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arXiv:1508.07187 · Physical Review A

Incoherent ensemble dynamics in disordered systems

Clemens Gneiting, F. R. Anger, Andreas Buchleitner

Published Mar 31, 2016• 24 citations
arXivHTMLPDF

Abstract

We derive a quantum master equation which describes the dynamics of the ensemble-averaged state of homogeneous disorder models at short times, and mediates a transition from coherent superpositions into classical mixtures. While each single realization follows unitary dynamics, this decoherencelike behavior arises as a consequence of the ensemble average. The master equation manifestly reflects the translational invariance of the disorder correlations and allows us to relate the disorder-induced dynamics to a collisional decoherence process, where the disorder correlations determine the spatial decay of coherences. We apply our theory to the (one-dimensional) Anderson model.

DOI
10.1103/physreva.93.032139

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We derive a quantum master equation which describes the dynamics of the ensemble-averaged state of homogeneous disorder models at short times, and mediates a transition from coherent superpositions into classical mixtures. While each single realization follows unitary dynamics, this decoherencelike behavior arises as a consequence of the ensemble average. The master equation manifestly reflects the translational invariance of the disorder correlations and allows us to relate the disorder-induced dynamics to a collisional decoherence process, where the disorder correlations determine the spatial decay of coherences. We apply our theory to the (one-dimensional) Anderson model.
Quantum coherences are paramount resources for applications, such as quantum-enhanced light-harvesting or quantum computing, which are fragile against environmental noise.We here derive generalized quantum master equations using perturbation theory in order to describe the effective time evolution of finite-size quantum systems subject to static noise on all time scales.We then analyze the time evolution of the coherences under energy broadening noise in a variety of systems characterized by both short-and long-range interactions and by strongly correlated and fully uncorrelated noise -a single qubit, a lattice model with on-site disorder and a potential ladder, and a Bose-Hubbard dimer with random interaction strength-and show that couplings can partially protect the system from the ensemble-averaging induced loss of coherence.Our work suggests that suitably tuned couplings could be employed to counteract part of the dephasing detrimental to quantum applications.Conversely, tailored noise distributions can be utilized to reach target nonequilibrium quantum states.
It is commonly known that the dephasing in open quantum systems is due to the establishment of bipartite correlations with ambient environments, which are typically difficult to be fully characterized. Recently, a new approach of average over disordered Hamiltonian ensemble is developed and shown to be capable of describing the nonclassicality of incoherent dynamics based on inferring the nonclassical nature of the correlations. Here we further extend the approach of Hamiltonian ensemble in the canonical form to the realm of structural disorder. Under the variable separation of the probability distribution within the Hamiltonian ensemble, the geometrical structure is easily visualized and can be characterized according to the degree of symmetry. We demonstrate four degrees and investigate the effects of different types of symmetry breaking on the incoherent dynamics. We show that these effects are easily understood from the emergences of additional terms in the master equations, leading to rather general master equations and, consequently, going beyond the previous frameworks of pure dephasing or isotropic depolarization.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
In the quantum trajectory approach an open quantum system is represented by a stochastic pure-state wave function. The mixed-state density operator that satisfies the usual master equation is recovered from the quantum trajectories by performing either an ensemble average or, for stationary systems, an average over time. This unraveling of the master equation dynamics into pure-state trajectories provides new insight into the decay of quantum coherence in systems that are open to the environment. Under some conditions macroscopic superposition states are preserved as macroscopic superposition states along individual trajectories, but are reduced to mixtures by the trajectory average. Under other conditions one of the states in a macroscopic superposition becomes dominant (in amplitude) over the other along each quantum trajectory. We discuss the mechanisms that produce these dynamics and assess the implications for the observation of Schrodinger cat states in optics experiments. We present results for some specific examples of macroscopic superposition states generated by a cavity QED system. The system involves a small collection of atoms in an optical cavity driven by a coherent laser field. Under strong-coupling conditions this system produces a variety of Schrodinger cat states whose precise form depends on the number of atoms, the method of excitation (through a cavity mirror or from the side), the initial state of the atoms, and the position of the atoms in the cavity.
In this thesis quantum-classical dynamics is applied to the study of quantum condensed phase processes. This approach is based on the quantum-classical Liouville equation where the dynamics of a small subset of the degrees of freedom are treated quantum mechanically while the remaining degrees of freedom are treated by classical mechanics to a good approximation. We use this approach as it is computationally tractable, and the resulting equation of motion accurately accounts for the quantum and classical dynamics, as well as the coupling between these two components of the system. \n\nBy recasting the quantum-classical Liouville equation into the form of a generalized master equation we investigate connections to surface-hopping. The link between these approaches is decoherence arising from interaction of the subsystem with the environment. We derive an evolution equation for the subsystem which contains terms accounting for the effects of the environment. One of these terms involves a memory kernel that accounts for the coherent dynamics. If this term decays rapidly, a Markovian approximation can be made. By lifting the resulting subsystem master equation into the full phase space, we obtain a Markovian master equation that prescribes surface-hopping-like dynamics. Our analysis outlines the conditions under which such a description is valid.\n\nNext, we consider the calculation of the rate constant for a quantum mechanical barrier crossing process. Starting from the reactive-flux autocorrelation function, we derive a quantum-classical expression for the rate kernel. This expression involves quantum-classical evolution of a species operator averaged over the initial quantum equilibrium structure of the system making it possible to compute the rate constant via computer simulation. Using a simple model for a proton transfer reaction we compare the results of the rate calculation obtained by quantum-classical Liouville dynamics with that of master equation dynamics. The master equation provides a good approximation to the full quantum-classical Liouville calculation for our model and a more stable algorithm results due to the elimination of oscillating phase factors in the simulation. \n\nFinally, we make use of the theoretical framework established in this thesis to analyze some aspects of decoherence used in popular surface-hopping techniques.
An open quantum system is usually characterized by a reduced ensemble density matrix, the dynamics of which is governed by a generalized Master equation. Transforming this equation of motion into the instantaneous diagonal basis of the corresponding reduced density matrix, we can separate the coherent and incoherent part of the dynamics: The coherent dynamics is incorporated in the time development of the diagonal basis states, while the coupling to the reservoirs leads to simple rate equations. Interpreting these rate equations as a stochastic point process allows one to simulate the stochastic time evolution (random telegraph signals, ``quantum jumps'') of single-quantum systems. The diagonal representation can be considered as a generalization of the dressed-state picture of open quantum systems. Numerical simulations (``quantum Monte Carlo'') allow one to derive various dynamical properties (including correlation functions) of single-quantum systems. This concept is applied to different two- and three-level scenarios (\ensuremath{\lambda} and \ensuremath{\nu} configuration), and its limitations are discussed.