Classical Bifurcation at the Transition from Rabi to Josephson Dynamics

We report on the experimental demonstration of the internal bosonic Josephson effect in a rubidium spinor Bose-Einstein condensate. The measurement of the full time dynamics in phase space allows the characterization of the theoretically predicted $\pi$-phase modes and quantitatively confirms analytical predictions, revealing a classical bifurcation. Our results suggest that this system is a model system which can be tuned from classical to the quantum regime and thus is an important step towards the experimental investigation of entanglement generation close to critical points.

Figure 1

Interacting many-particle system as a model system for bifurcation physics. (a) ${}^{87}\mathrm{Rb}$ offers two hyperfine states $|a⟩$ (blue) and $|b⟩$ (red) which are linearly coupled via a two-photon transition with Rabi frequency $\Omega$ and which allow for adjusting the interparticle interaction $\chi$ via a Feshbach resonance. (b) The many-particle state is represented on a generalized Bloch sphere, and its uncertainty area for our experimental parameters is shown, revealing that a mean field description is adequate. Points on the sphere represent population difference $z$ ($z$ direction) and relative phase $\varphi$ between the two internal states in the same spatial mode. (c) Trajectories on the Bloch sphere below and above the bifurcation value of the ratio $\Lambda =\chi N/\Omega$. The typical supercritical pitchfork bifurcation scenario occurs; i.e., a stable fixed point bifurcates in two new stable fixed points while the original becomes unstable. The arrows indicate the direction of flow close to these points.

Figure 2

Direct observation of the symmetry breaking in the dynamics due to the bifurcation. Two initial states symmetric in the upper and lower hemisphere (see the inset) lead to qualitatively different dynamics in the Rabi and Josephson regime, respectively. In the Rabi regime both initial states share the same trajectory around the stable fixed point $F_{\pi }$, and the temporal mean imbalance vanishes in both cases. By increasing $\Lambda$ exceeding the critical value, a separatrix is formed and the chosen initial preparations lead to two distinct trajectories separated by this separatrix. The dynamical modes are characterized by a nonvanishing mean population imbalance. The solid line represents the theoretical prediction.

Figure 3

Experimentally observed phase portraits of the dynamics showing all possible types of trajectories. The experimental data for three different values of $\Lambda$ in different panels are compared to the theoretical model with no free parameter (solid lines). The theoretical curves are shown additionally on the Bloch spheres. (a) Phase portrait in the Rabi regime for $\Lambda =0.78$. Plasma (blue) and $\pi$ oscillations (red) can be clearly identified. The deformation relative to noninteracting Rabi oscillations due to the interaction is clearly visible on the Bloch sphere. (b) The Josephson regime is entered by reducing the coupling strength $\Omega$ ($\Lambda =1.55$). Here the bifurcation leads to new stable solutions showing macroscopic quantum self-trapping with mean phase $\pi$. This is verified by the experimental data (green squares and green crosses). The black solid line on the Bloch sphere corresponds to the separatrix defining the green shaded area of macroscopic quantum self-trapping. (c) For $\Lambda >2$ the separatrix encloses the two poles, and self-trapped trajectories with running phase behavior are found experimentally for $\Lambda =3.1$ (orange squares and circles). These trajectories resemble the ac Josephson effect found in superconducting Josephson junctions.

Figure 4

Exemplary time dynamics of population imbalance and relative phase. (a) Plasma (blue) and $\pi$ oscillations (red) in the Rabi regime ($\Lambda =0.78$). The dynamics about the two fixed points is qualitatively the same. However, the oscillations with mean phase $\pi$ are slower than their counterparts with vanishing mean phase. In the former case the atom interaction counteracts, whereas in the latter it assists the linear coupling. The shaded area corresponds to the theoretical prediction including the experimental uncertainty in $\Lambda$. (b) Macroscopic quantum self-trapping for $\Lambda =1.55$. Going beyond the critical value $\Lambda$ for bifurcation, the Josephson regime is entered. Time traces show oscillations around the northern fixed point $F_{+}$, while the mean phase varies around $\pi$. (c) Running phase mode found at $\Lambda =3.1$. The system is prepared on a trajectory that encircles the north pole of the sphere. Dynamics in the population imbalance is similar to (b), but the phase runs without bound as is the case for the ac Josephson effect in superconductors.