# Pump–Probe Magnetometry

In a "pump–probe" magnetometer, atoms are optically pumped with a pump laser beam, and then optical rotation is measured in a weak probe beam. The pumping and probing steps may happen simultaneously or in succession. In the latter case, separate sets of equations are formed and solved for evolution in the pumping and probing steps (or pump and probe regions, if they are spatially separated). Even if the pumping and probing steps happen at the same time, it still may be useful conceptually to perform the calculation as if they occur sequentially. Another simplification can be made if the probe beam is always weak enough so that power broadening can be neglected: an analytical, perturbative expression for the optical rotation of the probe beam can be found, allowing the signal to be obtained directly from the solution for the pump region.

Here we consider a pump–probe magnetometer using a system, such as described in [Auzinsh2004]. The pump light is circularly polarized, propagating in the direction; precession occurs around a -directed magnetic field; optical rotation is measured in probe light propagating along and linearly polarized along . We calculate the optical rotation assuming a separated pump–probe arrangement. We then show that a simpler solution can be obtained by further separating the optical rotation process into three stages: pump, precession, and probe.

## Pump-Probe Calculation

Set DensityMatrix options to consider the steady state. |

The pump-region equations are too complicated to allow a convenient analytical solution, so we will solve them numerically after finding the formula for the optical rotation in the probe region in terms of the ground-state pump-region density matrix.

The probe-region relaxation matrix is the same as that for the pump region. The repopulation matrix, on the other hand is different. The probe-region is repopulated by atoms traveling from the pump region, so the transit repopulation matrix is the pump-region ground-state density matrix times the transit rate. We can ignore repopulation due to spontaneous decay in the probe region, since we will solve the probe-region equations to first order only.

Find the optical rotation of the probe light, in terms of the probe-region density-matrix elements; note that the parameters supplied to the Observables function must match those of the probe field. |

## Pump–Precession–Probe Calculation

The mechanism of optical rotation can be further simplified conceptually and computationally by separating the pump and precession steps (or regions) in order to model a three-stage (pump–precession–probe) process.

### References

[Auzinsh2004] M. Auzinsh, D. Budker, D.F. Kimball, S.M. Rochester, J.E. Stalnaker, A.O. Sushkov, V.V. Yashchuk, "Can a Quantum Nondemolition Measurement Improve the Sensitivity of an Atomic Magnetometer?," Phys. Rev. Lett. **93**, 173002 (2004). https://doi.org/10.1103/PhysRevLett.93.173002.