Thermofield Dynamics and its Applications to Quantum Optics

Abstract

Thermofield dynamics (TFD) arose out of efforts to define operators at finite temperatures in the theory of superconductivity.1 Since then it has developed into a powerful formalism for dealing with quantum field theories at finite temperatures2-4 and has found numerous applications in non-equilibrium phenomena5 and in quantum optics.6-15 The central idea in TFD is to represent a density operator as a vector in an extended Hilbert space. The same idea, in fact, also forms the basis of another formalism known as the Liouville space representation.16-19 The difference between the two formalisms is as follows. Consider a density operator ρ on a Hilbert space ℌ. Any operator on ℌ and, in particular, the density operator ρ can be expanded in terms of the operators |M > constitutes a complete orthonormal basis in ℌ. In TFD the operators |M > . In Liouville space representation,16-19 on the other hand, |M >< N| are regarded as the basis vectors in the Hilbert space ℌ of linear operators on ℌ. In what follows, we shall confine ourselves to the TFD representation and consider its application to quantum optics. In particular, we shall highlight its usefulness for exact or approximate solution of master equations encountered in quantum optics.