Non-Hermitian and Zeno limit of quantum systems under rapid measurements

Abstract

We investigate in depth the relation between the first detection time of an isolated quantum system that is repeatedly perturbed by strong local measurements with a large fixed frequency $1/\tau$, determining whether it is in some given state $|{\psi }_{\text{d}}⟩$, and the time of absorption to the same state of the same system with the added imaginary potential $2i\hslash |{\psi }_{\text{d}}⟩⟨{\psi }_{\text{d}}|/\tau$. As opposed to previous works, we compare directly the solutions of both problems in the small $\tau$, i.e., Zeno, limit. We find a scaling collapse in $F(t)$ with respect to $\tau$ and compute the total detection probability as well as the moments of the first detection time probability density $F(t)$ in the Zeno limit. We show that both solutions approach the same result in this small $\tau$ limit, as long as the initial state $|{\psi }_{\text{in}}⟩$ is not parallel to the detection state, i.e. as long as $|⟨{\psi }_{\text{d}}|{\psi }_{\text{in}}⟩|<1$. However, when this condition is violated, the small probability density to detect the state on time scales much larger than $\tau$ is precisely a factor of four different for all such times. We express the solution of the Zeno limit of both problems formally in terms of an electrostatic analogy. Our results are corroborated with numerical simulations.

arXiv comment22 pages, 8 Figures

References in corpus (60)

• Quantum Mechanics helps in searching for a needle in a haystack
• The Quantum Jump Approach to Dissipative Dynamics in Quantum Optics
• Intermittent search strategies
• Spatial search by quantum walk
• Highly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transport
• Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms
• Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas
• Topological Transition in a Non-Hermitian Quantum Walk
• Quantum Zeno dynamics: mathematical and physical aspects
• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• A Review of Perfect State Transfer and its Application as a Constructive Tool
• Search via Quantum Walk
• A simple model of quantum trajectories
• Observation of critical phenomena in parity-time-symmetric quantum dynamics
• Experimental realization of quantum zeno dynamics
• Spatial search by quantum walk is optimal for almost all graphs
• Hitting time for quantum walks on the hypercube
• Recurrence for discrete time unitary evolutions
• Recurrence and Pólya number of quantum walks
• Constant-pressure sound waves in non-Hermitian disordered media
• Quantum dynamics with stochastic reset
• A measurement-based approach to quantum arrival times
• Quantum walks with infinite hitting times
• Detection of a quantum particle on a lattice under repeated projective measurements
• Quantum walks: the first detected passage time problem
• Systematic Dimensionality Reduction for Quantum Walks: Optimal Spatial Search and Transport on Non-Regular Graphs
• Engineering tunable local loss in a synthetic lattice of momentum states
• Quantum walks on quotient graphs
• First detected arrival of a quantum walker on an infinite line
• Spectral properties of simple classical and quantum reset processes
• Quantum recurrence of a subspace and operator-valued Schur functions
• Hitting time for the continuous quantum walk
• Transition from discrete to continuous time of arrival distribution for a quantum particle
• Survival of classical and quantum particles in the presence of traps
• Interlacing Relaxation and First-Passage Phenomena in Reversible Discrete and Continuous Space Markovian Dynamics
• Probing Measurement Induced Effects in Quantum Walks via Recurrence
• Thermalization and Heating Dynamics in Open Generic Many-Body Systems
• Quantum Zeno Dynamics through stochastic protocols
• Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit
• Large fluctuations of the first detected quantum return time
• Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks: validity and effectiveness
• Disclosing hidden information in the quantum Zeno effect: Pulsed measurement of the quantum time of arrival
• Quantum Renewal Equation for the first detection time of a quantum walk
• Survival probability in a one-dimensional quantum walk on a trapped lattice
• Emergence of Randomness and Arrow of Time in Quantum Walks
• Experimental proof of Quantum Zeno-assisted Noise Sensing
• Return to origin problem for particle on a one-dimensional lattice with quasi-Zeno dynamics
• Continuous-Time Quantum Walks and Trapping
• Generalized Kac's Lemma for Recurrence Time in Iterated Open Quantum Systems
• Continuous Time Quantum Walks in finite Dimensions
• Generalized relation between pulsed and continuous measurements in the quantum Zeno effect
• Quantized recurrence time in iterated open quantum dynamics
• The spectral dimension controls the decay of the quantum first detection probability
• Uncertainty and symmetry bounds for the quantum total detection probability
• Beyond Zeno: Approaching Infinite Temperature upon Repeated Measurements
• Quantum Quasi-Zeno Dynamics: Transitions mediated by frequent projective measurements near the Zeno regime
• Suppression of Zeno effect for distant detectors
• Does the measurement take place when nobody observes it?
• Quantum total detection probability from repeated measurements II. Exploiting symmetry
• Quantization of the mean decay time for non-Hermitian quantum systems

Cited by in corpus (9)

• Restart expedites quantum walk hitting times
• Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit
• Dark states of quantum search cause imperfect detection
• The first detection time of a quantum state under random probing
• Driving quantum systems with repeated conditional measurements
• Quest for optimal quantum resetting: protocols for a particle on a chain
• Thermodynamic phases in first detected return times of quantum many-body systems
• Quantization of the mean decay time for non-Hermitian quantum systems
• Phase transitions induced by standard and predetermined measurements in transmon arrays