Decoherence Versus Disentanglement for two qubits in a squeezed bath.

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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
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Outline

• Introduction
• Some Previous Concepts
• The Problem
• The Model
• Results
• Analysis

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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the Schrödinger equation is no longer applicable,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 non-unitary
dynamics that is a consequence of the
system-environment coupling.
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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
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Entanglement
Entanglement
Suppose we are given a quantum system S,
described by a state vector ?gt , that is
composed of two subsystems S1 and S2 ( S is
therefore called a bipartite quantum system).
The state vector ?gt of S 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 ?gt1 of S1 and Fgt2 of S2 such that
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Entanglement
Examples
? S
01gt ? S
and
S1 in
S1 in
S2 in
and
and S2 in
? S
Maximally Entangled State
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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 non-Hermitian matrix
and is defined as
? being the complex conjugate of ? and sy is the
usual Pauli matrix. The concurrence C varies from
C0, for unentangled state to C1 for a maximally
entangled state.
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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
• introducing a novel definition of observer
  systems in quantum mechanical terms (this is
  usually done tacitly in classical terms), and
• postulating the required probability measure
  (according to the Hilbert space norm).

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Decoherence Free Subspace
Lidar et al. Introduced the term
Decoherence-free subspace, referring to robust
states against perturbations, in the context of
Markovian Master Equations. One uses the symmetry
of the system-environment 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.
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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 The
effect of the dephasing bath over these states
is the following one Where phi is a random
phase
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dfs
This transformation can be written as a
matrix Acting on the0gt,1gt 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 gtj The average density matrix over all
possible phases with a probability distribution
p()is
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dfs
Assume the distribution to be a Gaussian, then it
is simple to show that the average density
matrix over all phases is
Basically showing an exponential decay of the
non Diagonal elements of the density matrix
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Dfs EXAMPLE

• Two Particles
• In this case we have 4 basis states

Transform with the same phase,so any
linear Combination will have a GLOBAL irrelevant
phase
The states
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MODEL DFS

• Consider the Hamiltonian of a system
• (living in a Hilbert space H) interacting with a
  bath

where
Are the system, bath and system-bath interaction
respectively. The Interaction Hamiltonian can be
written quite generally as
Are system and bath operators respectively.
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(Hamiltonian Approach)

• Zanardi et al has shown that that there exists a
  set of states in the DFS such that

These are degenerate eigenvectors of the
system Operators whose eigenvalue depend only on
alpha But not on the state index k
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LINDBLAD APPROACH
General Lindblad form of Master Eq
System Hamiltonian
Lindblad operators in an M dimensional space
Positive hermitian matrix
DFS condition (semisimple case (Fs forming a Lie
Algebra)
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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 p/2. The
uncertainty relation for the two amplitudes is
?X ?Y ¼,
A squeezed state of the radiation field is
obtained if
(?Xi)2 lt ¼, (i X o Y)
An ideal squeezed state is obtained if in
addition to above eq. the relation ?X
?Y ¼, also holds.
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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 two-dimensional plane.
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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 is the spontaneous emission rate and
N, are the squeeze parameters of the bath
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The Model
Master Equation
It is simple to show that the above master
equation can also be written in the Lindblad form
with a single Lindblad operator S.
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
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The Model
DFS
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
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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.
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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
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Results
Concurrence
We can also write Ca and Cb in terms of the
density matrix in the basis as
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Analysis
In both cases, we vary e between 0 and 1 for a
fixed value of the parameter N.
After a finite period of time during which
concurrence stays null, it revives at a time tr
reaching asymptotically its steady state value.
The initial entanglement decays to zero in a
finite time td
0 e lt ec
e ec
td tr
ec
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Time Evolution of the Concurrenceversus time
Sudden death And revival
e ltec
e gtec
No sudden death
0.1
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Analysis
ec lt e 1
When ec lt e 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
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Analysis
Sudden Death Dissapears
We have Sudden Death
Entanglement Generated
e gtec
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Analysis
Sudden Death Dissapears
e gtec
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Analysis
Another way of seeing the same effect, is shown
in that graphic, where we plot, in the ?1gt case,
the SD and SR times versus e, for various values
of N.
In the case N0, we notice a steady increase of
the death time up to ec, where the death time
becomes infinite.
On the other hand, for N0.1, 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 N0.
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Analysis
The physical explanation of the before effect is
the following one
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.
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.
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Analysis
Lets write in terms of the standard
basis
We see that initially k1 increases with e, thus
favoring the coupling with the reservoir, or
equivalently, producing a decrease in the death
time. This is up to e0.288, where the curve
shows a maxima. (N0.1) 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.
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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

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Interaction between the atoms
It is interesting to study the effect of this
interaction on the DFS
Distance between atoms(mod)
Angle bet. Separation Between atoms and d
A state initially in the DFS STAYS in the DFS The
same conclusion is true for Ising- type
interaction
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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
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Decoherence and Disentanglement for two qubits in
a common squeezed reservoir, M.Hernandez,
M.Orszag (PRA, to appear)
PRA,78,21114(2008)
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The End