Single-Photon Added Coherent States

[doublemint]

Hello!
So this is my question:
The single-photon added coherent states (SPACS) are obtained from the action of a creation operator onto to a coherent state. $$\left|\alpha,1\right\rangle$$ = $$\hat{a}^{\dagger}\left|\alpha\right\rangle$$

So I need to find the decomposition of this state into number basis and the wavefunction of SPACS for a real $$\alpha$$.
$$a^{\dagger}$$ is the creation operator.
a is the annihilation operator.
The first part:
$$|\alpha> = \sum \alpha_{n}|n>$$
$$a|\alpha> = \sum \alpha_{n}\sqrt{n}|n-1>$$
$$a^{\dagger} a |\alpha> = \sum \alpha_{n}\sqrt{n}\sqrt{n-1}|n>$$

The second part:
This is the one that I don't understand...
so far I this:
$$a^{\dagger}|\alpha> =\frac{1}{\sqrt{2}}(X-\frac{d}{dX})\Psi_{\alpha}(x)=(\frac{d}{d\alpha}+\frac{\alpha}{2})\Psi_{alpha}(x)$$
$$(X-\frac{\alpha \sqrt{2}}{2}-\frac{d}{dX}-\frac{d}{d\alpha})\Psi_{\alpha}(x)=0$$

Any help would be appreciated!
Thank You
DoubleMint

 

[mark726]

, thank you for your question. To find the decomposition of the single-photon added coherent state (SPACS) into the number basis, we can use the following expression:

\left|\alpha,1\right\rangle = \hat{a}^{\dagger}\left|\alpha\right\rangle = \sum_{n=0}^{\infty} \alpha_n \sqrt{n+1} \left|n+1\right\rangle

where \alpha_n = \langle n|\alpha\rangle is the coefficient of the number state \left|n\right\rangle in the coherent state \left|\alpha\right\rangle. Using this expression, we can find the wavefunction of the SPACS for a real \alpha by substituting the expression for \alpha_n into the above equation:

\left|\alpha,1\right\rangle = \sum_{n=0}^{\infty} \langle n|\alpha\rangle \sqrt{n+1} \left|n+1\right\rangle = \sum_{n=0}^{\infty} \Psi_{\alpha}(x) \sqrt{n+1} \left|n+1\right\rangle

where \Psi_{\alpha}(x) = \langle x|\alpha\rangle is the wavefunction of the coherent state. We can then use the number basis \left|n\right\rangle to express the wavefunction of the SPACS as:

\Psi_{\alpha,1}(x) = \sum_{n=0}^{\infty} \Psi_{\alpha}(x) \sqrt{n+1} \langle x|n+1\rangle

To find the expression for \langle x|n+1\rangle, we can use the fact that the coherent state \left|\alpha\right\rangle can be written as a superposition of number states:

\left|\alpha\right\rangle = \sum_{n=0}^{\infty} \alpha_n \left|n\right\rangle

which gives us the following expression for the wavefunction:

\Psi_{\alpha}(x) = \langle x|\alpha\rangle = \sum_{n=0}^{\infty} \alpha_n \langle x|n\rangle = \sum_{n=0}^{\infty} \alpha_n \frac{1}{\