Solving quantum stochastic differential equations with unbounded coefficients

Fagnola, Franco; Wills, Stephen J.

doi:10.1016/s0022-1236(02)00089-7

Cited by 30 publications

(25 citation statements)

References 16 publications

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“…Here, H is a fixed self-adjoint operator representing the free Hamiltonian of the system, and L and S are system operators determining the coupling of the system to the field, with S unitary. For the sake of simplicity, we assume that the parameters S , L, H are bounded; however, we remark that under some suitable additional conditions the results should also apply to unbounded parameters [23,24].…”

Section: (A) the Filtering Problemmentioning

confidence: 99%

Single photon quantum filtering using non-Markovian embeddings

Gough

James

Nurdin

2012

Phil. Trans. R. Soc. A.

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Section: (A) the Filtering Problemmentioning

confidence: 99%

Single photon quantum filtering using non-Markovian embeddings

Gough

James

Nurdin

2012

Phil. Trans. R. Soc. A.

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“…Now, we carry out the next three stages of this procedure in the proof of Theorem 3.1. Stage 4 rests on modifications of ideas presented in [8].…”

Section: S and There Is A Constant K Such Thatmentioning

confidence: 99%

“…Inspired by the proof of Theorem 2.2 of [8], we now consider the variant of the Yosida approximation C ε = (I + εC) −1 C(I + εC) −1 . Combining (3.5) and…”

Section: S and There Is A Constant K Such Thatmentioning

confidence: 99%

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

Mora¹

2003

Math. Comp.

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Abstract. We address the problem of approximating numerically the solutions (Xt : t ∈ [0, T ]) of stochastic evolution equations on Hilbert spaces (h, ·, · ), with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of Xt, AXt , where A is a linear operator. First, we introduce estimates on the behavior of Xt. Then we characterize the error induced by the substitution of Xt with the solution Xt,n of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of E X t,n, AXt,n to E Xt, AXt , whereXt,n denotes the explicit Euler method. Finally, we consider an extrapolation method based on the Euler scheme. An application to the quantum harmonic oscillator system is included.

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“…We consider only the type of observables relevant for the description of homodyne/heterodyne detection and we make the mathematical simplification of introducing only bounded operators on the Hilbert space of the quantum system of interest and a finite number of noises; for the case of unbounded operators see [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Quantum continuous measurements: The stochastic Schrödinger equations and the spectrum of the output

Barchielli

Gregoratti

2013

Quantum Measurements and Quantum Metrology

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The stochastic Schrödinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diffusive case. Firstly, we discuss the quantum stochastic Schrödinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schrödinger equation by diagonalization of suitable commuting quantum observables. Then, we give the a posteriori state, the conditional system state at time given the output up to that time, and we link its evolution to the classical stochastic Schrödinger equation. Moreover, the relation with quantum filtering theory is shortly discussed. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a two-level atom stimulated by a laser, to discuss the differences between homodyne and heterodyne detection and to explicitly show squeezing and anti-squeezing of the homodyne spectrum and the Mollow triplet in the fluorescence spectrum.

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Solving quantum stochastic differential equations with unbounded coefficients

Cited by 30 publications

References 16 publications

Single photon quantum filtering using non-Markovian embeddings

Single photon quantum filtering using non-Markovian embeddings

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

Quantum continuous measurements: The stochastic Schrödinger equations and the spectrum of the output

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