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

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

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

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

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”.

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

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)

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

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

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

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

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

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.

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

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

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

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

The matrix elements are determined from the master equation

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

Dynamics of the state Markovian dynamics Non Markovian dynamics

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.

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

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.

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.

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

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) 285-290. Misbah Qurban, Tasawar Abbas, Rameez-ul- Islam and Manzoor Ikram “Quantum Teleportation of High-Dimensional Atomic Momenta State". Int J Theor Phys. DOI 10.1007/s10773-016-2930-1 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)575-579

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