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Journal of Applied Physics

Theory of dark optical solitons

J. C. Martinez, Paul M.E. Shutler, A. Toon

Published Aug 15, 2008• 3 citations
Publisher

Abstract

We study the quantum fluctuations about a dark soliton generated in a photon fluid confined in a Kerr-effect nonlinear microcavity. Semiclassical quantization imposes quantization conditions that lend a particle interpretation to the fluctuations. A Bogoliubov–de Gennes analysis complements this view by yielding the single-particle energies and wave functions. The zero modes are found to correspond to the background while the scattering solutions to the notch. We compute the mass associated with the soliton.

DOI
10.1063/1.2970152

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We study the quantum fluctuations about a dark soliton generated in a photon fluid confined in a Kerr-effect nonlinear microcavity. Semiclassical quantization imposes quantization conditions that lend a particle interpretation to the fluctuations. A Bogoliubov–de Gennes analysis complements this view by yielding the single-particle energies and wave functions. The zero modes are found to correspond to the background while the scattering solutions to the notch. We compute the mass associated with the soliton.
Quantum fluctuations of coupled dark solitons in a trapped BosendashndashndashEinstein condensate
The dark soliton solution of the Gross-Pitaevskii equation in one dimension has two parameters that do not change the energy of the solution: the global phase of the condensate wave function and the position of the soliton. These degeneracies appear in the Bogoliubov theory as Bogoliubov modes with zero frequencies and zero norms. These ``zero modes'' cannot be quantized as the usual Bogoliubov quasiparticle harmonic oscillators. They must be treated in a nonperturbative way. In this paper I develop a nonperturbative theory of zero modes. This theory provides a nonperturbative description of quantum phase diffusion and quantum diffusion of soliton position. An initially well localized wave packet for soliton position is predicted to disperse beyond the width of the soliton.
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Quantum depletion from an atomic quasi-one-dimensional Bose-Einstein condensate with a dark soliton is studied in a framework of the Bogoliubov theory. Depletion is dominated by an anomalous mode localized in a notch of the condensate wave function. Depletion in an anomalous mode requires a different treatment than depletion without anomalous modes. In particular, quantum depletion in the Bogoliubov vacuum of the anomalous mode is experimentally irrelevant. A dark soliton is initially prepared in a state with minimal depletion, which is not a stationary state of the Bogoliubov theory. The notch fills up with incoherent atoms depleted from the condensate. For realistic parameters the filling time can be as short as 10 ms.
Dark solitons are thought to always repel so that bound states of dark solitons do not form. We show that quantum fluctuations of the background induce attractive forces between dark optical solitons. We clarify the nature of these fluctuations. Together with the repulsive potential, this attractive force creates a potential similar to that in a diatomic molecule which allows for the formation of bound pairs of dark solitons. Such pairs can be useful in the creation of notch arrays and crystal-like structures.
Abstract Optical solitons classically are stationary solutions of the nonlinear Schrödinger equation. We perform a quantum field theoretic treatment by quantising a linearised fluctuation field around the classical soliton solution which can be seen as providing a background spacetime for the field. The linearised fluctuation modifies the soliton background, which is often neglected, reminiscent of the nondepleted-pump approximation. Going beyond this approximation and by using a number-conserving Bogoliubov approach, we find unstable modes that grow as the soliton propagates. Eventually, these unstable modes induce a considerable (backreaction) effect in the soliton. We calculate the backreaction in the classical field fully analytically in the leading second order. The result is a quadratic local decrease of the soliton photon number in propagation due to the backreaction effect of the unstable mode. Provided the initial pulse is close to the classical soliton solution, the unstable mode contributions always become dominant. We also consider practical scenarios for observing this quantum-induced soliton distortion, in the spectral domain. The backreaction, which we expect to be present in bright and dark, discrete and continuous solitons and other nonlinear pulses plays an important role in future optical analogue gravity experiments, for soliton lasers, and optical communications.