SMALLEST REFRIGERATOR WITHOUT MOVING PARTS Lajos Disi KFKI - PowerPoint PPT Presentation

SMALLEST REFRIGERATOR WITHOUT MOVING PARTS Lajos Diósi KFKI Research Institute for Particle and Nuclear Physics H-1525 Budapest 114, POB 49, Hungary

CONTENTS Linden, Popescu, Skrzypczyk: How small thermal machines can be? LSP: The smallest possible refrigerator, PRL 105 , 130401 (2010) SBrunnerLP: On the efficiency of very small refrigerators, arXiv:1009.0865 P: Maximally efficient quantum thermal machines: The basic principles, arXiv:1010.2536 LSP: The smallest possible heat engines, arXiv:1010.6029 Also L.D.: Short Course on Quantun Information Theory (Springer, 2nd, to appear) SMALLEST REFRIGERATOR: 3-LEVEL-SYSTEM 2nd SMALLEST REFRIGERATOR: 2xTLS TLS THERMALIZATION DYNAMICS 2nd SMALLEST Q-REFRIGERATOR DYNAMICS 2nd SMALLEST HEAT ENGINE: 2xTLS 2nd SMALLEST HEAT ENGINE DYNAMICS

SMALLEST REFRIGERATOR: 3-LEVEL-SYSTEM Hot and cold reservoirs: T h > T c . Refrigerator will yield T 0 < T c . Transition | 0 � → | 1 � is heated by T h , | 0 � → | 2 � is cooled by T c . Let ǫ c > ǫ h ! −− ǫ c −− exp ( − ǫ c / k B T c ) −− ǫ c −− exp [ − ( ǫ c / k B T c ) + ( ǫ h / k B T h )] −− ǫ h −− exp ( − ǫ h / k B T h ) −− ǫ h −− 1 −− 0 −− − 1 Make exp [ − ( ǫ c / k B T c ) + ( ǫ h / k B T h )] = exp [ − ( ǫ c − ǫ h ) / k B T 0 ] ⇒ Effective temperature of the TLS | 1 h � , | 1 c � : 1 − ǫ h ǫ c T 0 = ( < T c ) . T c 1 − ǫ h T c ǫ c T h

2nd SMALLEST REFRIGERATOR: 2xTLS Hot and cold reservoirs: T h > T c . Refrigerator will yield T 0 < T c . Transition | 0 � → | 1 � is heated by T h , | 0 � → | 2 � is cooled by T c . Let ǫ c > ǫ h ! | 1 c � −− ǫ c −− exp ( − ǫ c / k B T c ) | 1 h � −− ǫ h −− exp ( − ǫ h / k B T h ) | 0 c � −− 0 c −− 1 | 0 h � −− 0 h −− 1 Make exp [ − ( ǫ c / k B T h ) + ( ǫ h / k B T c )] = exp [ − ( ǫ c − ǫ h ) / k B T 0 ] ⇒ Effective temperature of the TLS | 1 h � , | 1 c � : 1 − ǫ h ǫ c T 0 = ( < T c ) . T c 1 − ǫ h T c ǫ c T h

TLS THERMALIZATION DYNAMICS a † = | 1 � � 0 | , ˆ a † ˆ TLS: ˆ a = | 0 � � 1 | , ˆ H = ǫ ˆ a ; Heat bath: β = 1 / k B T . Thermalization master equation: d ˆ dt = − i ρ � a † − 1 � � a − 1 � a † ˆ a † ˆ + e − βǫ Γ a † ˆ a † , ˆ � ǫ [ˆ a , ˆ ρ ]+Γ ˆ a ˆ ρ ˆ 2 { ˆ a , ˆ ρ } ˆ ρ ˆ 2 { ˆ a ˆ ρ } . 2nd term: spontaneous decay | 1 � → | 0 � at rate Γ . 3rd term: thermal excitation | 0 � → | 1 � at rate Γ × Boltzmann factor. Competition ⇒ Gibbs stationary state at (inverse) temperature β : ( t ≫ e βǫ / Γ) . ρ − → | 0 � � 0 | + exp ( − βǫ ) | 1 � � 1 | MLS: Any TL subspace may likewise be thermalized. ˆ a = | n � � m | , ǫ = ǫ n − ǫ m > 0 , Γ = Γ nm Each TL subspace may have different temperatures T nm . If some are equilibrated by reservoirs, the rest obtains calculable ‘effective temperatures’.

2nd SMALLEST Q-REFRIGERATOR DYNAMICS No external resources of energy just heat flow T h → T c . Refrigerator: 2xTLS, in contact with T h , T c where T h > T c and ǫ c > ǫ h . Develops a temperature T 0 < T c for the TLS subspace | 1 h � , | 1 c � . Can cool a ‘thermometer’ to temperature T 0 < T c . a † Thermometer: third TLS ˆ a 3 , ˆ 3 , ǫ 3 = ǫ 0 = ǫ c − ǫ h . a † Coupled to ˆ a 0 = | 1 h � � 1 c | , ˆ 0 = | 1 c � � 1 h | of the refrigerated subspace. Master eq. in interaction picture: � � � � d ˆ ρ c − 1 a c − 1 a † a † + e − β c ǫ c Γ c a † a † = Γ c ˆ a c ˆ ρ ˆ 2 { ˆ c ˆ a c , ˆ ρ } ˆ c ˆ ρ ˆ 2 { ˆ a c ˆ c , ˆ ρ } + dt � h − 1 � � a h − 1 � a † a † a † a † + e − β h ǫ h Γ h +Γ h ˆ a h ˆ ρ ˆ 2 { ˆ h ˆ a h , ˆ ρ } ˆ h ˆ ρ ˆ 2 { ˆ a h ˆ h , ˆ ρ } + − i g � � a † a † ˆ 3 ˆ a 0 + ˆ 0 ˆ a 3 , ˆ ρ � If coupling g ≪ Γ c , Γ h then ˆ ρ 3 → | 0 3 � � 0 3 | + exp ( − ǫ 3 / k B T 0 ) | 1 3 � � 1 3 | .

2nd SMALLEST HEAT ENGINE: 2xTLS We want a negative T 0 (population inversion). Change the role of T h and T c in refrigerator: ⇒ T 0 may be negative! Reorganized refrigerator becomes heat engine. Transition | 0 h � → | 1 h � is heated by T h , | 0 c � → | 1 c � is cooled by T c . Let ǫ h > ǫ c now (opposite than for refrigerator)! | 1 h � −− ǫ h −− exp ( − ǫ h / k B T h ) | 1 c � −− ǫ c −− exp ( − ǫ c / k B T c ) | 0 h � −− 0 h −− 1 | 0 c � −− 0 c −− 1 Make exp [ − ( ǫ h / k B T h ) + ( ǫ c / k B T c )] = exp [ − ( ǫ h − ǫ c ) / k B T 0 ] ⇒ Negative effective temperature of the TLS | 1 h � , | 1 c � : 1 − ǫ c < 0 if T h > ǫ h ǫ h T 0 = > 1 . T h 1 − ǫ c T h T c ǫ c ǫ h T c Negative T 0 means population inversion between | 1 c � and | 1 h � . It can ’lift a weight’ at constant speed!

2nd SMALLEST HEAT ENGINE DYNAMICS Resource: heat flow T h → T c . Engine: 2xTLS, in contact with T h , T c where T h / T c > ǫ h /ǫ c > 1. Develops population inversion T 0 < 0 for the TLS subspace | 1 c � , | 1 h � . Can ‘lift a weight’ at stationary power. a † Weight: harmonic ocillator ˆ a 3 , ˆ 3 , ǫ 3 = ǫ 0 = ǫ h − ǫ c . a † Coupled to ˆ a 0 = | 1 c � � 1 h | , ˆ 0 = | 1 h � � 1 c | of the population inverted TLS. Master eq. in interaction picture (formally same as refrigerator’s, just a † [ˆ a 3 , ˆ 3 ] = 1): d ˆ ρ � c − 1 � � a c − 1 � a † a † + e − β c ǫ c Γ c a † a † = Γ c a c ˆ ˆ ρ ˆ 2 { ˆ c ˆ a c , ˆ ρ } ˆ c ˆ ρ ˆ 2 { ˆ a c ˆ c , ˆ ρ } + dt � h − 1 � � a h − 1 � a † a † a † a † + e − β h ǫ h Γ h +Γ h a h ˆ ˆ ρ ˆ 2 { ˆ h ˆ a h , ˆ ρ } ˆ h ˆ ρ ˆ 2 { ˆ a h ˆ h , ˆ ρ } + − i g � � a † a † ˆ 3 ˆ a 0 + ˆ 0 ˆ a 3 , ˆ ρ � If coupling g ≪ Γ c , Γ h then, for T h / T c > ǫ h /ǫ c , the oscillator energy a † a 3 � grows like ∼ g 2 t . Carnot-efficiency is reached at g → 0 and � ǫ 3 ˆ 3 ˆ T h / T c → ǫ h /ǫ c .