Transcript Decoherence Versus Disentanglement For Two Qubits In A

Decoherence Versus Disentanglement
for two qubits in a squeezed bath.
M.Orszag ;
M.Hernandez
Facultad de Física
Pontificia Universidad Católica de Chile.
GRENOBLE-JUNE 2009
Outline

Introduction
• Some Previous Concepts
The Problem
 The Model
 Results
 Analysis

Introduction
An important factor is that macroscopic
systems
are
coupled
to
the
environment, and therefore, we are
dealing, in general, with open systems
where the Schrödinger equation is no
longer applicable, or, to put it in a
different way, the coherence leaks out
of the system into the environment, and,
as a result, we have Decoherence.
So, Decoherence is a consequence of
the inevitable coupling of any quantum
system to its environment, causing
information loss from the system to
the environment. In other words, we
consider the decoherence as a nonunitary
dynamics
that
is
a
consequence
of
the
systemenvironment coupling.
Introduction
Quantum
Mechanics
Closed systems
Reversible Dynamics
Unitary dynamics
Open systems
The theory of open quantum systems describes the interaction of a quantum
system with its environment
Reduced density operator
Master Equation
Non-Unitary and
Irreversible dynamics
Entanglement
Entanglement
Suppose we are given a quantum system S, described
by a state vector │Ψ> , that is composed of two
subsystems S1 and S2 ( S is therefore called a bipartite The state vector │Ψ> of S
quantum system).
is called entangled with
respect to S1 and S2 if it
CANNOT be written as a
tensor product of state
vectors of these two
subsystems, i.e., if there
do not exist any state
vectors │Ψ>1 of S1 and
│Φ>2 of S2 such that
Entanglement
Examples
ЄS
│01> Є S
S1 in
and S2 in
S1 in
and S2 in
and
ЄS
Maximally
Entangled
State
Measurement of Entanglement
A popular measure of entanglement is
the Concurrence. This measure
was proposed by Wootters in 1998
and is defined by
where the
are the eigenvalues
(
being the largest one) of a nonHermitian matrix
and
is
defined as:
ρ* being the complex conjugate of ρ
and σy is the usual Pauli matrix. The
concurrence C varies from C=0, for
unentangled state to C=1 for a
maximally entangled state.
Decoherence...
• is a consequence of quantum theory that affects virtually all physical systems.
• arises from unavoidable interaction of these systems with their natural environment
• explains why macroscopic systems seem to possess their familiar classical properties
• explains why certain microscopic objects ("particles") seem to be localized in space.
Decoherence can not explain quantum
probabilities without
(a) introducing a novel definition of
observer
systems
in
quantum
mechanical terms (this is usually done
tacitly in classical terms), and
(b) postulating the required probability
measure (according to the Hilbert
space norm).
Decoherence Free Subspace
Lidar et al. Introduced the term ‘Decoherencefree subspace’, referring to robust states
against perturbations, in the context of
Markovian Master Equations.
One uses the symmetry of the systemenvironment coupling to find a ‘quiet corner’ in
the Hilbert
Space not experiencing this
interaction.
A more formal definition of the DFS is as
follows:
A system with a Hilbert space
is said to have
a decoherence free subspace
if the
evolution inside
is purely unitary.
Simple example of dfs
Collective dephasing
Consider F two-level systems coupled to a
collective bath, whose effect is dephasing
Define a qubit written as

j
a0
j
b1
j
The effect of the dephasing bath over these
states is the following one
0  0
1  exp(i ) 1
Where phi is a random phase
dfs
This transformation can be written as a matrix
0 
1
Rz  

0 exp(i )
0 
1
Rz  

0 exp(i )
Acting on the{|0>,|1>} basis
We assume in this particular example that this
Transformation is collective, implying the same
Phase phi for all the 2-level systems. Now we study the Effect
of the bath over an initial state | >j
The average density matrix over all possible phases with
a probability distribution p()is
dfs
p( ) 
1
2
exp(
)
4
4
Assume the distribution to be a Gaussian, then it is simple to
show that the average density matrix over all phases is
 j   Rz 
j
 Rz  p( )d

| a |2
ab* exp( )
 *

2
a
b
exp(


)
|
b
|


Basically showing an exponential decay of the non
Diagonal elements of the density matrix
Dfs EXAMPLE


Two Particles
In this case we have 4 basis states
0  0  00  00
0  1  01  exp(i ) 01
1  0  10  exp(i ) 10
The states
1  1  11  exp(2i ) 11
Transform with the same phase,so any linear
01 , 10
Combination will have a GLOBAL irrelevant
phase
dfs dim 2{ 01 , 10 }
MODEL DFS
•Consider the Hamiltonian of a system
•(living in a Hilbert space H) interacting with a bath:
H  HS  I B  IS  HB  HI
where
HS , HB , HI
Are the system, bath and system-bath interaction respectively.
The Interaction Hamiltonian can be written quite generally as
H I   S  B
S , B
Are system and bath operators respectively.
(Hamiltonian Approach)

Zanardi et al has shown that that there exists
a set of states in the DFS such that
S k  c k
 , k
These are degenerate eigenvectors of the system
Operators whose eigenvalue depend only on alpha
But not on the state index k
LINDBLAD APPROACH
General Lindblad form of Master Eq
d
i
  H S ,    LD (  )
dt

1 M


LD (  )   ,  1 d ,  ( F , F  F  , F )
2

 

H S System Hamiltonian
F
d , 
Lindblad operators in an M dimensional space
Positive hermitian matrix
DFS condition
(semisimple case
(Fs forming a Lie
Algebra)
F k  0
 , k
Squeezed States
The Hermitian operators X and Y are
now readily seen to be the
amplitudes of the two quadratures of
the field having a phase difference
π/2. The uncertainty relation for the
two amplitudes is
X Y ≥ ¼,
A squeezed state of the radiation field
is obtained if
(Xi)2 < ¼, (i =X o Y)
An ideal squeezed state is obtained if in addition to above eq. the relation
X Y= ¼, also holds.
The Problem...
The Problem
If the environment would act on the various parties the same way it acts on single
system, one would expect that a measure of entanglement, would also decay
exponentially in time. However, Yu and Eberly had showed that under certain
conditions, the dynamics could be completely different and the quantum
entanglement may vanish in a finite time. They called this effect “Entanglement
Sudden Death".
In this work we explore the relation
between the Sudden Death (and
revival) of the entanglement of two
two-level atoms in a common
squeezed bath and the Normal
Decoherence, making use of the
decoherence free subspace (DFS),
which in this case is a twodimensional plane.
The Model
Here, we consider two two-level atoms that interact with a common squeezed
reservoir, and we will focus on the evolution of the entanglement between them,
using as a basis, the Decoherence Free Subspace states.
The master equation, in the Interaction Picture, for a two-level system in a
broadband squeezed vacuum bath is given by
Where
the bath
is the spontaneous emission rate and N,
are the squeeze parameters of
Master
Equation
The Model
It is simple to show that the above master equation can also be written in the Lindblad form
with a single Lindblad operator S.
S  N  1  exp(i ) N 
1 atom
For a two two-level system, the master equation has the same structure, but now
the S operator becomes(common squeezed bath)
2 atoms
, where
The Decoherence Free Subspace for this model was found by M.Orszag and Douglas,
and consists of the eigenstates of S with zero eigenvalue. The states defined in this
way, form a two-dimensional plane in Hilbert Space. Two orthogonal vectors in this
plane are:
DFS
The Model
We can also define the states
and
orthogonal to the
plane:
We solved analytically the master equation by using the
basis.
The various components of the time dependent density matrix depend on the initial state as
well as the squeezing parameters. For simplicity, we assumed
The Model
The Initial
State
In order to study the relation between Decoherence and Disentanglement,
we consider as initial states, superpositions of the form
where
is a variable amplitude of one of the states belonging to the DFS.
We would like to study the effect of varying on the sudden death and
revival times.
Results
Concurrence
For both
and
as initial states, the solution of the Master equation,
written in the standard basis has the following form
one easily finds that the concurrence is given by:
Results
We can also write Ca and Cb in terms of the density matrix
as
Concurrence
in the
basis
Analysis
In both cases, we vary ε between 0 and 1 for a fixed value of the parameter N.
0 ≤ ε < εc
The initial entanglement
decays to zero in a finite
time td
ε = εc
εc =
After a finite period of time
during which concurrence
stays null, it revives at a time
tr reaching asymptotically its
steady state value.
td = tr
Time Evolution of the Concurrence
versus time
ε <εc
ε >εc
Sudden death
And revival
No sudden
death
0.1
Analysis
εc < ε ≤ 1
When εc < ε ≤ 1 , that is when we get “near” the DFS, the
whole phenomena of sudden death and revival disapears for
both initial conditions, and the system shows no
disentanglement sudden death
Analysis
Sudden Death Dissapears
We have Sudden
Death
Entanglement
ε >εGenerated
c
Analysis
Sudden Death Dissapears
ε >εc
Analysis
Another way of seeing the same effect, is
shown in that graphic, where we plot, in the
│Ψ1> case, the SD and SR times versus ε, for
various values of N.
In the case N=0, we notice a steady increase of
the death time up to εc, where the death time
becomes infinite.
On the other hand, for N={0.1, 0.2}, we see
that the effect of the squeezed reservoir is to
increase the disentanglement, and the death
time shows an initial decrease up to the value
And for larger values, it shows a steady
increase, similar to the N=0.
Analysis
The physical explanation of the before effect is the following one:
The
squeezed
vacuum
reservoir has only nonzero
components for an even
number of photons, so the
interaction between the qubits
and the reservoir goes by
pairs of photons.
Now, for a very small N, the average
photon number is also small, so the
predominant interaction with the
reservoir will be with the doubly
excited state via two photon
spontaneous emission.
Analysis
Lets write
in terms of the standard basis:
We see that initially k1 increases with ε, thus
favoring the coupling with the reservoir, or
equivalently, producing a decrease in the death
time. This is up to ε=0.288, where the curve
shows a maxima. (N=0.1)
Beyond this point, k1 starts to decrease and
therefore our system is slowly decoupling from
the bath and therefore the death time shows a
steady increase.
Common Bath Effects


In general, in order to have the atoms in
a common bath, they will have to be
quite near, at a distance no bigger that
the correlation length of the bath. Thus,
one cannot avoid the interaction
between the atoms, which in principle
could affect the DFS
Take for example a dipole-dipole
interaction of the form
Interaction between the atoms
It is interesting to study the effect of this interaction on the DFS
(1  3 cos2  )
 d
,
3
R
R  Distance between atoms(mod)
2

H D  ( 1 2   1 2 )

( 1 2   1 2 ) 1  0

Angle bet. Separation
Between atoms and d

( 1 2   1 2 ) 2   2
A state initially in the DFS STAYS in the DFS
The same conclusion is true for Ising- type interaction
Summary
In summary, we found a simple quantum system
where we establish a direct connection between the
local decoherence property and the non-local
entanglement between two qubits sharing a
common squeezed reservoir.
Finally, the DFS is robust to Ising-like interactions
Decoherence and Disentanglement for two qubits in a
common squeezed reservoir,
M.Hernandez, M.Orszag (PRA,
to appear)
PRA,78,21114(2008)
The End