In this work, we propose a new approach to excite a Bose-Einstein condensate by introducing perturbations to the trapping potential through the separate fields technique, originally introduced by Norman F. Ramsey as an enhancement of I. I. Rabi method used for studying spin magnetic resonances. We employed the variational method based on the Thomas-Fermi wave function Ansatz to obtain dynamical equations governing the radii of a Bose-Einstein condensate when under external perturbations. The obtained dynamical equations allow us to construct the resonance curves for the collective modes of Bose- Einstein condensates, specifically focusing on the dipole, quadrupole and breathing modes. We observed that depending on whether the external signal is modulated by the Rabi or Ramsey procedure, the system behaves differently due to its inherent coherence, that reflects on the resonance curve of the collective modes, giving rise to the Ramsey fringes in a clear interference pattern, enabling a more precise frequency measurement. With the insights offered by the variational method, we introduced an alternative model to describe the dynamics of collective modes based on a three-level system, which proved to be consistent with the variational model. Moreover, the three-level model provided a framework for fostering the idea of coherent control by manipulating the population associated with collective modes through the application of perturbation pulses, such as a π-pulse and a π/2-pulse. This concept draws an analogy with interferometry in atoms. As a complementary study, we numerically solved the three-dimensional Gross-Pitaevskii equation in order to take into account other types of potential fluctuations not considered in both variational and three-level models. Finally, to present ongoing analyses and their perspectives, we used the Rabi excitation protocol to study how the external modulation leads to specific routes in terms of generation of quantum turbulence. By examining the behavior of the system under various excitations, this research offers new perspectives in the non-equilibrium BEC field, fostering a deeper understanding of BEC dynamics and its potential applications in quantum interference and turbulence studies.
In this work, we propose a new approach to excite a Bose-Einstein condensate by introducing perturbations to the trapping potential through the separate fields technique, originally introduced by Norman F. Ramsey as an enhancement of I. I. Rabi method used for studying spin magnetic resonances. We employed the variational method based on the Thomas-Fermi wave function Ansatz to obtain dynamical equations governing the radii of a Bose-Einstein condensate when under external perturbations. The obtained dynamical equations allow us to construct the resonance curves for the collective modes of Bose- Einstein condensates, specifically focusing on the dipole, quadrupole and breathing modes. We observed that depending on whether the external signal is modulated by the Rabi or Ramsey procedure, the system behaves differently due to its inherent coherence, that reflects on the resonance curve of the collective modes, giving rise to the Ramsey fringes in a clear interference pattern, enabling a more precise frequency measurement. With the insights offered by the variational method, we introduced an alternative model to describe the dynamics of collective modes based on a three-level system, which proved to be consistent with the variational model. Moreover, the three-level model provided a framework for fostering the idea of coherent control by manipulating the population associated with collective modes through the application of perturbation pulses, such as a π-pulse and a π/2-pulse. This concept draws an analogy with interferometry in atoms. As a complementary study, we numerically solved the three-dimensional Gross-Pitaevskii equation in order to take into account other types of potential fluctuations not considered in both variational and three-level models. Finally, to present ongoing analyses and their perspectives, we used the Rabi excitation protocol to study how the external modulation leads to specific routes in terms of generation of quantum turbulence. By examining the behavior of the system under various excitations, this research offers new perspectives in the non-equilibrium BEC field, fostering a deeper understanding of BEC dynamics and its potential applications in quantum interference and turbulence studies.
Collective modes are coherent excitations in Bose–Einstein condensates (BECs), and their study provides insight into the macroscopic quantum phenomena that govern these systems. Collective mode frequencies can be used to probe the properties of BECs, such as the trap geometry, the interatomic interactions, and the presence of defects; hence, it is essential to develop methods for high-resolution determination of collective mode frequencies. A standard technique consists of a single pulse of an external oscillatory field, which the authors denote Rabi-like due to the analogy with the field of nuclear magnetic resonances. In this work, the authors propose a method to achieve a better resolution than the Rabi-like protocol, which consists of two oscillating fields separated in time, which the authors call Ramsey-like. The authors focus on BECs in harmonic traps, considering mainly the quadrupole and breathing modes for different trap anisotropy values and comparing the results using single and double excitation periods. First, the authors employ a variational approach with a Thomas–Fermi ansatz, which gives rise to dynamical equations that are numerically solved. Then, the authors modeled the problem as a three-level system, offering a concise and alternative description of the same dynamics, enabling the idea of coherent control of these collective modes. Both approaches show that the Ramsey-like protocol provides a better resolution than the Rabi-like. This offers the possibility of measuring collective mode frequencies with higher precision, which is important in experiments where they are used as indirect measurements of properties that are difficult to probe.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.
In this thesis, we present the generation and studies of a 87Rb Bose-Einstein condensate (BEC) perturbed by an oscillatory excitation. The atoms are trapped in a harmonic magnetic trap where, after an evaporative cooling process, we produce the BEC. In order to study the effect caused by oscillatory excitations, a quadrupole magnetic field time oscillatory is superimposed to the trapping potential. Through this perturbation, collective modes were observed. The dipole mode is excited even for low excitation amplitudes. However, a minimum excitation energy is needed to excite the condensate quadrupole mode. Observing the excited cloud in TOF expansion, we note that for excitation amplitude in which the quadrupole mode is excited, the cloud expands without invert its aspect ratio. By looking these clouds, after long time-of-flight, it was possible to see vortices and, sometimes, a turbulent state in the condensed cloud. We calculated the momentum distribution of the perturbed BECs and a power law behavior, like the law to Kolmogorov turbulence, was observed. Furthermore, we show that using the method that we have developed to calculate the momentum distribution, the distribution curve (including the power law exponent) exhibits a dependence on the quadrupole mode oscillation of the cloud. The randomness distribution of peaks and depletions in density distribution image of an expanded turbulent BEC, remind us to the intensity profile of a speckle light beam. The analogy between matter-wave speckle and light speckle is justified by showing the similarities in the spatial propagation (or time expansion) of the waves. In addition, the second order correlation function is evaluated and the same dependence with distance was observed for the both waves. This creates the possibility to understand the properties of quantum matter in a disordered state. The propagation of a three-dimensional speckle field (as the matter-wave speckle described here) creates an opportunity to investigate the speckle phenomenon existing in dimensions higher than 2D (the case of light speckle).
The extension of quantum phenomena into macroscopic scales is responsible for a whole class of effects such as superconductivity, superfluidity, and Bose-Einstein condensation, which played central roles in physics throughout the last century.The production of the first Bose-Einstein condensates made possible the realization of experiments involving macroscopic quantum phenomena with an unprecedented level of control of the external parameters.The persistent currents in condensates are intimately related to the nucleation of quantized vortices, which are topological defects as response to transference of quanta of angular momentum.A conventional method for generation of such defects consists in confining the condensed atomic cloud into a rotating trap.It turns out that, for angular velocities higher than a critical value, vortex states become energetically favorable, thus inducing the creation of quantized vortices.Experimental realizations of condensed alkali-metal atoms confined by more general time-dependent potentials allowed the observation not only of vortex lattices but also of quantum turbulence.Since quantum turbulence is characterized by the presence of a self-interacting tangle of quantized vortices, the correct understanding of dynamics, formation, and stability of vortices has shown to be of paramount importance being the subject of many theoretical works.In particular, the role of acoustic excitations generated by decaying multi-charged vortices in the development of turbulence is still an open question.This work aims to provide a set of tools that helps to identify the presence as well as the charge of vortices in non-turbulent clouds observed using time-of-flight pictures.We have done a detailed study of condensates containing multi-charged vortices placed at its center where time-of-flight dynamics is only one point of our interest.Due to the control that this system provides experimentally, the collective modes become an important description since they can be excited using well stablished experimental methods as such as modulation of the s-wave scattering length, and they can also be responsible to vortex decaying.For such purposes we have used the semi-analytical variational method, and the fully numerical calculation of Gross-Pitaevskii equation.Thus we have describes the collective modes that couples dynamics of vortex with the oscillation of external components of condensed atomic cloud as well as the effects in time-of-flight.The angular momentum acts increasing the kinetic energy around the vortex core, which results in a faster expansion of perpendicular direction to it.This situation shifts the frequencies of collective oscillations of a vortex-free state, and generates richer collective modes due the coupling.Now there are four possible modes, being two types of monopole mode and two types of quadrupole modes.The difference among these types is the phase of vortex oscillation.When one considers fluctuations without polar symmetry, their collective modes result in the vortex decaying.In order to control and prevent these processes we have proposed three dynamical mechanisms such as modulation of s-wave scattering length, modulation of frequencies of harmonic trap, and modulation of the amplitude of Laguerre-Gauss potential.The last one has proven to be more effective.
In this thesis, we present the generation and studies of a 87 Rb Bose-Einstein condensate (BEC) perturbed by an oscillatory excitation. The atoms are trapped in a harmonic magnetic trap where, after an evaporative cooling process, we produce the BEC. In order to study the effect caused by oscillatory excitations, a quadrupole magnetic field time oscillatory is superimposed to the trapping potential. Through this perturbation, collective modes were observed. The dipole mode is excited even for low excitation amplitudes. However, a minimum excitation energy is needed to excite the condensate quadrupole mode. Observing the excited cloud in TOF expansion, we note that for excitation amplitude in which the quadrupole mode is excited, the cloud expands without invert its aspect ratio. By looking these clouds, after long time-of-flight, it was possible to see vortices and, sometimes, a turbulent state in the condensed cloud. We calculated the momentum distribution of the perturbed BECs and a power law behavior, like the law to Kolmogorov turbulence, was observed. Furthermore, we show that using the method that we have developed to calculate the momentum distribution, the distribution curve (including the power law exponent) exhibits a dependence on the quadrupole mode oscillation of the cloud. The randomness distribution of peaks and depletions in density distribution image of an expanded turbulent BEC, remind us to the intensity profile of a speckle light beam. The analogy between matter-wave speckle and light speckle is justified by showing the similarities in the spatial propagation (or time expansion) of the waves. In addition, the second order correlation function is evaluated and the same dependence with distance was observed for the both waves. This creates the possibility to understand the properties of quantum matter in a disordered state. The propagation of a three-dimensional speckle field (as the matter-wave speckle described here) creates an opportunity to investigate the speckle phenomenon existing in dimensions higher than 2D (the case of light speckle).
Multiple mode couplings in topological coherent modes of Bose–Einstein condensate are considered, by introducing an external alternating (resonating) field in the system. This analysis is based on the analytical solutions of nonlinear Gross–Pitaevskii equation for a trapped Bose gas at nearly absolute zero temperature. The dynamics of fractional populations of the generated coherent modes are analysed, particularly for a three-level system in the limit of small to large detuning of the intermediate state. These coupled topological modes, though nonlinear, are analogous to a resonant atom and exhibit a variety of significant non-trivial phenomena (effects), like: dynamic phase transitions, interference patterns, critical phenomena, mode-locking and chaotic motion.