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Physical Review A

Quantum theory of self-induced transparency solitons: A linearization approach

Yinchieh Lai, H. A. Haus

Published Sep 1, 1990• 35 citations
Publisher

Abstract

Linearization of the self-induced transparency equations reduces the problem to a single linear integro-differential equation for the perturbation field of a soliton. The adjoint system is introduced to establish orthogonality relations that lead to the relations between the perturbed field and perturbed scattering data. The commutation relations of the perturbation field provide the quantization rules for the perturbed ``scattering data.'' The quantized solitons are described by operators for photon number, phase, momentum, and position. The evolution and detection of these fluctuations are studied. The contribution of the continuum part can be suppressed by suitable detection schemes.

DOI
10.1103/physreva.42.2925

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Linearization of the self-induced transparency equations reduces the problem to a single linear integro-differential equation for the perturbation field of a soliton. The adjoint system is introduced to establish orthogonality relations that lead to the relations between the perturbed field and perturbed scattering data. The commutation relations of the perturbation field provide the quantization rules for the perturbed ``scattering data.'' The quantized solitons are described by operators for photon number, phase, momentum, and position. The evolution and detection of these fluctuations are studied. The contribution of the continuum part can be suppressed by suitable detection schemes.
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 general quantum theory of self-induced transparency (SIT) solitons is developed with nonlinear quantum effects of atoms taken into account. The quantization starts from the coarse-grain-averaged light-atom interaction Hamiltonian by which the quantum effects of ensemble atoms are modeled. The calculation of quantum properties is performed by the backpropagation method, which takes into account the contribution of the field continuum parts and the atomic fluctuations. The detectable squeezing ratio with a general homodyne local oscillator can be simulated for practical experimental realizations. Schemes for generating quantum correlation through SIT soliton interaction are also proposed and analyzed.
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.
Solitons on fibers are ideal for all optical switching operations.1 Squeezed solitons promise to provide improved measurement capabilities.2 The self-induced transparency (SIT) in CdS has been proposed as a particularly convenient means of squeezing.3 For this reason it is important to understand the quantization problem of solitons. Several authors have attacked the problem in the case of the nonlinear Schrodinger equation (NSE) both numerically4 and analytically.5 A particularly simple approach to quantization can be worked out in the limit of a linearized approximation. Since all switching and squeezing operations employ pulses with many photons (n > 10), the linearization is always legitimate.
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.
Based on the linearization approximation and the conservation of commutation brackets, a general, self-consistent scheme is developed to quantize nonlinear optical-pulse propagation problems. A general computation procedure is developed to calculate the quantum uncertainties of the inner product between any given function and the (perturbed) field operator. As an illustration, a self-consistent quantum theory of the self-Raman effect in optical fibers is presented. The influence of the self-Raman effect on soliton squeezing is examined.