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arXiv:1404.2454 · Russian Journal of Mathematical Physics

Zeno dynamics for open quantum systems

J. Gough

Published Jul 1, 2014• 4 citations
arXivHTMLPDF

Abstract

In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics.

DOI
10.1134/s1061920814030066

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We provide a general dynamical approach for the quantum Zeno and anti-Zeno effects in an open quantum system under repeated non-demolition measurements. In our approach the repeated measurements are described by a general dynamical model without the wave function collapse postulation. Based on that model, we further study both the short-time and long-time evolutions of the open quantum system under repeated non-demolition measurements, and derive the measurement-modified decay rates of the excited state. In the cases with frequent ideal measurements at zero-temperature, we re-obtain the same decay rate as that from the wave function collapse postulation (Nature {\bf 405}, 546 (2000)). The correction to the ideal decay rate is also obtained under the non-ideal measurements. Especially, we find that the quantum Zeno and anti-Zeno effects are possibly {\it enhanced} by the non-ideal natures of measurements. For the open system under measurements with arbitrary period, we generally derive the rate equation for the long-time evolution for the cases with {\it arbitrary} temperature and noise spectrum, and show that in the long-time evolution the noise spectrum is effectively tuned by the repeated measurements. Our approach is also able to describe the quantum Zeno and anti-Zeno effects given by the phase modulation pulses, as well as the relevant quantum control schemes.
A comprehensive approach to modeling open quantum systems consistent with thermodynamics is presented. The theory of open quantum systems is employed to define system bath partitions. The Markovian master equation defines an isothermal partition between the system and bath. Two methods to derive the quantum master equation are described: the weak coupling limit and the repeated collision model. The role of the eigenoperators of the free system dynamics is highlighted, in particular, for driven systems. The thermodynamical relations are pointed out. Models that lead to loss of coherence, i.e., dephasing are described. The implication of the laws of thermodynamics to simulating transport and spectroscopy is described. The indications for self-averaging in large quantum systems and thus its importance in modeling are described. Basic modeling by the surrogate Hamiltonian is described, as well as thermal boundary conditions using the repeated collision model and their use in the stochastic surrogate Hamiltonian. The problem of modeling with explicitly time dependent driving is analyzed. Finally, the use of the stochastic surrogate Hamiltonian for modeling ultrafast spectroscopy and quantum control is reviewed.
We iteratively apply a recently formulated adiabatic theorem for the strong-coupling limit in finite-dimensional closed and open quantum systems. This allows us to improve approximations to a perturbed dynamics, beyond the standard approximation based on quantum Zeno dynamics and adiabatic elimination. The effective generators describing the approximate evolutions are endowed with the same block structure as the unperturbed part of the generator, and exhibit adiabatic evolutions. This iterative adiabatic theorem reveals that adiabaticity holds eternally, that is, the system evolves within each eigenspace of the unperturbed part of the generator, with an error bounded by $O(1/\ensuremath{\gamma})$ uniformly in time, where $\ensuremath{\gamma}$ is the strength of the unperturbed part of the generator. We prove that the iterative adiabatic theorem reproduces Bloch's perturbation theory in the unitary case, and is therefore a full generalization to open systems. We furthermore prove the equivalence of the Schrieffer-Wolff and des Cloizeaux approaches in the unitary case and generalize both to arbitrary open systems, showing that they share the eternal adiabaticity, and providing explicit error bounds. Finally we discuss the physical structure of the effective adiabatic generators and show that ideal effective generators for open systems do not exist in general.
Abstract Quantum trajectory techniques have been used in the theory of open systems as a starting point for numerical computations and to describe the monitoring of a quantum system in continuous time. We extend this technique to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equations, we can describe a classical component and a quantum one which have their own intrinsic dynamics and which interact with each other. A mathematically rigorous construction is given, under the restriction of having a Markovian joint dynamics and of involving only bounded operators on the Hilbert space of the quantum component. An important feature is that, if the interaction allows for a flow of information from the quantum component to the classical one, necessarily the dynamics is dissipative. We show also how this theory is connected to a suitable hybrid dynamical semigroup, which reduces to a quantum dynamical semigroup in the purely quantum case and includes Liouville and Kolmogorov–Fokker–Planck equations in the purely classical case. Moreover, this semigroup allows to compare the proposed stochastic dynamics with various other proposals based on hybrid master equations. Some simple examples are constructed in order to show the variety of physical behaviors which can be described; in particular, a model presenting hidden entanglement is introduced.
Open quantum systems are a rich area of research in the intersection of quantum mechanics and stochastic analysis. By considering a variety of master equations, we unify multiple views of autonomous and controlled open quantum systems and, through considering their measurement dynamics, connect them to classical linear and bilinear system identification theory. This allows us to formulate corresponding notions of quantum state identifiability for these systems which, in particular, applies to quantum state tomography, providing conditions under which the probed quantum system is reconstructible. Interestingly, the dynamical representation of the system lends itself to considering two types of identifiability: the full master equation recovery and the recovery of the corresponding system matrices of the linear and bilinear systems. These concepts are discussed in detail, and conditions under which reconstruction is possible are given. We set the groundwork for a number of constructive approaches to the identification of open quantum systems.