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.
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.
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 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.
The variational JWKB method is used to determine experimentally accessible macroscopic quantum tunneling regimes of quasibound Bose-Einstein condensates in two quasi-one-dimensional trap configurations. The potentials can be created by magnetic and optical traps: a symmetric trap from two offset Gaussian barriers and a tilt trap from a linear gradient and Gaussian barrier. Scaling laws in barrier parameters, ranging from inverse polynomial to square root times exponential, are calculated and used to elucidate different dynamical regimes, such as when classical oscillations dominate tunneling rates in the symmetric trap. The symmetric trap is found to be versatile, with tunneling times at and below 1 s, able to hold 103−−104 atoms, and realizable for atoms ranging from rubidium to lithium, with unadjusted scattering lengths. The tilt trap produces subsecond tunneling times, is able to hold a few hundred atoms of lighter elements such as lithium, and requires the use of Feshbach resonance to reduce scattering lengths. To explore a large parameter space, an extended Gaussian variational ansatz is used, which can approximate large traps with Thomas-Fermi profiles. Nonlinear interactions in the Gross-Pitaevskii equation are shown to produce additional effective mean-field barriers, affecting scaling laws.
A Bose-Einstein condensate with attractive interaction can be metastable if it is spatially confined and if the number of condensate bosons N0 is below a certain critical value Nc. By applying a variational method and the instanton technique to the Gross-Pitaevskii energy functional, we find analytically the frequency of the collective excitation and the rate of macroscopic quantum tunneling. We show that near the critical point the tunneling exponent vanishes according to $({1\ensuremath{-}N}_{0}/{N}_{\mathrm{c}}{)}^{5/4}$ and that macroscopic quantum tunneling can be a dominant decay mechanism of the condensate for N0 very close to Nc.
We show theoretically the existence of a metastable state and the possibility of decay to the ground state through macroscopic quantum tunneling in two-component Bose-Einstein condensates with repulsive interactions. Numerical analysis of the coupled Gross-Pitaevskii equations clarifies the metastable states whose configuration preserves or breaks the symmetry of the trapping potential, depending on the interspecies interaction and the particle number. We calculate the tunneling decay rate of the metastable state by using the collective coordinate method under the WKB approximation. Then the height of the energy barrier is estimated by the saddle point solution. It is found that macroscopic quantum tunneling is observable in a wide range of particle numbers. Macroscopic quantum coherence between two distinct states is discussed; this might give an additional coherent property of two-component Bose condensed systems. Thermal effects on the decay rate are estimated.
We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.