We apply transition state theory to coupled Gaussian wave packets and calculate thermal decay rates of Bose–Einstein condensates with additional long-range interaction. The ground state of such a condensate is metastable if the contact interaction is attractive and a sufficient thermal excitation may lead to its collapse. The use of transition state theory is made possible by describing the condensate within a variational framework and locally mapping the variational parameters to classical phase space as has been demonstrated in the preceding paper (Junginger et al 2012 J. Phys. A: Math. Theor. 45 155201). We apply this procedure to Gaussian wave packets and present results for condensates with monopolar 1/ r -interaction comparing decay rates obtained by using different numbers of coupled Gaussian trial wavefunctions as well as different normal form orders.
We apply transition state theory to coupled Gaussian wave packets and calculate thermal decay rates of Bose–Einstein condensates with additional long-range interaction. The ground state of such a condensate is metastable if the contact interaction is attractive and a sufficient thermal excitation may lead to its collapse. The use of transition state theory is made possible by describing the condensate within a variational framework and locally mapping the variational parameters to classical phase space as has been demonstrated in the preceding paper (Junginger et al 2012 J. Phys. A: Math. Theor. 45 155201). We apply this procedure to Gaussian wave packets and present results for condensates with monopolar 1/ r -interaction comparing decay rates obtained by using different numbers of coupled Gaussian trial wavefunctions as well as different normal form orders.
Bose-Einstein condensates are bosonic many-body systems in which the quantum mechanical ground state is macroscopically occupied. If the interparticle interaction between the single bosons is attractive, this ground state is metastable and separated from the collapse by a barrier. A sufficient thermal excitation of the condensate can enable the crossing of this barrier and induce its collapse. In this thesis, the corresponding decay mechanism of the thermally induced coherent collapse of Bose-Einstein condensates with additional long-range interaction is investigated for the first time. For the description of the condensate’s dynamics a variational approach to the quantum wave function is used, which allows for the precise determination of the system’s transition state that governs the reaction dynamics. In order to quantitatively calculate the decay rate and lifetime of the condensate, respectively, a novel transition state theory for quantum wave packets is developed. This is based on a variational approach as well as normal form expansions of the dynamical equations and the energy functional in the vicinity of the transition state. The dynamical equations obtained from the variational approach induce a symplectic structure onto the space of variational parameters. Furthermore, the normal form expansions serve as a tool to locally construct canonical coordinates on this symplectic manifold. For this purpose, the dynamical equations as well as the energy functional are expanded and diagonalized. The eigenvectors of the linearized equations of motion thereby define a symplectic basis of the system. Afterwards, the transformations of the higher-order terms are carried out in two steps via nonlinear near-identity transformations: In the first step, the nonresonant terms of the expansions are eliminated and in the second step the remaining resonant terms are transformed in such a way that they fulfill canonical equations. The combination of these methods finally allows for the application of the well-established classical transition state theory. The procedure is valid for a wide variety of trial wave functions, for arbitrary dimensions of the variational space, in arbitrary order of the expansions, and it is independent of the precise form of the Hamilton operator. In addition, a procedure to handle bifurcations in the transition state is developed using a uniform rate formula in connection with an appropriate normal form of the potential energy surface. In contrast to the usual treatment which breaks down at the point of the bifurcation, the new method allows to cross the bifurcation smoothly, because it is valid on both sides of the bifurcation as well as arbitrarily close to it. At the same time, one can calculate the flux over the whole configuration of saddles which emerge in the bifurcation. The applicability of the newly developed transition state theory for wave packets is demonstrated for a cubic potential well. The thermal decay rates of a point particle and a frozen Gaussian wave packet, respectively, are calculated within the classical and quantum normal form approaches as well as the variational one. It is shown that the variational approach is capable to perfectly reproduce both the classical and quantum normal form results of the point particle in different limits. In the limit of a narrow trial wave function it agrees perfectly with the results using the classical normal form. The results with a broader trial wave function go even beyond the classical approach, i.e. they coincide with those of the quantum normal form. In the field of Bose-Einstein condensates with long-range interaction the application of transition state theory for quantum wave packets is extended to the field of nonlinear Schrödinger systems. For condensates with attractive 1/r-interaction, the thermal decay rates are calculated for different numbers of coupled Gaussian wave functions as well as different normal form orders. Significant corrections of the decay rate are obtained with increasing order of the normal forms. In the low-temperature regime convergence is observed in eighth order of the expansions. Finally, the thermally induced coherent collapse of dipolar condensates is investigated. These exhibit the feature of a biconcave ground state density distribution for certain values of the physical parameters. It is shown that such condensates reveal the astonishing effect of a symmetry-breaking collapse dynamics. This effect is directly related to the occurrence of the structured ground state, and it depends on the geometry of the external trapping potential. In case of a “conventional” density distribution, the collapse possesses an s-wave symmetry, while it has a d-wave symmetry in case of a blood-cell shaped ground state. The thermally induced coherent collapse plays a major role close to the stability threshold where the energy barrier is small and the decay rates are high. In case of Cr atoms, dipolar condensate have been first realized with, the region of interest is on the order of a Bohr radius above the critical scattering length. There, the lifetime of the condensate can be reduced to a millisecond. For the elements Dy and Er which have also been condensed in the laboratory, the effects become even stronger, i.e. the parameter region of importance becomes larger, the energy barriers become smaller, and the decay rates are higher.
We demonstrate the application of transition state theory to wave packet dynamics in metastable Schrödinger systems which are approached by means of a variational ansatz for the wavefunction and whose dynamics is described within the framework of a time-dependent variational principle. The application of classical transition state theory, which requires knowledge of a classical Hamilton function, is made possible by mapping the variational parameters to classical phase space coordinates and constructing an appropriate Hamiltonian in action variables. This mapping, which is performed by a normal form expansion of the equations of motion and an additional adaptation to the energy functional, as well as the requirements to the variational ansatz are discussed in detail. The applicability of the procedure is demonstrated for a cubic model potential for which we calculate thermal decay rates of a frozen Gaussian wavefunction. The decay rate obtained with a narrow trial wavefunction agrees perfectly with the results using the classical normal form of the corresponding point particle. The results with a broader trial wavefunction go even beyond the classical approach, i.e. they agree with those using the quantum normal form. The method presented here will be applied to Bose–Einstein condensates in the following paper (Junginger et al 2012 J. Phys. A: Math. Theor. 45 155202).
The ground state of Bose-Einstein condensates with attractive particle interactions is metastable. One of the decay mechanisms of the condensate is a collapse by macroscopic quantum tunneling, which can be described by the bounce trajectory as a solution of the time-dependent Gross-Pitaevskii equation in imaginary time. For condensates with an electromagnetically induced gravity-like interaction, the bounce trajectory is computed with an extended variational approach using coupled Gaussian functions and is simulated numerically exact within the mean-field approach on a spacetime lattice. It is shown that the variational computations converge very rapidly to the numerically exact result with increasing number of Gaussians. The tunneling rate of the condensate is obtained from the classical action and additional parameters of the bounce trajectory. The converged variational and numerically exact results drastically improve by several orders of magnitude the decay rates obtained previously with a simple variational approach using a single Gaussian-type orbital for the condensate wave function.
Bose-Einstein condensates with an attractive 1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov--de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.
The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this article will be applied to Bose-Einstein condensates with monopolar 1/r and dipolar 1/r3 interaction in the subsequent article [S. Rau et al., Phys. Rev. A 82, 023611 (2010)], where we will present a wealth of phenomena obtained using the ansatz with coupled Gaussian functions.
The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.