Experimental observation of the Poincaré-Birkhoff scenario in a driven many-body quantum system

Accessing the connection between classical chaos and quantum many-body systems has been a long-standing experimental challenge. Here, we investigate the onset of chaos in periodically driven two-component Bose-Einstein condensates, whose small quantum uncertainties allow for exploring the phase space with high resolution. By analyzing the uncertainties of time-evolved many-body states, we find signatures of elliptic and hyperbolic periodic orbits generated according to the Poincaré-Birkhoff theorem, and the formation of a chaotic region at increasing driving strengths. The employed fluctuation analysis allows for probing the phase-space structure by use of only short-time quantum dynamics.

Figure 1

Poincaré-Birkhoff scenario in a many-body quantum system. (a) A two-component Bose-Einstein condensate ($N\sim 700$ atoms) with interaction $\chi$ and linear coupling $\mathrm{\Omega }$ realizes an anharmonic oscillator with the population imbalance $z$ and the relative phase $\phi$ as the canonical variables. State-selective absorption imaging allows for the precise determination of the populations of the two components. (b) The minimal uncertainty state is a coherent spin state whose extension in the phase-space representation $(z,\phi )$ corresponds to ${\hslash }_{\text{eff}}=2/N$. (c) Classical orbits for the undriven case ($A=0$) at $\mathrm{\Lambda }=N\chi /{\mathrm{\Omega }}_0=0.7$ (green square: elliptic fixed point at $\phi =\pi$). (d) Stroboscopic Poincaré sections, showing the formation of an alternating sequence of elliptic (blue triangles) and hyperbolic (red dots) periodic orbits according to the Poincaré-Birkhoff theorem in the case of weak driving ($A=0.03$), the emergence of a chaotic layer for $A=0.2$, and a larger chaotic region at even higher modulation amplitude.

Figure 2

Detecting the Poincaré-Birkhoff scenario via variance analysis. (a) Stroboscopic Poincaré sections for $\mathrm{\Lambda }=0.7$ and $A=0.2$ for various times $t\in [0,2T]$, showing one full counterclockwise rotation of the elliptic and hyperbolic periodic orbits around the central elliptic fixed point. Red and blue points: Time evolution for $(z_0=0.55,{\phi }_0=\pi )$ and driving time offsets $t_0=0.2T$ (red) and $t_0=0.6T$ (blue). (b) The experimentally obtained mean values of $z$ exhibit oscillations with period $2T$ for preparations close to the hyperbolic orbit (left, red) and the elliptic orbit (right, blue). Error bars are the rms deviation. Black lines and shaded regions: Mean and rms uncertainty for classical time evolution of an ensemble corresponding to a CSS. (c) Variances of $z$ in units of the corresponding coherent state (see the Appendix). Preparation near the hyperbolic orbit (left) results in a strong increase compared to the undriven dynamics (gray), whereas suppressed uncertainties can be observed close to the elliptic orbit (right, blue). The results of classical simulations are shown as black lines. The oscillations of the variance result from the rotation of the corresponding manifolds around the central elliptic fixed point (insets).

Figure 3

Signatures of a mixed phase space in the short-time quantum evolution. (a) The undriven system ($\mathrm{\Lambda }=1.5$) exhibits one hyperbolic and two elliptic fixed points at $\phi =\pi$. For $A=0.07$, the strongly driven regime with a mixed phase space is reached, which is probed by different initial preparations (depicted by $2\sigma$ areas) with the time offsets $t_0=0.9T$ (center, red) and $t_0=0.4T$ (bottom, blue). (b) Time evolution of the $z$ distributions in gray scale. Compared to the undriven case (top), we observe increased fluctuations in the case of a preparation deep in the chaotic region (middle) and reduced fluctuations close to a moving island (bottom). Mean and rms deviations of the classical simulations are shown as solid and dashed lines. (c) Similar signatures of the underlying phase space are observed in $y=\sqrt{1-z^2}sin\left(\phi \right)$.

Figure 4

Due to the small size of ${\hslash }_{\text{eff}}$ in our system, we can experimentally probe phase-space structures on small scales. The initial position is adjusted by varying the time offset $t_0$ of the driving, and is indicated in the Poincaré section for $t_0=T$ along with the size of the effective Planck constant (left, blue points with $2\sigma$ area in red). (a) Weakly perturbed system: After a fixed evolution time ($t=48$ ms), we find enhanced variances near the hyperbolic periodic orbit and reduced variances near the elliptic orbit. Our findings are compared to the undriven case (gray) and the classical simulation (black line). (b) Strongly perturbed regime: The variance after an evolution time of $t=61$ ms reveals a strong dependence on the initial preparation, indicating the high resolution in phase space.