Phase diffusion and fluctuations in a dissipative Bose-Josephson junction

We analyze the phase diffusion, quantum fluctuations and their spectral features of a one-dimensional Bose-Josephson junction (BJJ) nonlinearly coupled to a bosonic heat bath. The phase diffusion is considered by taking into account of random modulations of the BJJ modes causing a phase loss of initial coherence between the ground and excited states, whereby the frequency modulation is incorporated in the system-reservoir Hamiltonian by an interaction term linear in bath operators but nonlinear in system (BJJ) operators. We examine the dependence of the phase diffusion coefficient on the on-site interaction and temperature in the zero- and $\pi$-phase modes and demonstrate its phase transition-like behavior between the Josephson oscillation and the macroscopic quantum self-trapping (MQST) regimes in the $\pi$-phase mode. Based on the thermal canonical Wigner distribution, which is the equilibrium solution of the associated quantum Langevin equation for phase, coherence factor is calculated to study phase diffusion for the zero- and $\pi$-phase modes. We investigate the quantum fluctuations of the relative phase and population imbalance in terms of fluctuation spectra which capture an interesting shift in Josephson frequency induced by frequency fluctuation due to nonlinear system-reservoir coupling, as well as the on-site interaction-induced splitting in the weak dissipative regime.

Figure 1

(a) Variation of the coherence factor $\langle cos\mathrm{\Psi }\rangle$ as a function of $U/K$ at $k_{B}T/\hslash \mathrm{\Omega }=1/10$ in zero-phase mode. (b) Variation of $\langle cos\mathrm{\Psi }\rangle$ as a function of $2k_{B}T/\hslash \mathrm{\Omega }$ for $U/K=1$ (blue solid line), $U/K=2$ (red solid line), and $U/K=3$ (black solid line). Note the logarithmic scale on the horizontal axis.

Figure 2

(a) Variation of $\langle cos\mathrm{\Psi }\rangle$ as a function of $U/K$ at $k_{B}T/\hslash \mathrm{\Omega }=1/10$ in the $\pi$-phase self-trapping regime. The vertical dashed-dotted magenta line corresponds to $U/K=0.004$. (b) Variation of $\langle cos\mathrm{\Psi }\rangle$ as a function of $2k_{B}T/\hslash \mathrm{\Omega }$ for $U/K=0.006$ (blue solid line) and $U/K=0.01$ (black solid line). Note the logarithmic scale on the horizontal axis.

Figure 3

Variation of the phase diffusion coefficient $\mathcal{D}$ as a function of $2k_{B}T/\hslash \mathrm{\Omega }$ for different values of $\tilde{\gamma }$ with many-body interaction parameter ${\mathrm{\Lambda }}_0=0.5$ in the zero-phase mode.

Figure 4

Variation of $\mathcal{D}$ as a function of ${\mathrm{\Lambda }}_0$ at $k_{B}T/\hslash \mathrm{\Omega }=1/10$ with $\tilde{\gamma }=1$ in the $\pi$-phase mode. Magenta solid line depicts $\mathcal{D}$ vs ${\mathrm{\Lambda }}_0$ curve for $z_s=0,{\theta }_s=\pi$ and for $z_s=\pm \sqrt{1-\frac{1}{{\mathrm{\Lambda }}_0^2}},{\theta }_s=\pi$; the $\mathcal{D}$ vs ${\mathrm{\Lambda }}_0$ curve is depicted by solid blue and red dashed lines. The vertical black dashed line corresponds to the critical ${\mathrm{\Lambda }}_0$ for the transition between Josephson oscillation and MQST in the $\pi$-phase mode. Inset: Variation of the steady-state population imbalance $z_s$ as a function of ${\mathrm{\Lambda }}_0$. The three different colors signify three different steady states of the population imbalance. The vertical black dashed line corresponds to ${\mathrm{\Lambda }}_0$ where the bifurcation of $z_s$ occurs.

Figure 5

Variations of the number fluctuation spectrum ${\mathcal{S}}_z\left(\mathrm{\Delta }\right)$ (a) and phase fluctuation spectrum ${\mathcal{S}}_{\theta }\left(\mathrm{\Delta }\right)$ (b) as a function of $\mathrm{\Delta }$ for zero-phase (solid blue line) and $\pi$-phase mode (solid black line) for $U/K=0.001$, $\tilde{\gamma }=1$ with $N=1000$, and $k_{B}T/\hslash \mathrm{\Omega }=1/10$. The blue and black dashed-dotted lines indicate the Josephson frequency in the absence of dissipation for zero and $\pi$-phase mode, respectively.

Figure 6

Variations of ${\mathcal{S}}_z\left(\mathrm{\Delta }\right)$ (a) and ${\mathcal{S}}_{\theta }\left(\mathrm{\Delta }\right)$ (b) as a function of $\mathrm{\Delta }$ for different values of $\tilde{\gamma }$ in zero-phase mode for $U/K=0.001$, with $N=1000$ and $k_{B}T/\hslash \mathrm{\Omega }=1/10$. The green dashed-dotted line indicates the Josephson frequency in zero-phase mode in the absence of dissipation.

Figure 7

Variations of ${\mathcal{S}}_z\left(\mathrm{\Delta }\right)$ (a) and ${\mathcal{S}}_{\theta }\left(\mathrm{\Delta }\right)$ (b) as a function of $\mathrm{\Delta }$ for different values of $\tilde{\gamma }$ in the $\pi$-phase Josephson oscillation regime for $U/K=0.001$, with $N=1000$ and $k_{B}T/\hslash \mathrm{\Omega }=1/10$. The green dashed-dotted line indicates the Josephson frequency in the $\pi$-phase mode in the absence of dissipation.

Figure 8

Variations of ${\mathcal{S}}_z\left(\mathrm{\Delta }\right)$ (a) and ${\mathcal{S}}_{\theta }\left(\mathrm{\Delta }\right)$ (b) as a function of $\mathrm{\Delta }$ for $\tilde{\gamma }=1$ (solid blue line) and $\tilde{\gamma }=8$ (solid black line) in the $\pi$-phase mode self-trapping regime for $U/K=0.003$, with $N=1000$ and $k_{B}T/\hslash \mathrm{\Omega }=1/10$. The blue dashed-dotted line indicates the self-trapping frequency in the $\pi$-phase mode in the absence of dissipation.