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

Buenos Aires

Mar del Plata

non equilibrium dynamics — relaxation & universality

thermalization

Isolated Quantum systems

Eigenstate thermalization hypothesis

Ann = hn|A|ni smooth (approx. constant)

Anm = hn|A|mi very small

hAit ⇡ hAiMC

Other mechanism: see C. Gogolin’s thesis

No Yes

NoYes

H(�) ! H(� + ��)quench

equilibrate?

thermalize?

⌧

H(�+ ��)

0

tH(�)

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

equilibrate?

equilibrate?

how? chaos?

von Neumann?

quantum Entropy

SvN(⇢) = �Tr(⇢ ln ⇢)

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

von Neumann entropy

SvN(⇢) = �Tr(⇢ ln ⇢)

diagonal entropy

SD(⇢) = �X

n

⇢nn ln ⇢nn

equilibrium

non-equilibrium

external operations

consistent with second law

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

diagonal entropy

SD(⇢) = �X

n

⇢nn ln ⇢nn

consistent with second law

increases

conserved for adiabatic process

uniquely related to P(E)

additive

Prob[SD(⇢0)] SD(⇢⌧ )] ⇠ 1

Sdec � SD(⌧) 1� �

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

Sdec = SD(⇢)� = 0.5772 . . .

Prob[SD(⇢0)] SD(⇢⌧ )] ⇠ 1

Sdec = SD(⇢)

Sdec � SD(⌧) 1� �

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

⌧

H(�+ ��)

0 tH(�)

Cyclic process

Quench dynamics

⇢(⌧) = e�H0⌧⇢0eiH0⌧⇢0 = |n0ihn0|

SD = �X

n

Cn(⌧) lnCn(⌧)

⌧

H(�+ ��)

0 tH(�)

Sdec = SD(⇢)

Sdec � SD(⌧)

�SD(⌧)/SD(⌧)

equilibrium

Sdec � SD(⌧) ! 1� �

�SD(⌧)/SD(⌧)

equilibrium

�SD(⌧)/SD(⌧) ⌧ 1

two different models

field

Dicke model

H(�) = !0Jz + !a†a+�p2j

(a† + a)(J+ + J�)

superradiant transition

�c =1

2

p!0!

!0 = ! = ~ = 1 �c = 0.5

Dicke model

Emary & Brandes, PRL 90, 044101 (2003)

Spin system

H(�) = H0 + �V

H0 =L�1X

i=1

J(Sx

i

Sx

i+1 + Sy

i

Sy

i+1 + µSz

i

Sz

i+1)

V =L�2X

i=0

J(Sx

i

Sx

i+2 + Sy

i

Sy

i+2 + µSz

i

Sz

i+2)

Spin system

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

7550250

6

4

2

0τ

1− γSD

10.80.60.40.2

0.1

0.01

λ0

∆SD(τ)/SD(τ)

0.5

0.4

0.3

0.2

0.1

0

Sdec−SD(τ)

�� = 0.1

Dicke

j = 20, N = 250

�c = 0.5

|10i

|100i

|500i

|1000i

|2000i

�0

|n0i

Sdec � SD

�SD/SD

�0

|n0i

Sdec � SD

�SD/SD

structure of initial state

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

… … … …

⇠ =1P

m |hn(�)|m(�+ ��)i|4

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

basis

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

251

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

251

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

251

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

251

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

251

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

100101

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

100101

0.1

0.01

0.001

Energy

3000200010000

150

100

50

0

-50

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 1

(1� �)⇠ � 1

⇠ + 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

Energy

3000200010000

150

100

50

0

-50

ξ

∆SD(τ)/SD(τ)

100101

0.1

0.01

0.001

0.5

0.25

0

ξ

Sdec − SD(τ )

150

125

100

75

50

25

0

0.5

0.4

0.3

0.2

0.10

�0 0.5

� < �c

(1� �)⇠ � 1

⇠ + 1

Dicke model

ξ

Sdec−

SD(τ)

1501251007550250

0.5

0.4

0.3

0.2

0.1

0

ξ

∆SD(τ)/SD(τ)

100101

0.1

0.01

0.001

Dicke model & spin system

(1� �)⇠ � 1

⇠ + 1

two different models

universal behavior

complexity is key

equilibration and chaos isolated systems

LDOS

to be done

Sdec � SD(⌧) = (1� �)⇠ � 1

⇠ + 1

δλ

ξ

10.10.010.0010.0001

100

10

1

to be done

δλ

ξ

10.10.010.0010.0001

100

10

1

10.10.010.0010.0001

100

10

1

δλ

ξ

to be done

δλ

ξ

10.10.010.0010.0001

100

10

1

10.10.010.0010.0001

100

10

1

δλ

ξ

to be done

B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. 79, 4365 (1997).

PRE 91, 010902 (R) (2015)

UBACYTDiego Wisniacki Augusto RoncagliaIGM &

UBACYTThank you

Thank you

questions

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Energy

6000400020000

400

200

0

�

j = 20, N = 250