Continuous Quantum Measurement and Filtering

Overview

Measurements have always held an odd stature in quantum mechanics. On the one hand, they provide the only experimentally relevant features of quantum mechanics ( data!!! ), which are used to verify the theory of quantum mechanics to excruciating accuracy. On the other hand, there is much confusion about how best to describe the details of the measurement process. As technology allows for increasingly sophisticated interactions with quantum mechanical systems, this confusion has practical implications. What is the best measurement to make? Is the probe system also quantum mechanical? How long does it take to measure? If it takes a long time to measure, can I change the system while I'm measuring it?

Projective Measurements

The usual story of measurement in quantum mechanics goes as follows. We have a quantum state $|\psi\rangle$ and are interested in measuring an observable $\hat{O}$ . This observable takes on several possible values, labeled $o_1,o_2,\ldots,o_n$ and each value has an associated projector $\hat{P}_1,\hat{P}_2,\ldots,\hat{P}_n$ onto the eigenstate(s) which correspond to the given value. The Born rule in quantum mechanics tells us that we will measure outcome $i$ with probability $p_i =\langle\psi|\hat{O}|\psi\rangle$ . Afterwards, the new quantum state is $|\psi'_i\rangle = \frac{1}{\sqrt{p_i}}\hat{P}_i|\psi\rangle$ .

The abridged version of the measurement story is:

1. Decide to measure $\hat{O}$

2. ?

3. Get outcome $i$ and now have state $|\psi'_i\rangle$

This version makes it clear that this story lacks any description of how the measurement process occurs; step 2 is missing. What is the device the experimenter measures with? How does this device extract the outcome from the system? Is the process instantaneous? If not, at what point can the experimenter declare that value $o_i$ was observed? We need a quantum mechanical description of the probe system and how it interacts with the system we are measuring.

Continuous Measurement

A typical continuous measurement setup from our lab. See Atomic Magnetometry

One approach is that of continuous quantum measurement, which draws on quantum probability theory, quantum stochastic calculus and quantum trajectory theory. Although the framework is abstract enough to be applied to any quantum system, it is strongly motivated by the use of lasers to probe atomic systems. The basic idea is that the quantum probe system, e.g. electromagnetic field modes of the laser, interacts very weakly with the quantum system, e.g. a cloud of atoms. After this interaction, the probe is itself measured by a much larger, effectively classical, pointer system (e.g. photodetection). These probe measurement outcomes reveal information about the quantum system of interest and the corresponding measurement back-action changes the state of the this system to be consistent with these results. Repeating this process using new probes should continue to reveal more information until the system of interest is projected into an eigenstate of the continuous measurement.

Quantum Filtering

The continuous measurement outcomes are random and thus correspond to a continuous-time stochastic process. Fortunately, the noise statistics that arise from coupling via a laser are essentially white, which make the problem tractable. The goal of quantum filtering is to infer properties about the quantum system from the noisy photocurrent. This is very analogous to the classical filtering problem, in which we want the best estimate of a random system, given noisy observations of the system.

Using methods from quantum probability theory and stochastic calculus^{[QSC 1]}, a mathematically rigorous approach allows one to derive the optimal filter to estimate an arbitrary quantum observable. Fortunately, the filter is recursive and can be cast as a stochastic differential equation, which allows easy numerical (and occasionally analytical) solutions. Rather than propagating a filter for each observable of interest, we can instead propagate a single density operator which is then used to calculate any observable. The quantum filtering equation in this adjoint form, for a continuous measurement of $\hat{O}$ with measurement strength $M$ is:

$d\rho_t = M(\hat{O}\rho_t\hat{O}-\frac{1}{2}\hat{O}^{\dagger}\hat{O}\rho_t - \frac{1}{2}\rho_t\hat{O}^{\dagger}\hat{O})dt + \sqrt{M}(\hat{O}\rho_t + \rho_t\hat{O}^{\dagger} - \langle \hat{O} + \hat{O}^{\dagger}\rangle\rho_t)(dY_t - \sqrt{M}\langle \hat{O} + \hat{O}^{\dagger}\rangle dt)$

with the innovations process based on the measurement current $dY_t$ given by

$dW_t = dY_t - \sqrt{M}\langle \hat{O} + \hat{O}^{\dagger}\rangle dt$

The innovations process is equivalent to a Wiener increment, which is the generator of Brownian motion. In some sense, the innovations process captures how surprised we are by a new measurement result and appropriately weights our update of the conditional density matrix.

As depicted in the figure below, the noisy continuous-measurement current is integrated via the quantum filter, which in turn allows us to estimate a system observable. For the continuous-measurement of a standard observable, we view this as a random walk which eventually collapses onto an eigenvalue of the measurement.

(a) Noisy measurement current and (b) filtered system observable

Given this picture, one can consider using feedback, in which the current estimate of observables can be used to influence the measurement outcome, or even stabilize the system.

[QSC 1]

From QMCGroup