Aspects of non-equilibrium dynamics in closed quantum syste - PowerPoint Presentation

Slide1

Aspects of non-equilibrium dynamics in closed quantum systems

K. SenguptaIndian Association for the Cultivation of Science, Kolkata

Collaborators:

Diptiman

Sen

,

Shreyoshi

Mondal

, Christian

Trefzger

Anatoli

Polkovnikov

,

Mukund

Vengalattore

,

Alsessandro

Silva,

Arnab

Das,

Bikas

K

Chakraborti

,

Sei

Suzuki, Takumi

Hikichi

.Slide2

Overview

Introduction: Why dynamics? 2.

Nearly adiabatic dynamics: defect production

3.

Correlation functions and entanglement generation

4

. Non-

integrable

systems: a specific case study

5

. Experiments

7. Conclusion: Where to from here? Slide3

Introduction: Why dynamics

Progress with experiments: ultracold atoms can be used to study dynamics of closed interacting quantum systems.

Finding systematic ways of understanding dynamics of model systems and

understanding its relation with dynamics of more complex models: concepts

of universality out of equilibrium?

Understanding similarities and differences of different ways of taking systems

out of equilibrium: reservoir versus closed dynamics and protocol dependence.

4. Key questions to

be answered:

What is universal in the dynamics of a system following a quantum quench ?

What are the characteristics of the asymptotic, steady state reached after a

quench ? When is it thermal ?Slide4

Nearly adiabatic dynamics: defect productionSlide5

Landau-Zenner dynamics in two-level systems

Consider a generic time-dependent

Hamiltonian for a two level system

The instantaneous energy levels

have an avoided level crossing at

t=0, where the diagonal terms vanish.

The dynamics of the system

can be exactly solved.

The probability of the system to

make a transition to the excited state

of the final Hamiltonian starting from

The ground state of the initial

Hamiltonian

can be exactly computedSlide6

Defect production and quench dynamics

Kibble and Zurek:

Quenching a system across a thermal phase

transition

:

Defect production in early universe.

Ideas can be carried over to

T=0

quantum phase transition.

The

variation of a system parameter

which

takes the system

across a quantum critical point

at

QCP

The simplest model to demonstrate

such defect production is

Describes many well-known

1D and 2D models.

For adiabatic evolution

, the

system would stay in the

ground states of the phases

on both sides of the critical

point.Slide7

Jordan Wigner transformation :

Hamiltonian in term of

fermion

in momentum space: [J=1]

Specific Example: Ising model in transverse field

Ising Model in a transverse field:Slide8

Defining

We can write,

where

The

eigen

values are,

The energy gap vanishes at g=1 and k=0: Quantum Critical point.Slide9

Let us now vary g as

2

∆

ε

k

g

The probability of defect formation is the probability of the system

to remain at the “wrong” state at the end of the quench.

Defect formation occurs mostly between a finite interval near the

quantum critical point. Slide10

While quenching,

the gap vanishes at g=1 and k=0.Even for a very slow quench, the

process becomes

non adiabatic at g = 1.

Thus while quenching after crossing g = 1

there is a finite probability of the system to remain in the excited state

.

The portion of the system which remains in the excited state is known as defect.

Final state with defects,

xSlide11

In the

Fermion representation, computation of defect probability amounts to solving a Landau-Zenner

dynamics for each k.

p

k

:

probability

of the system to be in the excited state

Total defect :

From

Landau

Zener

problem

if

the Hamiltonian of the system isThen

Scaling law for defect density in a linear quenchSlide12

For the Ising model

Here p

k

is maximum when sin(k) = 0

Linearising about k = 0 and making a transformation of variable

For a critical point with arbitrary

dynamical and correlation length

exponents, one can generalize

Ref: A

Polkovnikov

, Phys. Rev. B 72, 161201(R) (2005)

Scaling of defect density

-Slide13

Generic critical points: A phase space argument

The system enters the impulse region whenrate of change of the gap is the same orderas the square of the gap.

For slow dynamics, the impulse region is a

small region near the critical point where

scaling works

The system thus spends a time T in the impulse region which depends on the quench time

In this region, the energy gap scales as

Thus the scaling law for the defect density turns out to beSlide14

Critical surface: Kitaev

Model in d=2

Jordan-Wigner

transformation

a and b represents

Majorana

Fermions

living at the

midpoints of the vertical

bonds of the lattice.

D

n

is independent of a

and b and hence

commutes with H

F

: Special property of

the Kitaev modelGround statecorresponds toDn=1 on all links.Slide15

Solution in momentum space

z

z

Off-diagonal

element

Diagonal

elementSlide16

J

1

J

2

Quenching J

3

linearly

takes the system

through a critical line in

parameter space and

hence through the line

in momentum space.

In general a

quench of d dimensional

system can take the system through a

d-m dimensional gapless surface

in

momentum space.

For

Kitaev model: d=2, m=1

For quench through critical point: m=d

Gapless phase when

J

3

lies

between(J

1

+J

2

) and |J

1

-J

2

|.

The

bands touch each other at

special points in the

Brillouin

zone

whose location depend

on values of J

i

s.

Question: How would the defect density scale with quench rate?

J

3Slide17

Defect density for the Kitaev model

Solve the Landau-Zenner problem corresponding to

H

F

by taking

For slow quench, contribution to n

d

comes from momenta near the line

.

For the general case where quench of d dimensional system can take the system through a d-m dimensional

gapless surface with z= =1

It can be shown that if the surface

has arbitrary

dynamical and correlation

length exponents

, then the

defect

density

scales

as

Generalization of

Polkovnikov’s

result

for critical surfaces

Phys. Rev.

Lett

. 100, 077204 (2008)Slide18

Non-linear power-law quench across quantum critical points

For general power law quenches, the Schrodinger equation time evolution describing the time evolution can not be solved analytically.

Models with

z= D=1: Ising

, XY,

D=2 Extended Kitaev

l

can be a function of k

Two Possibilities

Quench term vanishes

at the QCP.

Novel universal exponent

for scaling of defect density

as a function of quench rate

Quench term does not

vanish at the QCP.

Scaling for the defect density is

same as in linear quench but

with a non-universal effective rateSlide19

Quench term vanishes at the QCP

Schrodinger equationScale

Defect probability must be a

generic function

Contribution to n

d

comes when

D

k

vanishes as |k|.

For a generic critical point

with exponents z and Slide20

Comparison with

numerics of model systems (1D Kitaev)

Plot of

ln

(n)

vs

ln

(t) for the 1D Kitaev model Slide21

Quench term does not vanish at the QCP

Consider a slow quench ofg as a power law in timesuch that the QCP is reachedat t=t

0

.

At the QCP, the instantaneous

energy gap must vanish.

If the quench is sufficiently slow,

then the contribution to the defect

production comes from the

neighborhood of t=t

0

and |k|=k

0.

Effective linear quench withSlide22

Ising model in a transverse field at d=1

10

15

20

There are two critical points

at

g=1 and -1

For both the critical points

Expect n to scale as

a

0.5Slide23

Anisotropic critical point in the Kitaev model

J

1

J

2

J

3

Linear slow dynamics which takes the system

from the gapless phase to the border of the

gapless phase ( take J1=J2=1 and vary J3)

The energy gap scales with momentum

in an anisotropic manner.

Both the analytical solution of the quench

problem and numerical solution of the

Kitaev model finds different scaling exponent

For the defect density and the residual energy

Different from the expected scaling n ~ 1/

t

Other models: See

Divakaran

, Singh and

Dutta

EPL (2009). Slide24

Phase space argument and generalization

Anisotropic critical point in d dimensions

for m momentum

components

i

=1..m

for the rest d-m momentum

components

i

=m+1..d with z’ > z

Need to generalize the phase space argument for defect production

Scaling of the energy gap still

remains the same

The phase space for defect

production also remains the same

However now different momentum components scales differently with the gap

Reproduces the expected

scaling for isotropic critical

points for z=z’

Kitaev model scaling is

obtained for z’=d=2

and Slide25

Deviation from and extensions of generic results: A brief survey

Topological sector of integrable model may lead to different scaling laws. Example of Kitaev chain studied in Sen

and

Visesheswara

(EPL 2010). In these models, the

effective low-energy theory may lead to emergent non-linear dynamics.

If disorder does not destroy QCP via Harris criterion, one can study dynamics across

disordered QCP. This might lead to different scaling laws for defect density and residual

energy originating from disorder averaging. Study of Kitaev model by

Hikichi

, Suzuki

and KS ( PRB 2010).

Presence of external bath leads to noise and dissipation: defect production and loss

due to noise and dissipation. This leads to a temperature ( that of external bath

assumed to be in equilibrium) dependent contribution to defect production

( Patane et al PRL 2008, PRB 2009).

Analogous results for scaling dynamics for fast quenches: Here one starts from the QCPand suddenly changes a system parameter to by a small amount. In this limit one has

the scaling behavior for residual energy, entropy, and defect density (de Grandi andPolkovnikov , Springer Lecture Notes 2010)Slide26

Correlation functions and entanglement generationSlide27

Defect Correlation in Kitaev model

The defect correlation as a

function of spatial distance

r

is given in terms of

Majorana

fermion

operators

For the Kitaev model

Plot of the defect correlation

function sans the delta function

peak for J

1

=J and J

t

=5 as a

function of J

2

=J. Note the change

in the anisotropy direction as a function of J2.

Only non-trivial

correlator

of the modelSlide28

Entanglement generation in transverse field anisotropic XY model

Quench the magnetic field h from

large negative to large positive values.

PM

PM

FM

h

One can compute all correlation functions for this dynamics in this model. (

Cherng

and

Levitov

).

1

-1

No non-trivial correlation between

the odd neighbors.

What’s the bipartite entanglement generated

due to the quench between spins at

i

and

i+n

?

Single-site entanglement:

the linear entropy or the

Single site concurrenceSlide29

Measures of bipartite entanglement in spin ½ systems

Concurrence

Negativity

(Hill and

Wootters

)

(Peres)

Consider a wave function for two spins

and its spin-flipped counterpart

C is 1 for singlet and 0 for separable states

Could be a measure of entanglement

Use this idea to get a measure for mixed state

of two spins : need to use density matrices

Consider a mixed state of two

spin ½ particles and write the

density matrix for the state.

Take partial transpose with respect

to one of the spins and check for

negative

eigenvalues

.

Note: For separable density matrices,

negativity is zero by constructionSlide30

Steps:1. Compute the two-body density matrix

2.

Compute concurrence and

negativity as measures of two-site

entanglement from this density matrix

3.

Properties of bipartite entanglement

Finite only between even neighbors

Requires a critical quench rate above which

it is zero.

c. Ratio of entanglement between even neighbors

can be tuned by tuning the quench rate.

The entanglement generated is entirely multipartite for reasonably fast quenches

For the 2D

Kitaev

model, one can show that the entire entanglement is always multipartiteSlide31

Evolution of entanglement after a quench: anisotropic XY model

Prepare the system in thermal mixed state and change the transverse field to zero from

it’s initial value denoted by a.

The long-time evolution of the system shows ( for T=0) a clear separation into two regimes

distinguished by finite/zero value of log negativity denoted by E

N

.

The study at finite time shows non-monotonic variation of E

N

at short time scales while

displaying monotonic behavior at longer time scales.

For a starting finite temperature T, the variation of E

N

could be either monotonic or

non-monotonic depending on starting transverse field. Typically non-monotonic behavior

is seen for starting transverse field in the critical region.

T=0

t=10

T=0

a=0.78t=1Sen-De, Sen,

Lewenstein

(2006)Slide32

Non-integrable

systems: a specific case studySlide33

Dynamics of the Bose-Hubbard model

Bloch 2001

Transition described by the

Bose-Hubbard model:Slide34

Mott-Superfluid transition: preliminary analysis

Mott state with 1 boson per site

Stable ground state for 0 <

m

< U

Adding a particle to the Mott state

Mott state is destabilized when

the excitation energy touches 0

.

Removing a particle from the Mott state

Destabilization of the Mott state via addition of particles/hole: onset of

superfluiditySlide35

Beyond this simple picture

Higher order energy calculation

by

Freericks

and

Monien

: Inclusion

of up to O(t

3

/U

3

) virtual processes.

Mean-field theory (Fisher

89,

Seshadri

93)

Phase diagram for n=1 and d=3

MFT

O(t2/U2) theories

Predicts a quantum phase transition with z=2 (except atthe tip of the Mott lobe where z=1).

Mott

Superfluid

Quantum Monte Carlo studies for

2D

& 3D systems

:

Trivedi

and

Krauth

,

B.

Sansone-Capponegro

No

method for studying dynamics beyond mean-field theorySlide36

A more accurate phase diagram: building fluctuations over MFT

Distinguish between two

types of hopping processes

using a projection operator

technique

(

A

)

(B)

Define a projection operator

Divide the hopping to classes A and B

Design a transformation

which eliminate hopping

processes of class A

perturbatively

in J/U. Slide37

Equilibrium phase diagram

Use the effective Hamiltonian to compute the ground state

energy and hence the phase

diagram

Reproduction of the phase

diagram with remarkable

accuracy in d=3: much better

than standard mean-field

or strong coupling expansion

in d=2 and 3.

Allows for straightforward generalization for treatment of dynamicsSlide38

We were not that lazy……Slide39

Non-equilibrium dynamics

Consider a linear ramp of J(t)=Ji +(Jf-J

i

) t/

t.

For dynamics, one needs to solve the Sch. Eq.

Make a time dependent transformation

to address the dynamics by projecting on

the instantaneous low-energy sector.

The method provides an accurate description

of the ramp if J(t)/U <<1 and hence can

treat slow and fast ramps at equal footing

.

Takes care of particle/hole production

due to finite ramp rate Slide40

Absence of critical scaling: may

be understood as the inability of

the system to access the critical

(k=0) modes.

Fast quench from the Mott to the SF

phase; study of superfluid dynamics.

Single frequency pattern near the critical

Point; more complicated deeper in the SF

phase.

Strong quantum fluctuations near the QCP;

justification of going beyond

mft

.Slide41

ExperimentsSlide42

Experimental Systems: Spin one ultracold bosons

Spin one bosons are loaded in an optical trap with mz

=0 and

subjected to a Zeeman magnetic

field.

Second order QPT at q= 2c

2

n

between the FM (q << 2c

2

n) and

the scalar (q>> 2c

2

n) phases.

One can study quench dynamics

by quenching the magnetic field

L. Sadler

et al.

Nature (2006).

In the experiment, quench was done from the scalar to the FM phase. Defects are absence of ferromagnetic domains. For a rapid quench, one starts with very small

domain density which then develop in time.

Perform

a slow power law quench

of

the magnetic field

across

the QCP from the FM to

the

scalar phase and measure

magnetization at the end of the quench.

Slide43

Experiments with

ultracold bosons on a lattice: finite rate dynamics

2D BEC confined in a trap and in the presence

of an optical lattice.

Single site imaging done by light-assisted collision

which can reliably detect even/odd occupation

of a site. In the present experiment they detect

sites with n=1.

Ramp from the SF side near the QCP to deep inside

the Mott phase in a linear ramp with different

ramp rates.

The no. of sites with odd n displays plateau like

behavior and approaches the adiabatic limit

when the ramp time is increased asymptotically.

No signature of scaling behavior. Interesting

spatial patterns.

W.

Bakr et al. arXiv:1006.0754Slide44

Conclusion

1. We are only beginning to understand the nature of quantum dynamics in some model systems: tip of the iceberg.Many issues remain to settled:

i

) specific calculations

a) dynamics of non-

integrable

systems

b) correlation function and entanglement generation

c) open systems: role of noise and dissipation.

d) applicability of scaling results in complicated realistic systems

3. Issues to be settled: ii) concepts and ideas

a) Role of the protocol used: concept of non-

equlibrium

universality?

b) Relation between complex real-life systems and simple models c) Relationship between

integrability and quantum dynamics.