Collective modes of two-species Bose-Einstein condensates in a Josephson junction barrier

TL;DR

The paper investigates how a central repulsive Josephson barrier alters the low-energy collective modes of quasi-1D Bose-Einstein condensates, including binary mixtures in miscible and immiscible phases. It combines a variational Gaussian approach with self-consistent Hartree-Fock-Bogoliubov-Popov theory to derive analytic mode frequencies and solve the coupled BdG equations, revealing barrier-induced mode softening and the emergence of Goldstone modes when symmetry is broken by the barrier. In miscible TBECs, two in-phase/out-of-phase dipole modes soften and bifurcate into two additional zero-energy modes, resulting in four Goldstone modes at sufficient barrier strength; immiscible TBECs show geometry-dependent spectra with barrier-driven decoupling and possible extra Goldstone modes, particularly in sandwich-type ground states. Overall, the barrier effect is shown to be strongly shaped by interspecies correlations and ground-state density topology, offering insight into Josephson-barrier physics in ultracold mixtures and potential parallels with superconducting systems. The findings advance understanding of barrier-induced quasiparticle spectra and may guide experiments exploring barrier control of collective modes in binary quantum gases.

Abstract

The ultracold atoms are an ideal platform to implement atomtronics and Josephson junctions analogous to superconducting circuits. The collective modes of a Bose gas split by a potential barrier have been known. However, the role of barriers on the collective excitation spectra of ultracold atomic mixtures has not been examined. Here, we examine the low-lying collective modes of (an)harmonically trapped quasi-one-dimensional Bose-Einstein condensates in a Josephson barrier by employing the variational approach and Bogoliubov theory. We first show that the anharmonicity of the external potential leads to an increase in the critical barrier strength of mode softening in a single-species condensate. The Josephson barrier drives the softening of in-phase and out-of-phase dipole modes of two-species Bose-Einstein condensates, and consequently leads to two additional zero-energy Goldstone modes in the miscible phase, in agreement with the variational approach. Furthermore, the sandwich immiscible state results in an additional Goldstone mode due to the barrier, in contrast to the spatially symmetry-broken side-by-side profile. Our results unveil the distinct collective response of the Josephson barrier in binary mixtures owing to interspecies atomic correlations.

Figures (8)

• Figure 1: The change of the energies of the low-lying quasiparticle modes as a function of the barrier strength $U_{0}$ introduced to quasi-1D condensate. The excitation spectrum shown in the main figure is obtained by the numerical diagonalization of BdG equations and the top left inset plot presents the two lowest excited mode frequencies obtained using analytical variational analysis. The corresponding ground-state density profiles for $U_{0} = 0, 10,$ and $30$ are shown in the inset plot, where the solid red lines are the numerical solution and dashed black lines are obtained by the analytical variational ansatz. The quasiparticle frequencies are scaled to trap frequency while barrier strengths are in terms of $\hbar\omega_{x}$.
• Figure 2: The evolution of quasiparticle amplitudes corresponding to Kohn mode (top row) and higher symmetric and antisymmetric modes (middle and bottom rows) as barrier strength increases. At $U > U^{\rm cr}_{0}$, these low-lying modes transformed to the modes of two off-trap-centered quasi-1D BECs and thus represent an additional zero-energy mode and out-of-phase degenerate dipole modes. The shown mode amplitudes are obtained by the numerically diagonalization of BdG equations.
• Figure 3: (a) The evolution of low-lying quasiparticle mode energies as a function of barrier strength $U_0$ for different anharmonicity parameters $\Omega = 0.0, 0.05$, and $0.1$. The main figure shows numerical results while inset of the figure shows the mode energies obtained using the analytical variational approach. (b) The variation of critical barrier strength $U^{\rm cr}_{0}$ at which the first excited (dipole) mode gets softened as a function of the anharmonicity parameter. Here, $\Omega$ is a dimensionless parameter.
• Figure 4: The evolution of low-lying quasiparticle mode energies as a function of repulsive barrier strength $U_0$ in the miscible phase of two-species Bose-Einstein condensates: $^{85}\mathrm{Rb}$–$^{87}\mathrm{Rb}$ (top panel) and $^{23}\mathrm{Na}$–$^{87}\mathrm{Rb}$ (bottom panel). Barrier results in the mode softening and degeneracy of low-lying quasiparticle modes. The main figures are numerical results and insets show results obtained using the variational approach discussed in Section \ref{['varr_app']} [Eq. \ref{['mode_2s_varr']}]. The change in the density profiles for different $U_{0}$ values are shown in the inset plot. The numerical density of first (second) species is represented by solid black (red) color lines, while the corresponding dashed lines are the variational ansatz solution.
• Figure 5: Shown here are the evolutions of the quasiparticle amplitudes for the miscible phase of the $^{85}\mathrm{Rb}$-$^{87} \mathrm{Rb}$ two-species condensate as $U_0$ increases from zero to $30$ (in units of $\hbar\omega_{x}$). The first (second) row corresponds to the evolution of an out-of-phase (in-phase) Kohn mode, and the third row depicts the evolution of an out-of-phase quadrupole mode. The Kohn mode gets softened and transforms to two additional Goldstone modes (with already existing two Goldstone modes of spontaneous symmetry breaking of Bose-Einstein condensation of two-species condensates). Moreover, the quadrupole mode transforms to out-of-phase Kohn mode of two isolated off-centered condensates above $U^{\rm cr}_{0}$. Here, the interspecies scattering length $a_{12}=10a_{0}$. The mode amplitudes $u_{k}$ and $v_{k}$ are in units of $1/\sqrt{a_{\rm osc}}$. The mode amplitudes are scaled with appropriate factors for better visualization.