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.
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 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 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.
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 quantum-jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in the case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in the same sequence is vanishingly small, which we can achieve in a weak-reset-rate limit. Our results extend previous findings on classical stochastic processes [N. R. Smith et al., Europhys. Lett. 142, 51002 (2023)0295-507510.1209/0295-5075/acd79e] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.
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
Abstract We consider a closed quantum system subject to a stochastic resetting process. The generic expression for the resulting density operator is formulated for arbitrary resetting dynamics, fully characterised by the distribution of times between consecutive reset events. We analyse the behaviour of the state in the long-time regime, as well as the evolution of relevant quantities in the study of quantum coherence and closed- versus open-system dynamics. Our general results are complemented with examples involving particular resetting distributions, and special attention is paid to the two-level (qubit) system, in which we elucidate the effects of the renewal process on the speed of evolution toward an orthogonal state, and gain insight into the resetting applied to open systems.