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arXiv:1412.7895 · Physical Review A

Quantum trajectories under frequent measurements in a non-Markovian environment

Luting Xu, Xin-Qi Li

Published Sep 30, 2016• 8 citations
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Abstract

In this work we generalize the quantum trajectory (QT) theory from Markovian to non-Markovian environments. We model the non-Markovian environment by using a Lorentzian spectral density function with bandwidth $(\mathrm{\ensuremath{\Lambda}})$, and find a perfect ``scaling'' property with the measurement frequency $({\ensuremath{\tau}}^{\ensuremath{-}1})$ in terms of the scaling variable $x=\mathrm{\ensuremath{\Lambda}}\ensuremath{\tau}$. Our result bridges the gap between the existing QT theory and the Zeno effect, by rendering them as two extremes corresponding to $x\ensuremath{\rightarrow}\ensuremath{\infty}$ and $x\ensuremath{\rightarrow}0$, respectively. This $x$-dependent criterion improves the idea of using $\ensuremath{\tau}$ alone and quantitatively identifies the validity condition of the conventional QT theory.

DOI
10.1103/physreva.94.032130

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In this work we generalize the quantum trajectory (QT) theory from Markovian to non-Markovian environments. We model the non-Markovian environment by using a Lorentzian spectral density function with bandwidth (\ensuremathΛ)(\mathrm{\ensuremath{\Lambda}})(\ensuremathΛ), and find a perfect ``scaling'' property with the measurement frequency (\ensuremathτ\ensuremath−1)({\ensuremath{\tau}}^{\ensuremath{-}1})(\ensuremathτ\ensuremath−1) in terms of the scaling variable x=\ensuremathΛ\ensuremathτx=\mathrm{\ensuremath{\Lambda}}\ensuremath{\tau}x=\ensuremathΛ\ensuremathτ. Our result bridges the gap between the existing QT theory and the Zeno effect, by rendering them as two extremes corresponding to x\ensuremath→\ensuremath∞x\ensuremath{\rightarrow}\ensuremath{\infty}x\ensuremath→\ensuremath∞ and x\ensuremath→0x\ensuremath{\rightarrow}0x\ensuremath→0, respectively. This xxx-dependent criterion improves the idea of using \ensuremathτ\ensuremath{\tau}\ensuremathτ alone and quantitatively identifies the validity condition of the conventional QT theory.
Diagonalization of the effective Hamiltonian describing an open quantum system is the usual method of tracking its exceptional points (EPs). Although such a method is successful for tracking EPs in Markovian systems, it may be problematic in non-Markovian systems where a closed expression of the effective Hamiltonian describing the open system may not exist. In this work we provide an alternative method of tracking EPs in open quantum systems, using an experimentally measurable quantity, namely, the effective decay rate of a qubit. The quantum system under consideration consists of two nonidentical interacting qubits, one of which is coupled to an external environment. We develop a theoretical framework in terms of the time-dependent Schr\"odinger equations of motion, which provides analytical closed-form solutions of the Laplace transforms of the qubit amplitudes, enabling the study of various cases of environmental spectral densities. The link between the peaked structure of the effective decay rate of the qubit that interacts indirectly with the environment and the onset of the quantum Zeno effect is investigated, revealing the connections between the latter and the presence of exceptional points. Our treatment and results in addition reveal an intricate interplay between non-Markovian dynamics, the quantum Zeno effect, and non-Hermitian physics.
Starting from a generalization of the quantum trajectory theory [based on the stochastic Schr\"odinger equation (SSE)], non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows a consistent theory of quantum measurement in continuous time to be developed for these non-Markovian quantum trajectory models. In this context, the notions of ``instrument,'' ``a priori,'' and ``a posteriori'' states can be introduced. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non-Markovian effects come from the random environment, colored noises, randomness in the stimulating light, and delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied, and we show how these quantities are sensitive to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescent light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.
We present a formulation of non-Markovian quantum trajectories for open systems from a measurement theory perspective. In our treatment there are three distinct ways in which non-Markovian behavior can arise: a mode-dependent coupling between the bath (reservoir) and system, a dispersive bath, and spectral detection of the output into the bath. In the first two cases the non-Markovian behavior is intrinsic to the interaction; in the third case the non-Markovian behavior arises from the method of detection. We focus in detail on the trajectories that simulate real-time spectral detection of the light emitted from a localized system. In this case the non-Markovian behavior arises from the uncertainty in the time of emission of particles that are later detected. The results of computer simulations of the spectral detection of the spontaneous emission from a strongly driven two-level atom are presented.
We analyze the non-Markovian stochastic Schrödinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case (H=p(2)/2m) and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.
We present a theory of non-Markovian quantum trajectories that describe the dynamics of a quantum system coupled to an environment of harmonic oscillators. When averaged over the noise, they reproduce the standard reduced density matrix without any approximation, in particular without Markov approximation. The time evolution of the environment can also be determined from these non-Markovian quantum trajectories. We present a simple example to demonstrate the power of our approach. As a remarkable feature of non-Markovian quantum trajectories, we show that for strong coupling to a dissipative environment with memory, the asymptotic state can be reached in a finite time. Further examples are described in the text.
Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. In practice an experimenter has access to an output filtered through various electronic devices, rather than the microscopic states of the detector. This introduces several imperfections into the measurement process, of which only inefficiency has previously been incorporated into quantum trajectory theory. However, all electronic devices have finite bandwidths, and the consequent delay in conveying the output signal to the observer implies that the evolution of the conditional state of the quantum system must be non-Markovian. We present a general method of describing this evolution and apply it to avalanche photodiodes and to photoreceivers. We include the effects of efficiency, dead time, bandwidth, electronic noise and dark counts. The essential idea is to treat the quantum system and classical detector jointly, and to average over the latter to obtain the conditional quantum state. The significance of our theory is that quantum trajectories for realistic detection are necessary for sophisticated approaches to quantum feedback, and our approach could be applied in many areas of physics.