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arXiv:1607.08871 · Annalen der Physik

Quantum Zeno Dynamics Through Stochastic Protocols

Matthias M. Müller, Stefano Gherardini, Filippo Caruso

Published Jul 21, 2017• 48 citations
arXivHTMLPDF

Abstract

Abstract Quantum measurements play a crucial role in quantum mechanics since they perturb, unavoidably and irreversibly, the state of the measured quantum system. More extremely, the constant observation of a quantum system can even freeze its dynamics to a subspace, effectively truncating the Hilbert space of the system. It represents the quantum version of the famous flying arrow Zeno paradox, and is called quantum Zeno dynamics. In general, it can be obtained by applying frequent consecutive quantum measurements that are equally spaced in time. Here, we introduce time disorder in the measurement sequence, and analytically investigate how this temporal stochasticity may affect the confinement probability of the system in the subspace. As main result, we then exploit how different dissipative and coherent Zeno protocols can be generalized to this stochastic scenario. Finally, our analytical predictions are numerically tested on a paradigmatic spin chain where we find a trade‐off between a probabilistic scheme with high fidelity (compared to perfect subspace dynamics) and a deterministic one with a slightly lower fidelity, moving further steps towards new schemes of Zeno‐based control for future quantum technologies.

DOI
10.1002/andp.201600206

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Abstract Quantum measurements play a crucial role in quantum mechanics since they perturb, unavoidably and irreversibly, the state of the measured quantum system. More extremely, the constant observation of a quantum system can even freeze its dynamics to a subspace, effectively truncating the Hilbert space of the system. It represents the quantum version of the famous flying arrow Zeno paradox, and is called quantum Zeno dynamics. In general, it can be obtained by applying frequent consecutive quantum measurements that are equally spaced in time. Here, we introduce time disorder in the measurement sequence, and analytically investigate how this temporal stochasticity may affect the confinement probability of the system in the subspace. As main result, we then exploit how different dissipative and coherent Zeno protocols can be generalized to this stochastic scenario. Finally, our analytical predictions are numerically tested on a paradigmatic spin chain where we find a trade‐off between a probabilistic scheme with high fidelity (compared to perfect subspace dynamics) and a deterministic one with a slightly lower fidelity, moving further steps towards new schemes of Zeno‐based control for future quantum technologies.
Quantum measurements are crucial for observing the properties of a quantum system, which, however, unavoidably perturb its state and dynamics in an irreversible way. Here we study the dynamics of a quantum system being subjected to a sequence of projective measurements applied at random times. In the case of independent and identically distributed intervals of time between consecutive measurements, we analytically demonstrate that the survival probability of the system to remain in the projected state assumes a large deviation (exponentially decaying) form in the limit of an infinite number of measurements. This allows us to estimate the typical value of the survival probability, which can therefore be tuned by controlling the probability distribution of the random time intervals. Our analytical results are numerically tested for Zeno-protected entangled states, which also demonstrate that the presence of disorder in the measurement sequence further enhances the survival probability when the Zeno limit is not reached (as it happens in experiments). Our studies provide a new tool for protecting and controlling the amount of quantum coherence in open complex quantum systems by means of tunable stochastic measurements.
Abstract. If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected “quantum Zeno subspace ” is called “quantum Zeno dynamics: ” for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.
A system under constant observation is practically freezed to the measurement subspace. If the system driving is a random classical field, the survival probability of the system in the subspace becomes a random variable described by the Stochastic Quantum Zeno Dynamics (SQZD) formalism. Here, we study the time and ensemble average of this random survival probability and demonstrate how time correlations in the noisy environment determine whether the two averages do coincide or not. These environment time correlations can potentially generate non-Markovian dynamics of the quantum system depending on the structure and energy scale of the system Hamiltonian. We thus propose a way to detect time correlations of the environment by coupling a quantum probe system to it and observing the survival probability of the quantum probe in a measurement subspace. This will further contribute to the development of new schemes for quantum sensing technologies, where nanodevices may be exploited to image external structures or biological molecules via the surface field they generate.
The unavoidable interaction between a quantum system and the external noisy environment can be mimicked by a sequence of stochastic measurements whose outcomes are neglected. Here we investigate how this stochasticity is reflected in the survival probability to find the system in a given Hilbert subspace at the end of the dynamical evolution. In particular, we analytically study the distinguishability of two different stochastic measurement sequences in terms of a Fisher information measure depending on the variation of a function, instead of a finite set of parameters. We find a characterization of Zeno phenomena as the physical result of the random observation of the quantum system, linked to the sensitivity of the survival probability to an arbitrary small perturbation of the measurement stochasticity. Finally, the implications on the Cram\'er-Rao bound and the Zeno time are discussed, together with a numerical example.
We address here the issue of quantum measurement in which the interaction between the system at hand and the measuring apparatus causes a collapse of the wave function or the density operator. The analysis is carried out for the prototypical tight-binding chain involving a quantum particle hopping between nearest-neighbour sites on a lattice, with the system subject to repeated projective measurements to a detector site. However, unlike previous work, the measurements are performed at random times. In the absence of measurement, the particle while starting from a site is known to spread out to farther sites as time progresses, leading to complete delocalization of the particle in time. In presence of measurements, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We consider the representative case in which the time gap between successive measurements are sampled independently from an exponential distribution. The novel result of our findings is the revelation of the striking effect of Stochastic Localization, whereby the particle has at long times time-independent probabilities to be on different sites. This implies localization of the particle at long times, achieved through classical stochasticity involved with the random times for measurements. An extreme case of such localization is attained if the measurements are done very frequently, when the particle is completely localized at the detector site! We also point out the stochastic version of the Zeno effect, whereby dynamical evolution of a quantum system is arrested through repeated measurements at frequent-enough times. In the wake of recent experiments on projective measurements at random times achieved through randomly-pulsed sequences, our proposition is amenable to experimental realization.