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A generalized notion of higher order nonclassicality (in terms of higherorder moments) is introduced. Under this generalized framework of higher ordernonclassicality, conditions of higher order squeezing and higher ordersubpoissonian photon statistics are derived. A simpler form of the Hong-Mandelhigher order squeezing criterion is derived under this framework by using anoperator ordering theorem introduced by us in [J. Phys. A. 33 (2000) 5607]. Itis also generalized for multi-photon Bose operators of Brandt and Greenberg.Similarly, condition for higher order subpoissonian photon statistics isderived by normal ordering of higher powers of number operator. Further, withthe help of simple density matrices, it is shown that the higher orderantibunching (HOA) and higher order subpoissonian photon statistics (HOSPS) arenot the manifestation of the same phenomenon and consequently it is incorrectto use the condition of HOA as a test of HOSPS. It is also shown that the HOAand HOSPS may exist even in absence of the corresponding lower orderphenomenon. Binomial state, nonlinear first order excited squeezed state(NLESS) and nonlinear vacuum squeezed state (NLVSS) are used as examples ofquantum state and it is shown that these states may show higher ordernonclssical characteristics. It is observed that the Binomial state which isalways antibunched, is not always higher order squeezed and NLVSS which showshigher order squeezing does not show HOSPS and HOA. The opposite is observed inNLESS and consequently it is established that the HOSPS and HOS are twoindependent signatures of higher order nonclassicality

Generalized structure of higher order nonclassicality