1. 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

2. Outline Introduction Some Previous Concepts The Problem The Model Results Analysis

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to the environment, and therefore, we are dealing, in general, with open systems where the Schrödinger equation is no longer applicable, or, to put it in a different way, the coherence leaks out of the system into the environment, and, as a result, we have Decoherence.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

4. Introduction Quantum Mechanics The theory of open quantum systems describes the interaction of a quantum system with its environment Reduced density operatorMaster Equation Non-Unitary and Irreversible dynamics Unitary dynamics Reversible Dynamics Closed systems Open systems

5. Entanglement Suppose we are given a quantum system S, described by a state vector │Ψ>, that is composed of two subsystems S 1 and S 2 ( S is therefore called a bipartite quantum system). Entanglement The state vector │Ψ> of S is called entangled with respect to S 1 and S 2 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 S 1 and │Φ> 2 of S 2 such that

6. Entanglement Maximally Entangled State S 1 in S 2 inand S 1 in and S 2 in Examples and │01> Є S Є S

7. 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 σ y 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

8. 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).

9. 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.

10. 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 Simple example of dfs

11. dfs This transformation can be written as a matrix 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

12. 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

13. Dfs EXAMPLE Two Particles In this case we have 4 basis states The states Transform with the same phase,so any linear Combination will have a GLOBAL irrelevant phase

14. 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. MODEL DFS

15. 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 (Hamiltonian Approach)

16. 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)

17. 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 (  X i ) 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

18. 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.

19. 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

20. 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 Model Master Equation 1 atom 2 atoms

21. 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 DFS

22. 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 Model The Initial State

23. 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 Concurrence

24. We can also write Ca and Cb in terms of the density matrix in the basis as Results Concurrence

25. Analysis 0 ≤ ε < ε c 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 t d After a finite period of time during which concurrence stays null, it revives at a time t r reaching asymptotically its steady state value. ε = ε c εc =εc = t d = t r

26. Time Evolution of the Concurrence versus time ε <εcε <εc ε >εcε >εc Sudden death And revival No sudden death 0.1

27. Analysis ε c < ε ≤ 1 When ε c < 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

28. We have Sudden Death Entanglement Generatedε >εcε >εc Sudden Death Dissapears Analysis

29. Sudden Death Dissapears Analysis ε >εcε >εc

30. Another way of seeing the same effect, is shown in that graphic, where we plot, in the │Ψ 1 > 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 ε 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 value And for larger values, it shows a steady increase, similar to the N=0.

31. 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.

32. Lets write in terms of the standard basis: Analysis We see that initially k 1 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, k 1 starts to decrease and therefore our system is slowly decoupling from the bath and therefore the death time shows a steady increase.

33. 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

34. 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

35. 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

36. Decoherence and Disentanglement for two qubits in a common squeezed reservoir, M.Hernandez, M.Orszag (PRA, to appear) PRA,78,21114(2008)

37. The End