Feedback Control of Nonlinear Quantum Systems: A Rule of Thumb

We show that in the regime in which feedback control is most effective—when measurements are relatively efficient, and feedback is relatively strong—then, in the absence of any sharp inhomogeneity in the noise, it is always best to measure in a basis that does not commute with the system density matrix than one that does. That is, it is optimal to make measurements that disturb the state one is attempting to stabilize.

Figure 1

(a) The steady-state probability density for $P$, being the squared overlap with the target state, in the absence of feedback. Solid line: $k/\beta =2$; dashed line: $k/\beta =1/2$; dash-dotted line: $k/\beta =0.1$. (b) The average steady-state probability that the system is found in the target state under two feedback algorithms. Solid line: feedback control using an unbiased measurement with feedback strength $\mu \to \infty$; circles: same algorithm with $\mu =100k$. Dashed line: feedback control using a measurement that commutes with the density matrix for $\mu \to \infty$. Squares and dashed line: same algorithm with $\mu =100k$.