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Physical Review A

Quantum theory of solitons in optical fibers. II. Exact solution

Yinchieh Lai, H. A. Haus

Published Jul 1, 1989• 207 citations

Abstract

In the preceding paper [paper I of a two-part study; Lai and Haus, Phys. Rev. A 40, 844 (1989)] we have used the time-dependent Hartree approximation to solve the quantum nonlinear Schr\"odinger equation. In the present paper, the eigenstates of the Hamiltonian are constructed exactly by Bethe's ansatz method and are superimposed to construct exact soliton states. Both fundamental and higher-order soliton states are constructed and their mean fields are calculated. The quantum effects of soliton propagation and soliton collisions are studied in the framework of this construction. It is shown that a soliton experiences dispersion as well as phase spreading. The magnitude of this dispersion is estimated and is shown to be very small when the average photon number of the soliton is much larger than unity. The phase and position shifts due to a collision and the uncertainty of these shifts are also calculated.

DOI: 10.1103/physreva.40.854

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In the preceding paper [paper I of a two-part study; Lai and Haus, Phys. Rev. A 40, 844 (1989)] we have used the time-dependent Hartree approximation to solve the quantum nonlinear Schr"odinger equation. In the present paper, the eigenstates of the Hamiltonian are constructed exactly by Bethe's ansatz method and are superimposed to construct exact soliton states. Both fundamental and higher-order soliton states are constructed and their mean fields are calculated. The quantum effects of soliton propagation and soliton collisions are studied in the framework of this construction. It is shown that a soliton experiences dispersion as well as phase spreading. The magnitude of this dispersion is estimated and is shown to be very small when the average photon number of the soliton is much larger than unity. The phase and position shifts due to a collision and the uncertainty of these shifts are also calculated.

The annihilation and creation operators of photons in solitons are defined (according) to the features of solitons in this paper. And then the operators of amplitudes for the mode's two quadrature phases are given. Using the time-dependent Hartree approximation, bound states can be constructed to satisfy the quantum nonlinear Schrdinger equation. Describing the propagation of solitons in fibers by means of soliton states which are superimposed by the approximation bound states, we find solitons will present squeezed states periodically.

The quantum theory of pulse propagation in a nonlinear optical fiber is presented using the time-dependent Hartree approximation. This formulation clarifies the connections between the quantum theory of soliton propagation and single-mode theories that have been used to describe the effects of self-phase modulation. An approximate solution is obtained for coherent-state soliton pulses that gives excellent agreement with numerical calculations for the quadrature phase amplitudes of the field. These amplitudes are found to undergo a series of collapses and revivals with propagation; the first collapse is related to the appearance of interference fringes in the field Q function.

The dynamics of a quantum soliton propagating in nonlinear optical fibers is analyzed in the second-quantized picture in a fully canonical way. The quantum nonlinear Schr\"odinger equation with equal-time commutation relations for the field operators is solved using the formalism developed by Haus and Lai [J. Opt. Soc. Am. B

7,

386 (1990)]. The canonical approach is compared with the treatment of Haus and Lai based on phenomenological equal-space commutation relations. The two theories are shown to provide similar outcomes of a homodyne measurement. This result justifies the use of equal-space commutation relations in quantum studies of soliton propagation.

Quantum fluctuations of optical solitons in fibers are discussed on the basis of the exact quantum states in the frame of the quantum nonlinear Schr\"odinger equation. Position and momentum fluctuations as well as quadrature phase fluctuations are calculated, and explicit results are expressed in terms of the intrinsic quantum parameters. Soliton wave-packet spreading and phase-diffusion phenomena are clearly demonstrated, and the corresponding squeezing effects are analyzed. In particular, soliton amplitude squeezing is investigated in an experimental setup of a Mach-Zehnder interferometer containing an optical fiber. Coherent states for quantum solitons are discussed with attention given to their dynamical preservability. Compared with previous linearized approaches, the present study may provide a more fundamental treatment to quantum fluctuations of solitons.