7.2 Dipolar BECs
7.2.1 Dipolar BECs within a simple variational approach
In analogy to the case of a monopolar BEC, the single Gaussian approach to the dipolar condensate can be mapped to a canonical formalism globally by introducing the canonical coordinates qη = [−8Arη]−1/2 and their conjugate momenta pη =Aiη[−2/Arη]1/2 for η ∈ {x,y,z}. As it is shown in Appx. D.2, this definition transforms the energy functional into the Hamilton function
H = p2
2 +V(q) (7.5)
with the potential (cf. Ref. [120]) V(q) = 1
8q2x + 1 8qy2 + 1
8qz2 + 2γρ2(q2x+qy2) + 2γz2qz2 +
s2 π
a
8qxqyqz + 1 24√
2πqz3
"
RD q2x qz2,qy2
qz2, 1
!
− qz2 qxqy
#
. (7.6)
Here, RD(x,y,z) = 32R0∞[(x+t)(y+t)(z+t)3]−1/2dt is an elliptic integral of second kind in Carlson form [121]. For given values of the scattering length and the trap frequencies the potential (7.6) fully describes the dynamics of the BEC in the Hilbert subspace of the variational ansatz (6.24a) with Ng = 1.
In what follows the trap frequencies are fixed by their geometric mean value ¯γ = (γρ2γz)1/3 = 3.4×104 and the trap aspect ratio λ=γz/γρ= 50, if not stated otherwise.
Because of the large aspect ratio of the trap, the dipoles are predominantly aligned in
7 Thermally induced coherent collapse of BECs with long-range interaction
side-by-side configuration where they repel each other and stabilize the BEC against collapse. Therefore, only the regime of a negative s-wave scattering length (a < 0) is considered in the following which counteracts this effect.
Bifurcation in the transition state
Figure 7.9 shows contour plots of the potential (7.6) for several values of the scattering lengtha and a fixed coordinate qz, which corresponds to the value of the cylindrically symmetric excited state. Below a critical value, a < acrit [Fig. 7.9(a)], there exists no stationary point of the potential. Two of these emerge in a tangent bifurcation at acrit ≈ −0.22723, and both are located at the angle bisector, so that they are cylindrically symmetric. One represents the stable ground state of the BEC, and the other is an unstable, collectively excited state [Fig. 7.9(b)]. At a scattering lengthapb ≈ −0.22657 two additional and nonaxisymmetric states emerge from the central saddle in a pitchfork bifurcation, forming two satellite saddles [Figs. 7.9(c)–(d)] and turning the central one into a rank-2 saddle.
In analogy to the monopolar BEC, the potential (7.6) again allows for a direct interpre-tation in terms of reaction dynamics of thermally excited dipolar condensates: In the case acrit < a < apb, i. e. in the region where only the center saddle exists [Fig. 7.9(b)], a sufficient thermal excitation of the BEC may allow the system to cross the center saddle, and to escape to qx,qy → 0, which means the collapse of the condensate. In this case the reaction path will always be located on the angle bisector, and, thus, this represents a condensate which collapses in a cylindrically symmetric way. The situation changes qualitatively when the parameter regiona > apb [Fig. 7.9(c)–(d)] is reached: Since the two satellite saddles are of lower energy than the central one the reaction path now breaks the cylindrical symmetry and crosses one of the satellite saddles. Because the case qx6=qy describes a quadrupole mode of the condensate, this means that the condensate collapses with an m= 2 symmetry.
Thermal decay rate
In order to calculate the reaction rate over the rank-2-rank-1 saddle configuration, the procedure is applied as presented in Sec. 4.5: First, the stationary points of the Hamiltonian (7.5), their energy, and their eigenvalue spectrum are determined on both sides of the bifurcation. On that side where the satellite saddles exist, all states are purely real. On the other side of the bifurcation, only the state corresponding to the central saddle is real, while those corresponding to the satellite saddles are shifted onto the
110
7.2 Dipolar BECs
(a)
0.005 0.006 0.007 0.008 qx
0.005 0.006 0.007 0.008
qy
(b)
0.005 0.006 0.007 0.008 qx
0.005 0.006 0.007 0.008
qy
(c)
0.005 0.006 0.007 0.008 qx
0.005 0.006 0.007 0.008
qy
(d)
0.005 0.006 0.007 0.008 qx
0.005 0.006 0.007 0.008
qy
a<acrit acrit<a<apb
a&apb aapb
Fig. 7.9: Contour plots of the potential (7.6) in dependence of the generalized coordinates qx,qy for different values of the scattering length (a)a < acrit, (b) acrit< a < apb, (c)
a&apb, and (d)aapb. The third coordinateqz is fixed to the value corresponding
to the cylindrically symmetric excited state. The color scale is defined with respect to the central saddle and different in every plot. Orange corresponds to the energy of the central saddle, blue to lower, and red to higher values ofV. See text for further description.
7 Thermally induced coherent collapse of BECs with long-range interaction
10-2 10-1 100 101 102 103 104 105
-0.2275 -0.2270 -0.2265 -0.2260 -0.2255
Decay rate Γ
Scattering length a apb
N-1β = 0.03 N-1β = 0.04 N-1β = 0.05 N-1β = 0.06
cylindrically symmetric
broken symmetry
Fig. 7.10: Decay rate Γ of the dipolar BEC in dependence of the scattering length a and at different temperatures N−1β. The lines show the results calculated from the conventional TST rate formula (4.34) and the dots show the corresponding reaction rate obtained from the uniform rate formula (4.41). While a peak in the decay rate atapb ≈ −0.22657 occurs due to the failure of the harmonic approximation, the point of the bifurcation is crossed smoothly using the uniform rate formula. The inverse temperatures 0.03≤N−1β ≤0.06 correspond to temperatures 65 nK≤T ≤130 nK for a 52Cr BEC with a particle number of N = 50 000.
imaginary axis. Second, the normal form (4.42) is chosen which is able to describe this bifurcation scenario, and the unfolding parameter u is calculated via Eq. (4.43). Third, the approximation to the Jacobi determinant is determined from Eqs. (4.44)–(4.46), and finally the reaction rate is given by Eq. (4.41), where the integral (4.40) is calculated numerically.
Figure 7.10 shows the thermal decay rates of the dipolar BEC in leading-order TST obtained by the uniform rate formula for the rank-2-rank-1 saddle configuration according to Eq. (4.41) (dots). For comparison, the results of the conventional approach (4.34) (solid lines) are shown. In the calculations using the conventional TST rate formula (lines), the divergence of the decay rate at the bifurcation,apb ≈ −0.22657, is obvious.
By contrast, the uniform solution (dots) passes the bifurcation smoothly. It is, again, emphasized that the collapse of the BEC will be cylindrically symmetric on one side of the bifurcation, and symmetry-breaking on the other side. Near the bifurcation, however, a clear distinction cannot be made.
112
7.2 Dipolar BECs
In Fig. 7.10, the particle number scaled inverse temperatures N−1β have been chosen in a way that the temperature regime (7.1) is fulfilled for a dipolar condensate of 52Cr atoms with a magnetic moment of µ= 6µB and a particle number ofN = 50 000 as it has been realized experimentally by Griesmaier et al. [9]. For this number of bosons the values 0.03 ≤N−1β ≤0.06 correspond to temperatures 65 nK≤T ≤130 nK which is clearly below the critical temperature of Tc ≈700 nK. Thus, the treatment within the mean-field framework is justified.
On the other hand, these temperatures are high enough to activate collective oscillations of the BEC: In the relevant region of the scattering length, the particle number scaled oscillation frequencies of the monopole and the quadrupole mode are both on the order of ∼10 000 oscillations per unit time. For the above mentioned particle number, this means an oscillation frequency of ω = 107 s−1. Assigning to this frequency the temperature (6.22), one finds a value of T = 0.8 nK to determine the order on which collective oscillations are activated. Hence, for the temperatures given above the latter are sufficiently present.
Occurrence of the satellite saddles
For experiments it will be of great interest in which region of the physical parameters (trap frequency and scattering length) a symmetry-breaking collapse is to be expected. Figure 7.11 shows that the existence of the symmetry-breaking states and the corresponding regions of the scattering length crucially depend on the trap aspect ratio. While for small λ .2.8 (including prolately trapped condensatesλ <1; not shown) only the cylindrically symmetric excited states exist, the additional symmetry-breaking states appear for oblate condensates with λ &2.8. The more oblate the BEC is, the larger becomes the region in which these states are present.
By contrast, with increasing trap aspect ratio, the parameter region of the scattering length withacrit < a < apb becomes smaller and vanishes for λ→ ∞. Therefore, the trap aspect ratio is expected to be the decisive tool to switch between the two scenarios in an experiment. Note that the curve in Fig. 7.11 for the critical scattering length corresponds to the one published by Kochet al. [32].
Discussion
The investigations of the dipolar BEC within the single Gaussian approach can be summarized as follows: Due to the partially attractive nature of the DDI, such quantum
7 Thermally induced coherent collapse of BECs with long-range interaction
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
1 10 100
Scattering length a
Trap aspect ratio λ broken symmetry cylindrical symmetry
no stationary state acrit
apb
Fig. 7.11: Stability diagram of the activated complex of dipolar BECs in dependence of the scattering length a and the trap aspect ratio λ. There exists either no excited state below the critical scattering lengthacrit (blue region), only the axisymmetric activated complex (yellow) or, in addition, two symmetry-breaking states (red). The latter only exist for prolately trapped condensates at λ&2.8, and the region where they exist becomes larger with increasing trap aspect ratio.
gases are metastable and exhibit the decay mechanism of the TICC. However, because of the anisotropy of the DDI, qualitative changes can be observed compared to the spherically symmetrical interaction in Sec. 7.1. The simple variational ansatz reveals a bifurcation in which additional transition states emerge, and these lead to a symmetry-breaking collapse scenario, i. e. the collapse dynamics can feature a different symmetry than the physical system does because of the external trapping potential and the interparticle interaction. The decisive quantity by which one can switch between the two scenarios is the trap aspect ratio λ of the external trap. The bifurcation itself can be handled with the uniform rate formula developed in Sec. 4.5.