Cat-state generation and stabilization for a nuclear spin through electric quadrupole interaction

Spin cat states are superpositions of two or more coherent spin states (CSSs) that are distinctly separated over the Bloch sphere. Additionally, the nuclei with angular momenta greater than 1/2 possess a quadrupolar charge distribution. At the intersection of these two phenomena, we devise a simple scheme for generating various types of nuclear-spin cat states. The native biaxial electric quadrupole interaction that is readily available in strained solid-state systems plays a key role here. However, the fact that built-in strain cannot be switched off poses a challenge for the stabilization of target cat states once they are prepared. We remedy this by abruptly diverting via a single rotation pulse the state evolution to the neighborhood of the fixed points of the underlying classical Hamiltonian flow. Optimal process parameters are obtained as a function of electric field gradient biaxiality and nuclear-spin angular momentum. The overall procedure is seen to be robust under 5% deviations from optimal values. We show that higher-level cat states with four superposed CSS can also be formed using three rotation pulses. Finally, for open systems subject to decoherence we extract the scaling of cat-state fidelity damping with respect to the spin quantum number. This reveals rates greater than the dephasing of individual CSSs. Yet, our results affirm that these cat states can preserve their fidelities for practically useful durations under the currently attainable decoherence levels.

Figure 1

Phase portraits obtained from Eqs. (10) and (11) for three different $\eta$ values. Dashed red lines mark the equator on the Bloch sphere. The thickness of the lines is proportional to the speed of the flows.

Figure 2

The spin Wigner quasiprobability distributions at the main stages of the procedure starting from (a) an initial CSS, (b) just before the rotation instant $t=t_R^{-}$, (c)+(e) just after rotation to pole and equator planes $t=t_R^{+}$, (d)+(f) much later at $t=10t_R$. Items (g) and (h) show the fidelity (solid red line) and normalized rQFI [see, Eq. (6)] (blue dashed line) for the polar- and equator-bound evolutions. A 5/2-spin with $\eta =1$ is considered. Vertical dotted lines mark the instances of the rotation pulses.

Figure 3

Rotation instants and angles (in degrees) for the optimal cat-state generation and stabilization as quantified by maximum fidelity $F_{\max }$ [Eq. (12)] and its ripple $F_{\mathrm{ripple}}$ [Eq. (13)] for polar- and equator-bound cases.

Figure 4

For polar- and equator-bound cases, sensitivity of optimal fidelities (red solid line) under 5% (blue dashed line) and 10% (black dotted line) deviation in the parameters of (a) rotation instant, (b) rotation angle, (c) polar, and (d) azimuthal offsets in the initial CSS. $I=3/2$, and $\eta =0.3$ is considered.

Figure 5

Fidelity evolution without (black dashed line) and with (red solid line) rotation pulse, the latter resulting in a $N=4$ cat state for $\eta =1$ case of $I=5/2$. Also shown is the fidelity with respect to a one-leg rotating target (blue dash-dot line) described by Eq. (14). The vertical gray line marks the instant of the rotation pulse. (a) Without any decoherence included, (b) and (c) with phase damping rates of $\gamma ={10}^{-4}{f}_Q$ and ${10}^{-2}{f}_Q$, respectively.

Figure 6

The ratio of the second-harmonic content with respect to fundamental in the evolution of fidelity as a function of the initial CSS location over the Bloch sphere for $I=5/2$.

Figure 7

For $I=5/2$, $\eta =1$, and polar-bound $N=2$ cat state the effect of decoherence. Left panel: Fidelity (solid red line) and normalized rQFI given by Eq. (6) (blue dash line). Right panel: Spin Wigner quasiprobability distributions at $10t_R$. Three different dephasing rates are considered, $\gamma /f_Q={10}^{-4},{10}^{-3},\text{and}{10}^{-2}$ from top to bottom, respectively.

Figure 8

The variation of fidelity decay time constant $\tau$ (in units of reciprocal dephasing rate, $1/\gamma$) with spin $I$. Polar-bound $N=2$ cat states are considered. Dashed lines mark the $I^{-5/2}$ and $I^{-3}$ scalings. Inset shows the damping of fidelity after the rotation pulse for the case of $I=5/2$, $\eta =0.5$, and $\gamma ={10}^{-2}{f}_Q$; dashed line is the exponential fit to extract the decay time constant.