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arXiv:1409.3527 · Physical Review A

Scattering processes in quantum optics

John Edward Gough

Published Jan 5, 2015• 3 citations
arXivHTMLPDF

Abstract

The derivation of a quantum Markovian model for an optomechanical system consisting of a quantum mechanical mirror interacting with quantum optical input fields via radiation pressure is a difficult problem which ultimately involves the scattering process of quantum stochastic calculus. We show that, while the scattering process may be approximated in a singular limit by regular processes using different schemes, the limit model is highly sensitive to the mathematical interpretation of the approximation scheme. We find two main types of stochastic limits of regular models, and illustrate the origin of this difference at the level of one-particle scattering. As an alternative modeling scheme, we consider models of mirrors as nontrivial dielectric media with boundaries that are themselves quantized. Rather than treating the plane waves for the electromagnetic field, we take the actual physical modes and quantize these. The input-output formalism is then obtained in the far zone where the plane-wave approximation is valid. Several examples are considered, and the quantum stochastic model is derived. We also consider the quantum trajectories problem for continual measurement of the reflected output fields, and derive the stochastic master equations for homodyning and photon-counting detection to estimate the mirror observables.

DOI
10.1103/physreva.91.013802

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The derivation of a quantum Markovian model for an optomechanical system consisting of a quantum mechanical mirror interacting with quantum optical input fields via radiation pressure is a difficult problem which ultimately involves the scattering process of quantum stochastic calculus. We show that, while the scattering process may be approximated in a singular limit by regular processes using different schemes, the limit model is highly sensitive to the mathematical interpretation of the approximation scheme. We find two main types of stochastic limits of regular models, and illustrate the origin of this difference at the level of one-particle scattering. As an alternative modeling scheme, we consider models of mirrors as nontrivial dielectric media with boundaries that are themselves quantized. Rather than treating the plane waves for the electromagnetic field, we take the actual physical modes and quantize these. The input-output formalism is then obtained in the far zone where the plane-wave approximation is valid. Several examples are considered, and the quantum stochastic model is derived. We also consider the quantum trajectories problem for continual measurement of the reflected output fields, and derive the stochastic master equations for homodyning and photon-counting detection to estimate the mirror observables.
The interaction between matter and squeezed light has mostly been treated within the approximation that the field-correlation time is small. Methods for treating squeezed light with more general correlations currently involve explicitly modeling the systems producing the light. We develop a general purpose input-output theory for a particular form of narrowband squeezed light, a squeezed wave-packet mode, that only concerns the statistics of the squeezed field and the shape of the wave packet. This formalism allows us to derive the input-output relations and the master equation. We also consider detecting the scattered field using photon counting and homodyne measurements, which necessitates the derivation of the stochastic master equation. The non-Markovian nature of the field manifests itself in the master equation as a coupled hierarchy of equations. We illustrate these with consequences for the decay and resonance fluorescence of two-level atoms in the presence of such fields.
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We consider an oscillating micromirror replacing one of the two fixed mirrors of a Mach-Zehnder interferometer. In this ideal optical set-up the quantum oscillator is subjected to the radiation pressure interaction of traveling light waves, no cavity is involved. The aim of this configuration is to show that squeezed light can be generated by pure scattering on a quantum system, without involving a cavity. The squeezing can be detected at the output ports of the interferometer either by direct detection or by measuring the spectrum of the difference current. We use the quantum-stochastic Schr\"odinger equation (Hudson-Parthasarathy equation) to model the global evolution. Indeed, it can describe the scattering of photons and the resulting radiation pressure interaction on the quantum oscillator. Moreover, it allows us to consider also the interaction with a thermal bath, so that it can describe also the damping of the harmonic oscillator and non-Markovian thermal effects. In this way we have a unitary dynamics giving the evolution of oscillator and fields. The Bose fields of quantum stochastic calculus and the related generalized Weyl operators allow us to describe the whole optical circuit. By working in the Heisenberg picture, the quantum Langevin equations for position and momentum and the output fields arise, which are used to describe the monitoring in continuous time of the light at the output ports. In the case of strong laser and weak radiation pressure interaction highly nonclassical light is produced, and this can be revealed either by direct detection (a negative Mandel QQQ-parameter is found) or by the intensity spectrum of the difference current of two photodetector; in the second case a nearly complete cancellation of the shot noise can be reached. In this last case it appears that the Mach-Zehnder configuration together with the detection of the difference current corresponds to an homodyne detection scheme, so that we can say that the apparatus is measuring the ``spectrum of squeezing.''
Abstract This chapter deals with a presentation of continuous measurement theory on the basis of the microscopic equations of quantum electrodynamics. The emphasis lies on the derivation of the quantum operations and the corresponding stochastic processes for various detection schemes directly from the Hamiltonian, which describes the interaction of the matter degrees of freedom with the quantized electromagnetic field. The chapter treats many examples and applications to atomic physics and quantum optics, such as dark state resonances and laser cooling of atoms. This example illustrates the interplay between incoherent processes and quantum interference effects which leads to the emergence of Lévy-type distributions for the atomic waiting time. Finally, dissipative phenomena in the dynamics of open quantum systems in strong driving fields are studied, for which appropriate master equations and stochastic wave function methods can be derived by employing a representation in terms of Floquet states.
We study a microscopic model, the Mirror-Oscillator-Field (MOF) model proposed in [1], for describing optomechanical interactions. In contrast with the conventional approach where the mirror-field interaction is understood as arising from the radiation pressure of an optical field inducing the motion of the mirror’s CoM, the MOF model incorporates the dynamics of the internal degrees of freedom of the mirror that couple to the optical field directly. Considering the mirror’s internal and mechanical degrees of freedom as two separate degrees of freedom we derive the optomechanical properties of the coupled mirror and field system. The major advantage in this approach is that it provides a self-consistent treatment of the three relevant subsystems (the mirror’s motion, its internal degrees of freedom and the field) including their back-actions on each other, thereby giving a more accurate account of the coupled internal and external dynamics. The optical and the mechanical properties of a mirror arising from its dynamical interaction with the field are obtained without imposing any boundary conditions on the field additionally, as is done in the conventional way. We find that our results agree with those from the boundary condition approach in the appropriate limits and more generally the model provides a framework within which one can study optomechanical elements with different internal structures and mechanical properties, which makes it suited for studying hybrid systems. Considering the quantum dynamics of the coupled subsystems we look at the entanglement between the mirror’s motion and the field, showing that the internal degrees of the mirror, in the appropriate parameter regimes, can act as a means to coherently transfer quantum correlations between the field and the mechanics thus leading to a larger optomechanical entanglement. We then use the MOF model to study the entanglement between the motion of an atom and a field for the setup in [95] and find a larger optomechanical entanglement when the field is closer to the internal resonance. We also study the interaction between two mirrors as described by the MOF model, specifically looking at the entanglement between the motion of their centers of mass. Conclusively, we see that including the dynamics of the internal degrees of freedom of a mirror, which is the quintessential mediator of interaction between the mirror center of mass and the field, leads to qualitatively different physics, specifically in the quantum regime, thus giving a physically more complete treatment of mirror-field interactions.
Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.