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arXiv:1311.2673 · New Journal of Physics

Inverse counting statistics for stochastic and open quantum systems: the characteristic polynomial approach

M. Bruderer, L. Debora Contreras-Pulido, M Thaller, L Sironi, Danail Obreschkow, Martin Bodo Plenio

Published Mar 26, 2014• 51 citations
arXivHTMLPDF

Abstract

We consider stochastic and open quantum systems with a finite number of states, where a stochastic transition between two specific states is monitored by a detector.The long-time counting statistics of the observed realizations of the transition, parametrized by cumulants, is the only available information about the system.We present an analytical method for reconstructing generators of the time evolution of the system compatible with the observations.The practicality of the reconstruction method is demonstrated by the examples of a laser-driven atom and the kinetics of enzyme-catalyzed reactions.Moreover, we propose cumulant-based criteria for testing the non-classicality and non-Markovianity of the time evolution, and lower bounds for the system dimension.Our analytical results rely on the close connection between the cumulants of the counting statistics and the characteristic polynomial of the generator, which takes the role of a cumulant generating function.

DOI
10.1088/1367-2630/16/3/033030

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We consider stochastic and open quantum systems with a finite number of states, where a stochastic transition between two specific states is monitored by a detector.The long-time counting statistics of the observed realizations of the transition, parametrized by cumulants, is the only available information about the system.We present an analytical method for reconstructing generators of the time evolution of the system compatible with the observations.The practicality of the reconstruction method is demonstrated by the examples of a laser-driven atom and the kinetics of enzyme-catalyzed reactions.Moreover, we propose cumulant-based criteria for testing the non-classicality and non-Markovianity of the time evolution, and lower bounds for the system dimension.Our analytical results rely on the close connection between the cumulants of the counting statistics and the characteristic polynomial of the generator, which takes the role of a cumulant generating function.
The time-evolution operator of open quantum systems is determined generally via a stochastic operator sum representation. Quantum trajectories originating from any quantum trajectory method applicable to the system can be used for the construction. The result does apply to non-Markovian systems as well. The computational time reducement is in the order of the dimension of the Hilbert space compared to conventional quantum trajectory methods combined with algebraic reconstruction of the map. To demonstrate the efficiency of the method, the effects of decoherence are analysed for a C-NOT gate realized by two atoms in a cavity.
The state of an open quantum system undergoing an adiabatic process evolves by following the instantaneous stationary state of its time-dependent generator. This observation allows one to characterize, for a generic adiabatic evolution, the average dynamics of the open system. However, information about fluctuations of dynamical observables, such as the number of photons emitted or the time-integrated stochastic entropy production in single experimental runs, requires controlling the whole spectrum of the generator and not only the stationary state. Here, we show how such information can be obtained in adiabatic open quantum dynamics by exploiting tools from large deviation theory. We prove an adiabatic theorem for deformed generators, which allows us to encode, in a biased quantum state, the full counting statistics of generic time-integrated dynamical observables. We further compute the probability associated with an arbitrary "rare" time history of the observable and derive a dynamics which realizes it in its typical behavior. Our results provide a way to characterize and engineer adiabatic open quantum dynamics and to control their fluctuations.
The state of an open quantum system undergoing an adiabatic process evolves by following the instantaneous stationary state of its time-dependent generator. This observation allows one to characterize, for a generic adiabatic evolution, the average dynamics of the open system. However, information about fluctuations of dynamical observables, such as the number of photons emitted or the time-integrated stochastic entropy production in single experimental runs, requires controlling the whole spectrum of the generator and not only the stationary state. Here, we show how such information can be obtained in adiabatic open quantum dynamics by exploiting tools from large deviation theory. We prove an adiabatic theorem for deformed generators, which allows us to encode, in a biased quantum state, the full counting statistics of generic time-integrated dynamical observables. We further compute the probability associated with an arbitrary "rare" time-history of the observable and derive a dynamics which realizes it in its typical behavior. Our results provide a way to characterize and engineer adiabatic open quantum dynamics and to control their fluctuations.
We derive a general expression for the cumulant generating function (CGF) of non-Markovian quantum stochastic transport processes. The long-time limit of the CGF is determined by a single dominating pole of the resolvent of the memory kernel from which we extract the zero-frequency cumulants of the current using a recursive scheme. The finite-frequency noise is expressed not only in terms of the resolvent, but also initial system-environment correlations. As an illustrative example we consider electron transport through a dissipative double quantum dot for which we study the effects of dissipation on the zero-frequency cumulants of high orders and the finite-frequency noise.
We propose a numerical method to efficiently compute quantum generating functions for a wide class of observables in one-dimensional quantum systems at high temperature. We obtain high-accuracy estimates for the cumulants and reconstruct full counting statistics from the quantum generating functions. We demonstrate its potential on spin S=1/2 anisotropic Heisenberg chain, where we can reach timescales hitherto inaccessible to state-of-the-art classical and quantum simulations. Our results challenge the conjecture of the Kardar-Parisi-Zhang universality for isotropic integrable quantum spin chains.
The dynamical behavior of many-body systems is often richer than what can be anticipated from their static properties. Here we show that in closed quantum systems this becomes evident by considering the statistics of time-integrated observables. In particular, the analytic properties of their generating functions, as estimated by full counting statistics (FCS), allow one to identify FCS phases, i.e., phases with specific fluctuation properties of time-integrated observables, and to locate transitions between these phases. We discuss in detail the case of the quantum Ising chain in a transverse field. We show that this model displays a continuum of full counting statistics transitions, of which the static transition is just an end point. These singularities are not a consequence of particular choices of initial conditions or other external nonequilibrium protocols such as quenches in coupling constants. They can be probed generically through quantum jump statistics of an associated open problem, and for the case of the quantum Ising chain we outline a possible experimental realization of this scheme by digital quantum simulation with cold ions.