Expectation value of field operator acting on vacuum state

[Nusc]

Homework Statement

I need to find the expectation value for E but I don't know how b acts on the vacuum state.

Homework Equations

$$b = \int dt \phi^{*}(t) \hat{{\cal E}}_{in}(t)$$
$$| \psi(t)\rangle = b^\dagger| 0\rangle$$

The Attempt at a Solution

$$\langle \psi(t) | \hat{{\cal E}}^\dagger\hat{{\cal E}}| \psi(t)\rangle =<br />$$

 

[Nusc]

What did I not make clear?

 

[Avodyne]

What did I not make clear?

Everything! What is the system supposed to be? How are your operators and other variables defined? Are you integrating over time? Then how is it that the state is time dependent? What exactly are you supposed to compute the expectation value of?

 

[Nusc]

This represents the single photon output level and I'm supposed to determine the
mean value and standard deviation of the single photon amplitude.
$$\hat{{\cal E}} = e^{-\kappa \tau}+ e^{-\kappa t}\int^{t}_{0}e^{\kappa \tau} \sqrt{2\kappa}\, \hat{{\cal E}}_{in}(\tau)dt$$

I'm integrating with respect to time.

$${\cal E}$$is an operator in Heisenberg picture.

b^+ creates a photon in the temporal mode $$\phi(t)$$

Does that make sense?

 

[Avodyne]

Sorry, it does not make sense to me. Perhaps someone else will be able to help.