The quantum nonlinear schrödinger equation (QNSE) is exactly solved by Beth's ansatz method and we give a reasonable definition of the quantum soliton states. From the definition we construct the soliton states of the QNSE from its bound-state solutions. The dispersion effect of the quantum soliton is also exactly analysed.
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 consider the integrability problem for a quantum version of the perturbed nonlinear Schr\"odinger (NS) equation, including a higher spatial dispersion and nonlinear dispersion of the group velocity (the corresponding classical equations are well known in the nonlinear fiber optics and in other applications). Employing the Bethe ansatz (BA) technique, which is known to yield a complete spectrum including the so-called quantum solitons (multiparticle bound states) of the unperturbed NS system, we find that the particular cases of the model which correspond to integrable classical equations, viz., the derivative NS and Hirota equations, are also fully integrable at the quantum level. In the generic (nonintegrable) case, the model remains integrable in the two-particle sector. In the three-particle sector, the BA produces unphysical states with complex energy. It turns out that the Hamiltonian in this case becomes non-Hermitian. We propose a procedure for finding the physical eigenstates of the system. We build an example of such a state and we show that it describes inelastic scattering.
The Wigner function, a kind of wavelet transform is studied for one- and two-soliton solutions to the nonlinear Schrödinger equation. The exact expression of the Wigner transform for the one-soliton solution is obtained. The features of soliton-soliton interaction of different two-soliton states, the states of coherence analogous to that of two-localized wave packets in quantum mechanics, are displayed. Suitability of this transform for analyzing “ soliton ” in open systems is also discussed.
The Hamiltonian of the process of cascaded second harmonic generation is found from Maxwell equations. In the double-gap model and under rotating-wave and effective-mass approximations, it is quantized and the generalized quantum nonlinear Shrödinger equation (GQNSE) is obtained. Tri-photon and quadri-photon bound states are found based on general solutions of GQNSE solved via Bethe's Ansatz method. Quantum parametric gap soliton (QPGS) solution is constructed consequently, and the existence of the double-gap QPGS is predicted for the first time.