Role of the relative phase in the merging of two independent Bose-Einstein condensates

We study the merging of two independent Bose-Einstein condensates with arbitrary initial phase difference, in the framework of a one-dimensional time-dependent Gross-Pitaevskii model. The role of the initial phase difference in the process is discussed, and various types of phase-sensitive excitations are identified.

Figure 1

Initial conditions used in the numerical simulations. Symbols: atomic densities ${\mid \Psi (x,0)\mid }^2$ corresponding to the ground state of Eq. (3) for the double-well potential $V(x,0)$. Lines: atomic density distribution for two identical independent condensates trapped in harmonic potentials centered at $x=\pm 4$ [see Eq. (7)]. Harmonic oscillator units are used for $x$.

Figure 2

(Color online) (a) Time evolution of the total energy of an interacting gas during a merging process, for various initial phase differences. (b) The same for an ideal gas. (c) Excess energy at the end of the merging as a function of the initial phase difference $\Delta {\phi }_{\mathrm{in}}$. The dashed line through the crosses is the theoretical result for ideal gas, given by Eq. (9). For symmetry reasons, all the plots can be extended to $\Delta {\phi }_{\mathrm{in}}<0$ by the transformation $\Delta {\phi }_{\mathrm{in}}\to -\Delta {\phi }_{\mathrm{in}}$. Merging times: $T_m=22.5$ (a), (b) and $T_m=23$ (c). Energy and time are in units of $\hslash \omega$ and ${\omega }^{-1}$, respectively.

Figure 3

(a) Time evolution of the atomic density distribution ${\mid \Psi (x,t)\mid }^2$ in arbitrary units after the merging of two identical condensates with $\Delta {\phi }_{\mathrm{in}}=3\pi ∕4$. Space and time are measured in HO units. (b) Relative depth of the dip in the final atomic density distribution at the center of the trap, for various values of $\Delta {\phi }_{\mathrm{in}}$. (c)–(f) The entire distribution is depicted only for $\Delta {\phi }_{\mathrm{in}}=0,\pi ∕3,2\pi ∕3$, and $\pi$.

Figure 4

(Color online) (a) Displacement of the center-of-mass in HO units with no interatomic interactions for $T_m=25{\omega }^{-1}$. (b) Same as (a) but with interactions $g=8.5$. (c) Amplitude of the dipole oscillation in dependence of the initial phase difference $\Delta {\phi }_{\mathrm{in}}$.

Figure 5

(a) The phase of the final condensate ${\phi }_f$, as a function of the initial phase difference $\Delta {\phi }_{\mathrm{in}}$ in the noninteracting case. Merging times are $T_m=21.52{\omega }^{-1}$ (crosses) and $T_m=26.52{\omega }^{-1}$ (squares). (b) Same as (a) but with interactions and merging times $T_m=23{\omega }^{-1}$ (crosses) and $T_m=23.4{\omega }^{-1}$ (squares). The line is Eq. (12) applied to the upper curve.

Figure 6

The function $\alpha$ in dependence of the merging time for interacting (squares) and noninteracting (crosses) gases.

Figure 7

Phase shift experienced by a condensate trapped in a moving harmonic potential described by Eq. (13), as a function of the movement time for $l=4$. (Crosses) Numerical solution obtained by propagating the GPE. (Squares) An estimation is given by Eq. (17).