# The Emergence of Chaos in Quantum Mechanics

## Abstract

**:**

## 1. Introduction

## 2. Two Routes to Chaos

- (i)
- For our purposes, the first, fundamental step is to plot selected final physical quantities as a function of initial physical quantities. By inspection we infer if they can be caused by a chaotic behaviour. Of course we are aware that this is a qualitative method that can lead to misjudgements.
- (ii)
- In general, a dense power spectrum of a physical quantity is considered an indication. In our analysis we use the power spectrum of the electric dipole moment, the choice is motivated by the fact that the dipole moment is related to the electromagnetic radiation from an atom.
- (iii)
- In the search for chaos, the Poincaré section is a very important tool. We use it as a flag of chaos.
- (iv)
- Since all the previous points provide only a qualitative characterisation of chaos, as a final step we calculate the Lyapunov exponent of the orbits of the Bloch vector.

#### 2.1. First Hamiltonian

#### 2.2. Second Hamiltonian

## 3. Final Remarks and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**First Hamiltonian. First row: final population of the upper level ${P}_{1}\left({t}_{f}\right)={\left|{c}_{1}\left({t}_{f}\right)\right|}^{2}$ as a function of its initial population ${P}_{1}\left(0\right)$ and relative phase between the two states $\varphi \left({t}_{f}\right)/\pi =[{\varphi}_{1}\left({t}_{f}\right)-{\varphi}_{0}\left({t}_{f}\right)]/\pi $ vs. ${P}_{1}\left(0\right)$. Second row ${\varphi}_{0}\left({t}_{f}\right)/\pi $ and ${\varphi}_{1}\left({t}_{f}\right)$ vs. ${P}_{1}\left(0\right)$. Third row ${P}_{1}\left({t}_{f}\right)$ vs. $\varphi \left({t}_{f}\right)/\pi $. All plots contain 100 points.

**Figure 2.**First Hamiltonian. First row: ${P}_{1}\left(n{T}_{L}\right)$ vs. $\varphi \left(n{T}_{L}\right)/\pi $ where n is an integer and ${T}_{L}$ the laser period. Second row, spectrum emitted by the atom. The atom starts from the ground state.

**Figure 3.**First Hamiltonian. The vector $\langle \mathit{\sigma}\rangle $ stroboscopically plotted at $t=n{T}_{L}$.

**Figure 4.**First Hamiltonian. First row: ${P}_{1}\left(n{T}_{L}\right)$ vs. $\varphi \left(n{T}_{L}\right)/\pi $ where n is an integer and ${T}_{L}$ the laser period. Second row, spectrum emitted by the atom. The initial population of the ground state is ${P}_{0}=0.6$.

**Figure 5.**First Hamiltonian. The vector $\langle \mathit{\sigma}\rangle $ stroboscopically plotted at $t=n{T}_{L}$.

**Figure 6.**First Hamiltonian. The vector $\langle \mathit{\sigma}\rangle $ stroboscopically plotted at $t=n{T}_{L}$.

**Figure 7.**First Hamiltonian. Plot of ${\mathrm{log}}_{2}\left(D\left(t\right)/D\left(0\right)\right)$ vs. $T/{T}_{L}$ of two orbits that start with close initial conditions. For the first orbit ${P}_{0}\left(0\right)=0.6$ and for the second orbit ${P}_{0}\left(0\right)=0.6001$. The value of the Lyapunov exponent is $\lambda \cong 1.4$ oc${}^{-1}$. In the inset, the time evolution of $D\left(t\right)$ is shown in linear scale for the first ${2}^{8}$ oc.

**Figure 8.**Second Hamiltonian. The vector $\langle \mathit{\sigma}\rangle $ stroboscopically plotted at $t=n{T}_{L}$.

**Figure 9.**Second Hamiltonian. First row: ${P}_{1}\left(n{T}_{L}\right)$ vs. $\varphi \left(n{T}_{L}\right)/\pi $ where n is an integer and ${T}_{L}$ the laser period. Second row, spectrum emitted by the atom. The initial population of the ground state is ${P}_{0}=0.5$.

**Figure 10.**Second Hamiltonian. First row: final population of the upper level ${P}_{1}\left({t}_{f}\right)$ as a function of the its initial population ${P}_{1}\left(0\right)$ and relative phase between the two states $\varphi \left({t}_{f}\right)/\pi =[{\varphi}_{1}\left({t}_{f}\right)-{\varphi}_{0}\left({t}_{f}\right)]/\pi $ vs. ${P}_{1}\left(0\right)$. Second row ${\varphi}_{0}\left({t}_{f}\right)/\pi $ and ${\varphi}_{1}\left({t}_{f}\right)$ vs. ${P}_{1}\left(0\right)$. Third row ${P}_{1}\left({t}_{f}\right)$ vs. $\varphi \left({t}_{f}\right)/\pi $. All plots contain 100 points.

**Figure 11.**Second Hamiltonian. Plot of ${\mathrm{log}}_{2}\left(D\left(t\right)/D\left(0\right)\right)$ vs. $T/{T}_{L}$ of two orbits that start with close initial condition. For the first orbit ${P}_{0}\left(0\right)=0.5$ and for the second orbit ${P}_{0}\left(0\right)=0.5001$. The value of the Lyapunov exponent is $\lambda \cong 1.9$ oc${}^{-1}$. In the inset the full time evolution of $D\left(t\right)$ is shown in linear scale.

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**MDPI and ACS Style**

Fiordilino, E.
The Emergence of Chaos in Quantum Mechanics. *Symmetry* **2020**, *12*, 785.
https://doi.org/10.3390/sym12050785

**AMA Style**

Fiordilino E.
The Emergence of Chaos in Quantum Mechanics. *Symmetry*. 2020; 12(5):785.
https://doi.org/10.3390/sym12050785

**Chicago/Turabian Style**

Fiordilino, Emilio.
2020. "The Emergence of Chaos in Quantum Mechanics" *Symmetry* 12, no. 5: 785.
https://doi.org/10.3390/sym12050785