Fractional revivals, multiple-Schrödinger-cat states, and quantum carpets in the interaction of a qubit with $N$ qubits

We study the dynamics of a system comprised of a single qubit interacting equally with $N$ qubits (a “spin star” system). Although this model can be solved exactly, the exact solution does not give much intuition for the dynamics of the model. Here, we find an approximation that gives some insight into the dynamics for a particular class of initial spin-coherent states of the $N$ qubits. We find an effective Hamiltonian for the system that is a finite Kerr (one-axis twisting) Hamiltonian for the $N+1$ qubits. The initial spin-coherent state evolves to spin-squeezed states on short time scales, and to “multiple-Schrödinger-cat” states (superpositions of many spin-coherent states) on longer time scales, a manifestation of the phenomenon of fractional revivals of the initial state. The evolution of the system is visualized with phase-space plots ($Q$ functions) that, when plotted against time, reveal a “quantum carpet” pattern. Of particular interest is the fact that our approximation captures the qualitative features of the model even for small values of $N$. This suggests the possibility of observing the phenomenon of fractional revival in this model for systems of few qubits.

Figure 1

Top: $|\langle {\zeta }_0|\widehat{E}|{\zeta }_0\rangle |$ plotted against ${\theta }_0=2arctan\left|\zeta \right|$ and $N$ for ${\varphi }_0=0$. Bottom: $|\langle {\zeta }_0|{\widehat{E}}^2|{\zeta }_0\rangle |$ plotted against ${\theta }_0$ and $N$ for ${\varphi }_0=0$. The black lines show $|{\zeta }_0{|}^2=1/N$ for $|{\zeta }_0|<1$ (or ${\theta }_0<\pi /2$) and $|{\zeta }_0{|}^2=N$ for $|{\zeta }_0|>1$ (or ${\theta }_0>\pi /2$).

Figure 2

Top: Fidelity of approximation to the exact state plotted against $N$ and $t/T$ with ${\theta }_0=\pi /2$ and ${\varphi }_0=0$. Bottom: Fidelity plotted against ${\theta }_0$ and $t/T$ with $N=40$ and ${\varphi }_0=0$. For ${\theta }_0\approx \pi /2$ (or $|{\zeta }_0|\approx 1$) and $\lambda t\ll N$ (or $t\ll T$) the fidelity is high (red).

Figure 3

$Q$ functions [Eq. (32)] and their stereographic projections at various times for the (exact) state $|{\Psi }^{+}\left(t\right)\rangle$ with $N=100$. (a) The initial state $|{\Psi }^{+}\left(0\right)\rangle =[\frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle +|\downarrow \rangle {\left)\right]}^{\otimes N+1}$, a spin-coherent state. (b) The state $|{\Psi }^{+}(T/50)\rangle$. (c) The state $|{\Psi }^{+}(T/10)\rangle$. (d) The state $|{\Psi }^{+}(T/2)\rangle$, a GHZ state. (e) The state $|{\Psi }^{+}(4T/5)\rangle$. (f) The state $|{\Psi }^{+}\left(T\right)\rangle$ at the revival time.

Figure 4

Plotted is $Q(\frac{\pi }{2},\varphi )$, the $Q$-function slice at $\theta =\pi /2$. Here $N=168$. We see a “quantum carpet.”

Figure 5

$Q$ functions (and their stereographic projections) at various times and for small values of $N$. Each time the initial state is $|{\Psi }^{+}\left(0\right)\rangle =[\frac{1}{\sqrt{2}}\left(\right|\uparrow \rangle +|\downarrow \rangle {\left)\right]}^{\otimes N+1}$. (a) $|{\Psi }^{+}(T/2)\rangle$ for $N=8$. (b) $|{\Psi }^{+}(T/3)\rangle$ for $N=10$. (c) $|{\Psi }^{+}(T/4)\rangle$ for $N=16$. (d) $|{\Psi }^{+}(T/2)\rangle$ for $N=16$.

Figure 6

Plotted is $Q(\frac{\pi }{2},\varphi )$ for $N=16$. Even for this small value of $N$ the “carpet” pattern is conspicuous.