Aspects of non-equilibrium dynamics in closed quantum systems

K. Sengupta

Indian 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.

Overview 1. 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?

Introduction: Why dynamics

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

2. 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?

3. 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 ?

Nearly adiabatic dynamics: defect production

Landau-Zenner dynamics in two-level systems

Consider a generic time-dependent Hamiltonian for a two level system

The instantaneous energy levelshave an avoided level crossing att=0, where the diagonal terms vanish.

The dynamics of the system can be exactly solved.

The probability of the system tomake a transition to the excited stateof the final Hamiltonian starting fromThe ground state of the initial Hamiltonian can be exactly computed

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 systemacross a quantum critical pointat

QCP

The simplest model to demonstratesuch defect production is

Describes many well-known1D and 2D models.

For adiabatic evolution, the system would stay in theground states of the phases on both sides of the criticalpoint.

Jordan Wigner transformation :

Hamiltonian in term of fermion in momentum space: [J=1]

jjzi

ijjjii

yi

ijjjii

xi

c2c1s

c2c1ccs

c2c1ccs

k

kkkkkk cccckasincckacosg2H

Specific Example: Ising model in transverse field

Ising Model in a transverse field:

Defining

We can write,

where

The eigen values are,

k

kk c

cψ

kkk

k ψHψH

kcosg2k2sink2sinkcosg2

Hk

22k sin(k)cos(k)g2ε

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

Let us now vary g as

k1

2∆εk

g

k1 k2

k2

The probability of defect formation is the probability of the systemto remain at the “wrong” state at the end of the quench.

Defect formation occurs mostly between a finite interval near thequantum critical point.

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,

x

In the Fermion representation, computation of defect probability amounts to solving a Landau-Zenner dynamics for each k. pk : probability of the system to be in the excited state

Total defect :

From Landau Zener problem if the Hamiltonian of the system is

Then

k1

kk

pdk kpn

tε*ΔΔtε

H2

1k

tεtεdtd

2πexpp21

2

k

Scaling law for defect density in a linear quench

For the Ising model

Here pk 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

-

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 wherescaling 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 be

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.

Dn is independent of aand b and hence commutes with HF: Special property of the Kitaev model

Ground statecorresponds toDn=1 on all links.

Solution in momentum space

z z3J 3J

Off-diagonal element

Diagonalelement

J1 J2

Quenching J3 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 J3 lies between(J1+J2) and |J1-J2|. The bands touch each other at special points in the Brillouin zone whose location depend on values of Ji s.

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

J3

Defect density for the Kitaev modelSolve the Landau-Zenner problem corresponding to HF by taking

For slow quench, contribution to nd comes from momenta near the line .

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

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 withz= D=1: Ising, XY,D=2 Extended Kitaevl can be a function of k

Two Possibilities

Quench term vanishesat the QCP.

Novel universal exponentfor scaling of defect densityas 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 rate

Quench term vanishes at the QCP

Schrodinger equation

Scale

Defect probability must be a generic function

Contribution to nd comes whenDk vanishes as |k|.

For a generic critical point with exponents z and

Comparison with numerics of model systems (1D Kitaev)

Plot of ln(n) vs ln(t) for the 1D Kitaev model

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=t0.

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=t0 and |k|=k0.

Effective linear quench with

Ising model in a transverse field at d=1

1015

20

There are two critical points at g=1 and -1

For both the critical points

Expect n to scale as a0.5

Anisotropic critical point in the Kitaev model

J1 J2

J3

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).

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 stillremains 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=2and

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 lossdue 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)

Correlation functions and entanglement generation

Defect Correlation in Kitaev model

The defect correlation as a function of spatial distance r is given in terms of Majoranafermion operators

For the Kitaev model

Plot of the defect correlation function sans the delta function peak for J1=J and Jt =5 as a function of J2=J. Note the change in the anisotropy direction as a function of J2.

Only non-trivial correlator of the model

Entanglement generation in transverse field anisotropic XY model

Quench the magnetic field h from large negative to large positive values. PM PMFM

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 concurrence

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 statesCould 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 respectto one of the spins and check for negative eigenvalues.

Note: For separable density matrices,negativity is zero by construction

Steps:1. Compute the two-body density matrix

2. Compute concurrence and negativity as measures of two-siteentanglement from this density matrix

3. Properties of bipartite entanglement

a. Finite only between even neighbors

b. 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 multipartite

Evolution of entanglement after a quench: anisotropic XY model

Prepare the system in thermal mixed state and change the transverse field to zero fromit’s initial value denoted by a.

The long-time evolution of the system shows ( for T=0) a clear separation into two regimesdistinguished by finite/zero value of log negativity denoted by EN.

The study at finite time shows non-monotonic variation of EN at short time scales while displaying monotonic behavior at longer time scales.

For a starting finite temperature T, the variation of EN could be either monotonic or non-monotonic depending on starting transverse field. Typically non-monotonic behavioris seen for starting transverse field in the critical region.

T=0t=10

T=0

a=0.78t=1

Sen-De, Sen, Lewenstein (2006)

Non-integrable systems: a specific case study

Dynamics of the Bose-Hubbard model Bloch 2001

Transition described by the Bose-Hubbard model:

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 stateMott 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 superfluidity

Beyond this simple picture

Higher order energy calculationby Freericks and Monien: Inclusionof up to O(t3/U3) 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 theory

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 hoppingprocesses of class A perturbatively in J/U.

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 betterthan standard mean-fieldor strong coupling expansion in d=2 and 3.

Allows for straightforward generalization for treatment of dynamics

We were not that lazy……

Non-equilibrium dynamics

Consider a linear ramp of J(t)=Ji +(Jf-Ji) t/t. For dynamics, one needs to solve the Sch. Eq.

Make a time dependent transformation to address the dynamics by projecting onthe instantaneous low-energy sector.

The method provides an accurate descriptionof 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

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 SFphase.

Strong quantum fluctuations near the QCP;justification of going beyond mft.

Experiments

Experimental Systems: Spin one ultracold bosons

Spin one bosons are loaded in an optical trap with mz=0 andsubjected to a Zeeman magnetic field.

Second order QPT at q= 2c2n between the FM (q << 2c2n) and the scalar (q>> 2c2n) phases.

One can study quench dynamics by quenching the magnetic fieldL. 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.

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 collisionwhich can reliably detect even/odd occupation of a site. In the present experiment they detectsites with n=1.

Ramp from the SF side near the QCP to deep insidethe 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:1006.0754

Conclusion1. We are only beginning to understand the nature of quantum dynamics in some model systems: tip of the iceberg.

2. 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.