Decoherence Versus Disentanglement for two qubits in a
squeezed bath.
Facultad de Física
Pontificia Universidad Católica de Chile.
M.Orszag ; M.Hernandez
GRENOBLE-JUNE 2009
OutlineIntroduction
• Some Previous Concepts
The ProblemThe ModelResults Analysis
IntroductionAn 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to the environment, and therefore, we are dealing, in general, with open systems where the Schrödinger equation is no longer applicable, or, to put it in a different way, the coherence leaks out of the system into the environment, and, as a result, we have Decoherence.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.
Introduction Quantum
Mechanics
The theory of open quantum systems describes the interaction of a quantum system with its environment
Reduced density operator Master EquationNon-Unitary and Irreversible dynamics
Unitary dynamics
Reversible Dynamics
Closed systems
Open systems
EntanglementSuppose 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 quantum system).
Entanglement
The state vector │Ψ> 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 │Ψ>1 of S1 and │Φ>2 of S2 such that
Entanglement
Maximally
Entangled
State
S1 in S2 inand S1 in and S2 in
Examples
and
│01> Є S Є S
Є S
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:
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.
Measurement of Entanglement
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 ‘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.
Collective dephasingConsider 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
Simple example of dfs
jjjba 10
1)exp(1
00
i
dfs
)exp(0
01
iRz
This transformation can be written as a matrix
Acting on the{|0>,|1>} basisWe assume in this particular example that thisTransformation 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 witha probability distribution p()is
)exp(0
01
iRz
dfs
)4
exp(4
1)(
2
p
Assume the distribution to be a Gaussian, then it is simple to show that the average density matrix over all phases is
2*
*2
||)exp(
)exp(||
)(
bba
aba
dpRR zjzj
Basically showing an exponential decay of the nonDiagonal elements of the density matrix
Dfs EXAMPLE
Two Particles In this case we have 4 basis states
}10,01{2dim
10,01
11)2exp(1111
10)exp(1001
01)exp(0110
000000
dfs
i
i
i
The statesTransform with the same phase,so any linearCombination will have a GLOBAL irrelevant
phase
•Consider the Hamiltonian of a system •(living in a Hilbert space H) interacting with a bath:
IBSBS HHIIHH
where IBS HHH ,,
Are the system, bath and system-bath interaction respectively.The Interaction Hamiltonian can be written quite generally as
BSH I
BS , Are system and bath operators respectively.
MODEL DFS
Zanardi et al has shown that that there exists a set of states in the DFS such that
k
kckS
,
These are degenerate eigenvectors of the systemOperators whose eigenvalue depend only on alpha
But not on the state index k
(Hamiltonian Approach)
M
D
DS
FFFFdL
LHi
dt
d
1, , ),,(2
1)(
)(,
LINDBLAD APPROACH
General Lindblad form of Master Eq
,d
F
H SSystem Hamiltonian
Lindblad operators in an M dimensional space
Positive hermitian matrix
k
kF
,
0
DFS condition(semisimple case(Fs forming a Lie
Algebra)
A squeezed state of the radiation field is obtained if
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
(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.
X Y ≥ ¼,
Squeezed States
The ProblemThe 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.
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
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.
For a two two-level system, the master equation has the same structure, but now the S operator becomes(common squeezed bath)
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:
, where
The ModelMaster
Equation
NiNS )exp(1
1 atom
2 atoms
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 ModelDFS
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.
The ModelThe Initial State
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:
ResultsConcurrence
We can also write Ca and Cb in terms of the density matrix in the basis as
ResultsConcurrence
Analysis
0 ≤ ε < εc
In both cases, we vary ε between 0 and 1 for a fixed value of the parameter N.
The initial entanglement decays to zero in a finite time td
After a finite period of time during which concurrence stays null, it revives at a time tr reaching asymptotically its steady state value.
ε = εc
εc =
td = tr
Time Evolution of the Concurrenceversus time
ε <εc
ε >εc
Sudden deathAnd 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
We have Sudden DeathEntanglement Generatedε >εc
Sudden Death Dissapears
Analysis
Sudden Death Dissapears
Analysis
ε >εc
AnalysisAnother 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.
Lets write in terms of the standard basis:
Analysis
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
222121
12121
2121
)(
0)(
)(
DH
It is interesting to study the effect of this interaction on the DFS
RR
d ,)cos31(
3
22
Distance between atoms(mod)
Angle bet. Separation Between atoms and d
A state initially in the DFS STAYS in the DFSThe same conclusion is true for Ising- type interaction
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
Summary
Decoherence and Disentanglement for two qubits in a common squeezed reservoir,
M.Hernandez, M.Orszag (PRA, to appear)PRA,78,21114(2008)
The End