Bright and dark solitons in a photonic nonlinear quantum walk: lessons from the continuum

We propose a nonlinear quantum walk model inspired in a photonic implementation in which the polarization state of the light field plays the role of the coin-qubit. In particular, we take profit of the nonlinear polarization rotation occurring in optical media with Kerr nonlinearity, which allows to implement a nonlinear coin operator, one that depends on the state of the coin-qubit. We consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us to gain some insight into the existence of different solitonic structures, such as bright and dark solitons. We illustrate several properties of these solitons with numerical calculations, including the effect on them of an additional phase simulating an external electric field.


Introduction
The quantum walk (QW) is a powerful toolbox with many applications.It can be shown to constitute a universal model of computation [1][2][3] with algorithmic applications, such as search problems [4][5][6][7][8][9] or element distinctness [10].QWs manifest into two main categories.Continuous-time QWs (CQWs) are described by a local Hamiltonian originated from the adjacency matrix on some graph with a time evolution which is dictated by the Schrödinger equation, while discrete-time QWs (DQWs) are defined by a unitary evolution operator which relates two consecutive time instants in a stroboscopic way.Another important difference is that, in the case of DQWs, the Hilbert space associated to the graph needs to be enlarged with an additional degree of freedom (the so called 'coin' space).In spite of this different formulation, it is possible to establish a connection between CQWs and DQWs [11][12][13][14][15].In this work, we will concentrate on DQWs.
In this work, we analyze a variant of the DQW which introduces nonlinearities on the angle of the coin operator, and shows some similar phenomena as in the Non-Linear Optical Galton Board (NLOGB) model introduced in [39], where such nonlinearities appeared as phases on the different components of the dynamical map.The main result in [39] was the appearance of soliton-like structures with a rich phenomenology that can be controlled by varying the coupling strength to the nonlinear Kerr medium.In the model we propose below, we observe the formation of bright solitons, as in the NLOGB, and also of dark solitons.We are able to connect these solutions with the continuum space-time limit of the QW, which can be easily obtained.We analyze numerically some aspects of the dynamics of these solitonic structures, including the effect of an additional electric field, and we also show that these solitons do not appear in the two-dimensional case.It is worth mentioning that we are considering localization leading to soliton-like structures due to nonlinearities on the evolution operator.This is to be distinguished from other kinds of localization effects appearing on QWs, which is a different and interesting topic in its own right.Localization may appear as a consequence of spatial disorder (see e.g.[40]) or via the introduction of phase defects [41][42][43].
This paper is organized as follows.In section 2 we first recall the setup for the linear DQW, and review the different proposals to account for a nonlinear DQW.In section 3 we introduce our own proposal, and we discuss its experimental implementation based on nonlinear Kerr optical media.Section 4 is devoted to the analysis of the continuum space-time limit, which is afterwards illustrated by our numerical calculations in section 5. We conclude in section 6 by summarizing our main findings.

Overview on linear and nonlinear DQWs
We start by briefly revisiting the standard (linear) and nonlinear models to describe the DQW for a walker on a one-dimensional lattice.

Linear DQW
Let us consider a particle (the walker) which can move along a discrete lattice with positions x = jϵ, j ∈ Z, with ϵ the lattice spacing.A position Hilbert space H x is associated to this system, which is spanned by the basis {|x⟩}, with x the lattice positions.As mentioned in the Introduction, we also need an additional degree of freedom that defines the coin Hilbert space H c , and will be spanned by two orthogonal states {|↑⟩ , |↓⟩}.The total Hilbert space is, therefore, the tensor product H = H x ⊗ H c , and the basis that spans the whole space is {|x⟩ ⊗ |↑↓⟩} j∈Z .For reasons that will be explained in section 4, we define a time step evolution of the walker using the same amount ϵ, i.e. the state |ψ t ⟩ at a given time t evolves as with U the evolution operator.The operator U is the composition of two unitary operators, where C = I ⊗ R is the coin operator acting on H c .In the latter equation, S represents the conditional displacement operator, which can be formally written as with p the lattice quasi-momentum operator, and σ i , i = x, y, z the Pauli matrices acting on the {|↑⟩ , |↓⟩} states.As for the operator R, it will be represented by a 2 × 2 unitary matrix.An example is given by In what follows, we will set ϵ = 1, so that x = j.However, we will need to restore this parameter in section 4, in order to derive the continuum limit of equation (1).
In terms of the tensor basis in H, one can expand |ψ t ⟩ as follows: In other words, the corresponding spinor is Finally, the operator S takes the form

Nonlinear DQW
The discrete nonlinear QW was not first introduced as such, but as a NLOGB [39], mainly because 'nonlinear QW' is close to be an oxymoron, being quantum mechanics a linear theory; however, the term Non-linear Quantum Walk (NLQW) has made its way through the literature and we adhere to it, but we must keep in mind that the waves used in a NLQW cannot be true quantum wave-functions but some other type of waves.
The NLOGB is a coined DQW on the line in which the wavefunction acquires an additional coin-state-dependent nonlinear phase ϕ c,NL depending on the probability as ϕ c,NL = i2πα |c t,x | 2 with c = u, d and α the nonlinearity strength.This is equivalent to either (i) replacing the standard QW coin operator R by the inhomogeneous nonlinear coin operator or to (ii) generalizing the conditional displacement operator as In [39], the NLOGB was numerically studied and the existence of solitons and of rich spatio-temporal dynamics, including chaotic behavior, was shown.The NLOGB was later experimentally implemented by Wimmer et al [44] in a system involving the propagation of light pulses in optical fibers, an implementation in which the displacement operation consists in delaying or advancing the pulses, so that the QW occurs along the physical time dimension.More recently, the same group made a proposal of NLQW in optical mesh lattices [45], see also the related paper [46], a system that has been recently revisited by Jana et al [47].The NLOGB model has also been the subject of several theoretical studies, including the study of its continuous limit as a nonlinear Dirac equation [48][49][50].Further numerical studies by Buarque and coworkers centred on self-trapping [51], breathing dynamics [52], and rogue waves [53].There has also been made a rigorous mathematical study of the discrete model [54,55] including the demonstration of long term soliton stability [56].Recently, the NLOGB has been extended to three-state coins [57], and generalized to include the effect of perturbing potential barriers [58].Moreover, NLQWs different from the NLOGB have been proposed.Shikano et al [59] proposed a NLQW in which the nonlinearity is due to a feed-forward quantum-coin mechanism such that the coin elements  2 , and concentrated in the study of the influence of zero modes on the formation of solitonic structures in the continuum limit.In [61] the work in [60] was generalized by using mathematical techniques appropriate to Floquet systems, which allowed for the finding of new bifurcations.Another alternative is that of Mendonça et al [62], who propose a nonlinear displacement operator of the form with They numerically find and describe a variety of nonlinear phenomena, which were further studied in [63].
As for Mallick and Flach [64], they use the nonlinear map with ϕ x = γP x + η x , where η x is a noise term, and study the breakdown of Anderson localization induced by the nonlinearity.Finally, in [65] single atoms are proposed as nonlinear beam-splitters in their proposal of a NLQW.
Closely related studies are those by Solntsev et al [66], who incorporate biphoton generation-an intrinsic nonlinear process-in a photonic wave-guide array and study the potential of the system to generate entangled light, but their QW is linear; Verga [67] studies edge-states in a QW with both linear and nonlinear disorder; Bisio et al [68] analytically diagonalize a discrete-time on-site interacting fermionic cellular automaton in the two-particle sector; Adami et al [69] study a NLQW naturally induced by a quantum graph with nonlinear delta potentials; Templeman et al [70] study topological protection in a strongly nonlinear interface lattice; and Held et al [71] introduce Gaussian QWs, which are NLQWs in which the coins are substituted by two-mode squeezers.As stressed by the authors, this kind of NLQWs directly lead to accessible quantum phenomena, rendering possible the quantum simulation of nonlinear processes.
We must also mention works on continuous time NLQWs.In [72] the destruction of Anderson localization by nonlinearity is studied through discrete Anderson nonlinear Schrödinger equations that correctly describe the one-dimensional disordered waveguide lattices used in the experiments of Lahini et al [73].But most studies are related to the problem of database searching.Ebrahimi Kahou and Feder study this problem with coupled discrete nonlinear Schrödinger equations, and discuss the implementability of the model with BECs [74].Meyer and Gong study quantum search with the Gross-Pitaevskii equation [75,76] concluding that it solves the unstructured search problem more efficiently than does the Schrödinger equation, because it includes a cubic nonlinearity, and Chiew et al [77] demonstrate that the nonlinear quantum search can be more efficient than quantum search for graph comparison.Di Molfetta and Herzog [78] generalize the Meyer-Gross algorithm to two dimensions finding a clear advantage over the linear QW.Finally, in [79] the thresholds between modulational stability, rogue waves and soliton regimes are studied with coupled nonlinear Schrödinger equations with on site saturating nonlinearity.
In the present paper, we introduce an alternative formulation of the NLQW appropriate for light polarization qubits propagating in Kerr media.Specifically, we introduce a nonlinear coin in which the rotation angle is given by θ = θ 0 + θ NL with θ 0 constant and θ NL depending on the light polarization state, hence in the coin state.

NLQW coin and map
The NLQW we propose introduces the non-linearity in the coin operator.The unitary operator R is now defined in a way that depends on the state of the walker where the angle of rotation is given by where we explicitly expressed the upper and lower components given by their modules and complex angle as x is the phase difference.If we write how the components evolve explicitly after each time step, we get Notice that the nonlinearity in the coin operator equation ( 13) depends on a single angle θ defined in equation (14).This is at variance with the model used in [39], which can be defined by a coin operator as in equation ( 8), with different phases appearing in front of each matrix element.This anticipates a different behavior, even if both models give rise to soliton-like structures.In fact, using elementary algebra one can easily prove that no unitary transformation can be used to map both coin operators, and that this remains true when one is allowed to include an extra space-time dependent global phase.This shows that both operations are not equivalent.Also, the procedure to reach the continuum limits is different.In [48], the authors require two time steps and an additional change of basis to obtain the continuum limit.In our case, a simple first order expansion in the parameter ϵ allows to obtain the corresponding limit, as shown in appendix A. The mass term in the nonlinear Dirac equation also differs from our equation (22).

Experimental proposal
In proposing the nonlinear rotation term in equation ( 14), we are thinking in a QW photonic platform, using light-polarization qubits [80][81][82], and including optical media with Kerr-type nonlinearity.It is well known that in an isotropic Kerr medium the normal modes of propagation are circularly polarized, and their corresponding indexes of refraction are given by [83] n where the subscripts + and − make reference to right-and left-circular polarization, respectively, n 0 is the linear refractive index, and A and B are the Maker-Terhune coefficients for the nonlinear medium, whose ratio depends on the specific physical mechanism responsible for the Kerr effect (e.g.A = B for nonlinear electronic response).In such a medium, the phenomenon of nonlinear polarization rotation occurs, by means of which a polarized monochromatic-wave that propagates a distance z along the nonlinear medium has the expression where k m = (n + + n − ) ω/2c is the mean propagation constant, σ± = (x ± iŷ) / √ 2 are the unit circular polarization vectors, and This expression shows that the polarization state of the light undergoes a rotation θ NL after the propagation, the circular components changing from σ± to σ ′ ± = σ± e ±iθNL , and the linear components of the polarization passing from (x, y) at the entrance to Now, we take the linear polarization components as the coin state basis, so that the displacement operator acts on these linear components.It is then necessary the use of the additional standard coin rotation θ 0 in equation ( 14), because, after the action of the displacement operator, the field is linearly polarized at the displaced positions, and we have seen that there is no nonlinear rotation for linear polarizations, which means that the nonlinear coin would not act but in the first step.Finally, by writing θ NL in terms of the linear polarization components of the field, one arrives at the expression in equation (14).
Hence, the nonlinear QW we are proposing could be experimentally implemented in a photonic platform that should consist of: (i), a quarter wave-plate, in order to implement the linear coin operator (or, more generally, an electro-optic modulator implementing a general unitary transformation of the light beam polarization state); (ii), a section including a suitable cubic nonlinear material, in order to implement the nonlinear polarization rotation; and (iii), a conditional displacement operation.This last conditional operation could be implemented in several ways, e.g. with an electro-optic modulator that increases/ decreases the light beam frequency depending on the polarization state, as in [80,81], or with a design that forces, again depending on the polarization state, the following of paths with different lengths, as in [82].In any case, it is most convenient that the elements be arranged within an optical cavity, which allows for the implementation of a large number of steps of the walk, as gain can be also introduced in the cavity in order to compensate for losses, see [80].Of course, all this could be actually implemented in a number of different ways.For example, if one thinks in an implementation within an optical cavity made of optical fibers, the nonlinear polarization rotation will naturally occur during the propagation of the light field along the fibers when the light beam has enough intensity.If, on the contrary, the cavity is such that the two orthogonal polarizations follow different paths, then one must ensure that there is a section of the paths which is common for the two polarizations, and placing in it the Kerr medium in which the nonlinear polarization will occur.

Continuum limit
The continuum limit of the QW is obtained by retaining the lowest order, i.e.O(ϵ), in the unitary evolution defined by equation (1).To this purpose, we need to restore the parameter ϵ both in the lattice spacing and in the time step.We also need to appropriately rescale the coin angle in equation ( 4), so that the expansion is performed around θ = 0.This condition is needed to cancel the zero order term (i.e. the identity operator) in the expansion.The continuum limit of this QW can then be obtained following the standard method.Details and definitions are given in appendix A.
The non-linear Dirac equation obtained from this limit reads where the Dirac matrices are γ 0 = σ y and γ 1 = iσ x and the mass term is given by where we defined the rescaled angle ϵ θ0 = θ 0 and nonlinearity parameter ϵα = α.We notice that this mass term is different from that obtained from the NLOGB in [48].In analogy with the discrete case, equation ( 6), we define the continuum spinor components as Ψ (t, x) = (u (t, x) , d (t, x)) T and, in what follows, we drop the explicit temporal and spatial dependence of the spinor components and the quantities defined form them.We can write equation ( 21) in terms of the spinor components as and we alleviated the notation by not writing the explicit spatial and time dependence of each component.This system of equations can be rewritten in terms of the modulus and phases of the spinor components u = |u|e iφu and d = |d|e iφ d .After defining one easily gets with θ ≡ θ0 + α|u||d| sin δ .(28)

Homogeneous stationary solutions
In order to gain some insight into the solutions of the system, we first study the homogeneous stationary solutions and their stability.Let us focus on the last two equations of equation ( 27), which can be related as which for stationary solutions 3 , and from the condition of normalization of the wavefunction, implies that |u| = |d|, and hence ∂ t |u| = ∂ t |d| = 0, which implies in the last two equations of (27) that δ is also time independent.
Hence, there is a clear distinction between solution δ = π/2, which is neutrally stable, and solution δ = −π/2, which is unstable versus perturbations with non-null wave-number.This is reminiscent of the modulational instability occurring in optical fibers [83].

Solitons as stationary solution of the continuum equation
We now aim to look for localized stationary solutions of the system of differential equations defined in equations ( 31) and (32).Let us first consider the case in which δ is close to −π/2 where the modulational instability is expected.Consider the case of a bright localized structure, such as a bright soliton.Notice first that this type of structure tends assymptotically towards the trivial solution far from its center.In particular, far from the structure center it is verified that ∂ x I → 0 and ∂ x δ → 0, so that we can assume that it reaches a nearly homogeneous solution with very small, but non null, intensity, i.e. solution (iii) above.We conclude that far from the structure δ = −π/2 and I → 0, which allows us to conclude from equation ( 31) that which will be valid in any region of space since σ is homogeneous (∂ x σ = 0).Finally, let us assume that the stationary solutions presents small variations of the phase difference around −π/2, i.e. δ ≈ −π/2 + ∆, where ∆ is a small perturbation.We can rewrite equations ( 31) and ( 32) for the new variable ∆, and taking into account equation (33) we get Approximating the trigonometric functions up to first order in ∆ these equations reduce to where in the last equation we considered that the term αI 2 ∆ is negligible.The solution of this system of equations is now exact and gives the following solution, where we imposed the normalization condition to obtain the constants of integration.This solution represents the usual shape of bright solitons.In the following section we numerically investigate if the predictions made for the continuous limit still hold for the discrete model.As for the case δ = π/2, even if there is not a modulational instability, we can still expect the formation of dark solitons.A dark soliton is nothing but a domain wall connecting two domains in which there is a π phase difference in the field components, a sign change, which manifests in the intensity as a dark line at the center of the domain wall separating two domains with homogeneous intensity.We can expect the formation of such structures because equations ( 30)-( 32) only depend on the intensity I, which means that the field amplitudes can take any of the two values ± √ I, thus allowing for the formation of domain walls.We numerically show below that this is actually the case.

Numerical
The continuum limit of the NLQW map proposed in equation ( 15) and its stability of analysis of homogeneous stationary solutions predicts the formation of bright and dark solitons.In this section we numerically investigate if these predictions are obeyed by the discrete NLQW and compare the structure of bright solitons with the analytical prediction.
We first explore in figure 1 the evolution of an extended initial condition where the spin of the walker is uniformly distributed, and consider three distinct types of stable or unstable regimes.In the left panel of figure 1, the walker is uniformly distributed with components phase difference δ = π/2, which according to the stability analysis of section 4.1 is a neutrally stable configuration, i.e. perturbations are not enhanced nor diminished.It can be seen that the probability density of the walker with this phase difference is mostly uniform after some initial interactions.When δ = −π/2 the stability was dependent on the value of the intensity I. On the one hand, when Iα/θ 0 < 1 only perturbations with small wave number (0 < k < θ 0 ) are enhanced.In the central panel of figure 1 we can see the appearance of soliton-like structures that have an extended (low k) stable probability distribution.On the other hand, when Iα/θ 0 > 1 perturbations of any wave number are enhanced.In the right panel of figure 1 it can be seen that, for a higher value of the intensity, only very narrow structures, with high wavenumber k, are formed.

Bright solitons
When the soliton-like structures of the central panel of figure 1 are formed, we obtained that the probability distribution of the walker components are well described by the typical sech 2 (x) function, which was also predicted for stationary solutions of the continuum limit.If we consider this distribution as an initial condition with relative phase between walker components δ = −π/2 ⟨x|ψ soliton 0,x where N β is a normalization constant that depends on β, we observed that the associated probability distribution remains stationary at different times.In figure 2 we show the probability distribution of the walker , and the difference of the walker phases δ t,x after t = 500 and 4 subsequent steps.
We also plot the stationary solution obtained in the continuum limit equation (38).The probability distribution is stationary and nicely fits the analytical solution.The phase differences have oscillating values around the boundary of the soliton, but the behavior around the center of the soliton is well described by the approximate analytical solution of the continuum model.We also observed that the phase sum is constant along the x direction, while it has a linear dependence in time.This dependence is observed to be where σ 0 is the initial value of the phases sum, and we notice that this expression is valid in the regions inside the soliton.This observation is in agreement with the results obtained in section 4.2.
When considering an initial condition of the form   we did not observe a stationary soliton, but a soliton that propagates at a constant velocity.We observed that if ν is positive, the initial soliton-like structure propagates to the right (positive x) and, if it is negative, it would propagate to the left, i.e. ν plays the role of velocity on this initial condition.In figure 3 we present the evolution of three initial solitons propagating with different values of ν: two that propagate with different velocities, and another one with ν = 0 that remains stationary.The probability distribution and relative phases are the same as in the static soliton, but with the center displaced at a constant velocity.This initial condition produces a kick, after which he soliton propagates at a constant velocity.Another feature that is characteristic of solitons is that the interaction between them leave the shape of their wave packets unaltered.This effect is also showcased by the solitons generated in this QW.In figure 4 we show the collision of two solitons propagating in opposite directions, and it can be observed that they cross each other without any significant modification after the crossing.

Dark solitons
We saw in the continuum limit that for δ = π/2 there is not any modulational instability, as homogeneous solutions are marginally stable.Hence, the formation of bright solitons is not expected to occur in this case.However, as the equations allow for homogeneous states of intensity I with amplitudes ± √ I, one can expect the formation of dark solitons, in the regions that connect these two possible amplitudes or solutions.In figure 5 we represent the stationary probability and phase difference for an initial condition with a smooth transition between the two regions with opposite amplitude sign.It can be observed that the left and right regions remain constant and keep the initial phase difference of δ = π/2.The valley of the central part represents the dark soliton, and right in the centre, where the probability distribution is null, the phase difference is not well-defined.
In figure 6 the formation of many propagating dark solitons, from two homogeneous regions that are initially spatially separated by a dark region, is observed.After some initial interaction around the boundary regions of the initial walker, some domains of constant intensity are formed.These domains are delimited by regions of near null probability density, which we already saw are stable; these are the dark solitions that have the same characteristics as the ones observed in figure 5.

Solitons in electric fields
We now explore whether these structures are robust against the presence of electric fields [84,85].To include the effect of an electric field we modify the step evolution defined in equation ( 1) by where X is the position operator, and Φ plays the role of an electric field intensity.In the limit where the non-linearity parameter α is null, this unitary evolution corresponds to a Dirac equation with constant electric field in the continuum limit.For the linear DQW it was pointed out in [84] that if Φ is an irrational multiple of 2π, the walker exhibits localization.If Φ is a rational fraction of 2π, i.e.Φ = 2π n/m, the walker can exhibit oscillations around the initial position but it will eventually become ballistic after a time that will depend on m.In figure 7 we explore the dynamics of a soliton of NLQW under the effect of the electric field in three regimes: Φ = 2π/φ, Φ = 2π/5 and Φ = 2π × 51/256, with φ = (1 + √ 5)/2 the golden ratio.The first case is known to correspond to the most irrational number, whereas the last two cases give a very close value Φ, the only difference being that the denominator is much larger for the last case.It can be observed that in the irrational case the walker remains localized, but the smooth structure of the initial soliton is lost.For the second case, the walker undergoes some oscillations but quickly becomes ballistic.In the last case the soliton is seen to split into two components that undergo oscillations and present some interference patterns.
We have observed (not shown) that the effect of the electric field dominates over the nonlinear rotation angle.The dynamics of the soliton is very similar to the dynamics of a linear walker with an extended probability distribution subject to an electric field.The effect of the nonlinear angle is only apparent at longer time scales where interferences become dominant.We have also observed (not shown) that the same phenomenology is displayed by dark solitons in the presence of electric fields.

No solitons in 2D
This NLQW can be extended to a two-dimensional spatial Hilbert space H x ⊗ H y with basis {|x⟩ ⊗ |y⟩}.We will make use of the split-step evolution for this QW [86,87], so that the same coin space and coin operators can be used with spinor components u t,x,y , d t,x,y .The time step is therefore defined as where S i = e −ip i σz is the conditional shift operator in the direction i = {x, y}, and C is the same coin operators as before, with the rotation angle similarly defined as where there is a dependence on the values of the walker in both dimensions.It was discussed in [78] that non-linear QWs, that introduce the nonlinearity in form of phases on the walker components, can be exploited to perform efficient search tasks on the two-dimensional grid.In line with those findings, we observed that the NLQW that introduces nonlinearities in the coin rotation operator angle produces ballistic dispersion, indicating that no soliton-like structures are formed in the two-dimensional case.

Conclusions
In this work, we have proposed and analyzed a nonlinear QW model which can be experimentally implemented using the components of the electric field on an optical nonlinear Kerr medium.Differently to the NLOGB model proposed in [39], where nonlinearities manifest as a set of different phases of the coin operator (or, equivalently, of the displacement operator), here they give rise to a rotation in the coin operator, with a single angle which depends (in a nonlinear fashion) on the state of the walker.This simple dependence makes it easy to consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation.The analysis of this continuum limit allows us, under some approximations, to gain some insight into the nature of the soliton structure, which is illustrated by our numerical calculations.
These solitons are stable structures whose trajectories can be modulated by choosing the appropriate initial condition.From the continuum limit stability analysis, we were able to predict the existence of both bright and dark solitons, which were numerically characterized.We have also studied the stability of solitons when they are subject to an additional phase that simulates an external electric field, for different rational and irrational values of the field strength.Finally, we also explored a 2D version of this model, where no evidence of soliton formation was found.
To summarize, nonlinear QWs constitute an interesting field with a rich phenomenology that can be used for a better control of its algorithmic and simulation properties.We also remark that the continuum limit of the DQW provided invaluable insight on the properties of the discrete model, which allowed us to predict the existence of both bright and dark solitons.
at the exit.Notice that for linearly polarized light |E − | 2 = |E + | 2 , and hence θ NL = 0. Notice also that the rotation does not change the proportion |E − | / |E + |.

Figure 1 .
Figure 1.Evolution of an extended initial walker, where the spin states in locations x = [−30, 30] all have the same initial coin state.The coin angle is θ0 = π/3, and the non-linearity parameter is α = 2π in all panels.In the first panel, the initial coin state is |ψ0⟩ = (1, −i) T / √ 2 so that δ = π/2, while in the last two panels the initial coin state is |ψ0⟩ = (1, i) T / √ 2 with a corresponding δ = −π/2.The initial intensity of each initial condition is given in the title of each panel.The intensity or probability density of the walker, defined as Pt,x = |ut,x| 2 + |dt,x| 2 , is given by a heatmap, where black indicates a low probability density and brighter/hotter colors indicate a higher probability density.

Figure 2 .
Figure 2. (Top) Probability distribution of the walker.(Bottom) Phase difference between walker components at different consecutive time steps.The red dashed line represents the analytical solution obtained in equation (38) for both quantities.The initial condition is (39) evaluated after t = 500 steps and 4 subsequent steps.The parameters of the quantum coin are α = 1 and θ0 = π/3 and the initial width of the walker is given by β = αθ 0/2, for a spacing ϵ = 0.5.The spatial coordinate has also been scaled as x = ϵx.

Figure 5 .
Figure 5. Probability distribution of the walker (upper panel), and phase difference between components (lower panel) with initial condition (42) evaluated after t = 500 steps.The parameters of the quantum coin are α = 1 and θ0 = π/3, the initial width of the walker is given by β = αθ 0/2, and the intensity I = β, for a spacing ϵ = 0.5.The spatial coordinate has also been scaled as x = ϵx.

Figure 6 .
Figure 6.Evolution of the probability distribution of the walker.The probability of finding the walker in locations x = [−10, 10] is initially null, while it is constant everywhere else.The initial spin state, where the intensity is constant, is |ψ0⟩ = (1, −i) T / √ 2 so that δ = π/2, the coin angle is θ0 = π/3 and the non-linearity parameter is α = 2π.

Figure 7 .
Figure 7. Evolution of the initial soliton considered in figure 2 which is subject to a constant electric field after t = 100.The probability density of the electric field is given in the title of each panel.The intensity of the walker is given by a heatmap, black indicates low probability density and brighter/hotter colors indicate higher probability density.