Annihilation operator phase states

The phase states can also be used in construction of a discrete Wigner function as will be described in Section III. The FD annihilation and creation operators in are defined by... 

Creation and annihilation of phase quanta in FD Hilbert space can be defined in a close analogy to the creation and annihilation of photons, as given by Eq. (8). Phase annihilation, < )v, and phase creation, <[>]., operators can be introduced with the help of the relation <1>V = (j)]4>s. for the Pegg-Barnett phase operator, given by (7). The FD phase annihilation and creation operators in the phase-state basis, have the following form [16]... 

The phase annihilation and creation operators act on the phase states in a similar way (particularly for 0O = 0) as the conventional (photon number) annihilation and creation operators act on number states. 

The operation with this set of input parameters serves to change only the phase of the input state 0y 2 ). The input state ly 1 h) is "annihilated" by this operation [Hong 1987], This means that for that input state the condition of having exactly one horizontally polarized photon in mode 4 never occurs. The NS operation using this particular BS reflectivity is important for a related "quantum filter" protocol [Hofmann 2002],... 

The polarization and quantum phase properties of multipole photons change with the distance from the source. This dependence can be adequately described with the aid of the local representation of the photon operators proposed in Ref. 91 and discussed in Section V.D. In this representation, the photon operators of creation and annihilation correspond to the states with given spin (polarization) at any point. This representation may be useful in the quantum near-field optics. As we know, so far near-field optics is based mainly on the classical picture of the field [106]. 

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