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arXiv:2507.01934 · Physical Review Letters

Deterministic Equations for Feedback Control of Open Quantum Systems

Alberto J. B. Rosal, Patrick P. Potts, Gabriel Teixeira Landi

Published Jun 25, 2026• 3 citations
arXivHTMLPDF

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.

DOI
10.1103/pg2p-h529

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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.
In this work, we consider a general feedback protocol based on quantum-jump detections, where the last detected jump channel is stored in a memory and subsequently used to implement a feedback action, such as modifying the system Hamiltonian conditioned on the last jump. We show that the time evolution of this general protocol can be described by a Lindblad master equation defined in a hybrid classical-quantum space, where the classical part encodes the stored measurement record (memory) and the quantum part represents the monitored system. Moreover, we show that this new representation can be used to fully characterize the counting statistics of a system subject to a general jump-based feedback protocol. We apply the formalism to a three-level system coupled to two thermal baths operating as a thermal machine, and we show that jump-based feedback can be used to convert the information obtained from the jump detections into work. Our framework provides analytical tools that enable the characterization of key statistical properties of any counting observable under jump-based feedback, such as the average current, noise, correlation functions, and power spectrum.
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.
Measurement and control of quantum systems play an important role in science and technology, from the foundations of quantum mechanics to practical applications. This thesis presents theoretical and experimental work that exploits quantum measurement and measurement-based feedback control to study some new problems in the contexts of solid state physics and optical quantum systems. In the first part, we consider the dynamics of open quantum systems conditioned on continuous measurements. We revisit the fact that dynamical quantum events are detector dependent by framing the problem as a quantum correlation test of the Einstein-Podolsky-Rosen (EPR) steering type. In this regards, a no-go theorem for quantum diusion is derived. In order to study a realistic experimental situation we scrutinise dierent physical systems, Einstein-Podolsky-Rosen steering inequalities and measurement protocol. We show that the no-go result for quantum diusion is not universal and does not apply to quantum jumps by devising a novel adaptive measurement scheme. Since the objectiveness of pure-state dynamical models is ruled out by quantum entanglement, then it is reasonable to say that quantum jumps are more quantum than quantum diusion. In the rst part we also study a stochastic feedback control of quantum transport. We show that using a feedback control strategy it is possible to restrict the dynamics of a double quantum dot system to a non-orthogonal two-state ensemble. The feedback control approach can be less experimentally challenging than an adaptive measurement scheme, as the latter requires implementing local oscillators, which might be dicult to realise in mesoscopic physics.
We consider open quantum systems weakly coupled to thermal reservoirs and subjected to quantum feedback operations triggered with or without delay by monitored quantum jumps. We establish a thermodynamic description of such systems and analyze how the first and second law of thermodynamics are modified by the feedback. We apply our formalism to study the efficiency of a qubit subjected to a quantum feedback control and operating as a heat pump between two reservoirs. We also demonstrate that quantum feedbacks can be used to stabilize coherences in nonequilibrium stationary states which in some cases may even become pure quantum states.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
The no-knowledge quantum feedback was proposed by Szigeti et al. [Phys. Rev. Lett. 113, 020407 (2014)] as a measurement-based feedback protocol for decoherence suppression for an open quantum system. By continuously measuring environmental noises and feeding back controls on the system, the protocol can completely reverse the measurement back-action and therefore suppress the system's decoherence. However, the complete decoherence cancellation was shown only for the instantaneous feedback, which is impractical in real experiments. Therefore, in this work, we generalize the original work and investigate how the decoherence suppression can be degraded with unavoidable delay times, by analyzing non-Markovian average dynamics. We present analytical expressions for the average dynamics and numerically analyze the effects of the delayed feedback for a coherently driven two-level system, coupled to a bosonic bath via a Hermitian coupling operator. We also find that, when the qubit's unitary dynamics does not commute with the measurement and feedback controls, the decoherence rate can be either suppressed or amplified, depending on the delay time.