1. International Scientific Spring 2016
Disentanglement in two qubit system subjected to dissipative environment :
Exact analysis
Misbah Qurban
Supervisor: Dr. Manzoor Ikram
Pakistan Institute of Engineering and Applied Sciences
National Institute of Lasers and Optronics
International Scientific Spring 2016

2. Outline Objectives Single Atom Dynamics Entanglement
Quantifiying Entanglement
Markovian Process
Non Markovian Process
Entanglement dynamics in Structured reservoir
Conclusion

3. Objectives
To Investigate the entanglement dynamics of close and separated atomic systems under nonmarkovian coupling
To enhance and prolong the entanglement in different environments

4. LOCAL DYNAMICS Two-level system coupled to vacuum
Reservoir
I a >
I b >
(Vacuum)
In the spontaneous decay of two-level atom in vacuum under Markovian approximation

5. Quantum Entanglement
A quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable properties of the systems.
Einstein called this “Spooky action at a distance”.

6. Entangled and separable states
For entangled state
Example
Example
For separable state
it violates locality principle

7. Quantitative measurement of entanglement
(1) Concurrence
λi’s eigenvalues
unentangled states
maximally entangled states
partially entangled states
W. K. Wooters, Phys. Rev. Lett. 80, 2245 (1998)

8. Quantitative measurement of entanglement
(2) Negativity
λi’s eigenvalues
(3) Von- Neumann Entropy

9. System environment interaction
Markovian Process
Nonmarkovian Process
System env. Int, can b categorized in 2 possible ways

10. Markovian Process Weak coupling process No feedback Memory effects are negligible Short correlation time No restoration of superposition
Non Markovian Process
strong coupling process
feedback
Memory effects are not negligible
Flow of energy and information from the system to the environment can be momentarily reversed
Recoherrence and restoration of lost superposition

12. Lorentenzian Spectral density
Leaky cavities
Lorentenzian Spectral density
Bipartite system, trapped in two different cavities, containing structured vacuum reservoir. also there is no direct inTeration between atoms.
This vacuum is created due to interaction of cavities with the vacuum outside

13. Model
The Hamiltonian of the system in interaction picture is
Approximations
Rotating wave approximation
Dipole approximation

14. Single atom dynamics in a leaky cavity
The field inside the cavity is
=spectral width of field distribution
= reservoir correlation time
In strong coupling regime
We focus on the case in which the structured reservoir is the electromagnetic field inside the lossy cavity. It means that the cavity modes can be neglected in favour of an effective spectral density. We consider a case when the atom is interacting resonantly with the cavity field reservoir with Lorentzian spectral density that characterizes the coupling strength of the reservoir to the qubit as.

15. Quantum theory of damping
System reservoir interaction
Density matrix for system
Using Markove and Born approximation

16. Single atom dynamics in a leaky cavity
Wave function of the system
The dynamics obeys
For markovian coupling
The correlation function is
The decay rate for non markovian system

17. Entanglement Dynamics in structured reservoir
Equation of motion for the reduced density matrix
assuming

18. using the basis The X-Matrix
Entanglement dynamics for X type density matrix
where

19. The matrix elements are determined from the master equation

20. Dynamics of the state asymptotic decay
and sudden death of entanglement (SDE) when
is the spontaneous decay of two-level atom

21. Dynamics of the state
Markovian dynamics
Non Markovian dynamics

22. Case I: The initial state Markovian dynamics Non markovian dynamics
maximum amount of entanglement when mixing a=0
Markovian dynamics
Non markovian dynamics
we can see that as a→1 concurrence becomes C(t)=2P²(t)(1-P²(t)). It means that although the state has become separable but it still has some entanglement that depends on P²(t). This only happens due to the memory effects of the environment as comparison to the Markovian case where separable state shows no entanglement.

23. Case II: Markovian dynamics Non markovian dynamics
SDT only when P(t) becomes zero or when mixing a=4(1-(1/(P²(t))))

24. Case III: Non markovian dynamics Markovian dynamics
he plot of concurrence against initial mixing a and time is shown in Fig. 5. When a=0, we have a four equally weighted state and the concurrence is zero. The concurrence increases and SDT decreases as a increases until a=1 where we have maximally entangled state.

25. Case IV: Non markovian dynamics Markovian dynamics
The plot of concurrence against initial mixing a and time is shown in Fig. 6. The concurrence is maximum at a=0 and a=1, at a=0 the doubly excited component is zero, concurrence decreases as a increases. After a value a=(1/2) when both mix states becomes equally weighted, a increases with time until we get another maximally entangled state.

26. Conclusion The dynamics in strong coupling regime is modified
Dynamics are slow
Sudden death time is delayed in each case
Shows oscillatory behavior due to the feedback from the environment

27. Publications
Misbah Qurban, Rabia Tahira, Rameez-ul- Islam and Manzoor Ikram “Disentanglement in a two qubit system subjected to dissipative environment: Exact analysis” Optics Communications 366, (2016)
Misbah Qurban, Tasawar Abbas, Rameez-ul- Islam and Manzoor Ikram “Quantum Teleportation of High-Dimensional Atomic Momenta State". Int J Theor Phys. DOI /s
Tasawar Abbas, Misbah Qurban, Rameez-ul- Islam and Manzoor Ikram. “Engineering distant cavity fields entanglement through Bragg diffraction of neutral atoms” Optics Communications 355,(2015)

28. Thank you!!!