Atomic Magnetometry with Effective Nonlinear Interactions

Overview

As we learn in freshman physics, a field is usually be detected by observing its effect on a test particle. We are interested in detecting magnetic fields by observing the precession of a small magnetic dipole, which experiences a torque from an external B field. In our experiment, the "test particle" is the collective spin angular momentum of a cloud of laser cooled Cs atoms. However, measurements in quantum mechanics are subject to Heisenberg uncertainty. This fundamental randomness affects measurements that are performed to determine how much the atomic magnetic moment has rotated, and therefore propagates into the determination of the strength of the applied field.

Theory

In our magnetometer we will measure the magnetic field by "watching" a collective spin Larmor precess. If we allow the spin, aligned initially along the x-axis, to precess for some time in an external field applied along the y-axis, and then make a measurement of the z-component, geometry allows us to find the angle through which the spin has rotated. With the knowledge of the time allowed to evolve and the angle, we can deduce the strength of the external field applied. However, quantum measurement puts limits on how well we can measure that z-component. Every measurement is going to be noisy, which complicates magnetic field estimation. We need to have a way to reduce this noise and/or extract better information from the noisy data.

If we want to measure a quantity such as a component of spin, without using entanglement, the lowest uncertainty we can reach is the "shotnoise uncertainty", which comes about as a result of projection measurements along the "uncertain" axes. Typically uncertainty can be reduced by increasing the system size N. By increasing the number of particles, and effectively the spin, the uncertainty scales as . By exploiting entanglement, this uncertainty can be further improved to scale as . This scaling is conventionally known as the "Heisenberg limit" and was long believed to be a fundamental lower bound for quantum uncertainty. However, it was recently realized that in addition to exploiting entanglement, a change in the structure of the evolution could also affect this lower bound. It was found that for a Hamiltonian evolution that is proportional to a k-body probe operator, the uncertainty could potentially scale as .

Experimental Design

How do we do this in practice? That is, how do we incorporate entanglement and enhanced dynamics into an actual experiment? Our proposed experimental procedure is a double-passed magnetometer with a polarimetric measurement of the spin z-component. We begin with the collective spin polarized along the x-axis, and apply an external magnetic field that causes the spins to precess in the x-z plane. A laser beam which is linearly polarized along the x-axis passes through the atoms. This interaction causes the laser beam to be rotated by an amount proportional to the current z-component of the spin.This Faraday rotation is converted to ellipticity by passing through a quarter-wave plate. The light passes again through the atoms, this time along the y-axis. This second pass has the effect of an additional magnetic field on the atoms in the same direction as the external magnetic field. The atoms, feeling this increased effective field, will Larmor precess faster and faster, as the higher z-component induces a greater Faraday rotation, which induces a greater effective magnetic field, which induces even faster Larmor precession, etc. In effect, the second pass causes the Larmor precession to accelerate.

Experimental Proposal & Progress

Old-school movie of how the fountain works, shown alongside a photo of the actual apparatus where the experiment will take place.

We hope to implement this procedure using a fountain of cold atoms. We will form a cloud of atoms in a MOT in the lower section of our vacuum chamber, and by tuning the lower beams, throw the ball of atoms up into the upper part of our chamber. Our experiment will take place at the top of the classical trajectory. Within the space of a few hundred microseconds, we will pump the atomic spins all along one direction, turn on a magnetic field, and send the double-passing probe beam through the atoms. We will then make our polarimetric measurement (which yields a value for the z-component) and propagate our stochastic filter to estimate the field as well as the uncertainty in that field estimate. We have chosen to use a fountain configuration because the experiment will take place away from the MOT, inside a magnetic shield that should eliminate most external magnetic influences.