Adiabatic theorems in quantum statistical mechanics and Landauer - PowerPoint PPT Presentation

Adiabatic theorems in quantum statistical mechanics and Landauer principle Vojkan Jaksic McGill University Joint work with T. Benoist, M. Fraas, and C-A. Pillet October 9, 2016

ADIABATIC THEOREMS IN QSM • Hilbert space H , dim H < ∞ , H ( t ) = H + V ( t ) , t ∈ [0 , 1] , V (0) = 0 . ρ i = e − βH (0) /Z, ρ f = e − βH (1) /Z. • T > 0 adiabatic parameter, U T ( t ) time-evolution gener- ated by H ( t/T ) over the time interval [0 , T ] . ρ i ( T ) = U ∗ T ( T ) ρ i U T ( T ) . • Before taking the adiabatic limit T → ∞ we need to take first the TD (thermodynamic) limit. The limiting objects are denoted by the superscript ( ∞ ) . 1

• Adiabatic theorem for thermal states (Araki-Avron-Elgart): T →∞ � ρ ( ∞ ) ( T ) − ρ ( ∞ ) lim � = 0 . i f Proof: Combination of the Avron-Elgart gapless adiabatic theorem and Araki’s theory of perturbation of KMS struc- ture. Assumption: Ergodicity of TD limit quantum dynamical sys- tem w.r.t. instantaneous dynamics. • Adiabatic theorem for relative entropy: T →∞ S ( ρ ( ∞ ) ( T ) | ρ ( ∞ i / f ) = S ( ρ ( ∞ ) | ρ ( ∞ ) lim i / f ) . i f S ( A | B ) = tr( A (log A − log B )) . 2

• Adiabatic theorem for Renyi’s relative entropy: T →∞ S i α ( ρ ( ∞ ) ( T ) | ρ ( ∞ ) = S i α ( ρ ( ∞ ) | ρ ( ∞ ) lim ) . i i f i S i α ( A | B ) = tr( A 1 − i α B i α ) . • Adiabatic theorem for FCS. Let P ( ∞ ) be the probability mea- T sure on R describing the statistics of energy differences ∆ E in two times measurement protocol of the total energy (initially and at the time T ). � R e i α ∆ E d P ( ∞ ) lim (∆ E ) = S − i α/β . T T →∞ 3

LANDAUER PRINCIPLE • Finite level quantum system S coupled to a thermal reser- voir R ( H R , H R ). dim H S = d , ρ S , i = I /d , ρ S , f > 0 the final (target state). Landauer principle concerns energetic cost of the state transition ρ S , i → ρ S , f mediated by R . • Coupled system: H = H S ⊗ H R , H = H R , V ( t ) local interaction, V (0) = 0 , V (1) = − 1 β log ρ S , f , H ( t ) = H R + V ( t ) , ρ i / f = e − βH (0 / 1) /Z = ρ S , i / f ⊗ e − βH R /Z. 4

• First TD limit, then adiabatic limit. The transition ρ S , i → ρ S , f follows from lim T →∞ � ρ ( ∞ ) ( T ) − ρ ( ∞ ) � = 0 . i f • Landauer bound: The balance equation ∆ S T + σ T = β ∆ Q T where, with S ( σ ) = − tr( σ log σ ) , ∆ S T = S ( ρ S , i ( T )) − S ( ρ S , i ) , ∆ Q T = tr( ρ i ( T ) H R ) − tr( ρ i H R ) , σ T = S ( ρ i ( T ) | ρ S , i ( T ) ⊗ e − βH R /Z ) . σ T ≥ 0 is the entropy production term, and the Landauer bound follows ∆ S T ≥ β ∆ Q T . 5

• After the TD limit, the adiabatic theorem for relative entropy T →∞ S ( ρ ( ∞ ) ( T ) | ρ ( ∞ ) = S ( ρ ( ∞ ) | ρ ( ∞ ) lim ) i i f i gives the saturation of the Landauer bound in the adiabatic limit: lim T →∞ σ T = 0 , T →∞ ∆ S ( ∞ ) T →∞ β ∆ Q ( ∞ ) S ( ρ S , i ) − S ( ρ S , f ) = lim = lim . T T • Additional limit ρ S , f → | ψ �� ψ | gives the familiar form Q ( ∞ ) . log d = β ∆ ¯ • Full Counting Statistics goes beyond mean values and cap- tures fluctuations. 6

• The adiabatic theorem for FCS gives ρ f e i α � β (log d +log ρ f ) � � R e i α ∆ E d P ( ∞ ) lim = S − i α/β = tr T T →∞ • If ρ S , f = � p k | k �� k | , then lim T →∞ P ( ∞ ) = ¯ P ( ∞ ) , where T � � 1 P ∞ ¯ β (log d + log p k ) = p k . The heat is a discrete random variable, and each allowed quanta of heat corresponds to a transition to a certain level of the final state. P ( ∞ ) describes the heat fluctuations • The atomic measure ¯ around the mean value given by the Landauer bound � P ( ∞ ) (∆ E ) = S ( ρ S , i ) − S ( ρ S , f ) . R ∆ E d¯ 7

P ( ∞ ) → δ β − 1 log d , together with • In the limit ρ S , f → | ψ �� ψ | , ¯ convergence of all momenta. • At the same time α  β log d e if α > − β,    � P ( ∞ ) = R e α ∆ E d¯  lim 1 α = − β, if ρ S , f →| ψ �� ψ |   ∞ α < − β. if   • We expect that this divergence is experimentally observable via recently proposed interferometry and control protocols for measuring FCS using an ancilla coupled to the joint sys- tem S + R . 8