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arXiv:2012.01763 · Physical Review A

First-detection time of a quantum state under random probing

David A. Kessler, Eli Barkai, K. Ziegler

Published Feb 25, 2021• 23 citations
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Abstract

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n\mathrm{th}$ attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\ensuremath{\delta}$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.

DOI
10.1103/physreva.103.022222

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We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the nthn\mathrm{th}nth attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size LLL, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow (\ensuremathδ\ensuremath{\delta}\ensuremathδ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.
Abstract We provide a general framework to compute the probability distribution F r ( t ) of the first detection time of a ‘state of interest’ in a closed quantum system subjected to random projective measurements. In our ‘quantum resetting’ protocol, resetting of a state is not implemented by an additional classical stochastic move, but rather by the random projective measurement. We then apply this general framework to Poissonian measurement protocol with a constant rate r and demonstrate that exact results for F r ( t ) can be obtained for a generic two level system. Interestingly, the result depends crucially on the detection schemes involved and we have studied two complementary schemes, where the state of interest either coincides or differs from the initial state. We show that F r ( t ) at short times vanishes universally as F r ( t ) ∼ t 2 as t → 0 in the first scheme, while it approaches a constant as t → 0 in the second scheme. The mean first detection time, as a function of the measurement rate r, also shows rather different behaviors in the two schemes. In the former, the mean detection time is a non-monotonic function of r with a single minimum at an optimal value r ∗ , while in the later, it is a monotonically decreasing function of r, signaling the absence of a finite optimal value. These general predictions for arbitrary two level systems are then verified via explicit computation in the Jaynes–Cummings model of light–matter interaction. We also generalize our results to non-Poissonian measurement protocols with a renewal structure where the intervals between successive independent measurements are distributed via a general distribution p ( τ ) and show that the short time behavior of F r ( t ) ∼ p ( 0 ) t 2 is universal as long as p ( 0 ) ≠ 0 . This universal t 2 law emerges from purely quantum dynamics that dominates at early times.
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\mathcal{N}N increases, with two plateaus corresponding to integers, where Fn{F}_{n}Fn​ is the first detected return probability at the nthn\mathrm{th}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{F}_{n}Fn​ decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of Fn{F}_{n}Fn​. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.
A quantum walker on a graph, prepared in the state ∣ψin⟩| ψ_{\rm in} \rangle∣ψin​⟩, e.g. initially localized at node rinr_{\rm in}rin​, is repeatedly probed, with fixed frequency 1/τ1/τ1/τ, to test its presence at some target node rdr_{\rm d}rd​ until the first successful detection. This is a quantum version of the first-passage problem. We investigate the total detection probability PdetP_{\rm det}Pdet​, i.e. the probability to eventually detect the particle after an arbitrary number of detection attempts. It is demonstrated that this total detection probability is less than unity in symmetric systems, where it is possible to find initial states which are shielded from the detector by destructive interference, so-called dark states. The identification of physically equivalent initial states yields an upper bound for PdetP_{\rm det}Pdet​ in terms of the reciprocal of the number ννν of physically equivalent states. The relevant subgroup of the system's symmetry operations is found to be the stabilizer of the detection state. Using this, we prove that all bright, i.e. surely detectable, states are symmetric with respect to the stabilizer. This implies that PdetP_{\rm det}Pdet​ can be obtained from a diagonalization of the "symmetrized" Hamiltonian, instead of having to find all eigenstates of the Hamiltonian.
Abstract Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements (MNM) until the first detection is an integer, namely the dimensionality of the accessible Hilbert space. Moreover, the mean first detected return (FDR) time is equal to the average time step between successive measurements times the MNM. Thus, the mean FDR time scales linearly with the dimensionality of the accessible Hilbert space. The main goal of this work is to explain the quantization of the mean return time in terms of a quantized Berry phase.
Abstract We consider the quantum dynamical evolution of a single particle hopping on a graph of N nodes with a Hamiltonian H = ∑ x , y = 1 N | x ⟩ ⟨ y | that allows it to hop between any pair of sites with equal amplitudes. We try to detect the particle in a subset of sites A applying projective measurements at random times. We study analytically the distribution of the time at which the detection of the particle in A is successful for the first time. Exact analytical expressions are obtained both for the mean first detection time as well as its full distribution as functionals of the initial state | Ψ 0 ⟩ and the measurement rate r . In particular, we elucidate the role of the presence of dark components in the initial state that affect the statistics of the first detection time. Exact enumeration agrees perfectly with our analytical predictions.
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