NMR studies of quantum chaos in a two-qubit kicked top

Quantum chaotic kicked top model is implemented experimentally in a two-qubit system comprising of a pair of spin-1/2 nuclei using nuclear magnetic resonance techniques. The essential nonlinear interaction was realized using indirect spin-spin coupling, while the linear kicks were realized using radio-frequency pulses. After a variable number of kicks, quantum state tomography was employed to reconstruct the single-qubit reduced density matrices, using which we could extract von Neumann entropies and Husimi distributions. These measures enabled the study of correspondence with classical phase space as well as probing distinct features of quantum chaos, such as symmetries and temporal periodicity in the two-qubit kicked top.

Figure 1

Linear kicks and nonlinear evolutions for simulating a QKT using two qubits.

Figure 2

Classical trajectories of the kicked top in $|\theta ,\varphi \rangle$ phase space for chaoticity parameter values (a) $k=0.5$, (b) $k=2.5$, (c) $k=-2.5$, and (d) $k=2\pi +2.5$. The phase space points chosen for detailed analysis are marked by black dots.

Figure 3

NMR pulse sequence for simulating a QKT in a two-qubit system. The flip angles and phases (subscripts) are shown under each radio-frequency pulse. Here ${\mathrm{G}}_1$ and ${\mathrm{G}}_2$ correspond to pulsed-field-gradients (PFG).

Figure 4

Average fidelity of the experimental states for various kick numbers. The error bars indicate one standard deviation of distribution.

Figure 5

Contours represent experimental order parameter averaged over $n$ kicks ($s_n$) for chaoticity parameter $k=0.5$ [(a1)–(a8)], $k=2.5$ [(b1)–(b8)], $k=2\pi -2.5$ [(c1)–(c8)], and $k=2\pi +2.5$ [(d1)–(d8)]. Background color maps represent the corresponding simulated values. Average rms deviations $\langle \delta \rangle$ between the experimental and simulated values are 0.14 for $k=0.5,0.033$ for $k=2.5,0.031$ for $k=2\pi -2.5$, and 0.032 for $k=2\pi +2.5$.

Figure 6

Experimental (mesh grids) and simulated (color background) Husimi probability distributions (in units of $1/\pi$) for certain $k$ values and initial states (as indicated in the left of each row) after kick numbers $n=0$ to 8 (from left to right).

Figure 7

Distribution of maxima neighborhood of Husimi probability function on the Bloch sphere. Blue dots represent simulated data and red dots represent experimental data for the values of $k$ and initial angles (a) $k=0.5,|\theta ,\varphi \rangle =|\pi /2,\pi \rangle$, (b) $k=2.5,|\theta ,\varphi \rangle =|\pi /3,7\pi /6\rangle$, (c) $k=2.5,|\theta ,\varphi \rangle =|\pi /3,4\pi /3\rangle$, and (d) $k=2\pi -2.5,|\theta ,\varphi \rangle =|\pi /6,4\pi /3\rangle$.