Abstract
In this paper, we present numerical schemes for evaluating the matrix elements of Gaussian/non-Gaussian operators in the Fock state basis, which are identified as multivariate Hermite polynomials (MHPs). Using the integral transformation operator to perform the multimode Bogoliubov transformation, Husimi’s Q-functions of Gaussian/non-Gaussian operators are easily derived as the generating functions of MHPs.
Highlights
Multivariate Hermite polynomials (MHPs) play important roles in various field of research, including quantum optics [1, 2] and molecular spectroscopy [3, 4]
Willink [5] showed that the MHPs could be converted to multivariate Gaussian moments (MGMs)
We connect the matrix elements of non-Gaussian operators in the Fock state basis to the MHPs so that the evaluation can be conducted in an equal footing as the Gaussian case
Summary
Multivariate Hermite polynomials (MHPs) play important roles in various field of research, including quantum optics [1, 2] and molecular spectroscopy [3, 4]. We connect the matrix elements of non-Gaussian operators in the Fock state basis to the MHPs so that the evaluation can be conducted in an equal footing as the Gaussian case. [25] throughout the paper to obtain Husimi’s Q-function, which is used as a generating function of the MHPs for the Gaussian/nonGaussian matrix elements in the Fock state basis. With the aid of the integral operator method proposed by Fan et al, the explicit parameter dependence on the multimode Bogoliubov transformation matrices is made in Husimi’s Q-function. Our work is presented as follows: The relationship between the matrix elements of Gaussian/non-Gaussian operators in the Fock state basis, the MHPs, and the multivariate Gaussian distribution is given first (some contents are taken from the PhD thesis (Ch. 3) of the author [11] for the following section). Husimi’s Q-functions are presented as complex Gaussian functions via the integral operator method [25, 26]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
