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Register a new userContinuous monitoring measured signals bounded by past and future conditions in enlarged quantum systems
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Le Bin Ho
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In a quantum system that is bounded by past and future conditions, weak continuous monitoring forward-evolving and backward-evolving quantum states are usually carried out separately. Therefore, measured signals at a given time t cannot be monitored continuously. Here, we propose an enlarged-quantum-system method to combine these two processes together. Therein, we introduce an enlarged quantum state that contains both the forward- and backward-evolving quantum states. The enlarged state is governed by an enlarged master equation and propagates one-way forward in time. As a result, the measured signals at time t can be monitored continuously and can provide advantages in the signals amplification and signal processing techniques. Our proposal can be implemented on various physical systems, such as superconducting circuits, NMR systems, ion-traps, quantum photonics, and among others.
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We expand the time reversal symmetry arguments of quantum mechanics, originally proposed by Wigner in the context of unitary dynamics, to contain situations including generalized measurements for monitored quantum systems. We propose a scheme to derive the time reversed measurement operators by considering the Schr{o}dinger picture dynamics of a qubit coupled to a measuring device, and show that the time reversed measurement operators form a Positive Operator Valued Measure (POVM) set. We present three particular examples to illustrate time reversal of measurement operators: (1) the Gaussian spin measurement, (2) a dichotomous POVM for spin, and (3) the measurement of qubit fluorescence. We then propose a general rule to unravel any rank two qubit measurement, and show that the backward dynamics obeys the retrodicted equations of the forward dynamics starting from the time reversed final state. We demonstrate the time reversal invariance of dynamical equations using the example of qubit fluorescence. We also generalize the discussion of a statistical arrow of time for continuous quantum measurements introduced by Dressel et al. [Phys. Rev. Lett. 119, 220507 (2017)]: we show that the backward probabilities can be computed from a process similar to retrodiction from the time reversed final state, and extend the definition of an arrow of time to ensembles prepared with pre- and post-selections, where we obtain a non-vanishing arrow of time in general. We discuss sufficient conditions for when times arrow vanishes and show our method also captures the contributions to times arrow due to natural physical processes like relaxation of an atom to its ground state. As a special case, we recover the time reversibility of the weak value as its complex conjugate using our method, and discuss how our conclusions differ from the time-symmetry argument of Aharonov-Bergmann-Lebowitz (ABL) rule.
We analyze the problem of estimating past quantum states of a monitored system from a mathematical perspective in order to ensure self-consistency with the principle of quantum non-demolition. Despite several claims of ``measuring noncommuting observables in the physics literature, we show that we are always measuring commuting processes. Our main interest is in the notion of quantum smoothing or retrodiction. In particular, we examine proposals to estimate the result of an external measurement made on an open quantum systems during a period where it is also undergoing continuous monitoring. A full analysis shows that the non-demolition principle is not actually violated, and so a well-posed as a statistical inference problem can be formulated. We extend the formalism to consider multiple independent external measurements made on the system over the course of a continual period of monitoring.
We study the heat statistics of a multi-level $N$-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number $M$ of measurements $(M to infty)$. In this context, the conditions allowing for an Infinite-Temperature Thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the non-equilibrium evolution of the system and its initial state. Exceptions to ITT, to which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasi-commuting) with the Hamiltonian of the quantum system, or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces ($N to infty$), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits $Mtoinfty$ and $Ntoinfty$ matters: when $N$ is fixed and $M$ diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A non trivial result is obtained fixing $M/N^2$ where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.
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We consider a single copy of a quantum particle moving in a potential and show that it is possible to monitor its complete wave function by only continuously measuring its position. While we assume that the potential is known, no information is available about its state initially. In order to monitor the wave function, an estimate of the wave function is propagated due to the influence of the potential and continuously updated according to the results of the position measurement. We demonstrate by numerical simulations that the estimation reaches arbitrary values of accuracy below 100 percent within a finite time period for the potentials we study. In this way our method grants, a certain time after the beginning of the measurement, an accurate real-time record of the state evolution including the influence of the continuous measurement. Moreover, it is robust against sudden perturbations of the system as for example random momentum kicks from environmental particles, provided they occur not too frequently.
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