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Physica Scripta

Persistent oscillations in correlation function: from the perspective of past quantum states

Qing Yang Xu, Xiaobo Shen, 禹海军 Yu Haijun

Published Nov 22, 2024• 0 citations
Publisher

Abstract

Abstract The recent past quantum state formalism (2013 Phys. Rev. Lett. 111, 160401) indicates that the outcome probability of the measurement performed on a quantum system at time t shows dependence on the measurement events recorded until that time, and is also revised by the outcomes acquired after time t. We apply the past quantum state formalism to calculate the three-time correlation function of a radiation field and to analyze the conditional dynamics on the pre- and postselection by the first and last detection events. The persistent oscillations in the three-time field correlation function, which cannot be well understood by the usual quantum regression theorem, are straightforwardly explained by the probabilistic knowledge of the past and the future through the past quantum state reasoning.

DOI
10.1088/1402-4896/ad9652

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Abstract The recent past quantum state formalism (2013 Phys. Rev. Lett. 111, 160401) indicates that the outcome probability of the measurement performed on a quantum system at time t shows dependence on the measurement events recorded until that time, and is also revised by the outcomes acquired after time t. We apply the past quantum state formalism to calculate the three-time correlation function of a radiation field and to analyze the conditional dynamics on the pre- and postselection by the first and last detection events. The persistent oscillations in the three-time field correlation function, which cannot be well understood by the usual quantum regression theorem, are straightforwardly explained by the probabilistic knowledge of the past and the future through the past quantum state reasoning.
A density matrix ρ(t) yields probabilistic information about the outcome of measurements on a quantum system. We introduce here the past quantum state, which, at time T, accounts for the state of a quantum system at earlier times t<T. The past quantum state Ξ(t) is composed of two objects, ρ(t) and E(t), conditioned on the dynamics and the probing of the system until t and in the time interval [t, T], respectively. The past quantum state is characterized by its ability to make better predictions for the unknown outcome of any measurement at t than the conventional quantum state at that time. On the one hand, our formalism shows how smoothing procedures for estimation of past classical signals by a quantum probe [M. Tsang, Phys. Rev. Lett. 102, 250403 (2009)] apply also to describe the past state of the quantum system itself. On the other hand, it generalizes theories of pre- and postselected quantum states [Y. Aharonov and L. Vaidman, J. Phys. A 24, 2315 (1991)] to systems subject to any quantum measurement scenario, any coherent evolution, and any Markovian dissipation processes.
Correlations in photodetection signals from quantum light sources are conventionally calculated by application of the source master equation and the quantum regression theorem. In this article we show how the conditioned dynamics, associated with the quantum theory of measurements, allows calculations and offers interpretations of the behavior of the same quantities. Our theory is illustrated for photon counting and field-amplitude measurements, and we show, in particular, how transient correlations between field-amplitude measurements and later photon counting events can be accounted for by a recently developed theory of past quantum states of a monitored quantum system.
For classical Markovian stochastic systems, past and future events become statistically independent when conditioned to a given state at the present time. Memory non-Markovian effects break this condition, inducing a nonvanishing conditional past-future correlation. Here, this classical memory indicator is extended to a quantum regime, which provides an operational definition of quantum non-Markovianity based on a minimal set of three time-ordered quantum system measurements and postselection. The detection of memory effects through the measurement scheme is univocally related to departures from Born-Markov and white noise approximations in quantum and classical environments, respectively.
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Memory effects can be studied through a conditional past-future correlation, which measures departure with respect to a conditional past-future independence valid in a memoryless Markovian regime. In a quantum regime this property leads to an operational definition of quantum non-Markovianity based on three consecutive system measurement processes and postselection [Phys. Rev. Lett. 121, 240401 (2018)]. Here, we study the conditional past-future correlation for a qubit system coupled to different dephasing environments. Exact solutions are obtained for a quantum spin bath as well as for classically fluctuating random Hamiltonian models. The developing of memory effects and departures from Born-Markov or white-noise approximations are related to a measurement back action that changes the system dynamics between consecutive measurements. It is shown that this effect may develop even when the former system evolution is given by a time-independent Lindblad equation. This unusual non-Markovian case arises when the characteristic parameters of the dynamics become Lorentzian random distributed variables.
Conditional past-future correlations measure the lack of statistical independence between past and future system measurement outcomes when conditioned to a given state at a present time. Quantum non-Markovian memory effects are present whenever this correlation is not null. Conditional past-future correlations can also be used to detect initial system-environment correlated states. In this work, we provide theoretical and experimental evidence for these general properties. We build an optical setup that implements the dynamics of a qubit interacting with a dephasing spin bath. Both finite and infinite bath-size limits are observed. Confirmation of theoretical predictions for conditional past-future correlations is obtained. This work provides experimental support for quantum memory indicators based solely on outcomes of explicit system measurement processes.