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arXiv:quant-ph/9810018 · Physical Review A

Cumulant expansion for studying damped quantum solitons

Eduard Schmidt, L. Knöll, Dirk‐Gunnar Welsch

Published Mar 1, 1999• 16 citations
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Abstract

The quantum statistics of damped optical solitons is studied using cumulant-expansion techniques. The effect of absorption is described in terms of ordinary Markovian relaxation theory, by coupling the optical field to a continuum of reservoir modes. After introduction of local bosonic field operators and spatial discretization, pseudo-Fokker-Planck equations for multidimensional s-parametrized phase-space functions are derived. These partial differential equations are equivalent to an infinite set of ordinary differential equations for the cumulants of the phase-space functions. Introducing an appropriate truncation condition, the resulting finite set of cumulant evolution equations can be solved numerically. Solutions are presented in a Gaussian approximation and the quantum noise is calculated, with special emphasis on squeezing and the recently measured spectral photon-number correlations [Sp\"alter et al., Phys. Rev. Lett. 81, 786 (1998)].

DOI
10.1103/physreva.59.2442

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The quantum statistics of damped optical solitons is studied using cumulant-expansion techniques. The effect of absorption is described in terms of ordinary Markovian relaxation theory, by coupling the optical field to a continuum of reservoir modes. After introduction of local bosonic field operators and spatial discretization, pseudo-Fokker-Planck equations for multidimensional s-parametrized phase-space functions are derived. These partial differential equations are equivalent to an infinite set of ordinary differential equations for the cumulants of the phase-space functions. Introducing an appropriate truncation condition, the resulting finite set of cumulant evolution equations can be solved numerically. Solutions are presented in a Gaussian approximation and the quantum noise is calculated, with special emphasis on squeezing and the recently measured spectral photon-number correlations [Sp"alter et al., Phys. Rev. Lett. 81, 786 (1998)].
Using cumulant expansion in Gaussian approximation, the internal quantum statistics of damped soliton-like pulses in Kerr media are studied numerically, considering both narrow and finite bandwidth spectral pulse components. It is shown that the sub-Poissonian statistics can be enhanced, under certain circumstances, by absorption, which damps out some destructive interferences. Further, it is shown that both the photon-number correlation and the correlation of the photon-number variance between different pulse components can be highly nonclassical even for an absorbing fiber. Optimum frequency windows are determined in order to realize strong nonclassical behavior, which offers novel possibilities of using solitons in optical fibers as a source of nonclassically correlated light beams.
Using cumulant expansion in Gaussian approximation, the internal quantum statistics of damped soliton-like pulses in Kerr media are studied numerically, considering both narrow and finite bandwidth spectral pulse components. It is shown that the sub-Poissonian statistics can be enhanced, under certain circumstances, by absorption, which damps out some destructive interferences. Further, it is shown that both the photon-number correlation and the correlation of the photon-number variance between different pulse components can be highly nonclassical even for an absorbing fiber. Optimum frequency windows are determined in order to realize strong nonclassical behavior, which offers novel possibilities of using solitons in optical fibers as a source of nonclassically correlated light beams.
We introduce two approaches to modeling the quantum dynamics of optical χ(3)χ^{(3)}χ(3) solitons. Taking an open-system viewpoint, we project the underlying quantum field into system (soliton) and residual reservoir components. The reservoir is treated as either (i) a discrete ``Lanczos supermode'' (LSM) expansion which localizes dynamics to a few-supermode basis, or (ii) a non-local environment which can be traced out by deriving a Markovian master equation (ME). Using these methods, we analyze and identify the quantum structure of both the soliton's stability and its hierarchy of perturbations. Through numerical simulations, we confirm both methods effectively capture quantum-induced soliton phase shifts in a concise few-mode (single-mode for ME) basis, and the LSM approach also captures photon loss which arises from non-Markovian dispersive couplings. As neither method is limited to the linearized regime, our approaches provide powerful computational tools to analyze complex non-Gaussian quantum dynamics of solitons where other commonly-used methods fail, providing insight into such non-perturbative regimes. We also investigate radiation that occurs in the presence of higher-order dispersion with ultrashort pulses, deriving a ME that predicts photon loss consistent with classical theory, but find that both classical and ME theory dramatically underestimate the actual amount of dissipation, which we explain in terms of dispersive coupling-induced soliton broadening.
Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.
A cavity-damped Jaynes-Cummings model with a Kerr-like medium filling the cavity is investigated in the rotating-wave approximation. We introduce six operators with respect to the light field whose equations of motion are transformed to six coupled partial differential equations using the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)]. Equations of motion for expansion coefficients of the distribution functions are solved by a Runge-Kutta procedure for vector tridiagonal relations. Starting with an initial coherent state for the cavity field and the atom in its upper state, we find that revivals of the atomic inversion are more pronounced for a given damping constant compared to the case of no Kerr medium. Also, quadrature squeezing is less affected by weak cavity damping and thermal noise compared to the standard Jaynes-Cummings model. The effect of damping on interesting non-Gaussian structures is also discussed.