# The Effect of Spin Squeezing on the Entanglement Entropy of Kicked Tops

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## Abstract

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## 1. Introduction

## 2. Two-Coupled Quantum Kicked Tops

#### 2.1. Kicked Tops

#### 2.2. Initial States

#### 2.3. Quantum Kicked Top Evolution

#### 2.4. Spin Squeezing

## 3. Quantum-Classical Correspondence

## 4. Quantification of Entanglement

## 5. Results and Discussion

#### 5.1. Entanglement without Spin Squeezing

#### 5.2. Initial Spin Squeezing and Periodic Spin Squeezing

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The quasiprobability density of the coherent state based on the spin Q function is shown here mapped onto a sphere, which is its phase space. The state follows a Gaussian distribution, and the area of the plot indicates the uncertainty of the state. Note that $j=80$.

**Figure 2.**The quasiprobability density of the spin squeezed state after the coherent state has been acted on by the one-axis twisting squeeze operator with $\mu =0.2$. Note that the resulting state is elongated and stretched in both the x- and y-direction. Note that $j=80$.

**Figure 3.**The quasiprobability density of the state after the action of the one-axis twisting squeeze operator on the coherent state at $\mu =10$. The state has split into multiple islands, which is not suitable for our study of quantum-classical correspondence, since the integrity of a squeezed state has been destroyed. Note that $j=80$.

**Figure 4.**Phase space map for $k=3$ with six initial conditions marked in the plot. Note that the triangle markers represent the coordinates of the regular kicked tops, while the square markers represent the coordinates of the chaotic kicked top.

**Figure 5.**A plot of the entanglement dynamics of the kicked tops evolution without spin squeezing. The entanglement rate for the kicked tops with a regular-regular initial state is extremely low over the 100 time steps. The case where at least one of the tops is chaotic has a slight increase in entanglement rate, while the entanglement rate is highest for the chaotic-chaotic initial state.

**Figure 6.**A plot of the entanglement dynamics of the kicked tops for a total of $3\times {10}^{3}$ time steps. The entanglement rates are linear initially, but approach saturation subsequently. Our simulation results show that only the kicked tops starting with the chaotic-chaotic initial state are near to the maximum statistical bound of $4.58$ for the entanglement entropy.

**Figure 7.**A comparison between the entanglement dynamics of quantum kicked tops for the case of the non-squeezed and spin-squeezed initial regular-regular state with $\mu =0.2$. The effect of spin squeezing on the state leads to a substantial increase in entanglement that is comparable to the chaotic quantum kicked top without spin squeezing.

**Figure 8.**A comparison between the entanglement dynamics of quantum kicked tops for the case of the non-squeezed and spin-squeezed initial chaotic-chaotic state with $\mu =0.2$. While the kicked tops with initial spin squeezing is observed to entangle at a faster rate in the beginning, its entanglement eventually drops below the case without spin squeezing.

**Figure 9.**The entanglement dynamics of Figure 8 is extended to a total of $3\times {10}^{3}$ time steps. The asymptotic entanglement entropy of the kicked tops is observed to be lower for the initial spin-squeezed case compared to the case of no squeezing. The squeezing parameter is taken to be $\mu =0.2$, and the initial states $\varphi =0.63$ and $\theta =0.89$.

**Figure 10.**A comparison of the entanglement dynamics for the six different initial conditions indicated in the phase space map of Figure 4. The squeezing parameter is set to be $\mu =0.2$. The dynamical behavior for the entanglement dynamics with no squeezing, with periodic spin squeezing and with initial spin squeezing is consistent with respect to the various regions of the phase space map. In particular, periodic squeezing is again shown to be most effective in generating entanglement for the various initial conditions. Note that the blue line represents no squeezing, the red line initial squeezing and the green line periodic squeezing. (

**a**) $\varphi =0.63$, $\theta =2.25$; (

**b**) $\varphi =0.63$, $\theta =2.10$; (

**c**) $\varphi =0.63$, $\theta =0.89$; (

**d**) $\varphi $ = $-1.50$, $\theta =1.90$; (

**e**) $\varphi =0.63$, $\theta =2.00$; (

**f**) $\varphi =0.63$, $\theta =1.60$.

**Figure 11.**A comparison of the entanglement dynamics for the three cases of no squeezing, periodic spin squeezing and initial spin squeezing, when the kicked tops are in the full chaotic regime at $k\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}7.2$. The entanglement rate and asymptotic entanglement entropy for the three cases are found to match closely with each other and converge to the statistical bound of $4.58$ in the asymptotic limit. The points are plotted sporadically in order to better show the two curves. The squeezing parameter is set to be $\mu =0.2$, and the initial state chosen is $\varphi =0.63$ and $\theta =0.89$.

**Figure 12.**A comparison of the entanglement dynamics for systems of different j. The squeezing parameter is $\mu =0.2$ and t he initial state chosen here is $\varphi =0.63$ and $\theta =0.89$. Note that the blue line represents no squeezing, the red line initial squeezing and the green line periodic squeezing. (

**a**) $j=5$; (

**b**) $j=40$; (

**c**) $j=80$; (

**d**) $j=100$.

**Figure 13.**The quantum power density spectrum of the regular initial state ($\theta =2.25$, $\varphi =0.63$) and the chaotic initial state ($\theta =0.89$, $\varphi =0.63$) with no squeezing, with initial squeezing and with periodic squeezing. (

**a**) Regular state with no squeezing; (

**b**) chaotic state with no squeezing; (

**c**) regular state with initial squeezing; (

**d**) chaotic state with initial squeezing; (

**e**) regular state with periodic squeezing; (

**f**) chaotic state with periodic squeezing. The act of squeezing is observed to cause the quantum power density spectrum of the regular state in (

**c**,

**e**) to approach the spectrum for a chaotic state. In (

**d**), initial squeezing of a chaotic state leads to a spectrum that corresponds to that which arises from a more regular state, a result that is also observed with respect to the entanglement dynamics.

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**MDPI and ACS Style**

Ong, E.T.S.; Chew, L.Y.
The Effect of Spin Squeezing on the Entanglement Entropy of Kicked Tops. *Entropy* **2016**, *18*, 116.
https://doi.org/10.3390/e18040116

**AMA Style**

Ong ETS, Chew LY.
The Effect of Spin Squeezing on the Entanglement Entropy of Kicked Tops. *Entropy*. 2016; 18(4):116.
https://doi.org/10.3390/e18040116

**Chicago/Turabian Style**

Ong, Ernest Teng Siang, and Lock Yue Chew.
2016. "The Effect of Spin Squeezing on the Entanglement Entropy of Kicked Tops" *Entropy* 18, no. 4: 116.
https://doi.org/10.3390/e18040116