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

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum trajectories: physical interpretation

Miguel Orszag

Published Jul 6, 1999• 0 citations
Publisher

Abstract

A general description of the Quantum Trajectories is presented, both from the Montecarlo and Stochastic Schrodinger equation viewpoint. Also a physical interpretation of these trajectories is given. The example is a mode of the electromagnetic field in a cavity interacting with a beam of two or three level atoms, the field playing the role of the small system and the atomic beam standing for a reservoir at finite temperature. We show that the evolution of the field states, under continuous monitoring of the state of the atom which leave the cavity can be described with the Monte-Carlo method or a diffusion equation, depending on the configuration of the system.

DOI
10.1117/12.358423

Related papers

Observing different quantum trajectories in cavity QED

Marcelo França Santos, Andre R. R. Carvalho

The experimental observation of quantum jumps is an example of single open quantum systems that, when monitored, evolve in terms of stochastic trajectories conditioned on measurements results. Here we present a proposal that allows the experimental observation of a much larger class of quantum trajectories in cavity QED systems. In the microwave regime, three-level atoms probe the cavity field while in the optical case the field probes atomic states. In the first situation the cavity field evolves stochastically between highly non-classical states while the second case allows for the monitoring of engineered thermal baths that are crucial for recent proposals for probing entanglement decay and also for entanglement protection.

arXiv:1102.1047•Europhysics Letters (EPL)•Jun 1, 2011•18 citations

Quantum Monte Carlo calculations

Eric Dale Lieser

A review of the density operator and master equation leads to a derivation of the method of quantum jumps, or quantum trajectories. This method of quantum trajectories is used to describe the evolution of a two level atom driven by a laser field and damped by a quantized radiation field. The quantum trajectory method is then compared to the master equation via numerical simulations of this two level atom system.

•Jan 1, 2003•2 citations

Topics in Quantum Measurement and Quantum Noise

Kurt A. Jacobs

In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.

arXiv:quant-ph/9810015•Oct 5, 1998•5 citations

Existence, uniqueness and approximation of a stochastic Schrödinger equation: The diffusive case

Clément Pellegrini

Recent developments in quantum physics make heavy use of so-called “quantum trajectories.” Mathematically, this theory gives rise to “stochastic Schrödinger equations,” that is, perturbation of Schrödinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the literature. They pose a serious problem in terms of justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrödinger equation. We physically justify the equations by proving that they are a continuous-time limit of a concrete physical procedure for obtaining a quantum trajectory.

arXiv:0709.1703•The Annals of Probability•Nov 1, 2008•74 citations

Decay of quantum coherence using quantum trajectories

H. J. Carmichael, L. Tian, P. Kochan

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.

Annual Meeting Optical Society of America•Jan 1, 1992•0 citations

Quantum trajectories generalized

P.E. van Dorsselaer, Gerard Nienhuis

Summary form only given. Quantum trajectories of state vectors can be used to describe open systems that otherwise can only be described using a density matrix. Besides the numerical advantage, quantum trajectories provide insight in the actual evolution of a system under continuous observation. Two distinct methods are usually considered, the quantum-jump method, associated with direct photodetection, and the quantum state diffusion (QSD) method, associated with homodyne or heterodyne mixing. We derive a general stochastic equation for quantum trajectories.

Conference on Lasers and Electro-Optics•Nov 27, 2002•0 citations
A general description of the Quantum Trajectories is presented, both from the Montecarlo and Stochastic Schrodinger equation viewpoint. Also a physical interpretation of these trajectories is given. The example is a mode of the electromagnetic field in a cavity interacting with a beam of two or three level atoms, the field playing the role of the small system and the atomic beam standing for a reservoir at finite temperature. We show that the evolution of the field states, under continuous monitoring of the state of the atom which leave the cavity can be described with the Monte-Carlo method or a diffusion equation, depending on the configuration of the system.
The experimental observation of quantum jumps is an example of single open quantum systems that, when monitored, evolve in terms of stochastic trajectories conditioned on measurements results. Here we present a proposal that allows the experimental observation of a much larger class of quantum trajectories in cavity QED systems. In the microwave regime, three-level atoms probe the cavity field while in the optical case the field probes atomic states. In the first situation the cavity field evolves stochastically between highly non-classical states while the second case allows for the monitoring of engineered thermal baths that are crucial for recent proposals for probing entanglement decay and also for entanglement protection.
In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.
Recent developments in quantum physics make heavy use of so-called “quantum trajectories.” Mathematically, this theory gives rise to “stochastic Schrödinger equations,” that is, perturbation of Schrödinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the literature. They pose a serious problem in terms of justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrödinger equation. We physically justify the equations by proving that they are a continuous-time limit of a concrete physical procedure for obtaining a quantum trajectory.
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.
Summary form only given. Quantum trajectories of state vectors can be used to describe open systems that otherwise can only be described using a density matrix. Besides the numerical advantage, quantum trajectories provide insight in the actual evolution of a system under continuous observation. Two distinct methods are usually considered, the quantum-jump method, associated with direct photodetection, and the quantum state diffusion (QSD) method, associated with homodyne or heterodyne mixing. We derive a general stochastic equation for quantum trajectories.