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Josephson effect in multicomponent Bose-Einstein condensates

M. Melé-Messeguer

Published Jul 12, 2013• 1 citation
OpenAlex

Abstract

In this thesis we study and characterize the behavior of Bose-Einstein condensates in a BJJ, using two different theoretical formalisms: the mean-field approximation (with the Gross-Pitaevskii equation) and many-body calculations (based on Bose-Hubbard models). \n\t\t\t\t With single-component Bose-Einstein condensates, we have focused on the study of the structure of the ground state as a function of the system parameters. We have looked for strongly correlated states, that cannot be described with mean-field theories, and we have proposed a variational wave function that captures the structure of the ground state for a broad interval of the system parameters. We have also studied the nonlinear effects of the Gross-Pitaevskii equation, visible when atom-atom interactions are strong. \n\t\t\t\t In the case of binary mixtures of Bose-Einstein condensates we have performed an intensive study of the different regimes that can arise and in which conditions. The standard two-mode approximation is one of the most used in the study of the Josephson effect, as it gives simple analytic equations that capture, to a great extend, the behavior of the system. When the link between condensates is not weak enough, one has to consider a correction to this approximation, namely, the improved two-mode approximation. In this thesis, we have derived this last approximation for the binary mixture and we have checked its validity comparing it with numerical simulations of the three-dimensional Gross-Pitaevskii equation. Moreover, as the Josephson dynamics is almost one-dimensional, we have considered the two most common reductions of the dimensionality of the Gross-Pitaevskii equation. In this case, we have also compared, using simulations, these one-dimensional reductions with the three-dimensional equation. \n\t\t\t\t We have also studied spinor condensates in an external BJJ. We have focused on condensates formed by atoms with spin $F=1$, which can be in any of the three internal states $m_F = 0, \\pm 1$. Furthermore, in contrast to the binary mixture, spin interchange is allowed, so that the number of particles of each component becomes a dynamic variable. First, we have studied this system within the mean-field framework, using the Gross-Pitaevskii equation. We have derived the two-mode approximation equations and we have focused in studying the decoupling of the Josephson effect and the population transfer dynamics. In this case, we also compare the results with numerical simulations of the three dimensional Gross-Pitaevskii equation. Second, we have studied the spinor BJJ using the Bose-Hubbard formalism, because some features of quantum fluctuations are better captured than with the mean-field. We have characterized the ground state, paying special attention to the regions where it is strongly correlated. We have seen how the spin singlet formation (strongly correlated state between two particles) affects the structure of the ground state. \n\n\t\t\t\t Finally, we have studied finite temperature effects on spinor Bose-Einstein condensates in the presence of a magnetic field, for two different cases. First, we have analyzed a condensate formed by $F=1$ atoms with contact interactions. We have considered that the condensate was formed by atoms in the internal state $m_F=0$ and we have studied the dependence of the fluctuations of the other two components $m_F = \\pm 1$ as a function of temperature. We have used the Bogoliubov formalism applied to an homogeneous system, and then, we have generalized the result to an harmonic trap by using the local density approximation. Second, we have studied a condensate formed by particles with $F=3$ with contact interactions and moreover, dipolar interactions. In a similar way as the previous case, we consider that the condensate is formed by particles with $m_F =-3$ and we study the fluctuations in $m_F = -2$ and $m_F = -3$.

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This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov’s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schrödinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation.

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Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas of quantum mechanics have been realized and proven to be true in experiments, by studying the Bose-Einstein condensates. Amongst them, the observation of macroscopic matter wave interference, the creation of vortices, dark and bright solitons etc, find special mention. The dynamics of this system in presence of an optical lattice reveals a superfluid to Mott insulator quantum phase transition. This has opened a new avenue for research on BEC and its applications. From a theoretical standpoint, the dynamics of BEC can be described by means of an effective mean-field theory. This formalism is much simpler than the many-body Schr¨odinger equation and based on a classical nonlinear evolution equation, known as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc., which are closely related to nonlinear fiber optics. The present thesis deals with nonlinear excitations and different phases of BEC and is organized as follows: after a brief introduction, in chapter two, we start with BEC in a one-dimensional shallow optical lattice, experiencing both cubic and quintic nonlinearity. A superfluid density wave, of twice the periodicity of lattice potential, is identified. This indicates matter redistribution, where atoms from neighbouring sites have been depleted leading to a density wave behaviour. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In presence of pure two-body interaction, the superfluid density shows a periodicity commensurate with the lattice potential. Apart from the superfluid state, a density wave insulating phase is also found to exist, possessing two frequency modulations commensurate with the lattice potential. A dynamical phase transition connects the superfluid phase with an insulating phase, which in turn, is connected with the density wave insulating phase. The solutions are found to be marginally stable, as per the Vakhitov-Kolokolov (VK) criterion. Chapter three describes the dynamics of the atomic-molecular Bose-Einstein condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into account the atom-molecular interaction, as well as atom-molecule inter-conversion. The two-body atom-atom, atom-molecule and molecule-molecule interactions are analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic nonlinear term arising from the inter-conversion between atoms and molecules is identical to the scenario in strongly coupled BEC in one dimension. The presence of both weak and strong interactions adds further richness to this system. We find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed in order to see the effect of the both weak and strong interactions on Lieb mode profile and compare it with the pure atomic condensate profile. Another class of solutions exist in this system, where both atomic and molecular condensates possess bright solitons. In chapter four, we investigate the effect of a harmonic trap on the dynamics of grey solitons in Bose-Einstein condensates. 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A detailed study of the energy and momentum of the soliton is reported, in order to find the effect of the trap on the Lieb mode. In chapter five, we study a BEC-defect complex in a quasi-one dimensional scenario, where the defect atom moves in the condensate with constant velocity. Explicit solutions of the coupled mean-field equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can lower the grey soliton’s maximum velocity, as compared to the defect free case, where the maximum velocity is the sound velocity. We have also carried out the stability analysis of the obtained solutions using renormalized momentum formalism and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable through both the methods. The grey soliton’s energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics. Chapter six describes a parametrically forced Bose-Einstein condensate in the presence of a general moving optical lattice. The interaction between the atoms in the condensate and the time dependent lattice potential leads to a novel propagating superfluid matter wave, which can be controlled through chirp management. This system, when placed in a trap, accelerates and undergoes rapid nonlinear compression, controlled by the chirp. The density achieves its maximum, when the matter wave changes direction. It is also shown that this system exhibits dynamical superfluid insulator transition, where superfluidity breaks down and the condensate transits to an insulating phase. The stability has been investigated using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well as on the nature of the atom-atom interaction. The exact expression for energy is obtained and analyzed in detail to gain physical understanding of the chirp management of the sinusoidal excitations and also the dynamical phase transition.

•Jul 1, 2011•0 citations
In this thesis we study and characterize the behavior of Bose-Einstein condensates in a BJJ, using two different theoretical formalisms: the mean-field approximation (with the Gross-Pitaevskii equation) and many-body calculations (based on Bose-Hubbard models). \n\t\t\t\t With single-component Bose-Einstein condensates, we have focused on the study of the structure of the ground state as a function of the system parameters. We have looked for strongly correlated states, that cannot be described with mean-field theories, and we have proposed a variational wave function that captures the structure of the ground state for a broad interval of the system parameters. We have also studied the nonlinear effects of the Gross-Pitaevskii equation, visible when atom-atom interactions are strong. \n\t\t\t\t In the case of binary mixtures of Bose-Einstein condensates we have performed an intensive study of the different regimes that can arise and in which conditions. The standard two-mode approximation is one of the most used in the study of the Josephson effect, as it gives simple analytic equations that capture, to a great extend, the behavior of the system. When the link between condensates is not weak enough, one has to consider a correction to this approximation, namely, the improved two-mode approximation. In this thesis, we have derived this last approximation for the binary mixture and we have checked its validity comparing it with numerical simulations of the three-dimensional Gross-Pitaevskii equation. Moreover, as the Josephson dynamics is almost one-dimensional, we have considered the two most common reductions of the dimensionality of the Gross-Pitaevskii equation. In this case, we have also compared, using simulations, these one-dimensional reductions with the three-dimensional equation. \n\t\t\t\t We have also studied spinor condensates in an external BJJ. We have focused on condensates formed by atoms with spin F=1F=1F=1, which can be in any of the three internal states mF=0,pm1m_F = 0, \\pm 1mF​=0,pm1. Furthermore, in contrast to the binary mixture, spin interchange is allowed, so that the number of particles of each component becomes a dynamic variable. First, we have studied this system within the mean-field framework, using the Gross-Pitaevskii equation. We have derived the two-mode approximation equations and we have focused in studying the decoupling of the Josephson effect and the population transfer dynamics. In this case, we also compare the results with numerical simulations of the three dimensional Gross-Pitaevskii equation. Second, we have studied the spinor BJJ using the Bose-Hubbard formalism, because some features of quantum fluctuations are better captured than with the mean-field. We have characterized the ground state, paying special attention to the regions where it is strongly correlated. We have seen how the spin singlet formation (strongly correlated state between two particles) affects the structure of the ground state. \n\n\t\t\t\t Finally, we have studied finite temperature effects on spinor Bose-Einstein condensates in the presence of a magnetic field, for two different cases. First, we have analyzed a condensate formed by F=1F=1F=1 atoms with contact interactions. We have considered that the condensate was formed by atoms in the internal state mF=0m_F=0mF​=0 and we have studied the dependence of the fluctuations of the other two components mF=pm1m_F = \\pm 1mF​=pm1 as a function of temperature. We have used the Bogoliubov formalism applied to an homogeneous system, and then, we have generalized the result to an harmonic trap by using the local density approximation. Second, we have studied a condensate formed by particles with F=3F=3F=3 with contact interactions and moreover, dipolar interactions. In a similar way as the previous case, we consider that the condensate is formed by particles with mF=−3m_F =-3mF​=−3 and we study the fluctuations in mF=−2m_F = -2mF​=−2 and mF=−3m_F = -3mF​=−3.
This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov’s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schrödinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation.
In this thesis, the physics of trapped, interacting Bose-Einstein condensates is analyzed by solving the many-body Schrödinger equation. Particular emphasis is put on coherence, fragmentation and reduced density matrices. First, the ground state of a trapped Bose-Einstein condensate and its correlation functions are obtained. Then the dynamics of a bosonic Josephson junction is investigated by solving the time-dependent many-body Schrödinger equation numerically exactly. These are the first exact results in literature in this context. It is shown that the standard approximations of the field, Gross-Pitaevskii theory and the Bose-Hubbard model fail at weak interaction strength and within their range of expected validity. For stronger interactions the dynamics becomes strongly correlated and a new equilibration phenomenon is discovered. By comparison with exact results it is shown that a symmetry of the Bose-Hubbard model between attractive and repulsive interactions must be considered an artefact of the model. A conceptual innovation of this thesis are time-dependent Wannier functions. Equations of motion for time-dependent Wannier functions are derived from the variational principle. By comparison with exact results it is shown that lattice models can be greatly improved at little computational cost by letting the Wannier functions of a lattice model become time-dependent.
We present a study of binary mixtures of Bose–Einstein condensates confined in a double-well potential within the framework of the mean field Gross–Pitaevskii (GP) equation. We re-examine both the single component and the binary mixture cases for such a potential, and we investigate what are the situations in which a simpler two-mode approach leads to an accurate description of their dynamics. We also estimate the validity of the most usual dimensionality reductions used to solve the GP equations. To this end, we compare both the semi-analytical two-mode approaches and the numerical simulations of the one-dimensional (1D) reductions with the full 3D numerical solutions of the GP equation. Our analysis provides a guide to clarify the validity of several simplified models that describe mean-field nonlinear dynamics, using an experimentally feasible binary mixture of an F = 1 spinor condensate with two of its Zeeman manifolds populated, m = ±1.
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)
We study binary mixtures of Bose–Einstein condensates confined in a two-well potential within the framework of the Gross–Pitaevskii equation. We re-examine both the single component and the binary mixture cases for such a potential. We investigate the most usual dimensional reductions used to solve the Gross–Pitaevskii equations, including the one proposed by Reatto and collaborators. To this end, we compare numerical simulations of the 1D reductions with the full 3D numerical solutions of the Gross–Pitaevskii equation. Our analysis considers an experimentally feasible binary mixture of an F = 1 spinor condensate where two of its Zeeman manifolds are populated.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas of quantum mechanics have been realized and proven to be true in experiments, by studying the Bose-Einstein condensates. Amongst them, the observation of macroscopic matter wave interference, the creation of vortices, dark and bright solitons etc, find special mention. The dynamics of this system in presence of an optical lattice reveals a superfluid to Mott insulator quantum phase transition. This has opened a new avenue for research on BEC and its applications. From a theoretical standpoint, the dynamics of BEC can be described by means of an effective mean-field theory. This formalism is much simpler than the many-body Schr¨odinger equation and based on a classical nonlinear evolution equation, known as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc., which are closely related to nonlinear fiber optics. The present thesis deals with nonlinear excitations and different phases of BEC and is organized as follows: after a brief introduction, in chapter two, we start with BEC in a one-dimensional shallow optical lattice, experiencing both cubic and quintic nonlinearity. A superfluid density wave, of twice the periodicity of lattice potential, is identified. This indicates matter redistribution, where atoms from neighbouring sites have been depleted leading to a density wave behaviour. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In presence of pure two-body interaction, the superfluid density shows a periodicity commensurate with the lattice potential. Apart from the superfluid state, a density wave insulating phase is also found to exist, possessing two frequency modulations commensurate with the lattice potential. A dynamical phase transition connects the superfluid phase with an insulating phase, which in turn, is connected with the density wave insulating phase. The solutions are found to be marginally stable, as per the Vakhitov-Kolokolov (VK) criterion. Chapter three describes the dynamics of the atomic-molecular Bose-Einstein condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into account the atom-molecular interaction, as well as atom-molecule inter-conversion. The two-body atom-atom, atom-molecule and molecule-molecule interactions are analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic nonlinear term arising from the inter-conversion between atoms and molecules is identical to the scenario in strongly coupled BEC in one dimension. The presence of both weak and strong interactions adds further richness to this system. We find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed in order to see the effect of the both weak and strong interactions on Lieb mode profile and compare it with the pure atomic condensate profile. Another class of solutions exist in this system, where both atomic and molecular condensates possess bright solitons. In chapter four, we investigate the effect of a harmonic trap on the dynamics of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear excitations of the cigar-shaped Bose-Einstein condensate have strong coupling with the trap at low momenta and hence can be effectively isolated from the Bogoliubov sound modes, which respond weakly to harmonic confinement. This strong coupling with the trap also makes the grey soliton amenable for control and manipulation through trap modulation and temporal variation of the two-body interaction. We have shown that in presence of an expulsive oscillator trap, the grey soliton gets amplified, while having a compression in its width. Exact solutions are also obtained in case of the two component Bose-Einstein condensates (TBECs), where one component is a grey soliton and the other a bright one. The presence of the trap and inter-species interaction significantly affects the grey soliton dynamics. Similar to the single component, in the present case, we have shown that both grey and bright soliton can be compressed simultaneously. A detailed study of the energy and momentum of the soliton is reported, in order to find the effect of the trap on the Lieb mode. In chapter five, we study a BEC-defect complex in a quasi-one dimensional scenario, where the defect atom moves in the condensate with constant velocity. Explicit solutions of the coupled mean-field equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can lower the grey soliton’s maximum velocity, as compared to the defect free case, where the maximum velocity is the sound velocity. We have also carried out the stability analysis of the obtained solutions using renormalized momentum formalism and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable through both the methods. The grey soliton’s energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics. Chapter six describes a parametrically forced Bose-Einstein condensate in the presence of a general moving optical lattice. The interaction between the atoms in the condensate and the time dependent lattice potential leads to a novel propagating superfluid matter wave, which can be controlled through chirp management. This system, when placed in a trap, accelerates and undergoes rapid nonlinear compression, controlled by the chirp. The density achieves its maximum, when the matter wave changes direction. It is also shown that this system exhibits dynamical superfluid insulator transition, where superfluidity breaks down and the condensate transits to an insulating phase. The stability has been investigated using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well as on the nature of the atom-atom interaction. The exact expression for energy is obtained and analyzed in detail to gain physical understanding of the chirp management of the sinusoidal excitations and also the dynamical phase transition.