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arXiv:2302.00756 · Physical Review A

Quantum decay of an optical soliton

Stuart Ward, Rouzbeh Allahverdi, Arash Mafi

Published May 12, 2023• 5 citations
arXivHTMLPDF

Abstract

Optical solitons are known to be classically stable objects which are robust to perturbations. In this work we show that due to quantum-mechanical effects, an optical soliton that is initially in a classical soliton coherent state will shed photons into the continuum and hence decay. The standard formulation of the quantized soliton uses the linearized version of the quantum nonlinear Schr\"odinger equation in the background of the classical soliton, and the quantized soliton remains stable in this approximation. We show that if higher-order interaction terms are taken into account, the soliton is no longer stable, and its photon number decreases quadratically as a function of the number of soliton cycles. We compute the power spectrum for the continuum radiation and find a narrow band that is localized about the initial soliton momentum with a cutoff that is inversely proportional to the initial soliton width.

DOI
10.1103/physreva.107.053513

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Optical solitons are known to be classically stable objects which are robust to perturbations. In this work we show that due to quantum-mechanical effects, an optical soliton that is initially in a classical soliton coherent state will shed photons into the continuum and hence decay. The standard formulation of the quantized soliton uses the linearized version of the quantum nonlinear Schr"odinger equation in the background of the classical soliton, and the quantized soliton remains stable in this approximation. We show that if higher-order interaction terms are taken into account, the soliton is no longer stable, and its photon number decreases quadratically as a function of the number of soliton cycles. We compute the power spectrum for the continuum radiation and find a narrow band that is localized about the initial soliton momentum with a cutoff that is inversely proportional to the initial soliton width.
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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.
We analyse a nonlinear QW model which can be experimentally implemented using the components of the electric field on an optical nonlinear Kerr medium, which translates into a rotation in the coin operator, with an angle which depends (in a nonlinear fashion) on the state of the walker. This simple dependence makes it easy to consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us, under some approximations, to gain some insight into the nature of soliton structures, which is illustrated by our numerical calculations. These solitons are stable structures whose trajectories can be modulated by choosing the appropriate initial conditions. We have also studied the stability of solitons when they are subject to an additional phase that simulates an external electric field, and also explored if they are formed in higher dimensional spaces.
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Solitons of the classical nonlinear Schrödinger equation are stable to noise perturbations. One may ask whether the uncertainties associated with the quantum description of a soliton could cause the soliton to become unstable and to break up. It is not difficult to derive the fact that the first order soliton disperses, so that the probability of detecting the soliton at a specified position approaches zero as the time of propagation approaches infinity. This fact does not prove or disprove breakup. It can be shown by using the formalism of Ref. 1 that the intensity correlation function of a quantum soliton is time independent.2 In this paper we describe a form of a Hanbury, Brown, and Twiss experiment that could demonstrate the stability of the fundamental soliton (in principle), and we develop expressions for the hierarchy of correlation functions measured by such an experimental setup. If the soliton breaks up, the correlation functions should show a strong dependence on the propagation time of the soliton. The calculations demonstrate that all the measurable correlation functions do not depend on the propagation time; therefore the quantum soliton is stable.
The influence of high-order effects on the stability of the optical soliton in a semiconductor three-quantum-dot molecular system under the excitation of narrow pulse probe light is analyzed analytically by using the multi-scale method. The results show that optical soliton described by the standard nonlinear Schrödinger equation will have a large attenuation in the propagation process, while the optical soliton described by the high-order nonlinear Schrödinger equation has relatively good stability. In addition, numerical simulations of the interaction between optical solitons show that the amplitudes of the two optical solitons described by the standard nonlinear Schrödinger equation attenuate rapidly after the collisions and radiation of more serious dispersion waves, while the shapes of the two optical solitons described by the high-order nonlinear Schrödinger equation hardly changes after the collision. This is mainly because when the incident probe light pulse is narrow enough, the system must be described by a higher-order equation. The physical reason is that the higher-order effects in the equation, including non-instantaneous effects and third-order dispersion effects, cannot be ignored or treated as perturbations. This kind of stable optical soliton has potential application value for future optical information processing and transmission technology.