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Page 1: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 2: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

OutlineIntroduction

• Some Previous Concepts

The ProblemThe ModelResults Analysis

Page 3: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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to the environment, and therefore, we are dealing, in general, with open systems where the Schrödinger equation is no longer applicable, or, to put it in a different way, the coherence leaks out of the system into the environment, and, as a result, we have Decoherence.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.

Page 4: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 5: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 6: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

Entanglement

Maximally

Entangled

State

S1 in S2 inand S1 in and S2 in

Examples

and

│01> Є S Є S

Є S

Page 7: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 8: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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).

Page 9: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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.

Page 10: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 11: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 12: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 13: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 14: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

•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

Page 15: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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)

Page 16: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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)

Page 17: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 18: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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.

Page 19: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 20: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 21: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 22: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 23: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 24: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

We can also write Ca and Cb in terms of the density matrix in the basis as

ResultsConcurrence

Page 25: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 26: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

Time Evolution of the Concurrenceversus time

ε <εc

ε >εc

Sudden deathAnd revival

No sudden death

0.1

Page 27: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 28: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

We have Sudden DeathEntanglement Generatedε >εc

Sudden Death Dissapears

Analysis

Page 29: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

Sudden Death Dissapears

Analysis

ε >εc

Page 30: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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.

Page 31: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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.

Page 32: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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.

Page 33: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 34: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 35: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

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

Page 36: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

Decoherence and Disentanglement for two qubits in a common squeezed reservoir,

M.Hernandez, M.Orszag (PRA, to appear)PRA,78,21114(2008)

Page 37: Decoherence Versus Disentanglement  for two qubits in a squeezed bath .

The End


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