Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0401171 · Physical Review A

Jumplike unravelings for non-Markovian open quantum systems

Jay Gambetta, T. Askerud, Howard M. Wiseman

Published May 10, 2004• 22 citations
arXivHTMLPDF

Abstract

Non-Markovian evolution of an open quantum system can be ``unraveled'' into pure state trajectories generated by a non-Markovian stochastic (diffusive) Schr\"odinger equation, as introduced by Di\'osi, Gisin, and Strunz. Recently we have shown that such equations can be derived using the modal (hidden variable) interpretation of quantum mechanics. In this paper we generalize this theory to treat jumplike unravelings. To illustrate the jumplike behavior we consider a simple system: a classically driven (at Rabi frequency $\ensuremath{\Omega}$) two-level atom coupled linearly to a three mode optical bath, with a central frequency equal to the frequency of the atom, ${\ensuremath{\omega}}_{0}$, and the two side bands have frequencies ${\ensuremath{\omega}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}$. In the large $\ensuremath{\Omega}$ limit we observed that the jumplike behavior is similar to that observed in this system with a Markovian (broad band) bath. This is expected as in the Markovian limit the fluorescence spectrum for a strongly driven two level atom takes the form of a Mollow triplet. However, the length of time for which the Markovian-like behavior persists depends upon which jumplike unraveling is used.

DOI
10.1103/physreva.69.052104

Related papers

A non-Markovian stochastic Schrodinger equation developed from a hidden variable interpretation

Jay Gambetta, Howard M. Wiseman

Do non-Markovian diffusive stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE at time t represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we show that such trajectories be interpreted using a non-standard interpretation of quantum mechanics, namely hidden variables theory. The simplest example is a non-Markovian SSE, which we call a position unraveling, which results from a description of the bath using Bohmian mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a weighted sum of trajectories for the positions of the bath modes. The solution of the non-Markovian SSE is the state of the system 'conditioned' on values of the bath positions. It also determines the guiding wave for these positions. We also discuss previously used unravelings which have the Markovian limits of homodyne and heterodyne detection.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 16, 2003•3 citations

The non-Markovian stochastic Schrödinger equation for the position unravelling

Jay Gambetta, Howard M. Wiseman

An important and well established area of quantum optics is the theory of Markovian stochastic Schr\"odinger equations (or by another name quantum trajectory theory). Recently stochastic Schr\"odinger equations have been developed for non-Markovian systems. In this paper we extend the current known stochastic Schr\"odinger equations for non-Markovian systems to include the position unraveling. We also discuss and illustrate that this stochastic Schr\"odinger equation can have an interpretation under both the orthodox and the de Broglie-Bohm hidden variable interpretation of quantum mechanics. We conclude that only the de Broglie-Bohm hidden variable theory provides a continuous-in-time interpretation of the non-Markovian stochastic Schr\"odinger equation.

arXiv:quant-ph/0305047•Journal of Optics B Quantum and Semiclassical Optics•Jul 28, 2004•6 citations

Interpretation of non-Markovian stochastic Schrödinger equations as a hidden-variable theory

Jay Gambetta, Howard M. Wiseman

Do diffusive non-Markovian stochastic Schr\"odinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function $z(t)$ appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value $z(t)$ even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system ``conditioned'' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit.

arXiv:quant-ph/0307078•Physical Review A•Dec 9, 2003•64 citations

Non-Markovian stochastic Schrödinger equations: Generalization to real-valued noise using quantum-measurement theory

Jay Gambetta, Howard M. Wiseman

Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schr\"odinger equation introduced by Strunz, Di\'osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.

arXiv:quant-ph/0202117•Physical Review A•Jul 29, 2002•82 citations

On the Unraveling of Open Quantum Dynamics

Brecht Donvil, Paolo Muratore-Ginanneschi

It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.

arXiv:2309.13408•Open Systems & Information Dynamics•Sep 1, 2023•5 citations

Time-local quantum-state-diffusion equation for multilevel quantum systems

Jun Jing, Xinyu Zhao, J. Q. You +1 more

An open quantum system with multiple levels coupled to a bosonic environment at zero temperature is investigated systematically using the non-Markovian quantum-state-diffusion (QSD) method [W. T. Strunz, L. Di\'osi, and N. Gisin, Phys. Rev. Lett. 82, 1801 (1999)]. We have established exact time-local QSD equations for a set of interesting multilevel open systems, including high-spin systems, multiple-transition atomic models, and multilevel atomic models driven by time-dependent external fields. These exact QSD equations have paved the way to evaluate the dynamics of open multilevel atomic systems in the general non-Markovian regimes without any approximation.

arXiv:1201.6558•Physical Review A•Apr 6, 2012•41 citations
Non-Markovian evolution of an open quantum system can be ``unraveled'' into pure state trajectories generated by a non-Markovian stochastic (diffusive) Schr"odinger equation, as introduced by Di'osi, Gisin, and Strunz. Recently we have shown that such equations can be derived using the modal (hidden variable) interpretation of quantum mechanics. In this paper we generalize this theory to treat jumplike unravelings. To illustrate the jumplike behavior we consider a simple system: a classically driven (at Rabi frequency \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ) two-level atom coupled linearly to a three mode optical bath, with a central frequency equal to the frequency of the atom, \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​, and the two side bands have frequencies \ensuremathω0\ifmmode±\else\textpm\fi\ensuremathΩ{\ensuremath{\omega}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}\ensuremathω0​\ifmmode±\else\textpm\fi\ensuremathΩ. In the large \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ limit we observed that the jumplike behavior is similar to that observed in this system with a Markovian (broad band) bath. This is expected as in the Markovian limit the fluorescence spectrum for a strongly driven two level atom takes the form of a Mollow triplet. However, the length of time for which the Markovian-like behavior persists depends upon which jumplike unraveling is used.
Do non-Markovian diffusive stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE at time t represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we show that such trajectories be interpreted using a non-standard interpretation of quantum mechanics, namely hidden variables theory. The simplest example is a non-Markovian SSE, which we call a position unraveling, which results from a description of the bath using Bohmian mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a weighted sum of trajectories for the positions of the bath modes. The solution of the non-Markovian SSE is the state of the system 'conditioned' on values of the bath positions. It also determines the guiding wave for these positions. We also discuss previously used unravelings which have the Markovian limits of homodyne and heterodyne detection.
An important and well established area of quantum optics is the theory of Markovian stochastic Schr\"odinger equations (or by another name quantum trajectory theory). Recently stochastic Schr\"odinger equations have been developed for non-Markovian systems. In this paper we extend the current known stochastic Schr\"odinger equations for non-Markovian systems to include the position unraveling. We also discuss and illustrate that this stochastic Schr\"odinger equation can have an interpretation under both the orthodox and the de Broglie-Bohm hidden variable interpretation of quantum mechanics. We conclude that only the de Broglie-Bohm hidden variable theory provides a continuous-in-time interpretation of the non-Markovian stochastic Schr\"odinger equation.
Do diffusive non-Markovian stochastic Schr\"odinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t)z(t)z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t)z(t)z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system ``conditioned'' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit.
Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schr\"odinger equation introduced by Strunz, Di\'osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.
An open quantum system with multiple levels coupled to a bosonic environment at zero temperature is investigated systematically using the non-Markovian quantum-state-diffusion (QSD) method [W. T. Strunz, L. Di\'osi, and N. Gisin, Phys. Rev. Lett. 82, 1801 (1999)]. We have established exact time-local QSD equations for a set of interesting multilevel open systems, including high-spin systems, multiple-transition atomic models, and multilevel atomic models driven by time-dependent external fields. These exact QSD equations have paved the way to evaluate the dynamics of open multilevel atomic systems in the general non-Markovian regimes without any approximation.