Josephson effect in multicomponent Bose-Einstein condensates

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). 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. 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. 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.

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 $m_F = -3$.

En aquesta Tesi s'estudia i es caracteritza el comportament dels condensats de Bose-Einstein en una junció bosònica de Josephson (BJJ), tot utilitzant dos formalismes teòrics diferents: l'aproximació de camp mig (amb l'equació de Gross-Pitaevskii) i càlculs de molts cossos (basats en models de Bose-Hubbard). En condensats d'una sola component, ens hem centrat en l'estudi de l'estructura de l'estat fonamental en funció dels paràmetres del sistema. Hem identificat estats altament correlacionats que no es poden descriure amb teories de camp mig, i hem proposat una funció d'ona variacional que captura l'estructura de l'estat fonamental en un ampli ventall de valors d'aquests paràmetres. També hem estudiat els efectes no lineals de l'equació de Gross-Pitaevskii, visibles quan les interaccions entre àtoms són fortes. Per condensats formats per dues components hem fet un estudi intensiu dels diferents règims que es poden formar i en quines condicions. Hem utiltizat el formalisme de camp mig i hem derivat l'aproximació bimodal estàndard millorada (I2M) tot comprovant-ne la seva validesa, comparant-la amb simulacions numèriques de l'equació de Gross-Pitaevskii tridimensional. També hem estudiat condensats espinorials en una BJJ externa. Ens hem centrat en condensats formats per àtoms amb spin $F=1$, que poden estar en qualsevol dels tres estats interns $m_F=0,\pm 1$. Primer, hem estudiat aquest sistema dins la teoria de camp mig, tot utilitzant l'equació de Gross-Pitaevskii. Hem derivat les equacions de l'aproximació bimodal, i ens hem centrat en estudiar com es desacobla l'efecte Josephson de la dinàmica d'intercanvi de partícules. Segon, hem utilitzant el formalisme de Bose-Hubbard i hem caracteritzat l'estat fonamental, tot fixant-nos en els effectes de la creació de singlets. Finalment, hem estudiat l'efecte de temperatura finita en condensats de Bose-Einstein espinorials en presència d'un camp magnètic, per dos casos ben diferenciats: 1) un condensat amb $F=1$ i interaccions de contacte i 2) un condensat amb $F=3$ i interaccions de contacte i dipolars. Per a tots dos cassos, proposem un mètode per fer termometria a molt baixes temperatures, i un mètode per refredar el sistema tot variant el camp magnètic extern.