# Calculations on Coherent States and Squeezed States (1)

Quantum Optics

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

#### Basic formulas and relationships

where ----- (1.1)

when is commutable with and ----- (1.2)

#### 2.3 Coherent States

Displacement operator

1.
2. is Unitary. i.e.

1. is obvious. For 2., using anti-Hermite (i.e ), . Hence .

3.
4.

For 3., . Hence, from (1.1), . 4. is conjugate of 3.

5.
6.

For 5., since . 6. is conjugate of 5.

7.
8.

For 7., . 8. is conjugate of 7.

Coherent state

9.

From 3., . Hence

10.
11.
12.

For 10., using (1.2), since . Then .

11. directly follows from 10. For 12., using 7. .

From 11. and 12., you can see that the distribution of photon numbers follows the Poisson distribution, and the average is

#### 2.4 Squeezed States

20.
21.

From 3. and 4., .

22.
23.

For 22., using 5. to 8.,

23. is the same as 22.

24.
25.

Directly follows from 20. to 23.

Squeeze operator 　where

26.
27. is Unitary. i.e.

26. is obvious. For 27., using anti-Hermie . Hence

28.
29.

For 28., we use (1.1) with and .

By mathematical induction, in general,

Hence,

29. is conjugate of 28.

30.
31.
32.

The same as 5., 6., 7.

33.
34.
35.
36.

For 33., using 30.,

34. is conjugate of 33.

For 35., using 32.,

For 36.,

Squeezed state

Rotated amplitude , equivalently,

37.
38.

Since ,

(from 3.)
(from 28., .)
Hence, , and .

39.
40.

(Using 3. to 8.)
(Using 33. to 36. and )

41.
42.

Directly from 37. to 40.

#### 2.5 Two-Photon Coherent States

50. 　where

(where )

On the other hand,

(since .)

Hence , and by multiplying from the right side, you get 50.