Discrete Time Quantum Walks and Dirac fermions · Discrete Time Quantum Walks (DTQWs) Introduced by Feynmann (1965) and reintroduced by Aharonov (1993) DTQWs = Formal quantum analogues

Discrete Time Quantum Walksand Dirac fermions

P. Arnault, G. Di Molfetta, M. Brachet, F. DebbaschUPMC, ENS, Paris

Fort Lauderdale, 18 December 2014

ANR ProbaGeo (2009-2013)

– Typeset by FoilTEX –

Why a talk on Discrete Time Quantum Walks (DTQWs)

• My original interest: relativistic hydrodynamics and statistical physics

• Two main topics:

– Proper modelization of irreversible phenomena in the relativisticframework: Relativistic Stochastic Processes, ...

– Mean field theory for relativistic gravitation: back-reaction problem,BH thermodynamics, ...

– Non quantum matter

• Interest in DTQWs comes from the wish to incorporate quantum aspectsof matter

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Why a talk on Discrete Time Quantum Walks (DTQWs)

DTQWs

• are discrete, hence simple systems

• can be realized experimentally in optical systems and condensed matter

• are inherently relativistic

• incorporate a coupling to (artificial) gauge fields, including gravity

• seem to promise a new and natural unification of interactions (!)

• can exhibit various degrees of coherence i.e. can be really quantum(reversible dynamics), or classical (irreversible dynamics)

• pave the way for new laboratory astrophysics and cosmology

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Discrete Time Quantum Walks (DTQWs)

• Introduced by Feynmann (1965) and reintroduced by Aharonov (1993)

• DTQWs = Formal quantum analogues of classical discrete time randomwalks

• DTQWs traditionally useful in

– Quantum information and computing: Ambainis’ algorithm for elementdistinctness(Ambainis, SIAM Journal of Computing, 2007)

– Fundamental physics: study of decoherence(Giulini et al, 1996; Perets et al, Phys. Rev. Lett, 2008)

– Applied physics: Transport in solids, disordered media(Bose, Phys. Rev. Lett, 2003; Burgarth, 2006; Westermann et al, Eur. Phys. J

D, 2006; Bose, Contemp. Phys., 2007)

– Biology: Phototransport in complexes of algae(Engel et al, Nature, 2007; Collini et al, Nature, 2010)

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Discrete Time Quantum Walks (DTQWs)

• DTQWs have been realized experimentally:

– Trapped ions(Schmitz et al, Phys. Rev. Lett., 2009; Zahringer et al, Phys. Rev. Lett., 2010)

– Photons in wave guide lattices or optical networks(Perets et al, Phys. Rev. Lett., 2008; Schreiber et al, Phys. Rev. Lett., 2010)

– Atoms in optical lattices(Karski et al, Science, 2009)

This talk:

Continuous limit of DTQWs in (1 + 1) and (1 + 2) dimensions

→ Propagation of Dirac fermions in artificial gauge fields

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DTQWs in (1 + 1) space-time dimensions

[ψLj+1,m

ψRj+1,m

]= B (αj,mθj,m, ξj,m, ζj,m)

[ψLj,m+1

ψRj,m−1

](1)

where

B(α, θ, ξ, ζ) = eiα[eiξ cos θ eiζ sin θ−e−iζ sin θ e−iξ cos θ

](2)

B = U(2) operator acting on ‘spinor’ Ψ =

[ψL

ψR

]θ, ξ, ζ = 3 Euler angles

DTQW defined by {αj,m, θj,m, ξj,m, ζj,m, (j,m) ∈ N× Z} + initialcondition

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How to obtain the formal continuous limit (in (1 + 1)D)

(Di Molfetta and Debbasch, J. Math. Phys., 2011)

• Suppose there exist two regular functions ψL(t, x) and ψR(t, x) such that

ψL/Rj,m = ψL/R(tj, xm)

• Idem for the angles α, θ, ξ, ζ

• Then: [ψL(tj + ∆t, xm)ψR(tj + ∆t, xm)

]= B(tj, xm)

[ψL(tj, xm + ∆x)ψR(tj, xm −∆x)

]

B(tj, xm) = B (α(tj, xm), θ(tj, xm), ξ(tj, xm), ζ(tj, xm))

• Formal continuous limit ← expansion in ∆t and ∆x at fixed tj and xm.

• ∀(j,m), B(tj, xm) must tend to 1 when ∆t and ∆x tend to 0.

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How to obtain the formal continuous limit

• Thus

∆t = τε

∆x = λεδ, δ > 0. (3)

•

αε(t, x) = α0(t, x) + α(t, x)εβ

θε(t, x) = θ0(t, x) + θ(t, x)εγ

ξε(t, x) = ξ0(t, x) + ξ(t, x)εη (4)

ζε(t, x) = ζ0(t, x) + ζ(t, x)εν.

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Formal continuous limit in (1 + 1)D

• B → 1 as ε→ 0 ⇒ θ0 = kπ

ξ0 = (k+ + k−)π (5)

α0 = (k + k+ − k−)π

ζ0 arbitrary

• Richest scaling: η = β = γ = δ = ν = 1

• Equation of motion in the continuous limit:

(∂T − ∂X)ψL = i(α+ ξ

)ψL + θei(θ0+α0+ζ0)ψR

(∂T + ∂X)ψR = i(α− ξ

)ψR + θei(θ0+α0−ζ0)ψL, (6)

where T = t/τ , X = x/λ

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Continuous limit = Dirac dynamics with electric coupling

• x0 = T , x1 = X

• (ηµν) = diag(1,−1)

• A0(x0, x1) = α(x0, x1), A1(x

0, x1) = −ξ(x0, x1)

• Dµ = ∂µ − iAµ

• γ0 = σ1 =

[0 11 0

], γ1 = −iσ2 =

[0 −11 0

]• {γµ, γν} = 2ηµν

•(iγ0D0 + iγ1D1 −M

)Ψ = 0

M = diag(mL,mR)

mL(x0, x1) =(mR(x0, x1)

)∗= −iθei(θ0+α0+ζ0(x

0,x1))

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Generalized continuous limit in (1 + 1)D

(Di Molfetta, Brachet and Debbasch, 2012, 2013)

• Consider only one step out of n, n ≥ 1

• Continuous limit exists iff Bn → 1 as ε→ 0

• Systematic study = surprisingly complicated problem !

• Complete treatment of the case n = 2 by Di Molfetta, Brachet, Debbasch(2013)

• Here: two examples for n= 2

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Generalized continuous limit: example 1

Example 1

• B =

[− cos θ i sin θ−i sin θ cos θ

]• Admits for n = 2 a continuous limit for all choices of θjm (i.e. θ(t, x)):

∂TψL − (cos2 θ)∂Xψ

L +i

2(sin 2θ)∂Xψ

R =

− sin 2θ

2(∂Xθ)ψ

L +i

2((∂Tθ)− (cos 2θ)(∂Xθ))ψ

R

∂TψR + (cos2 θ)∂Xψ

R − i

2(sin 2θ)∂Xψ

L =

+sin 2θ

2(∂Xθ)ψ

R +i

2((∂Tθ) + (cos 2θ)(∂Xθ))ψ

R

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Generalized continuous limit: example 1

• New basis:

b− = i

(cos

θ

2

)bL −

(sin

θ

2

)bR,

b+ = i

(sin

θ

2

)bL +

(cos

θ

2

)bR. (7)

• Equations in the new basis:

∂Tψ− − (cos θ)∂Xψ

− +∂Xθ

2(sin θ)ψ− = 0

∂Tψ+ + (cos θ)∂Xψ

+ − ∂Xθ2

(sin θ)ψ− = 0

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Generalized continuous limit: example 1

• Metric (gµν) = diag

(1,− 1

cos2 θ

)• ‘True’ spinor Ψg = Ψ(cos θ)1/2

• Ψg obeys the massless Dirac dynamics in the metric g

• Here, θ determines the gravitational field

For n = 1, θ determines the mass (see above).

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A QW propagating radially in and around a black hole

• Lemaıtre coordinates for a Schwarzschild black hole:

ds2 = dτ2 − rgrdρ2 − r2dΓ2

r(τ, ρ) = r1/3g

[32 (ρ− τ)

]2/3• Domain of the Lemaıtre coodinates: ρ ≥ τ

• Singularity: r = 0 i.e. ρ = τ

• Event horizon: r = rg i.e. ρ = τ + 23 rg

• T = τ

X = λρ, λ > 0

• ds2 = dT 2 − rgλ2r

dX2 − r2dΓ2

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A QW propagating radially in and around a black hole

• ρ ≥ τ ⇔ X ≥ λT

• Singularity: X = λT

• Horizon: X = λT + 2λ3 rg

• λ2r ≥ rg ⇔ X ≤ λT + 23λ2

rg: domain D

• In D:

ds2 = dT 2 − 1

cos2 θdX2 − (r(T,X))

2dΓ2

cos (θ(T,X)) = λ

√r(T,X)

rg

r(T,X) = r1/3g

[32

(Xλ − T

)]2/3

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A QW propagating radially in and around a black hole

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Generalized continuous limit: example 2

Example 2

• B = eiα[eiξ cos θ i sin θi sin θ e−iξ cos θ

]• Admits for n = 2 a continuous limit for αε(t, x) = (2k + 1)π2 + ε α(t, x)

ξε(t, x) = (2k′ + 1)π2 + ε ξ(t, x)

θ(t, x) arbitrary

(∂T − iα)ψ− − (cos θ)(∂X + iξ)ψ− +∂Xθ

2(sin θ)ψ− = 0

(∂T − iα)ψ+ + (cos θ)(∂X + iξ)ψ+ − ∂Xθ2

(sin θ)ψ− = 0

• AT = α, Ax = −ξ

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Generalized continuous limit: example 2

• Example 2 = Dirac fermion coupled to both an electric and a gravitationalfield

• Electric and gravitational fields appear as different aspects of the samefield (B operator)

• There exists an exact discrete gauge invariance generating ‘discreteelectromagnetism’

• Problem more complicated for ‘discrete gravitation’

• Do these results extend to higher dimensions and other gauge fields?

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Generalized continuous limit in (1 + 2)D: example 3

(Arnault and Debbasch, 2014)

[ψLj+1/2,p,q

ψRj+1/2,p,q

]= U(αp(ν,B), θ+(ν,m))

[ψLj,p+1,q

ψRj,p−1,q

][ψLj+1,p,q

ψRj+1,p,q

]= V(αp(ν,B), θ−(ν,m))

[ψLj+1/2,p,q+1

ψRj+1/2,p,q−1

]

with

U(α, θ) =

[eiα cos θ ieiα sin θie−iα sin θ e−iα cos θ

], V(α, θ) =

[eiα cos θ ie−iα sin θieiα sin θ e−iα cos θ

]

αp(ν,B) = ν2Bp

2

θ±(ν,m) = ±π4− ν m

2

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Generalized continuous limit: example 3

• Tj = j∆T , Xp = p∆X, Yq = q∆Y

• ∆T = ∆X = ∆Y = ε

• Continuous limit exists iff UV → 1 as ε→ 0 ⇐ ν = ε

• (iγµDµ −m)Ψ = 0 with X0 = T , X1 = X, X2 = Y

{γµ, γν} = 2ηµν (with ηµν = diag(1,−1,−1))

γ0 = σ1 =

[0 11 0

], γ1 = iσ2 =

[0 1−1 0

], γ2 = iσ3 =

[i 00 −i

]

Dµ = ∂µ − iAµ with A0 = A1 = 0, A2 = −BX

B = uniform magnetic field perpendicular to (X,Y )

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Generalized continuous limit: example 3

• Relativistic Landau levels = energy eigenstates of a (1 + 2)D Diracparticle immersed in a uniform perpendicular magnetic field

• Define formally a Hamiltonian for the DTQW by

Ψ(T + ∆T ) = BεΨ(T ) = exp (iHε∆T ) Ψ(T ) (with ε = ∆T !)

• Hε = − 1

iεlnBε =

∞∑k=0

εkHk with H0 = HDirac

• Relativistic Landau levels = eigenstates of H0

• At any order in ε, eigenstates of Hε found by standard perturbation theory

⇒

Relativistic Landau levels for DTQWs

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A few things left to do

• Experiments!

• Systematic extension to higher dimensional space-times

• Extension to other gauge theories/interactions

• Extension to other spins

• Effect of random gauge fields. Link with Relativistic Stochastic Processes

• Extension to graphs. Link with graph geometry

• Extension to non-linear quantum walks

• Consequences for quantum computing, quantum simulation andfundamental physics?

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