Effect of quantum fluctuations on soliton regimes in microlasers

We present a theoretical investigation of effect of quantum fluctuations on laser solitons. Derivation of the stochastic equation, linearized with respect to quantum perturbations is carried out and the solutions are found. Explicit expressions are obtained for the time dependence of the soliton coordinates and momentum dispersion (variance) for perturbations averaged over the reservoir. It is shown that the dispersion of the soliton momentum becomes constant. It is shown that the dispersion of quantum perturbations tends to infinity near the Andronov-Hopf bifurcation threshold. The magnitude of quantum perturbations near the threshold of the appearance of hysteresis is estimated. It is shown that quantum perturbations do not significantly noise the soliton profile even with a very low intensity tending to zero. The number of photons in such solitons without supporting radiation, can reach unity.


Introduction
The effect of quantum fluctuations on the position and spectral characteristics of solitons in microlasers is one of the topical problems in set-up experiments [1,2], the purpose of which is to use them in optical memory circuits, [3]. These efforts are focused on the use of conservative solitons with Kerr nonlinearity. However, the properties of conservative solitons, even in the classical limit, imply a drift of both spectral and spatial characteristics for an arbitrarily small perturbation, not necessarily quantum. Recently, a general theory has been developed that describes the effect of quantum fluctuations on the position and spectrum of laser solitons, the equilibrium and stability of which is associated with dissipative nonlinearity at the centre of the lasing line, see [4,5]. The development of alternative schemes for controlling solitons based on dissipative nonlinearity, which makes it possible to more accurately control the discrete spectrum of their characteristics, can also provide the prospect of their miniaturization. In our works [6,7], direct numerical simulation of two-dimensional laser solitons was carried out taking into account quantum pumping fluctuations. It is shown that quantum fluctuations do not significantly change the stability region of laser solitons, and the use of a supporting beam is possible down to very low intensities, six orders of magnitude less than the saturation intensity of the passive medium, which makes it possible to control the drift of the soliton position and phase. Controlling the frequency and amplitude of the small supporting radiation allows you to switch between different states of stable vortex combinations. Direct numerical simulation within the framework of solving the stochastic equation shows the stability of two-dimensional solitons against quantum pumping perturbations, including the stability of the trajectories of the classical motion of a soliton under the condition of the presence of coherent sustaining radiation.
The amplitude of the dimensionless stochastic force g E Q χ is specified by the reservoir-averaged values of the correlators 2 E Q , [8], which are proportional to the intensity, and by the dimensionless

Dispersion or variance of quantum fluctuations
A sufficiently small value of quantum perturbations allows one to obtain analytically the time dependence of the variance of the coordinate and momentum of a soliton. For this, the initial stochastic equation was linearized, and its solution was averaged over the reservoir of perturbations. The temporal asymptotes of the variances are different. If for the soliton coordinate it grows linearly (which indicates the presence of uncompensated Brownian motion of the soliton as a whole), then the dispersion of the momentum becomes constant. The latter circumstance can be used to realize the effect of quantum compression of the momentum perturbations of a dissipative soliton, see [8] and [9]. We consider weak fluctuations of the field in the vicinity of the classical soliton: , . Linearization of equation (1) where is a vector of functions, and a vector of stochastic sources: dissipative damping factor and detuning in the linear approximation: Finally, the matrix of differential operators depends from soliton intensity ( ) S I x : The fundamental difference between the type of optical nonlinearity in the considered laser and the case of an interferometer with Kerr nonlinearity lies in the dissipative character of the media with nonlinear laser amplification and absorption. In view of this, in our case, a number of symmetries are absent, indicated in [9] for an interferometer with nonlinearity of only the refractive index. At the same time, symmetry (2) is preserved to coordinate inversion x x → − .
Following [8], we will seek solution (2) by expanding in eigenfunctions of the homogeneous equation corresponding to (2). An eigenfunction with an eigenvalue has the form Here Then for the operator L and the Hermitian conjugate operator † Whence it follows that for each eigenvalue L there is a second eigenvector: When Re 0 λ = these two vectors coincide, † U U λ λ = . Unlike [9], where the medium possessed only "conservative" Kerr nonlinearity, the operator 3 L σ is not Hermitian, therefore its eigenvectors are not orthogonal (here the Pauli matrixes 1,3 σ are used). Figure 1 shows the spatial distributions of the amplitude (figure 1a-d) and phase (figure 1e-h) of the classical fundamental soliton (figure 1a, e) and the components of an odd eigenfunction with eigenvalues. It can be seen from figure 1a that the sustaining radiation weakly changes the amplitude profile in the central region of the soliton. The eigenfunctions of the discrete spectrum are localized and exponentially decrease at the periphery of the soliton, while the phase in this region increases linearly with distance from the center. For a laser with its dissipative nonlinearity, the relationship between the eigenvalues of the operators L and † L , which is valid for an interferometer with Kerr nonlinearity, is violated. This circumstance decreases the rate of damping of perturbations of a stable soliton, and when crossing the boundary of the region of its stability due to changes in the parameters of the scheme, the growth of perturbations with simultaneous oscillations with frequency 1 ω serves as a sign of Andronov-Hopf bifurcation. The approach developed in [8] makes it possible to find the dynamic changes of the perturbation of the coordinates of the center x δ and momentum p δ of a soliton, which can be defined as follows: When deriving (9), (10), we used definitions for scalar products of eigenvectors, which are no longer orthogonal to each other ( )

The number of photons in a soliton near the bistability threshold
Let us analyze the possibility of realizing solitons with a small number of photons. This possibility appears if the value of the absorption coefficient of a weak signal is chosen near the threshold of the appearance of hysteresis in a "free" laser without a supporting beam. In this case, the bistability interval of two homogeneous solutions (without generation and with the intensity of stable generation) is compressed to a point and the intensity of stable generation tends to zero. Bistability threshold: → . The soliton is stable in the interval within the hysteresis region; therefore, estimating the number of photons in the soliton, we can assume that its intensity is ( )