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Leaky Repeated Interaction Quantum Systems

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57 pages
Leaky Repeated Interaction Quantum Systems? Alain JOYE FOURIER INSTITUTfi ? Joint work with Laurent BRUNEAU (Universite de Cergy) & Marco MERKLI (Memorial University) CIRM, January 22nd 2009 – p.1/20

  • fermionic reservoir

  • he1 ?

  • hc ?

  • interaction quantum

  • hamiltonian hs

  • quantum sub-systems

  • quantum system

  • finite dimensional

  • universite de cergy


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Leaky Repeated Interaction Quantum Systems
Alain JOYE
INSTITUT
i f
FOURIER

Joint work with
Laurent BRUNEAU (Universite´ de Cergy) & Marco MERKLI (Memorial University)
nd
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
nd
CIRM, January 22 2009 – p.2/20The Formal Model
Complete system S +R+C
Formal Hilbert space H ⊗”H ”⊗H
S R E
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
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
nd
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
nd
CIRM, January 22 2009 – p.3/20Leaky Repeated Interactions Quantum Systems
Pictorially
nd
CIRM, January 22 2009 – p.4/20