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Characteristic function, being complex Fourier transform of the Wigner function, admit \begin{equation} \chi(r) = \mathbf{Tr}\left[\rho \hat{D}_{r} \right] \ \ \ , \end{equation} where $\rho$ is density matrix and $\hat{D}_{r}$ is displacement operator. Knowing characteristic function has a lot of advantages, including relative easier calculation of partial trace. This can be seen from the Fourier-Weyl relation, which establishes connection between operator and phase space function, \begin{equation} \rho = \frac{1}{\pi}\int dr \chi(r)\hat{D}_{-r} \ \ \ , \end{equation} and note the partial trace will act on the displacement operator and yield a delta function. This is the motivation for me looking in particular the characteristic function in my research.

The Wigner function of a Fock state $|n\rangle \langle n|$ has been studied, and there are answers already in this site What is the Wigner function of $|n\rangle\langle m|$?. I have been searching through the literature and find that there are not much analytical descriptions of characteristics functions of Fock state $|n\rangle \langle n|$, expect knowing that they are complex Fourier transform of their Wigner functions. While the Wigner function of a Fock state $|n\rangle \langle n|$ is analytical, their complex Fourier transform seems lack a close form solution.

Any help, hint or useful discussions on this topic is appropriated!

Qmechanic
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user1475985
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1 Answers1

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The trick is to compute a generating function for the Wigner functions of the Fock states. This generating function is a Gaussian function and therefore easy to use in calculations such as to compute the characteristic function.

To compute the generating function, we use the original way to compute Wigner functions $$ W(q,p) = \int \langle q+x/2|\hat{\rho}|q-x/2\rangle \exp(-ixp)\ \text{d} x , $$ and insert identities resolved in terms of coherent states $$ \mathbb{1} = \frac{1}{\pi} \int |\alpha\rangle\langle\alpha|\ \text{d} \alpha , $$ on either side of the density operator. It then contains the overlaps $\langle q+x/2|\alpha_1\rangle$ and $\langle\alpha_2|q-x/2\rangle$ which can be evaluated. The overlap with the density operator produces $$ \langle\alpha_1|n\rangle\langle n|\alpha_2\rangle = \exp(-|\alpha_1|^2-|\alpha_2|^2) \frac{\alpha_1^{* n}\alpha_2^n}{n!} . $$ Here, we introduce the generating function $$ \mathcal{G} = \exp(-|\alpha_1|^2-|\alpha_2|^2 +J\alpha_1^*\alpha_2) . $$ It produces the overlap with $$ \frac{1}{n!} \partial_J^n \mathcal{G} |_{J=0} = \langle\alpha_1|n\rangle\langle n|\alpha_2\rangle . $$ Now we substitute this generating function with the other overlaps into the integral and evaluate all the integrals to get the generating function $$ \mathcal{W}(q,p;J) = \frac{2}{1+J} \exp\left(-\frac{1-J}{1+J}\left(q^2+p^2\right)\right) . $$ The individual Wigner functions for the Fock states are produced with $$ W_n(q,p) = \frac{1}{n!} \partial_J^n \mathcal{W}(q,p;J) |_{J=0} . $$ The generating function for the Fock state Wigner functions can now be used to compute a generating function for the characteristic functions of the Fock states simply by evaluating its symplectic Fourier transform.

flippiefanus
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