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arXiv:1305.0681 · Physical Review Letters

Past Quantum States of a Monitored System

Søren Gammelmark, Brian Julsgaard, Klaus Mølmer

Published Oct 15, 2013• 147 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.111.160401

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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.
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.
State smoothing is a technique to estimate a state at a particular time, conditioned on information obtained both before (past) and after (future) that time.For a classical system, the smoothed state is a normalized product of the filtered state (a state conditioned only on the past measurement information and the initial preparation) and the retrofiltered effect (depending only on the future measurement information).For the quantum case, while there are well-established analogues of the filtered state (ρ F ) and the retrofiltered effect ( ÊR ), their product does not, in general, provide a valid quantum state for smoothing.However, this procedure does seem to work when ρ F and ÊR are mutually diagonalizable.This fact has been used to obtain smoothed quantum states-purer than the filtered states-in a number of experiments on continuously monitored quantum systems, in cavity QED and atomic systems.In this paper we show that there is an implicit assumption underlying this technique: that if all the information were known to the observer, the true system state would be one of the diagonal basis states.This assumption does not necessarily hold, as the missing information is quantum information.It could be known to the observer only if it were turned into a classical measurement record, but then its nature would depend on the choice of measurement.We show by a simple model that, depending on that measurement choice, the smoothed quantum state can: agree with that from the classical method, disagree with it but still be co-diagonal with it, or not even be co-diagonal with it.That is, just because filtering and retrofiltering appear classical does not mean classical smoothing theory is applicable in quantum experiments.
Wave functions and density matrices represent our knowledge about a quantum system and give probabilities for the outcomes of measurements. If the combined dynamics and measurements on a system lead to a density matrix $\ensuremath{\rho}(t)$ with only diagonal elements in a given basis ${|n\ensuremath{\rangle}}$, it may be treated as a classical mixture, i.e., a system which randomly occupies the basis states $|n\ensuremath{\rangle}$ with probabilities ${\ensuremath{\rho}}_{nn}(t)$. Equivalent to so-called smoothing in classical probability theory, subsequent probing of the occupation of the states $|n\ensuremath{\rangle}$ may improve our ability to retrodict what was the outcome of a projective state measurement at time ttt. Here, we show with experiments on a superconducting qubit that the smoothed probabilities do not, in the same way as the diagonal elements of $\ensuremath{\rho}(t)$, permit a classical mixture interpretation of the state of the system at the past time ttt.
Gaussian states of quantum oscillators are fully characterized by the mean values and the covariance matrix of their quadrature observables. We consider the dynamics of a system of oscillators subject to interactions, damping, and continuous probing which maintain their Gaussian state property. Such dynamics is found in many physical systems that can therefore be efficiently described by the ensuing effective representation of the density matrix $\ensuremath{\rho}(t)$. Our probabilistic knowledge about the outcome of measurements on a quantum system at time ttt is not only governed by $\ensuremath{\rho}(t)$ conditioned on the evolution and measurement outcomes obtained until time ttt but is also modified by any information acquired after ttt. It was shown [S. Gammelmark, B. Julsgaard, and K. M\o{}lmer, Phys. Rev. Lett. 111, 160401 (2013)] that this information is represented by a supplementary matrix, E(t)E(t)E(t). We show here that the restriction of the dynamics of $\ensuremath{\rho}(t)$ to Gaussian states implies that the matrix E(t)E(t)E(t) is also fully characterized by a vector of mean values and a covariance matrix. We derive the dynamical equations for these quantities and we illustrate their use in the retrodiction of measurements on Gaussian systems.
A quantum system can be monitored through repeated interactions with meter systems. The state of the system at time ttt, represented by the density matrix $\ensuremath{\rho}(t)$, then becomes conditioned on the information obtained by the meters until that time. More insight in the state of the system at any time ttt is provided, however, by taking into account the full detection of all meters interacting with the system both in the past and in the future of ttt. We present experiments that use near-resonant atomic probes to monitor the evolution of the quantized field in a cavity. The application of the forward-backward smoothing method to this quantum problem, justified by the past quantum state formalism [S. Gammelmark et al., Phys. Rev. Lett. 111, 160401 (2013)], allows us to resolve a posteriori dynamics, which is not reflected by the usual quantum state $\ensuremath{\rho}(t)$.
We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.