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arXiv:quant-ph/9907014 · International Journal of Modern Physics B

q-SYMMETRIES IN DNLS-AL CHAINS AND EXACT SOLUTIONS OF QUANTUM DIMERS

Demosthenes Ellinas, Panagiotis Maniadis

Published Oct 20, 1999• 1 citation
arXivHTMLPDF

Abstract

Dynamical symmetries of Hamiltonians quantized models of discrete nonlinear Schrödinger chain (DNLS) and of Ablowitz–Ladik chain (AL) are studied. It is shown that for n-sites the dynamical algebra of DNLS Hamilton operator is given by the su(n) algebra, while the respective symmetry for the AL case is the quantum algebra suq(n). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site. Invariants of motions are found in terms of Casimir central elements of su(n) and suq(n) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the n=2 quantum dimer case and formulate the eigenvalue problem of each dimer as a nonlinear (q)-spin model. Analytic investigations of the ensuing three-term nonlinear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined. The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of nonlinearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the nonlinearity parameter near the classical bifurcation point.

DOI
10.1142/s0217979299002873

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Dynamical symmetries of Hamiltonians quantized models of discrete nonlinear Schrödinger chain (DNLS) and of Ablowitz–Ladik chain (AL) are studied. It is shown that for n-sites the dynamical algebra of DNLS Hamilton operator is given by the su(n) algebra, while the respective symmetry for the AL case is the quantum algebra suq(n). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site. Invariants of motions are found in terms of Casimir central elements of su(n) and suq(n) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the n=2 quantum dimer case and formulate the eigenvalue problem of each dimer as a nonlinear (q)-spin model. Analytic investigations of the ensuing three-term nonlinear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined. The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of nonlinearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the nonlinearity parameter near the classical bifurcation point.
The investigation of the dynamics of nonlinear systems becomes a hot topic in modern science and engineering. One of the most challenging problem is the quantum evolution of nonlinear excitations which has many applications in condensed matter physics, quantum optics, quantum information and other areas of physics. Many different methods were suggested and implemented in this direction, both analytical and numerical. Among the most fruitful approaches are the so called phase space representations of quantum dynamics. In this method the evolution equation for quantum density operator is converted to the partial differential equation for some classical quasi-distribution function, which in some cases can be reduced to the system of ordinary stochastic differential equations. In this paper we are presenting the results of the investigation of quantum Ablowitz-Ladik model by the phase space representation methods. The classical Ablowitz-Ladik equation is the integrable discretization of Nonlinear Schrodinger Equation. The quantum dynamics of solitons and their interaction will be discussed, based on the results of the numerical simulation. The possibility of the quantum entanglement and squeezing in quantum Ablowitz-Ladik model will be investigated.
We show that the one-dimensional discrete nonlinear Schrödinger chain (DNLS) at finite temperature has three different dynamical regimes (ultralow-, low-, and high-temperature regimes). This has been established via (i) one-point macroscopic thermodynamic observables (temperature T, energy density ε, and the relationship between them), (ii) emergence and disappearance of an additional almost conserved quantity (total phase difference), and (iii) classical out-of-time-ordered correlators and related quantities (butterfly speed and Lyapunov exponents). The crossover temperatures T_{l-ul} (between low- and ultra-low-temperature regimes) and T_{h-l} (between the high- and low-temperature regimes) extracted from these three different approaches are consistent with each other. The analysis presented here is an important step forward toward the understanding of DNLS which is ubiquitous in many fields and has a nonseparable Hamiltonian form. Our work also shows that the different methods used here can serve as important tools to identify dynamical regimes in other interacting many-body systems.
The Hamiltonian of one-dimensional chain of n-level atoms is represented in terms of Boson operators by using the Dyson-Maleev transformation and it is shown that the finite-ladder effect disappears when n tends toward infinity. In this way, it is found that the Heisenberg equation of motion of this system is exactly described in the coherent state representation by the dark discrete nonlinear Schrödinger (DNLS) equation. It is also briefly shown that the DNLS equation has some general soliton solutions. This indicates that this simple system has richness of nonlinear waves.
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.
We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su q (2) algebra. The corresponding matrix elements of finite rotations (the q -deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su q (2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.