3.3 Oscillations in a 1D optical superlattice
3.3.3 Phonon modes in optical superlattices
Figure 3.2 (a) Illustration of how a 1D optical lattice with an n-point basis is formed by two laser beams and their reflected beams. Frame (b) and (c) show the resulting potential energies (thick solid lines) as a function of lattice spacing of the n = 2 and n = 3 case. c denotes the lattice constant and J’s denote the Josephson tunnelling couplings. In frame (b), d denotes the spacing between any potential maximum and its nearby minimum.
a z-direction 1D optical lattice with an n-point basis. Since the ratio of amplitude A1/A2 can be adjusted and the factor n can be chosen, 1D optical superlattice can be created for a great flexibility. Figure 3.2 illustrates how the 1D optical superlattice with a 2 and 3-point basis is formed and their resulting potential energies (A1/A2 =
√5 is taken). Josephson coupling J’s that differ due to condensates tunnelling across different potential barriers are also indicated.
3.3.3 Phonon modes in optical superlattices
At sufficiently low temperatures, dynamics of Bose condensates is governed by the time-dependent Gross-Pitaveskii (GP) energy functional
E [Ψ∗, Ψ] =
where Ψ = Ψ(r, t) is the time-dependent wave function of the condensate, µ is the chemical potential, and g = 4π¯h2a/m with a the s-wave scattering length of the two-body interaction. In the present case of interest, the external trap potential has two contributions, Vtrap(r) = Vm(r) + Vo(r), where Vm(r) = 12m[ωρ2ρ2 + ωz2z2] with ρ2 ≡ x2+ y2 is due to the magnetic trap, while Vo(r) = Vo(z) = V1cos2³2πzc ´+ V2cos2³πzc ´ is due to the optical trap. When ωρ À ωz for the magnetic trap, optical potential then results a 1D optical superlattice (along z axis) with a two-point basis. Here c corresponds to the lattice constant which is related to the wavelength (λ) or the wavevector (k) of the laser beam with frequency ω by c = λ/2 = π/k (see Fig. 3.2).
It is convenient to work with dimensionless quantities and rescale the energy, time, and length as E/(¯hωρ) → E, tωρ → t, and r/`ρ→ r (`ρ≡q¯h/mωρ). Eq. (3.17) thus
The frequency ωz is completely dropped in (3.18).
For longitudinal phonon modes to be considered under the harmonic approxi-mation, tight-binding (TB) limit is applied. This means that anharmonic effect is negligibly small. This also means that optical potential barriers need to be large, so condensate wave functions are strongly localized in z-direction around the potential minimums. In this limit, Ψ(r, t) can be taken to be the form
Ψ(r, t) =X
`
{w1[z − `c − d]Φ1(x, y, `; t) + w2[z − (` + 1)c + d]Φ2(x, y, `; t)} ,(3.19) where ` sums over all unit cells and d is the spacing between any potential maximum and its nearby minimum (see Fig. 3.2). Pertaining to site 1 and site 2, w1 and w2 are the strongly localized Wannier functions depending on z only, while Φ1 and Φ2 are the ones associated with x, y coordinates which in turn contain the periodic factor
∼ exp(i`kc) for any propagating wave (along the lattice direction) of wave vector k.
It is noted that w1(z) = w2(−z) for the reflection symmetry. Moreover, for simplicity, fluctuations (time dependence) are considered to be through Φi(x, y, `; t) only.
Substitution of (3.19) reduces (3.18) into
E =X
In (3.20) and (3.21), only nearest-neighbor intersite couplings are considered. J1 and J2 are thus the two different nearest-neighbor Josephson couplings responsible for condensate tunneling. In principle, w1, w2 can be solved numerically, which in turn solve J1, J2, and ¯U. Typical value of J1 or J2 is less than 0.1 in the TB limit.
In the following, as mentioned before, we will focus on the breathing and phonon modes. Similar to Martikainen and Stoof [51], a Gaussian ansatz
Φi(x, y, `; t) =
is assumed for the wave functions, where N represents the average number of atoms per site and B0 represents the condensate size in the equilibrium state. In fact, when N is fixed, the value of B0 can be calculated through minimizing the GP energy in (3.20). This was done in Ref. [51] where B0 = 1/√
1 + 2U0 with U0 ≡ 4N ¯U/π is
given. No cross (xy) term is considered in (3.22) because breathing mode is isotropic in the x-y plane. For each site (`, i), dimensionless dynamical variable ²0`,i(t), ²00`,i(t), δ`,i(t), and ν`,i(t) corresponds respectively to fluctuations of the local amplitude, the local phase, the number of atoms, and the global phase of the condensate.
One crucial aspect on (3.22) is however that it enables naturally the coupling between the transverse breathing and longitudinal phonon modes [51]. In a one-component Bose-condensed system with repulsive interaction, the fluctuation of densed atom number is coupled to the fluctuation of condensate size. When con-densates are distributed in an optical lattice, the fluctuating degrees of freedom are complicated by the intersite coupling of Josephson tunnelling. For the present optical superlattice, the case is further complicated by the possible out-of-phase motion in addition to usual in-phase one. How important are the various couplings and how the condensates are fluctuating in such a system are thus the subject of the following studies.
By variational approach, one starts from the Lagrangian of the system, L = T −E.
Here T =R dr i¯2h(Ψ∗∂Ψ/∂t − Ψ∂Ψ∗/∂t) and E is given by (3.18) [and hence (3.20)].
Applying (3.22) in T and E and expanding to second order in dynamical variables, one obtains
where the Josephson-tunneling term linear in δ`i terms vanish. In this case, only second order terms are left with EJ in (3.25) and it justifies the smallness of EJ in accordance with TB.
With (3.23)-(3.25), one can then derive the Euler-Lagrange equations of motion for all eight dynamical variables. They can be linearized and written as
˙²0`,1 = 2B0²00`,1+ J1(²00`,1− ²00`−1,2) + J2(²00`,1− ²00`,2), Note that the above eight linear first-order differential equations can be cast into four second-order ones in terms of any two sets of variables ²0`,i(t), ²00`,i(t), δ`,i(t), and
kc /π
Figure 3.3 Dispersions of four-branch modes resulting from ( 3.26). The parameters J1 = 0.09 and J2 = 0.1 are the same for all three frames, while U0 = 1, 10, and 100 respectively for frame (a), (b), and (c). Here IB, OB, AP, and OP denote for in-phase breathing, out-of-phase breathing, acoustic phonon, and optical phonon modes. See text for more description.
ν`,i(t) (i = 1, 2) only. (That is in the lowest-order limit, the other two sets will share the same excitation spectra.) Considering the coupled equations between ²0`,i(t) and δ`,i(t), for example, and searching for solutions of the type: ²0`,i(t) ≡ ²0iei(`kc−ωt) and δ`,i(t) ≡ δiei(`kc−ωt), a 4 × 4 dynamical matrix (not shown) will be attained, which is then diagonalized to obtain four branches of modes. In Fig. 3.3, dispersions of the four branches of modes are shown for three cases: with same J1 = 0.09 and J2 = 0.1, but different U0 = 1, 10, and 100. In the study of Bose-Hubbard model on the SF-Mott insulator quantum phase transition in 1D, it has been established that (U/J)c= 2.2N is a critical point for the case N À 1. Below (above) it the system is in the SF (insulating) phase [36, 79]. In our case, U = U¯√2πB0 = U0
8N√4
1+πU0/4N [see Eq. (3.14)]
with the scale `ρ = 1/√
B0. Besides, we are interested in collective excitation of condensates with average atom number per site being N ∼ 10−100. Thus the system
is justified to be in the SF phase when U0/J < √4 U0
1+πU0/4N/J < (17.2N2) ≈ 2 × 103. Consequently the cases U0 = 1, 10, and 100 studied in Fig. 3.3 are considered to be in the SF phase.
The four branches of modes are labelled by in-phase breathing (IB), out-of-phase breathing (OB), acoustic phonon (AP), and optical phonon (OP) modes respectively (see Fig. 3.3). Strictly speaking, these descriptions are exact only when phonon modes are completely decoupled from breathing modes and at the same time, the propagating wave comprised of ²0`,1(t), for instance, has a phase angle π shift (out-of-phase) by the the propagating wave comprised of ²0`,2(t). The coupling effect, which is proportional to the magnitude of tunnelling strength J, is indeed small in the TB limit. One should be noted that the coupling effect is also accompanied by the factor of B0U0 (recall that B0 corresponds to the size of condensate), to which it becomes more and more important in the large U0 limit.
At the long-wavelength limit (kc ¿ 1), dispersions of the four branches of modes can be analytically solved and uniformly written as
ωi2 = ω20i+ u2ik2, (3.27) where, at the TB limit (keeping to J linear order), one obtains
ω0i2 =
As shown in (3.27)–(3.29), the slopes (ui) and the k = 0 intercept (ω0i) of the four branches of modes depend crucially on the values of J1, J2, and U0. Therefore ob-servation of these modes will be a direct way to access them. Some features worth noting are as follows. For IB mode, the k = 0 frequency (ω = 2ωρ) is robust regard-less of the values of J1 and J2. It is a clear result to the uniform (single-condensate) limit. For AP mode, on the other hand, the phonon velocity at small k behaves like u ∼√
B0U0Jc, an expected result for the current system of a bulk modulus ∼ B0U0J.
In the large U0 case [see Fig. 3.3(c)], IB and OP modes start to hybridize, to which a small gap (not visible in the scale potted) develops where the two branches cross. This feature may be a signature when the system undergoes a quantum phase transition from SF into insulating phase.
3.4 Summary
This chapter studies the BEC condensates in optical lattices. In Sec. 3.1, I consider the relevant energy scales in an optical lattice system. In Sec. 3.2, I discuss the tight-binding model in an optical lattice. Detailed derivations are given for a 1D system. In Sec. 3.3, oscillations of Bose condensates in a 1D optical lattice with a two-point basis is investigated. Focuses are made to the transverse breathing modes and longitudinal phonon modes. Pertaining to condensates in site 1 and site 2, in-phase and out-of-in-phase modes are obtained. The mode dispersion relations depend crucially on the sizes of the two intersite Josephson tunnelling strengths J1 and J2 and the on-site repulsion U. For the phonons, there are optical as well as acoustic modes, in a close resemblance to those in a 1D crystalline solid. While direct observation of these in-phase and out-of-phase oscillations of condensates may be resolution limited by current instruments, dispersion relations of these modes should be probable by
inelastic scattering measurements.
BEC with dipolar interaction
4.1 Introduction
In recent years dipolar gases become a fast growing field of theoretical and exper-imental interests in the studies of ultracold atoms and molecules. Among several dipolar systems, chromium (52Cr) atoms have been successfully realized and studied to great extents [11, 80–84]. The dipolar interaction effects for 87Rb atoms as well as 39K atoms are also observed in different groups [85, 86]. Several polar molecule systems such as CO [87], ND3 [88], RbCs [89], LiCs [90], and CsCl have also been trapped, cooled, and studied [91,92]. More recently, a high phase-space density gas of polar40K87Rb molecules have also been produced [93]. These stimulate great interest in the studies of dipolar systems at low temperatures. Within the first Born approx-imation (FBA), Yi and You [94] proposed a pseudopotential to study the long-range dipolar interaction. Based on this approximation, which is justified only in the weak dipole moment limit, various theoretical studies of the excitations, collapses, insta-bility, etc. of the dipolar BEC system have been carried out in these years. However these results do not become justified when applied to systems of polar molecules,
41
which can have large electric dipole moments. So developing a correct and widely ap-plicable pseudo-potential and the associated many-body theory for systems of dipolar atoms/molecules is a very important and crucial step for future theoretical and exper-imental studies. Thus in this chapter, we shall study the ground-state wavefunction and dynamics of dipolar condensates going beyond the FBA.