Quantum estimation of a time-dependent perturbation

We analyze the estimation of a time-dependent perturbation acting on a continuously monitored quantum system. We describe the temporal fluctuations of the perturbation by a hidden Markov model, and we combine quantum measurement theory and classical filter theory into a time-evolving hybrid quantum and classical trajectory. The forward-backward analysis that permits smoothed estimates of classical hidden Markov models has a counterpart in the theory of retrodiction and past quantum states. As a specific example, we apply our hybrid trajectory and past-quantum-state theory to the sensing of a fluctuating magnetic field by microwave interrogation of a single quantum spin.

Figure 1

Schematic setup for detection of a time-varying magnetic field $B_z\left(t\right)$. A single spin, e.g., a dopant spin $\frac{1}{2}$ in a solid-state material, interacts near resonantly with a microwave cavity field. When a magnetic field shifts the energy difference between the spin eigenstates, it changes the amplitude and phase of the reflected field from the cavity. See [17] for an implementation geometry that maximizes the spin-cavity coupling.

Figure 2

Conditional probability distribution for the time-dependent detuning shifts experienced by a spin qubit system subject to a fluctuating magnetic field. The red curve shows the true simulated variation, while the bright blue area and the central dark blue curve show the distribution and most likely value of the field inferred by probing of the spin dynamics. The rms deviation of the maximum-likelihood estimate from the true value of the detuning is $0.26\gamma$. The parameters are given in the text.

Figure 3

Past-quantum-state probability distribution for the detuning of the spin transition imposed by a randomly varying magnetic field. The red curve shows the true simulated variation, while the bright blue area and the central dark blue curve show the inferred distribution and most likely value. The parameters are given in the text. The rms deviation of the maximum-likelihood value from the actual value, averaged over time, is $0.20\gamma$, in good agreement with the standard deviation of the conditional probability distribution.

Figure 4

Error estimates defined as the square root of the time-averaged variances of the forward-filter and PQS distributions (upper and lower solid curves, respectively) and of the mean-squared errors of the maximum-likelihood estimators with respect to the true value of the fluctuating detuning. The results are shown as functions of the probe-field amplitude $\beta$.