Quantum Trajectories for Brownian Motion

Strunz, Walter T.; Diósi, Lajos; Gisin, Nicolas; Yu, Ting

doi:10.1103/physrevlett.83.4909

Cited by 81 publications

(90 citation statements)

References 34 publications

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“…It is interesting to note that a more general three-level system with different energy spacings such as V-type or λ-type atoms, a universal effective control via UUD combined with non-Markovian QSD are still possible, but it becomes much more complicated technically and multi-nesting sequences with different control operators have to be employed. Moreover, the exact QSD will be difficult to find, but still we use approximate nonMarkovian QSD to simulate the Uhrig control dynamics [21,37,41].…”

Section: Discussionmentioning

confidence: 99%

“…In this paper, without the above assumption, we treat the total system plus environment with control as a integrated and consistent entirety to solve its exact dynamics by using the non-Markovian QSD equation initially proposed in [36]. Derived directly from an underlying microscopic model irrespective of environment memory time and coupling strength, the stochastic QSD equation is a useful approach for solving several models exactly [37][38][39][40][41]. As shown below, the QSD equation can provide a systematic tool to deal with the non-Markovian quantum open system under the UDD control fields.…”

Section: Introductionmentioning

confidence: 99%

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Uhrig dynamical control of a three-level system via non-Markovian quantum state diffusion

Shu

Zhao

Jing

et al. 2013

J. Phys. B: At. Mol. Opt. Phys.

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In this paper, we use the quantum state diffusion (QSD) equation to implement the Uhrig dynamical decoupling (UDD) to a three-level quantum system coupled to a non-Markovian reservoir comprising of infinite numbers of degrees of freedom. For this purpose, we first reformulate the nonMarkovian QSD to incorporate the effect of the external control fields. With this stochastic QSD approach, we demonstrate that an unknown state of the three-level quantum system can be universally protected against both colored phase and amplitude noises when the control-pulse sequences and control operators are properly designed. The advantage of using non-Markovian quantum state diffusion equations is that the control dynamics of open quantum systems can be treated exactly without using Trotter product formula and be efficiently simulated even when the environment comprise of infinite numbers of degrees of freedom. We also show how the control efficacy depends on the environment memory time and the designed time points of applied control pulses.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uhrig dynamical control of a three-level system via non-Markovian quantum state diffusion

Shu

Zhao

Jing

et al. 2013

J. Phys. B: At. Mol. Opt. Phys.

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“…A method of treating larger systems in the framework of the QME is the stochastic unraveling of QME (also known as quantum jump, quantum trajectory, and Monte Carlo wave function method) [1,2,34,35,36,37,38,39,41,42,40,5,10,11]. In this method, since one treats a wave function instead of a density matrix, one can investigate systems larger than those in the direct methods.…”

Section: Introductionmentioning

confidence: 99%

A Perturbative Method for Nonequilibrium Steady State of Open Quantum Systems

Yuge

Sugita

2015

J. Phys. Soc. Jpn.

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We develop a method of calculating the nonequilibrium steady state (NESS) of an open quantum system that is weakly coupled to reservoirs in different equilibrium states. We describe the system using a Redfield-type quantum master equation (QME). We decompose the Redfield QME into a Lindblad-type QME and the remaining part R. Regarding the steady state of the Lindblad QME as the unperturbed solution, we perform a perturbative calculation with respect to R to obtain the NESS of the Redfield QME. The NESS thus determined is exact up to the first order in the system-reservoir coupling strength (pump/loss rate), which is the same as the order of validity of the QME. An advantage of the proposed method in numerical computation is its applicability to systems larger than those in methods of directly solving the original Redfield QME. We apply the method to a noninteracting fermion system to obtain an analytical expression of the NESS density matrix. We also numerically demonstrate the method in a nonequilibrium quantum spin chain.

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“…Nevertheless there are many physical meaningful QMEs which result in positive-or almost positive-definite RDMs although they are not of Lindblad form. The increasing interest in descriptions beyond the Lindblad class such as the quantum Brownian motion [10,11], the Redfield formalism [12], non-Markovian schemes [13,14,15], etc. resulted in various efforts to develop new stochastic wave function algorithms.…”

mentioning

confidence: 99%

“…Nevertheless there are many physical meaningful QMEs which result in positive-or almost positive-definite RDMs although they are not of Lindblad form. The increasing interest in descriptions beyond the Lindblad class such as the quantum Brownian motion [10,11] developed a method how to exactly represent the RDM of a system coupled to a linear heat bath in terms of SSEs. The numerical properties of this approach need to be explored.…”

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confidence: 99%

Stochastic unraveling of time-local quantum master equations beyond the Lindblad class

2002

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A new method for stochastic unraveling of general time-local quantum master equations (QME) which involve the reduced density operator at time t only is proposed. The present kind of jump algorithm enables a numerically efficient treatment of QMEs which are not of Lindblad form. So it opens new large fields of application for stochastic methods. The unraveling can be achieved by allowing for trajectories with negative weight. We present results for the quantum Brownian motion and the Redfield QMEs as test examples. The algorithm can also unravel non-Markovian QMEs when they are in a time-local form like in the time-convolutionless formalism. 1 for a number of typical examples). These QMEs describe the time evolution of density matrices which are used in order to represent the mixed nature of the states. Stochastic unraveling is an efficient numerical tool for solving such equations. This method allows one to simulate much larger and more complex systems with many degrees of freedom. It can, for example, be used to accurately describe femtochemical experiments in the liquid phase whose description has been limited until now to models with one or two effective interaction coordinates. In the unraveling scheme one considers an ensemble of stochastic Schrödinger equations (SSEs) which in the limit of a large ensemble resembles the respective QME. The numerical effort scales much more favorably with the size of the basis since one is now dealing with wave functions and not density matrices anymore (for a comparison of direct integrators, see Ref. 2). Another aspect of the stochastic methods is the possible physical interpretation of experiments detecting macroscopic fluctuations (e.g. photon counting) in various quantum systems [3]. Most of the unraveling schemes [3,4,5,6,7,8] have been restricted to QMEs of Lindblad form [9] which ensures that the reduced density matrix (RDM) stays positive semi-definite for all times and all parameters. Nevertheless there are many physical meaningful QMEs which result in positive-or almost positive-definite RDMs although they are not of Lindblad form. The increasing interest in descriptions beyond the Lindblad class such as the quantum Brownian motion [10,11] developed a method how to exactly represent the RDM of a system coupled to a linear heat bath in terms of SSEs. The numerical properties of this approach need to be explored. Breuer et al.[15] extended a scheme which they had used to calculate multi-time correlation functions [19] to the unraveling of QMEs. Their technique is based on doubling the Hilbert space. Instead of a single stochastic wave function one has a pair of them [15]. This scheme conserves Hermiticity of the RDM only on average and not for every single realization. Thus, the deviation from Hermiticity is a quantity with statistical error and one has to perform a huge number of realizations in order to achieve good convergence. Since stability and efficiency are crucial issues for unraveling algorithms we propose in this article an alternative approach which fu...

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Quantum Trajectories for Brownian Motion

Cited by 81 publications

References 34 publications

Uhrig dynamical control of a three-level system via non-Markovian quantum state diffusion

Uhrig dynamical control of a three-level system via non-Markovian quantum state diffusion

A Perturbative Method for Nonequilibrium Steady State of Open Quantum Systems

Stochastic unraveling of time-local quantum master equations beyond the Lindblad class

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