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Physical Review A

Agreement of stochastic soliton formalism with second-quantized and configuration-space models

John M. Fini, Peter L. Hagelstein, H. A. Haus

Published Jun 1, 1998• 8 citations
Publisher

Abstract

The stochastic theory presented by Drummond, Gardiner, and Walls [Phys. Rev. A 24, 914 (1981)] is an interesting approach to problems in quantum optics. In this theory, an exact, quantum evolution is written in terms of classical functions (not operators) driven by explicit, quantum noise. We examine the origin of uncertainty in the formalism through the simple example of a single, nonlinear oscillator. We then test the stochastic theory applied to the problem of soliton propagation. We extend the linearized stochastic model by computing analytically quantum uncertainties in the four basic soliton parameters: photon number, momentum, phase, and position. Agreement with second-quantized and configuration-space soliton theories verifies the stochastic formalism.

DOI
10.1103/physreva.57.4842

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The stochastic theory presented by Drummond, Gardiner, and Walls [Phys. Rev. A 24, 914 (1981)] is an interesting approach to problems in quantum optics. In this theory, an exact, quantum evolution is written in terms of classical functions (not operators) driven by explicit, quantum noise. We examine the origin of uncertainty in the formalism through the simple example of a single, nonlinear oscillator. We then test the stochastic theory applied to the problem of soliton propagation. We extend the linearized stochastic model by computing analytically quantum uncertainties in the four basic soliton parameters: photon number, momentum, phase, and position. Agreement with second-quantized and configuration-space soliton theories verifies the stochastic formalism.
Making use of c-number stochastic theory and soliton perturbation theory, we study the quantum fluctuations of a self-induced transparency (SIT) soliton propagating through a lossless two-level medium. Considering the fluctuations as small corrections to the classical soliton, we are able to construct and solve four stochastic equations that govern the evolution of four soliton parameters: photon number (intensity), phase, timing, and momentum (frequency). We find excellent agreement between our stochastic theory of SIT solitons and the second-quantized theory of Lai Haus [Phys. Rev. A42, 2925 (1990)].
A linearized perturbation quantum theory of second-order soliton propagation is developed. The theory shows that the quantum fluctuations of photon number, phase, momentum, and position at an arbitrary propagation distance are linear combinations of these fluctuations at zero distance. The evolutions of second-order soliton quantum fluctuations are evaluated and compared with the quantum-fluctuation evolutions of a fundamental soliton. Based on this theory, the squeezing effect of a second-order soliton is studied. It is shown that, like a fundamental soliton, a second-order soliton also exhibits squeezing along propagation when a proper combination of the number and phase operators is detected.
A linearized quantum theory of soliton squeezing and detection is presented. The linearization reduces the quantum problem to a classical one. The classical formulation provides physical insight. It is shown that a quantized soliton exhibits uncertainties in photon number and phase, position (time), and momentum (frequency). Detectors for the measurement of all four operators are discussed. The squeezing of the soliton in the fiber is analyzed. An optimal homodyne detector for detection of the squeezing is presented that suppresses the noise associated with the continuum and the uncertainties in position and momentum.
We investigate the noise in a fundamental soliton state constructed from the exact energy eigenstates of the quantized nonlinear Schr\"odinger equation. It is shown that such a state has statistical properties that are substantially different from those of a coherent state with the same mean field. We can construct, for example, a fundamental soliton state that has initially minimum fluctuations for the four linearized soliton operators (particle number, phase, momentum, and position). We also show that the linearized noise analysis is not valid for nonlinear phase shifts larger than n01/4{\mathit{n}}_{0}^{1/4}n01/4​, where n0{\mathit{n}}_{0}n0​ is the average number of particles in the fundamental soliton. \textcopyright{} 1996 The American Physical Society.
Quantization of optical solitons is discussed, using the nonlinear Fourier-transform (inverse-scattering) method. A quantum soliton is described in terms of two sets of conjugate variables such as the photon number and the phase, and the momentum and the center position. The theory of a soliton collision is extended to describe the quantum-nondemolition measurement of soliton photon number and momentum. These two variables are indeed quantum-nondemolition observables and can be measured by means of the phase and the center position of the probe soliton. It is demonstrated that the measurement error and backaction noise on the conjugate variable satisfy an uncertainty product.
Starting with the projective interpretation of the Hilbert space special stochastic representation of the wave function in Quantum Mechanics (QM), based on soliton realization of extended particles, is considered with the aim to model quantum states via classical computer. Entangled solitons construction having been earlier introduced in the nonlinear spinor field model for the calculation of the Einstein-Podolsky-Rosen (EPR) spin correlation for the spin-1/2 particles in the singlet state, the other example is now studied. The latter concerns the entangled envelope solitons in Kerr dielectric with cubic nonlinearity, where we use two-solitons configurations for modeling the entangled states of photons. Finally, the concept of stochastic qubits is used for quantum computing modeling.