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

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

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ignacio garca-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),

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Relaxation of isolated quantum systems beyond chaos

ignacio garcía-mata

IFIMAR (CONICET-UNMdP)

Mar del Plata, Argentina

School for advanced sciences of Luchon
 Quantum chaos: fundamentals and applications


Session Workshop I (W1), March 14 - 21, 2015

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Buenos Aires Mar del Plata

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non equilibrium dynamics — relaxation & universality thermalization Isolated Quantum systems

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Eigenstate thermalization hypothesis Ann = hn| ˆ A|ni smooth (approx. constant) Anm = hn| ˆ A|mi very small hAit ⇡ hAiMC

Other mechanism: see C. Gogolin’s thesis

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No Yes No Yes

H(λ) → H(λ + δλ)

quench equilibrate? thermalize?

τ H(λ + δλ)

t

H(λ)

subsystem state independence bath state independence diagonal form of eq. state thermal (Gibbs) state

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equilibrate?

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equilibrate? how? chaos?

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

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Polkovnikov, Ann. Phys. 326, 486 (2011) Santos, Polkovnikov & Rigol, PRL 107, 040601 (2011)

von Neumann entropy SvN(ρ) = −Tr(ρ ln ρ) diagonal entropy SD(ρ) = − X

n

ρnn ln ρnn equilibrium non-equilibrium external operations

consistent with second law

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Polkovnikov, Ann. Phys. 326, 486 (2011) Santos, Polkovnikov & Rigol, PRL 107, 040601 (2011)

diagonal entropy SD(ρ) = − X

n

ρnn ln ρnn

consistent with second law

increases conserved for adiabatic process uniquely related to P(E) additive

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Prob[SD(ρ0)] ≤ SD(ρτ)] ∼ 1 Sdec − SD(τ) ≤ 1 − γ

Ikeda, Sakumichi, Polkovnikov, & Ueda. Ann. Phys 354 (2015) 338-352

Sdec = SD(ρ)

γ = 0.5772 . . .

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Prob[SD(ρ0)] ≤ SD(ρτ)] ∼ 1

Sdec = SD(ρ)

Sdec − SD(τ) ≤ 1 − γ

Ikeda, Sakumichi, Polkovnikov, & Ueda. Ann. Phys 354 (2015) 338-352

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τ H(λ + δλ)

t

H(λ)

Cyclic process Quench dynamics

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ρ(τ) = e−H0τρ0eiH0τ

ρ0 = |n0ihn0|

SD = − X

n

Cn(τ) ln Cn(τ)

τ H(λ + δλ)

t

H(λ)

Sdec = SD(ρ)

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Sdec − SD(τ) ∆SD(τ)/SD(τ) equilibrium

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Sdec − SD(τ) → 1 − γ ∆SD(τ)/SD(τ) equilibrium ∆SD(τ)/SD(τ) ⌧ 1

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two different models

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field Dicke model

H(λ) = ω0Jz + ωa†a + λ √2j (a† + a)(J+ + J−)

superradiant transition

λc = 1 2 √ω0ω

ω0 = ω = ~ = 1

λc = 0.5

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Dicke model

Emary & Brandes, PRL 90, 044101 (2003)

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Spin system

H(λ) = H0 + λV H0 =

L−1

X

i=1

J(Sx

i Sx i+1 + Sy i Sy i+1 + µSz i Sz i+1)

V =

L−2

X

i=0

J(Sx

i Sx i+2 + Sy i Sy i+2 + µSz i Sz i+2)

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Spin system

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

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75 50 25 6 4 2 τ 1 − γ SD 1 0.8 0.6 0.4 0.2 0.1 0.01 λ0 ∆SD(τ)/SD(τ) 0.5 0.4 0.3 0.2 0.1 Sdec − SD(τ)

δλ = 0.1

Dicke j = 20, N = 250

λc = 0.5

|10i |100i

|500i |1000i

|2000i

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λ0

|n0i

Sdec − SD ∆SD/SD

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λ0

|n0i

Sdec − SD ∆SD/SD structure

  • f initial state
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IPR chaos Localization complexity of eigenstates

B.V. Chirikov,F.M Izrailev and D.L. Shepelyansky Physica D 33 (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).

  • 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
  • M. G. Kozlov, Phys. Rev. A 50, 267 (19


… … … …

ξ = 1 P

m |hn(λ)|m(λ + δλ)i|4

  • L. F. Santos F. Borgonovi, and F. M. Izrailev, PRE 85, 036209 (2012)

basis

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 25 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1 Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 25 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1 Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 25 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1 Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 25 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1 Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 25 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1 Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 100 10 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1 Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 100 10 1 0.1 0.01 0.001

Energy 3000 2000 1000 150 100 50

  • 50

λ 1

(1 − γ)ξ − 1 ξ + 1

Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

Energy 3000 2000 1000 150 100 50

  • 50

ξ ∆SD(τ)/SD(τ) 100 10 1 0.1 0.01 0.001

λ 0.5

λ < λc

(1 − γ)ξ − 1 ξ + 1

Dicke model

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ξ Sdec − SD(τ) 150 125 100 75 50 25 0.5 0.4 0.3 0.2 0.1

ξ ∆SD(τ)/SD(τ) 100 10 1 0.1 0.01 0.001

Dicke model & spin system

(1 − γ)ξ − 1 ξ + 1

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two different models universal behavior

complexity is key

equilibration and chaos isolated systems

LDOS

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to be done Sdec − SD(τ) = (1 − γ)ξ − 1 ξ + 1

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δλ ξ 1 0.1 0.01 0.001 0.0001 100 10 1

to be done

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δλ ξ 1 0.1 0.01 0.001 0.0001 100 10 1 1 0.1 0.01 0.001 0.0001 100 10 1 δλ ξ

to be done

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δλ ξ 1 0.1 0.01 0.001 0.0001 100 10 1 1 0.1 0.01 0.001 0.0001 100 10 1 δλ ξ

to be done

  • B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. 79, 4365 (1997).
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PRE 91, 010902 (R) (2015) UBACYT

Diego Wisniacki Augusto Roncaglia IGM &

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UBACYT

Thank you

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Thank you

questions

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0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Energy 6000 4000 2000 400 200

λ

j = 20, N = 250