Quantum corrections for Work Olivier Brodier - L.M.P.T., Tours, - PowerPoint PPT Presentation

Quantum corrections for Work Olivier Brodier - L.M.P.T., Tours, France in collaboration with Kirone Mallick - Saclay, Paris. Alfredo Ozorio de Almeida - C.B.P.F., Rio de Janeiro, Br´ esil

Plan ➙ Work in thermodynamics ➙ Jarzynski approach ➙ Quantum problem ➙ Different scenarii

Work in thermodynamics P 6 A 5 4 3 2 1 W B min V T = T 0 constant + second principle : W � W min = ∆F ( T 0 , V )

Jarzynski approach {v n } 8 same path in microvariables space P W 8 2 1 3 13 A 7 5 W 1 specific path 8’ not necessarily reversible 2’ 1’ {x } n B 7’ 5’ V Thermodynamics is intrinsically statistical and N ” W ” = 1 � W n = � W � N n = 1 Jarzynski states that, if the initial state is a thermal state, then � e − βH 0 ( x 0 ) e − βW ( x 0 → x τ ) dx 0 = e − β∆F = Z B � e − βW � ≡ Z 0 Z A [ Jarzynski 1996]

Some definitions Time perturbation of a Hamiltonian : H t ( x ) = H 0 − Φ t · q with Φ τ = Φ 0 = 0, x = ( p , q ) and Φ t is a force. � τ W = Φ t · ˙ q t dt 0 Integration by parts � τ ˙ W = 0 − Φ t · q t dt 0 � τ ∂H t W = ∂t ( x t ) dt 0

General scheme of the proofs ( [ Jarzynski PRE 1997] ) ( 1 ) d � K τ ( x , x ′ ) � e − βW � x τ = x ′ dx ′ − β∂H τ dτ � e − βW � x τ = x = ∂τ � e − βW � x τ = x Thermal equilibrium state Π τ = e − βHτ ( x ) verifies detailed balance : Z τ � K τ ( x , x ′ ) Π τ ( x ′ ) dx ′ = 0 Hence Z τ Z 0 Π τ ( x ) is a solution of (1) and 1 � � � e − βW � x τ = x dx Z τ Π τ ( x ) dx = Z 0

Quantum problem W ? + 2 ➙ Problem in defining a single work operator for the whole process. ➙ Work as the difference between final and initial energy in an adiabatical process. ➙ Work as a difference between final and initial energy of the operator ? ➙ Master equation approach to generalize the notion of classical path in a non-adiabatical process

Master equation approach � � � � � � † � � † � � � † � ∂ � ρ t = − i 1 � � 2 � � � � H t , � L n , t � L n , t � ρ t − � = L t ( � ρ t + ρ t L n , t − L n , t ρ t L n , t L n , t ρ t ) dt h 2 � h � n as L t is time dependent, � Π t is not solution, that is ∂ � Π t � = L t ( � Π t ) dt Find a superoperator W t such that d Π t = ( L t + W t ) ( � � Π t ) dt With the assumption that � Π t is ”balanced” by L t , that is L t ( � Π t ) = 0, then a brute force solution is then � d � � � − 1 � � W t ( � ρ ) = � Π t Π t ρ dt

Expansion in W t To obtain a Jarzynski-like equation one uses a Schwinger-Dyson expansion in W t of the solution to the modified master equation � � � n U L + W � = U 0, t 1 W t 1 U t 1 , t 2 . . . W t n U t n , t i = 1 dt i Π 0, τ 0 � t 1 � ... � t n � t n � � � � Π 0 − → � τ t dt ( � Π τ � � � 0 W L Tr A = Tr e A τ ) [ R.Chetrite and K.Mallick 2011]

Quantum work corrections From Baker-Campbell-Hausdorff � � �� � � � � − 1 = − β∂ � ∂ � ∂ � ∂t − β 2 − β 3 d H t H t H t dt ( Z t � Z t � ∂t , � ∂t , � � Π t ) Π t H t H t H t − . . . 2! 3! From Moyal expansion in Weyl representation � ∂H t hβ ) 3 �� � ∂H t �� ∂H t � hβ ) 2 ∂t ( x ) + i � hβ � + ( i � � � W t ( x ) = − β ∂t ( x ) , H t ( x ) ∂t ( x ) , H t ( x ) , H t ( x ) + O ( � 2 6

Harmonic oscillator p 2 H t = � 2 m + k t � q 2 2 � In Weyl representation � � p 2 hω + ( 1 + f ( θ )) mωq 2 + ig ( θ ) pq W t ( p , q ) = − ˙ θ ( 1 − f ( θ )) m � h h � � f ( t ) = sinh ( 4 t ) / ( 4 t ) g ( t ) = sinh ( 2 t ) 2 /t θ = β � hω

h expansion � � ˙ � � hω ) 3 � hω ) ˙ ω ( p 2 ˙ k t ω hω ) 2 2 ω 2 m − k q 2 + i ( β � 2 q 2 ) + O W t ( p , q ) = − β 2 � ωωpq + ( β � ( β � 3

Quantum trajectory during a time step δt the state | ψ � can chose between a jump with Lindblad operator (proba pδt ) → � | Ψ t � − L k | Ψ t � or a pseudo-unitary evolution (proba 1 − pδt ) with effective non-Hermitian Hamiltonian � H L t h � → e − iδt H L t | Ψ t � | Ψ t � − � Then � ρ t = | Ψ t �� Ψ t |

Quantum trajectory for the thermal state � � | n t � e − βE n , t � n t | Π t = n Modify � t so that quantum trajectory follows � H L Π t � t + � H L → H L H π t − t � � d Π t = − i � � t , � + L t ( � H π Π t Π t ) dt h �

Quantum Work of the trajectory A possible expression is ˙ n t �� n t | − iβ � h E n , t | n t �� n t | − i � h Z t � � � ˙ H π t = i � h | ˙ 2 2 Z t n n H t = p 2 Example of the Harmonic oscillator � 2 m + k t 2 q 2 ˙ t = − ˙ � p � q + � q � ˙ ω p − iβ � h ω H t − i � h Z t � � H π 2 ω 2 2 ω 2 Z t First term makes evolution of | n t � , and second term makes evolution of E n , t . Third term is normalization.

PERSPECTIVES Find a more natural proof Unify the different approaches Treat a realistic system where work is an accessible quantity Give an experimental meaning to the ”quantum work”

A simple proof � ρ 0 ( x 0 ) e − βW ( x 0 , x t ) dx 0 � e − βW � = thermal initial state and adiabatic Hamiltonian system : � e − βH 0 ( x 0 ) e − β [ H t ( x t )− H 0 ( x 0 )] dx 0 � e − βW � = Z 0 � e − βW � = 1 � e − βH t ( x t ) dx 0 Z 0 simplecticity : � e − βW � = 1 � e − βH t ( x t ) dx t = Z t Z 0 Z 0 with � e − βH t ( x ) dx Z t = e − βF t =

General scheme of the proofs e −β H (x) t Π ( ) = x density t Z t p ideal path of the thermal states associated with H(t) real path dynamics of H(t) ρ ( ) = ρ ( ) t x 0 x −t e −β H (x) 0 q ρ ( ) = = x Π ( ) 0 x 0 Z 0 Transport of the real state : d dt [ ρ t ( x t )] = ˙ ρ t ( x t ) − { H t ( x t ) , ρ t ( x t ) } = 0 ρ τ ( x τ ) = ρ 0 ( x 0 ) Invariance by transport for the thermal equilibrium state : � � ˙ d Z t dt [ Π t ( x t )] = ˙ − β ˙ Π t ( x t )− { H t ( x t ) , Π t ( x t ) } = H t ( x t ) − Π t ( x t ) Z t Π τ ( x τ ) = Π 0 ( x 0 ) Z 0 � τ 0 ˙ e − β H t ( x t ) dt Z t

Fluctuation relations Jarzynski relation allows to derive some fluctuation relations H t = � � H 0 + λ t � V � � � � � τ t dt � 0 W L Π τ � � � Tr A = Tr Π 0 e A τ � � �� � � �� d d � τ t dt � Π τ � � � 0 W L Tr A = Tr Π 0 e A τ dλ t dλ t � � ∂ W t ∂ ( � � � 0 = � U 0, t U t , τ A τ ) � | λ = 0 − A τ � | λ = 0 ∂λ t ∂λ t

Quantum work W t interpreted as some work rate operator. A brute force solution is � d � � � − 1 � � W t ( � � ρ ) = Π t Π t ρ dt So that d Π t = W t ( � � Π t ) = ( L t + W t ) ( � Π t ) dt

Quantum trajectory ( δt ) [ γ ] � 1 ) † . . . ( � � � e − βE n ,0 � T γ N . . . � T γ 1 | n 0 �� n 0 | ( � T γ T γ N ) † Π t = n γ ( n ) with [ γ ] = number of jumps in trajectory γ The natural quantum trajectory is combined by episodes which track the thermal state e − iδt h � t | n t � ≃ e − βδt ˙ H π E n , t | n t + δt � � 2 � � † � � � † � � † � n L † L † � � dt i � U L + π tn , t � L tn . . . � U L + π � L t 1 � U L + π � U L + π � U L + π � . . . � U L + π � Π t = Π 0 Π t 1, t 2 0, t 1 0, t 1 t 1 t 1, t 2 tn tn , t i = 1 n 0 � t 1 � ... � tn � t

Adiabatical case where α n ( t ) is Berry’s phase. ( δt ) [ γ ] � 1 ) † . . . ( � � � e − βE n � T γ N . . . � T γ 1 | n �� n | ( � T γ T γ N ) † Π t = n γ ( n ) with [ γ ] = number of jumps in trajectory γ