Relaxation of isolated IFIMAR (CONICET-UNMdP) Mar del Plata, - PowerPoint PPT Presentation

ignacio garcía-mata Relaxation of isolated IFIMAR (CONICET-UNMdP) Mar del Plata, Argentina quantum systems School for advanced sciences of Luchon 
 beyond chaos Quantum chaos: fundamentals and applications 
 Session Workshop I (W1), March 14 - 21, 2015

Buenos Aires Mar del Plata

Isolated Quantum systems non equilibrium dynamics — relaxation & universality thermalization

Eigenstate thermalization hypothesis smooth A nn = h n | ˆ A | n i (approx. constant) h A i t ⇡ h A i MC A nm = h n | ˆ A | m i very small Other mechanism: see C. Gogolin’s thesis

H ( λ + δλ ) H ( λ ) → H ( λ + δλ ) quench H ( λ ) t τ 0 No Yes equilibrate? Yes No subsystem state independence bath state independence thermalize? diagonal form of eq. state thermal (Gibbs) state

equilibrate?

equilibrate? how? chaos?

quantum Entropy von Neumann? S vN ( ρ ) = − Tr( ρ ln ρ )

consistent with second law von Neumann entropy equilibrium S vN ( ρ ) = − Tr( ρ ln ρ ) diagonal entropy non-equilibrium X S D ( ρ ) = − ρ nn ln ρ nn external operations n Santos, Polkovnikov & Rigol, PRL 107 , 040601 (2011) Polkovnikov, Ann. Phys. 326 , 486 (2011)

consistent with second law increases diagonal entropy conserved for adiabatic process X S D ( ρ ) = − ρ nn ln ρ nn n uniquely related to P(E) additive Santos, Polkovnikov & Rigol, PRL 107 , 040601 (2011) Polkovnikov, Ann. Phys. 326 , 486 (2011)

Prob[ S D ( ρ 0 )] ≤ S D ( ρ τ )] ∼ 1 S dec − S D ( τ ) ≤ 1 − γ S dec = S D ( ρ ) γ = 0 . 5772 . . . Ikeda, Sakumichi, Polkovnikov, & Ueda. Ann. Phys 354 (2015) 338-352

Prob[ S D ( ρ 0 )] ≤ S D ( ρ τ )] ∼ 1 S dec − S D ( τ ) ≤ 1 − γ S dec = S D ( ρ ) Ikeda, Sakumichi, Polkovnikov, & Ueda. Ann. Phys 354 (2015) 338-352

Quench dynamics H ( λ ) H ( λ + δλ ) t τ 0 Cyclic process

ρ ( τ ) = e − H 0 τ ρ 0 e iH 0 τ ρ 0 = | n 0 ih n 0 | H ( λ ) H ( λ + δλ ) t τ 0 X S D = − C n ( τ ) ln C n ( τ ) n S dec = S D ( ρ )

equilibrium S dec − S D ( τ ) ∆ S D ( τ ) /S D ( τ )

equilibrium S dec − S D ( τ ) → 1 − γ ∆ S D ( τ ) /S D ( τ ) ∆ S D ( τ ) /S D ( τ ) ⌧ 1

two different models

Dicke model λ √ 2 j ( a † + a )( J + + J − ) H ( λ ) = ω 0 J z + ω a † a + field superradiant transition λ c = 1 √ ω 0 ω 2 λ c = 0 . 5 ω 0 = ω = ~ = 1

Dicke model Emary & Brandes, PRL 90 , 044101 (2003)

Spin system H ( λ ) = H 0 + λ V L − 1 X i +1 + S y i S y J ( S x i S x i +1 + µS z i S z H 0 = i +1 ) i =1 L − 2 i +2 + S y i S y X J ( S x i S x i +2 + µS z i S z V = i +2 ) i =0

Spin system Santos, Borgonovi & Izrailev, PRE 85 . 036209 (2012)

Dicke 0.5 0.4 S dec − S D ( τ ) 0.3 6 4 0.2 S D 1 − γ 2 0.1 0 0 25 50 75 τ 0 | 10 i ∆ S D ( τ ) /S D ( τ ) 0.1 | 100 i | 500 i 0.01 | 1000 i 0 . 2 0 . 4 0 . 6 0 . 8 1 | 2000 i λ 0 j = 20 , N = 250 δλ = 0 . 1 λ c = 0 . 5

λ 0 S dec − S D ∆ S D /S D | n 0 i

λ 0 S dec − S D structure of initial state ∆ S D /S D | n 0 i

1 ξ = P m | h n ( λ ) | m ( λ + δλ ) i | 4 IPR B.V. Chirikov,F.M Izrailev and D.L. Shepelyansky Physica D 33 basis (1988) 77-88 Y. V. Fyodorov and A. D. Mirlin, Phys. Rev. B 52, R11580 (1995). Ph. Jacquod and D. L. Shepelyansky, Phys. Rev. Lett. 75, 3501 (1995). complexity of eigenstates B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. 79, 4365 (1997). 
 R. Berkovits and Y. Avishai, Phys. Rev. Lett. 80, 568 (1998). 
 V. V. Flambaum, A. A. Gribakina, G. F. Gribakin, and Localization M. G. Kozlov, Phys. Rev. A 50, 267 (19 
 … … … chaos … L. F. Santos F. Borgonovi, and F. M. Izrailev, PRE 85, 036209 (2012)

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) 0.01 0.3 0.001 0.2 1 25 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) 0.01 0.3 0.001 0.2 1 25 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) 0.01 0.3 0.001 0.2 1 25 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) 0.01 0.3 0.001 0.2 1 25 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) 0.01 0.3 0.001 0.2 1 25 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) 0.01 0.3 0.001 0.2 1 10 100 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ 0 1 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) (1 − γ ) ξ − 1 0.01 0.3 ξ + 1 0.001 0.2 1 10 100 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model λ < λ c λ 0 0.5 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) (1 − γ ) ξ − 1 0.01 0.3 ξ + 1 0.001 0.2 1 10 100 ξ 0.1 150 0 100 0 25 50 75 100 125 150 Energy 50 ξ 0 -50 0 1000 2000 3000

Dicke model & spin system 0.5 0.1 ∆ S D ( τ ) /S D ( τ ) 0.4 S dec − S D ( τ ) (1 − γ ) ξ − 1 0.01 0.3 ξ + 1 0.001 0.2 1 10 100 ξ 0.1 0 0 25 50 75 100 125 150 ξ

two different models equilibration and chaos isolated systems universal behavior complexity is key LDOS

to be done S dec − S D ( τ ) = (1 − γ ) ξ − 1 ξ + 1

to be done 100 ξ 10 1 0.0001 0.001 0.01 0.1 1 δλ

to be done 100 100 ξ ξ 10 10 1 1 0.0001 0.0001 0.001 0.001 0.01 0.01 0.1 0.1 1 1 δλ δλ

to be done 100 100 ξ ξ 10 10 1 1 0.0001 0.0001 0.001 0.001 0.01 0.01 0.1 0.1 1 1 δλ δλ B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. 79, 4365 (1997).

PRE 91 , 010902 (R) (2015) UBACYT Diego Wisniacki IGM & Augusto Roncaglia

Thank you UBACYT

questions Thank you

0.9 0.8 400 0.7 λ 0.6 Energy 0.5 200 0.4 0.3 0.2 0 0.1 0 2000 4000 6000 j = 20 , N = 250