Bose-Einstein condensation of atomic gases; pair excitation
A significant advance in physics in 1995 was the first experimental observation
at MIT, University of Colorado and JILA of Bose-Einstein condensation -- in which particles of integer spin, called bosons,
occupy the same quantum state macroscopically at sufficiently low temperatures -- in dilute atomic gases.
The gases in these experiments
consist of weakly interacting atoms trapped by external
potentials, which made it possible to cool them down at low temperatures, of the order of nanodegrees Kelvin.
Because of the weak interparticle interactions, a systematic theoretical treatment is possible. On the other hand,
because of the presence of traps, the system of particles lacks translational invariance and cannot
be treated conveniently via
the momentum representation.
A continuum description of the trapped atomic gases usually relies on
a cubic nonlinear Schrödinger-type equation for the macroscopic wave function.
This equation, although adequate for many experimental situations, fails to
describe pair excitations, by which two particles initially occupying the same, single-particle macroscopic quantum state
scatter from each other at different positions. Pair creation is particularly important
in understanding phonon excitations.
My research has focused on the derivation and study of macroscopic equations of motion for dilute atomic gases that go beyond the ``mean-field''
description of the nonlinear Schrödinger-type equation. My purpose is to give a systematic,
analytical description of particle correlations. In particular, I am interested
in deriving continuum equations for extremely low and finite temperatures from
the microscopic Hamiltonian of the system with inclusion of pair
excitations, and studying particular solutions of these equations in an effort to quantify
the time scales involved. In the simplest nontrivial case, the equations are integro-differential.
Topics and papers:
Study of hydrodynamic aspects of many-particle interacting Boson system.
A goal is to connect
analytically the description of many-body effects such as pair-excitation to BBKGY-type
hierarchies for appropriate correlation functions.
Related paper:
1. M. G. Grillakis and D. Margetis,
Apriori estimates for
many-body Hamiltonian evolution of interacting Boson system
(PDF),
Journal of Hyperbolic Differential Equations, 26 pages, accepted for publication, 2008.
Particular solutions of integro-differential equation for the pair-excitation
function at extremely low temperatures:
Related papers:
1. D. Margetis,
Solvable model for
pair excitation
in trapped Boson gas at zero temperature
(PDF), 20 pages, submitted to Journal of Physics A: Mathematical and Theoretical,
2008.
Derivation and analysis of continuum equations at finite temperatures from the N-body Hamiltonian;
coupled integro-differential equations for macroscopic wave function and
pair-excitation function:
Related papers:
1. D. Margetis and T. T. Wu,
Bose-Einstein condensation in an external potential at finite temperatures
(PDF), preprint.
Connections of the 2nd Painleve transcendent to ``Josephson-type'' atomic currents
flowing between traps:
Related papers:
1. D. Margetis,
Asymptotic formula for the condensate wave function of a trapped Bose gas
(PDF),
Phys. Rev. A, Vol. 61, art. 055601, pp. 1-2 (2000).
Solitary-wave solutions of the equations of motion of trapped atomic gases at extremely
low temperatures:
Related papers:
1. D. Margetis,
Bose-Einstein condensation in an external potential at zero temperature:
Solitary-wave theory
(PDF), J. Math. Phys., Vol. 40, pp. 5522-5543 (1999).