The same theorem applies for magnetic multipole distributions.

Sir James-Jeans, in 1925, pointed out a way around this limit. If the matter is polarizible, this can change the electric fields.

File:Techniques for cooling to ultralow temperatures-jeans-earnshaw

\noindent The displacement of the trapped particle or of the field generated by a particle which are energetically favorable are forbidden due to constraints. This leads to a loophole, as exemplified by magnet stabilized above a superconductor:

File:Techniques for cooling to ultralow temperatures-superc-trap

The constraint that no magnetic field may enter a superconducot causes shielding currents to be created which stabilize the trapping configuraiton for the levitated particle. This happens in a configuration which is not a minimum of the magnetic energies. Of course, electronic feedback can be used to create a stable trapping potential, providing time varying fields.

Another exaple is provided by diamagnetic matter. Such matter expels an external field.

Another example is if ${\vec {\mu }}\cdot {\vec {B}}$ is constant. This constrains angular momentum; the dipole moment is not allowed to flip over to move into what might be a more energetically favorable configuration.

Let us consider when a magnetic moment might be constrained such that the potential depends only on the absolute value of the magnetic field, $U\sim |{\vec {B}}|$. Wing's theorem (1983) says that in a region dvoid of charges and currents, quasistatic electric of magnetic fields can have local minima, but not local maxima. The proof of this fact is by non-existence of a maximum by contradition. Assume $E(0)$ is a max. Then

${\vec {e}}(r)={\vec {E}}(0)+\delta {\vec {E}}$

It follows that

$|E(r)|^{2}=E^{2}(0)+2{\vec {E}}(0)\delta {\vec {E}}(r)+(\delta {\vec {E}}(r))^{2}$

and the cross-term must be always negative. Without loss of generality, this requires, for example, $\delta E_{z}(r)<0$. However, that presents a contraditction with the Laplace equation (recall Earnshaw's theorem). A minimum, on the other hand, is possible because of the $(\delta {\vec {E}}(r))^{2}$ term. There can thus be local minima, but not local maxima, in an electric field configuration. By Wing's theorem, since there can only be local minima (and not maxima) in a static magnetic field, to have a stable magnetic trap, ${\vec {\mu }}$ must be antiparallel to ${\vec {B}}$. The limits to stability of magnetic traps include spin realaxation (inolving nuclear magnetic moment changes), dipolar relaxation (involving angular momenta changes of atomic motion), and Majorana flops.

More specifically, ${\vec {\mu }}$ must be antiparallel to ${\vec {B}}$ only adiabatically, as follows. Consider the problem of trapping a classical magnetic moment. Such a magnetic moment precesses in a magnetic field:

File:Techniques for cooling to ultralow temperatures-umom-precession

\noindent What is needed is for the precession of the magnetic moment to follow and maintain the same angle with respect to the field, as the moment moves in an inhomogeneous field. If the field changes slowly, the precession will follow the field, but if the field changes too rapidly, the spin may no longer be antiparallel with respect to the field:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{rot} = \frac{d\theta}{dt} \ll \frac{U_i-U_j}{\hbar} = \frac{g\mu_B B}{\bar} = \omega_{larmor} \,. }

In the spherical quadrupole magnetic trap, made by anti-Helmholz coils, spins can flip in the following way, known as Majorana flops. As an atom moves through the field configuration:

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\noindent it should flip its moment as it moves through the center of the field; however, there is a zero in the field at the center, at which point the spin can easily find itself aligned parallel with the field upon the atom's exit. This causes atoms to leak from the trap. (show Levitron movie)

Counterintuitively, spining the top faster does not lead to a more stable trap. Rather, it causes increased instability; this is due to a process we may think of as being a classical analog of Majorana flops. The levitron works in the following way. At $r=0$,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(z) = B_0 + B' z + B" z^2/2 }

etc. The magnetic field above the center of the spinning magnet is thus

File:Techniques for cooling to ultralow temperatures-levitron-field

The second derviative of the magnetic field must give positive curvature along both the radial and axial directions, for the trap to be stable. This plot shows there is only a very small stability range for the trap.

Recall our trapping criterion: ${\vec {\mu }}$ must stay anti-parallel to ${\vec {B}}$. This happens due to precession, as long as $\omega _{prec}>{\mbox{rate of change of }}{\hat {B}}\approx \omega _{osc}$. The precession frequency of a gyroscope is given by the torque, so

$\omega _{osc}={\frac {\mbox{max torque}}{\mbox{ang. momentum}}}={\frac {\mu B}{I\omega _{spin}}}\,,$

where $I$ is the moment of intertia. $\omega _{prec}>\omega _{osc}$ is thus violated for large $\omega _{osc}$ or $B\approx 0$. Around $B\approx 0$, one gets spin flips, referred to as Majorana flops in magnetic trapping of atoms.

The quantum equivalent of this scenario is reduction of the $g$ factor: spining the magnet faster corresponds to reduction of $g$. In magnetic traps, $U=\mu _{B}Bgm_{F}$. $B$ can only have local minima, and not maxima, so we must ensure the angle $\theta$ of the field is such that the prefactors are positive. For Rubidium, we have these energy level depenencies on the field:

File:Techniques for cooling to ultralow temperatures-rb-weak-field-seekers

\noindent The only states which are trapped are thus which have positive energy versus magnetic field slope.

The spherical quadrupole trap has a linear field dependency versus postiion going through the center:

File:Techniques for cooling to ultralow temperatures-s-quad-trap

\noindent A plug is thus needed to close the $B=0$ hole at ${\vec {r}}=0$. Two strategies are (1) to use a repulsive dipole force, or (2) use a rapidly rotating magnetic field -- the TOP trap. The TOP trap adds a linear bias field to displace the origin of the trap, such that the time-average potential produces no zero in the magnetic field. The Ioffe-Pritchard trap does this, with a constant field and a curvature created by four "Ioffe" bars, adding to the two "Pinch" coils:

File:Techniques for cooling to ultralow temperatures-io-trap

The Ioffe bars create the quadrature ield, and the pinch field creates a linear field. This is a photograph of coils typically used:

File:Techniques for cooling to ultralow temperatures-io-coil-photo

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