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Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Discrete-Time Quantum Walk in Momentum Space

#### 2.1. Quantum Walk Evolution

#### 2.2. Lattice Vibrations

#### 2.3. Numerical Simulations: Quantum-to-Classical Transitions

## 3. Double Kicked Rotor Evolution

#### 3.1. Proposal

#### 3.2. Stability with Respect to Noise

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Momentum distributions of quantum walks with kick strength $k\equiv {k}_{0}=1.45$ after a series of $T=15$ steps for different noise strengths ${\mathsf{\Delta}}_{k}$ as given in the corresponding legends. Amplitude noise only: (

**a**,

**b**). Amplitude and phase noise: (

**c**,

**d**), with equal noise strength ${\mathsf{\Delta}}_{\varphi}={\mathsf{\Delta}}_{k}$. Only phase noise in (

**e**) with values of ${\mathsf{\Delta}}_{\varphi}$ as given in the legend. The light-shift phase was compensated perfectly in (a,c) and only according to Equation (13) in (b,d). In (e), the compensation was assumed to be perfect since k was constant.

**Figure 2.**Numerical simulations of the quantum walk with amplitude and phase noise to observe the quantum-to-classical transition. Amplitude noise with (

**a**,

**b**) and without perfect light-shift compensation (

**c**,

**d**). Phase noise in (

**e**,

**f**). (

**a**,

**c**,

**e**) for $T=15$ as a function of the respective noise strength. (

**b**,

**d**,

**f**) as a function of the number of walk steps for ${\mathsf{\Delta}}_{k}=0.3{k}_{0}$ in (b,d) and ${\mathsf{\Delta}}_{\varphi}=0.3$ in (f). Note the similarity of the transitions in the time and noise-strength domain and the disparity of the limiting distributions for both noise effects. Momentum n is given in the experimental units of two photon recoils $2\hslash {k}_{L}$, with ${k}_{L}$ being the wave vector of the kicking beam.

**Figure 3.**Behaviour of the averaged (over 20 steps) mean chiral displacement ${\overline{C}}_{1}(t)$ for the time evolution under ${\widehat{U}}_{1}$ (in blue) and ${\overline{C}}_{2}(t)$ for the time evolution under ${\widehat{U}}_{2}$ (in red). To check the convergence of ${\overline{C}}_{1}(t)$ towards $\frac{{\nu}_{1}}{2}$ and ${\overline{C}}_{2}(t)$ towards $\frac{{\nu}_{2}}{2}$ as stated in Equation (19), $\frac{{\nu}_{1}}{2}$ (green) and $\frac{{\nu}_{2}}{2}$ (yellow) are plotted as well. The parameter ${k}_{1}=\frac{\pi}{2}$ is kept constant as ${k}_{2}$ is varied. The four panels show the effect of different types of perturbation. All data sets represent averages taken over 1000 realisations (each of which with a fixed initial quasimonentum).

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**MDPI and ACS Style**

Groiseau, C.; Wagner, A.; Summy, G.S.; Wimberger, S.
Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor. *Condens. Matter* **2019**, *4*, 10.
https://doi.org/10.3390/condmat4010010

**AMA Style**

Groiseau C, Wagner A, Summy GS, Wimberger S.
Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor. *Condensed Matter*. 2019; 4(1):10.
https://doi.org/10.3390/condmat4010010

**Chicago/Turabian Style**

Groiseau, Caspar, Alexander Wagner, Gil S. Summy, and Sandro Wimberger.
2019. "Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor" *Condensed Matter* 4, no. 1: 10.
https://doi.org/10.3390/condmat4010010