FUNCTIONAL APPROACH TO HEAT EXCHANGE APPLICATION TO THE SPIN BOSON - PowerPoint PPT Presentation

CNR-SPIN (GENOVA) FUNCTIONAL APPROACH TO HEAT EXCHANGE APPLICATION TO THE SPIN BOSON MODEL: FROM MARKOV TO QUANTUM NOISE REGIME Matteo Carrega In collaboration with : Dr. P. Solinas Dr. A. Braggio Prof. M. Sassetti Prof. U. Weiss

Outline ◮ Quantum Thermodynamics ◮ Path-integral approach to energy exchange ◮ Application to the spin-boson model ◮ Results for average heat and heat power

Introduction Esposito RMP ’09, Campisi RMP ’11 ◮ Thermodynamics of small devices ◮ Definitions of work and heat at quantum level ◮ Precise measurement protocols

Recent experiments Batalhao PRl ’14 • Measurement of work distribution • NMR study with RF field Work distribution − Closed system Verification of Jarzynski equality � e − β W � = Z ( τ ) Z (0)

Recent experiments Pekola PRL ’13, Gasparinetti Phys. Applied ’15, Pekola Nat. Phys. ’15 Measurement of dissipated heat Hybrid electronic circuit → Temperature measurement of the environment

Measurement protocol Tasaki ArXiv ’00, Talkner PRE ’07, Gasparinetti NJP ’14 H tot = H S ( t ) + H R + H I • Double measurement protocol ◮ initial state ρ tot ( t = 0) ◮ first projective measurement � E 1 p E 1 ◮ time evolution U ( t ) generated by H tot ◮ second projective measurement � E 2 p E 2

Heat statistics P ( Q , t ) Probability distribution of energy exchange � P ( Q , t ) = δ ( E 2 − E 1 − Q ) P [ E 2 ; E 1 ] P [ E 1 ] E 1 , E 2 Conditional probability P [ E 2 ; E 1 ] U † ( t ) p E 2 U ( t ) p E 1 ρ tot (0) p E 1 � � P [ E 2 ; E 1 ] P [ E 1 ] = Tr � + ∞ −∞ dQe iQ ν P ( Q , t ) • Characteristic function G ν ( t ) = U † ( t ) e i ν H R U ( t ) e − i ν H R ρ tot (0) � � G ν ( t ) = Tr

Heat statistics G ν ( t ) • G ν ( t ) Moment generating function ρ ( ν ) ν =0 = ( − i ) n d n Tr � � � Q n ( t ) � = ( − i ) n d n G ν ( t ) tot ( t ) � � � � d ν n d ν n ν =0 Generalized time evolution ρ ( ν ) tot ( t ) = U ν/ 2 ( t ) ρ tot (0) U † ν/ 2 ( t ) with U ν ( t ) = e i ν H R U ( t ) e − i ν H R Factorized initial condition ρ tot (0) = ρ S (0) ⊗ ρ R (0) = ρ S (0) ⊗ e − β H R Z R Standard approach: Master equation → Born-Markov approximation

Functional integral Feynman Ann. Phys. ’63, Caldeira Phys. A ’83, Weiss ’99 Path integral approach − Reduced system dynamics � � � � D ξ e i S S [ η,ξ ] F FV [ η, ξ ] · e i ∆Φ ( ν ) [ η,ξ ] G ν ( t ) = d η i � η i | ρ S (0) | η i � d η f D η Generalization of Feynman-Vernon influence functional for heat exchange • n − th moment � Q n ( t ) � d n � d ν n e i ∆ φ ( ν ) [ η,ξ ] Φ ( n ) [ η, ξ ] = ( − i ) n � � � ν =0 Carrega NJP ’15

Spin - boson model • Dissipative two level system ∆ ϵ H S = − ∆ 2 σ x − ǫ ( t ) 2 σ z state basis | R / L � with σ z | R / L � = ±| R / L � • low energy state of a double well potential v ( q ) q = q 0 σ z ∆ tunneling amplitude external bias ǫ = ǫ 0 + ǫ 1 ( t )

Weak coupling Spectral function j ( ω ) ∝ K ω K coupling strength • Weak damping regime K ≪ 1 constant bias ǫ 0 • Total transferred heat Q ∞ ≡ lim t →∞ � Q ( t ) � = � E ini �−� E eq � � � δ 2 + ǫ 2 δ 2 + ǫ 2 Q ∞ = ( P R − P L ) ǫ 0 0 0 � � + tanh 2 2 2 T

Markov regime � ∆ 2 + ǫ 2 • High temperature regime T > 0 Average heat power � P ( t ) � = � ˙ Q ( t ) � � P ( t ) � = π 2 K δ 2 − π KT ∆ [ � σ x ( t ) � s − ( P R − P L ) � σ x ( t ) � a ] 0 . 20 0 . 20 P R − P L = 1 T = 5∆ 0 . 15 P R − P L = 0 T = 10∆ P R − P L = − 1 T = 20∆ 0 . 10 0 . 15 0 . 05 � P ( t ) � � P ( t ) � 0 . 00 0 . 10 − 0 . 05 − 0 . 10 0 . 05 − 0 . 15 − 0 . 20 0 . 00 0 2 4 6 8 10 0 2 4 6 8 10 t t K = 0 . 02 ∆ = ǫ 0 = 1

Quantum noise regime � ∆ 2 + ǫ 2 • Low temperature T < 0 Average heat power − quantum noise contributions � t � P ( t ) � = − ∆ � � d τ L ′ ( τ ) � σ x ( t − τ ) � s � σ z ( τ ) � s − � σ z ( t − τ ) � a � σ x ( τ ) � a 2 0 � t +∆( P R − P L ) � � d τ L ′ ( τ ) � σ x ( t − τ ) � a � σ z ( τ ) � s − � σ z ( t − τ ) � s � σ x ( τ ) � a + 2 0 0 . 06 → 0 . 2 T = 0 . 1∆ 0 . 04 T = 1∆ 0 . 1 T = 3∆ 0 . 02 0 . 0 0 . 00 → � P ( t ) � � Q ( t ) � − 0 . 1 − 0 . 02 − 0 . 2 − 0 . 04 − 0 . 3 − 0 . 06 → T = 0 . 1∆ T = 1∆ − 0 . 4 − 0 . 08 T = 3∆ − 0 . 10 − 0 . 5 0 5 10 15 20 0 5 10 15 20 25 30 35 40 t t K = 0 . 02 ∆ = ǫ 0 = 1

Conclusions ◮ Functional integral approach to energy exchange ◮ Application to the spin-boson model ◮ Average heat and heat power ◮ Quantum noise contribution at low temperature