Deterministic Equations for Feedback Control of Open Quantum Systems II: Properties of the memory function

TL;DR

The paper develops a deterministic, memory–based framework for open quantum systems under feedback, showing that the memory function behaves as a classical process coupled to the quantum system via a hybrid classical–quantum state. It introduces memory statistics, including moments, cumulants, and time correlations, through a memory‑resolved state and characteristic function, and defines information measures such as the system–memory mutual information. The authors present explicit memory function examples (jump‑based and current‑resolved) and apply the framework to two control schemes: an inversion protocol driven by quantum jump detections and a Rabi‑stabilization protocol with projective measurements. This work unifies discrete and continuous feedback in a single analytical formalism, enabling analytical insight into memory statistics and information flow for robust quantum control and metrology.

Abstract

Feedback uses past detection outcomes to dynamically modify a quantum system and is central to quantum control. These outcomes can be stored in a memory, defined as a stochastic function of past measurements. In this work, we investigate the main properties of a general memory function subject to arbitrary feedback dynamics. We show that the memory can be treated as a classical system coupled to the monitored quantum system, and that their joint evolution is described by a hybrid bipartite state. This framework allows us to introduce information-theoretic measures that quantify the correlations between the system and the memory. Furthermore, we develop a general framework to characterize the statistics of the memory -- such as moments, cumulants, and correlation functions -- which can be applied both to general feedback-control protocols and to monitored systems without feedback. As an application, we analyze feedback schemes based on detection events in a two-level system coupled to a thermal bath, focusing on protocols that stabilize either the excited-state population or Rabi oscillations against thermal dissipation.

Figures (8)

• Figure 1: Schematics of the inversion protocol. After detecting a decay event $\ket{e}\!\to\!\ket{g}$, the feedback-activated drive acts to steer the system back toward the excited state $\ket{e}$.
• Figure 2: Probability distribution $P_{\text{ss}}(k)$ of the jump memory $k_t$ in the steady state. The jump memory $k_t$ can take the values $-1$ (decay $\ket{e}\to\ket{g}$) and $+1$ (excitation $\ket{g}\to\ket{e}$), so that $P_{\text{ss}}(-1)$ is the probability that the last jump in the steady state corresponds to a decay, and $P_{\text{ss}}(+1) = 1 - P_{\text{ss}}(-1)$. The no-feedback case corresponds to an external drive that is always on, with dynamics described by the Liouvillian in Eq. \ref{['eq: thermal dynamics']}. We consider $\tau_1 = \tau_1^\text{opt}$ [Eq. \ref{['eq: optimal drive time']}], and $\bar{N} = 0.5$.
• Figure 3: Covariance between the Pauli operators $\sigma_z$, $\sigma_y$ and the jump memory $k_t$ in the steady state, as defined in Eq. \ref{['eq: def memory covariance']}. The blue lines correspond to the covariance between $\sigma_z$ and $k_t$, whereas the red lines correspond to the covariance between $\sigma_y$ and the jump memory. The parameters used are $\bar{N} = 0.5$, with $\tau_0 = 0$ for solid lines and $\tau_0 = 1/(2\lambda)$ for dashed lines, and $\tau_1 = \tau_1^\text{opt}$ [Eq. \ref{['eq: optimal drive time']}].
• Figure 4: Mutual information between the system and the jump memory in the steady state as a function of $\lambda\tau_0$, for $\gamma/\lambda = 0.5$ and different bath temperatures.
• Figure 5: Diagram of the stabilization of Rabi oscillations. A qubit is coupled to both a thermal bath and a coherent external drive, and is subjected to projective measurements of its states $\ket{e}$ and $\ket{g}$ at regular intervals $\delta t$. Whenever the qubit is found in the excited state $\ket{e}$, a $\pi$-rotation around the $x$-axis of the Bloch sphere is applied, bringing the system to the ground state $\ket{g}$.

Theorems & Definitions (1)

• Definition 1