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Quantum Theory of Soliton Squeezing: Time-Dependent Hartree Approximation

Bojun Yang

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

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.

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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.
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.
We study the quantum-noise properties of spectrally filtered solitons in optical fibers. Perturbation theory, including a quantum description of the continuum, is used to derive a complete analytical expression for the second-order correlator of the amplitude quadrature. This correlator is subsequently used to optimize the frequency response of the filter numerically in order to achieve the minimum photon-number noise. For propagation distances up to three soliton periods, the length at which the best noise reduction occurs, a square filter is found to be approximately optimum. For longer distances, more-complicated filter shapes are predicted for the best noise reduction.
We study the time-domain quantum-noise properties of solitons in optical fibers. Perturbation theory, including quantum description of the continuum, is used to derive a complete analytical expression for the second-order correlator of the amplitude quadrature in time domain. This correlator is subsequently used to calculate the photon-number noise reduction below the standard quantum limit by means of a time-gating filter. For propagation distances up to three soliton periods, the length at which the best noise reduction occurs, a square band-reject time filter is found to produce approximately 3.2 dB of squeezing.
We develop a quantum theory of propagation in dispersive nonlinear media from the foundations of a correctly quantized field theory. Quantum fluctuations are handled by coherent-state expansions of localized field states. A stochastic nonlinear Schrödinger equation in the field variables is obtained for media with an intensity-dependent refractive index. This predicts squeezing for a continuous-wave input, over a wide bandwidth with anomalous dispersion, and over a gradually reducing bandwidth with normal dispersion. The equation is easily modified to include thermal-noise sources as well. For solitons, fluctuations are reduced over the soliton bandwidth. This leads to quantum solitons that have quadrature fluctuations less than the level of vacuum fluctuations. The complementary quadrature has a correspondingly increased fluctuation level.