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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 Bose-Einstein condensate is loaded into 1D, 2D and 3D lattice potentials. These periodic optical dipole potentials are formed by counter-propagating, 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.

2D and 3D lattice geometries
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: V0 up to 30 Erecoil, radial trapping frequencies up to 35 kHz, 10-20 Atoms per tube 
3D lattice: V0 up to 22 Erecoil, radial trapping frequencies up to 30 kHz, 1-4 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

TOF of a superfluid lattice

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.

A 3D-lattice from 2 directions 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

Lattice-image with a Pi phase difference 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 quasi-momentum 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.

Brilloin zones in optical lattices

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

M. Greiner, I. Bloch, O. Mandel, T.W. Hänsch and T. Esslinger
Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates
Phys. Rev. Lett. 87, 160405 (2001)