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arXiv:1010.6004 · Infinite Dimensional Analysis Quantum Probability and Related Topics

Quantum stochastic dynamics in multi-photon optics

Ricardo Castro Santis

Published Mar 1, 2014• 0 citations
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Abstract

Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20). In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).

DOI
10.1142/s0219025714500076

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Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20). In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).
Quantum Open System Theory is nowadays a highly developed field which involves, on the mathematical side, many mathematical concepts such as operator theory, semigroups of operators on certain Banach spaces (algebras), classical and quantum stochastic differential equations. Inside Quantum Open System Theory there is the theory of quantum continual measurements: a quantum system is monitored with continuity in time. This is surely an open system because, apart from possible interactions with the external environment like a thermal bath, the system is interacting with the measuring apparatus. The typical field in which a theory of continual measurements is needed is quantum optics: when light emitted by some system is observed (by counting of photons - direct detection - or through some interference mechanism - heterodyne and homodyne detection) a continual measurement is performed.
We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.
Abstract We determine filtering and master equations for a quantum system interacting with wave packet of light in a continuous-mode squeezed number state. We formulate the problem of conditional evolution of a quantum system making use of model of repeated interactions and measurements. In this approach, the quantum system undergoes a sequence of interactions with an environment defined by a chain of harmonic oscillators. We assume that the environment is prepared in an entangled state being a discrete analogue of a continuous-mode squeezed number state. We present a derivation of a discrete stochastic dynamics that depends on the results of measurement performed on the field after its interaction with the system. In this paper, we consider a photon counting measurement scheme. By taking a continuous time limit, we finally obtain differential stochastic equations for the system. Analytical formulae for quantum trajectories and exclusive probability densities that allow to fully characterize the statistics of photons in the output field are given.
The purpose of this paper is to derive filters for an arbitrary open quantum system driven by a light wavepacket prepared in a continuous-mode multi-photon state. A continuous-mode multi-photon state is a state of a travelling light wavepacket that contains a definite number of photons and is characterised by a temporal (or equivalently spectral) profile. After the interaction with the system, the outgoing light can be monitored by means of homodyne detection or photodetection. Filters for both measurement schemes are derived in this paper. Unlike the vacuum or the coherent state case, the annihilation operator of the light field acting on a multi-photon state changes the state by annihilating a photon, and this makes the traditional filtering techniques inapplicable. To circumvent this difficulty, we adopt a non-Markovian embedding technique proposed in \cite{gough:2013} for the study of the single-photon filtering problem. However, the multi-photon nature of the problem addressed in this paper makes the study much more mathematically involved. Moreover, as demonstrated by an example --- a two-level system driven by a continuous-mode two-photon state, multi-photon filters can reveal interesting strong nonlinear optical phenomena absent in both the single-photon state case and the continuous-mode Fock state case.
Abstract This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.
Nonlinear multiphoton conversion processes are theoretically and experimentally challenging systems in quantum optics and related areas. These phenomena bring remarkable quantum properties and can be of fundamental relevance in quantum information and computation applications. Based on recent experimental advances and the possible uses in generalized three-photon squeezing, we investigate degenerate three-photon parametric down-conversion generated inside a cavity fed by a classical pump. In addition, we consider that the process is stimulated by a coherent driving mode and subjected to dissipation by spontaneous emission of the cavity. A treatment using phase-space methods and stochastic differential equations is applied and the stability conditions of its steady states are found. We find that the system does not exhibit multistability, instead it has a limited parameter region leading to a single stable steady branch, alongside multiple unstable branches. Focusing on the stable steady states and linearizing the dynamics around them, we calculate the spectral densities of the quadrature variables. We identify and characterize the squeezing of three-photon down-conversion. Moreover, since the quasi-probability differential equations of the higher-order multiphoton processes have higher-order derivatives, hindering their mapping into stochastic differential equations, we make use of a positive Wigner function approach to calculate two-dimensional spectral densities associated with three-time correlation functions, with the goal of studying non-Gaussian properties of the system.