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Optics and Spectroscopy

Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: II. Two-dimensional light solitons

Leonid A. Nesterov, Nikolay A. Veretenov, Nikolay N. Rosanov

Published May 1, 2015• 5 citations
Publisher

Abstract

Quantum fluctuations of two-dimensional dark dissipative solitons in an interferometer filled with a Kerr nonlinear medium are analyzed theoretically. The boundaries of the stability region of classical solitons excited in the considered interferometer by an external radiation are refined. The solution of the quantum problem is studied in the linear approximation. For this purpose, the formalism of expansion of the sought solution in discrete spectral functions is extended to the two-dimensional case. Using this expansion made it possible to construct a solution that describes fluctuations of collective variables of a two-dimensional soliton, namely, the coordinate of the center and momentum. Mean squares of quantum fluctuations of these quantities are calculated. Parameter domains in which there exist squeezed states of solitons with respect to each of the collective variables are determined. Comparative analysis of squeezing with respect to these variables for light and dark solitons is performed.

DOI
10.1134/s0030400x1505015x

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Quantum fluctuations of two-dimensional dark dissipative solitons in an interferometer filled with a Kerr nonlinear medium are analyzed theoretically. The boundaries of the stability region of classical solitons excited in the considered interferometer by an external radiation are refined. The solution of the quantum problem is studied in the linear approximation. For this purpose, the formalism of expansion of the sought solution in discrete spectral functions is extended to the two-dimensional case. Using this expansion made it possible to construct a solution that describes fluctuations of collective variables of a two-dimensional soliton, namely, the coordinate of the center and momentum. Mean squares of quantum fluctuations of these quantities are calculated. Parameter domains in which there exist squeezed states of solitons with respect to each of the collective variables are determined. Comparative analysis of squeezing with respect to these variables for light and dark solitons is performed.
Quantum fluctuations of one-dimensional dark dissipative solitons sustained by an external radiation in an interferometer with a Kerr nonlinearity are analyzed theoretically. The stability region of classical solitons in this interferometer is studied. The boundaries of this region are determined, and types of excited solitons are classified. Quantum fluctuations of solitons are analyzed in an approximation linear in fluctuations. This problem was solved by linearizing the quantum Langevin equation in a neighborhood of a classical solution for the main type of a soliton from the obtained stability region. The main attention has been paid to studying quantum fluctuations of collective variables of dissipative solitons, namely, the coordinate of the center and momentum of the soliton. Based on the expansion of solutions of the linearized equation in eigenfunctions of the discrete spectrum of this equation, a solution describing quantum fluctuations of these variables is constructed. Using this expansion scheme made it possible to give a rigorous definition of the dissipative soliton position fluctuation operator. The study performed based on this scheme has made it also possible to construct a solution for a one-dimensional dark relaxing dissipative soliton. This soliton generalizes the stationary soliton with allowance for the shift of its center and deformation of its profile followed by the recovery of its initial shape. Average squares of quantum fluctuations of collective variables are calculated. A domain of parameters in which there exist quantum states of solitons with an initially high degree of squeezing with respect to the momentum is found. It is shown that such states are in correspondence with significantly higher velocities of soliton center drift. An experiment that could detect the relative squeezing with respect to the momentum due to the soliton center drift is discussed.
A quantum Langevin equation is derived that makes it possible to study the radiation field in a large-aperture nonlinear interferometer excited by external classical radiation. This equation is linearized in the vicinity of the solution for a stationary soliton. A mathematical formalism for obtaining a spectral representation of the solution to the linearized problem is constructed. It is shown that, in general, the excitation spectrum of a soliton consists of three branches, two of which belong to a continuous spectrum, while the third branch is discrete. The spectral representation obtained makes it possible to rigorously define the operator of soliton coordinate fluctuations, since, as is shown in the study, the traditional definition of this operator leads to a divergence in the vicinity of the solution. A new type of dissipative soliton is found, which is a natural generalization of a stationary soliton and takes into account its motion. A relation is found between this soliton and the contribution to the solution for field fluctuations from the discrete spectrum expansion. The mean squares of fluctuations of the soliton coordinate and momentum are calculated. A range of parameters is determined where the momentum of the soliton can always be measured with a spread smaller than the standard quantum limit. This possibility is related to the occurrence of states squeezed with respect to the soliton momentum. A scheme is proposed for the experimental observation of these states.
We present a method for the study of quantum fluctuations of dissipative structures forming in nonlinear optical cavities, which we illustrate in the case of a degenerate, type I optical parametric oscillator. The method consists in (i) taking into account explicitly, through a collective variable description, the drift of the dissipative structure caused by the quantum noise, and (ii) expanding the remaining---internal---fluctuations in the biorthonormal basis associated to the linear operator governing the evolution of fluctuations in the linearized Langevin equations. We obtain general expressions for the squeezing and intensity fluctuations spectra. Then we theoretically study the squeezing properties of a special dissipative structure, namely, the bright cavity soliton. After reviewing our previous result that in the linear approximation there is a perfectly squeezed mode irrespective of the values of the system parameters, we study which are the better squeezed modes at the different bifurcations that the cavity soliton can suffer, as well as the squeezing level attainable when the homodyne detection is made with a plane-wave local oscillator field. In this last case, the squeezing level is quite large in most of the existence domain of the cavity soliton. We then take into account the effect of the detector size and find that, in general, squeezing degrades as the detector is made smaller, except for detector sizes around the cavity soliton size, in which case there is an interesting deviation from the general trend.
Abstract An analysis of weak quantum fluctuations of a spatial soliton in a laser with fast saturable absorption in the possible presence of weak holding radiation is carried out. In the c-number representation with adiabatic elimination of atomic variables, the master Heisenberg–Langevin equation is linearized in the vicinity of a stable classical (in the absence of fluctuations) laser soliton. The linearized equation’s solution is constructed on the basis of the spectral expansion in eigenfunctions of the evolution operator unperturbed by fluctuations. The discrete spectrum of this operator is found and used to determine the mean values of the squared fluctuations of the coordinates of the soliton center and its momentum. A comparison is made with similar fluctuations in driven interferometers with Kerr non-linearity of the medium.
Quantum fluctuations of optical solitons in fibers are discussed on the basis of the exact quantum states in the frame of the quantum nonlinear Schr\"odinger equation. Position and momentum fluctuations as well as quadrature phase fluctuations are calculated, and explicit results are expressed in terms of the intrinsic quantum parameters. Soliton wave-packet spreading and phase-diffusion phenomena are clearly demonstrated, and the corresponding squeezing effects are analyzed. In particular, soliton amplitude squeezing is investigated in an experimental setup of a Mach-Zehnder interferometer containing an optical fiber. Coherent states for quantum solitons are discussed with attention given to their dynamical preservability. Compared with previous linearized approaches, the present study may provide a more fundamental treatment to quantum fluctuations of solitons.