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arXiv:nlin/0607055 · Journal of Physics A Mathematical and Theoretical

Evolution operators for quantum chains

С. М. Сергеев

Published Feb 6, 2007• 3 citations
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Abstract

Discrete-time evolution operators in integrable quantum lattice models are sometimes more fundamental objects then Hamiltonians. In this paper we study an evolution operator for the one-dimensional integrable q-deformed Bose gas with XXZ-type impurities and find its spectrum. Evolution operators give a new interpretation of known integrable systems, for instance our system describes apparently a simplest laser with a clear resonance peak in the spectrum.

DOI
10.1088/1751-8113/40/8/f03

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Discrete-time evolution operators in integrable quantum lattice models are sometimes more fundamental objects then Hamiltonians. In this paper we study an evolution operator for the one-dimensional integrable q-deformed Bose gas with XXZ-type impurities and find its spectrum. Evolution operators give a new interpretation of known integrable systems, for instance our system describes apparently a simplest laser with a clear resonance peak in the spectrum.
The authors construct the Hamiltonian of a new quantum integrable 'q-boson' lattice model in 1+1 dimensions which has q-bosons as dynamical variables and solve it for its energy eigenstates and energy eigenvalues under periodic boundary conditions of finite period. This model can be regarded as the q-deformation of the integrable lattice Bose gas (the quantum lattice nonlinear Schrodinger (quantum lattice NLS)) model. In appropriate continuum limits both of these lattice models become the quantum repulsive NLS model (the Bose gas). In the lattice case the distinction between the q-bosons and ordinary bosons leads to significant new features-particularly in the energy spectrum for which the q-boson lattice model has an optical branch and bound state spectra as well as the (one) acoustic branch of the quantum lattice NLS model. It is argued that this sort of distinction is generic for discrete systems with a finite number of degrees of freedom as opposed to continuum field theories like the Bose gas where quantization by quantum group quantization or by canonical quantization leads to the same results. Various properties of the q-bosons are given and used. In particular, a q-boson Primakov-Holstein transformation is given which connects the q-boson lattice model with the XXZ models of arbitrary spin.
In this paper we study nonequilibrium dynamics of one dimensional Bose gas from the general perspective of dynamics of integrable systems. After outlining and critically reviewing methods based on inverse scattering transform, intertwining operators, q-deformed objects, and extended dynamical conformal symmetry, we focus on the form-factor based approach. Motivated by possible applications in nonlinear quantum optics and experiments with ultracold atoms, we concentrate on the regime of strong repulsive interactions. We consider dynamical evolution starting from two initial states: a condensate of particles in a state with zero momentum and a condensate of particles in a gaussian wavepacket in real space. Combining the form-factor approach with the method of intertwining operator we develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density-density correlation functions. In both cases we observe a tendency toward formation of crystal-like correlations at intermediate time scales.
In this paper we study nonequilibrium dynamics of one dimensional Bose gas from the general perspective of dynamics of integrable systems. After outlining and critically reviewing methods based on inverse scattering transform, intertwining operators, q-deformed objects, and extended dynamical conformal symmetry, we focus on the form-factor based approach. Motivated by possible applications in nonlinear quantum optics and experiments with ultracold atoms, we concentrate on the regime of strong repulsive interactions. We consider dynamical evolution starting from two initial states: a condensate of particles in a state with zero momentum and a condensate of particles in a gaussian wavepacket in real space. Combining the form-factor approach with the method of intertwining operator we develop a numerical procedure which allows explicit summation over intermediate states and analysis of the time evolution of non-local density-density correlation functions. In both cases we observe a tendency toward formation of crystal-like correlations at intermediate time scales.
A composite quantum model on a lattice which describes the system of q-bosons interacting with U q (su(2)) spin impurities is introduced and solved exactly under periodic boundary conditions. In one limit the model is shown to become a new exactly soluble quantum system on a lattice which can be interpreted as a q-deformed version of the quantum Dicke model. In the limit of infinitesimal spacing the model is further reduced to a multilevel version of the previously introduced continuum-limit Dicke model. For spin 1/2 the previous results for this particular case are confirmed.