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.
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 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.
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.
Feedback can convert continuously monitored quantum jumps into a thermodynamic resource. We formulate full counting statistics for open quantum systems under unital jump feedback by incorporating the feedback maps into the tilted generator. The resulting trajectory ensemble determines both current fluctuations and the Fisher information of the measurement record. We show that feedback can enhance reservoir-parameter estimation and clock precision without necessarily changing average thermodynamic currents. This enhanced precision is not bounded by reservoir entropy production alone. By embedding the reduced dynamics in an enlarged measurement-feedback process, we derive a feedback-modified thermodynamic uncertainty relation in which the information entropy production of the feedback apparatus supplies the missing cost. A charge-monitored double quantum dot illustrates the framework: jump-conditioned feedback improves thermometry and chemical-potential sensing, and stabilizes a quantum clock defined by output-current ticks.
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.
Quantum systems are the foundation of quantum technology development. To harness these systems, one must extract information from them and be able to control their dynamics. This thesis presents a body of work, containing both published and unpublished papers, that focuses on two key research areas: quantum continuous measurement and quantum bath sensing. While these topics may seem distinct, they are fundamentally connected through the framework of open quantum systems. As such, this thesis is structured into two parts. The first part of this thesis is dedicated to diffusive continuous quantum measurement, where the system of interest is continuously monitored through a coupled Markovian bath or meter. A key challenge in this context is the numerical errors introduced in quantum trajectory simulations caused by the finite time resolution ( 𝑡) in real experiments. To address this, we propose a systematic framework to assess theoretical tools used in processing measurement records. Focusing on a single Lindblad channel, we analyze the errors of existing measurement (or Kraus) operators in literature, in orders of 𝑡 under conditions that ensure valid average evolution (complete positivity, convex linearity, and trace preservation). We also analyze a stronger condition: finding the order at which these existing measurement operators match the Lindblad equation. Based on these analyzes, we propose a Kraus operator that satisfies the conditions for valid quantum evolution and matches the Lindblad master equation to second order in 𝑡, surpassing all existing approaches. Moreover, to evaluate how well our operator reproduces exact quantum trajectories, we analyze two qubit measurement examples where exact maps can be derived: 𝑧-basis measurement and fluorescence (dissipative) measurement. We show analytically that our proposed operator achieves the smallest average trace distance to the exact quantum trajectories compared to existing methods. Finally, we extend this Kraus operator to multiple Lindblad channels with a driving Hamiltonian. We test our analytical findings numerically using the two qubit examples, considering both single and multiple Lindblad channels with a driving Hamiltonian. The numerical results confirm our theoretical predictions. The second part of this thesis focuses on quantum bath sensing, where we use a quantum system (qubit) to extract information about its surrounding baths. Specifically, we develop protocols to detect nonclassical pure-dephasing noise in a single-qubit system coupled to temporally correlated (non-Markovian) baths. Inspired by the Ramsey protocol, we introduce a coherent intervention (𝜋 2 -pulse) within the protocol, which can reveal the quantum noise - noise correlations affecting the probe qubit that originate from quantum degrees of freedom. Our protocols are simple, experimentally viable, and rigorously derived for general bath models. Using an exactly solvable bosonic bath model, we analyse how key noise parameters respond to fixed control schemes, identifying the optimal strategy for detecting the quantum noise. From the perspective of bath dynamics, we further decompose the qubit-bath state, demonstrating that only Gaussian components of the bath contribute to quantum noise, highlighting the bath's role in system dynamics. Finally, we extend the modified Ramsey protocol by incorporating multi-axis 𝜋 2 pulse and demonstrate that it can emulate the modified Ramsey protocol with any completely positive trace-preserving intervention. This applies to both Gaussian and non-Gaussian baths under the pure-dephasing assumption.
Generalized master equations (GMEs)---time-local but generally neither trace-preserving nor Hermiticity-preserving---are convenient tools to compute properties of the environment of an open or continuously monitored quantum system. A two-sided master equation yields the fidelity and quantum Fisher information (QFI) of environment states, thereby setting fundamental limits for hypothesis testing and parameter estimation under continuous monitoring. For unmonitored noise or inefficient detection, the QFI of the detectable part of the environment may be obtained from a recently derived GME acting on multiple system replicas. Tilted master equations provide the full counting statistics of quantum jumps and diffusive measurements, enabling, e.g., studies of quantum thermodynamics beyond average values. Here we focus on bosonic linear systems, governed by a quadratic Hamiltonian and linear jump operators, whose dynamics preserves Gaussianity. For Gaussian initial operators, we recast a generic GME as a compact set of ordinary differential equations for the covariance matrix (a Riccati-type equation), first moments, and normalization. These equations can be integrated efficiently without Hilbert-space truncation, and admit analytical results in simple settings. We also provide specialized forms for fidelity/QFI and full counting statistics. We illustrate the formalism with a continuously monitored optical parametric oscillator, using it to determine sensitivity limits for frequency estimation and to benchmark Hasegawa's thermodynamic uncertainty relations.