Chaos-assisted many-body tunnelling

We study the interplay of chaos and tunneling between two weakly coupled Bose-Josephson junctions. The classical phase space of the composite system has a mixed structure including quasi-integrable self-trapping islands for particles and excitations, separated by a chaotic sea. We show that the many-body dynamical tunneling gap between macroscopic Schrödinger cat states supported by these islands is chaos-enhanced. The many-body tunneling rate fluctuates over several orders of magnitude with small variations of the system parameters or the particle number.

Figure 1

Spectrum and level spacing statistics. (a) The unperturbed ($\omega =0$) eigenenergies, classified according to the good quantum numbers $n,j$ and colored according to their participation number in the exact eigenstate basis for $\omega =0.082$. (b) The exact eigenenergies for the same $\omega$, classified according to the ${\sigma }_n,{\sigma }_j$ standard deviations and colored according to the Shannon entropy of the corresponding eigenstates. (c) Level spacing statistics for $\omega =0.01$ (black) and $\omega =0.082$ (red). For comparison, Poisson (dashed black) and Wigner surmise of GOE class (dotted red) are overlaid. (d) The spacing correlation measure $r$ throughout the parameter space. The expected values for a GOE and Poisson statistics are marked on the color bar. The $+$ marker marks the parameters used in panels (a)–(c). The interaction parameter for panels (a)–(c) is $u=0.4$. Here and in other figures we set particle number $N=21$ unless it is otherwise mentioned.

Figure 2

Quantum-classical correspondence. (a) The $J=9$ surface of the spectrum in Fig. 1 over representative chaotic (gray, center) and self-trapped (green, right-hand side) classical trajectories launched at the points marked by filled triangles. Contour lines correspond to the energy $E(N,J,n,j)$. The adiabatic invariance of $J$ dissects the energy surface into self-trapped and chaotic regions. High quantum participation ratios correspond to chaotic classical dynamics since the exact chaotic eigenstates are spread throughout the entire chaotic region. The dynamics of the particle imbalance $n$ and action imbalance $j$ for the same chaotic and regular trajectories are plotted in (b) and (c), respectively.

Figure 3

Chaos assisted many-body tunneling. (i) Section of the eigenenergy spectrum with $u=0.4$ and $N=21$, plotted as a function of the coupling frequency $\omega$. Lines are colored according to the symmetry class of the corresponding eigenstates: $A$ (dash-dotted green), $B_1$ (dashed red), $B_2$ (solid blue), and $B_3$ (dotted magenta). (ii) Zoom on the $\{B_1,B_2\}$ chaos-assisted MBT resonance with eigenenergy lines colored according to the Shannon entropy of the three participating eigenstates. (iii) The resulting chaos-enhanced tunneling gap between the doublet states. (iv) The $p_{n,j}$ distribution of the pertinent states, marked in (i) at (a), (d), and (g) $\omega =0.07$, (b), (e), and (h) 0.0778, (c), (f), and (i) and 0.085. Bars are colored according to the sign of the (real) amplitudes.

Figure 4

Tunnelling rate fluctuations. (a) Tunnel gaps $\mathrm{\Delta }E/\mathrm{\Omega }N$ between doublet states, belonging to $\{B_1,B_2\}$ symmetry classes, are plotted against ${\sigma }_n/N$ for $\omega =0.03$. Dashed line marks the island-chaos border. (b) The points represent one of the states in $\{B_1,B_2\}$ doublet lying on the $J/N\approx 1/2$ surface and having the larger ${\sigma }_n/N$ of the two, arranged according to ${\sigma }_n/N$ with varying $N$, and colored according to their ${\sigma }_j/J$. (c), (d) The same as in (a), (b) for $\omega =0.08$. (e) $N$ dependence of the doublet spacing. Point color corresponds to the color code in panels (b), (d) at the marked places. Triangles and circles correspond to the encircled group in panels (b), (d), exhibiting smooth exponential decay and fluctuations, respectively. (f) Statistics of fluctuations $\mathrm{\Delta }{E}^{\prime }$ around the mean spacing $\overline{\mathrm{\Delta }E}$ corresponding to doublets encircled in (d) for each $N$, varied up to $N=46$.

Figure 5

Signature of chaos assisted tunneling in the population imbalance dynamics Panels (a)–(c) depict the tunneling dynamics for $\omega =\text{(a)}0.07$, (b) 0.0778, and (c) 0.85 for an ideal local preparation using only the states involved in the resonance shown in Fig. 3. Panels (d)–(f) show a more realistic scenario where the system is prepared at the unperturbed eigenstate $|N,J,n,j\rangle =|21,9,-15,-3\rangle$ in (d), (e) or $|21,9,-15,-5\rangle$ in (f) and $\omega$ is then quenched to the stated value. The insets in (b), (e) show the dual frequency power spectra on resonance.