Generalized Q-functions

Smith, T.B. (2006). Generalized Q-functions. Journal of Physics A: Mathematical and General, 39(44) pp. 13747–13756.



The modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter lambda specifies orderings in a mapping from the operator psi)(sigma to the corresponding phase space wavefunction, where sigma is a given fiducial vector. The choice lambda = 0 specifies the Weyl mapping and the Q-function so obtained is the usual one when sigma is the vacuum state. More generally, any choice of of lambda in the range (-1,1) corresponds to orderings varying between standard and anti-standard. For all such orderings the generalized Q-functions are non-negative by construction. They are shown to be proportional to expectation of the system state rho with respect to a generalized displaced squeezed state which depends on lambda and position (p,q) in phase space. Thus, when a system has been prepared in the state rho, a generalized Q-function is proportional to the probability of finding it in the generalized squeezed state. Any such Q-function can also be written as the smoothing of the Wigner function for the system state rho by convolution with the Wigner function for the generalized squeezed state.

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