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

Chapter 10
Array of Josephson junctions
In a deep lattice, the time-evolution of the system can be described in terms of the dynamics
of the number of particles and the condensate phase at each lattice site. This formulation
incorporates a particularly clear physical picture: The system is considered as an array of
weakly linked condensates, each of which contains a time-dependent number of particles and
is characterized by a time-dependent phase. The coupling between the condensates is provided
by the tunneling of atoms between neighbouring lattice wells. From this point of view, the
system constitutes a particular realization of an array of Josephson junctions. This approach
is particularly valuable because it allows for a link between the regime of validity of GP-theory
with a regime, described by a quantum Josephson Hamiltonian, where quantum fluctuations of
phases and site occupations are important. The link is established by quantizing the Josephson
Hamiltonian obtained from GP-theory.
Within the regime of validity of GP-theory, the occupation and phase of each site is welldefined
at any time. We derive the dynamical equations governing their time evolution in the
purely periodic potential (see section 10.1). In the most general case, they are characterized
by the appearance of time-dependent tunneling parameters.
We show how to reproduce the tight binding expression for the lowest Bogoliubov band
found in chapter 7.3 above (see section 10.2). The Josephson junction array description allows
for a particularly clear physical interpretation: Bogoliubov excitations are associated with the
exchange of a small amount of atoms between the lattice sites. The quasi-momentum ¯hq of
the excitation is a measure of the number of lattice sites over which this exchange takes place.
At the maximal value q = π/d atoms move back and forth only between neighbouring sites.
We show that in the limit of very deep lattices, the spectrum can be determined neglecting
the difference in the occupation of neighbouring sites and retaining only the phase difference.
Under certain simplifying assumptions the dynamical equations for phases and site occupations
of the array can be recast in the form of Hamiltonian equations (see section 10.3).
We present the corresponding Josephson Hamiltonian and discuss the governing Josephson
parameters. The plasma oscillation frequency of a single Josephson junction is compared with
the corresponding Bogoliubov excitation in an array. The equations of motion governing a
single Josephson junction bear certain analogies with those for the center-of-mass motion in
the combined trap of lattice and harmonic trap in the tight binding regime (see chapter 9.6
above).
133