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Physical Review A

Nonperturbative studies of a quantum higher-order nonlinear Schrödinger model using the Bethe ansatz

Alexander Shnirman, Boris A. Malomed, Eshel Ben‐Jacob

Published Oct 1, 1994• 35 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.50.3453

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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.
We have reformulated the Bethe Ansatz approach to a Derivative Nonlinear Schrödinger equation, known to be a non-ultralocal system. In this present version we have shown that an approach exactly similar to the usual NLS case is possible. Furthermore the Yang-Baxter equation and factorization is shown to be valid. Afterwards the R (μ, v ) matrix has been constructed from the Bethe Ansatz result (which is impossible otherwise, as the Faddeev approach is not applicable) and the commutation rules for the scattering data is deduced. These are then utilised to construct the algebraic Bethe ansatz eigen states. Next the periodicity condition is invoked and the eigenvalues are determined via some algebraic equations which are also seen to be reproduced via the algebraic approach. Lastly we consider the possibility of exceptional states in the algebraic Bethe states.
The authors continue their analysis of the extended derivative nonlinear Schrodinger equation with the help of the Bethe-ansatz technique. In the previous analysis by the QISM approach it was not possible to consider the reduction of the extended system to the derivative NLSE, as the theory then becomes non-ultralocal. Therefore, they have applied the approach of the Bethe ansatz to both the extended DNLSE and the usual DNLSE, and show that it is possible to construct multiparticle quantum states in both the cases.
We have formulated the quantum inverse problem for an alternative version of the nonlinear Schr\"odinger equation (NLS), which is a variant of the derivative NLS (DNLS) equation. While the DNLS equation is nonultralocal and cannot be formulated as a problem of the quantum inverse scattering method, our equation is ultralocal, and hence it is possible to construct the quantum R matrix and the commutation rules for the scattering data. Next we have constructed the algebraic Bethe ansatz, from which the eigenvalue equation for the excited states is derived. This eigenvalue equation admits stringlike configurations, and the density of states corresponding to real and complex eigenvalues is explicitly derived in the thermodynamic limit.
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 quantum inverse scattering method relates the Bethe Ansatz to the theory of classical completely integrable differential equations. These are sometimes called soliton equations. The modern way to solve them is called the classical inverse scattering method. In a sense this is a nonlinear generalization of the Fourier transform. In this Part the quantum inverse scattering method is expounded. The main statements of the classical inverse scattering method necessary for quantization are given in Chapter V where the Lax representation is introduced. The Hamiltonian structure of integrable models is also discussed along with the infinite number of integrals of motion. The most convenient method of analyzing the Hamiltonian structure relies on the classical r -matrix. Some concrete models will be considered. Chapter VI is devoted in particular to the quantum inverse scattering method. The R-matrix, which is the main object of this method, is introduced. The Yang-Baxter equation for the R-matrix is discussed. The main statements of the method are given and a number of examples are presented. The algebraic formulation of the Bethe Ansatz, one of the main achievements of the quantum inverse scattering method, is presented in Chapter VII. The notion of the determinant of the transition matrix in the quantum case is introduced in this chapter. (This is closely related to the concept of the antipode in quantum groups.)
We study differential and integral relations for the quantum Jost solutions associated with an integrable derivative nonlinear Schrodinger (DNLS) model. By using commutation relations between such Jost solutions and the basic field operators of DNLS model, we explicitly construct first few quantum conserved quantities of this system including its Hamiltonian. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This modified coupling constant plays a crucial role in our comparison between the results of algebraic and coordinate Bethe ansatz for the case of DNLS model. We also find out the range of modified coupling constant for which the quantum NNN-soliton state of DNLS model has a positive binding energy.