General quantum quenches in the transverse field Ising chain

Tatjana Puškarov and Dirk Schuricht

Novi Sad, 23.12.2015

Institute for Theoretical PhysicsUtrecht University

Non-equilibrium dynamics of isolated quantum systems

1. Consider an isolated many-particle quantum system

2. Take it out of equilibrium

3. Let the system evolve

4. Observe what happens:

• does the system thermalise/does it relax?

• how can we describe the steady state?

• how do the observables behave?

Ultracold atoms in optical lattices

Bloch, Dalibard & Zwerger, RMP, 2008

Simulation of solid-state models:

atoms

optical lattice

electrons

crystal lattice

• control of tuneable parameters: lattice type, dimensionality, interactions

• parameters tuneable in time • almost perfect isolation from the environment

JILA BEC87Rb BEC SF to MI transition Greiner et al., Nature, 2002

Quantum Newton’s cradle

Kinoshita, Wenger & Weiss, Nature, 2006

• 1D tubes with 40-250 87Rb atoms

• initially momentum eigenstate with

• essentially unitary time evolution

±k

Quantum Newton’s cradle

Kinoshita, Wenger & Weiss, Nature, 2006

3D1D

• no thermalisation in 1D • thermalisation after ~3 collisions in 3D

� = 18t = 15⌧

� = 3.2t = 25⌧

� = 14t = 25⌧

t = 0⌧

t = 2⌧

t = 4⌧

t = 9⌧

Relaxation following sudden quenches

• pre quench:

• quench:

• post quench:t0

g

gi

gfH(gi), | (0)i

H(gf), | (t)i = e�iH(gf )t| (0)i

gi ! gf

decomposition in terms of post-quench Hamiltonian eigenstates

| (t)i = e�iH(gf )t| (0)i =X

n

hn| (0)ie�iEnt|ni

Observables:

h (t)|O| (t)i =X

m,n

h (0)|mihn| (0)ihn|O|miei(En�Em)t

interference of oscillating phases

Consider a system with a tuning parameter gH(g)

Relaxation following quenches

Full system AUB:

• initial state is a pure state the system never relaxes as a whole

• density matrix

• observables AB

| i

⇢(t) = | (t)ih (t)|

h (t)|O| (t)i = tr [⇢(t)O]

lim

t!1lim

L!1

h (t)|OB | (t)ih (t)| (t)i = const

Subsystem B:

• A (infinite) acts as a bath for B (finite) the system relaxes locally

• density matrix

• observables

⇢B(t) = trA⇢(t)

Relaxation following quenches

• energy is the only local integral of motion

• Gibbs ensemble

A

BCan we express in terms of a statistical ensemble?⇢B(1)

⇢G =

1

ZGexp

�� �H

�

Generic systems

Deutsch, PRA, 1991

thermalisation

Integrable systems

• additional local integrals of motion

• generalised Gibbs ensemble Rigol et al., PRL, 2007

⇢GGE =

1

ZGGEexp

�X

n

�nIn

!

Lagrange multipliers full set of local integrals of motion[H, In] = [Im, In] = 0tr (⇢GGEIn) = h 0|In| 0i

relaxation

General quantum quenches

t⌧0

g

gi

gf

Consider a system with a tuning parameter gH(g)

Initial system: t < 0

H(gi)

| 0i

Post-quench system:H(gf)

| (t)i = e�iH(gf )(t�⌧)| (⌧)i

t > ⌧

General quantum quenches

t⌧0

g

gi

gf

Consider a system with a tuning parameter gH(g)

Initial system: t < 0

H(gi)

| 0i

Post-quench system:H(gf)

| (t)i = e�iH(gf )(t�⌧)| (⌧)i

t > ⌧

General quantum quenches

t⌧0

g

gi

gf

Consider a system with a tuning parameter gH(g)

Initial system: t < 0

H(gi)

| 0i

Post-quench system:H(gf)

| (t)i = e�iH(gf )(t�⌧)| (⌧)i

t > ⌧

General quantum quenches

t⌧0

g

gi

gfsudden quench

cosine quench

linear quench

“general” quench

Consider a system with a tuning parameter gH(g)

Initial system: t < 0

H(gi)

| 0i

Post-quench system:H(gf)

| (t)i = e�iH(gf )(t�⌧)| (⌧)i

t > ⌧

General quantum quenches

t0

g

gi

gf

⌧ t0

g

gi

gf

⌧

⌧ ! 0 ⌧ ! 1

Compare the duration of quench to the timescales in the system: gap in the system, interaction energies

⌧

• sudden limit • the change is too fast for the system

to react in intermediate times • it only notices the final value of the

quench parameter

• adiabatic limit • the change is slow enough for the

system to follow in the ground state of each intermediate Hamiltonian

Transverse Field Ising chain

H = �J

NX

i=1

��x

i

�x

i+1 + g�z

i

�

g = 0 : g = 1 :

| ### . . . #i| """ . . . "i

| !!! · · · !i

g

0 1 1FM PM

PM to AFM transition in an ultracold atom system

Simon et al., Nature, 2011

Sachdev, Sengupta & Girvin, PRB, 2002

Response of the system to a quantum quench

Meinert et al., PRL, 2013

Transverse Field Ising chain

H = �J

NX

i=1

��x

i

�x

i+1 + g�z

i

�

1) Jordan-Wigner transformation

2) Fourier transformation

3) Bogoliubov transformation

�zi = 1� 2c†i ci

�x

i

=Y

j<i

(1� 2c†j

cj

)(c†i

+ ci

)

⌘k = ukck � ivkc†�k

⌘†k = ukc†k + ivkc�k

H =X

k

"k

✓⌘†k⌘k � 1

2

◆

"k = 2Jp

1 + g2 � 2g cos k

"k

k� = 2J |1� g|

Global general quench in TFI chain

H(t) = �J

NX

i=1

��x

i

�x

i+1 + g(t)�z

i

�

t0

g

gi

gf

⌧

Each instantaneous Hamiltonian can be diagonalised as

⌘k,t = uk,tck,t � ivk,tc†�k,t

H(t) =X

k

"k,t

✓⌘†k,t⌘k,t �

1

2

◆

"k,t = 2Jp

1 + (g(t))2 � 2g(t) cos k

How to relate the infinitely many instantaneous Hamiltonians?

Global general quench in TFI chain

Cast the dynamics into the Bogoliubov coefficients: ck(t) = uk(t)⌘k + ivk(t)⌘†�k

i@

@tck(t) = [ck, H]

i@

@tc†k(t) = [c†k, H]

Explicit time dependance from Heisenberg equations of motion:

i@

@t

✓uk(t)v⇤�k(t)

◆=

✓Ak(t) Bk

Bk �Ak(t)

◆✓uk(t)v⇤�k(t)

◆

Ak(t) = 2

�g(t)� cos k

�

Bk = 2 sin kDuring quench

@2

@t2y(t) +

Ak(t)

2 +B2k ± i

@

@tAk(t)

�y(t) = 0

Post quench@2

@t2y(t) + !2

ky(t) = 0

sudden vs linear quench

Transverse magnetisation Mz(t) = h�z(t)i

During quench Post quench

⇠ t�3/2

Stationary values • behaviour during the quench depends on the quench duration

• behaviour after the quench general - algebraic decay with internal oscillations

• GGE constructed from the post-quench mode occupation numbers

Two-point correlation functionsC(x, t) = hO(x, t)O(0, t)i � hO(t)i2

• initially correlations only within the system correlation length

• state acts as a source of entangled quasiparticles

• quasiparticles propagate classically through the system

• onset of correlations at t⇤ ⇡ x

2v

⇠0

| 0i

light-conet

0x

C(x, t) 6= 0

C(x, t) = 0x > 2vt

x < 2vt

Cheneu et al, Nature, 2012

Calabrese & Cardy, PRL, 2006

Two-point correlation functionsC(x, t) = hO(x, t)O(0, t)i � hO(t)i2

Cheneu et al, Nature, 2012Langen et al, Nature Phys 2013

Transverse two-point correlation function⇢zn(t) =

⌦�zi (t)�

zi+n(t)

↵

⇠ t�3

• algebraic decay with internal oscillations

• light-cone effect and Lieb-Robinson bound observable

n = 10 n = 20 n = 30 n = 40 n = 50

n = 60 n = 70 n = 80 n = 90 n = 100

Time evolution of the transverse two-point correlation function for various separations

Transverse two-point correlation function⇢zn(t) =

⌦�zi (t)�

zi+n(t)

↵

C(x, t) = hO(x, t)O(0, t)i � hO(t)i2

t

0x

C(x, t) 6= 0

C(x, t) = 0x > 2vt

x < 2vt

20 40 60 80 100

5

10

15

20

25

30

0.0002

0.0004

0.0006

0.0008

0.0010

Modification of the quasiparticle picture:

• at any time during the quench the quasiparticles propagate with their instantaneous velocity

• for general quenches the quasiparticles will not be created just at but over the quench time

t

t = 0⌧

• Isolated 1D systems show non-trivial non-equilibrium dynamics

• Local observables in the system relax (GGE)

• Two-point functions allow for observation of the light-cone effect

• General quenches add additional features compared to the sudden case

• TFIC: quenching across the critical point (Kibble-Zurek mechanism, DPT)

Summary and outlook

Karra

sch

& Sc

huric

ht,

PRB

, 201

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