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 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.
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.
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 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 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.
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.
We study a quantum walk of a single particle that is subject to stroboscopic projective measurements on a graph with two sites. This two-level system is the minimal model of a measurement-induced quantum walk. The mean first detected transition and return time are computed on an IBM quantum computer as a function of the hopping matrix element between the sites and the on-site potential. The experimentally monitored quantum walk reveals the theoretically predicted behavior, such as the quantization of the first detected return time and the strong increase of the mean first detected transition time near degenerate points, with high accuracy.
We consider a continuous-time quantum walk stroboscopically subjected to projective measurements on two distinct targets with sampling time period τ. We construct a general renewal framework for the first-detection amplitude at each target, from which we obtain the associated splitting probabilities. For Hamiltonians with reflection symmetry, we further show that each dual-target first-detection amplitude can be viewed as an interference effect between a pair of single-target first-detection amplitudes. This formulation yields exact expressions for the splitting probabilities and gives special sampling times at which they undergo pointwise discontinuous jumps due to a form of ergodicity breaking associated with dark states. As an illustrative example, we apply the framework to a one-dimensional tight-binding lattice chain with nearest-neighbor hopping, and study the splitting probabilities as functions of the initial position and sampling time. Below a critical sampling time, 0<τ<τc where τc=2π/ΔE with ΔE being the energy bandwidth, the splitting probabilities for large system sizes become equal to 1/2, independent of the initial position, as long as this position does not coincide with target sites. Above the critical sampling time, τ≥τc, the splitting probabilities deviate from 1/2 and develop oscillations as a function of initial position. The wavenumber associated with these oscillations is Q=2sin−1(rτc/τ), where r is a positive integer. We also explore the effects of relative phase and spatial separation between different components of a coherent initial state on the splitting probabilities. Finally, we study how a finite number of measurements modifies the splitting probabilities and broadens the resonances at special sampling times.
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.