Effect of interaction

In this section, the effect of the interaction (g) is studied by the perturbation approach.

Similar to previous section, we consider also the three cases: λ = 0.03, 0.01, and 0.005. For the interaction, ng = 0.5 and 1.0 are considered respectively. In Fig. 5.4, the number of N_c[terms involved in the summation (5.18)] is determined and plotted as the function of ω₀/Ω for six cases, namely (ng, λ) = (1, 0.03), (1, 0.01), (1, 0.005), (0.5, 0.03), (0.5, 0.01), and (0.5, 0.005). It turns out that the results of N_c are quite fascinating in terms of the change of Ω and the strength of the interaction. One common feature of all frames in Fig. 5.4 is that N_c shifts from 1 (at lower Ω) to 2 eventually (at higher Ω). In between of the 1 to 2 period, transition states occur for

a large Ω span. For example, N_c≥ 3 transition states can occur in all frames except in frame (d). Besides, N_cincreases in a trend as Ω increases for the transition states.

The reason why N_c all shift from 1 (at lower Ω) to 2 eventually (at higher Ω) can be understood as follows. One recalls the N_c results for g = 0 in Fig. 5.2(d), where all three cases are observed to exhibit the N_c = 1 → 2 transition (at different critical Ω though). When the interaction g is turned on and still valid in the perturbative regime, g will only play little role in the large Ω limit. It means that when Ω is large enough, transition states, which arise due to the effect of g, will disappear. As an extreme example shown in Fig. 5.4(d), because g is relatively small (ng = 0.5) and the quartic coupling is relatively large (λ = 0.03), consequently no transition state occur.

When λ is reduced [see Fig. 5.4(e) and (f)], or when g is increased [see Fig. 5.4(a)], transition states will occur.

Moreover, it is found that the critical value of Ω to which N_c changes from 1 to 2 at the lower Ω side is smaller when λ is smaller (if ng is fixed) or when ng is larger (if λ is fixed). It implies that the system will enter the vortex state earlier if g is relatively larger or λ is relatively smaller. Furthermore, with the same λ, transition states will sustain for a larger span of Ω if g is larger and can go up to a higher value of N_c. The latter means that vortex lattice can have a higher row number. One sees in Fig. 5.4(c) that N_c can go up to 5, although for a small period of Ω.

The normalized density profile in the xy plane, |Ψ|²/n, is shown in Fig. 5.5 for (ng, λ) = (1, 0.005) [corresponding to the case in Fig. 5.4(c)]. As seen in Fig. 5.4(c), the change of N_cin the transition states is quite rich for this case. Six angular veloci-ties are studied, namely ω0/Ω = 1.02, 1.0, 0.984, 0.9815, 0.98 and 0.9785 respectively for Fig. 5.5(a)–(f). With these values of Ω, Nc corresponds to 2, 3, 4, 5, 4, and 2 respectively. One sees in Fig. 5.5(a)–(d) that when Ω is increased from ω0/Ω = 1.02 to 0.9815, atoms are pushed to the two sides and the number of vortices becomes

more and more. The row number of vortex line also increases from 1 to 4. While in Fig. 5.5(e), the vortex row number is reduced to 3 (N_c = 4) again. In the case of Fig. 5.5(f), although N_c is reduced to 2, but vortex lattice vanishes (melts) due to the large centrifugal force. Similar vortex lattice melting transition (atoms are completely pushed to the two sides) at large Ω has already been seen in the previous section of no interaction.

5.4 Summary

In this chapter, we discuss the basic physics for a fast rotating BEC in Secs. 5.1 and 5.2. In Sec. 5.3, we investigates the effect of a quartic potential on a fast rotating BEC system under the extreme elongation. In contrast to the harmonic trap alone case where system is unstable when the angular velocity Ω is larger than the radial trap oscillator frequency ω₀, it is shown that the quartic trap can lead the system to remain stable at higher rotation velocity (Ω > ω₀). The interplay between the weak s-wave interaction and the quartic trap potential can result rich vortex lattice transition states as a function of Ω. At large Ω, atoms are eventually push to the two sides along the elongated potential well.

x

Figure 5.5 Normalized atom number density distribution, |Ψ|²/n, is plotted in the xy plane for (a) ω0/Ω = 1.02, (b) 1.0, (c) 0.984, (d) 0.9815, (e) 0.98, and (f) 0.9785. Here ng = 1 and λ = 0.005 for all frames [corresponding to the case in Fig. 5.4(c)]. Vortex lattices with the transitions of row number and lattice constant are observed as the change of Ω.

Conditions of energetic stability

In this Appendix, for a two-component BEC system, I show the details of expanding the thermodynamic energy G = E − µ1N1 − µ2N2 in terms of the fluctuations of the ground-state wavefunctions Ψ1 and Ψ2. As the GP energy function E is given in (2.7), G is then given by

G =

Z

dr



 X

i=1,2

Ψ^∗_i

"

−¯h²∇²

2m + V_i(r) + gi

2 |Ψ_i|²− µ_i

#

Ψ_i+ g₁₂|Ψ₁|²|Ψ₂|²



, (A.1) where µ_i is the chemical potential for species i. To minimize G, one can write Ψ_i ≡ Ψ_i,0+ δΨ_i with Ψ_i,0 the equilibrium condensate wavefunction and δΨ_i the fluctuation around Ψ_i,0. One then expands G to the second order in δΨ_i. Involving in this expansion, there are two key terms. One involves the kinetic energy:

Ψ^∗_ih_iΨ_i → (Ψ^∗_i,0+ δΨ^∗_i)h_i(Ψ_i,0+ δΨ_i)

= Ψ^∗_i,0h_iΨ_i,0+ Ψ^∗_i,0h_iδΨ_i+ δΨ^∗_ih_iΨ_i,0+ δΨ^∗_ih_iδΨ_i, (A.2) where h_i ≡ −^¯h_2m^2∇2

i + V_i(r) − µ_i. Another involves the interaction energy:

g_ij|Ψ_i|²|Ψ_j|² → g_ijⁿ|Ψ_i,0|²|Ψ_j,0|²+ |Ψ_i,0|²(Ψ^∗_j,0δΨ_j+ δΨ^∗_jΨ_j,0) + |Ψ_j,0|²(Ψ^∗_i,0δΨ_i+ δΨ^∗_iΨ_i,0) + (Ψ^∗_j,0δΨ_j+ δΨ^∗_jΨ_j,0)(Ψ^∗_i,0δΨ_i+ δΨ^∗_iΨ_i,0) + |Ψ_i|²δΨ^∗_jδΨ_j+ |Ψ_j|²δΨ^∗_iδΨ_i

+ higher order terms } . (A.3)

77

It is instructive to show that when i = j the above long expansion leads to

|Ψi,0|⁴+ 2|Ψi,0|²(Ψ^∗_i,0δΨi+ δΨ^∗_iΨi,0) (A.4) + (Ψ^∗_i,0)²(δΨ_i)²+ (Ψ_i,0)²(δΨ^∗_i)²+ 4|Ψ_i,0|²δΨ_iδΨ^∗_i + higher order term . Collecting the same order terms in (A.2)– (A.4), one can write G = G₀[Ψ_i,0] + δG₁ + δG₂, where δG₁ is linear in δΨ_i while δG₂ is quadratic in δΨ_i. Consequently, the zeroth-order term G₀ is

G₀ =

The first-order term δG₁ is given by δG₁ = ^X

while the second order δG₂ is given by δG₂ = δG2 can be rewritten in a more concise way in terms of a matrix equation

δG₂ = 1

If G₀corresponds to stationary values, δG₁must vanish while δG₂must be positive for any fluctuations of δΨ_i and δΨ^∗_i. Therefore, if the system is in the minimum-energy ground state, the necessary conditions of energetic stability are δG₁ = 0 and all eigenvalues of matrix M are positive. The δG₁ = 0 condition actually leads to

"

−¯h²∇²

2m₁ + V₁(r) + g₁₁|Ψ_1,0|²+ g₁₂|Ψ_2,0|²

#

Ψ_1,0 = µ₁Ψ_1,0

"

−¯h²∇²

2m₂ + V2(r) + g22|Ψ2,0|²+ g12|Ψ1,0|²

#

Ψ2,0 = µ2Ψ2,0, (A.10)

which are just the time-independent coupled GP equations.

Variational method

In this appendix, we show how to apply the variational method to study the dynamics of an ultracold atom system and how to obtain the minimum-energy state for it. As an example, we consider a one-component BEC system in a magnetic trap. The corresponding energy functional is given by [see also Eq. (2.5)]

E =

Z

drΨ^∗(r)

"

−¯h²∇²

2m + Vext(r) + g

2|Ψ(r)|²

¸

Ψ(r), (B.1)

where Vext(r) = ¹₂m(w_x²x²+ w_y²y²+ w²_zz²) is the anisotropic trapping potential and g is the interaction between atoms. If the interaction vanishes (g = 0), the minimum-energy (ground) state wave function Ψ(r) of (B.1) is exactly the simple Gaussian function. Thus it is convenient to use the simple Gaussian

Ψ(r) = N^1/2

π^3/4(R_xR_yR_z)^1/2e^{−12(x2/R2x+y2/R2y+z2/R2z)} (B.2) as a trial wave function for (B.1).

Using Eq. (B.2), the energy functional (B.1) is then solved to be E(R_x, R_y, R_z) = ^X

i=x,y,z

( N 4m

1 R²_i +N

4mω_i²R²_i

)

+ N²

2(2π)^3/2R_xR_yR_z. (B.3) Firstly, one is seeking global minimum of the energy functional (B.3) upon the changes of R_x, R_y, R_z. For this purpose, one can use the Newton’s Method to determine the

81

values of R_x, R_y, R_z which lead to extrema and then use the Hessian matrix to deter-mine whether they give a maximum or minimum. The corresponding mathematical theorem is the following. Given a real-valued function F (x₁, x₂, ..., x_n) of n energy state, J(x) = 0 and the eigenvalues of the matrix H(x) need to be all positive.

To discuss the dynamics of the system, one can incorporate appropriate dynamical variables into the wavefunction [Ψ(r) → Ψ(r, t)], which are sensible to the dynamics under studies. For example, if one wants to consider the breathing modes, the time-dependent trial wave function Ψ(r, t) can be used as

Ψ(r, t) = N^1/2((1 + ²_x(t))(1 + ²_y(t))(1 + ²_z(t)))^1/4

π^3/4(R_xR_yR_z)^1/2 (B.6)

× e^{−12(x2(1+²x(t)+iζx(t))/R2x+y2(1+²y(t)+iζy(t))/R2y+z2(1+²y(t)+iζy(t))/R2z)},

where parameter ²_i(t) and ζ_i(t) correspond to the fluctuations of local amplitude and local phase of the atom cloud associated with the i-direction. The next step is to derive the corresponding Lagrangian: L =^R drT − E, where

T = i¯h

and E is the energy functional (B.1) with time-dependent wave functional Ψ(r, t).

Finally a set of coupled dynamical equations of motion can be derived through the Euler-Lagrange equations:

∂L

∂β = d dt

∂L

∂ ˙β, (B.8)

where β is any one of the dynamical variables chosen. The corresponding dynamics, in particular the collective mode dispersion relation, can be obtained by solving the roots of the dynamical matrix (diagonalization).

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在文檔中 在一個和二個組成的玻色-愛因斯坦凝結系統的研究主題 (頁 82-0)