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arXiv:1609.05501 · Physical Review A

Quantum evolution in the stroboscopic limit of repeated measurements

I. A. Luchnikov, Sergey N. Filippov

Published Feb 10, 2017• 30 citations
arXivHTMLPDF

Abstract

We consider a quantum system dynamics caused by successive selective and nonselective measurements of the probe coupled to the system. For the finite measurement rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$ and the system-probe interaction strength $\ensuremath{\gamma}$ we derive analytical evolution equations in the stroboscopic limit $\ensuremath{\tau}\ensuremath{\rightarrow}0$ and ${\ensuremath{\gamma}}^{2}\ensuremath{\tau}=\mathrm{const}$, which can be considered as a deviation from the Zeno subspace dynamics on a longer time scale $T\ensuremath{\sim}{({\ensuremath{\gamma}}^{2}\ensuremath{\tau})}^{\ensuremath{-}1}\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}1}$. Nonlinear quantum dynamics is analyzed for selective stroboscopic projective measurements of an arbitrary rank. Nonselective measurements are shown to induce the semigroup dynamics of the system-probe aggregate. Both nonlinear and decoherent effects become significant at the time scale $T\ensuremath{\sim}{({\ensuremath{\gamma}}^{2}\ensuremath{\tau})}^{\ensuremath{-}1}$, which is illustrated by a number of examples.

DOI
10.1103/physreva.95.022113

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We consider a quantum system dynamics caused by successive selective and nonselective measurements of the probe coupled to the system. For the finite measurement rate \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1 and the system-probe interaction strength \ensuremathγ\ensuremath{\gamma}\ensuremathγ we derive analytical evolution equations in the stroboscopic limit \ensuremathτ\ensuremath→0\ensuremath{\tau}\ensuremath{\rightarrow}0\ensuremathτ\ensuremath→0 and \ensuremathγ2\ensuremathτ=const{\ensuremath{\gamma}}^{2}\ensuremath{\tau}=\mathrm{const}\ensuremathγ2\ensuremathτ=const, which can be considered as a deviation from the Zeno subspace dynamics on a longer time scale T\ensuremath∼(\ensuremathγ2\ensuremathτ)\ensuremath−1\ensuremath≫\ensuremathγ\ensuremath−1T\ensuremath{\sim}{({\ensuremath{\gamma}}^{2}\ensuremath{\tau})}^{\ensuremath{-}1}\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}1}T\ensuremath∼(\ensuremathγ2\ensuremathτ)\ensuremath−1\ensuremath≫\ensuremathγ\ensuremath−1. Nonlinear quantum dynamics is analyzed for selective stroboscopic projective measurements of an arbitrary rank. Nonselective measurements are shown to induce the semigroup dynamics of the system-probe aggregate. Both nonlinear and decoherent effects become significant at the time scale T\ensuremath∼(\ensuremathγ2\ensuremathτ)\ensuremath−1T\ensuremath{\sim}{({\ensuremath{\gamma}}^{2}\ensuremath{\tau})}^{\ensuremath{-}1}T\ensuremath∼(\ensuremathγ2\ensuremathτ)\ensuremath−1, which is illustrated by a number of examples.
In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective projective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such nonselective measurements. Importantly, we show that there is a difference between using non-selective projective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective projective measurements can differ very significantly as compared to the repeated selective measurement scenario.
We investigate the evolution of a single qubit subject to a continuous unitary ``free'' dynamics and an additional interrupting influence which occurs periodically. One may imagine a dynamically evolving closed quantum system which becomes open at certain times. The interrupting influence is represented by an operation, which is assumed to equivalently describe a nonselective weak measurement. It may be decomposed into the action of a positive operator, which in the case of a measurement represents the pure measurement part, followed by a unitary backaction depending on the result of the measurement (feedback). Equations of motion for the state evolution are derived in the form of difference equations. In the lowest order, the stochastic feedbacks cause a modification of the ``free'' Hamiltonian by an additional appropriately averaged Hamiltonian. The positive operator specifies a decoherence rate and results in a decoherence term. Two higher-order terms are discussed. One shows decoherence induced by the stochastic feedbacks and the other represents generalized friction. The selective evolution is investigated. In order to bridge the gap between sequential and continuous measurements, the continuum limit to a master equation is performed. Additional correcting higher-order terms are worked out in the Appendixes.
We present a model of continuous (in time) position measurements on a quantum system using a single pseudoclassical meter. The nonselective evolution of the system is described by a master equation which is identical to that obtained from previous models. The selective evolution is described by a stochastic nonlinear Schr\"odinger equation. The significance of this equation is that the stochastic term has a physical interpretaion. By carefully choosing the parameters which define the meter and the system-meter coupling, we obtain a meter pointer with well-defined position which undergoes fluctuations. This ``jitter'' in the pointer position gives rise to the stochastic dynamical collapse of the system wave function. By the inclusion of feedback on the meter, the pointer is made to relax towards an appropriate readout. We apply this model to the selective measurement of the position of a particle in a double-well potential. In contrast to a recent claim [H. Fearn and W. E. Lamb, Jr., Phys. Rev. A 46, 1199 (1992)] we show that truly continuous position measurements lead to a quantum Zeno effect in certain parameter regimes. This is manifest by the changing of the particle dynamics from coherent tunneling between the well minima to incoherent flipping, as in a random telegraph. As the measurement strength increases, the average length of time the particle remains stuck in one well increases proportionally.
In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider rapid projective measurements with equal time intervals being performed on the system to check whether or not the system is in the initial state. These projective measurements are selective measurements in the sense that the measurement results are read out and only the case where all the measurement results correspond to the initial state is considered in the analysis of the effect of the measurements. In this paper, we extend such a treatment to consider the effect of repeated non-selective measurements - only the final measurement is required to correspond to the initial state, while we do not know the results of the intermediate measurements. We present a general formalism to derive the effective decay rate of the initial quantum state with such non-selective measurements. Importantly, we show that there is a difference between using non-selective measurements and the usual approach of considering only selective measurements only if we go beyond the weak system-environment coupling regime in models other than the usual population decay models. As such, we then apply our formalism to investigate the quantum Zeno and anti-Zeno effects for three exactly solvable system-environment models: a single two-level system undergoing dephasing, a single two-level system interacting with an environment of two-level systems, and a large spin undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno effects in the presence of non-selective measurements can differ very significantly as compared to the repeated selective measurement scenario.
We study the dynamics of an open quantum system linearly coupled to a bosonic reservoir. We show that, in the ultrastrong coupling limit, the system undergoes a nonselective measurement and then evolves unitarily according to an effective Zeno Hamiltonian. This dynamical process is largely independent of the reservoir state. We examine the entanglement breaking effect of the ultrastrong coupling on the system. We also derive the evolution equation for systems in contact with several reservoirs when one coupling is ultrastrong. The effective system dynamics displays a rich structure and, contrarily to the single reservoir case, it is generally non-Markovian. Our approach is based on a Dyson series expansion, in which we can take the ultrastrong limit termwise, and a subsequent resummation of the series. Our derivation is mathematically rigorous and uncomplicated.