Discrete Time Quantum Walks and Dirac fermions P. Arnault, G. Di Molfetta, M. Brachet, F. Debbasch UPMC, ENS, Paris Fort Lauderdale, 18 December 2014 ANR ProbaGeo (2009-2013)
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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
1
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
2
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)
3
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
4
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
5
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.
6
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.
7
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
8
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))
9
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
10
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
11
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
12
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).
13
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
14
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
15
A QW propagating radially in and around a black hole
16
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
17
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?
18
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
19
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 )
20
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
21
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?
22
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