The quantum first-passage problem involves stochastic trajectories conditioned on measurement outcomes. The timing statistics of such trajectories remain largely unexplored in open quantum systems. Here, we investigate the first-passage time to an energy threshold for a ubiquitous model: a harmonic oscillator driven by classical additive noise. We find that projective measurements and energy quantization lead to substantial differences between the quantum and classical first-passage-time distributions at low thresholds, while these differences gradually diminish as the threshold energy increases. We treat the problem using both ensemble-averaged conditioned density-matrix dynamics and trajectory-resolved stochastic pure-state dynamics. The two descriptions yield indistinguishable timing statistics. Quantization effects appear in the ensemble-level phase-space distributions of the surviving states and vanish at larger threshold energies. Individual trajectories reveal emergent quantum signatures from the repeated measurements, such as persistent Wigner negativity. Our results provide a framework for using first-passage processes to create measurement-induced nonclassical resource states and to study the quantum-classical crossover of monitored systems.
The quantum first-passage problem involves stochastic trajectories conditioned on measurement outcomes. The timing statistics of such trajectories remain largely unexplored in open quantum systems. Here, we investigate the first-passage time to an energy threshold for a ubiquitous model: a harmonic oscillator driven by classical additive noise. We find that projective measurements and energy quantization lead to substantial differences between the quantum and classical first-passage-time distributions at low thresholds, while these differences gradually diminish as the threshold energy increases. We treat the problem using both ensemble-averaged conditioned density-matrix dynamics and trajectory-resolved stochastic pure-state dynamics. The two descriptions yield indistinguishable timing statistics. Quantization effects appear in the ensemble-level phase-space distributions of the surviving states and vanish at larger threshold energies. Individual trajectories reveal emergent quantum signatures from the repeated measurements, such as persistent Wigner negativity. Our results provide a framework for using first-passage processes to create measurement-induced nonclassical resource states and to study the quantum-classical crossover of monitored systems.
First-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.
The quantum first-detection problem concerns the statistics of the time at which a system, subject to repeated measurements, is observed in a prescribed target state for the first time. Unlike its classical counterpart, the measurement back action intrinsic to quantum mechanics may profoundly alter the system dynamics. Here we show that it induces a distinct change in the statistics of the first-detection time. For a quantum particle in one spatial dimension subject to stroboscopic measurements, we observe an algebraic decay of the probability of the first-detection time if the particle is free, an exponential decay in the presence of a confining potential, and a time-dependent crossover between these behaviors if the particle is partially confined. This crossover reflects the purely quantum nature of the detection process, which fundamentally distinguishes it from the first-passage problem in classical systems.
The evolution of a quantum system subject to measurements can be described by stochastic quantum trajectories of pure states. Instead, the ensemble average over trajectories is a mixed state evolving via a master equation. Both descriptions lead to the same expectation values for linear observables. Recently, there is growing interest in the average entanglement appearing during quantum trajectories. The entanglement is a nonlinear observable that is sensitive to so-called measurement-induced phase transitions, namely, transitions from a system-size dependent phase to a quantum Zeno phase with area-law entanglement. Intriguingly, the mixed steady-state description of these systems is insensitive to this phase transition. Together with the difficulty of quantifying the mixed state entanglement, this favors quantum trajectories for the description of the quantum measurement process. Here, we study the entanglement of a single particle under continuous measurements (using the newly developed configuration coherence) in both the mixed state and the quantum trajectories descriptions. In both descriptions, we find that the entanglement at intermediate time scales shows the same qualitative behavior as a function of the measurement strength. The entanglement engenders a notion of coherence length, whose dependence on the measurement strength is explained by a cascade of underdamped-to-overdamped transitions. This demonstrates that measurement-induced entanglement dynamics can be captured by mixed states.
Classical first-passage time distributions have been extensively studied both theoretically and experimentally. Their quantum counterparts—quantum first-passage time distributions (QFPTDs)—remain largely unexplored and have deep implications for both fundamental physics and the development of emerging quantum technologies. We measure the first QFPTDs using a motional mode of a single trapped ion. We develop a composite-phase laser pulse sequence to perform tunable stroboscopic single-shot projective measurements of the motional state of a trapped ion. We measure QFPTDs of the ion energy when coupled to electric-field noise. The measurement protocol developed here is broadly applicable to other quantum systems and provides a powerful method for exploring a broad range of QFPTD phenomena. With these results, we open a field of experimental investigations of QFPT processes with potential future relevance to quantum search algorithms, unraveling connections between classical and quantum dynamics, and study of the quantum measurement problem.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.