Deterministic Equations for Feedback Control of Open Quantum Systems

TL;DR

This work presents a deterministic, memory-resolved framework for feedback control in open quantum systems, unifying disparate feedback master equations under a single update rule for a memory variable $y_n$ and a memory-resolved state ${\varrho}_n(y)$. The central Result 1 provides a general equation ${\varrho}_{n+1}(y)=\sum_{x',y'}\delta_{y, f_{n+1}(x',y')}\mathcal{M}_{x'}(y')\, {\varrho}_n(y')$, from which prior approaches are recovered as special cases. Result 2 specializes the framework to quantum-jump dynamics by introducing a two-component memory $(k_n,\tau_n)$, yielding a deterministic, jump-conditioned evolution ${\varrho}_t(k,\tau)$ with a no-jump propagator $G(k,\tau)$ and enabling time-dependent feedback actions. The authors illustrate two experimentally relevant protocols—population inversion in a qubit coupled to a thermal bath and real-time reversal of quantum transitions—providing analytical steady-state solutions and jump statistics, and demonstrate that the framework recovers classic results while enabling new, analytically tractable strategies. Overall, the method offers computational efficiency, conceptual clarity, and direct links to measurable jump statistics, with broad applicability to quantum control, thermodynamics, and information processing.

Abstract

Feedback control in open quantum dynamics is crucial for the advancement of various coherent platforms. However, currently only a handful of feedback master equations exist in the literature, which are restricted to specific types of feedback. In this letter we first introduce a unifying framework, based on a single general equation, that describes all possible feedback schemes in sequentially (and continuously) measured systems, and from which all previous results follow. Next, we specialize it to the case of quantum jumps and introduce a new type of feedback based on the channel of the last detected jump, as well as the time elapsed since it occurred. Our description is experimentally grounded, and naturally allows for the introduction of realistic effects, such as time-delays in the feedback loop. We illustrate our results with two time-dependent feedback protocols conditioned on quantum-jump detections: one achieving population inversion of a two-level system against a thermal bath, and another enabling real-time reversal of quantum transitions, both admitting steady-state solutions.

Figures (6)

• Figure 1: General feedback diagram. A detection outcome is stored in memory and used in a feedback action that can modify the system dynamics (e.g., the Hamiltonian $H$), the environment (e.g., bath temperature), or the measurement process itself.
• Figure 2: Conditional evolution under feedback. The instruments $\mathcal{M}_x$ describe both the system dynamics and the measurement process at each step. The quantities $\rho_{x_{1:n}}$ and $y_n$ denote, respectively, the system state and memory after $n$ detections given the dataset $(x_1, \dots, x_n)$. Feedback is implemented by using the memory $y_n$ to modify the instrument, while the measurement yields the outcome $x_{n+1}$ and updates the memory as $y_{n+1} = f(x_{n+1}, y_n)$.
• Figure 3: Population inversion via time-dependent feedback with quantum jumps. (a) Upon detection of a decay event, an external drive is applied to return the qubit to the excited state. This feedback includes a possible time delay $\tau_0$ before the drive is turned on, and $\tau_1$ is the drive duration. (b) Population of the excited state without feedback delay ($\tau_0 = 0$), and considering the optimal drive duration $\tau_1^{\text{opt}}$ (Eq. \ref{['eq: results of the example - optimal time']}). The no-feedback line corresponds to a qubit coupled to a thermal bath without an external drive. (c) Effect of feedback delay on the excited state population in the low-temperature limit. (d) Maximum allowable feedback delay in the low-temperature limit and the optimal drive duration.
• Figure 4: Reverting quantum transitions. (a) Three-level system with a hidden transition Minev2019CatchingReverseQuantumJump between the ground $\ket{G}$ and dark $\ket{D}$ states (shaded region). The states $\ket{B}$ and $\ket{G}$ are coupled to an effective thermal bath, analogous to the qubit setup in Example 1, with coupling rate $\gamma_B$ and Bose–Einstein occupation $\bar{N}_B$, so that $\Gamma = \gamma_B(\bar{N}_B + 1)$. The protected states $\ket{D}$ and $\ket{G}$ are only weakly coupled to a second bath characterized by $\gamma_D$ and $\bar{N}_D$. (b) Populations of the feedback steady state as a function of the waiting time $\tau_c$. The inset shows the average time $\braket{\Delta t}$ between jumps $\ket{B}\!\to\!\ket{G}$ in the stationary regime. The limit $\tau_c \!\to\! \infty$ corresponds to the no-feedback case, where no rotations are applied. Parameters: $\Omega_{BG}/(2\pi) = 0.9$ MHz, $\gamma_B = 10~ \Omega_{BG}$, $\Omega_{DG}/(2\pi) = 14$ kHz, $\gamma_D = 0.875~\Omega_{DG}$, $\bar{N}_B = 0.01$, $\bar{N}_D = 0.05$, $\theta = \phi = \pi/2$.
• Figure S1: Populations of the system's state immediately after the pulse. (a) The vertical arrow on the left-hand side represents the last detected jump $\ket{B}\to\ket{G}$, and the horizontal axis indicates the time elapsed since its detection. The system is driven by $\Omega_{BG}$ (blue line) and $\Omega_{DG}$ (red line). After a time $\tau_c$ has elapsed since the last jump, a pulse $R(\theta,\phi)$ is applied in the $\{\ket{G}, \ket{D}\}$ subspace. The drive $\Omega_{DG}$ can either remain on during the full interval $\tau_c$ or be turned off after a time $\Delta t_{\text{on}}$. (b) Populations of the ground $\ket{G}$ ($P_G$) and dark $\ket{D}$ ($P_D$) states after the pulse for $\theta = \pi/2$, as a function of the direction angle $\phi$, for $\tau_{c} = 4.2~\mu\text{s}$. (c) Population $P_G$ of the ground state as a function of the waiting time $\tau_c$. Solid lines correspond to the case where the drive $\Omega_{DG}$ remains on during $\tau_c$, while dashed lines correspond to the case where $\Omega_{DG}$ is turned off after a time $\Delta t_{\text{on}}$, as illustrated in (a). Blue curves represent the feedback case (rotation applied), while red curves correspond to the no-feedback case (no rotation after $\tau_c$). In the feedback case, for each $\tau_c$ we determine the optimal pulse angles $\{\theta(\tau_c), \phi(\tau_c)\}$ that maximize $P_G$, and use them to compute $P_G(\tau_c)$. The parameters $\gamma_{B,D}$, $\bar{N}_{B,D}$, $\Omega_{BG}$, and $\Omega_{DG}$ are the same as those used in Fig. \ref{['fig:nature-example']}.