A quantum walk simulation of extra dimensions with warped geometry

We investigate the properties of a quantum walk which can simulate the behavior of a spin 1/2 particle in a model with an ordinary spatial dimension, and one extra dimension with warped geometry between two branes. Such a setup constitutes a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+1$$\end{document}1+1 dimensional version of the Randall–Sundrum model, which plays an important role in high energy physics. In the continuum spacetime limit, the quantum walk reproduces the Dirac equation corresponding to the model, which allows to anticipate some of the properties that can be reproduced by the quantum walk. In particular, we observe that the probability distribution becomes, at large time steps, concentrated near the “low energy” brane, and can be approximated as the lowest eigenstate of the continuum Hamiltonian that is compatible with the symmetries of the model. In this way, we obtain a localization effect whose strength is controlled by a warp coefficient. In other words, here localization arises from the geometry of the model, at variance with the usual effect that is originated from random irregularities, as in Anderson localization. In summary, we establish an interesting correspondence between a high energy physics model and localization in quantum walks.

The extrema of this action gives the following Einstein equations T hid δ (y)g µν δ After computing the second derivative of A(y) from Eq. (S.4), and taking into account the periodicity of the metric (1), yields which allows us to identify, from Eq. (S.6), the values of the tensions The results obtained in this section indicate that the bulk geometry has to be Anti-de Sitter, with a negative bulk cosmological constant, and that the visible brane has negative tension, while the hidden one is positive. These results differ, from standard works on Randall-Sundrum, on the constant coefficient appearing in the expression for k, Eq. (S.4) because we are considering a one dimensional ordinary space, so that the computations of the curvature tensor yield different constant factors.

Hamiltonian eigenstates
In order to solve the eigenvalue problem (30), it is convenient to perform the change of basis ξ (q, y) = Hφ (q, y), with the Hadamard matrix, so that the eigenvalue equation becomes where ξ ± are the components of ξ = (ξ + , ξ − ) T . This system of equations can be decoupled, giving which is a second order differential equation that can be solved for the appropriate boundary conditions. We solve this equation both in the positive [ξ ± (0 < y < L)] P and negative domain [ξ ± (−L < y < 0)] N , delivering ξ ± n (y) P =Ae ky 2 cos e ky α n + Be ky 2 sin e ky α n , (S.13) ξ ± n (y) N =Ce − ky 2 cos e −ky α n + De ky 2 sin e −ky α n , (S.14) where we defined These solutions are related by the continuity conditions where in the last one the periodicity of the wavefunctions, Eq. (20), has been used, and imply that the solutions are related by A = C and B = D. The discontinuity introduced by the delta terms at y = 0 and y = ±L, coming from A (y), imposes B = A tan α n , and the following restrictions to the energies tan α n = tan e kL α n , (S.17) which yields the spectrum in Eq. (32). After taking into account these conditions, the eigenstates become ξ ± n (y) = Ae k|y| 2 cos e k|y| α n + tan α n sin e k|y| α n . (S.18) However, these solutions come from the second order differential equation (S.12), whereas the original equations were first order, and relate ξ + (y) to ξ − (y). To find the appropriate solution of the eigenfunctions, we need to take into account these relations. Since any lineal combination of solutions is also a solution of the equations, we consider the solution [ξ + n (y)] 2 of Eq. (S.12) to obtain [ξ − n (y)] 1 from Eq. (S.11), where [ · ] i denotes whether the solution comes from a first (i = 1) or second (i = 2) order differential equation. Similarly, from [ξ − n (y)] 2 we obtain [ξ + n (y)] 1 , so that The general solution for the eigenstates is a lineal combination of this pair of solutions where the relation between the constants K 1 and K 2 is set by Eq. (28), which, depending on the possible values of η, implies the restrictions The solution for the particular case of n = 0 has only a lower component, and is given by

QW explicit time step
In 9 the following QW operator was proposed as a way to reproduce the continuous limit of the Dirac equation in a curved space time with 2 spatial dimensions where the angles {θ ab /a, b = 1, 2} are allowed to depend, in general, both on the time index j and the spatial coordinates x, y. In our case, the metric does not depend on time, therefore we have dropped the subindex j appearing in the above reference.
We have also omitted a factor which depends on the mass, since we are interested in the massless case. The matrix Π reads For a given θ , the operator W k (θ ), for each k = x, y, is defined by with matrices r(θ ) = i cos θ /2 i sin θ /2 − sin θ /2 cos θ /2 , u(θ ) = − cos θ i sin θ −i sin θ cos θ , (S.31) and S k (ε) = exp(−iε p k σ z ) the spin-dependent shift operator along the k direction. We have checked that, in the continuum limit, this QW reproduces the Dirac equation (16)  which allows us to simplify the operators W k (θ ), resulting in W x (θ 12 ) = W y (θ 21 ) = I, and W x (θ 11 ) = S x (−ε). With these simplifications we finally obtain Eq. (38). Making use of the equations that define the QW, Eqs. (37,38,39) and (40), one can recast the evolution of |χ j as a recurrence relation relating the spinor components Eq. (41) at two consecutive time steps. We arrive at for the upper component, where we recall that y = εs, and we defined e ±iθ (y) = c(y) ± is(y). For the lower component one finds We notice that the upper components are displaced in one direction along the x dimension, while the lower components are displaced in the opposite direction.

Mode decomposition of the freely propagating distribution
The stationary states found above form an orthonormal basis, in the continuum limit, that allow for a decomposition of any function along the y coordinate, for a given value of q. They can also be used, after a proprer discretization, in the lattice on which the QW is defined. Following this idea, we introduced the decomposition in Eq. (47), which is a function in the space of q, the lattice quasimomentum along the x coordinate. For this quasimomentum space, the spinor components are related to Eq. High kL limit of the QW time step and limiting entropy In the limit of a high warp factor kL, the exponential e −A(L) becomes very small, so that the QW discrete time recursive evolution Eqs. (S.38,S.39) can be expanded up to the lowest order in this factor, giving χ ↑ j+1,r,s =χ ↑ j,r+1,s , χ ↓ j+1,r,s =χ ↓ j,r−1,s . (S.49) Although this expansion is only valid for values of y close to L, it is still accurate enough for the initial condition located at y = L/2. As discussed in the main text, the asymptotic value of the entanglement entropy decreases as kL is increased. Therefore, the minimum value of the entropy is reached in the limit e −A(L) ≈ 0. The initial condition χ 0,r,s = δ r,0 δ s,s 0 C 0 can be iterated with the help of Eqs. (S.49) to produce the explicit time evolution χ ↑ j,r,s =C ↑ 0 δ r, j δ s,s 0 , χ ↓ j,r,s =C ↓ 0 δ r,− j δ s,s 0 . (S.50) The corresponding reduced density matrix becomes time-independent and diagonal: ρ c (t) = diag(|C ↑ 0 | 2 , |C ↓ 0 | 2 ) (S.51) from which the minimum value of the entropy can finally be obtained: S min = −|C ↑ 0 | 2 log 2 |C ↑ 0 | 2 − |C ↓ 0 | 2 log 2 |C ↓ 0 | 2 . (S.52) 5/5