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
Jan 23, 2016
Decoherence Versus Disentanglement for two qubits in a squeezed bath.Facultad de FsicaPontificia Universidad Catlica de Chile.M.Orszag ; M.HernandezGRENOBLE-JUNE 2009
OutlineIntroductionSome 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to the environment, and therefore, we are dealing, in general, with open systems where the Schrdinger equation is no longer applicable, or, to put it in a different way, the coherence leaks out of the system into the environment, and, as a result, we have Decoherence.So, Decoherence is a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.
IntroductionQuantumMechanicsThe theory of open quantum systems describes the interaction of a quantum system with its environmentReduced density operatorMaster EquationNon-Unitary and Irreversible dynamicsUnitary dynamicsReversible DynamicsClosed systemsOpen systems
EntanglementSuppose we are given a quantum system S, described by a state vector > , that is composed of two subsystems S1 and S2 ( S is therefore called a bipartite quantum system).EntanglementThe state vector > 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 >1 of S1 and >2 of S2 such that
EntanglementMaximallyEntangledStateS1 in S2 inand S1 inand S2 in Examplesand01> S S S
Measurement of EntanglementA popular measure of entanglement is the Concurrence. This measure was proposed by Wootters in 1998 and is defined bywhere the are the eigenvalues ( being the largest one) of a non-Hermitian matrix and is defined as:* being the complex conjugate of and y 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 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).
Decoherence Free SubspaceLidar 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.
Simple example of dfsCollective dephasingConsider F two-level systems coupled to a collective bath, whose effect is dephasingDefine a qubit written as
The effect of the dephasing bath over these states is the following one
Where phi is a random phase
dfs
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 samePhase phi for all the 2-level systems. Now we study the Effect of the bath over an initial state | >jThe average density matrix over all possible phases witha probability distribution p()is
dfsAssume the distribution to be a Gaussian, then it is simple to show that the average density matrix over all phases isBasically showing an exponential decay of the nonDiagonal elements of the density matrix
Dfs EXAMPLETwo ParticlesIn this case we have 4 basis states
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: where Are the system, bath and system-bath interaction respectively.The Interaction Hamiltonian can be written quite generally asAre system and bath operators respectively.MODEL DFS
Zanardi et al has shown that that there exists a set of states in the DFS such thatThese are degenerate eigenvectors of the systemOperators whose eigenvalue depend only on alphaBut not on the state index k (Hamiltonian Approach)
LINDBLAD APPROACHGeneral Lindblad form of Master EqSystem HamiltonianLindblad operators in an M dimensional spacePositive hermitian matrixDFS 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 /2. The uncertainty relation for the two amplitudes is (Xi)2 < , (i =X o Y)An ideal squeezed state is obtained if in addition to above eq. the relation X Y= , also holds. X Y ,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.
The ModelHere, 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
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:
, whereThe ModelMaster Equation1 atom2 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 formwhere 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 asResultsConcurrence
Analysis0 < cIn both cases, we vary between 0 and 1 for a fixed value of the parameter N.The initial entanglement decays to zero in a finite time tdAfter a finite period of time during which concurrence stays null, it revives at a time tr reaching asymptotically its steady state value. = cc = td = tr
Time Evolution of the Concurrenceversus time cSudden deathAnd revivalNo sudden death0.1
Analysis c < 1When c < 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 Generated >cSudden Death DissapearsAnalysis
Sudden Death DissapearsAnalysis >c
AnalysisAnother way of seeing the same effect, is shown in that graphic, where we plot, in the 1> case, the SD and SR times versus , for various values of N. In the case N=0, we notice a steady increase of the death time up to c, 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 valueAnd for larger values, it shows a steady increase, similar to the N=0.
AnalysisThe 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:AnalysisWe see that initially k1 increases with , thus favoring the coupling with the reservoir, or equivalently, producing a decrease in the death time. This is up to =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 EffectsIn 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 DFSTake for example a dipole-dipole interaction of the form
Interaction between the atomsIt is interesting to study the effect of this interaction on the DFSDistance 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
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
.In both cases, we vary between 0 and 1 for a fixed value of the parameter N. We observe that for the interval the initial entanglement decays to zero in a finite time, td.
After a finite period of time during which the concurrence stays null, it revives at a time tr reaching asymptotically its steady state value.