- #1

- 488

- 4

## Main Question or Discussion Point

I seen in some paper, there is an matrix whose element has a square root of number operator, e.g.

[tex]

A = \left(

\begin{matrix}

\alpha & \gamma \sqrt{\hat{a}\hat{a}^\dagger} \\

-\gamma \sqrt{\hat{a}^\dagger\hat{a} & \beta

\end{matrix}

\right)

[/tex]

where [tex]\alpha, \beta, \gamma[/tex] are real number.

What is [tex]A^\dagger[/tex]? Can I write it as the following?

[tex]

A^\dagger = \left(

\begin{matrix}

\alpha & -\gamma \sqrt{\hat{a}^\dagger\hat{a}} \\

\gamma \sqrt{\hat{a}\hat{a}^\dagger & \beta

\end{matrix}

\right)

[/tex]

By the way, if I have it operate on any Fock state, how could the operators in the matrix operating those states?

[tex]

A = \left(

\begin{matrix}

\alpha & \gamma \sqrt{\hat{a}\hat{a}^\dagger} \\

-\gamma \sqrt{\hat{a}^\dagger\hat{a} & \beta

\end{matrix}

\right)

[/tex]

where [tex]\alpha, \beta, \gamma[/tex] are real number.

What is [tex]A^\dagger[/tex]? Can I write it as the following?

[tex]

A^\dagger = \left(

\begin{matrix}

\alpha & -\gamma \sqrt{\hat{a}^\dagger\hat{a}} \\

\gamma \sqrt{\hat{a}\hat{a}^\dagger & \beta

\end{matrix}

\right)

[/tex]

By the way, if I have it operate on any Fock state, how could the operators in the matrix operating those states?