Continuous-variable cat state - phonon number mean and variance

[matteo137]

Homework Statement

calculate the mean and variance of the number operator for a coherent-superposition of two coherent states (cat state)

Relevant Equations

$$(\sqrt{2(1+e^{-2\alpha^2})})^{-1}(\vert\alpha\rangle+\vert -\alpha\rangle)$$

I found the mean to be $$\langle n\rangle=\vert\alpha\vert^2 \tanh(\alpha^2)$ and $\langle n^2\rangle=\vert\alpha\vert^2 \left( \alpha^2\sech(\alpha^2)^2 + \tanh(\alpha^2) \right)$$.

Do you know if there is any reference where I can check if this is correct?

 

[Nate810]

I am not aware of any direct reference that contains the exact expressions you have provided. However, there are several references that discuss photon number distributions and/or moments in the context of coherent states. In particular, Ref. 1 discusses the first and second moments of a coherent state in the context of Heisenberg's uncertainty relation. Ref. 2 provides an example of computing the average photon number for a coherent state in the context of the Glauber-Sudarshan P representation. Additionally, Ref. 3 provides a more general discussion on computing the moments of photon number in arbitrary probability distributions, including the case of coherent states. References1. B. M. Garraway, “The Density Matrix and Uncertainty in Coherent States,” J. Mod. Opt. 48, 1743–1751 (2001).2. J. J. Sakurai, Modern Quantum Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1994), p. 230.3. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997), p. 255.