Stochastic Master Equation Research Articles

Stochastic Master Equations in Thermal Environment

We derive stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. It is well-known that at zero temperature one obtains stochastic differential equations of jump-diffusion type. We show that only pure diffusion type equations are relevant at strictly positive temperatures.

Markov chains approximation of jump–diffusion stochastic master equations

Les trajectoires quantiques sont des solutions d’équations différentielles stochastiques décrivant des phénomènes aléatoires associés aux prinicpes de mesure (quantique) des systèmes quantiques ouverts. Ces équations, également appelées équations de Belavkin ou équations maîtresses stochastiques sont habituellement de deux types: soit diffusif soit de type saut. Dans cet article, nous considérons des modèles plus avancés où des équations de type saut-diffusion apparaissent. Ces équations sont obtenues comme solutions de problèmes de martingales. Ces problèmes de martingales sont obtenus comme limites continus (en temps) à partir de chaînes de Markov classiques décrivant des trajectoires quantiques pour des modèles à temps discret. Les résultats de cet article généralisent ceux de [Ann. Probab. 36 (2008) 2332–2353] et [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. Ici, les techniques probabilistes utilisés sont complétement différentes afin de pouvoir mixer les deux types d’évolutions: diffusives et poissoniennes.

Reading the Secrets of Biological Fluctuations

AbstractI explore tools and applications of the van Kampen expansion of the stochastic master equation in several areas of population biology. I highlight fluctuation-dissipation and fluctuation-enhancement regimes, using information from noise to help model choice, and examples where noise changes the macroscopic dynamics of a system.

Time-delayed quantum feedback for traveling optical fields

Quantum nonlinear feedback control is developed for traveling optical fields. We first describe the discretization of the traveling optical fields. The discrete-time formulation is used to describe the stochastic master equation subject to homodyne measurement. Nonlinear feedback is formulated by directly feeding the measurement outcomes back to the traveling field through a multiplicative action. Since the measurement outcomes have a correlation with the system, the multiplicative feedback control can create nonlinear effects in the traveling field. In this formulation, a time delay is naturally introduced in the feedback loop. This is essentially different from instantaneous feedback in a continuous-time setting. As an example of the feedback scheme, a quantum nondemolition sum gate is considered. Numerical results show that quantum superposition state can be created by applying the feedback to a squeezed state.

Continuous quantum nondemolition measurement of Fock states of a nanoresonator using feedback-controlled circuit QED

We propose a scheme for the quantum non-demolition (QND) measurement of Fock states of a nanomechanical resonator via feedback control of a coupled circuit QED system. A Cooper pair box (CPB) is coupled to both the nanoresonator and microwave cavity. The CPB is read-out via homodyne detection on the cavity and feedback control is used to effect a non-dissipative measurement of the CPB. This realizes an indirect QND measurement of the nanoresonator via a second-order coupling of the CPB to the nanoresonator number operator. The phonon number of the Fock state may be determined by integrating the stochastic master equation derived, or by processing of the measurement signal.

Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket. II. The eigenvalue spectrum

We continue the study of a particle (atom, molecule) undergoing an unbiased random walk on the Sierpinski gasket, and obtain for the gasket and tower the eigenvalue spectrum of the associated stochastic master equation. Analytic expressions for recurrence relations among the eigenvalues are derived. The recurrence relations obtained are compared with those determined for two Euclidean lattices, the closed chain with an absorbing site and a finite chain with an absorbing site at one end. We check and confirm the internal consistency between the smallest eigenvalue and the mean walklength in each of the cases studied. Attention is drawn to the relevance of the results obtained to a problem of electron transfer in proteins.

Characterizing the Effect of Coarse-Scale PBN Modeling on Dynamics and Intervention Performance of Genetic Regulatory Networks Represented by Stochastic Master Equation Models

Fine-scale models based on stochastic master equations can provide the most detailed description of the dynamics of gene expression and imbed, in principle, all the information about the biochemical reactions involved in gene interactions. However, there is limited time-series experimental data available for inference of such fine-scale models. Furthermore, the computational complexity involved in the design of optimal intervention strategies to favorably effect system dynamics for such detailed models is enormous. Thus, there is a need to design mappings from fine-scale models to coarse-scale models while maintaining sufficient structure for the problem at hand and to study the effect of intervention policies designed using coarse-scale models when applied to fine-scale models. In this paper, we propose a mapping from a fine-scale model represented by a stochastic master equation to a coarse-scale model represented by a probabilistic Boolean network that maintains the collapsed steady state probability distribution of the detailed model. We also derive bounds on the performance of the intervention strategy designed using the coarse-scale model when applied to the fine-scale model.

Noncommutative dynamics and generalized master equations

We review the basic concepts of quantum probability and stochastics using the universal Ito B*-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Levy process is defined in terms of the modular B*-Ito algebra, and the corresponding quantum stochastic master equation on the predual space of theW*-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.

Tunable joint measurements in the dispersive regime of cavity QED

Joint measurements of multiple qubits have been shown to open new possibilities for quantum information processing. Here, we present an approach based on homodyne detection to realize such measurements in the dispersive regime of cavity/circuit QED. By changing details of the measurement, the readout can be tuned from extracting only single-qubit to only multi-qubit properties. We obtain a reduced stochastic master equation describing this measurement and its effect on the qubits. As an example, we present results showing parity measurements of two qubits. In this situation, measurement of an initially unentangled state can yield with near unit probability a state of significant concurrence.

The diffusive finite state projection algorithm for efficient simulation of the stochastic reaction-diffusion master equation.

We have developed a computational framework for accurate and efficient simulation of stochastic spatially inhomogeneous biochemical systems. The new computational method employs a fractional step hybrid strategy. A novel formulation of the finite state projection (FSP) method, called the diffusive FSP method, is introduced for the efficient and accurate simulation of diffusive transport. Reactions are handled by the stochastic simulation algorithm.

Diffusion approximation of stochastic master equations with jumps

In the presence of quantum measurements with direct photon detection, the evolution of open quantum systems is usually described by stochastic master equations with jumps. Heuristically, diffusion models can be obtained from these equations as an approximation. A condition for a general diffusion approximation for jump master equations is presented. This approximation is rigorously proved by using techniques for Markov processes, which are based on the convergence of Markov generators and martingale problems. This result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection.

On the quantum master equation under feedback control

The nature of the quantum trajectories, described by stochastic master equations, may be jump-like or diffusive, depending upon different measurement processes. There are many different unravelings corresponding to different types of stochastic master equations for a given master equation. In this paper, we study the relationship between the quantum stochastic master equations and the quantum master equations in the Markovian case under feedback control. We show that the corresponding unraveling no longer exists when we further consider feedback control besides measurement. It is due to the fact that the information gained by the measurement plays an important role in the control process. The master equation governing the evolution of ensemble average cannot be restored simply by eliminating the noise term unlike the case without a control term. By establishing a fundamental limit on performance of the master equation with feedback control, we demonstrate the differences between the stochastic master equation and the master equation via theoretical proof and simulation, and show the superiority of the stochastic master equation for feedback control.

Quantum filter reduction for measurement-feedback control via unsupervised manifold learning

We derive simple models for the dynamics of a single atom coupled to a cavity field mode in the absorptive bistable parameter regime by projecting the time evolution of the state of the system onto a suitably chosen nonlinear low-dimensional manifold, which is found by use of local tangent space alignment. The output field from the cavity is detected with a homodyne detector allowing observation of quantum jumps of the system between states with different average numbers of photons in the cavity. We find that the models, which are significantly faster to integrate numerically than the full stochastic master equation, largely reproduce the dynamics of the system, and we demonstrate that they are sufficiently accurate to facilitate feedback control of the state of the system based on the predictions of the models alone.

Non-Markovian quantum repeated interactions and measurements

A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutionless master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non-Markovian stochastic master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed.

QUANTUM TRAJECTORIES IN RANDOM ENVIRONMENT: THE STATISTICAL MODEL FOR A HEAT BATH

In this article, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in \cite{A1P1}, where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

The information about the state of a qubit gained by a weakly coupled detector

We analyze the information that one can learn about the state of a quantum two-level system, i.e. a qubit, when probed weakly by a nearby detector. In particular, we focus on the case when the qubit Hamiltonian and the qubit's operator being probed by the detector do not commute. Because the qubit's state keeps evolving while being probed and because the measurement data is mixed with detector-related background noise, one might expect the detector to fail in this case. We show, however, that under suitable conditions and by proper analysis of the measurement data useful information about the state of the qubit can be extracted. It turns out that the measurement basis is stochastically determined every time the experiment is repeated. We analyze in detail the probability distributions that govern the choice of measurement bases. We also analyze the information acquisition rate and show that it is largely unaffected by the apparent conflict between the measurement and intrinsic qubit dynamics. We discuss the relation between our analysis and the stochastic master equation that describes the evolution of the qubit's state under the influence of measurement and decoherence. In particular, we write down a stochastic equation that encompasses the usual stochastic master equation for the evolution of the qubit's density matrix and additionally contains the measurement information that can be extracted from the observed signal.

Continuous measurement feedback control of a Bose-Einstein condensate using phase-contrast imaging

We consider the theory of feedback control of a Bose-Einstein condensate (BEC) confined in a harmonic trap under a continuous measurement constructed via nondestructive imaging. A filtering theory approach is used to derive a stochastic master equation (SME) for the system from a general Hamiltonian based upon system-bath coupling. Numerical solutions for this SME in the limit of a single atom show that the final steady-state energy is dependent upon the measurement strength, the ratio of photon kinetic energy to atomic kinetic energy, and the feedback strength. Simulations indicate that for a weak measurement strength, feedback can be used to overcome heating introduced by the scattering of light, thereby allowing the atom to be driven toward the ground state.

Poisson and Diffusion Approximation of Stochastic Master Equations with Control

"Quantum trajectories" are solutions of stochastic differential equations of non-usual type. Such equations are called "Belavkin" or "Stochastic Schr\"odinger Equations" and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called "Quantum Repeated Measurements". In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics.

A stochastic master equation description of laserlike systems

We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model. Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P(n), the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of (n) and log[(n2)/(n)2] versus the pumping parameter show close agreement with the microscopic results of the Casa-grande-Lugiato model. A similar discussion of the fluctuations can also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.

Fluctuation Induced Homochirality in an Open System

We study a possibility for chiral symmetry breaking to take place without autocatalytic process in an open system. Our model consists of an inflow of enantiomers and outflow of heterogeneous dimers from the system, which corresponds to the Frank model without autocatalysis. It is demonstrated analytically as well as numerically that the rate equations for this system does not predict any kind of symmetry breaking but the stochastic master equation predicts that the system approaches asymptotically a homochiral state. This is caused by the fluctuations induced by the discreteness of population numbers of enantiomers in their perpetual flow. It is argued that this scenario can be effective for a very small open system and could be a candidate for an initial chiral imbalance.