Discrete-time quantum walks in random artificial gauge fields

Abstract

Discrete-time quantum walks (DTQWs) in random artificial electric and gravitational fields are studied analytically and numerically. The analytical computations are carried by a new method which allows a direct exact analytical determination of the equations of motion obeyed by the average density operator. It is proven that randomness induces decoherence and that the quantum walks behave asymptotically like classical random walks. Asymptotic diffusion coefficients are computed exactly. The continuous limit is also obtained and discussed.

Appendices

Appendix 1: Interpretation in terms of artificial gauge fields

It has been proven in [20–22] that quantum walks in $(1 + 1)$ dimensional space-times can be viewed as modeling the transport of a Dirac fermion in artificial electric and gravitational fields generated by the time-dependance of the angles $\theta$ and $\xi$. We recall here some basic conclusions obtained in [20–22] and also offer new developments useful in interpreting the results of the present article.

The DTQWs defined by (1) are part of a larger family whose dynamics reads:

$$\begin{aligned} \begin{bmatrix} \psi ^{\mathrm{L}}_{j+1, m }\\ \psi ^{\mathrm{R}}_{j+1, m} \end{bmatrix} \ = {\tilde{\mathcal B}}\left( \theta _{j, m} ,\xi _{j, m}, \zeta _{j, m}, \alpha _{j, m} \right) \begin{bmatrix} \psi ^{\mathrm{L}}_{j, m+1} \\ \psi ^{\mathrm{R}}_{j, m-1} \end{bmatrix}, \end{aligned}$$

(35)

where

$$\begin{aligned} {\tilde{\mathcal B}}(\theta ,\xi , \zeta , \alpha ) = \mathrm{e}^{i \alpha } \begin{bmatrix} \mathrm{e}^{i\xi } \cos \theta&\mathrm{e}^{i \zeta } \sin \theta \\ - \mathrm{e}^{i \zeta } \sin \theta&\mathrm{e}^{-i\xi } \cos \theta \end{bmatrix}. \end{aligned}$$

(36)

The walks in this larger family are characterized by three time- and space-dependent Euler angles $(\theta , \xi , \zeta )$ and by a global, also time- and space-dependent phase $\alpha$. They have been shown to model the transport of Dirac fermions in artificial electric and relativistic gravitational fields generated by the time-dependence of the three Euler angles and of the global phase. In a $(1 + 1)$ dimensional space-time, an electric field derives from a 2-potential $A_{j, m} = \left( V_{j, m}, {\mathcal A}_{, mj} \right)$ and a relativistic gravitational field is represented by 2D metrics $G_{j, m}$. The walks considered in this article correspond to

$$\begin{aligned} \xi= & {} \frac{\pi }{2} + {\bar{\xi }}_j \nonumber \\ \theta= & {} \frac{\pi }{4} + {\bar{\theta }}_j \nonumber \\ \alpha= & {} \frac{\pi }{2} + {\bar{\alpha }} \nonumber \\ \zeta= & {} 0 \end{aligned}$$

(37)

where ${\bar{\xi }}_j$ and ${\bar{\theta }}_j$ are random variables which depend on the time j and ${\bar{\alpha }} = 3 \pi /2$. According to [22], these walks model the transport of a Dirac fermion in an electric field generated by the 2-potential

$$\begin{aligned} A_{j} = \left( V_{j}, {\mathcal A}_{j} \right) = \left( {\bar{\alpha }}_{j}, - {\bar{\xi }}_{j}\right) = \left( \pi /2, - {\bar{\xi }}_{j}\right) \end{aligned}$$

(38)

and in a gravitational field characterized by the metric

$$\begin{aligned} G_{ij} = \text{ diag } (1, - \cos ^{-2} (\theta _j)). \end{aligned}$$

(39)

Since relativistic gravitational fields are represented by space-time metrics [56], making the angle $\theta$ a time-dependent random variable is equivalent to imposing a time-dependent random gravitational field. To better understand the electric aspects of the problem, let us recall that the DTQWs defined by (35) exhibit the following exact discrete gauge invariance [22]:

$$\begin{aligned}&\Psi '_{j, m} = \Psi _{j, m} \mathrm{e}^{- i \phi _{j, m}} \nonumber \\&\xi '_{j, m} = \xi _{j, m} + \delta _{j, m} \nonumber \\&\theta '_{j, m} = \theta _{j, m} \nonumber \\&{\alpha }'_{j, m} = \alpha _{j, m} + \frac{\sigma _{j, m}}{2} \nonumber \\&\zeta '_{j, m} = = \zeta _{j, m} - \delta _{j, m} \end{aligned}$$

(40)

where

$$\begin{aligned} \sigma _{j, m}= & {} \phi _{j, m+1} + \phi _{j, m-1} - 2 \phi _{j +1, m} \nonumber \\ \delta _{j, m}= & {} \frac{\phi _{j, m+1} - \phi _{j, m-1} }{2} \end{aligned}$$

(41)

and $\phi$ is the arbitrary time- and space-dependent phase shift. Let us now define a new quantity $E_{j, m}$ by

$$\begin{aligned} E_{j, m} = - \left( {\mathcal D}_s V\right) _{j, m} + \left( {\mathcal D}_t {\mathcal A}\right) _{j, m} \end{aligned}$$

(42)

where the actions of the operators ${\mathcal D}_s$ and ${\mathcal D}_s$ on an arbitrary time- and space-dependent quantity $u_{j, m}$ are

$$\begin{aligned} \left( {\mathcal D}_s u\right) _{j, m} = \frac{u_{j, m+1} - u_{j, m-1}}{2} \end{aligned}$$

(43)

and

$$\begin{aligned} \left( {\mathcal D}_t u\right) _{j, m} = \frac{2 u_{j+1, m} - u_{j, m+1} - u_{j, m-1}}{2}. \end{aligned}$$

(44)

The operators ${\mathcal D}_s$ and ${\mathcal D}_t$ are discrete counterparts of space- and time-derivatives. It is straightforward to check that the quantity $E_{j, m}$ is a gauge invariant, and coincides in the continuous limit with the standard electric field E(t, x), defined by $E(t, x) = - \partial _x V + \partial _t {\mathcal A}$. The quantity $E_{j. m }$ is thus a bona fide electric field in discrete space-time. For the DTQWs considered in this article, this electric field depends only on the time j and is related to the angle ${\bar{\xi }}$ by $E_{j} = - \left( {\bar{\xi }}_{j+1} - {\bar{\xi }}_j\right)$. Making this angle a time-dependent random variable is thus equivalent to imposing a random electric field.

Appendix 2: Asymptotic computation of the eigenvalues and eigenvectors of the averaged transport operators

Let us here compute the eigenvalues $\lambda ^{\mathrm{e/g}}_r$ and eigenvectors $w^{\mathrm{e/g}}_r$, $r = 1, 2, 3, 4$ only for values of K much smaller than unity. We do not perform an expansion in p because the initial condition is uniform in p and the average evolution does not localize the density operator around $p = 0$. Indeed, the initial condition is localized at $x' = x$ , i.e., does not exhibit any spatial correlation and the dynamics does not create spatial correlations.

The second order expansions of the operators ${\bar{\mathcal R}}^\mathrm{e}$ and ${\bar{\mathcal R}}^\mathrm{g}$ in K read:

$$\begin{aligned} \displaystyle { {\bar{\mathcal R}}^\mathrm{e}_2(K, p, \sigma ) = \begin{bmatrix} 1 - 2K^2&\quad 2iK&\quad 0&\quad 0 \\ 0&\quad 0&\quad \text {sinc}(\sigma /2) \cos (2p)&\quad - i\, \text {sinc}(\sigma /2) \sin (2p) \\ 2i \text {sinc}(\sigma /2) K&\quad \text {sinc}(\sigma /2) \left( 1 - 2 K^2 \right)&\quad \frac{ (1 - \text {sinc}(\sigma ))}{2} \cos (2p)&\quad i \frac{( \text {sinc}(\sigma ) - 1)}{2} \sin (2p)\\ 0&\quad 0&\quad i \frac{(1 + \text {sinc}(\sigma )) }{2}\sin (2p)&\quad - \frac{( \text {sinc}(\sigma ) + 1)}{2} \cos (2p) \end{bmatrix},} \end{aligned}$$

(45)

and

$$\begin{aligned} {\bar{\mathcal R}}^\mathrm{g}_2(K, p, \sigma ) = \begin{bmatrix} 1 - 2K^2&\quad 2i K&\quad 0&\quad 0 \\ 0&\quad 0&\quad \text {sinc}(\sigma ) \cos (2p)&\quad - i\, \text {sinc}(\sigma ) \sin (2p) \\ 2 i \text {sinc}(\sigma ) K&\quad \text {sinc}(\sigma )&\quad 0&\quad 0\\ 0&\quad 0&\quad i \sin (2p)&\quad - \cos (2p) \end{bmatrix}. \end{aligned}$$

(46)

For $K = 0$, these two matrices are both block diagonal and we write ${\bar{\mathcal R}}^{\mathrm{e/g}}_2(K = 0, p, \sigma ) = \text{ diag } (1, M^{\mathrm{e/g}}(p, \sigma ))$, where $M^{\mathrm{e/g}}(p, \sigma )$ are $3 \times 3$ matrices acting in the space spanned by $(u_2, u_3, u_4)$. The matrices ${\bar{\mathcal R}}^{\mathrm{e/g}}_2(K = 0, p, \sigma )$ share $u_1$ as common eigenvector, which we identify as $w^{\mathrm{e/g}}_1(K = 0, p, \sigma )$; the associated eigenvalue is $\lambda _1^{\mathrm{e/g}}(K = 0, p, \sigma ) = 1$. The other eigenvectors and eigenvalues, at zeroth order in K, are those of $M^{\mathrm{e/g}}(p, \sigma )$. These eigenvalues can be computed analytically by solving the third-order characteristic polynomials associated to these matrices. The explicit expressions of these eignevalues are quite involved and need not be replicated here. What is important is how the moduli of these eigenvalues compares to unity. Direct inspection reveals that the moduli of all three $\lambda ^\mathrm{e}_r(0, p, \sigma )$, $r = 2, 3, 4$ are strictly inferior to unity if $\sigma$ is not vanishing. The same goes for all three eigenvalues in the gravitational case, except for one of them which reaches $\pm 1$ independently of $\sigma$ for $p = \pm \pi$ and is also equal to $+1$ for $p = 0$; the eigenspaces corresponding to $\lambda _4^\mathrm{g} (\pm \pi , \sigma )$ and $\lambda _4^\mathrm{g} (0, \sigma )$ are identical and generated by $u_4$, which we choose as $w_4^\mathrm{g}(p = \pm \pi , \sigma ) = w_4^\mathrm{g}(0, \sigma )$. For other values of p, the eigenvalue $\lambda _4^\mathrm{g}(p, \sigma )$ and the eigenvector $w_4^\mathrm{g}(p, \sigma )$ are defined by continuity. All other eigenvectors need not be specified for what follows.

Let us now turn to nonvanishing values of K. The characteristic polynomials of ${\bar{\mathcal R}}^{\mathrm{e/g}}_2(K, p, \sigma )$ contain terms of order 2 and 4 in K; at lowest order in K, the corrections to the eigenvalues $\lambda _j^{\mathrm{e/g}}(K = 0, p, \sigma )$ thus scale generically as $K^2$. Let $\lambda$ be the variable of the characteristic polynomials. At second order in K, the K-dependent correction to each of the zeroth order eigenvalues $\lambda _r^{\mathrm{e/g}}(K = 0, p, \sigma )$ can be found by expanding the characteristic polynomial of ${\bar{\mathcal R}}^{\mathrm{e/g}}_2(K, p, \sigma )$ at first order in $(\lambda - \lambda _r^{\mathrm{e/g}}(K = 0, p, \sigma ))$ and by keeping only the terms scaling as $K^2$. This gives rational expressions for the corrections to the eigenvalues; these rational expressions can be further simplified by a final expansion around $K = 0$ if p is treated as a finite, non-infinitesimal quantity , i.e., $\mid K \mid \ll \mid p \mid$. One then finds:

$$\begin{aligned} \lambda _1^{\mathrm{e/g}}(K, p, \sigma ) = 1 - \alpha ^{\mathrm{e/g}} (p, \sigma ) K^2 + O(K^4) \end{aligned}$$

(47)

with

$$\begin{aligned} \alpha ^{\mathrm{e}}(p, \sigma ) = 2\, \frac{3 + \left( \text {sinc}(\sigma ) \right) ^2 + 2 \left( \text {sinc}(\sigma /2) \right) ^2 \left( 1 + \text {sinc}(\sigma ) \right) + 4 \cos (2p) \left( \text {sinc}(\sigma ) + \left( \text {sinc}(\sigma /2) \right) ^2 \right) }{3 + \left( \text {sinc}(\sigma ) \right) ^2 - 2 \left( \text {sinc}(\sigma /2) \right) ^2 \left( 1 + \text {sinc}(\sigma ) \right) + 4 \cos (2p) \left( \text {sinc}(\sigma ) - \left( \text {sinc}(\sigma /2) \right) ^2 \right) } \end{aligned}$$

(48)

and

$$\begin{aligned} \alpha ^{\mathrm{g}}(p, \sigma ) = 2 \, \frac{1 + \left( \text {sinc}(\sigma )\right) ^2}{1 - \left( \text {sinc}(\sigma )\right) ^2}. \end{aligned}$$

(49)

Note that $\alpha ^\mathrm{g}$ is actually independent of p. Note also that the condition $\mid K \mid \ll \mid p \mid$ does not hinder asymptotic computations, at least on the infinite line. Indeed, as time increases, the density operator becomes more and more localized around $K = 0$, but it does not localize in p-space.² If one works on the infinite line, both K and p are continuous variables and the localization of the density operator around $K = 0$ implies that the size of the region in p-space where the condition $\mid K \mid \ll \mid p \mid$ does not apply actually shrinks to zero with time. For dynamics taking place on a finite circle (finite value of M), computations are a little more involved but can nevertheless be carried out. We feel a detailed analysis of the problem for finite values of M does not bring any valuable insight on interesting physics or mathematics, and we thus restrict the analytical discussion of the asymptotic dynamics to DTQWs on the infinite line, where expressions (48) and (49) suffice.

A direct computation shows that the corrections to the eigenvectors are first order in K. By convention, we fix to unity the value of the first component of $w_1^{\mathrm{e/g}}(K, p, \sigma )$ in the basis $(u_1, u_2, u_3, u_4)$. One thus gets, for example,

$$\begin{aligned} w_1^{\mathrm{g}}(K, p, \sigma ) = \ {\begin{matrix} 1,&\frac{2 i K \text {sinc}(\sigma )^2}{1-\text {sinc}(\sigma )^2},&\frac{2 i K \text {sinc}(\sigma )}{1-\text {sinc}(\sigma )^2},&\frac{- 2 K \text {sinc}(\sigma ) \tan (p)}{1-\text {sinc}(\sigma )^2} \end{matrix}}. \end{aligned}$$

(50)

The expression of $w_1^\mathrm{e}$ is substantially more complicated and need not be reproduced here.

Appendix 3: Asymptotic expression of the density operator in Fourier space

Let us now use the above results to determine the time evolution of the average density operator in both cases under consideration. The first step is to express the initial condition, ${\hat{\rho }}_{j = 0} (K, p) = (u_1 - u_4)/2$ for all (K, p), as a linear combination of the eigenvectors $w^{\mathrm{e/g}}_r(K, p, \sigma )$. We thus write, for $a = 1, 2, 3, 4$

$$\begin{aligned} u_ a= \sum _{r = 1}^4 u^{\mathrm{e/g}}_{ar} (K, p, \sigma ) w^{\mathrm{e/g}}_r(K, p, \sigma ) \end{aligned}$$

(51)

and, conversely,

$$\begin{aligned} w^{\mathrm{e/g}}_r(K, p, \sigma )= \sum _{a = 1}^4 w^{\mathrm{e/g}}_{ra} (K, p, \sigma ) u_a. \end{aligned}$$

(52)

By the above discussion of the eigenvalues and eigenvectors of ${\bar{\mathcal R}}^{\mathrm{e/g}}_2$, one has notably $u^{\mathrm{e/g}}_{11} (K, p, \sigma ) = 1$, $u^{\mathrm{e/g}}_{1r} (K, p, \sigma ) = O(K)$ for $r = 2, 3, 4$, $w^{\mathrm{e/g}}_{11} (K, p, \sigma ) = 1 + O(K)$.

One then writes, for all K and p:

$$\begin{aligned} {\hat{\rho }}_{j = 0} (K, p) = \frac{1}{2} \sum _{r = 1}^4 \left( u^{\mathrm{e/g}}_{1r} (K, p, \sigma ) - u^{\mathrm{e/g}}_{4r} (K, p, \sigma ) \right) w^{\mathrm{e/g}}_r(K, p, \sigma ). \end{aligned}$$

(53)

which leads to

$$\begin{aligned} {\hat{\rho }}_{j = J} (K, p) = \frac{1}{2} \sum _{r = 1}^4 \left( \lambda _r^{\mathrm{e/g}} (K, p, \sigma ) \right) ^J \left( u^{\mathrm{e/g}}_{1r} (K, p, \sigma ) - u^{\mathrm{e/g}}_{4r} (K, p, \sigma ) \right) w^{\mathrm{e/g}}_r(K, p, \sigma ) \end{aligned}$$

(54)

or, expressing the eigenvectors $w^{\mathrm{e/g}}_r(K, p, \sigma )$ in terms of the original basis vectors $(u_1, u_2, u_3, u_4)$:

$$\begin{aligned} {\hat{\rho }}_{j = J} (K, p) = \frac{1}{2} \sum _{r = 1}^4 \sum _{a = 1}^4 \left( \lambda _r^{\mathrm{e/g}} (K, p, \sigma ) \right) ^J \left( u^{\mathrm{e/g}}_{1r} (K, p, \sigma ) - u^{\mathrm{e/g}}_{4r} (K, p, \sigma ) \right) w^{\mathrm{e/g}}_{ra} (K, p, \sigma ) u_a \end{aligned}$$

(55)

Now, for all r,

$$\begin{aligned} \lambda _r^{\mathrm{e/g}} (K, p, \sigma )/ \lambda _1^{\mathrm{e/g}} (K, p, \sigma ) = \lambda _r^{\mathrm{e/g}}(K =0, p, \sigma ) ( 1 + O(K^2)), \end{aligned}$$

(56)

since $\lambda _r^{\mathrm{e/g}}(K =0, p, \sigma ) = 1$. It follows that for small enough K, the contributions to (55) proportional to $( \lambda _r^{\mathrm{e/g}} (K, p, \sigma ))^J$ are much smaller than the contribution proportional to $(\lambda _1^{\mathrm{e/g}} (K, p, \sigma ))^J$ for all values of p and $\sigma$ such that $\mid \lambda _r^{\mathrm{e/g}}(K =0, p, \sigma )\mid < 1$. According to the above discussion, this is realized for all $r \ne 1$ and for all values of p and $\sigma$, except in case 2 (random gravitational field) for $r = 4$, $p = \pm \pi$ or $p = 0$ and all values of $\sigma$. What happens at $p = \pm \pi$ has no incidence on the computation of the density operator in physical space. Indeed, for finite values of M, the maximum value $p_{\text{ max }}$ of $\mid p \mid$ is $p_{\text{ max }} = (2M/(2M + 1)) \pi < \pi$. Thus $\pm \pi$ is only reached in the limiting case of infinite M , i.e., for quantum walks in the infinite line. However, $\pm p_{\text{ max }} = \pm \pi$ then only appear as upper and lower bounds for integrals over p, and the values taken by ${\hat{\rho }} (J, K, p)$ at points $\pm \pi$ does not modify the values of the integrals. Moreover, all current computations are only valid for $\mid p \mid \ll \mid K \mid$, and are thus a priori invalid for $p = 0$. What happens around $p = 0$ has however no relevance to asymptotic computations on the infinite line because, as time increases, the density operator becomes more and more localized around $K = 0$ (see discussion below (23)). For large enough J and small enough K, the double sum in (55) thus simplifies into:

$$\begin{aligned} {\hat{\rho }}^{\mathrm{e/g}}_{j = J}(K, p) = \frac{1}{2} \sum _{a = 1}^4 \left( \lambda _1^{\mathrm{e/g}} (K, p, \sigma ) \right) ^J \left( u^{\mathrm{e/g}}_{11} (K, p, \sigma ) - u^{\mathrm{g/e}}_{41} (K, p, \sigma ) \right) w^{\mathrm{e/g}}_{1a} (K, p, \sigma ) u_a \end{aligned}$$

(57)

Now, $u_{11}^{\mathrm{e/g}} (K, p, \sigma ) = 1 + O(K)$, $u_{41}^{\mathrm{e/g}} (K, p, \sigma ) = O(K)$, $w_{11}^{\mathrm{e/g}} (K, p, \sigma ) = 1 + O(K)$ and $w_{1b}^{\mathrm{e/g}} (K, p, \sigma ) = O(K)$ for $b = 2, 3, 4$. As far as orders of magnitude are concerned, Eq. (57) gives:

$$\begin{aligned} {\hat{\rho }}^{\mathrm{e/g}}_{j = J} (K, p) = \frac{1}{2} \left( 1 - \alpha ^{\mathrm{e/g}} (p, \sigma ) K^2 \right) ^N \left( 1 + O(K)\right) u_1 + \sum _{b = 2}^4 O(K) u_b. \end{aligned}$$

(58)

At lowest order in K, $( 1 - \alpha ^{\mathrm{e/g}} (K, p, \sigma ) K^2 )^J = 1 - \alpha ^{\mathrm{e/g}} (K, p, \sigma ) J K^2$. We will now restrict the discussion to scales K and times J obeying $JK^2 \gg K$ i.e., $J K \gg 1$. Note that the maximum spatial spread of ${\bar{\rho }}$ at time J is $L_{\text {max}}(J) = 2J$, so that the minimum value of K for which ${\hat{\rho }}$ takes non-negligible values at time J is of order $K_{\text {min}}(N) = 1/J$. The condition $J K \gg 1$ thus restricts the discussion to length scales much smaller than $L_{\text {max}}(N)$. In particular, consider the time-dependent scale $K_J= K_*/\sqrt{J}$, where $K_*$ is an arbitrary time-independent wave-vector. The wave-vector $K_J$ obeys $JK_J^2 = K_*^2 \gg K_J$ for sufficiently large J. Thus, the possible diffusive behavior of the averaged transport is encompassed by the present discussion.

With the above assumption, Eq. (57) implies the following approximate, but very simple expression for the long time (large J) density operator in Fourier space:

$$\begin{aligned} {\hat{\rho }}^{\mathrm{e/g}}_{j = J} (K, p) = \frac{1}{2} \left( 1 - \alpha ^{\mathrm{e/g}} (p, \sigma ) K^2 \right) ^J u_1. \end{aligned}$$

(59)

In particular, for $K_J= K_*/\sqrt{J}$ (where $K_*$ is an arbitrary but J-independent wave number) and large enough J,

$$\begin{aligned} {\hat{\rho }}^{\mathrm{e/g}}_{j = J} (K_J, p) = \frac{1}{2} \left( 1 - \alpha ^{\mathrm{e/g}} (p, \sigma ) \frac{K_*^2}{J} \right) ^J u_1 \sim \frac{1}{2} \exp \left( - \alpha ^{\mathrm{e/g}} (p, \sigma ) K_*^2\right) u_1 = \frac{1}{2} \exp \left( - \alpha ^{\mathrm{e/g}} (p, \sigma ) J K_J^2\right) u_1. \end{aligned}$$

(60)

This is an approximate expression for the asymptotic density operator presented in the main body of this article.