# Calculations on Coherent States and Squeezed States (2)

Quantum Optics

This is intended to fill some gaps in calculation details in the book above.

#### 2.6 Variance in the Electric Field

Hamiltonian

Electric field operator

Time evolution (in the Heisenberg picture)

1.

2.

3.

For 1., , hence .
2. is conjugate of 1. 3. follows directly from 1. and 2.

4.

Rewriting 3. with

Time dependent variance of E(t)

5.

Hence, results in 5.
where

6. 　when

Apply the Schroedinger's uncertainty relation

Especially when , under the condition that , you get the next result.

7.

Hence, for (unsqueezed coherent state), the variance becomes constant whereas for (squeezed coherent state), the variance flactuates with the angular velocity .

8.

This follows from 3. and .

Especially for , it results in

The following shows graphs of for various s.

#### Schematic view of oscillation

Electric field operator

Time evolution

Expected value

In terms of , it corresponds to:

Define the rotated coordinates for some fixed .
Then at , . Hence,

These results can be summarized in the following diagram.

#### 3.7.1 Squeezed State

Photon number fluctuation for the squeezed state .

1.
2. 　where

For 1.,

For 2.,
since

Hence,

since

As a result,

since

From these results, you can see that for (non-squeezed coherent state) since it obeys the Poisson distribution.

On the other hand, when :

for , . It becomes the sub-Poissson.

for , . It becomes the super-Poissson.

Especially, for (Squeezed vacuum)], . It's always the super-Poission.