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SCIENCE CHINA Information Sciences, Volume 64 , Issue 2 : 122502(2021) https://doi.org/10.1007/s11432-020-3075-2

Generation of two-axis countertwisting squeezed spin states via Uhrig dynamical decoupling

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  • ReceivedMay 19, 2020
  • AcceptedSep 3, 2020
  • PublishedJan 4, 2021

Abstract


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 11547244, 11547208, 11974334), and the Foundation of Collaborative Innovation Team of Discipline Characteristics of Jianghan University (Grant No. 03100061).


References

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  • Figure 1

    (Color online) Schematic illustrations of the alternate evolution under the Hamiltonian $\chi~J_x^2$ and $-\chi~J_y^2$ according to UDD. (a) The sequence of $\chi~J_x^2$ and $-\chi~J_y^2$ corresponding to ${\rm~UDD}_M^J$ for $M=1,~2,~\ldots,~6$, and for simplicity $\chi$ is omitted. protect łinebreak (b) ${\rm~PS}_{{\rm~UDD}_M^{J}}$, the whole series of $\chi~J_x^2$ and $-\chi~J_y^2$ from $t=0$ to $t=t_{\rm~opt}$.

  • Figure 2

    (Color online) An illustration of the pulse sequences for generating TACT-type Hamiltonian $H_2~\propto~\chi~(J_x^2-J_y^2)$. (a) The pulse sequence of ${\rm~UDD}_M^T$ for $M=1,~2,~\ldots,~6$, here the red and green rectangles correspond to the $\frac{\pi}{2}$ and $-\frac{\pi}{2}$ pulses around $y$ axis, respectively. (b) ${\rm~PS}_{{\rm~UDD}_M^{T}}$, the whole pulse sequence from $t=0$ to $t=t_{\rm~opt}$.

  • Figure 3

    (Color online) (a) The sequences of $\chi~J_x^2$ and $-\chi~J_y^2$ based on ${\rm~UDD}_1^J$ and ${\rm~UDD}_2^J$. (b) The pulse sequences for one single period used in [25]and in our previous work [27], which can be obtained via ${\rm~UDD}_1^J$ and ${\rm~UDD}_2^J$ respectively.

  • Figure 4

    (Color online) The spin squeezing vs. the evolution time of our ${\rm~PS}_{{\rm~UDD}_M^T}$ scheme while $M=2,~\,~4,~\,~6,~\,8,~\,10$ for the system with $N=1250$ spins. The solid grey, the dotted (solid) cyan, the dotted (solid) black, the dotted (solid) yellow, the dotted (solid) red and the dotted (solid) green line display the result for the ideal TACT Hamiltonian $H_2$, ${\rm~PS}_{{\rm~UDD}_2^T}$ with $N_c=3~\,~(6)$, ${\rm~PS}_{{\rm~UDD}_4^T}$ with $N_c=3~\,~(6)$, ${\rm~PS}_{{\rm~UDD}_6^T}$ with $N_c=3~\,~(6)$, ${\rm~PS}_{{\rm~UDD}_8^T}$ with $N_c=3~\,~(6)$, and ${\rm~PS}_{{\rm~UDD}_{10}^T}$ with $N_c=3~\,~(6)$, respectively. $N_c$ is the cycle number of the pulse sequences.

  • Figure 5

    (Color online) The spin squeezing vs. the evolution time for $N=1250$. The solid grey, magenta, cyan, black and yellow line and the dotted yellow line denote the result of the ideal TACT Hamiltonian $H_2$, the TS2 scheme with $N_c=50$, ${\rm~PS}_{{\rm~UDD}_2^T}$ with $N_c=35$, ${\rm~PS}_{{\rm~UDD}_4^T}$ with $N_c=9$, and ${\rm~PS}_{{\rm~UDD}_6^T}$ with $N_c=7$ and $N_c=6$, respectively. $N_c$ is the cycle number of the pulse sequences. A zoom-in around the optimal squeezing time is shown in the inset.

  • Figure 6

    (Color online) The variation of the relative error $E_r=\frac{\xi^2_{{\rm~TS2}({\rm~PS}_{{\rm~UDD}_M^T})}-\xi^2_{\rm~eff}}{\xi^2_{\rm~eff}}$ for the corresponding pulse schemes shown in Figure (5).

  • Figure 7

    (Color online) The minimum pulse number $N_p$ for different spin number $N$. The inset shows the corresponding pulse cycle number $N_c$ for different spin number $N$.