Isolating the Hexadecapole Moment

The hexadecapole (rank 4) polarization moment can be supported by atomic states with angular momentum J=2 or greater. One method to produce the hexadecapole moment without creating an appreciable amount of the quadrupole moment is to apply a magnetic field and then optically pump synchronously at four times the Larmor frequency. However, the efficiency of this method falls rapidly with the strength of the magnetic field, because a multi-photon interaction must take place during a single Larmor cycle in order to create the hexadecapole moment. Synchronous optical pumping at twice the Larmor frequency is a more efficient method of creating the hexadecapole moment at higher magnetic fields, but large quantities of the quadrupole moment are also created with this method. In Ref. [Acosta2008], it is demonstrated that by synchronously pumping at twice the Larmor frequency and then changing the phase of the pumping, the hexadecapole moment can be created efficiently while removing the quadrupole moment from the ensemble.

This example models this method in a simple case: a J=2J=1 transition, neglecting Doppler broadening. Illuminated and non-illuminated experimental regions are included, with continuous exchange of atoms between the regions, as for an antirelaxation-coated vapor cell.

This loads the package.
Define the atomic system. State 1 is a J=2 ground state. State 2 is a J=1 excited state.
Define the experimental system. ExperimentalRegion a is the illuminated pumping region and region b is the non-illuminated precession region. Atoms can travel back and forth between these two regions. The light is linearly x-polarized, and its amplitude corresponds to a reduced Rabi frequency ΩR. The z-directed magnetic field strength corresponds to a Larmor frequency ΩL.
The Hamiltonian for each region.
Apply the rotating-wave approximation to the Hamiltonians.
Relaxation matrices for each region, describing spontaneous decay, transit relaxation, and other isotropic relaxation (e.g., due to wall collisions).
Repopulation matrices, accounting for atoms that have relaxed.
The evolution equations for the two regions.
Initial conditions, assuming an unpolarized initial state.
Define a square-wave function.
Choose parameters for the calculation, including a time-dependent Rabi frequency. We choose a relative Larmor frequency much slower than the one used in the experiment, in order to be able to easily plot the Larmor precession.
A plot of the light-field amplitude as a function of time.
Solve the system of differential equations.
Define the density matrix for the ground state in region b (the unilluminated region).
The tensor decomposition of the ground-state density matrix, with the solutions found above substituted in.
A plot of the real part of the q=2 component of the quadrupole moment as a function of time.
A plot of the real part of the q=4 component of the hexadecapole moment as a function of time.
A combined plot showing the time-evolution of the system in response to the synchronous pumping.

We see that, initially, both quadrupole and hexadecapole moments are created by the synchronous pumping. After the phase of the pumping cycle is flipped, the pumping acts to reduce the quadrupole moment, while the hexadecapole continues to increase. We can visualize why this happens by plotting the angular-momentum probability surface for the ground state.

Calculate the formula for the angular-momentum probability surface, subtracting a fraction of the moment, in order to be able to examine the higher polarization moments.
The following generates one frame of an animation of the probability surface, superimposed over the time-evolution plot.

Unfortunately, the complete animation is too large to conveniently assemble inside a Mathematica notebook. Here is a link to the finished animation on our website (also, a version for a J=2J=2 transition).

We see that the quadrupole moment has two-fold symmetry about the magnetic field direction. The initial pumping phase establishes the phase of this symmetry with respect to the Larmor precession. When the pumping phase is flipped, polarization is pumped at right angles to the quadrupole polarization. Since the quadrupole moment cannot support a four-fold symmetry, the newly pumped quadrupole polarization cancels out the previous quadrupole moment. The hexadecapole moment, on the other hand, can support a four-fold symmetry, so the new polarization adds to the previously existing hexadecapole polarization.


[Acosta2008] Acosta, V. M., M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. J. Kimball, L. Krzemien, M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker. "Production and detection of atomic hexadecapole at Earth's magnetic field" Opt. Express 16 (2008): 11423-11430. arXiv:0709.4283