## Bose-Einstein condensation of atomic gases; pair excitation

A significant advance in physics in 1995 was the first experimental observation
at MIT and at the
University of Colorado and JILA of Bose-Einstein condensation (BEC) in dilute atomic gases.
In BEC particles of integer spin, called bosons,
occupy the same quantum state macroscopically at sufficiently low temperatures.
The gases in these recent experiments
consist of weakly interacting atoms trapped by external
potentials, which made it possible to cool the atoms
down to extremely
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 translation
invariance and hence cannot
be treated conveniently via
the momentum representation.

A continuum description of the trapped atomic gases usually relies on
a cubic nonlinear Schrödinger-type (``Gross-Pitaevskii'') equation for the macroscopic wave function.
This PDE,
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, and computing the condensate depletion, i.e., the fraction of particles out of the lowest single-particle state.

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-partial-differential, therefore including nonlocal effects. The kernel responsible for such nonlocal effects is not known a priori but must be computed as part of the solution.

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Research leading to some of the publications listed on this page was
supported by the National Science Foundation through grants
DMS-1517162 (2015-18), DMS-0847587 (2009-14)
and DMR-0520471 (2007-09).
### Topics and papers:

Effect of spatially varying scattering length on fraction of particles depleting the condensate:
Derivation of equations of motion from microscopic Hamiltonian; and periodic homogenization.
#### Related papers:

1. D. Margetis (2012),
Bose-Einstein condensation beyond mean field:
Many-body bound state of periodic microstructure,
(SIAM) Multiscale Modeling & Simulation, Vol. 10, pp. 383-417;
Erratum (2013), Vol. 11, p. 410.

2. 1. D. Margetis,
``Bose-Einstein condensation at finite temperatures: Mean field laws with periodic microstructure''
(PDF), submitted to (SIAM) Multiscale Model. Simul. (26pp).

Rigorous study of corrections to mean field limit
for evolution of Boson system via coherent states (inspired by works of T.T. Wu, and I. Rodnianski and B. Schlein).
#### Related papers:

1. M. G. Grillakis, M. Machedon, and D. Margetis (2010),
Second-order corrections to mean field evolution of weakly interacting Bosons. I.(PDF),
29pp,
Communications in Mathematical Physics, Vol. 294, pp. 273-301.

2. M. G. Grillakis,
M. Machedon, and D. Margetis (2011),
``Second-order corrections to mean field evolution of weakly interacting Bosons. II.''
Advances in Mathematics, Vol. 228, pp. 1788-1815.

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
hi

erarchies for appropriate correlation functions.
#### Related paper:

1. M. G. Grillakis and D. Margetis (2008),
A priori estimates for
many-body Hamiltonian evolution of interacting Boson system
(PDF),
Journal of Hyperbolic Differential Equations, Vol. 5, pp. 857-883.

Particular solutions of integro-differential equation for the pair-excitation
function at extremely low temperatures:
#### Related papers:

1. D. Margetis (2008),
Solvable model for
pair excitation
in trapped Boson gas at zero temperature
(PDF),
Journal of Physics A: Mathematical and Theoretical, Vol. 41, art. 385002, pp. 1-18;
*Corrigendum*, J. Phys. A: Math. Theor., Vol. 41, 459801, 1p.

Connections of the 2nd Painleve transcendent to ``Josephson-type'' atomic currents
flowing between traps:
#### Related papers:

1. D. Margetis (2000),
Asymptotic formula for the condensate wave function of a trapped Bose gas
(PDF),
Phys. Rev. A, Vol. 61, art. 055601, pp. 1-2.

Solitary-wave solutions of the equations of motion of trapped atomic gases at extremely
low temperatures:
#### Related papers:

1. D. Margetis (1999),
Bose-Einstein condensation in an external potential at zero temperature:
Solitary-wave theory
(PDF), J. Math. Phys., Vol. 40, pp. 5522-5543.