Elementary excitations

This section devotes to the elementary excitations of a dipolar condensate beyond the first Born approximation. As sketched in Fig. 4.3, two types of collective modes, namely the in-phase and out-of-phase breathing modes, are considered in this paper.

They are of particular importance because they are the lowest two excitation modes in a trapped condensate [104, 105]. In the paper of Goral and Santos [103], these two modes were also called breathing and quadrupole modes. For simplicity of presenta-tion, we will show the results of the breathing modes (of lower energy) only in this paper, although both of them can be calculated easily in the same method.

Here we will use a variational method to study the collective excitation of a system.

It corresponds to solving the stationary point of the action S = ^R dtL, where the Lagrangian L = T − E with T = ^R dr(i¯h/2) [Ψ^∗(r)∂Ψ(r)/∂t − Ψ(r)∂Ψ^∗(r)/∂t] and E is given in Eq. (4.21). For the study of breathing modes, the MG function in

Eq. (4.20) will be generalized to include dynamical variables as follows:

Here, in a cylindrically symmetric trap, εi and ε⁰_i (i = 0, z) correspond to the fluctua-tions of local amplitude and local phase of the dipole cloud associated with the ρ and z directions. As mentioned before, the values of R0, Rz, a0, and az are determined by minimizing the energy functional. After some lengthy derivations, we obtain the dispersions for the breathing modes:

ω² = {f₁f₄+ f₂f₅± [(f₁f₄− f₂f₅)²+ 4f₃²f₄f₅]^1/2}/2,

(4.25)

where ± correspond to the in-phase or out-of-phase modes and f₁ = 1

To determine which one is the in-phase or out-of-phase mode in (4.25) requires solving explicitly the time dependence of the dynamical variables. In any event, we will focus on the one of lower energy.

In the following, a case study is given first, which will lead to an explicit idea how to extract quantitatively the ζ_0,2 term in real experiments. We first study the case of ζ_d + ζ_0,2 = 20 to which three combinations of (ζ_d, ζ_0,2) = (10, 10), (20, 0), and (40, −20) are considered. In Fig. 4.4(a) and (b), the lower breathing modes for these three combinations are presented and compared as a function of trapping

3 4 5 6 7

Figure 4.4 Frame (a) and (b): Lower breathing mode frequencies are plotted as a function of λ for three combinations, (ζd, ζ0,2) = (10, 10), (20, 0), and (40, −20). Here ζ_s= 0 for (a) and ζ_s= 2 for (b). Frame (c) and (d): Parallel to frame (a) and (b), γ ratios are plotted as a function of λ.

aspect ratio λ. Fig. 4.4(a) corresponds to the case of ζ_s = 0. In view of Fig. 4.4(a), two important features are revealed. Firstly, all three curves merge in the large λ limit (i.e. a pancake like trapping potential). Secondly, in contract, when λ = 5.9, for example, to which the system is close to the collapsed regime, ω = 1.338ω₀ for (ζ_d, ζ_0,2) = (20, 0), while ω = 1.044ω₀ for (ζ_d, ζ_0,2) = (40, −20). The relative frequency difference, ∆ω ≡ |ω₁ − ω₂|/ω₂, can become at least 21% large. ∆ω will increase more significantly when the system is even approaching the collapsed regime. Similar results can be also observed in Fig. 4.4(b), where a finite value of s-wave scattering is included (ζ_s = 2).

In Fig. 4.4(c) and (d), we also show the condensate aspect ratio, γ, as a function of λ. In view of Fig. 4.4(c) with ζ_s = 0, one finds that all three γ curves also merge in the large λ limit. Besides, when λ = 5.9 close to the collapsed regime, γ = 0.353 and 0.392 respectively for the case of (ζ_d, ζ_0,2) = (20, 0) and (40, −20). The relative γ difference, ∆γ ≡ |γ₁− γ₂|/γ₂, is about 11%. In Fig. 4.4(d) with ζ_s= 2, when λ = 3.8,

γ ratio is 0.528 and 0.593 for the case of (ζ_d, ζ_0,2) = (20, 0) and (40, −20). This gives a relatively smaller ∆γ = 2%. Similar to the ∆ω case, ∆γ will increase significantly when the system is even more close to the collapsed regime.

It should be emphasized that in the case that ζ_s is large or the sum of ζ_d+ ζ_0,2 is small, the system will tend to stabilize over a large span of λ. This means that whatever combinations of ζ_d and ζ_0,2 under ζ_d+ ζ_0,2 = const will roughly lead to the same curve. Thus, ∆ω or ∆γ studied above, will always be small, i.e. the effect of the effects beyond the FBA cannot be observed easily.

In the following, we propose an idea how to extract quantitatively the value of ζ_0,2 by measuring the lower breathing mode frequency or the condensate aspect ratio γ of the system. Firstly, we assume that the value of ζ_s is relatively small such that the behaviors of breathing mode frequency and γ ratio resemble those in Fig. 4.4(a)-(d).

Suppose that one does not know the value of any one of ζ_s, ζ_d, and ζ_0,2. One can first perform the measurement of the lower breathing mode frequency and/or γ ratio at large λ case, say λ = 10. As shown in Fig. 4.4(a)-(d), all three curves merge at large λ limit. This means that one can unambiguously determine the values of both ζ_s and the sum of ζ_dand ζ_0,2 simply by carrying out a theoretical fitting to the experimental data. The next task is to separate the values of ζ_d and ζ_0,2. For this purpose, one can redo the experiment and decrease λ to approach the collapsed regime. Since at this regime, the fitting will be quite sensitive to both values of ζ_d and ζ_0,2, one can then unambiguously determine the value of ζ_0,2 by comparing with our theoretical calculation result.

Now we provide a short discussion on the effect of the next order effect beyond the FBA by including finite value of ∆a⁽⁰⁾_2,2. Same as before, we define ζ2,2 ≡ N∆a⁽⁰⁾_2,2 for the convenience. In Fig. 4.5 we show the breathing mode frequency and the aspect ratio, γ, as a function of ζ2,2 for a fixed trapping aspect ratio, λ = 6. Besides, we

-0.02 0.00 0.02 0.8

1.0 1.2 1.4

-0.02 0.00 0.02 0.32

0.36 0.40 0.44

2,2 (40, -20)

(10, 10) (a)

( d

, 0,2

)

(b)

2,2

Figure 4.5 (a) Lower breathing mode and (b) γ ratio of the dipolar system are plotted as a function of ζ2,2. In both frames, λ = 6 and two combinations of (ζ_d, ζ_0,2) = (10, 10) and (40, −20) are chosen. Here we set ζ_s = 0.

choose two combinations of (ζ_d, ζ_0,2) = (10, 10) and (40, −20), and assume ζ_s = 0.

When ζ_2,2is turned on to be positive, we find that the system becomes more cigar-like shape (i.e. γ becomes larger), while it becomes more pancake-like shape when ζ_2,2 is negative. The positive value of ζ_2,2 will lead the system to be more attractive along the z direction and easily collapsed, but its effect is relatively weaker compared to the terms we discussed before.

4.4 Summary

This chapter concerns about the physics of Bose condensates of strong dipole interac-tions. In Sec. 4.1 we introduce the pseudopotential V_ps for dipole-dipole interaction.

In Sec. 4.2, we introduce a basic theory beyond the FBA. In Sec 4.3, an attempt is made to study the ground state and elementary excitations of a strongly interacting dipolar bosonic gas based on a theory going beyond the first Born approximation (FBA). By using an appropriate trial wave function in the variational method, the

leading higher-order corrections to the FBA are studied in details for in the condensate aspect ratio and in the elementary excitations. An idea for extracting quantitatively such leading-order effect beyond the FBA in real experiments is provided.

Vortex state of fast rotating BEC

5.1 Introduction

One of the most remarkable characteristics of a Bose-Einstein condensed system is its response to rotation. As was first understood in the context of superfluid helium-4, a Bose-Einstein condensate does not rotate in the manner of a conventional fluid, which undergoes rigid body rotation. In recent years vortices in a rotating Bose-Einstein condensate have been extensively studied theoretically and experimentally.

Rotating condensates are typically confined in a harmonic potential such that large vortex arrays are obtained when angular velocity Ω is smaller than but approaching the radial trap oscillator frequency ω⊥. For instances in Refs. [106–112], a single vortex state or a state of several hundreds of vortices have been successfully created.

Theoretically, the behaviors of the rapid rotating BEC systems are usually studied based on the lowest Landau level (LLL) approximation [113–118]. When angular velocity is greater than the radial trap oscillator frequency, Ω > ω⊥, the system becomes unstable due to the ineffectiveness of confinement.

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