Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: I. One-dimensional dark solitons

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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.