Probing two-particle exchange processes in two-mode Bose-Einstein condensates

We study the fidelity decay and its freeze for an initial coherent state of two-mode Bose-Einstein condensates in the Fock regime considering a Bose-Hubbard model that includes two-particle tunneling terms. By using linear-response theory we find scaling properties of the fidelity as a function of the particle number that prove the existence of two-particle mode exchange when a nondegeneracy condition is fulfilled. Tuning the energy difference of the two modes serves to distinguish the presence of two-particle mode-exchange terms through the appearance of certain singularities. We present numerical calculations that illustrate our findings, and propose exploiting a Feshbach resonance to verify experimentally our predictions.

Figure 1

Time dependence (in Heisenberg time units) of the fidelity obtained numerically for the extended Bose-Hubbard model in log-log scale, using an initial coherent state Eq. (9) with $n_1=n_2=n/2=64$ and $\varphi =\pi /4$. The parameters for the Hamiltonian are $U=1, J_1={10}^{-6}, J_2={10}^{-8}, {\epsilon }_1=0.76$, and ${\epsilon }_2=0.93$. The horizontal line corresponds to the value of the fidelity freeze $F_{\mathrm{Fr}}$ obtained from Eq. (15). In the inset we present the result of the second-order linear-response theory.

Figure 2

Log-log plot of $1-F_{\mathrm{Fr}}$ as a function of the number of particles, for an initial symmetric coherent state $n_1=n_2=n/2$. Triangles (green, dotted line) correspond to the parameters $J_1={10}^{-6}$ and $J_2=0$, squares (blue, dashed line) to $J_1=0$ and $J_2={10}^{-8}$, and circles (red, continuous line) to $J_1={10}^{-6}$ and $J_2={10}^{-8}$; the remaining parameters are those of Fig. 1.

Figure 3

Behavior of ${log}_{10}(1-F_{Fr})$ as a function of the energy difference between modes $\mathrm{\Delta }\epsilon$ scaled by the two-particle interaction coefficient $U$ of the Bose-Hubbard model, Eq. (2). The 3D plot depicts the appearance of a peak around $\mathrm{\Delta }\epsilon /U=0,2$ which becomes noticeable as the particle number $n$ increases.

Figure 4

Computed tunneling parameters of the two-mode BEC model as a function of the potential-barrier height, where the trap considered corresponds to that of Ref. [8]. The red curves correspond to ${}^{87}\mathrm{Rb}$, and the blue, green, and purple to ${}^{85}\mathrm{Rb}$ with $s$-wave scattering lengths of $20a, 50a$, and $100a$, respectively, where $a=5.32\mathrm{nm}$ is the $s$-wave scattering length of ${}^{87}\mathrm{Rb}$. The upper curves correspond to $J_1/U$ and the lower to $J_2/U$.