Quantum signatures of chaos, thermalization, and tunneling in the exactly solvable few-body kicked top

Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important—more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to nonintegrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the three- and four-qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons-based experiment realizing the three-qubit case, we find agreement for short times, including a peculiar steplike behavior in correlations of some states. In the case of four qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.

Figure 1

(a) Regular and (b) mixed phase-space structures resulting from the classical chaotic dynamics. Points labeled with red square and red circle correspond to initial states $\mathrm{\Theta }=0,\mathrm{\Phi }=0$ on a period-4 orbit and $\mathrm{\Theta }=\pi /2,\mathrm{\Phi }=-\pi /2$ at the center of a regular island, respectively.

Figure 2

Husimi (quasiprobability distribution, $|\langle {\varphi }_i|{\theta }_0,{\varphi }_0\rangle {|}^2$) plots for a set of 4 three-qubit bases states ($|{\varphi }_i\rangle$), where $|{\theta }_0,{\varphi }_0\rangle$ is an arbitrary three-qubit, parametrized by (${\theta }_0,{\varphi }_0$).

Figure 3

Linear entropy of a single-qubit reduced state versus $n$ is plotted for initial state $|000\rangle$ at different values of ${\kappa }_0=0.1,0.4,0.8$, and 1.2 as labeled on the right end of each curve.

Figure 4

Concurrence of a two-qubit reduced state versus $n$ is plotted for ${\kappa }_0=0.1,0.4,0.8$, and 1.2 as labeled on the right end of each curve.

Figure 5

Solid curve with circles and dashed curve with squares show the variation of entropy of a single-qubit reduced state and concurrence between a pair of two qubits, respectively, with $n$ as three-qubit initial state $|000\rangle$ evolves under ${\mathcal{U}}^n$. Parts (a), (b), (c), and (d) correspond to different values of chaoticity parameter (${\kappa }_0$) as mentioned.

Figure 6

Linear entropy of a single-qubit reduced state versus $n$ is plotted for different values of ${\kappa }_0$. Curves correspond to ${\kappa }_0=0.4,0.8,1.2$, and 3.0 are shown by a solid line with diamonds, a solid line with circles, a dashed line with triangles, and a dashed line with squares, respectively.

Figure 7

(a) Time-averaged linear entropy, obtained over $n=1000$ periods, of a single qubit vs. the parameter ${\kappa }_0$, for three initial coherent states $|{\theta }_0,{\varphi }_0\rangle$. The Eqs. (29) and (40) apply to the curves labeled (1) and (3), as for ${\theta }_0=0$ the value of ${\varphi }_0$ is immaterial on the sphere. Inset shows the entanglement periodicity in the parameter at ${\kappa }_0=3\pi$. Part (b) displays the time-averaged linear entropy across all initial coherent states for the value ${\kappa }_0=3\pi /2$ and is described by Eq. (42).

Figure 8

Plots showing analytical (dashed curves with markers), experimental (solid curves with markers), and numerical (dashed) curves of linear entropy and concurrence as a function of the number of kicks, as the initial state $|{\psi }_0\rangle$ is evolved under repeated applications of operator $\mathcal{U}$. Parameters of the initial state, $({\theta }_0,{\varphi }_0)$, and chaoticity parameter, ${\kappa }_0$, are specified in each figure. Analytical (wherever plotted) and numerical curves exactly overlap and hence cannot be seen separately.

Figure 9

Husimi (quasiprobability distribution, $|\langle {\varphi }_i|{\theta }_0,{\varphi }_0\rangle {|}^2$) plots for a set of 5 four-qubit bases states ($|{\varphi }_i\rangle$), where $|{\theta }_0,{\varphi }_0\rangle$ is an arbitrary four-qubit, parametrized by (${\theta }_0,{\varphi }_0$).

Figure 10

Linear entropy of a single-qubit reduced state versus $n$ is plotted at different values of ${\kappa }_0$ shown in parts (a) and (b), corresponding to a four-qubit initial state, $|0000\rangle$.

Figure 11

Analytically obtained expressions for time-averaged linear entropy for initial states $|0000\rangle$ [Eq. (55)] and ${|++++\rangle }_y$ [Eq. (59)] are plotted for ${\kappa }_0\in (0,4\pi )$. Extreme values are presented as horizontal lines, with their respective values ($S_{(\theta ,\varphi )}^{\left(4\right)}\left({\kappa }_0\right)=\mathrm{constant}$) specified on the right side. Solid red curve and dashed black curve correspond to initial states ${|++++\rangle }_y$ and $|0000\rangle$, respectively.

Figure 12

Linear entropy of a single-qubit reduced state versus $n$ is plotted at different values of ${\kappa }_0$ shown in parts (a), (b), (c), and (d), corresponding to a four-qubit initial state, ${|++++\rangle }_y$.

Figure 13

Simulated time-average linear entropy $\left[\langle S_{(\frac{\pi }{2},\pm \frac{\pi }{2})}^{\left(4\right)}\left({\kappa }_0\right)\rangle \right]$ subject to initial state, $\left|\right(\pi /2,\pm \pi /2)\rangle$, plotted versus ${\kappa }_0$, is shown for different values of $n$ (as given in the inset) in the interval ${\kappa }_0\in [0,4\pi ]$. Inset shows the blown-up horizontal scale for ${\kappa }_0\in [0,\mathrm{\Delta }{\kappa }_0]$, where $\mathrm{\Delta }{\kappa }_0=2\pi /5$, that clearly presents the curves approaching the solid curve curve of Fig. 11.

Figure 14

Husimi (quasiprobability distribution) plots for the four-qubit initial state, ${\otimes }^4|+\rangle$, evolving under $n$ implementations of $\mathcal{U}$, and leading to tunneling to the state, ${\otimes }^4|-\rangle$, at time $n_{*}\approx 128\pi /{\kappa }_0^3\approx 402124$ (${\kappa }_0=0.1$).

Figure 15

Normalized average single-qubit entanglement when the initial state is ${\otimes }^{2j}{|+\rangle }_y$ for increasing number of qubits (except multiples of 4 where there is tunneling for $p=\pi /2$).

Figure 16

Normalized average single-qubit entanglement when the initial state is ${\otimes }^{2j}|0\rangle$ for increasing number of qubits.

Figure 17

[(a) and (b)] Time-averaged linear entropy $\left[\langle S_{({\theta }_0,-\pi /2)}^{\left(3\right)}\rangle \right]$ of a single-party reduced state vs. chaoticity parameter ${\kappa }_0$. Different curves correspond to different initial states, $|{\theta }_0,-\pi /2\rangle$ as labeled 1 to 5, along with explicit values of ${\theta }_0$ given in the plot legends. These corresponding initial states $|{\theta }_0,{\varphi }_0\rangle$ are also marked as numbered circles in the contour plot given in part (d). Part (c) contains the plots for $\langle S_{(2\pi /3,{\varphi }_0)}^{\left(3\right)}\rangle$ vs. chaoticity parameter ${\kappa }_0$ for a fixed value of ${\theta }_0=2\pi /3$. Different curves correspond to different initial states, labeled by numbers 1 to 4, along with explicit values of ${\varphi }_0$ given in the plot legends. Respective initial states $|{\theta }_0,{\varphi }_0\rangle$ are also marked as numbered squares (with a green border) in the contour plot given in part (d). Contour plot shown in part (d) corresponds to ${\kappa }_0=3\pi /2$.