A new optical field generated as an output of the displaced Fock state in an amplitude dissipative channel

We propose a new optical field and show that such an optical field can be generated as an output of a displaced Fock state in an amplitude dissipative channel. We derive new generating function formulas and binomial formula involving two-variable Hermite polynomials to reach this result. The photon number average in this new optical field is (m+|α|²)e^-2κt, which indicates that controlling the photon number can be realized by adjusting the value of m or |α|² or κ. The time evolution law of displaced Fock state in a thermo reservoir is thus revealed.