Macroscopic superpositions in Bose-Josephson junctions: Controlling decoherence due to atom losses

We study how macroscopic superpositions of coherent states produced by the nondissipative dynamics of binary mixtures of ultracold atoms are affected by atom losses. We identify different decoherence scenarios for symmetric or asymmetric loss rates and interaction energies in the two modes. In the symmetric case the quantum coherence in the superposition is lost after a single loss event. By tuning appropriately the energies we show that the superposition can be protected, leading to quantum correlations useful for atom interferometry even after many loss events.

Figure 1

Modulus of the density matrix ${\widehat{\rho }}_{N_0-2}\left(t_2\right)$ in the subspace with $N_0-2$ atoms in the Fock basis at time $t_2$, obtained by an exact diagonalization of Eq. (3). Upper panels: symmetric losses rates [${\gamma }_1={\gamma }_2=\chi /\left(200\pi \right)$]; lower panels: asymmetric losses [${\gamma }_1=4\chi /\left(300\pi \right)$, ${\gamma }_2=0$]. Left column: symmetric energies (${\chi }_1=-{\chi }_2=\chi$); right column: asymmetric energies (${\chi }_1=2\chi$, ${\chi }_2=0$). Other parameters: ${\gamma }_{12}=E_i=0$, $N_0=100$.

Figure 2

Fisher information in the subspace with $N_0-2$ atoms as a function of the loss rate ${\gamma }_1$ (in semilogarithmic scale) at time $t_2=T/4$, with $T=2\pi /\chi$. Solid black line: ${\gamma }_1={\gamma }_2$, ${\chi }_1=-{\chi }_2=\chi$; dot-dashed brown line: ${\gamma }_1={\gamma }_2$, ${\chi }_1=0$, ${\chi }_2=-2\chi$; dotted red line: ${\gamma }_2=0$, ${\chi }_1=-{\chi }_2=\chi$; dashed blue line: ${\gamma }_2={\chi }_1=0$, ${\chi }_2=-2\chi$. The optimization over the interferometer directions is done independently in the $(N_0-2)$ subspace. Insets: Husimi functions for the values of ${\gamma }_i$ and ${\chi }_i$ corresponding to the circles on the curves. Other parameters as in Fig. 1.

Figure 3

Total quantum Fisher information $F_{\mathrm{tot}}\left(t\right)$ vs time $t$ (in units of $T$). From top to bottom: ${\gamma }_{1}T=8/300$, ${\gamma }_2={\chi }_1=0$, ${\chi }_2=-2\chi$ (blue dashed line); ${\gamma }_{1}T=8/300$, ${\gamma }_2=0$, ${\chi }_1=-{\chi }_2=\chi$ (red dotted line); ${\gamma }_{1}T={\gamma }_{2}T=1/100$, ${\chi }_1=0$, ${\chi }_2=-2\chi$ (brown dot-dashed line); ${\gamma }_{1}T={\gamma }_{2}T=1/100$, ${\chi }_1=-{\chi }_2=\chi$ (black solid line). Other parameters as in Fig. 1. Dashed horizontal line: $F_{\mathrm{tot}}$ for the highest squeezed state in the lossless case. Inset: average number of atoms vs time for the same parameters.