This chapter mainly applies the variational method to study the properties of one and two-component BEC systems. Our approach can be applied to other two-component systems such as boson-fermion and fermion-fermion mixtures. In Sec. 2.2, the coupled GP energy functional for a two-component BEC system is introduced. By using the perturbation expansion around the equilibrium ground-state wavefunction, one is able to derive the time-independent GP equation for studying the dynamics. The formalism is easily reduced to a single-component BEC system. In Sec. 2.3, I consider the condition of energetic stability by using the second-derivative test. Together with
the Lagrangian of the system, one obtains the same time-dependent GP equations.
In Sec. 2.4, the density profile, the ground-state energy, and the breathing modes of a two-component Bose condensed system are studied in details using the variational method. We propose a more suitable trial wavefunction, called “modified Gaussian (MG) distribution function”, which is shown to give a better description for the two-component as well as one-two-component BEC systems when the nonlinear effect is not too large. The MG trial wavefunction can be used to study the vortices of a two-component system, the dipolar system (see Chapter 4), and possibly the system of imbalanced spin population.
BEC in optical superlattice
3.1 Introduction
In past few years, Bose-Einstein condensates (BECs) in optical lattices have been the subject of extremely intense and rewarding research, both theoretically and ex-perimentally. It is an active research topic in the cross-disciplined field of atomic molecular and optics physics (AMO) and condensed-matter physics (CMP). From the condensed matter point of view, because (i) laser power can be tuned to vary the ratio of on-site repulsion to intersite coupling of cold atoms, (ii) different geometry of laser beams can be taken to manipulate different dimensions and configurations of optical lattices, and (iii) different types of atoms (fermions, bosons, ions, or their mix-tures) can be loaded into the optical lattice, consequently the simulation of condensed matter environment is considerably easy and flexible. Recent investigations on the cold-atom system with an optical lattice have been made in various aspects, including the superfluid -Mott insulator quantum phase transition [53–57], the band-structure phenomena [58, 59], 2D and 1D systems [55, 60–63], fermionic systems [64–69], and the quantum informatics [70–73].
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3.1.1 Optical lattice
An optical lattice is a periodic potential which is generated by overlapping two coun-terpropagating laser beams. Due to the interference between the two laser beams, an optical standing wave with a period of half wavelength λ is formed, in which atoms can be trapped. The physical origin of the confinement of cold atoms with laser light is the dipole force,
F = 1
2α(ωL)∇[|E(r)|2], (3.1)
due to a spatially varying ac Stark shift to which atoms experience in an off-resonant light field. Since the time scale for the center-of-mass motion of atoms is much slower than the inverse laser frequency ωL, only the time-averaged intensity |E(r)|2 enters.
The direction of the force depends on the sign of the polarizability α(ωL). In the vicinity of an atomic resonance from the ground state |gi to an excited state |ei at frequency ω0, the polarizability has the form α(ωL) ≈¯¯¯he| ˆdE|gi¯¯¯2/¯h(ω0−ωL), with ˆdE the dipole operator in the direction of the field. Atoms are thus attracted to the nodes or to the antinodes of the laser intensity for blue-detuned (ω0 > ωL) or red-detuned (ω0 < ωL) laser light, respectively.
For example in a 1D optical lattice the potential seen by the atoms is given by
V (x) = V0cos2(πx/c) , (3.2)
where the lattice spacing c = λ/2 and V0 is the lattice depth. Similarly periodic potentials in two or three dimensions can be formed by overlapping two or three optical standing waves along different (usually orthogonal) directions, like those in Fig 3.1. In addition, by controlling the polarizations of the lattice beams, it is also possible to create state-dependent potentials that can be shifted relative to each other.
This kind potential is called spin-dependent optical lattice potentials.
Figure 3.1 (a) Two- and (b) three- dimensional optical lattice potentials formed by superimposing two or three orthogonal standing waves. The figure is borrowed from Ref. [36].
3.1.2 Relevant energy scales
A basic theory of optical lattice systems can be classified to different regimes depend-ing on the relative importance of the nonlinear interaction and the lattice parameters.
In general, there are four important energy scales (takeing 1D case as an example):
(i) Bandwidth Ew : the energy difference between q = π/c and q = 0 of the lowest band.
(ii) On site interaction U: gives the on-site interaction energy per atom on a single lattice site.
(iii) Energy gap Eg: represents the energy difference between the bands at q = π/c.
In a deep optical lattice, this corresponds to the energy difference between the lowest and first vibrational state in a single potential well of the lattice.
(iv) Tunneling rate J : the nearest-neighbor Josephson coupling responsible for
con-densate tunneling.
In the deep periodic potential limit, there is also a direct connection between Ew and J; that is, Ew = 4J. In this regime, the nonlinearity will be the smallest energy scale if U is smaller than the bandwidth Ew and the band gap Eg. When Ew < U < Eg, the nonlinear energy scale is in the intermediate range. The nonlinear energy scale is dominant when U is larger than Ew and Eg. We shall focus on the nonlinear energy scale being the smallest one and the optical lattice is in the deep limit, called the “tight-binding limit”. For other regimes, a detailed account can be found in the reviews [35, 36].
In the tight-binding limit mentioned above, the condensate wave function can be described by localized Wannier function associated with the lowest band. The condensate energy functional in this regime is derived in next section.