2.4 Properties of two-component BEC system
2.4.4 Equilibrium and dynamical properties
The analyses in previous section are limited to single-component systems. In the case of a two-component mixture, the system could undergo a phase transition from a miscible state to a phase-separate state when the inter-species coupling is strong.
Considering an isotropic system, when phase separation occurs one component of the atoms will have maximum density deviated from the origin, while the density of the other component remains maximum at the origin. As pointed out before, a simple Gaussian is ineffective to describe such phenomenon. Here we show how the phase separation occurs and their related physics when the modified Gaussian distributions are in use. At the same time, the dynamics (breathing modes) of the two-component system will also be studied starting from the same modified Gaussian distribution.
Equilibrium Property
The ground-state properties of the two-component bosonic atoms are first studied.
Considering that both components are described by the modified Gaussian distribu-tion
Φi(r) = Ciexp
µ
−1 2Air2
¶
cosh(Airir), (2.16)
where i = 1, 2 and the normalization factor Ci is dependent of the total number Ni of individual component i. The ground-state energy E of the system can be calculated
7
Figure 2.4 The dependence of ground-state energy E, divided by N¯hω1, on the inter-component scattering length a12 of a two-component BEC system.
Modified Gaussian distribution and numerical calculation are undertaken.
The slopes of ∂E/∂a12vs. a12are also shown [E0 ≡ E/(N¯hω1)]. Left column corresponds to N1 = N2 ≡ N = 103, while right column corresponds to N = 104. See text for the values of other parameters.
by substitution of (2.16) into Eq. (2.7). Given three coupling strength a1, a2, and a12, and the particle number N1 and N2, final result of E can be obtained by minimizing E against the variation of A1 and A2, and r1 and r2. In Fig. 2.4, the results of E are shown as a function of the inter-component scattering length a12. Two cases of equal population N1 = N2 ≡ N = 103 and N = 104 are presented. The slopes,
∂E/∂a12 vs. a12 are also shown. For easy comparison, throughout this section the parameters are taken to be the same as those in Ref. [46]. Based on the experiment of Na-Rb mixture, the scattering lengths of Na and Rb atoms are chosen to be 3 nm and 6 nm respectively. Regarding the external magnetic trap, the corresponding trap frequencies are ω1 = 2π × 310Hz for Na and ω2 = 2π × 160Hz for Rb respectively.
In both plots in Fig. 2.4 (especially ∂E/∂a12 vs. a12), one sees clear evidence that the system undergoes a transition from one phase to another at a12 ∼ 2 − 2.5 when N = 104 and at a12 ∼ 4 − 5 when N = 103. Numerical results are also shown
for comparison. It is found that MG results are very close to those of numerical calculations for the (intermediate) N = 103 case.
Fig. 2.5 shows how this transition behaves in the density profiles of the two-component system and we have presented both MG and numerical results for com-parison. The figures are plotted in a variety of interspecies scattering lengths (a12) with fixed a1 = 3 nm and a2 = 6 nm and by varying the atom number N. In the case of N = 104, it is found that both |Φ1(r)|2 and |Φ2(r)|2 have maximum away from the center for the MG cases when a12 = 0. It has to be emphasized again, that modified Gaussian distribution has a feature that the maximum density could deviate from the origin due to large nonlinear effect. Even though a12 = 0, the individual nonlinear effect is still large, which lead to such “unphysical” result. As mentioned before, it is the fact that modified Gaussian has an overall distribution more close to TFA, that one has to look into. When a12 becomes larger (a12 ∼ 2 − 2.5), the maximum of
|Φ1(r)|2 is pushed significantly away from the center, while the maximum of |Φ2(r)|2 is right at the center – the phenomenon of phase separation.
To see the phase-separation phenomenon in an alternative way, in Fig. 2.6 we plot the value of parameter r1 (for the modified Gaussian) versus a12. It is reminded that r1 is referred to the Na component in Fig. 2.5. In the case of N = 104, there is a sudden change for r1 at a12 ∼ 2 − 2.5. This is in great consistence with the ground-state energy shown in Fig. 2.4 where a sharp change also occurs at this range of a12. In view of the original definition of r1 in the first line of Eq. (2.15) (i.e., r0 there), a sudden change of r1 means a sudden change of the coordinate of density maximum (see also Fig. 2.1). This sudden change indeed corresponds to the transition of “phase separation”. In the case of N = 103, we also see a consistent picture among the results in Figs. 2.4–2.6. A phase separation occurs at a12∼ 4−5. When N = 200, the phenomenon is less obvious. However, one can still see a phase separation that
0 1 2 3 4 5 6
Figure 2.5 The density profile of individual component is plotted in a variety of the interspecies scattering length a12 for a two-component system. Both MG and numerical results are presented for comparison. From the top to the bottom are for N1 = N2 ≡ N = 104, 103 and 200 respectively. Note that in x axis, r is scaled to `2 =q¯h/m2ω2.
0 2 4 6 8 10 12 1.0
1.5 2.0 2.5 3.0 3.5 4.0
r 1 /l 2
a12 (nm)
N = 104 N = 103 N = 200
Figure 2.6 The value of the parameter r1 divided by `2 (r1 is referred to Na component in Fig. 2.5) for the modified Gaussian plotted as a function of the interspecies scattering length a12.
appears at a12∼ 9 − 10.
When MG results are compared to those of the numerical calculation, there is a good matching for them for the small N = 200 and intermediate N = 103 cases.
For the large N = 104 case, apart from the defect that density profile could show a unphysical maximum away from the origin (see Fig. 2.5 for example), MG results still match well with the numerical ones. Thus it is convincing that MG is a good trial wave function for one- and two-component BEC systems when the nonlinear effect is not too large.
Breathing Modes
Here we investigate how the phase separation affects the dynamics of a two-component BEC system. It was shown in Ref. [46] that for a two-component mixture, the collec-tive mode dispersion will have a drastic change near the occurrence of phase separa-tion. Since the property of collective mode is intimately connected to the equilibrium property of the system, the sharp transition in the ground-state energy of the system
could imply a drastic collective mode dispersion change in the same regime.
For the dynamics, we shall focus on the breathing modes of the system. We first extend the modified Gaussian distribution for each component [Eq. (2.16)] to include the dynamical variables [51, 52]:
Φi(r, t) = Ciexp
½
−1
2Ai[1 + εi(t) + iε0i(t)]r2
¾
× cosh(Airi[1 + εi(t)]1/2r). (2.17)
Here εi and ε0i correspond to the fluctuations of local amplitude and local phase of the atom cloud along the radial r direction. Ai and ri are determined earlier through minimizing the ground-state energy of the system (see previous subsection). Since we are interested in the breathing modes of the lowest energy, the fluctuation of local amplitude, εi, is coupled to the r and r2 terms in (2.17). When the value of ε1/ε2 is obtained to be positive (negative), the corresponding breathing mode is called in-phase (out-of-phase) mode.
Substitution of the above dynamical wave function into corresponding Lagrangian equation and Eq. (2.7), one can derive the coupled equations of motion [through the Euler-Lagrange equations] for εi and ε0i, and then solve a corresponding dynamical matrix to obtain the dispersions of relations for the breathing modes. In Fig. 2.7, we present the breathing mode dispersion as a function of the inter-component scattering length a12. Again, all parameters are the same as before. The lowest-energy mode results given by numerical calculations are also shown for comparison [46].
Much information is revealed in Fig. 2.7. Taking the N = 104 case as an example, when interspecies scattering length a12 is negative, the system is miscible. In-phase breathing mode has lower energy and is more easily excited. When a12 is positive, the results are divided into three regimes. When 0 ≤ a12≤ 2.5, the system is miscible and thus out-of-phase breathing mode has lower energy and more easily gets excited.
-2 0 2 4 6 8
Figure 2.7 Based on the variational method (V), in-phase and out-of-phase breathing mode dispersions are plotted as a function of a12 for a two-component boson system. In each frame, the turning point in lower curves corresponds to where the system starts to be phase separated. The lowest-energy mode result given by numerical calculation (N) are shown and compared. The crosse signs at a12 = 0 correspond to the breathing modes of individual component 1 and 2 (decoupled regime).
When the system starts to phase separate but most parts of the two components are still overlapping (2.5 ≤ a12 ≤ 4), the energy of the out-of-phase breathing mode remains lower than that of in-phase mode . When the two components are well separated (not much overlap) at a12 ≥ 4 (see also Fig. 2.5), in-phase breathing mode turns out to have lower energy again. The switch of the in-phase and out-of-phase breathing modes in the above latter two cases can be understood alternatively as follows. When phase separation occurs but the two components are still very much overlapped, out-of-phase mode is more easily excited because it corresponds to the decrease of the overlap (and the energy). However, when the two components are well separated, in-phase mode is in turn more easily excited because in this case out-of-phase mode simply increases the overlap (and the energy).
As seen in Fig. 2.7 and N = 104 case, there is one turning point in the dispersion curves. The turning point is the signature of the system being phase separated. In comparison with the numerical results (where only the lowest-energy mode is given), our result of lowest-energy mode matches theirs quantitatively before phase separa-tion arises. After the phase separasepara-tion occurs, our result is qualitatively similar to numerical result. It is worth noting that the two points of a12 = 0 correspond actually to the breathing modes of two decoupled components (lower-frequency one is for Rb, while higher-frequency one is for Na). From the goodness of the match of our results to numerical result, it strongly suggests that modify Gaussian is also a good function to describe the dynamics of a one-component system when the nonlinear effect is important. Importantly the in-phase and out-of phase breathing modes in numerical result are verified by a nonlinear response analysis [46]. So there is no the exchange of the in-phase and out-of-phase breathing modes by numerical calculation in the regime where the two component are well separated. It is because the definition is different in variation method and numerical calculation.