Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

Chapter 3
Introduction
Since the achievement of Bose-Einstein condensation in trapped ultra-cold dilute atomic gases
in 1995 [2, 3, 4], condensates have proven to be extraordinarily robust samples for the study
of a wide class of phenomena [1, 15]. A great advantage of these systems is provided by
the fact that they are very well isolated and can be controlled and manipulated with high
precision by means of electromagnetic fields. This opens up the possibility to design external
potentials which change the statics and the dynamics of the gas in a well-defined manner and
offer new ways of controlling its properties. Regular lattice potentials produced by light fields
provide a prominent example. They allow for the external control of the effect of interactions,
the transport properties and the dimensionality of the sample. Situations well known from
solid state and condensed matter physics can be mimicked and new types of systems can be
engineered.
The tailoring of optical potentials of various forms in space and time is based on the
efficient exploitation of the interaction of atoms with laser fields. In the dipole approximation,
the interaction Hamiltonian is given by
ˆV (r,t)=−ˆ d · E(r,t) , (3.1)
where ˆ d is the electric dipole operator of an atom and
E(r,t)=E(r)e −iωt + c.c. (3.2)
is a classical time-dependent electric field oscillating with frequency ω. The energy change
of each atom associated with the dipolar polarization can be accounted for by the effective
potential
V (r) =− 1
2 α(ω)〈E2 (r,t)〉 , (3.3)
where α(ω) is the dipole dynamic polarizability of an atom and the brackets 〈...〉 indicate a
time average. Here, the response of the atom is assumed to be linear and energy absorption is
excluded implying that α(ω) is real. This can be ensured by detuning the laser sufficiently far
away from the atomic resonance frequencies. The time averaging of the potential in (3.3) is
justified because the time variation of the laser field is much faster than the typical frequencies
of the atomic motion. If the response is dominated by a single resonance state |R〉 the
polarizability behaves like
α(ω) = |〈R| ˆ dE|0〉| 2
, (3.4)
¯h(ωR − ω)
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