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arXiv:cond-mat/0204230 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum vortices in systems obeying a generalized exclusion principle

G. Kaniadakis, Antonio Maria Scarfone

Published Jul 18, 2001• 2 citations
arXivHTMLPDF

Abstract

The paper deals with a planar particle system obeying a generalized exclusion principle (EP) and governed, in the mean field approximation, by a nonlinear Schrödinger equation. We show that the EP involves a mathematically simple and physically transparent mechanism, which allows the genesis of quantum vortices in the system. We obtain in a closed form the shape of the vortices and investigate its main physical properties.

DOI
10.1103/physreve.64.026106

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The paper deals with a planar particle system obeying a generalized exclusion principle (EP) and governed, in the mean field approximation, by a nonlinear Schrödinger equation. We show that the EP involves a mathematically simple and physically transparent mechanism, which allows the genesis of quantum vortices in the system. We obtain in a closed form the shape of the vortices and investigate its main physical properties.
In this work is studied a many body system obeying to a generalized Exclusion-Inclusion Principle (EIP) originated by collective effect, the dynamics, in mean field approximation, being ruled by a nonlinear Schroedinger equation. The EIP is introduced by a judicious generalization of the particle current. By means of variational principle is obtained a canonical nonlinear Schroedinger equation. We study the Lagrangian and Hamiltonian formulation. We study the symmetries of the nonlinear Schroedinger equation obeying to EIP and by means of Noether theorem we obtain and discuss conserved quantities. Successively, EIP-Schroedinger equation is coupled in a minimal way to an abelian gauge field which dynamics is described by the Maxwell-Chern-Simons Lagrangian. We show that the anyonic statistic behavior ascribed to the system by the Chern-Simons Lagrangian is not destroyed by the presence of EIP potential. Finally, we study special solutions of the system. Applications on the Bose-Einstein condensation and vortex-like solution in the picture of Chern-Simons interaction are considered.
Recently [G. Kaniadakis, Phys. Rev. A 55, 941 (1997)], we introduced a Schr\"odinger equation containing a complex nonlinearity $W(\ensuremath{\rho},\mathbf{j})+i\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ which describes the collective interaction introduced by an exclusion-inclusion principle (EIP). The EIP does not affect $W(\ensuremath{\rho},\mathbf{j})$ and determines $\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ univocally. In the above reference $W(\ensuremath{\rho},\mathbf{j})$ was deduced by means of a stochastic quantization approach, in this way obtaining a noncanonical quantum system. In this work we introduce a family of nonlinearities $W(\ensuremath{\rho},\mathbf{j})$ generating a family of nonlinear canonical quantum systems, and derive their Lagrangian and the Hamiltonian functions and the evolution equations of the fields. We derive also the Ehrenfest relations and study the soliton properties. The shape of the soliton, propagating in the system obeying the EIP, can be obtained by solving a first-order ordinary differential equation. We show that, in the case of soliton solutions, by means of a unitary transformation, the EIP potential is equivalent to a real algebraic nonlinear potential proportional to $\ensuremath{\kappa}{\ensuremath{\rho}}^{2}/(1+\ensuremath{\kappa}\ensuremath{\rho}).$
We consider a nonlinear Schr\"odinger equation for particles obeying an exclusion-inclusion principle recently proposed by us and derive, within the Wigner formalism, a generalized quantum phase-space transport equation. We calculate the $\ensuremath{\Elzxh}$ expansion of this equation and study the quantum corrections, up to the ${\ensuremath{\Elzxh}}^{2}$ order, of the finite temperature Thomas-Fermi phase-space distribution. The effects and the quantitative relevance of the exclusion principle on the distribution function are considered for a system of fermionic particles in a harmonic confining potential.
Quantum vortices are pivotal for understanding of phenomena in quantum hydrodynamics. Vortices were observed in different physical systems like trapped dilute Bose-Einstein condensates, liquid helium, exciton-polariton condensates and other types of systems. Foreign particles attached to the vortices often serve for a visualization of the vortex shape and kinematics in superfluid experiments. Fascinating discoveries were made in the field of cold quantum mixtures, where vortices created in one component may interact with the other component. This works raise the fundamental question of the interaction between quantum vortices and matter. The generalized nonlinear Schrodinger equation based formalism is applied here to model three different processes involving the interaction of quantum vortices with foreign particles: propagation of a fast classical particle in a superfluid under the influence of sound waves, scattering of a single fermion by a quantized vortex line and dynamics of vortex pairs doped with heavy bosonic matter. The obtained results allow to to clarify the details of recent experiments and acquire a better understanding of the multicomponent quantum turbulence.