6.5 Variational approach to BECs
7.1.2 Thermal decay rate
7 Thermally induced coherent collapse of BECs with long-range interaction
Variation
al space Metastable
ground state
Activated complex
Thermal deca
y
Fig. 7.3: Visualization of the structure of the variational space for a metastable condensate.
The ground state of the BEC is metastable and the condensate can classically cross the barrier formed by a rank-1 saddle point, which means the collapse of the condensate.
The reaction rate is given by the flux over the saddle, and it can be calculated via Eq.
(4.30) after canonical coordinates have been constructed.
7.1 Monopolar BECs
-0.150 -0.140 -0.130 -0.120
-1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 ξm , |m| = 0
Scattering length a
(a) GS EX
0 2 4 6
-1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 ξm , |m| = 1
Scattering length a (b)
-300 -200 -100 0 100 200
-1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 ξm , |m| = 2
Scattering length a (c)
-100000 -50000 0 50000 100000
-1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 ξm , |m| = 3
Scattering length a (d)
Fig. 7.4: Coefficients of the normal form HamiltonianH(J) =PmξmJm in dependence of the scattering lengtha, for Ng = 2 coupled Gaussian wave functions and a trap strength γ2 = 2.5×10−4. Shown are (a) the zeroth-order coefficients|m|= 0, i. e. the fixed point energies, (b) the linear coefficients|m|= 1, i. e. the stability eigenvalues, (c) the quadratic coefficients|m|= 2, and (d) the cubic coefficients|m|= 3. The red solid lines correspond to the expansion at the ground state (GS) and the dash-dotted blue lines to the excited state (EX).
7 Thermally induced coherent collapse of BECs with long-range interaction
10-40 10-30 10-20 10-10 100
-1.2 -1.1 -1.0 -0.9 -0.8 -0.7
Decay rate Γ
Scattering length a Ng = 1
Ng = 2
Fig. 7.5: Comparison of the thermal decay rate of a monopolar BEC described withNg= 1 (red solid line) andNg = 2 (blue dash-dotted line) Gaussian wave functions in dependence of the scattering lengtha. The decay rate has been calculated for a trap strength of γ2 = 2.5×10−4, a particle number scaled inverse temperature of N2β= 900 and in sixth order normal form of the variational parameters. See text for further description.
Comparison between different numbers of Gaussians
Figure 7.5 shows the thermal decay rate of a monopolar BEC calculated in sixth order normal form in the variational parameters (third order in the action variables) using a single Gaussian trial wave function (red solid line) andNg = 2 coupled Gaussians (blue dash-dotted line). For the calculation, a particle number scaled inverse temperature of N2β = 900 has been used.
Analogously to the discussion of the mean-field energy, Fig. 7.5 reveals a shift of the whole curve describing the decay rate Γ. It shows that the thermal decay rate calculated with a single Gaussian underestimates the result of the extended variational ansatz by several orders of magnitude for a fixed value of the scattering length a. Considering the respective values of the critical scattering length, the decay rates at these points differ only very little between the approaches with Ng = 1 and Ng = 2. The general dependence of the decay rate on the scattering length is retained exhibiting a rapid monotonic increase when decreasing the value a. This increase, however, becomes weaker when one approaches the critical value.
104
7.1 Monopolar BECs
10-12 10-10 10-8 10-6 10-4 10-2 100
-1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90
Decay rate Γ
Scattering length a NFO = 2
NFO = 4 NFO = 6 NFO = 8
Fig. 7.6: Thermal decay rate of a monopolar BEC described withNg = 2 coupled Gaussian wave functions in dependence of the scattering length a and normal form orders 2≤NFO≤8 of the variational parameters. Temperature and trap frequency are the same as in Fig. 7.5. Right inset: The thermal decay rates obtained from the sixth- and eighth-order normal form Hamiltonian cannot be distinguished any more, indicating convergence. Left inset: Ata≈ −0.999 the eigenvalues are close to “resonance” (cf.
Fig. 7.7) and the normal form expansion as well as the decay rate diverge.
Comparison between different normal form orders
Figure 7.6 shows the thermal decay rates for different normal form orders. The calculations have been performed for Ng = 2 coupled Gaussians and for the same physical parameters as used in Fig. 7.5. The second-order approximation in the variational parameters (first-order in the action variables; red dashed line) overestimates the decay rate over the whole range of the scattering length, whereas using the normal form Hamiltonian in fourth order (second order in action variables; green dash-dotted line), one observes the smallest values throughout. However, the results calculated within the sixth and eighth order (orange dotted and blue solid lines) cannot be distinguished within the line width of the plot (right inset in Fig. 7.6), indicating convergence.
At a scattering length of a≈ −0.999, one observes a strong deviation of the decay rate in the eighth-order approximation from all the other curves (left inset in Fig. 7.6), which is in contrast to the behavior all along the rest of the range investigated. However, the reason of this divergence is not of physical nature. As shown in Fig. 7.7, the eigenvalues λi of the linearized equations of motion which are used for the normal form expansion
7 Thermally induced coherent collapse of BECs with long-range interaction
-0.5 0.0 0.5 1.0 1.5
-1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 λ2 + 6 λ3 - λ1
Scattering length a
Fig. 7.7: Fulfillment of the condition of resonance (3.36b) for the ground state in dependence of the scattering length for the parameters used in Fig. 7.6. Ata≈ −0.999 the resonance λ2+ 6λ3 =λ1 is fulfilled within the numerical accuracy.
run into “resonance” for the ground state. More precisely, the equation λ2+ 6λ3 =λ1 and, therewith, Eq. (3.36b) is fulfilled within the numerical accuracy in seventh-order of the variational parameters in the dynamical equations, |m| = 7 (corresponding to the eighth-order of the Hamiltonian). This leads to the divergence of the eighth-order normal form Hamiltonian and, with it, the decay rate at a≈ −0.999.
Moreover, the convergence behavior of the decay rate with increasing normal form order strongly depends on the temperature of the system [see Fig. 7.8(a)]. While one observes fast converging results for low temperatures and large particle numbers, respectively, i. e.
large values N2β, the convergence becomes worse when decreasing this value. In the first case, the decay rates calculated from the sixth- and eighth-order normal forms match within the line width for N2β &800, the calculations even show a monotonic increase of the decay rate with higher normal form order forN2β .200. Consequently, even higher-order approximations are necessary to obtain converged results in the high-temperature regime.
In order to estimate the convergence of the results, the relative deviation δi = ΓNFO=2i
ΓNFO=2i−2
−1 (7.4)
is used which is shown in Fig. 7.8(b). The corrections to the decay rate obtained from the fourth- (i= 2) and the sixth- (i= 3) order normal form are significant throughout, while this is only true for low N2β for the eighth order (i= 4). For large values of N2β the corrections quickly shrink and in case ofN2β &1000 these are of the relative order of 10−4 to 10−5, clear evidence again of the convergence of the decay rate in eighth-order approximation in the low-temperature regime.
106
7.1 Monopolar BECs
10-8 10-6 10-4 10-2
100 1000
Decay rate Γ
Inverse temperature N2β (a)
NFO = 2 NFO = 4 NFO = 6 NFO = 8
-0.5 0.0 0.5 1.0 1.5
100 1000
δi
Inverse temperature N2β
(b) i = 2
i = 3 i = 4
Fig. 7.8: (a) Thermal decay rate of a monopolar BEC for a fixed scattering length ofa≈ −0.96, a trap strength ofγ2 = 2.5×10−4and normal form orders 2≤NFO≤8 in dependence of the particle number scaled temperatureN2β. The BEC is described withNg = 2 coupled Gaussian wave functions. (b) The relative deviationδi defined in Eq. (7.4) is used in order to estimate the convergence of the procedure.
7 Thermally induced coherent collapse of BECs with long-range interaction
Discussion
Recapitulating the above investigations of monopolar BECs, the following points can be listed: Thermally excited condensates with attractive monopolar interparticle interaction exhibit the decay mechanism of the TICC. The latter is mediated by a collectively excited, unstable state of the BEC which forms the activated complex of the system and which has the meaning of a certain density distribution of the condensate. The decay rate can be calculated within a variational approach using the application of TST to wave packet dynamics developed in Sec. 4.4. The decay mechanism is important near the critical value acrit of the s-wave scattering length, where the energy barrier is small and the rates are high. Far away from this point (aacrit) this process can be neglected.
Enhancing the trial wave function by increasing the number of coupled Gaussians yields a shift of the curve to higher values of the scattering length. Further significant corrections are obtained using higher normal form orders of the constructed Hamiltonian.
Convergence of the results is, in general, expected when increasing the number of Gaussians as well as the normal form order (Ng → ∞, NFO → ∞). For two coupled Gaussians and the parameters used above, convergence has been achieved in eighth order normal form for low temperatures and high particle numbers, respectively.