Quantum walk hydrodynamics

A simple Discrete-Time Quantum Walk on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results are presented which show that suitable initial conditions indeed produce hydrodynamical shocks. An analytical computation of the asymptotic quantum shock structure is presented. The non-relativistic limit is explored in the Supplementary Material (SM).


Results
New variables. The definition of j leads to so that the current j is necessarily timelike or null. We then introduce ϕ . In particular, the spinor ψ now reads In terms if these new variables, the Lagrangian density and the stress energy tensor read , where ε µν denotes the completely antisymmetric symbol of rank two, with the convention ε ε = − = 1 01 10 .

The dynamical equations derived from
Dirac quantum hydrodynamics. Since j is time-like or null, one can define the density n of the (1 + 1)D Dirac fluid by = µ µ n j j ( ) 1/2 . We now suppose that j is not null and define the vector = u j n / as the 2-velocity of the fluid, normed to unity. The two variables j 0 and j 1 can then be replaced by n and u 1 i.e. the density and the spatial part of the fluid 2-velocity. Equation (5) can then be re-written as and, in this form, brings to mind the standard relation ϕ = −∂ µ µ u w n which links the velocity u of a relativistic potential flow to its potential ϕ, the enthalpy per unit volume w and the particle density n. We thus retain ϕ = − w mn cos as the enthalpy per unit volume of the (1 + 1)D Dirac fluid. The velocity field u then derives from two potentials. One is ϕ + /2 i.e. the global phase of the spinor ψ and contributes to u in the standard way. The other potential is the phase differential ϕ − /2 and contributes to u in a non-standard way, by contraction of its gradient with the (1 + 1)D completely antisymmetric symbol.
Using ( (5)), one then finds that to be compared with the stress-energy tensor η = − µν µ ν µν T wu u p of a relativistic perfect fluid of pressure p. The pressure of the Dirac fluid thus vanishes. This is not surprising because classical pressure in spin-0 superfluids is generated by non-linearities [22][23][24] in the wave equation and the free Dirac Eq. (2) is linear.
The last two terms on the right-hand side of (7) depend on the gradient of ϕ − and, thus, on the gradient of w/n. Indeed, the definition of w leads to where σ is the sign of φ − sin . As for relativistic spin 0 superfluids 25 , the two extra-terms in the above expression of the stress-energy tensor thus depend on the gradient of a thermodynamic function (the enthalpy per particle w/n) and are therefore best viewed as generalized 'quantum pressure' terms. As shown in the SM the two component spinor which obeys Dirac equation degenerates, in the Galilean limit, into a single wave-function which obeys the Schrödinger equation and the relativistic hydrodynamics degenerates into the standard 26-28 Madelung hydrodynamics.
www.nature.com/scientificreports www.nature.com/scientificreports/ Numerical shock simulation. The above generalization of the Madelung transform strongly suggests that the original DTQW can be used to simulate quantum flows. First note that a general positive energy plane wave solution of (2) can be written as (see (3)(4)(5)(6) ). The spinor thus describes, at = t 0, a unit density fluid ( = n 1) in motion with constant velocity u 1 given by In order to simulate quantum flows, we now select the initial conditions of the form (9) but with . These initial conditions are inspired by the similar (but somewhat simpler) choice φ = q x m cos( )/ max which has already been used in the cosmological context to simulate the dynamics of (i) a non-quantum cosmological fluid?, (ii) a Bose-Einstein condensates of axions? Figure 1 shows the evolution of the initial condition (10) through the DTQW for various values of m and constant q max (the larger the mass, the less relativistic the propagation) and displays multiple shocks. The evolution of (11) is shown in Fig. 2 (compare Fig. 2a,b), which displays a single symmetric shock. Thus, both figures reveal that the DTQW can indeed be used to simulate hydrodynamical shocks in a quantum fluid 29,30 .
Let us stress that the shock is present, not only at high resolutions, but also for resolutions as low as = n 64 (see Fig. 2), which is well within current experimental limits 31,32 . Analytical shock computation in the Galilean regime. We now present an analytical computation which reproduces the shock solution in the Galilean limit where the DTQW becomes a continuous time quantum cos( ) 2 In the large-m limit, this integral can be computed by making use of methods that are standard in optics 33 and involve Pearcey's integral 34 defined by i Xy Ty y ( ) 2 4 To wit, we set in the large-m limit, where = a m/4! and ε = m 1/ . In this way, Pearcey's integral Eq. (13) alone can correctly reproduces the structure of the shock (see Fig. 2c).
Useful asymptotic expansions of  I are given in 35,36 and §36.2 of 37 . In particular, the steepest descent method can be directly used in zone I (see Fig. 2d) where  m t x / 2 . It yields the the following asymptotic form: www.nature.com/scientificreports www.nature.com/scientificreports/ . Well inside the caustic in zone III, the function can be written as the sum of 3 interfering contributions (see Fig. 2c,d).
Details of the evolution of the density ψ = | | n 2 and velocity φ = ∂ v m / x of the Schrödinger shock are presented in Fig. 3.

Conclusion
We have shown through a novel generalization of the Madelung transform that one of the simplest DTQWs on the line can be considered as a minimalist model of quantum fluids. This conclusion has been supported by numerical simulations which show that, even within the coherence limit of current experiments, the DTQW evolves an initial condition already considered in the literature into a quantum hydrodynamic shock 29,30 . Thus, under current experimental conditions, DTQW can exhibit hydrodynamical behaviour and, therefore, be used to simulate quantum fluid dynamics. We have also computed the asymptotic shock structure analytically in the non-relativistic limit and proposed an extensive discussion of this limit in the SM.  www.nature.com/scientificreports www.nature.com/scientificreports/ Quantum walks have already been linked to with hydrodynamics in 38,39 , but these earlier results address the quantum Boltzmann equation and transport phenomena, and are thus quite different from those presented in this Letter.
Note that the quantum fluid described in this article exhibits a non-vanishing quantum pressure but its traditional pressure vanishes identically because the underlying QW is linear. The results presented in this article can be generalised to quantum fluids with non vanishing traditional pressure by working with non-linear QWs such as those already considered in [40][41][42] . The non-linearity of these walks can be reconciled with the linearity of Quantum Physics by viewing the non-linear terms as an effective description of (self-)interaction, generated for example by the coupling of QWs through gauge fields 21,[43][44][45] . Let us note that non-linear QWs can in principle be realized experimentally, at least through non-linear optics experiments 46,47 .
The present work should be extended to higher dimensions and higher spins. Also, classical pressure terms should be added by considering non-linear DTQWs 41 , or DTQWs with site to site interactions. One should finally incorporate in the Madelung transform the natural coupling of DTQWs to gauge fields 21,44,45,48 , thus obtaining novel models of superconducting quantum fluids or of quantum fluids in relativistic gravitational fields.