# Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos

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## Abstract

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## 1. Introduction

## 2. Kicked-Top Model

#### 2.1. Classical Kicked Top

#### 2.2. Quantum Chaos of the Kicked-Top Model

#### 2.3. Coherent States

## 3. Multifractality of Coherent States

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Row (

**a**): Phase-space portraits of the classical kicked top. The classical variables $(\varphi ,\theta )$ are plotted for 289 random initial conditions, each evolved for 300 kicks. Row (

**b**): Color scaled plots of the largest Lyapunov exponent of the classical kicked top for different initial conditions. The largest Lyapunov exponents are calculated on a grid with $200\times 200$ initial conditions, each evolved for 5000 kicks. The different columns correspond to (from left to right): $\kappa =0.4,1.7,3$ and $\kappa =7$. Other parameter: $\alpha =4\pi /7$. All quantities are dimensionless.

**Figure 2.**(

**a**): Phase-space-averaged largest Lyapunov exponent ${\overline{\lambda}}_{+}$ as a function of $\kappa $ for several values of $\alpha $. (

**b**): ${\overline{\lambda}}_{+}$ as a function of $\kappa $ and $\alpha $. The averaged largest Lyapunov exponents are calculated by averaging ${\lambda}_{+}$ over 40,000 different initial conditions, each evolved for 5000 kicks. In (

**b**), the white dot-dashed curve corresponds to the values of ${\kappa}_{c}$ at which ${\overline{\lambda}}_{+}=0.002$. All quantities are dimensionless.

**Figure 3.**Level spacing distributions of the kicked top model for (

**a**) $\kappa =0.4$, (

**b**) $\kappa =1.7$, (

**c**) $\kappa =3$, and (

**d**) $\kappa =7$. The Poisson distribution is plotted as a blue solid curve, and the red dot-dashed curve denotes the Wigner–Dyson statistics. (

**e**) The level repulsion exponent $\beta $ as a function of $\kappa $. (

**f**) Averaged level spacing ratio $\langle r\rangle $ as a function of $\kappa $. The upper (bottom) red dashed line indicates ${\langle r\rangle}_{COE}\approx 0.527$ (${\langle r\rangle}_{P}\approx 0.386$). Other parameters: $j=1000$ and $\alpha =4\pi /7$. All quantities are dimensionless.

**Figure 4.**Color scaled plot of multifractal dimensions ${D}_{q}$ for (

**a1**–

**a4**) $q=1$, (

**b1**–

**b4**) $q=2$, and (

**c1**–

**c4**) $q=\infty $, calculated on a grid of $100\times 100$ coherent states. The different columns correspond to (from left to right): $\kappa =0.4$, $\kappa =1.7$, $\kappa =3$, and $\kappa =7$. Other parameters: $j=150$ and $\alpha =4\pi /7$. All quantities are dimensionless.

**Figure 5.**The variation in phase-space-averaged multifractal dimensions ${\overline{D}}_{1}$ (

**a**), ${\overline{D}}_{2}$ (

**b**), and ${\overline{D}}_{\infty}$ (

**c**) with kicking strength $\kappa $ for different j are denoted by color scales. The phase space average is performed over ${10}^{4}$ coherent states in phase space. Other parameters: $\alpha =4\pi /7$. All quantities are dimensionless.

**Figure 6.**Phase-space-averaged fractal dimensions ${\overline{D}}_{q}$ with $q=1,2,\infty $ versus $1/ln\mathcal{N}$ for $\kappa =0.4$ (

**a**) and $\kappa =7$ (

**b**). Here, $\mathcal{N}$ denotes the dimension of Hilbert space. ${\overline{D}}_{q}$ were calculated from ${10}^{4}$ coherent states in phase space. Dashed lines in panel (

**a**) are of the form $1/2-{f}_{q}/ln\mathcal{N}$, with ${f}_{1}=0.421,{f}_{2}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.267$ and ${f}_{\infty}=-0.0758$. In panel (

**b**), dashed lines for $q=1,2$ are of the form $1-{g}_{q}/ln\mathcal{N}$ with ${g}_{1}=0.484$ and ${g}_{2}=0.779$, while the dashed line for $q=\infty $ is given by $1\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}{g}_{\infty}ln(ln\mathcal{N})/ln\mathcal{N}$ with ${g}_{\infty}=1.097$. Other parameters: $\alpha =4\pi /7$. All quantities are dimensionless.

**Figure 7.**Histograms of $P(lnx)$ for $\kappa =0.4$ (

**a**), $\kappa =1.7$ (

**b**), $\kappa =3$ (

**c**), and $\kappa =7$ (

**d**). The purple solid lines in the main panels denote ${P}_{2}(lnx)$ [cf. Equation (30)]. The inset in each panel plots their cumulative distributions with blue solid curve corresponds to numerical result, while the red dashed curve represents ${F}_{2}\left(x\right)$ (cf. Equation (32)). $P(lnx)$ has been computed from ${10}^{4}$ coherent states in phase space. Other parameters: $j=150$ and $\alpha =4\pi /7$. All quantities are dimensionless.

**Figure 8.**Panel (

**a**): ${\mathcal{D}}_{KL}^{\left(2\right)}$ as a function of $\kappa $ for different system sizes. Inset: ${\mathcal{D}}_{KL}^{\left(2\right)}$ as a function of Hilbert space dimension $\mathcal{N}$ with $\kappa =8$. Panel (

**b**): ${\mathcal{R}}_{d}^{\left(2\right)}$ as a function of $\kappa $ for different j values. Inset: ${\mathcal{R}}_{d}^{\left(2\right)}$ versus Hilbert space dimension $\mathcal{N}$ for $\kappa =8$. Other parameter: $\alpha =4\pi /7$. All quantities are dimensionless.

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**MDPI and ACS Style**

Wang, Q.; Robnik, M.
Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos. *Entropy* **2021**, *23*, 1347.
https://doi.org/10.3390/e23101347

**AMA Style**

Wang Q, Robnik M.
Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos. *Entropy*. 2021; 23(10):1347.
https://doi.org/10.3390/e23101347

**Chicago/Turabian Style**

Wang, Qian, and Marko Robnik.
2021. "Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos" *Entropy* 23, no. 10: 1347.
https://doi.org/10.3390/e23101347