In recent work, the so-called quasi-Zeno dynamics of a system has been investigated in the context of the quantum first passage problem. This dynamics considers the time evolution of a system subjected to a sequence of selective projective measurements made at small but finite intervals of time. This means that one has a sequence of steps, with each step consisting of a unitary transformation followed by a projection. The dynamics is nonunitary and, in recent work, it has been shown that it can be effectively described by two different non-Hermitian Hamiltonians. Here we explore this connection by considering the problem of detecting a free quantum particle moving on a one-dimensional lattice, where the detector is placed at the origin and the particle is initially located at some specified lattice point. We find that results for distribution times for the first detection probability, obtained from the non-Hermitian Hamiltonians, are in excellent agreement with known exact results as well as exact numerics. Interesting finite-size effects are discussed. We also study the first detection problem for the example of a particle moving in a quasiperiodic potential, an example where the unperturbed particle's motion can be ballistic, localized, or diffusive.
In recent work, the so-called quasi-Zeno dynamics of a system has been investigated in the context of the quantum first passage problem. This dynamics considers the time evolution of a system subjected to a sequence of selective projective measurements made at small but finite intervals of time. This means that one has a sequence of steps, with each step consisting of a unitary transformation followed by a projection. The dynamics is nonunitary and, in recent work, it has been shown that it can be effectively described by two different non-Hermitian Hamiltonians. Here we explore this connection by considering the problem of detecting a free quantum particle moving on a one-dimensional lattice, where the detector is placed at the origin and the particle is initially located at some specified lattice point. We find that results for distribution times for the first detection probability, obtained from the non-Hermitian Hamiltonians, are in excellent agreement with known exact results as well as exact numerics. Interesting finite-size effects are discussed. We also study the first detection problem for the example of a particle moving in a quasiperiodic potential, an example where the unperturbed particle's motion can be ballistic, localized, or diffusive.
We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect its arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals $\ensuremath{\tau}$, and we consider the evolution of the wave function until the time a detection occurs. We study the probabilities of its first detection at some time and, conversely, the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, which consists of unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian. For some examples of a particle moving on one- and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results. A mean-field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and nontrivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.
The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave-function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schr\"odinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented.
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.
We consider a hypothetical experiment where a particle is released at time t=0 in a fixed initial state. A detector makes instantaneous quantum measurements at regular intervals of time τ, and keeps doing so till it detects the particle, say on the nth observation, at time t=nτ. We take τ to be small but do not take the limit of continuous measurements τ→0. We examine the survival probability P(t) that the particle is un-detected till time t. Here we present results for a simple model of a free particle in a one-dimensional box and find that interesting features such as power-law tails are obtained for the survival probability. We provide a perturbative calculation which compares very well with numerical results. Some exact results are obtained for a mean-field model. In the limit of continuous observations, as expected, one always gets a Zeno-like effect where the particle is never detected.
Abstract The problem of the detection statistics of a quantum walker has received increasing interest. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time , with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of , from a state where the probability of detection decreases exponentially in time and the total detection probability is very small, to a state with power-law decay and a significantly higher total probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical is 10/3, as opposed to 3 for every larger . In addition, the moving detector strongly modifies the Zeno effect.
Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the time of arrival (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider a setup where a detector keeps making instantaneous measurements at regular finite time intervals until it detects the particle at time t , which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics.