Weakly coherent collisional models

Weakly coherent collisional models

Franklin Luis S. Rodrigues, Instituto de Física, Universidade de São Paulo

Gabriele de Chiara, Center for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen’s University Belfast

Mauro Paternostro, Center for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen’s University Belfast

Gabriel T Landi, Instituto de Física, Universidade de São Paulo

Qu^an

tu^{m thermod}ynam ics

IFUSP

Collisional models

• Prepare system ρS and unit ρA uncor- related.

• Interact: U = exp{−iVτ}.

• Throw unit away.

• Repeat.

Global Map:

ρ⁰_SA = U(ρS ⊗ ρA)U^† For the system:

ρ⁰_S = Tr_A{U(ρS ⊗ ρA)U^†}.

Perturbative coherence

Introducing weakly coherent states:

ρA = ρ^th + √

τλχ.

Scaling for dissipative effect:

V → V/ √ τ For small τ:

ρ⁰_S = ρ^S − iτ[H_S + λG, ρ^S] + τD[ρ^S], G = Tr_A{Vχ},

D = −1

2Tr_A{[V, [V, ρS ⊗ ρ^th]]},

• Thermal state induces dissipation

• Weak coherence induces non- commutative unitary evolution

Continuum limit

Taking

ρ˙ = lim

τ→0

ρ⁰_S − ρ^S τ , then:

ρ˙^S = −i[H_S + λG, ρ^S] + D[ρ^S]

• Exact master equation

• Lindblad dissipator

Parametrization

Parametrization of energy preserving interac- tion:

V = X

k

g_k(L_kA^†_k + L^†_kA_k),

where the eigenoperators L_k and A_K are de- fined as

[H_S, L_k] = −ωkL_k, [H_S, A_k] = −ωkA_k.

G = X

k

g_k(L_khA^†_ki_χ + L^†_khA_ki_χ)

D[ρS] = X

k

|g_k|² (

hA_kA^†_ki_thD[L_k] + hA^†_kA_ki_thD[L^†_k] )

Example : Qubits

V = g(σ^S₊σ^A− + σ^S−σ^A₊), ρ^A = f 0

0 1 − f

!

+ 0 q q 0

! , then

G = gqσx, and

D(ρ^S) = γ⁻

"

σ^S−ρ^Sσ^S₊ − 1

2{σ^S₊σ^S−, ρ^S}

#

+ γ⁺

"

σ^S₊ρSσ^S− − 1

2{σ^S−σ^S₊, ρS}

# , where

γ⁻ = g²(1 − f ), γ⁺ = g² f. (1) For

√τλ = 0.7, q = 0.1, ω = 1, g = 0.25 and f = 0.4:

EME vs Exact Dynamics

0 10 20 30 40 50

−0.4

−0.35

−0.3

−0.25

−0.2

t hσzi

0 20 40 60 80 100

−0.2

−0.1 0 0.1 0.2

t hσxi

Energy Balance

H˙ _A ≡ Q˙

H˙ _S = iλ tr{[H_S, G]ρ^S} + tr{D[ρ]H_S}.

˙

W_c ≡ iλ tr{[H_S, G]ρ^S}

˙

Q_inc ≡ τ tr{D[ρ]H_S}. H˙ _S = −Q˙ = W˙ _c + Q˙ _inc.

• Transformation process: Disordered to ordered energy.

• Quantum coherence as a resource.

• No work! Interaction preserves en- ergy.¹

Entropy production

Defined as:²

Σ = I(S : A)⁰ + K(ρ⁰_A||ρ^A) ≥ 0,

where I and K are the mutual in- formation and Kullback-Leibler diver- gence respectively.

With the map above:

˙

W_c ≥ −T C˙ _A

• C_A is the coherence entropy, is a measure for coherence.

• Coherence consumption bounds ordered energy produced

Moreover:

Σ =˙ β( ˙W_c − F˙ _S) = S˙ _S − βQ˙ _inc.

• Modified second law

Next Steps

• Study thermal engines mixing classical and quantum resources

• Include external work

References

1 G. De Chiara, G. Landi, A. Hewgill, B. Reid, A. Ferraro, A. J.

Roncaglia, and M. Antezza, Reconciliation of quantum local master equations with thermodynamics, New Journal of Physics 20, 113024 (2018), 1808.10450.

2 P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and Information Thermodynamics: A Unifying Framework based on Repeated Interactions, arXiv 021003, 1 (2016), 1610.01829.