Fighting decoherence by feedback-controlled dissipation

Repeated closed-loop control operations acting as piecewise-constant Liouville superoperators conditioned on the outcomes of regularly performed measurements may effectively be described by a fixed-point iteration for the density matrix. Even when all Liouville superoperators point to the completely mixed state, feedback of the measurement result may lead to a pure state, which can be interpreted as selective dampening of undesired states. Using a microscopic model, we exemplify this for a single qubit, which can be purified in an arbitrary single-qubit state by tuning the measurement direction and two qubits that may be purified towards a Bell state by applying a special continuous two-local measurement. The method does not require precise knowledge of decoherence channels and works for large reservoir temperatures provided measurement, processing, and control can be implemented in a continuous fashion.

Figure 1

Comparison of the effective evolution of the $\langle {\sigma }_t^x\rangle$ expectation value under feedback control for finite measurement intervals (symbols) with an average of ${10}^2$, ${10}^3$, and ${10}^4$ trajectories (thin dotted, dashed, and solid curves in lighter colors, respectively; single trajectories would decay at different pace from $\pm 1$ towards 0 between measurements). In the continuous measurement limit (top bold curve), the final purity is maximal and would vanish without feedback. Parameters: ${\lambda }_{+}=1.0$, ${\lambda }_{-}=5.0$, $\Omega /\gamma =5.0$, and ${\theta }_{\pm }=\pi /2$.

Figure 2

Effective evolution of the nonvanishing $\langle {\Sigma }_t^{\alpha \beta }\rangle$ expectation values under feedback control for finite measurement intervals (symbols) and in the continuum limit (solid curves). Parameters: ${\lambda }_{\mathrm{B}}=1$, ${\lambda }_{\mathrm{R}}=5$, ${\omega }_A\Delta t={\omega }_B\Delta t=0$. A nonvanishing stationary concurrence [see inset extending Eq. (19) to finite measurement intervals, with contour lines ranging from 0.1 to 0.9 in steps of 0.1] requires short measurement intervals.