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−βW = 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

Htot = HS(t) + HR + HI

• Double measurement protocol

◮ initial state ρtot(t = 0) ◮ first projective measurement E1 pE1 ◮ time evolution U(t) generated by Htot ◮ second projective measurement E2 pE2

Heat statistics P(Q, t)

Probability distribution of energy exchange P(Q, t) =

• E1,E2

δ(E2 − E1 − Q)P[E2; E1]P[E1] Conditional probability P[E2; E1] P[E2; E1]P[E1] = Tr

• U†(t)pE2U(t)pE1ρtot(0)pE1
• Characteristic function Gν(t) =

+∞

−∞ dQeiQνP(Q, t)

Gν(t) = Tr

• U†(t)eiνHRU(t)e−iνHRρtot(0)

Heat statistics Gν(t)

• Gν(t) Moment generating function

Qn(t) = (−i)n dnGν(t) dνn

• ν=0 = (−i)n dnTr
• ρ(ν)

tot(t)

• dνn
• ν=0

Generalized time evolution ρ(ν)

tot(t) = Uν/2(t)ρtot(0)U† ν/2(t) with

Uν(t) = eiνHRU(t)e−iνHR Factorized initial condition ρtot(0) = ρS(0) ⊗ ρR(0) = ρS(0) ⊗ e−βHR ZR 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

Gν(t) =

• dηi ηi| ρS(0) |ηi
• dηf
• Dη
• Dξ ei SS[η,ξ] FFV[η, ξ]·ei∆Φ(ν)[η,ξ]

Generalization of Feynman-Vernon influence functional for heat exchange

• n−th moment Qn(t)

Φ(n)[η, ξ] = (−i)n dn dνn ei∆φ(ν)[η,ξ]

• ν=0

Carrega NJP ’15

Spin - boson model

• Dissipative two level system

ϵ ∆

HS = −∆ 2 σx − ǫ(t) 2 σz state basis |R/L with σz|R/L = ±|R/L

• low energy state of a double well potential v(q)

q = q0σz ∆ tunneling amplitude external bias ǫ = ǫ0 + ǫ1(t)

Weak coupling

Spectral function j(ω) ∝ Kω K coupling strength

• Weak damping regime K ≪ 1

constant bias ǫ0

• Total transferred heat

Q∞ ≡ lim

t→∞Q(t) = Eini−Eeq

Q∞ = (PR − PL)ǫ0 2 +

• δ2 + ǫ2

2 tanh

• δ2 + ǫ2

2T

Markov regime

• High temperature regime T >
• ∆2 + ǫ2

Average heat power P(t) = ˙ Q(t) P(t) = π 2 Kδ2 − πKT∆ [σx(t)s − (PR − PL)σx(t)a ]

2 4 6 8 10 t −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 P(t) PR − PL = 1 PR − PL = 0 PR − PL = −1 2 4 6 8 10 t 0.00 0.05 0.10 0.15 0.20 P(t) T = 5∆ T = 10∆ T = 20∆

K = 0.02 ∆ = ǫ0 = 1

Quantum noise regime

• Low temperature T <
• ∆2 + ǫ2

Average heat power − quantum noise contributions

P(t) = −∆ 2 t dτL′(τ)

• σx(t − τ)s σz(τ)s − σz(t − τ)a σx(τ)a
• +∆(PR − PL)

2 t dτL′(τ)

• σx(t − τ)a σz(τ)s − σz(t − τ)s σx(τ)a
• +

5 10 15 20 t −0.10 −0.08 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 P(t) T = 0.1∆ T = 1∆ T = 3∆ 5 10 15 20 25 30 35 40 t −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 Q(t) → → → T = 0.1∆ T = 1∆ T = 3∆

K = 0.02 ∆ = ǫ0 = 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