We present a model that unifies Gaussian wave-packet evolution with the so-called ``time-dependent Thomas-Fermi approximation'' for a Bose-Einstein condensate (BEC) of strongly interacting atoms. The unified simple intuitive formalism describes the stationary or time-dependent properties of single atoms, thermal clouds, or a BEC over a wide range of interaction strengths, including intermediate interaction regimes or scenarios where both interaction regimes coexist. Excellent agreement with precise numerical calculations (Gross-Pitaevskii equations) is obtained. The model is particularly suitable for describing three-dimensional evolution in free space or in time-dependent potentials for trapping, guiding, accelerating, focusing, or splitting matter waves in interferometers. We present a unified theory of spatial coherence in matter-wave interferometers and find good agreement between analytical expressions for the evolution of the coherence length of thermal atoms or a BEC and precise numerical calculations. In addition to the insight provided by the model, it supplies a simple and useful tool for the design and performance analysis of atom interferometers with many operational parameters, where a precise numerical calculation might exhaust unrealistic calculational resources.
We present a model that unifies Gaussian wave-packet evolution with the so-called ``time-dependent Thomas-Fermi approximation'' for a Bose-Einstein condensate (BEC) of strongly interacting atoms. The unified simple intuitive formalism describes the stationary or time-dependent properties of single atoms, thermal clouds, or a BEC over a wide range of interaction strengths, including intermediate interaction regimes or scenarios where both interaction regimes coexist. Excellent agreement with precise numerical calculations (Gross-Pitaevskii equations) is obtained. The model is particularly suitable for describing three-dimensional evolution in free space or in time-dependent potentials for trapping, guiding, accelerating, focusing, or splitting matter waves in interferometers. We present a unified theory of spatial coherence in matter-wave interferometers and find good agreement between analytical expressions for the evolution of the coherence length of thermal atoms or a BEC and precise numerical calculations. In addition to the insight provided by the model, it supplies a simple and useful tool for the design and performance analysis of atom interferometers with many operational parameters, where a precise numerical calculation might exhaust unrealistic calculational resources.
We present a generalized model for the evolution of atomic wave-packets in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers using atom clouds prepared in a trap as a Bose-Einstein condensate (BEC) or as a thermal ensemble. Predictions of the model for dynamic properties such as wave-packet size and phase are in excellent agreement with explicit numerical solutions of the non-linear Gross-Pitaevskii equations and enable fast calculations of interferometric performance in regimes where full numerical solutions become impractical. As a starting point, the static Thomas-Fermi (TF) approximation for a BEC in a harmonic trap is generalized to the whole range of atom-atom interaction strengths: from non-interacting atoms (low densities) to the standard TF limit (high atomic densities, as long as the condensate approximation still holds). In particular, this generalization allows a good estimation of atomic cloud properties along the transition from a three-dimensional to a quasi-one-dimensional BEC in an elongated trap. We then develop a theoretical model of wave-packet evolution in time-dependent conditions. The model is applicable for a wide range of dynamical problems involving evolution in time-dependent potentials and in a changing mean-field atomic repulsion due to splitting and separation of wave-packets. We use the model for studying two effects that influence interferometric coherence: imperfect spatial recombination in a two-state interferometer (the so-called "Humpty-Dumpty effect") and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential).
We present a general method for calculating wave-packet evolution in the context of ultracold atoms in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers based on atom clouds initially prepared in a trap in a Bose-Einstein condensate (BEC) state or in a thermal distribution. Without having to solve explicitely the Gross-Pitaevskii or the Schrodinger equation, this provides a good estimation of dynamic wave-packet properties such as size and phase across the wave-packet, and slows to predict the coherence of the interferometer. We develop a generalized Thomas-Fermi approximation for a BEC in a harmonic trap, which is valid for any atom-atom interaction strength. In particular, this approximation also provides a good estimation of the transition from a three-dimensional to a one-dimensional BEC in an elongated trap. The approximation is followed by a theory of dynamic wave-packet evolution when the trap is fully or partially switched off and the atoms are split and propagate, for example, in free space or in a matter waveguide. The method is applied for studying two examples of interferometric effects: reduction of coherence of two-state interferometers due to imperfect spatial recombination of the two interferometer arms (the Humpty-Dumpty effect) and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential). For both effects we extend the applicability of the theory to a wide range of interferometric scenarios that were not included in previous theories.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.
This work considers the semi-classical description of two applications involving cold atoms. This is, on one hand, the behavior of a BOSE-EINSTEIN condensate in hybrid systems, i.e. in contact with a microscopic object (carbon nanotubes, fullerenes, etc.). On the other, the evolution of phase space distributions in matter wave interferometers utilizing ray tracing methods was discussed. For describing condensates in hybrid systems, one can map the GROSS-PITAEVSKII equation, a differential equation in the complex-valued macroscopic wave function, onto a system of two differential equations in density and phase. Neglecting quantum dispersion, one obtains a semiclassical description which is easily modified to incorporate interactions between condensate and microscopical object. In our model, these interactions comprise attractive forces (CASIMIR-POLDER forces) and loss of condensed atoms due to inelastic collisions at the surface of the object. Our model exhibited the excitation of sound waves that are triggered by the object’s rapid immersion, and spread across the condensate thereafter. Moreover, local particle loss leads to a shrinking of the bulk condensate. We showed that the total number of condensed particles is decreasing potentially in the beginning (large condensate, strong mean field interaction), while it decays exponentially in the long-time limit (small condensate, mean field inetraction negligible). For representing the physics of matter wave interferometers in phase space, we utilized the WIGNER function. In semi-classical approximation, which again consists in ignoring the quantum dispersion, this representation is subject to the same equation of motion as classical phase space distributions, i.e. the LIOUVILLE equation. This implies that time evolution of theWIGNER function follows a phase space flow that consists of classical trajectories (classical transport). This means, for calculating a time-evolved distribution, one has know the initial distribution and one has to solve the classical equations of motion. Concerning the initial distribution, we have studied a stationary solution of the nonlinear LIOUVILLE equation, the LAMBERT density. We saw that it agrees very well with results from singleparticle quantum mechanics as well as the MAXWELL-BOLTZMANN distribution in the weakly interacting limit. Likewise, in the strongly interacting limit, familiar results of the THOMAS-FERMI approximation are recovered. A distribution that is first prepared in a trap and then released can be described quite conveniently in terms of WIGNER functions. However, propagation in optical potentials associated to the interferometer elements (beam splitter, Pi-half-pulse) do not satisfy the condition of the semiclassical approximation. Nevertheless, one finds discrete before-after mappings that describe the effect of these elements on incident distributions. This leads to several channels of phase space propagation which relate to the interferometer paths and interferences between them. The formalism for WIGNER functions in an interferometer can be translated straightforwardly into a ray tracing algorithm. As mentioned above, this algorithm solves the classical equations of motion and computes time-evolved distributions, using values of the initial distribution. This procedure, in contrast to most analytical solutions, does not require the HAMILTON function to be quadratic (e.g. free propagation, const. acceleration, harmonic oscillator). We compared simulation results to analytic expressions in case of freely propagating GAUSS distribution. They showed perfect agreement, especially for the functional dependence of wave length and contrast of interference fringes on the laser pulse timings.
An atom interferometer is a sensitive device that has potential for many useful applications. Atoms are sensitive to electromagnetic fields due to their electric and magnetic moments and their mass allows them to be deflected in a gravitational field, thereby making them attractive for measuring inertial forces. The narrow momentum distribution of Bose-Einstein condensate (BEC) is a great asset in realizing portable atom interferometers. An example is a guided-wave atom interferometer that uses a confining potential to guide the motion of the condensate. Despite the promise of guided-wave atom interferometry with BEC, spatial phase and phase diffusion limit the contrast of the interference fringes. The control of these phases is required for successful development of a BEC-based guided-wave atom interferometer. This thesis analyses the guided-wave atom interferometer, where an atomic BEC cloud at the center of a confining potential is split into two clouds that move along different arms of the interferometer. The clouds accumulate relative phase due to the environment, spatially inhomogeneous trapping potential and atom-atom interactions within the condensate. At the end of the interferometric cycle, the clouds are recombined producing a cloud at rest and moving clouds. The number of atoms in the clouds that emerge depends on the relative phase accumulated by the clouds during propagation. This is investigated by deriving an expression for the probability of finding any given number of atoms in the clouds that emerge after recombination. Characteristic features like mean, standard deviation and cross-correlation function of the probability density distribution are calculated and the contrast of the interference fringes is optimized. This thesis found that optimum contrast is achieved through the control of total population of atoms in the condensate, trap frequencies, s-wave scattering length, and the duration of the interferometric cycle.
The wave packet dynamics in atomic systems and in Bose-Einstein condensates is investigated by means of a time-dependent variational principle. The wave packets are assumed to be parametrized by a set of time-dependent parameters. The time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations, as obtained from the time-dependent variational principle. Unfortunately, the set of equations is ill-behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. A general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well-behaved is presented. The power of the method is demonstrated for a non-integrable system with two degrees of freedom. The Gaussian wave packet (GWP) method is applied to the three-dimensional hydrogen atom. The regularization based on the introduction of Kustaanheimo-Stiefel coordinates and a fictitious time is performed. The regularization implies a transition from the three-dimensional physical position space to a four-dimensional parameter space, spanned by the Kustaanheimo-Stiefel coordinates. The regularization is accompanied by a restriction on physically allowed wave functions. The Coulomb potential is transformed to a harmonic potential and GWPs are the exact solutions, provided they fulfill the restriction. The effect of the restriction on the four-dimensional GWP is discussed and it is shown that the GWPs can satisfy the restriction if the Gaussian parameter space is reduced in a certain way. The exact analytic evolution of the restricted GWP is presented, and the expansion of a localized initial wave function in the restricted Gaussian basis set and its analytic propagation in the fictitious time are shown. Symmetry subspaces with conserved magnetic quantum number m and with conserved angular momentum quantum numbers l,m are treated separately. The method is also applied to the non-integrable H atom in in a homogeneous magnetic field and in perpendicular external electric and magnetic fields. The evolution of the wave packets is determined by the constrained time-dependent variational principle. The numerical results are compared to numerically exact values and show excellent agreement. Another class of systems where the variational Gaussian wave packet method yields good results are cold gases. In this thesis Bose-Einstein condensates are investigated, where in addition to the common short-range contact interaction two different long-range interactions, viz. a laser induced gravity-like 1/r-interaction or a magnetic dipole-dipole interaction are present. The dynamics as resulting from the time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r-interaction is investigated with both the GWP method and numerically exact calculations. It is shown that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses. The dynamics of condensates with inter-atomic magnetic dipole-dipole interaction is investigated in the mean field limit using a Gaussian trial function. The anisotropy of the magnetic dipole-dipole interaction breaks the spherical symmetry and for ordered dipoles an effectively two-dimensional system with cylindrical symmetry is obtained. Special attention is payed to the regularity of the dynamics and it is shown that a transition from regular to chaotic motion takes place with increasing energy where regions of regular and chaotic motion coexist. It is shown that stable modes exist at energies high above the saddle point energy, i.e. at energies where a collapse of the condensate is expected.