BEC
in Higher Dimensional Optical Lattices
(Experiments
by
M. Greiner, O. Mandel, T. Esslinger, T.W.
Hänsch, and I. Bloch in Munich)
Lattice
Potentials
A
BoseEinstein condensate is loaded into 1D, 2D and 3D lattice
potentials. These periodic optical dipole potentials are formed by
counterpropagating, far red detuned laser beams. When we superimpose
two standing wave pairs, we obtain a 2D lattice. With three orthogonal
standing waves 3D lattice of light is created.

Wavelength: 
850 nm (approx. 60
nm detuning) 
Lattice
Spacing: 
425 nm 
Lattice
type: 
simple cubic 
Beam
waist: 
120 µm 
Polarization 
orthogonal between
standing wave pairs 
2D
lattice: 
V_{0} up
to 30 E_{recoil}, radial trapping frequencies up to 35 kHz,
1020 Atoms per tube 
3D
lattice: 
V_{0} up
to 22 E_{recoil}, radial trapping frequencies up to 30 kHz, 14
Atoms on average per site 

When we create a BEC in the magnetic trap
and ramp up the lattice
potential, the BEC splits up in up to 150.000 lattice sites. The
potential is ramped up slowly in about 80 ms. This ensures that we
always stay in the ground state of the system.
Time
of Flight Measurement
The
lattice spacing is just lambda over two, so we cannot resolve the
individual lattice sites with our absorption imaging. When we switch
off the lattice beams, the localized wavefunctions at each lattice site
expand and interfere with each other. They form a multiple matter wave
interference pattern which reveals the momentum distribution of the
system.
The sharp and discrete peaks we observe
directly prove the phase coherence across the entire lattice.

For
a 3D lattice the momentum distribution results in a three
dimensional
interference pattern. When we take absorption images after a time of
flight period, we observe the projection of this interference pattern
in one direction. 
Wavefunction
in the Superfluid Regime

In
the superfluid regime the system can be well described by a macroscopic
wave function which is the sum over localized wave packets at a lattice
site with a certain amplitude and phase at each lattice site:
Phases between neighboring lattice sites can be adjusted arbitrarily
by applying a magnetic field gradient (Bloch oscillation). When we
apply the magnetic field gradient for a long time we observe that the
system totally dephases. 
Detecting
the Band Population
When
the lattice potential is ramped down adiabatically, the quasimomentum
is mapped to real space momentum and the population of the energy bands
can be directly measured by observing the population of the
corresponding Brillouin zones. 
a) 
dephased
BEC (statistical mixture of all Bloch states in the lowest energy
band), populating the first brillouin zone isotropically which
indicates that the first energy band is also populated isotropically.
This corresponds to a state where all atoms are in the vibrational
ground state of the lattice sites, but where random phases are present
between lattice sites. 
b) 
population of higher bands,
induced by raman transitions 

