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arXiv:quant-ph/0605200

Continuous observation of a squeezed coherent state

Anita Magdalena Dąbrowska

Published May 23, 2006• 0 citations
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Abstract

The main aim of the paper is to present the analytical solution of the Belavkin quantum filtering equation for damped harmonic oscillator being initially in the squeezed coherent state for diffusion observation with complex white noise. The comparison of the a priori and a posteriori mean value of the optical quadrature operators and the photon number operator is given.

DOI
10.48550/arxiv.quant-ph/0605200

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The main aim of the paper is to present the analytical solution of the Belavkin quantum filtering equation for damped harmonic oscillator being initially in the squeezed coherent state for diffusion observation with complex white noise. The comparison of the a priori and a posteriori mean value of the optical quadrature operators and the photon number operator is given.
We present the Belavkin filtering equation for the intense balanced heterodyne detection in a unitary model of an indirect observation. The measuring apparatus modelled by a Bose field is initially prepared in a coherent state and the observed process is a diffusion one. We prove that this filtering equation is relaxing: any initial square-integrable function tends asymptotically to a coherent state with an amplitude depending on the coupling constant and the initial state of the apparatus. The time development of a squeezed coherent state is studied and compared with the previous results obtained for the measuring apparatus prepared initially in the vacuum state.
We derive the filtering equation for Markovian systems undergoing homodyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that is, quantum input processes in a coherent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filtered estimate π t (a) for the mode annihilator a based on the homodyne quadrature observations up to time t.'
A brief presentation of the basic concepts in quantum probability theory is given in comparison to the classical one. The notion of quantum white noise, its explicit representation in Fock space, and necessary results of noncommutative stochastic analysis and integration are outlined. Algebraic differential equations that unify the quantum non Markovian diffusion with continuous non demolition observation are derived. A stochastic equation of quantum diffusion filtering generalising the classical Markov filtering equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian quantum diffusion with one dimensional continuous observation is considered.The a posteriori quantum state difusion in this case is reduced to a linear quantum stochastic filter equation of Kalman-Bucy type and to the operator Riccati equation for quantum correlations. An example of continuous nondemolition observation of the coordinate of a free quantum particle is considered, describing a continuous collase to the stationary solution of the linear quantum filtering problem found in the paper.
We derive the filtering equation for Markovian systems undergoing ho-modyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that, quantum input processes in a co-herent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filtered estimate pit(a) for the mode annihilator a based on the homodyne quadrature observations up to time t. In memory of Slava Belavkin. 1
It has widely been accepted that the laser field is a coherent state with phase diffusion. For example, in quantum theory of the laser it is conventional that the initial field state is given as a coherent state. But strictly speaking, this initial condition is a hypothesis which should be examined through experiments in terms of the quantum fluctuations of the held. In this study a new steady-state solution for the laser field density matrix is suggested, whose initial photon state is a quadrature-phase squeezed state, and whose photon number statistics is just the same as the conventional ansatz.
Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.