Fast bit flipping based on stability transition of coupled spins

A bipartite spin system is proposed for which a fast transfer from one defined state into another exists. For sufficient coupling between the spins, this implements a bit-flipping mechanism, which is much faster than that induced by tunneling. The states correspond in the semiclassical limit to equilibrium points with a stability transition from elliptic-elliptic stability to complex instability for increased coupling. The fast transfer is due to the spiraling characteristics of the complex unstable dynamics. Based on the classical system we find an approximate scaling relation for the transfer time, which even applies in the deep quantum regime. By investigating a simple model system, we show that the classical stability transition is reflected in a fundamental change in the structure of the eigenfunctions.

Figure 1

Quantum transfer from $|0\rangle \equiv |\rightleftarrows \rangle$ to $|1\rangle \equiv |\rightleftarrows \rangle$ as measured by the fidelity (2) for different parameters $(j,k,\epsilon )$ as $(25,1.0,1.2)$ for the black dashed line, $(40,1.0,1.2)$ for the magenta dash-dotted line, and $(25,1.2,1.4)$ for the cyan dotted line. The transfer for different parameters follows the same curve by using the rescaled time $\tau$, see Eq. (9), which is the actual time divided by the time of the analytical approximation, explained in more detail in the main text below. The green dashed curve with $F_{\text{qm}}\approx 0$ is for $(j,k,\epsilon )=(15,1.0,0.8)$ where only dynamical tunneling occurs. The fast bit flipping for the first sets of parameters semiclassically corresponds to complex unstable dynamics (defined below), whereas, no transfer occurs for elliptic-elliptic stability.

Figure 2

(a) Relevant energy levels as function of the coupling $\epsilon$: Red lines highlight those levels, whose eigenfunctions have maximal overlap with $|\rightleftarrows \rangle$, and blue lines highlight those with $|\leftrightarrows \rangle$. (b) Dynamical tunneling time $t_{\mathrm{dyn}}$ (blue $+$) for $\epsilon <{\epsilon }_{\text{crit}}=1$ and the transfer time $t_{\mathrm{trans}}$ (yellow to orange $\times$) for $\epsilon >{\epsilon }_{\text{crit}}$, see Eq. (4) on the semilogarithmic scale. The fidelity at time $t_{\mathrm{trans}}$ is encoded by color. The green vertical dotted line indicates the critical coupling strength ${\epsilon }_{\text{crit}}=1.0$, and the red vertical dashed lines indicate the parameter regime ${\epsilon }_{\text{crit}}\mp 1/N$ for $N=11$ ($j=5$).

Figure 3

Classical motion of the initial state $|\rightleftarrows \rangle$ indicated by red arrows in the two Bloch spheres. A spiraling motion towards the antipodal point leads to the spin-flipped state $|\leftrightarrows \rangle$.

Figure 4

Numerically computed average time of an ensemble of orbits, which are a spherical distance $\delta$ on the Bloch sphere away from the equilibrium point corresponding to $|\rightleftarrows \rangle$ to reach the equator of the sphere. The solid lines show $\epsilon =1.05$, 1.1, 1.2 (red, blue, and green from top to bottom).

Figure 5

Energy levels of the quantized Hamiltonian (10) for $M=10$ (crosses), compared to the effective Hamiltonian (11) (lines) for a single cluster $\alpha =0$ and $\mathrm{\Delta }=0.02$. The red dashed vertical line indicates the EE-CU transition, yellow dashed lines are the boundaries imposed by the Geršgorin circle theorem (only upper bounds). The inset illustrates how the energy clusters arise for the uncoupled case.