Efficient generation of spin cat states with twist-and-turn dynamics via machine optimization

Spin cat states are promising candidates for achieving Heisenberg-limited quantum metrology. It is suggested that spin cat states can be generated by adiabatic evolution. However, due to the limited coherence time, the adiabatic process may be too slow to be practical. To speed up the state generation, we propose to use machine optimization to generate desired spin cat states. Our proposed scheme relies only on experimentally demonstrated one-axis twisting interactions with piecewise time-modulation of rotations designed via machine optimization. The required evolution time is much shorter than the one with adiabatic evolution and it does not make large modification to the existing experimental setups. Our protocol with machine optimization is efficient and easy to be implemented in state-of-the-art experiments.

Figure 1

Spin cat state generation via adiabatic evolution with adiabatic-parameter-fixed sweeping. Blue solid line is the variation of Rabi frequency $\mathrm{\Omega }\left(t\right)$ versus evolution time $t$. Here, the total atom number $N=100$ and $\chi <0$. For $\left|\chi \right|t=0.375, 0.55, 0.69, 0.96$, the corresponding evolved states are close to the spin cat states of ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}, |\mathrm{\Psi }\left(\frac{\pi }{6}\right){\rangle }_{\mathrm{CAT}}, |\mathrm{\Psi }\left(\frac{\pi }{4}\right){\rangle }_{\mathrm{CAT}}, |\mathrm{\Psi }\left(\frac{\pi }{3}\right){\rangle }_{\mathrm{CAT}}$ with fidelity $F=0.99, 0.98, 0.98, 0.91$, respectively.

Figure 2

Spin cat state generation via adiabatic- parameter-fixed sweeping with different $\epsilon$. Green (dash-dotted), red (dashed), and blue (solid) lines represent the results of $\epsilon =0.2, 0.1,$ and 0.05, respectively. (a) The variation of Rabi frequency $\mathrm{\Omega }\left(t\right)$ versus evolution time $t$. (b) The fidelity ${F\left(t\right)=|\langle \psi \left(t\right)|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}{|}^2$ between the instant evolved state and the target spin cat state ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ versus time $t$. The insets show the probability distribution $P\left(m\right)={\left|\langle J,m|\psi \left(T\right)\rangle \right|}^2$ of the final prepared states. Here, the total atom number $N=100$ and $\chi <0$. Conventionally, an OAT dynamics along with a $\pi /2$ pulse along the $y$ axis can generate the spin cat state ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ at $\chi T=\pi /2$ ($\chi >0$). For reference, the result of OAT is also shown, see the orange (dotted) line.

Figure 3

Spin cat state generation via machine optimization. The optimized time-sequence of Rabi frequency $\mathrm{\Omega }\left(t\right)$ for total evolution time (a) $\chi T=0.25$, (b) $\chi T=0.2$, and (c) $\chi T=0.15$. The corresponding fidelity ${F\left(t\right)=|\langle \psi \left(t\right)|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}{|}^2$ between the instant evolved state and the target spin cat state ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ versus time $t$ are shown in panels (d)–(f). The insets show the probability distribution $P\left(m\right)={\left|\langle J,m|\psi \left(T\right)\rangle \right|}^2$ of the final prepared states. Here, the segment number is chosen as $n=20$, the total atom number $N=100$ and $\chi >0$.

Figure 4

Spin cat state generation via machine optimization for a fixed total evolution time $T$ with different segment number $n$. Here, the total evolution time $\chi T=0.15$, the total atom number $N=100$ and $\chi >0$. The optimized time-sequence of the Rabi frequency $\mathrm{\Omega }\left(t\right)$ for segment number (a) $n=20$, (b) $n=10$, (c) $n=5$, and (d) $n=4$. The corresponding fidelity ${F\left(t\right)=|\langle \psi \left(t\right)|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}{|}^2$ between the instant evolved state and the target spin cat state ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ versus time $t$ are shown in panel (e). The inset shows the enlarged region near the final time $T$.

Figure 5

State generation via machine optimization for target spin cat state ${|\mathrm{\Psi }\left(\theta \right)\rangle }_{\mathrm{CAT}}$ with (a) $\theta =0$, (b) $\theta =\frac{\pi }{6}$, (c) $\theta =\frac{\pi }{4}$, and (d) $\theta =\frac{\pi }{3}$. The orange lines are the optimized time-sequence of Rabi frequency $\mathrm{\Omega }\left(t\right)$. The blue lines are the corresponding fidelity ${F\left(t\right)=|\langle \psi \left(t\right)|\mathrm{\Psi }\left(\theta \right)\rangle }_{\mathrm{CAT}}{|}^2$ between the instant evolved state and the target state versus time $t$. Here, the total atom number $N=100$ and $\chi >0$. The total evolution time for (a) $T=0.147$, (b) $T=0.134$, (c) $T=0.121$, and (d) $T=0.097$.

Figure 6

The optimal total evolution time $T_{\mathrm{opt}}$ for different spin cat state generation via machine optimization. Here, we fix the segment number $n=5$ and the fidelity between the prepared state and the target state is $F\left(T_{\mathrm{opt}}\right)\gtrsim 0.99$. Roughly, the optimal total evolution time $T_{\mathrm{opt}}$ exhibits linear relation versus $1/\sqrt{N}$.

Figure 7

The ultimate measurement precision $\mathrm{\Delta }{\varphi }_Q$ with prepared spin cat states versus total atom number $N$. The spin cat states are prepared via machine optimization with segment number $n=5$. The target states for inverted triangles, circles, squares, and triangles are ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}, |\mathrm{\Psi }\left(\frac{\pi }{6}\right){\rangle }_{\mathrm{CAT}}, |\mathrm{\Psi }\left(\frac{\pi }{4}\right){\rangle }_{\mathrm{CAT}}, |\mathrm{\Psi }\left(\frac{\pi }{3}\right){\rangle }_{\mathrm{CAT}}$, respectively. The lines represent the corresponding analytical bounds (2) for spin cat states.

Figure 8

The schematic of procedure via machine optimization.

Figure 9

GHZ state generation via machine optimization. The final fidelity (at the final time $T$) versus the total evolution time $T$.

Figure 10

GHZ state generation via machine optimization with different total evolution time. The state distributions on the Bloch sphere for different time are shown. The blue (solid) lines are the sequence of $\mathrm{\Lambda }\left(t\right)$ and the orange (dotted) lines are the corresponding fidelity.

Figure 11

Spin cat state generation via machine optimization for a fixed total evolution time $T$ with different segment number $n$. The state distributions on the generalized Bloch sphere for different time are shown. The blue (solid) lines are the sequence of $\mathrm{\Lambda }\left(t\right)$ and the orange (dotted) lines are the corresponding fidelity.