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Quantum Networks Via Quantum Optical Systems

Final Report Summary - QUANTNET (Quantum Networks Via Quantum Optical Systems)

According to the QUANTNET project objectives the following research activities have been accomplished:
- Analysis on dephasing in simple quantum systems based on analytic solution of the Bloch equation for the two-state Demkov model. Up to Dr.Vasilev knowledge there are very few exact solutions for the Bloch equation. The complexity of this problem ensues from the difficulty of deriving an exact solution for third order linear differential equations. For the two-state Demkov model an exact solution of the Bloch equation is derived in terms of generalized hypergeometric function (GHF). The exact solution allows for detailed analysis of the dephasing mechanism. A various analytical relations have been derived between the original two-state Demkov model and the solution for the Bloch equation for the Demkov model.
- Analysis on adiabatic and superadiabatic analytic methods and more specifically the Dykhne-Davis-Pechukas (DDP) approach. The accuracy of the approximate solutions derived with DDP approach has been widely discussed in DDP related articles. This was motivation for using DDP methods as adiabatic optimization techniques. According to the project objectives, DDP approach is examined as an optimization method in the second adiabatic basis, namely as a superadiabatic optimization technique.
- Analytic factorization methods and their applications to coherent quantum control have been studied based on the Morris-Shore (MS) transformation. This transformation is a powerful tool for studying multi-level quantum systems via factorization to set of two-state problems and set decouple states. Under the project objectives the MS transformation is generalized to handle the problems where the two subsets of levels have unequal detunings. For this generalized MS problem has been found perturbative solution under constrain of week detuning degeneracy.
-Analytic properties of coherent excitation of specific two-state quantum system have been investigated, where the Lorentzian model (pulsed excitation with Lorentzial temporal profile) has been studied. At present quantum information uses more complicate systems then two-level; nevertheless the simple two-state system is fundamental ingredient for the theory. Under the project objectives coherent excitations of two-level system with pulsed field with Lorentzian temporal profile, have been studied. For this problem has been pointed a way for exact analytical solution in terms of confluent Heun functions (CHF). Due to analytical limitation related to CHF an approximate solution has been also derived using Dykhne-Davis-Pechukas (DDP) approach.
- Under the general theory of stochastic processes and their analytical treatment, exact solution for the Heston model with time-dependent parameters has been derived. This model is primarily associated with stochastic framework of financial mathematics; nevertheless the Heston model is among the most popular stochastic processes. Exact solution to the generalized Heston model has been derived, where the model parameters are assumed to have linear time dependence. The solution for the model in expressed in terms of confluent hypergeometric functions.

According to the research work and research objectives the project has significant impact on the scientific collaborations between the Host institutions - Sofia University and other research groups.
Furthermore research works related to project objectives not only contributes to the scientific knowledge in the related fields but also have a strong potential to influence and give rise to further scientific studies.

According to the project plan, scientific results have been reported in relation to the transfer of knowledge to the Host Institution:
- Genko S. Vasilev, talk at Theoretical Physics Division; work-seminar, Sofia June 2011;
- Genko S. Vasilev, talk at Theoretical Physics Division; work-seminar, Sofia June 2012;
Scientific research motivated by the project has lead to creation of lecture curse devoted to common stochastic methods in financial mathematics and physics - Econophysics. The lecture curse Econophysics is intended for students in the Master's program at the Department of Theoretical Physics at Sofia University. Following the scientific ideas enshrined in the project plan, another lecture course dedicated to stochastic differential equations (SDE) will be announced in this calendar year. SDE will be also intended for students in the Master's program at the Department of Theoretical Physics at Sofia University.

The QUANTNET project was necessary step towards making QIP and stochastic processes among the main research areas in the Theoretical Physics Division of Sofia University. Since these are among the most recent and attractive areas in modern science, the research activities related to the project have significant impact of attracting more talented young people towards science;

Moreover, the research works devoted to stochastic processes have attracted attention beyond the academic community. Beyond the project objectives, collaborations with private companies, which are among the few in Bulgaria dealing with applications of stochastic modelling within the aria of financial derivatives, has been initiated.

From personal point of view Dr. Vasilev greatly benefited from the opportunity to continue his research ideas. Furthermore, the scientific work motivated by the project objectives have led to interdisciplinary work related to various stochastic processes. The results of these studies are beyond the time-frame of the project and are yet to come. The project has been stimulating the carrier development of Dr.Vasilev and as a result at preset time he holds permanent position as Assistant Professor at Sofia University.