The quantum first-detection problem concerns the statistics of the time at which a system, subject to repeated measurements, is observed in a prescribed target state for the first time. Unlike its classical counterpart, the measurement back action intrinsic to quantum mechanics may profoundly alter the system dynamics. Here we show that it induces a distinct change in the statistics of the first-detection time. For a quantum particle in one spatial dimension subject to stroboscopic measurements, we observe an algebraic decay of the probability of the first-detection time if the particle is free, an exponential decay in the presence of a confining potential, and a time-dependent crossover between these behaviors if the particle is partially confined. This crossover reflects the purely quantum nature of the detection process, which fundamentally distinguishes it from the first-passage problem in classical systems.
The quantum first-detection problem concerns the statistics of the time at which a system, subject to repeated measurements, is observed in a prescribed target state for the first time. Unlike its classical counterpart, the measurement back action intrinsic to quantum mechanics may profoundly alter the system dynamics. Here we show that it induces a distinct change in the statistics of the first-detection time. For a quantum particle in one spatial dimension subject to stroboscopic measurements, we observe an algebraic decay of the probability of the first-detection time if the particle is free, an exponential decay in the presence of a confining potential, and a time-dependent crossover between these behaviors if the particle is partially confined. This crossover reflects the purely quantum nature of the detection process, which fundamentally distinguishes it from the first-passage problem in classical systems.
We investigate a tight-binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory," and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one-dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates that cause remarkable changes in the behavior in the system, such as divergence of the mean detection time in finite systems and decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line, we show that in our system the first detection probability decays classically like (time)^{-3/2}. This is dramatically different compared to local measurements, which yield a decay rate of (time)^{-3}, indicating that the exponents of the first passage time depend on the type of measurements used.
Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.
The quantum first-passage problem involves stochastic trajectories conditioned on measurement outcomes. The timing statistics of such trajectories remain largely unexplored in open quantum systems. Here, we investigate the first-passage time to an energy threshold for a ubiquitous model: a harmonic oscillator driven by classical additive noise. We find that projective measurements and energy quantization lead to substantial differences between the quantum and classical first-passage-time distributions at low thresholds, while these differences gradually diminish as the threshold energy increases. We treat the problem using both ensemble-averaged conditioned density-matrix dynamics and trajectory-resolved stochastic pure-state dynamics. The two descriptions yield indistinguishable timing statistics. Quantization effects appear in the ensemble-level phase-space distributions of the surviving states and vanish at larger threshold energies. Individual trajectories reveal emergent quantum signatures from the repeated measurements, such as persistent Wigner negativity. Our results provide a framework for using first-passage processes to create measurement-induced nonclassical resource states and to study the quantum-classical crossover of monitored systems.
We consider the first detection problem for a one-dimensional quantum walk with repeated local measurements. Employing the stroboscopic projective measurement protocol and the renewal equation, we study the effect of tunneling on the detection time. Specifically, we study the continuous-time quantum walk on an infinite tight-binding lattice for two typical situations with physical reality. The first is the case of a quantum walk in the absence of tunneling with a Gaussian initial state. The second is the case where a barrier is added to the system. It is shown that the transition of the decay behavior of the first detection probability can be observed by modifying the initial condition, and in the presence of a tunneling barrier, the particle can be detected earlier than the impurity-free lattice. This suggests that the evolution of the walker is expedited when it tunnels through the barrier under repeated measurement. The first detection tunneling time is introduced to investigate the tunneling time of the quantum walk. In addition, we analyze the critical transitive point by deriving an asymptotic formula.
First-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.
The problem of quantum first detection time has been extensively investigated in recent years. Employing the stroboscopic measurement protocol, we consider such a monitored quantum walk on sequentially or periodically perturbed rings and focus on the statistics of first detected return time, namely, the time it takes a particle to return to the initial state for the first time. Using time-independent perturbation theory, we obtain the general form of the eigenvalues and eigenvectors of the Hamiltonian. For the case of a sequentially perturbed ring system, we find steplike behaviors of ${\ensuremath{\sum}}_{n=1}^{\mathcal{N}}n{F}_{n}$ ($\ensuremath{\rightarrow}\ensuremath{\langle}n\ensuremath{\rangle}$ as $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$) when N increases, with two plateaus corresponding to integers, where Fn is the first detected return probability at the nth detection attempt. Meanwhile, if the initial condition preserves the reflection symmetry, the mean return time is the same as the unperturbed system. For the periodically perturbed system, similar results can also appear in the case where the symmetry is preserved; however, the size of the ring, the interval between adjacent perturbations, and the initial position may change the mean return time in most cases. In addition, we find that the decay rate of the first detection probability Fn decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of Fn. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.