∗

Leaky Repeated Interaction Quantum Systems

Alain JOYE

INSTITUT

i f

FOURIER

∗

Joint work with

Laurent BRUNEAU (Universite´ de Cergy) & Marco MERKLI (Memorial University)

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CIRM, January 22 2009 – p.1/20The Formal Model

Quantum system S :

Finite dimensional system, driven by Hamiltonian H on H , s.t.

S S

σ(H ) ={e , ,e } .

S 1 d

nd

CIRM, January 22 2009 – p.2/20The Formal Model

Quantum system S :

Finite dimensional system, driven by Hamiltonian H on H , s.t.

S S

σ(H ) ={e , ,e } .

S 1 d

Chain C of identical quantum sub-systems E ≡E , k = 1,2, :

k

C =E +E +E +E +

1 2 3 4

nd

CIRM, January 22 2009 – p.2/20The Formal Model

Quantum system S :

Finite dimensional system, driven by Hamiltonian H on H , s.t.

S S

σ(H ) ={e , ,e } .

S 1 d

Chain C of identical quantum sub-systems E ≡E , k = 1,2, :

k

C =E +E +E +E +

1 2 3 4

Each E is driven by the Hamiltonian H ≡H on H ≡H ,

k E E E E

k

k

dimH ≤∞

E

The chain C is driven by H ≡H +H +

C E E

1 2

on H ≡H ⊗H ⊗ , with [H ,H ] = 0, ∀j,k.

C E E E E

j k

1 2

nd

CIRM, January 22 2009 – p.2/20The Formal Model

Quantum system S :

Finite dimensional system, driven by Hamiltonian H on H , s.t.

S S

σ(H ) ={e , ,e } .

S 1 d

Chain C of identical quantum sub-systems E ≡E , k = 1,2, :

k

C =E +E +E +E +

1 2 3 4

Each E is driven by the Hamiltonian H ≡H on H ≡H ,

k E E E E

k

k

dimH ≤∞

E

The chain C is driven by H ≡H +H +

C E E

1 2

on H ≡H ⊗H ⊗ , with [H ,H ] = 0, ∀j,k.

C E E E E

j k

1 2

Fermionic reservoir R :

∞ -ly extended gas of indep. fermions at temperature β , driven by ”H ”

R

on ”H ” .

R

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CIRM, January 22 2009 – p.2/20The Formal Model

Complete system S +R+C

Formal Hilbert space H ⊗”H ”⊗H

S R E

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CIRM, January 22 2009 – p.3/20The Formal Model

Complete system S +R+C

Formal Hilbert space H ⊗”H ”⊗H

S R E

Interaction S−C

W operator on H ⊗H , k = 1,2, .

SE S E

k

nd

CIRM, January 22 2009 – p.3/20The Formal Model

Complete system S +R+C

Formal Hilbert space H ⊗”H ”⊗H

S R E

Interaction S−C

W operator on H ⊗H , k = 1,2, .

SE S E

k

Interaction S−R

W operator on H ⊗”H ”.

SR S R

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CIRM, January 22 2009 – p.3/206

The Formal Model

Complete system S +R+C

Formal Hilbert space H ⊗”H ”⊗H

S R E

Interaction S−C

W operator on H ⊗H , k = 1,2, .

SE S E

k

Interaction S−R

W operator on H ⊗”H ”.

SR S R

2

Evolution Let τ > 0 be a duration, λ = (λ ,λ )∈R be couplings

R E

For t = (m−1)τ +s, 0≤s<τ ,

S , R and E are driven by H +”H ”+H +λ W +λ W

m S R E R SR E SE

E evolve freely with H , ∀k =m

k E

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CIRM, January 22 2009 – p.3/20Leaky Repeated Interactions Quantum Systems

Pictorially

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CIRM, January 22 2009 – p.4/20