Classical dynamics of a two-species condensate driven by a quantum field

We present a stability analysis of an interacting two-species Bose-Einstein condensate driven by a quantized field in the semiclassical limit. Transitions from Rabi to Josephson dynamics are identified depending on both the interatomic interaction to field-condensate coupling ratio and the ratio between the total excitation number and the condensate size. The quantized field is found to produce asymmetric dynamics for symmetric initial conditions for both Rabi and Josephson oscillations.

Figure 1

Fixed points $F_{\Phi }$, with $\Phi =0$,$\pi$, defined by the parameter set given by the dimensionless detuning, couplings ratio, fractional population difference, and excitation ratio, $\left\{\Delta ,\Lambda ,z,k\right\}$, respectively, for excitation ratio parameters $k=0.1$ (light blue) and $k=10$ (dark red).

Figure 2

Fractional population difference $z$ as a function of the coupling ratio $\Lambda$ for two excitation ratios $k=0.1$ (gray lines) and $k=10$ (black lines) for fixed points $F_0$ (solid lines) and $F_{\pi }$ (dotted lines). Rabi oscillations occur around the fixed points marked from $A$ to $C$, Josephson oscillations around fixed points $D$ and $F$, and plasma oscillations around fixed points marked $G$ and $H$. The dashed branch for the latter excitation ratio $k=10$ shows the fixed points acting as a separatrix of the two localized oscillations in the Josephson regime; in particular the fixed point $E$ acts as a separatrix of oscillations around fixed points $D$ and $F$.

Figure 3

On resonance $\Delta =0$, the (a) fractional population difference $z$ as a function of the excitation ratio $k$ for a given coupling ratio $\Lambda =0$. Alphabetical markers show fixed points for excitation ratios $k=0.1$ ($A$) and $k=10$ ($B$). Trajectories for $\pi$ oscillations around the fixed points (b) $A$ and (c) $B$ for initial conditions including a survey of the initial fractional population difference and an initial phase $\Phi \left(0\right)=0$; highlighted trajectories sharing the same tone correspond to symmetric initial conditions $z\left(0\right)=0.9$ (gray dotted line), $z\left(0\right)=0.5$ (black solid line), $z\left(0\right)=-0.5$ (black dotted line), and $z=-0.9$ (gray solid line). Mean value for the Hamiltonian showing the critical points defined by (d) $A$ and (e) $B$.

Figure 4

On resonance $\Delta =0$, the (a) fractional population difference $z$ as a function of the excitation ratio $k$ for a given coupling ratio $\Lambda =6$. Alphabetical markers show fixed points for excitation ratios $k=0.1$ ($C$ and $G$) and $k=10$ ($D$–$F$ and $H$). Trajectories for $\pi$ oscillations around the fixed points (b) $C$ (for low excitation ratio) and (c) $D$ and $F$ (for large excitation ratio) for initial conditions including a survey of the initial fractional population difference and an initial phase $\Phi \left(0\right)=0$; highlighted trajectories sharing the same color correspond to symmetric initial conditions $z\left(0\right)=0.9$ (gray dotted line), $z\left(0\right)=0.5$ (black solid line), $z\left(0\right)=-0.5$ (black dotted line), and $z=-0.9$ (gray solid line). Mean value for the Hamiltonian showing the critical points defined by (d) $C$ and $G$, (e) $D$, $E$, $F$, and $H$.

Figure 5

On resonance $\Delta =0$, the fractional population difference $z$ (solid line) as a function of the excitation ratio $k$ for a very large coupling ratio $\Lambda =5000$. A bifurcation point appears at the critical excitation ratio $k_{c+}\approx {\Lambda }^2/2=1.25\times {10}^7$. The separatrix for the Josephson regime is shown as a dashed line.