Mathematical Physics: Quantum Hamilton Jacobi, Exceptional Orthogonal Polynomials, Supersymmetric Quantum Mechanics.
Biology : A mathematical model of DNA mutation is proposed using SUSY Quantum Mechanics
Quantum mechanics : The equivalence between Stieltjes electrostatic problem and quantum Hamilton Jacobi (QHJ) formalism. Quantum hydrodynamics Kirchhoff equations.
Random Matrix Theory : Random matrices and its application to Quantum mechanics.
Random matrix applications in quantum information : Inseparability criterion for Gaussian states to the multi-mode Gaussian states using the Marchenko-Pastur theorem.
Optics : Mathematical aspects of Metamaterials such as a new kind of convergence known as Cesaro convergence
is used to define metamaterials. This convergence is demonstrated in Perfect lens, zero index materials.
Quantum optics and Quantum Information:
Applications of master equations in quantum optics:-
In Quantum Mechanics the evolution of the quantum state is described by the Schrodinger Equation
and it is a unitary evolution. In real word it is well known that it is not true. In general, when a
mixed state is appended with an environment evolve unitarily giving rise to dissipation.
A master equation is used to model the dissipation.
The master equation lies at the heart of quantum optics. It describes the evolution of the
dissipative systems. The master equation is used to model decoherence,
entanglement sudden death, has found many applications in quantum optics and open
quantum systems . In quantum information, they are used to model quantum channel
capacities.
features, the solving of master equation is reduced to solving a Schroedinger equation, thus all
the techniques available to solve the Schroedinger equation are applicable here. Hence, the
solving of master equation using TDF gives a very simple and elegant approach.
This methodology is applied to Entanglement in two-site Bose–Hubbard model with non-linear dissipation.
We are looking its applications to Cold atoms and Bose Einstein condensate (BEC).
I have being also interested in, Gisin’s theorem and quantum channel capacities.
Nonlinear coherent states : Interested in the Berry-Phase for coherent states of nonlinear su(2) and su(1, 1) algebras.
Thesis:
Baryogenesis (Cosmology) : The baryon asymmetry of the universe (BAU) problem is a challenging one in the
standard model of cosmology. It is characterised by the ratio η =nb/ nγ , where nb is the number
of baryons and nγ is the number of photons in the universe. The present value of the asymmetry
in the universe is η = 3 × 10−10 .
Three conditions postulated by Sakharov are sufficient to guarantee baryon asymmetry.
These are (i) Baryon number (B) violation, (ii) Charge (C) and Parity (CP) violation, (iii) the
presence of nonequilibrium processes.
There are several models explaining the value of η. I am interested in supersymmetric Affleck-Dine baryogenesis, gravitational baryogenesis and leptogenesis.