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From unitary dynamics to statistical mechanics in isolated quantum systems Marcos Rigol Department of Physics The Pennsylvania State University The Tony and Pat Houghton Conference on Non-Equilibrium Statistical Mechanics ICERM, Brown

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From unitary dynamics to statistical mechanics in isolated quantum systems

Marcos Rigol

Department of Physics The Pennsylvania State University

The Tony and Pat Houghton Conference on Non-Equilibrium Statistical Mechanics ICERM, Brown University May 4, 2015

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 1 / 29

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Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 2 / 29

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SLIDE 3

Experiments with ultracold gases in 1D

Effective one-dimensional δ potential

  • M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where g1D = 2asω⊥ 1 − Cas mω⊥

2

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 3 / 29

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Experiments with ultracold gases in 1D

Effective one-dimensional δ potential

  • M. Olshanii, PRL 81, 938 (1998).

U1D(x) = g1Dδ(x) where g1D = 2asω⊥ 1 − Cas mω⊥

2

Girardeau ’60, Lieb and Liniger ’63

  • T. Kinoshita, T. Wenger, and D. S. Weiss,

Science 305, 1125 (2004).

  • T. Kinoshita, T. Wenger, and D. S. Weiss,
  • Phys. Rev. Lett. 95, 190406 (2005).

γeff= mg1D

2ρ Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 3 / 29

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SLIDE 5

Absence of thermalization in 1D?

density position Density profile momentum distribution momentum Momentum profile

  • T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006).

MR, A. Muramatsu, and M. Olshanii, Phys. Rev. A 74, 053616 (2006).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 4 / 29

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SLIDE 6

Absence of thermalization in 1D?

density position Density profile τ=0 momentum distribution momentum Momentum profile τ=0

  • T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006).

MR, A. Muramatsu, and M. Olshanii, Phys. Rev. A 74, 053616 (2006).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 4 / 29

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SLIDE 7

Absence of thermalization in 1D?

Experiment

0.2 0.4 0.6 0.8 1

k τ nk

−π/2 π/2 −π/4 π/4 1000 2000 3000 4000

Theory

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 5 / 29

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SLIDE 8

Absence of thermalization in 1D?

  • T. Kinoshita, T. Wenger, and D. S. Weiss,

Nature 440, 900 (2006).

γ = mg1D 2ρ

g1D: Interaction strength ρ: One-dimensional density

If γ ≫ 1 the system is in the strongly correlated Tonks-Girardeau regime If γ ≪ 1 the system is in the weakly interacting regime

Gring et al., Science 337, 1318 (2012).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 6 / 29

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Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 7 / 29

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Exact results from quantum mechanics

If the initial state is not an eigenstate of H |ψ0 = |α where

  • H|α = Eα|α

and E0 = ψ0| H|ψ0, then a generic observable O will evolve in time following O(τ) ≡ ψ(τ)| O|ψ(τ) where |ψ(τ) = e−i

Hτ|ψ0.

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 8 / 29

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SLIDE 11

Exact results from quantum mechanics

If the initial state is not an eigenstate of H |ψ0 = |α where

  • H|α = Eα|α

and E0 = ψ0| H|ψ0, then a generic observable O will evolve in time following O(τ) ≡ ψ(τ)| O|ψ(τ) where |ψ(τ) = e−i

Hτ|ψ0.

What is it that we call thermalization? O(τ) = O(E0) = O(T) = O(T, µ).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 8 / 29

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SLIDE 12

Exact results from quantum mechanics

If the initial state is not an eigenstate of H |ψ0 = |α where

  • H|α = Eα|α

and E0 = ψ0| H|ψ0, then a generic observable O will evolve in time following O(τ) ≡ ψ(τ)| O|ψ(τ) where |ψ(τ) = e−i

Hτ|ψ0.

What is it that we call thermalization? O(τ) = O(E0) = O(T) = O(T, µ). One can rewrite O(τ) =

  • α′,α

C⋆

α′Cαei(Eα′−Eα)τOα′α

where |ψ0 =

  • α

Cα|α, Taking the infinite time average (diagonal ensemble ˆ ρDE ≡

α |Cα|2|αα|)

O(τ) = lim

τ→∞

1 τ τ dτ ′Ψ(τ ′)| ˆ O|Ψ(τ ′) =

  • α

|Cα|2Oαα ≡ ˆ Odiag, which depends on the initial conditions through Cα = α|ψ0.

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 8 / 29

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Width of the energy density after a sudden quench

Initial state |ψ0 =

α Cα|α is an eigenstate of

  • H0. At τ = 0
  • H0 →

H = H0 + W with

  • W =
  • j

ˆ w(j) and

  • H|α = Eα|α.

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 9 / 29

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Width of the energy density after a sudden quench

Initial state |ψ0 =

α Cα|α is an eigenstate of

  • H0. At τ = 0
  • H0 →

H = H0 + W with

  • W =
  • j

ˆ w(j) and

  • H|α = Eα|α.

The width of the weighted energy density ∆E is then ∆E =

  • α

E2

α|Cα|2 − (

  • α

Eα|Cα|2)2 =

  • ψ0|

W 2|ψ0 − ψ0| W|ψ02,

  • r

∆E =

j1,j2∈σ

[ψ0| ˆ w(j1) ˆ w(j2)|ψ0 − ψ0| ˆ w(j1)|ψ0ψ0| ˆ w(j2)|ψ0]

N→∞

∝ √ N, where N is the total number of lattice sites.

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 9 / 29

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SLIDE 15

Width of the energy density after a sudden quench

Initial state |ψ0 =

α Cα|α is an eigenstate of

  • H0. At τ = 0
  • H0 →

H = H0 + W with

  • W =
  • j

ˆ w(j) and

  • H|α = Eα|α.

The width of the weighted energy density ∆E is then ∆E =

  • α

E2

α|Cα|2 − (

  • α

Eα|Cα|2)2 =

  • ψ0|

W 2|ψ0 − ψ0| W|ψ02,

  • r

∆E =

j1,j2∈σ

[ψ0| ˆ w(j1) ˆ w(j2)|ψ0 − ψ0| ˆ w(j1)|ψ0ψ0| ˆ w(j2)|ψ0]

N→∞

∝ √ N, where N is the total number of lattice sites. Since the width W of the full energy spectrum is ∝ N ∆ǫ = ∆E W

N→∞

∝ 1 √ N , so, as in any thermal ensemble, ∆ǫ vanishes in the thermodynamic limit.

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 9 / 29

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Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 10 / 29

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Relaxation dynamics of hard-core bosons in 2D

Hard-core boson Hamiltonian

  • H = −J
  • i,j
  • ˆ

b†

iˆ

bj + H.c.

  • + U
  • i,j

ˆ niˆ nj, ˆ b†2

i

= ˆ b2

i = 0

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

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Relaxation dynamics of hard-core bosons in 2D

Hard-core boson Hamiltonian

  • H = −J
  • i,j
  • ˆ

b†

iˆ

bj + H.c.

  • + U
  • i,j

ˆ niˆ nj, ˆ b†2

i

= ˆ b2

i = 0

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Nonequilibrium dynamics in 2D

Initial

Weak n.n. U = 0.1J Nb = 5 bosons N = 21 lattice sites Hilbert space: D = 20349 All states are used!

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

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SLIDE 19

Relaxation dynamics of hard-core bosons in 2D

Hard-core boson Hamiltonian

  • H = −J
  • i,j
  • ˆ

b†

iˆ

bj + H.c.

  • + U
  • i,j

ˆ niˆ nj, ˆ b†2

i

= ˆ b2

i = 0

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

“One can rewrite O(τ) =

  • α′,α

C⋆

α′Cαei(Eα′−Eα)τOα′α

where |ψ0 =

  • α

Cα|α, and taking the infinite time average (diagonal ensemble) O(τ) = lim

τ→∞

1 τ τ dτ ′Ψ(τ ′)| ˆ O|Ψ(τ ′) =

  • α

|Cα|2Oαα ≡ ˆ Odiag, which depends on the initial conditions through Cα = α|ψ0.”

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

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SLIDE 20

Relaxation dynamics of hard-core bosons in 2D

Hard-core boson Hamiltonian

  • H = −J
  • i,j
  • ˆ

b†

iˆ

bj + H.c.

  • + U
  • i,j

ˆ niˆ nj, ˆ b†2

i

= ˆ b2

i = 0

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Nonequilibrium dynamics in 2D

0.5 1 1.5 2 −2 −1 1 2

n(kx)

kx[2π/(Lx d)]

Time evolution of n(kx) time average tJ=000

Weak n.n. U = 0.1J Nb = 5 bosons N = 21 lattice sites Hilbert space: D = 20349 All states are used!

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

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Relaxation dynamics of hard-core bosons in 2D

Hard-core boson Hamiltonian

  • H = −J
  • i,j
  • ˆ

b†

iˆ

bj + H.c.

  • + U
  • i,j

ˆ niˆ nj, ˆ b†2

i

= ˆ b2

i = 0

MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).

Nonequilibrium dynamics in 2D

50 100 150 200

tJ

1 1.2 1.4 1.6 1.8 2

n(kx=0)

relaxation dynamics time average

Weak n.n. U = 0.1J Nb = 5 bosons N = 21 lattice sites Hilbert space: D = 20349 All states are used!

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

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Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 12 / 29

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SLIDE 23

Statistical description after relaxation

Canonical calculation

O = Tr

  • ˆ

Oˆ ρ

  • ˆ

ρ = Z−1 exp

  • − ˆ

H/kBT

  • Z = Tr
  • exp
  • − ˆ

H/kBT

  • E0 = Tr
  • ˆ

H ˆ ρ

  • T = 1.9J

Momentum distribution

  • 2
  • 1

1 2

kx[2π/(Lxd)]

0.5 1 1.5 2

n(kx)

initial state time average canonical

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 13 / 29

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Statistical description after relaxation

Canonical calculation

O = Tr

  • ˆ

Oˆ ρ

  • ˆ

ρ = Z−1 exp

  • − ˆ

H/kBT

  • Z = Tr
  • exp
  • − ˆ

H/kBT

  • E0 = Tr
  • ˆ

H ˆ ρ

  • T = 1.9J

Momentum distribution

  • 2
  • 1

1 2

kx[2π/(Lxd)]

0.5 1 1.5 2

n(kx)

initial state time average canonical

Microcanonical calculation

O = 1 Nstates

  • α

Ψα| ˆ O|Ψα with E0 − ∆E < Eα < E0 + ∆E Nstates : # of states in the window

  • 2
  • 1

1 2

kx[2π/(Lxd)]

0.5 1 1.5 2

n(kx)

initial state time average microcanonical

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 13 / 29

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SLIDE 25

Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 14 / 29

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SLIDE 26

Eigenstate thermalization hypothesis

Paradox?

  • α

|Cα|2Oαα = Omicrocan.(E0) ≡ 1 NE0,∆E

  • |E0−Eα|<∆E

Oαα Left hand side: Depends on the initial conditions through Cα = Ψα|ψI Right hand side: Depends only on the initial energy

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 15 / 29

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SLIDE 27

Eigenstate thermalization hypothesis

Paradox?

  • α

|Cα|2Oαα = Omicrocan.(E0) ≡ 1 NE0,∆E

  • |E0−Eα|<∆E

Oαα Left hand side: Depends on the initial conditions through Cα = Ψα|ψI Right hand side: Depends only on the initial energy i) For physically relevant initial conditions, |Cα|2 practically do not fluctuate. ii) Large (and uncorrelated) fluctuations occur in both Oαα and |Cα|2. A physically relevant initial state performs an unbiased sampling of Oαα.

MR and M. Srednicki, PRL 108, 110601 (2012).

  • K. He and MR, Phys. Rev. A 87, 043615 (2013).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 15 / 29

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SLIDE 28

Eigenstate thermalization hypothesis

Paradox?

  • α

|Cα|2Oαα = Omicrocan.(E0) ≡ 1 NE0,∆E

  • |E0−Eα|<∆E

Oαα Left hand side: Depends on the initial conditions through Cα = Ψα|ψI Right hand side: Depends only on the initial energy

MR, PRA 82, 037601 (2010).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 15 / 29

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SLIDE 29

Eigenstate thermalization hypothesis

Paradox?

  • α

|Cα|2Oαα = Omicrocan.(E0) ≡ 1 NE0,∆E

  • |E0−Eα|<∆E

Oαα Left hand side: Depends on the initial conditions through Cα = Ψα|ψI Right hand side: Depends only on the initial energy i) For physically relevant initial conditions, |Cα|2 practically do not fluctuate. ii) Large (and uncorrelated) fluctuations occur in both Oαα and |Cα|2. A physically relevant initial state performs an unbiased sampling of Oαα.

MR and M. Srednicki, PRL 108, 110601 (2012).

  • K. He and MR, Phys. Rev. A 87, 043615 (2013).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 15 / 29

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SLIDE 30

Eigenstate thermalization hypothesis

Paradox?

  • α

|Cα|2Oαα = Omicrocan.(E0) ≡ 1 NE0,∆E

  • |E0−Eα|<∆E

Oαα Left hand side: Depends on the initial conditions through Cα = Ψα|ψI Right hand side: Depends only on the initial energy Eigenstate thermalization hypothesis (ETH)

[J. M. Deutsch, PRA 43 2046 (1991); M. Srednicki, PRE 50, 888 (1994); MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).]

iii) The expectation value Ψα| ˆ O|Ψα of a few-body observable O in an eigenstate of the Hamiltonian |Ψα, with energy Eα, of a large interacting many-body system equals the thermal average of O at the mean energy Eα: Ψα| ˆ O|Ψα = Omicrocan.(Eα)

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 15 / 29

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SLIDE 31

Eigenstate thermalization hypothesis

Momentum distribution

Eigenstates a − d are the ones with energies closest to E0

  • 2
  • 1

1 2

kx[2π/Lxa]

0.5 1 1.5 2

n(kx)

time average/microcan. eigenstate a eigenstate b eigenstate c eigenstate d Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 16 / 29

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SLIDE 32

Eigenstate thermalization hypothesis

Momentum distribution

Eigenstates a − d are the ones with energies closest to E0

  • 2
  • 1

1 2

kx[2π/Lxa]

0.5 1 1.5 2

n(kx)

time average/microcan. eigenstate a eigenstate b eigenstate c eigenstate d 1 2 3

n(kx=0)

  • 10
  • 8
  • 6
  • 4
  • 2

E[J]

1 2

ρ(E)[J

  • 1]

ρ(E) exact ρ(E) microcan. ρ(E) canonical

n(kx = 0) vs energy

ρ(E) = P(E) × dens. stat. P(E)exact → |Cα|2 P(E)mic → constant P(E)can → exp (−E/kBT)

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 16 / 29

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SLIDE 33

One-dimensional integrable case

Similar experiment in one dimension

Initial 13 sites 8 sites

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 17 / 29

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SLIDE 34

One-dimensional integrable case

Similar experiment in one dimension

Initial 13 sites 8 sites

Time average vs Stat. Mech.

No thermalization!

  • 10
  • 5

5 10

kx[2π/Lxa]

0.2 0.4 0.6

n(kx)

time average microcan. canonical Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 17 / 29

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SLIDE 35

Breakdown of eigenstate thermalization

Momentum distribution

Eigenstates a − d are the ones with energies closest to E0

  • 10
  • 5

5 10

kx[2π/Lxa]

0.2 0.4 0.6

n(kx)

eigenstate a eigenstate b eigenstate c eigenstate d 0.5 1 1.5

n(kx=0)

  • 8
  • 6
  • 4
  • 2

E[J]

0.5 1 1.5

ρ(E)[J

  • 1]

ρ(E) exact ρ(E) microcan. ρ(E) canonical

n(kx = 0) vs energy

ρ(E) = P(E) × dens. stat. P(E)exact → |Cα|2 P(E)mic → constant P(E)can → exp (−E/kBT)

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 18 / 29

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SLIDE 36

Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 19 / 29

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SLIDE 37

Relaxation dynamics of hard-core bosons in 2D

Hard-core boson Hamiltonian

  • H = −J
  • i,j
  • ˆ

b†

iˆ

bj + H.c.

  • + U
  • i,j

ˆ niˆ nj, ˆ b†2

i

= ˆ b2

i = 0

Nonequilibrium dynamics in 2D

50 100 150 200

tJ

1 1.2 1.4 1.6 1.8 2

n(kx=0)

relaxation dynamics time average

Weak n.n. U = 0.1J Nb = 5 bosons N = 21 lattice sites Hilbert space: D = 20349 All states are used!

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 20 / 29

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SLIDE 38

Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 21 / 29

slide-39
SLIDE 39

Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Time average of ˆ O

ˆ O =

  • α

|Cα|2Oαα ∼

  • α

1 Nstates Oαα ∼ Otypical

αα

One needs: Otypical

α′α

≪ Otypical

αα

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 21 / 29

slide-40
SLIDE 40

Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Time average of ˆ O

ˆ O =

  • α

|Cα|2Oαα ∼

  • α

1 Nstates Oαα ∼ Otypical

αα

One needs: Otypical

α′α

≪ Otypical

αα

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 21 / 29

slide-41
SLIDE 41

Time fluctuations

Are they small because of dephasing? ˆ O(t) − ˆ O(t) =

  • α′,α

α′=α

C⋆

α′Cαei(Eα′−Eα)tOα′α ∼

  • α′,α

α′=α

ei(Eα′−Eα)t Nstates Oα′α ∼

  • N 2

states

Nstates Otypical

α′α

∼ Otypical

α′α

Time average of ˆ O

ˆ O =

  • α

|Cα|2Oαα ∼

  • α

1 Nstates Oαα ∼ Otypical

αα

One needs: Otypical

α′α

≪ Otypical

αα

  • M. Srednicki, J. Phys. A 29, L75 (1996).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 21 / 29

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SLIDE 42

Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 22 / 29

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SLIDE 43

Bose-Fermi mapping

Hard-core boson Hamiltonian in an external potential ˆ H = −J

  • i
  • ˆ

b†

iˆ

bi+1 + H.c.

  • +
  • i

vi ˆ ni, constraints ˆ b†2

i

= ˆ b2

i = 0

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 23 / 29

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SLIDE 44

Bose-Fermi mapping

Hard-core boson Hamiltonian in an external potential ˆ H = −J

  • i
  • ˆ

b†

iˆ

bi+1 + H.c.

  • +
  • i

vi ˆ ni, constraints ˆ b†2

i

= ˆ b2

i = 0

⇓

Map to spins and then to fermions (Jordan-Wigner transformation) σ+

i = ˆ

f †

i i−1

  • β=1

e−iπ ˆ

f †

β ˆ

fβ, σ− i = i−1

  • β=1

eiπ ˆ

f †

β ˆ

fβ ˆ

fi

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 23 / 29

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SLIDE 45

Bose-Fermi mapping

Hard-core boson Hamiltonian in an external potential ˆ H = −J

  • i
  • ˆ

b†

iˆ

bi+1 + H.c.

  • +
  • i

vi ˆ ni, constraints ˆ b†2

i

= ˆ b2

i = 0

⇓

Map to spins and then to fermions (Jordan-Wigner transformation) σ+

i = ˆ

f †

i i−1

  • β=1

e−iπ ˆ

f †

β ˆ

fβ, σ− i = i−1

  • β=1

eiπ ˆ

f †

β ˆ

fβ ˆ

fi

⇓

Exact Green’s function Gij(τ) = det

  • Pl(τ)

† Pr(τ)

  • Computation time ∼ L2N 3

3000 lattice sites, 300 particles

MR and A. Muramatsu, PRL 93, 230404 (2004); PRL 94, 240403 (2005).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 23 / 29

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SLIDE 46

Relaxation dynamics in an integrable system

0.1 0.2 −300 −150 150 300 n x/a Density profile 0.1 0.2 0.3 0.4 π −π π/2 −π/2 nk ka Momentum profile

MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, 050405 (2007).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 24 / 29

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SLIDE 47

Outline

1

Introduction Experiments with ultracold gases Unitary evolution and thermalization

2

Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations

3

Integrable systems Time evolution Generalized Gibbs ensemble

4

Summary

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SLIDE 48

Statistical description after relaxation

Thermal equilibrium

ˆ ρ = Z−1 exp

  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • Z = Tr
  • exp
  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • E = Tr
  • ˆ

H ˆ ρ

  • ,

Nb = Tr

  • ˆ

Nbˆ ρ

  • MR, PRA 72, 063607 (2005).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 26 / 29

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SLIDE 49

Statistical description after relaxation

Thermal equilibrium

ˆ ρ = Z−1 exp

  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • Z = Tr
  • exp
  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • E = Tr
  • ˆ

H ˆ ρ

  • ,

Nb = Tr

  • ˆ

Nbˆ ρ

  • MR, PRA 72, 063607 (2005).

Evolution of nk=0

1000 2000 3000 4000

τ

0.5 1 1.5

nk=0

Time evolution Thermal

nk after relaxation

  • π
  • π/2

π/2 π

ka

0.25 0.5

nk

After relax. (Nb=30) After relax. (Nb=15) Thermal (Nb=30) Thermal (Nb=15)

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 26 / 29

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SLIDE 50

Statistical description after relaxation

Thermal equilibrium

ˆ ρ = Z−1 exp

  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • Z = Tr
  • exp
  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • E = Tr
  • ˆ

H ˆ ρ

  • ,

Nb = Tr

  • ˆ

Nbˆ ρ

  • MR, PRA 72, 063607 (2005).

Integrals of motion

(underlying noninteracting fermions)

ˆ HF ˆ γf†

m |0 = Emˆ

γf†

m |0

  • ˆ

If

m

  • =
  • ˆ

γf†

m ˆ

γf

m

  • Evolution of nk=0

1000 2000 3000 4000

τ

0.5 1 1.5

nk=0

Time evolution Thermal

nk after relaxation

  • π
  • π/2

π/2 π

ka

0.25 0.5

nk

After relax. (Nb=30) After relax. (Nb=15) Thermal (Nb=30) Thermal (Nb=15)

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 26 / 29

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SLIDE 51

Statistical description after relaxation

Thermal equilibrium

ˆ ρ = Z−1 exp

  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • Z = Tr
  • exp
  • −
  • ˆ

H − µ ˆ Nb

  • /kBT
  • E = Tr
  • ˆ

H ˆ ρ

  • ,

Nb = Tr

  • ˆ

Nbˆ ρ

  • MR, PRA 72, 063607 (2005).

Generalized Gibbs ensemble

ˆ ρc = Z−1

c

exp

  • −
  • m

λm ˆ Im

  • Zc = Tr
  • exp
  • −
  • m

λm ˆ Im

  • ˆ

Imτ=0 = Tr

  • ˆ

Imˆ ρc

  • Evolution of nk=0

1000 2000 3000 4000

τ

0.5 1 1.5

nk=0

Time evolution Thermal GGE

nk after relaxation

  • π
  • π/2

π/2 π

ka

0.25 0.5

nk

After relaxation Thermal GGE

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 26 / 29

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SLIDE 52

Summary

Thermalization occurs in generic isolated systems ⋆ Finite size effects

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 27 / 29

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SLIDE 53

Summary

Thermalization occurs in generic isolated systems ⋆ Finite size effects Eigenstate thermalization hypothesis ⋆ Ψα| ˆ O|Ψα = Omicrocan.(Eα)

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 27 / 29

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SLIDE 54

Summary

Thermalization occurs in generic isolated systems ⋆ Finite size effects Eigenstate thermalization hypothesis ⋆ Ψα| ˆ O|Ψα = Omicrocan.(Eα) Small time fluctuations ← smallness of

  • ff-diagonal elements

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 27 / 29

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SLIDE 55

Summary

Thermalization occurs in generic isolated systems ⋆ Finite size effects Eigenstate thermalization hypothesis ⋆ Ψα| ˆ O|Ψα = Omicrocan.(Eα) Small time fluctuations ← smallness of

  • ff-diagonal elements

Time plays only an auxiliary role

  • E a

E b E c E a E b E c

thermal thermal thermal coherence

Thermal state

dephasing time thermal thermal thermal coherence

EIGENSTATE THERMALIZATION Initial state

  • Marcos Rigol (Penn State)

Dynamics in quantum systems May 4, 2015 27 / 29

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SLIDE 56

Summary

Thermalization occurs in generic isolated systems ⋆ Finite size effects Eigenstate thermalization hypothesis ⋆ Ψα| ˆ O|Ψα = Omicrocan.(Eα) Small time fluctuations ← smallness of

  • ff-diagonal elements

Time plays only an auxiliary role Integrable systems are different (Generalized Gibbs ensemble)

  • E a

E b E c E a E b E c

thermal thermal thermal coherence

Thermal state

dephasing time thermal thermal thermal coherence

EIGENSTATE THERMALIZATION Initial state

  • Marcos Rigol (Penn State)

Dynamics in quantum systems May 4, 2015 27 / 29

slide-57
SLIDE 57

Summary

Thermalization occurs in generic isolated systems ⋆ Finite size effects Eigenstate thermalization hypothesis ⋆ Ψα| ˆ O|Ψα = Omicrocan.(Eα) Small time fluctuations ← smallness of

  • ff-diagonal elements

Time plays only an auxiliary role Integrable systems are different (Generalized Gibbs ensemble) Thermalization and ETH break down close integrability (finite system) ⋆ Quantum equivalent of KAM?

  • E a

E b E c E a E b E c

thermal thermal thermal coherence

Thermal state

dephasing time thermal thermal thermal coherence

EIGENSTATE THERMALIZATION Initial state

  • Marcos Rigol (Penn State)

Dynamics in quantum systems May 4, 2015 27 / 29

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SLIDE 58

Collaborators

Vanja Dunjko (U Mass Boston) Alejandro Muramatsu (Stuttgart U) Maxim Olshanii (U Mass Boston) Anatoli Polkovnikov (Boston U) Lea F . Santos (Yeshiva U) Mark Srednicki (UC Santa Barbara) Current group members: Deepak Iyer, Baoming Tang Former group members: Kai He (NOAA), Ehsan Khatami (SJSU)

Supported by:

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 28 / 29

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SLIDE 59

Information entropy (Sj = − D

k=1 |ck j|2 ln |ck j|2)

L.F . Santos and MR, PRE 81, 036206 (2010); PRE 82, 031130 (2010).

Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 29 / 29