A general framework for the Quantum Zeno and anti-Zeno effects.

Adam Zaman Chaudhry

doi:10.1038/srep29497

Abstract

Repeated measurements can slow down (the quantum Zeno effect) or speed up (the quantum anti-Zeno effect) the temporal evolution of a quantum system. In this paper, a general treatment of the quantum Zeno and anti-Zeno effects is presented which is valid for an arbitrary system-environment model in the weak system-environment coupling regime. It is shown that the effective lifetime of a quantum state that is subjected to repeated projective measurements depends on the overlap of the spectral density of the environment and a generalized ‘filter function’. This filter function depends on the system-environment Hamiltonian, the state of the environment, and the measurement being performed. Our general framework is then used to study explicitly the Zeno to anti-Zeno crossover behaviour for the spin-boson model where a single two-level system is coupled to a bosonic environment. It is possible to not only reproduce results for the usual population decay case as well as for the pure dephasing model, but to also study the regime where both decay and dephasing take place. These results are then extended to many two-level systems coupled collectively to the bosonic environment to further illustrate the importance of the correct evaluation of the effective decay rate.

Highlights

It is well-known that if a quantum system is subjected to rapidly repeated measurements, the temporal evolution of the quantum system slows down[1]
Assuming that the coupling V can be written as F ⊗B where F is an operator acting in space of the system and B is an operator acting in Hilbert space of the environment, we have found that the effective decay rate can be written as an overlap integral of the spectral density of the environment J(ω) and a generalized ‘filter function’ Q(ω, τ)
We have derived an expression for the effective decay rate of a quantum state in the presence of repeated measurements which is valid when the system-environment coupling is weak

Summary

Introduction

It is well-known that if a quantum system is subjected to rapidly repeated measurements, the temporal evolution of the quantum system slows down[1]. These results are limited in the sense that they are only applicable for exactly solvable pure dephasing models These differences motivate us to investigate the QZE and the QAZE for more arbitrary system-environment Hamiltonians and state preparations. An expression for the effective lifetime of a quantum state, subjected to repeated projective measurements, that is valid for weak system-environment coupling strength This effective lifetime depends on the overlap between the spectral density of the environment and a generalized ‘filter function’ that in turn depends on the state that is repeatedly prepared, the environment correlation function, the system Hamiltonian parameters, the measurement interval, and the system-environment coupling. We find that the effective decay rate is amplified depending on the number of particles coupled to the environment, thereby making the use of the correct filter function even more important

Methods

Results

Conclusion