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Quantum survival and first detection probabilities in projective measurements

Shrabanti Dhar, Subinay Dasgupta, Abhishek Dhar

Published Dec 20, 2013• 0 citations
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Abstract

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 $\tau$, and keeps doing so till it detects the particle, say on the $n^{\rm th}$ observation, at time $t=n\tau$. We take $\tau$ to be small but do not take the limit of continuous measurements $\tau\to 0$. 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.

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We consider a hypothetical experiment where a particle is released at time t=0t=0t=0 in a fixed initial state. A detector makes instantaneous quantum measurements at regular intervals of time τ\tauτ, and keeps doing so till it detects the particle, say on the nthn^{\rm th}nth observation, at time t=nτt=n\taut=nτ. We take τ\tauτ to be small but do not take the limit of continuous measurements τ→0\tau\to 0τ→0. We examine the survival probability P(t)P(t)P(t) that the particle is un-detected till time ttt. 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.
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 \emph{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 the setup where a detector keeps making instantaneous measurements at regular finite time intervals {\emph{till}} it detects the particle at some time ttt, 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.
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.
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.
What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: How does the survival probability SmS_mSm​, namely, the probability that an initial state survives even after mmm number of measurements, behave as a function of mmm? In the context of two paradigmatic quantum systems, one evolving in discrete and the other in continuous time, we present a wealth of numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in mmm for large mmm. By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability decays asymptotically as a power law in mmm with exponent 3/23/23/2. These results hold independently of the choice of the distribution of times between successive measurements, implying robustness and ubiquity of our derived results.
Abstract What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m ? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N . For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m . By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying m < m 1 ⋆ ( N ) , the decay of the survival probability is as m −2 , (ii) for m satisfying m 1 ⋆ ( N ) ≪ m < m 2 ⋆ ( N ) , the decay is as m −3/2 , while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our resu
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.