Eur. Phys. J. D 58, 311–325 (2010) DOI: 10.1140/epjd/e2010-00109-5 Regular Article T HE EUROPEAN P HYSICAL JOURNAL D Band structures of a slowly rotating dipolar Bose-Einstein condensate with a quantized vortex along a one-dimensional optical lattice H.C. Lee a and T.F. Jiang Institute of Physics, National Chiao Tung University, 30010 Hsinchu, Taiwan Received 31 August 2009 / Received in final form 22 December 2009 Published online 26 April 2010 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2010 Abstract. We derive the effective Gross-Pitaevskii equation for a slowly rotating dipolar Bose-Einstein condensate (BEC) with a quantized vortex along a one-dimensional optical lattice and calculate its band structures. The band structure of a slowly rotating BEC in a lattice becomes interesting when dipole-dipole interaction (DDI) is involved. Under rotation, a dipolar rotating term emerges from the DDI potential. The dipolar rotating term makes a BEC with an attractive DDI more stable than one with a repulsive DDI. The dipolar rotating term changes and generalizes the definition for the type of BEC, which cannot be simply determined by an s-wave scattering length or an effective contact interaction term. The dipolar rotating term also makes the band structure fascinating and tunable. A so-called swallowtail band structure, i.e., a multi-valued solution due to nonlinear interaction, can either elongate or shrink as the band index increases, in contrast to a non-rotating dipolar BEC system with a monotonic dependence. With the dipolar rotating term, various band structures as well as an attractive BEC without collapse can be easily achieved. We demonstrate that a rotating dipolar BEC system subject to an optical lattice combines features of a crystal and a superfluid and promises wide applications. 1 Introduction Rotation of a superfluid, which occurs only in the presence of quantized vortices, is one of the differences between su- perfluids and classical fluids, in addition to zero viscos- ity and two-fluid behavior (a mixture of superfluid and normal fluid) [1]. A quantized vortex is a singular point about which the velocity of the fluid is inversely propor- tional to its distance from the vortex; velocity integration along a contour encircling the vortex is quantized in units of h/m 0 , where m 0 is the particle mass of the fluid. The study of quantized vortices is closely connected with the quantum Hall effect [2–5] due to the equivalence of the ro- tation to an effective magnetic field in a Hamiltonian [6], which relates superfluids to a seemingly very different sys- tem: a two-dimensional electron gas in a magnetic field [7]. Furthermore, since turbulence can be regarded as a tan- gle of vortex lines, understanding vortex dynamics, such as reconnection and decay processes, is helpful for study- ing the difference between quantum and classical turbu- lences [8,9]. In particular, it is still unclear whether quan- tum turbulence obeys the Kolmogorov energy spectrum E(k v ) ∼ k -5/3 v as classical turbulence does [10–14], where k v is the wave number of the velocity field. a e-mail: hclee@mail.nctu.edu.tw A Bose-Einstein condensate (BEC) in optical lattices is a versatile system for exploring the quantum phase tran- sition from superfluid to Mott insulator [15], spin-wave excitation [16], and spinor condensates [17] such as those in the ferromagnetic, polar, and cyclic phases. In optical lattices, a BEC shares many of the properties of electrons in a crystal structure involving Bloch oscillation, Landau- Zener tunneling [18,19], intra (inter) band dynamics, etc. However, a BEC has a unique feature that never occurs in a solid state: while being periodically modulated by a one-dimensional (1D) optical lattice, the BEC density can rotate around a vortex line along the optical lattice, as shown in Figure 1. For comparison, although helium superfluid (He II) can also rotate around a vortex line, the helium density cannot be periodically modulated as that of a BEC in an optical lattice can. Otherwise, a BEC is well described by the mean-field theory, but He II is not, due to the strongly correlated interaction. Thus, its substantial differences from He II make a BEC more significant for exploring the quantum Hall effect in su- perfluids [2–5] and quantum turbulence [10–14]. Hence, knowledge regarding the band structure of a rotating BEC along an optical lattice is essential; however, current un- derstanding is still limited. Recently, a new species of dipolar BEC was experi- mentally realized with 52 Cr [20] and attracted consider- able interest [21–26]. 52 Cr is a spin-3 boson with electronic