Time-of-arrival distributions for continuous quantum systems and application to quantum backflow

Abstract

Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\hat{A}$ (position or otherwise), the distribution ${\pi }_a\left(t\right)$ of time measurement at a fixed state $a$ can be inferred from the distribution ${\rho }_t\left(a\right)$ of a state measurement at a fixed time $t$ via the transformation ${\pi }_a(t)\propto \left|\frac{\partial }{\partial t}{\int }_{-\infty }^a{\rho }_t(u)du\right|$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule, and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time-of-arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.

13 pages, 2 Figures, 1 Table. This new version contains a general formula for the current of a superposition of two waves and applications to a superposition of two Gaussian wave packets

References in corpus (24)

• Direct Measurement of the Quantum Wavefunction
• Coherent Backscattering of Ultracold Atoms
• Direct measurement of a nonlocal entangled quantum state
• Transition from discrete to continuous time of arrival distribution for a quantum particle
• Quantum arrival times and operator normalization
• Shouldn't there be an antithesis to quantization?
• Detecting quantum backflow by the density of a Bose-Einstein Condensate
• Realization of a distributed Bragg reflector for propagating guided matter waves
• What does one measure, when one measures the arrival times of a quantum particle?
• Demonstrating backflow in classical two beams' interference
• Shaping the distribution of vertical velocities of antihydrogen in GBAR
• Distribution of the Time at Which an Ideal Detector Clicks
• Relativistic Quantum Backflow
• $\delta$-Quench measurement of quantum wavefunction
• Interference of spin-orbit coupled Bose-Einstein condensates
• Experiment-friendly formulation of quantum backflow
• Quantum State Interferography
• On the experiment-friendly formulation of quantum backflow
• Quantum free fall motion and quantum violation of weak equivalence principle
• Quantum backflow for many-particle systems
• Unbounded quantum backflow in two dimensions
• A regular Hamiltonian halting ratchet for matter wave transport
• Analysis of the timing of freely falling antihydrogen
• Quantum delay in the time of arrival of free-falling atoms

Cited by in corpus (6)

• Time-of-Flow Distributions in Discrete Quantum Systems: From Operational Protocols to Quantum Speed Limits
• Quantum backflow for two identical particles
• Context-Dependent Time-Energy Uncertainty Relations from Projective Quantum Measurements
• Quantum arrival times in free fall
• Quantum stroboscopy for time measurements
• Quantum control of exciton motion in electric field