Four-level systems in quatnum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependant Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical , first order differential equations. The unitary integration method is applied to calculate the  time evolution of the density matrix of a four-level system which in this case is an  implementation of two-qubit gates with Josephson junctions.
      Abstract:(for  single photon source using chiral nematic liquid crystal)
                  
                   A good single photon source is important when it comes to implement a linear optical quantum computing circuit or for quantum cryptography. Here we use 4by4 bermann matrix method to  find out if chiral nemtic liquid crystal, which is a natural photonic band gap material, can be used to design a omnidirectional, on demand single photon source.