Analyzing the Quantum Zeno and anti-Zeno effects using optimal projective measurements

Muhammad Junaid Aftab,Adam Zaman Chaudhry

doi:10.1038/s41598-017-11787-9

Muhammad Junaid Aftab, Adam Zaman Chaudhry

Open Access

https://doi.org/10.1038/s41598-017-11787-9

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Journal: Scientific Reports	Publication Date: Sep 18, 2017
Citations: 9	License type: open-access

Affiliation: Lahore University of Management Sciences

Abstract

Measurements in quantum mechanics can not only effectively freeze the quantum system (the quantum Zeno effect) but also accelerate the time evolution of the system (the quantum anti-Zeno effect). In studies of these effects, a quantum state is prepared repeatedly by projecting the quantum state onto the initial state. In this paper, we repeatedly prepare the initial quantum state in a different manner. Instead of only performing projective measurements, we allow unitary operations to be performed, on a very short time-scale, after each measurement. We can then repeatedly prepare the initial state by performing some projective measurement and then, after each measurement, we perform a suitable unitary operation to end up with the same initial state as before. Our objective is to find the projective measurements that minimize the effective decay rate of the quantum state. We find such optimal measurements and the corresponding decay rates for a variety of system-environment models such as the pure dephasing model and the spin-boson model. We find that there can be considerable differences between this optimized effective decay rate and the usual decay rate obtained by repeatedly projecting onto the initial state. In particular, the Zeno and anti-Zeno regimes can be considerably modified.

Highlights

Rapid repeated measurements can slow down the time evolution of a quantum system, an effect known as the Quantum Zeno effect (QZE)[1]
The survival probability of the system state after one measurement is thens(τ) = TrS,B[( ψ ψ ⊗ )eiHSτe−iHτρ(0)eiHτe−iHSτ], where TrS,B denotes taking the trace over the system and the environment, ρ(0) is the initial combined state of the system and the environment, and the evolution of the system state due to the system Hamiltonian itself has been eliminated via a suitable unitary operation just before performing the measurement[28,38,42]
For the case of the central quantum system being a simple two-level system, we have derived an expression that optimizes the survival probability, or equivalently the effective decay rate. This expression implies that the optimal projective measurement at time τ corresponds to the projector that is parallel to the Bloch vector of the system’s density matrix at that time

Summary

Introduction

Rapid repeated measurements can slow down the time evolution of a quantum system, an effect known as the Quantum Zeno effect (QZE)[1]. Starting from an arbitrary initial state, our goal is to find the projective measurements followed by appropriate unitary operators that can be used to repeatedly prepare the initial state such that the survival probability of the quantum state is maximized. We consider a collection of two-level systems interacting with a common environment In this case, finding the optimal projective measurement is a very difficult problem. We can restrict ourselves to measurements that project onto spin coherent states since these are the measurements that can be performed relatively experimentally For this scenario, we again derive the optimal measurements and the optimized decay rate for the pure dephasing case as well as the more general scenario with both dephasing and relaxation present.

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