4.5 Bifurcations in the transition state
5.2.2 Thermal decay in the cubic potential well – Comparison with the
5.2 Results
5.2.2 Thermal decay in the cubic potential well – Comparison with
5 Thermal decay of a metastable state in a one-dimensional cubic potential
10-8 10-7 10-6 10-5 10-4
0 2 4 6 8 10 12
Decay rate Γ
γ
classical normal form quantum normal form variational (a = 1.9) variational (a = 3.0) variational (a = 10.0) variational (a → ∞ )
Fig. 5.4: Comparison of the thermal decay rates of a particle placed in the metastable cubic potential well (5.1). The rates are calculated by the classical normal form (solid line), the quantum normal form (dashed line) and the variational approach with frozen Gaussian wave functions of different widtha(dots). The data have been calculated in dependence of the parameterγ, for a temperature ofβ = 1/kBT = 1.5 and a barrier height ofα= 10. See text for further description.
Figure 5.4 shows a comparison of the thermal decay rates calculated for the cubic potential by the three methods in dependence of the parameter γ of the potential and for different width parameters a of the Gaussian trial wave function. The temperature is set to β = 1.5 and a barrier height ofα= 10 is used.
The classical (solid line) and quantum (dashed line) normal forms reveal a significantly differing decay rate which is especially pronounced for small γ due to the quantization effects in the narrow potential well. The quantum rate is higher because it takes into account tunneling through the barrier, so that reactions become possible already at excitation energies which are lower than the height of the barrier. Furthermore, the barrier heightE‡ =Eex−Egs itself becomes smaller because of the shift of the ground state’s energy to higher values and that of the excited state to lower ones as it has been discussed above.
The variational approach with a single frozen Gaussian trial wave function, however, is able to reproduce both curves. In the limit of a very narrow wave function (empty circles), i. e. a → ∞, one finds a perfect match of the latter with the classical result.
When the width of the Gaussian wave function is increased, the decay rate increases
80
5.2 Results
as well (triangles and filled circles). For a width of a = 1.9 (squares) the decay rate calculated from the quantum normal form of the corresponding point particle is finally recovered. These observations support the above discussion of the width parameter a as an effective inverse Planck constant ~eff =ωcl/(2a) of the system. The small deviation of the value a= 1.9 from the above estimated value a= 1.97 can be explained by the fact that the latter has been determined from a harmonic approximation of the potential that does not take into account its precise form.
One further observes this behavior at different temperatures as can be seen in Fig. 5.5 for some exemplary values. Again, the results using the classical normal form (solid lines) are perfectly recovered within the limit a→ ∞ of very narrow wave functions (empty circles). For broad ones (squares) the decay rates agree with those obtained from the quantum normal form (dashed lines).
With increasing temperature, the decay rates also increase rapidly by several orders of magnitude. However, the difference between the classical and the quantum normal forms becomes smaller as well as it does for the two limits of the wave function concerning their width. At high temperatures compared to the height of the potential barrier, kBT &E‡, the classical limit is reached. For the parameters used here, this is the case for β .0.1, where the difference of the decay rates calculated by the three methods vanishes and they cannot be distinguished any more.
5 Thermal decay of a metastable state in a one-dimensional cubic potential
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
0 2 4 6 8 10 12
Decay rate Γ
γ
β = 1.5 β = 1.0 β = 0.5 β = 0.1 classical normal form quantum normal form broad wave function narrow wave function
Fig. 5.5: Thermal decay rates for the cubic potential at different temperatures. As in Fig. 5.4 the results of the classical (solid lines) and quantum (dashed lines) normal forms as well as the variational approach in the limit of broad (squares) and narrow (empty circles) Gaussian trial wave functions are compared. With increasing temperature, the decay rates also increase and the difference between the classical and quantum normal forms becomes smaller in the same way as the two cases of narrow and broad wave functions do within the variational approach. Forβ .0.1 the decay rates calculated by the different methods cannot be distinguished any more.
82
6 Bose-Einstein condensates
The investigation of BECs has been a field of growing interest over the last decades on both the theoretical as well as the experimental side. As already mentioned in the introduction, these macroscopic quantum objects are of high interest because they form nearly ideal quantum laboratories which are extremely controllable and manipulatable.
The phenomenon of Bose-Einstein condensation can be observed in systems of indistin-guishable particles with integer spin, and it sets in when the thermal de Broglie wave length becomes comparable to the mean particle separation in a gas. In the laboratory, BECs are usually created in external trapping potentials that can typically be well approximated by a harmonic trap. Thus, the main aspects of this phenomenon can be understood in the simple treatment of an ideal Bose gas in a harmonic trap [98, 99]. In this framework, the single bosons are treated without interparticle interactions, so that the basic results can be widely obtained analytically. It can be shown that the phase transition sets in at a critical temperature
Tc= ~ω¯ kB
N ξ(3)
!1/3
≈0.94~ω¯
kB N1/3, (6.1)
where ¯ω = (ωxωyωz)1/3 is the mean strength of the external trap, N is the number of bosons and ξ(x) is the Riemann zeta function. Below this critical temperature (T ≤Tc), the occupation N0 of the ground state behaves as
N0
N = 1−
T Tc
3
. (6.2)
Although the presence of interparticle interactions modifies the results, Eqs. (6.1) and (6.2) can also be used as a rough estimate to these quantities in real BECs.
In this chapter, the theoretical description of BECs with long-range interaction is presented and their excitations are discussed. In Sec. 6.1, the general treatment of a Bose gas at zero and finite temperature is demonstrated briefly. At arbitrary high temperature, it can be described by the Hartree-Fock-Bogoliubov equations (HFBE) [22, 23] which reduce to the Gross-Pitaevskii equation (GPE) [24, 25] if the condensate depletion can be neglected. Section 6.2 introduces the scattering interaction, a “gravity-like”
monopolar 1/r-interaction, and the dipole-dipole interaction (DDI). In Sec. 6.4
single-6 Bose-Einstein condensates
and quasiparticle excitations of BECs are discussed, and the treatment of BECs within a variational framework is presented in Sec. 6.5.
6.1 Theoretical description of Bose-Einstein condensates at zero and finite temperature
In second quantization, a gas of interacting bosons with mass m is described by the Hamilton operator [22, 23, 100]
Hˆ =
Z
d3r Ψˆ†(r,t)
"
− ~2
2m∆ +Vext(r)
#
Ψ(r,ˆ t) +1
2
Z Z
d3r d3r0 Ψˆ†(r,t) ˆΨ†(r0,t)Vint(r,r0) ˆΨ(r0,t) ˆΨ(r,t) , (6.3) where Vext(r) is an external trapping potential and Vint(r,r0) describes the interparticle interaction between the bosons. Furthermore, the bosonic field operators ˆΨ(†)(r,t) obey the equal-time commutation relations
hΨ(r,ˆ t), ˆΨ(r0,t)i=hΨˆ†(r,t), ˆΨ†(r0,t)i= 0 , (6.4a)
hΨ(r,ˆ t), ˆΨ†(r0,t)i=δ(r−r0) . (6.4b) The dynamics of the system is then given by the Heisenberg equations of motion for the field operator
i~
∂Ψ(r)ˆ
∂t =hΨ(r), ˆˆ Hi
=
"
−~2
2m∆ +Vext(r)
#
Ψ(r) +ˆ
Z
d3r0 Ψˆ†(r0)Vint(r,r0) ˆΨ(r0) ˆΨ(r) , (6.5) where the time argument t has been omitted for brevity. Eq. (6.5) can handle BECs with arbitrary internal and external interactions as well as all relevant temperatures.