Band structures of a dipolar Bose-Einstein condensate in one-dimensional lattices
YuanYao Lin,
1
Ray-Kuang Lee,
1
Yee-Mou Kao,
2
and Tsin-Fu Jiang
3,
*
1
Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan
2
Department of Physics, National Changhua University of Education, Changhua 500, Taiwan
3
Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan
Received 14 February 2008; published 25 August 2008
We derive the effective Gross-Pitaevskii equation for a cigar-shaped dipolar Bose-Einstein condensate in
one-dimensional lattices and investigate the band structures numerically. Due to the anisotropic and the long-
ranged dipole-dipole interaction in addition to the known contact interaction, we elucidate the possibility of
modifying the band structures by changing the alignment of the dipoles with the axial direction. With the
considerations of the transverse parts and the practical physical parameters of a cigar-shaped trap, we show the
possibility to stabilize an attractive condensate simply by adjusting the orientation angle of dipoles. Some
interesting Bloch waves at several particle current densities are identified for possible experimental
observations.
DOI: 10.1103/PhysRevA.78.023629 PACS numbers: 03.75.Lm, 05.30.Jp
I. INTRODUCTION
The Bose-Einstein condensate BEC in an optical lattice
has provided a versatile and controllable platform to study
the condensed-matter-like properties by the atomic quantum
gases 1. As electrons in a crystal lattice, matter waves in
laser-induced optical lattices have many similar but ubiqui-
tous interesting features due to the nonlinear atom-atom in-
teractions in BECs. Bloch waves with discrete eigenenergy
are the stationary solutions for BEC in periodic potentials.
The resulting band structures are identified by the Brillouin
zones. With the nonlinear mean-field Gross-Pitaevskii equa-
tion, the swallow-tailed loop structure at the boundary of the
Brillouin zone was first predicted by a simple two-state
model 2. Later on, an exact solution of such loop behaviors
in the band structure was found for a particular kind of one-
dimensional lattice 3,4, and was further studied numeri-
cally by a detailed many-mode expansion method 5,6. The
atomic band structures are related to the dynamics and sta-
bility of the condensates. The new property has attracted
intensive investigations, including the nonlinear Landau-
Zener tunneling 2,7, the Bloch oscillation 8,9, and the
stability of Bloch waves 6,10.
In 2005, a new species of dipolar BEC was realized in
addition to the alkali-metal atoms BEC systems since the
first realization in 1995. This dipolar system uses chromium
atoms,
52
Cr. Each chromium atom has a magnetic dipole
moment of 6 Bohr magneton which is larger than that of the
alkaline atom 11–14. More recently,
52
Cr BEC was pro-
duced with an all-optical method 15. For the dipolar BEC,
there is then an extra dipole-dipole interaction between at-
oms in addition to the known contact interaction in the BECs
of alkali-metal atoms. The dipole-dipole interaction is aniso-
tropic and long-ranged. So, there are new tunable parameters
from this interaction. There are renewed interests in the di-
polar BEC due to the dipole force. Namely, an unusual prop-
erty of double-peak order parameter for the dipolar BEC un-
der certain environments was demonstrated 16; and the
effects of the dipolar interaction to the quantum phase tran-
sition temperature was also explored 17. The stability,
ground state, and excitations of the dipolar BEC in a trap
potential were already investigated in the literature 18–20.
It was found that the Luttinger-liquid phase persists for a
wide range of density in one-dimensioanl dipolar gas 21.
The solidlike to liquidlike phase change of the one-
dimensional dipolar system ground state with respect to the
linear density by the quantum Monte Carlo method was
shown 22. The signature of one-dimensional dipolar gas in
the Super-Tonks-Girardeau regime was studied 23. The
ground state phase diagram of the two-dimensional dipolar
gas was also investigated 24. Applying the Bose-Hubbard
model to the system of dipolar BEC in a two-dimensional
optical lattice, the possibility of several quantum phases for
the ground state with different aspect ratios was elucidated
25. An extension to dipolar spinor BEC was also explored
recently 26. More interestingly, a novel structure of dipolar
bosons in a planar array of one-dimensional tubes was pro-
posed 27.
The goal of this paper is to investigate the band structures
of the dipolar BEC in a quasi-one-dimensional optical lattice.
As shown in Ref. 25, in the case of extreme quantum re-
gime, there are several possible phases for the ground state
for dipolar Bosons in optical lattice. Instead of using the
Bose-Hubbard model with long-range force, mean-field
theory is applied here as the cases of nondipolar bosons in
lattice and our system parameters are within the range of
superfluid phase. The number density of atoms we consid-
ered will be quite large and without significant fluctuations.
In such a scenario, even though the effects of the transverse
confinement could wash out some of the ground states, but
effectively with the regime of a mean-field approach one
only obtains a modified one-dimensional Gross-Pitaevskii
equation with coefficients depending on the geometry of the
ground state. Instead of using the ideal pure one-dimensional
lattice model 2–6, in this work we use practical physical
parameters of a cigar-shaped trap and take into account the
effects of transverse parts. As mentioned above, with the
dipolar potential, the effects of the spatial distribution of the
*
tfjiang@faculty.nctu.edu.tw
PHYSICAL REVIEW A 78, 023629 2008
1050-2947/2008/782/0236298 ©2008 The American Physical Society 023629-1