Transcript Why dynamics?

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 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 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 system
across a quantum critical point
at
The simplest model to demonstrate
such defect production is
QCP
For adiabatic evolution, the
system would stay in the
ground states of the phases
on both sides of the critical
point.
Describes many well-known
1D and 2D models.
Specific Example: Ising model in transverse field
Ising Model in a transverse field:
Jordan Wigner transformation :

s ix  ci  ci

s iy  ci  ci
 1  2c c 

j
j
ji
 1  2c c 

j
j
ji
s iz  1  2c j c j
Hamiltonian in term of fermion in momentum space: [J=1]


H   2g  coska c k c k  sin ka  c k c  k  c k c k
k

Defining
We can write,
 ck 
ψ k    
 c k 
H   ψ k H k ψ k
k
where
2sin k  
 2g  cosk 

H k  
 2g  cosk 
 2sin k 
The eigen values are,
ε k  2
g  cos(k)  sin(k)  
2
2
The energy gap vanishes at g=1 and k=0: Quantum Critical point.
Let us now vary g as
1k 
2k 
εk
2∆
1k 
2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.
 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
Scaling law for defect density in a linear quench
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 1k 
Total defect :
n   pk    dk p k
k
From Landau Zener problem
if the Hamiltonian of the system is
Then
 ε1 t  Δ 

H k  
 Δ * ε 2 t 




 2π2

p k  exp 
 d ε t   ε t  


1
2
 dt

For the Ising model
Here pk is maximum when sin(k) = 0
Linearising about k = 0 and making a transformation of variable
Scaling of defect density
For a critical point with arbitrary
dynamical and correlation length
exponents, one can generalize
Ref: A Polkovnikov, Phys. Rev. B 72, 161201(R) (2005)
-
Generic critical points: A phase space argument
The system enters the impulse region when
rate of change of the gap is the same order
as 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 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 a
and b and hence
commutes with HF:
Special property of
the Kitaev model
Ground state
corresponds to
Dn=1 on all links.
Solution in momentum space
Off-diagonal
element
Diagonal
element
z
J 3  
z
J3  
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.
J1
J2
J3
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
Quenching J3 linearly
takes the system
through a critical line in
parameter space and
hence through the line
in momentum space.
For quench through critical point: m=d
Question: How would the defect density scale with quench rate?
Defect density for the Kitaev model
Solve 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 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)
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 rate
Quench term vanishes at the QCP
Schrodinger equation
Scale
Defect probability must be a
generic function
Contribution to nd comes when
Dk 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 of
g as a power law in time
such that the QCP is reached
at 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
There are two critical points
at g=1 and -1
For both the critical points
Expect n to scale as a0.5
15
10
20
Anisotropic critical point in the Kitaev model
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.
J1
J2
J3
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 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
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 QCP
and 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 and
Polkovnikov , 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 Majorana
fermion operators
Only non-trivial correlator of the model
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.
Entanglement generation in transverse field anisotropic XY model
Quench the magnetic field h from
large negative to large positive values.
-1
PM
1
FM
h
PM
One can compute all correlation functions for this dynamics in this model. (Cherng and Levitov).
No non-trivial correlation between
the odd neighbors.
Single-site entanglement:
the linear entropy or the
Single site concurrence
What’s the bipartite entanglement generated
due to the quench between spins at i and i+n?
Measures of bipartite entanglement in spin ½ systems
Concurrence (Hill and Wootters)
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
Negativity (Peres)
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 construction
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
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
T=0
T=0
t=10
a=0.78
t=1
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 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 behavior
is seen for starting transverse field in the critical region.
Sen-De, Sen, Lewenstein (2006)
Non-integrable systems: a specific case study
Dynamics of the Bose-Hubbard model
Transition described by the
Bose-Hubbard model:
Bloch 2001
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 superfluidity
Beyond this simple picture
Higher order energy calculation
by Freericks and Monien: Inclusion
of up to O(t3/U3) virtual processes.
Mean-field theory (Fisher 89,
Seshadri 93)
Quantum Monte Carlo studies for
2D & 3D systems: Trivedi and Krauth,
B. Sansone-Capponegro
Phase diagram for n=1 and d=3
O(t2/U2) theories
MFT
Superfluid
Predicts a quantum phase
transition with z=2 (except at
the tip of the Mott lobe where
z=1).
Mott
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
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.
(A)
(B)
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 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 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
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.
Experiments
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= 2c2n
between the FM (q << 2c2n) and
the scalar (q>> 2c2n) 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.
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:1006.0754
Conclusion
1. 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.