Displaced Fock states and their connection to quasiprobabilities

Abstract

The properties of the displaced Fock states mod a,n) identical to d(a,a*) mod n), (a complex numbers, D(a,a*) displacement operators, n=0,1,2,. . .) are systematically investigated with emphasis on the connections to the Heisenberg-Weyl group and to its irreducible representations. The displaced Fock states comprise the coherent states mod a) identical to mod a,0) as well as the Fock states mod n) identical to mod 0,n) as particular cases. An orthocompleteness relation for the displaced Fock states in the form of the area integral of the operators mod a,m)(a,n mod over the complex a-plane is derived. It generalizes for m=n the well known completeness relation for the coherent states and leads for m not=n to identities expressing the overcompleteness of the displaced Fock states. A basic formula is obtained for the convolution of the operators mod a,m)(a,n mod with the class of Gaussian functions that have exponents proportional to aa*. The connection of the displaced Fock states to the transition operators from the density operator for a single boson mode to quasiprobabilities is studied in general form and specified to the class of transition operators with the displaced Fock states as their eigenstates and connected by convolutions with Gaussian functions to the coherent-state quasiprobability, the Wigner quasiprobability, and the Glauber-Sudarshan quasiprobability.