Abstract. The ground states of two-component Bose–Einstein condensates with a Josephson junction can be described by minimizers of the problem [Formula: see text] where [Formula: see text] is the Gross–Pitaevskii energy functional [Formula: see text] [Formula: see text] [Formula: see text] By introducing some technical blow-up arguments, we classify how the Rabi frequency [Formula: see text] and potential [Formula: see text] affect the existence and nonexistence of minimizers at one of the endpoints [Formula: see text] or [Formula: see text], where [Formula: see text] and [Formula: see text] can be given explicitly. This improves the results of [W. Bao and Y. Cai, East Asian J. Appl. Math., 1 (2011), pp. 49–81] and [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183–230], where [Formula: see text], or both [Formula: see text] and [Formula: see text] are nonendpoints. We also investigate the uniqueness of minimizers for certain local homogeneous potentials as [Formula: see text]. Different from the case of [Formula: see text] in [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183–230], [Formula: see text] causes an uncontrolled error term in estimating the difference of Euler–Lagrange multipliers. To overcome this difficulty, we first establish the relationship between the [Formula: see text]-norm of minimizers and the partial derivatives of [Formula: see text] for almost all [Formula: see text] and then show that the Euler–Lagrange multipliers are independent of the choice of minimizers.
Abstract. The ground states of two-component Bose–Einstein condensates with a Josephson junction can be described by minimizers of the problem [Formula: see text] where [Formula: see text] is the Gross–Pitaevskii energy functional [Formula: see text] [Formula: see text] [Formula: see text] By introducing some technical blow-up arguments, we classify how the Rabi frequency [Formula: see text] and potential [Formula: see text] affect the existence and nonexistence of minimizers at one of the endpoints [Formula: see text] or [Formula: see text], where [Formula: see text] and [Formula: see text] can be given explicitly. This improves the results of [W. Bao and Y. Cai, East Asian J. Appl. Math., 1 (2011), pp. 49–81] and [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183–230], where [Formula: see text], or both [Formula: see text] and [Formula: see text] are nonendpoints. We also investigate the uniqueness of minimizers for certain local homogeneous potentials as [Formula: see text]. Different from the case of [Formula: see text] in [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183–230], [Formula: see text] causes an uncontrolled error term in estimating the difference of Euler–Lagrange multipliers. To overcome this difficulty, we first establish the relationship between the [Formula: see text]-norm of minimizers and the partial derivatives of [Formula: see text] for almost all [Formula: see text] and then show that the Euler–Lagrange multipliers are independent of the choice of minimizers.
In this paper, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an internal atomic Josephson junction, and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented, for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported, to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.
We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number [Formula: see text] becomes large. In the dilute regime, when the interaction potentials have the length scale of order [Formula: see text], we show that the leading order of the ground state energy is captured correctly by the Gross–Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross–Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is [Formula: see text], we are able to verify Bogoliubov’s approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaptation to the multi-component setting is non-trivial in various respects and the analysis will be presented in detail.
In this paper, we consider two-component Bose-Einstein condensates with an internal atomic Josephson junction in the general case, i.e., 0 < p < (d−2)+2 . We prove existence and uniqueness results for the ground states, and obtain some properties of the ground states with large parameters.
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 mF=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=1 atoms with contact interactions. We have considered that the condensate was formed by atoms in the internal state mF=0 and we have studied the dependence of the fluctuations of the other two components mF=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=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=−3 and we study the fluctuations in mF=−2 and mF=−3.
We investigate the existence of ground states for two-component Bose--Einstein condensates with intraspecies interactions a1,a2∈(0,a∗) and interspecies interaction β>0. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface γ=γ(a1,a2) between β∗:=(a∗−a1)(a∗−a2) and β∗:=a∗−2a1+a2. This surface provides a complete classification for the existence of minimizers: ground states exist for β∈(0,γ) and do not exist for β>γ. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for β=γ. Our results close the gap case β∈[β∗,β∗] when a1=a2 in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.
The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. We apply GFSI to the two-component scenario with Josephson junction and rotating term, which is one of the most important and topical models in multi-component Bose-Einstein condensates (MBECs), and rigorously establish the following fundamental results for the first time. By introducing a Lagrange multiplier to reformulate GFSI into an equivalent form, we prove its energy dissipation property and global convergence to stationary states. More significantly, we uncover an intrinsic connection between this classical numerical PDE discretization rooted in imaginary-time evolution and Riemannian optimization, a state-of-the-art mathematical framework for manifold-constrained optimization. This connection enables us to fully characterize the local convergence behavior of GFSI within the Riemannian optimization framework. Together with the aforementioned global convergence result, this yields a complete global--local convergence theory for GFSI. Finally, numerical experiments comprehensively validate the theoretically predicted energy dissipation and convergence properties.