Techniques for cooling to ultralow temperatures

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.

Earnshaw theorem

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.

Circumventing the Earnshaw theorem

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

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:

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 example is provided by diamagnetic matter. Such matter expels an external field.

Yet another, important, 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.

Trapping a magnetic moment

Now let us see how these ideas can allow trapping of an object with a magnetic moment. 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

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.

Classical magnetic moment in a trap

More specifically, to trap, ${\vec {\mu }}$ 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:

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:

\omega _{rot}={\frac {d\theta }{dt}}\ll {\frac {U_{i}-U_{j}}{\hbar }}={\frac {g\mu _{B}B}{\hbar }}=\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:

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.

Levitron example

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 $r=0$ ,

B(z)=B_{0}+B'z+B''z^{2}/2

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

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 $g$ .

Magnetic traps

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:

The only states which are trapped are thus which have positive energy versus magnetic field slopw.

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

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:

The Ioffe bars create the quadrature field,

Failed to parse (syntax error): {\displaystyle B_z(z) = B_0 + \frac{B"}{2} z^2 \,, }

and the pinch field creates a linear field,

Failed to parse (syntax error): {\displaystyle B = \sqrt{B_z^2 + B_\rho^2} \approx B_0 + \frac{B"}{2} z^2 + \frac{1}{2} \frac{{B'}^2}{B_0^2} (x^2+y^2) \,, }

where the last term on the right is recognized to be $B_{\rho }^{2}/2B_{0}$ . This configuration can be used with $B_{0}\neq 0$ :