# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 5)

#### 5.1

You can redefine with the same commutation relation .

So you can set without losing the generality of the discussion.

Now,

and,

Hence,

By scaling back with ,

For and , use the relationship

#### 5.3

Hence, the coherent state is an eigen state of .

So,

Now,

Likewise,

Hence,

#### 5.4

Assuming that the lowest energy can be realized with a coherent state , find the corresponding .

With this , the coherent state becomes an eigen state as below:

Note: this is not a rigorous proof because it's based on the assumption mentioned above. I'm not sure how I can justify the assumption.

#### 5.5

On the other hand,

Hence,

By the way, if is a constant ,

So, the state oscillates between and the ground state found in Exercise 5.4.

#### 5.6

(i) Coherent state

Definition

Coherent state:

Integration variable of the Fourier transformation:

Displacement operator:

( if )

Now,

So,

(ii) Number state

(iii) Schrödinger cat state

Note: strictly speaking, since , the normalization factor should be . This is approximately if is sufficiently large.

Thanks to the linearity of trace, the Wigner function is the sum of the ones for above four terms.

and are the same as ones for the coherent states:

For the third term,

Likewise,

Finally, we have:

Coherent state is Gaussian and saturates the uncertainty relation. Others are not.

#### 5.7

From the relationship , choose locations where the average flax becomes maximum.

(1) The both end for , and one end with the capacitor for .

(2) The both end and the middle point for , and and for .

(3) The points where the average flax is the same for and .

#### 5.8

(a) The systems's Hamiltonian is:

Take the external force as frame of reference, that is, set . Then, the interaction Hamiltonian is:

Using the relationship:

, where

we have,

Hence, we have a time-independent interaction Hamiltonian:

Suppose that the system is in a thermal bath with the zero-temperature , from (B.15), the interaction with the thermal bath is described as (in the interaction picture):

--- (1)

Also, the interaction with the external force is described as:

--- (2)

By combining (1) and (2), we have:

(b) For ,

Using the relationships , we have:

With the initial condition (i.e. ), we have:

--- (3)

For ,

Using (3), we have:

where

By setting , we have:

Using the initial condition ,

(c) Asymptotically (),

Hence, at , becomes half. So, FHW =

#### 5.9

Without the external driving:

--- (1)

--- (2)

Substituting (1) into (2), and using the relationships:

You can reorder each term in the normal order, that is, , and you get:

Since ,

or

For the decoherence, substituting (1) into , and using the relationships:

You get:

Assuming that all the off-diagonal elements decays at the similar pace , we have

Hence, it decays at a pace .

???