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arXiv:1912.08662 · Physical Review Research

Continuous quantum measurement for general Gaussian unravelings can exist

Nina Megier, Walter T. Strunz, Kimmo Luoma

Published Dec 15, 2020• 11 citations
arXivHTMLPDF

Abstract

Quantum measurements and the associated state changes are properly described in the language of instruments.We investigate the properties of a time continuous family of instruments associated with the recently introduced class of general Gaussian non-Markovian stochastic Schrödinger equations.In this article we find that when the covariance matrix for the Gaussian noise satisfies a particular δ-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator.This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions.Remarkably, due to quantum memory effects the reduced state, in general, does not obey a closed form master equation while the unraveling has a time continuous measurement interpretation.

DOI
10.1103/physrevresearch.2.043376

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Quantum measurements and the associated state changes are properly described in the language of instruments.We investigate the properties of a time continuous family of instruments associated with the recently introduced class of general Gaussian non-Markovian stochastic Schrödinger equations.In this article we find that when the covariance matrix for the Gaussian noise satisfies a particular δ-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator.This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions.Remarkably, due to quantum memory effects the reduced state, in general, does not obey a closed form master equation while the unraveling has a time continuous measurement interpretation.
Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schr\"odinger equation introduced by Strunz, Di\'osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.
By using a condition of average trace preservation we rederive a general class of non-Markovian Gaussian diffusive unravelings [L. Di\'osi and L. Ferialdi, Phys. Rev. Lett. 113, 200403 (2014)], here valid for arbitrary non-Hermitian system operators and noise correlations. The conditions under which the generalized stochastic Schr\"odinger equation has the same symmetry properties (invariance under unitary changes of operator base) as a microscopic system-bath Hamiltonian dynamics are determined. While the standard quantum diffusion model (standard noise correlations) always shares the same invariance symmetry, the generalized stochastic dynamics can be mapped with an arbitrary bosonic environment only if some specific correlation constraints are fulfilled. These features are analyzed for different non-Markovian unravelings equivalent on average. Results based on quantum measurement theory that leads to specific cases of the generalized dynamics [J. Gambetta and H. M. Wiseman, Phys. Rev. A 66, 012108 (2002)] are studied from the perspective of the present analysis.
General open quantum systems display memory features, their master equations are non-Markovian. We show that the subclass of Gaussian non-Markovian open system dynamics is tractable in a depth similar to the Markovian class. The structure of master equations exhibits a transparent generalization of the Lindblad structure. We find and parametrize the class of stochastic Schrödinger equations that unravel a given master equation, such a class was previously known for Markovian systems only. We show that particular non-Markovian unravelings known in the literature are special cases of our class.
The “correlated-projection technique” has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.
Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.