Autonomy of coupled quantum thermodynamic systems Lajos Di osi - PowerPoint PPT Presentation

Autonomy of coupled quantum thermodynamic systems Lajos Di´ osi Wigner Center, Budapest 1 Apr 2014, Kibbutz Ma’agan, Israel Acknowledgements go to: Hungarian Scientific Research Fund under Grant No. 75129 EU COST Action MP1006 ‘Fundamental Problems in Quantum Physics’ 1 Apr 2014, Kibbutz Ma’agan, Israel 1 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Quantum thermalization 1 Coupling of local quantum thermalized systems 2 Incoherent resonant coupling 3 Incoherent resonant coupling: steady state 4 Incoherent resonant coupling: heat flow 5 Summary 6 1 Apr 2014, Kibbutz Ma’agan, Israel 2 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Quantum thermalization Quantum thermalization TLS in heat bath β = 1 / k B T d ˆ ρ dt = − i [ ˆ H , ˆ ρ ] + L ˆ ρ a † ˆ ˆ a † } = 1 . H = ω ˆ n , n = ˆ ˆ a , { ˆ a , ˆ Thermalizer: a † − 1 ρ } ) + Γ e − βω ( ˆ a † ˆ L ˆ ρ = Γ( ˆ a ˆ ρ ˆ 2 { ˆ n , ˆ ρ ˆ a − 1 2 { 1 − ˆ n , ˆ ρ } ) 1 Stationary state: ρ s = 1 + e − βω e − β ˆ n ˆ Current operator: J = L ⋆ ˆ ˆ n + Γ ω e βω (1 − ˆ H = − Γ ω ˆ n ) Current vanishes in steady state: J = Tr ( ˆ ρ s ) = 0 J ˆ 1 Apr 2014, Kibbutz Ma’agan, Israel 3 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Coupling of local quantum thermalized systems Coupling of local quantum thermalized systems TLS A in hot bath β A , TLS B in cold bath β B > β A d ˆ ρ A d ˆ ρ B dt = − i [ ˆ = − i [ ˆ H A , ˆ ρ A ] + L A ˆ ρ A , H B , ˆ ρ B ] + L B ˆ ρ B dt Joint stationary state: ρ s ρ s ρ s ˆ AB = ˆ A ˆ B † ˆ a † ˆ Coupling: ˆ b + ˆ K AB = ǫ ( ˆ a ) b d ˆ ρ AB = − i [ ˆ H A + ˆ H B + ˆ K AB , ˆ ρ AB ] + ( L A + L B )ˆ ρ AB dt Levy-Kosloff arXiv:1402.3825: This equation is incorrect, even for weak coupling Its stationary state may violate 2nd Law Correct equation: joint thermalizer L AB instead of L A + L B L AB is intricate: autonomy of A and B is lost 1 Apr 2014, Kibbutz Ma’agan, Israel 4 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Incoherent resonant coupling Incoherent resonant coupling My points: Revive autonomy of A and B It would persist for weak stochastic coupling Let’s smash coherence of coupling ... and restrict to resonant coupling K AB = √ ǫ † ˆ a † ˆ � � Coupling: ˆ b + ξ ⋆ ( t ) ˆ ξ ( t ) ˆ b a √ ξ = ( w 1 + iw 2 ) / 2 with standard white-noises w 1 , w 2 In resonance ω A = ω B : [ ˆ K AB , ˆ H A + ˆ H B ] = 0 d ˆ ρ AB = − i [ ˆ H A + ˆ ρ AB ] + ( L A + L B + L K H B , ˆ AB )ˆ ρ AB dt † ˆ a † ˆ a † − 1 � † ˆ � ρ AB ˆ a ˆ ρ AB ˆ L K AB ˆ ρ = ǫ ˆ b ˆ a + ˆ b ˆ 2 { ˆ n A + ˆ n B − 2 ˆ n A ˆ n B , ˆ ρ AB } b b Incoherent coupling L K AB preserves autonomy of A and B. 1 Apr 2014, Kibbutz Ma’agan, Israel 5 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Incoherent resonant coupling: steady state Incoherent resonant coupling: steady state Full master equation: d ˆ ρ AB dt = − i ω [ ˆ n A + ˆ n B , ˆ ρ AB ] a † − 1 ρ AB } ) +Γ A e − β A ω ( ˆ a † ˆ +Γ A ( ˆ a ˆ ρ AB ˆ 2 { ˆ n A , ˆ ρ AB ˆ a − 1 2 { 1 − ˆ n A , ˆ ρ AB } ) † − 1 † ˆ +Γ B ( ˆ ρ AB ˆ ρ AB } )+Γ B e − β B ω ( ˆ ρ AB ˆ b ˆ 2 { ˆ n B , ˆ b − 1 2 { 1 − ˆ n B , ˆ ρ AB } ) b b † ˆ a † ˆ † ˆ ρ AB ˆ a ˆ ρ AB ˆ a † − 1 + ǫ ( ˆ b ˆ a +ˆ b ˆ 2 { ˆ n A + ˆ n B − 2 ˆ n A ˆ n B , ˆ ρ AB } ) b b Take (for simplicity) Γ A = Γ B = Γ ≫ ǫ B + ˆ D A ˆ D B + 2 ˆ D A + 2 ˆ D 2 ˆ ρ s ρ s B − ǫ A ˆ D B ρ s ρ s ρ s ˆ AB = ˆ A ˆ (2 + e − β A ω + e − β B ω )(1 + e − β A ω )(1 + e − β B ω ) Γ ˆ ˆ D A = e − ωβ A ˆ n A − e − ωβ B ˆ D B = e − ωβ B ˆ n B − e − ωβ A ˆ D = e − ωβ A − e − ωβ B , n A , n B 1 Apr 2014, Kibbutz Ma’agan, Israel 6 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Incoherent resonant coupling: heat flow Incoherent resonant coupling: heat flow Current operators: ˆ A ˆ J A = L ⋆ n A + Γ A e − β A ω A (1 − ˆ H A = − Γ A ˆ n A ) ˆ B ˆ J B = L ⋆ n B + Γ B e − β B ω B (1 − ˆ H B = − Γ B ˆ n B ) A = Tr ( ˆ B = Tr ( ˆ Currents in steady state: J s ρ s AB ) and J s ρ s J A ˆ J B ˆ AB ) J s A + J s B = 0 . In our case: A = Tr ( ˆ AB ) = ǫω ( e − ωβ A − e − ωβ B ) J s ρ s J A ˆ Γ cancels from J A in lowest order in ǫ/ Γ. Heat flows from hot to cold baths, 2nd Law respected ∗ . 1 Apr 2014, Kibbutz Ma’agan, Israel 7 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8

Summary Summary Autonomy of local Q-thermodynamic systems is lost for coherent coupling Autonomy of local Q-thermodynamic systems is preserved for incoherent coupling Resonant coupling seems also necessary for consistency Heat flows from hot to cold Proof is given for weak incoherent resonant coupling in special case Γ A = Γ B 1 Apr 2014, Kibbutz Ma’agan, Israel 8 / Lajos Di´ osi (Wigner Center, Budapest) Autonomy of coupled quantum thermodynamic systems 8