# Solutions of a Two-Particle Interacting Quantum Walk

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Dirac Quantum Walk

## 3. The Thirring Quantum Walk

## 4. Symmetries of the Thirring Quantum Walk

## 5. Review of the Solutions

#### 5.1. Scattering Solutions

#### 5.2. Bound States

#### 5.3. Solution for ${e}^{-i\omega}={e}^{\pm i2p}$

#### 5.4. Solutions for $p\in \{0,\phantom{\rule{0.166667em}{0ex}}\pi /2\}$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Notation

## References

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**Figure 1.**Continuous spectrum of the two-particle walk as a function of the total momentum $p\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}[0,\pi /2]$ with mass parameter $m=0.7$. The continuous spectrum is the same as in the free case. The solid blue curves are described by the functions $\omega =\pm 2\omega \left(p\right)$, and the red ones by $\omega =\pm (\pi -2Arccos(nsinp\left)\right)$. As one can notice, the light-red lines $\omega =\pm 2p$ lie entirely in the gaps between the solid curves, highlighting the fact that ${e}^{\pm i2p}$ is not in the range of ${e}^{-i{\omega}_{sr}(p,k)}$ for $p\ne 0,\phantom{\rule{0.166667em}{0ex}}\pi /2$ (see text).

**Figure 2.**Spectrum of the walk for $m=0.6$ and $p=\pi /6$ as a function of k. The colours highlight the different ranges of eigenvalues corresponding to the dispersion relation ${\omega}_{sr}(p,k)$. The range of ${\omega}_{sr}(p,k)$ is understood to be computed $\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}\left(2\pi \right)$. One can notice that there are four values of the relative momentum k having the same value of the dispersion relation ($\omega =2$ in the figure). This is in contrast to the Hamiltonian model for which there are only two solutions.

**Figure 3.**Complete spectrum of the two-particle Thirring walk as a function of the total momentum p with mass parameter $m=0.7$. The continuous spectrum is as in Figure 1. The solid lines in the gaps show the point spectrum for different values of the coupling constant: from top to bottom, $\chi =2\pi /3,\phantom{\rule{0.166667em}{0ex}}3\pi /7,\phantom{\rule{0.166667em}{0ex}}-3\pi /7,\phantom{\rule{0.166667em}{0ex}}-2\pi /3$. It is worth noticing that, for each pair $(\chi ,p),$ there is only one value in the discrete spectrum. The light-red lines $\omega =\pm 2p$ lie entirely in the gap between the continuous bands highlighting the fact that the ${e}^{\pm i2p}$ is not in the range of ${e}^{-i{\omega}_{sr}(p,k)}$ for $p\ne 0,\pi /2$; for a given coupling constant $\chi $, ${e}^{\pm i2p}$ is an eigenvalue for $p=\chi /2$.

**Figure 4.**We show for comparison the free evolution (

**a**) and the interacting one (

**b**) highlighting the appearance of bound states components along the diagonal, namely when the two particles are at the same site (i.e., ${x}_{1}={x}_{2}$), where ${x}_{1}$ and ${x}_{2}$ denote the positions of the two particles. The plots show the probability distribution $p({x}_{1},{x}_{2})$ in position space after $t=32$ time-steps. The chosen value of the mass parameter is $m=0.6$ and the coupling constant is $\chi =\pi /2$. The two particles are initially prepared in a singlet state located at the origin.

**Figure 5.**We show the evolution of a bound state of the two particles peaked around the value of the total momentum $p=0.035\pi $. The mass paramater is $m=0.6$ and the coupling constant $\chi =0.2\pi $. In (

**a**) is depicted the probability distribution of the initial state. In (

**b**) is depicted the probability distribution of the evolved state after $t=128$ time-steps. One can notice that, in the relative coordinate ${x}_{1}-{x}_{2}$, the probability distribution remains concentrated on the diagonal, highlighting the fact that the two particles are in a bound state. The diffusion of the state happens only in the centre of a mass coordinate.

**Figure 6.**We show the case of two proper eigenstates for $p=0$. In both cases the mass parameter is m = 0.6. (

**a**): probability distribution in the relative coordinate y of $\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}k\phantom{\rule{0.166667em}{0ex}}({v}_{k}^{+-}-{v}_{k}^{-+}){e}^{-iyk}$. (

**b**): probability distribution in the y-coordinate of $\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}k\phantom{\rule{0.166667em}{0ex}}({v}_{k}^{+-}+{v}_{k}^{-+}){e}^{-iyk}$.

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**MDPI and ACS Style**

Bisio, A.; D’Ariano, G.M.; Mosco, N.; Perinotti, P.; Tosini, A.
Solutions of a Two-Particle Interacting Quantum Walk. *Entropy* **2018**, *20*, 435.
https://doi.org/10.3390/e20060435

**AMA Style**

Bisio A, D’Ariano GM, Mosco N, Perinotti P, Tosini A.
Solutions of a Two-Particle Interacting Quantum Walk. *Entropy*. 2018; 20(6):435.
https://doi.org/10.3390/e20060435

**Chicago/Turabian Style**

Bisio, Alessandro, Giacomo Mauro D’Ariano, Nicola Mosco, Paolo Perinotti, and Alessandro Tosini.
2018. "Solutions of a Two-Particle Interacting Quantum Walk" *Entropy* 20, no. 6: 435.
https://doi.org/10.3390/e20060435