# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 7)

#### 7.1

Using the unit system and adding constant to the total energy, the Hamiltonian is:

The anzatz is:

where

Then,

Hence, from the Schroedinger equation , we have:

#### 7.2

From (5.61), we have a Langevin equation for the qubit's amplitude.

We ignore the input photons , and assume , then,

So, the decay rate is and no Lamb shift.

#### 7.3

Eigenvalues are

At and , we have . So .

Defining --- (1)

we have:

And,

Hence, becomes maximum at .

From (1), takes maximum when .

So, takes maximum at .

???

#### 7.5

I assume that is a typo of .

(1) When photons propagate both directions, we can use the relationship:

.

On the other hand, if photons propagate only one direction, the relationship above becomes:

.

So,

(2) From (1), . As the pre-factor of in (7.20) came from , we have:

(3) Since contains only right-moving waves ,

So from the definition of ,

with the assumption , we have:

Hence, from (7.16):

--- (7.16)

we have:

Defining as , we have:

For , we have:

Hence,

(4) When a qubit is placed at the end of a semi-infinite transmission line, we have only reflected photons. If we virtually (or mathematically) invert the direction of the reflected photons, it's the same as the chiral model discussed here. Note that we suppose that input photons are reflected not only by a qubit but also the end of the transmission line.

#### 7.6

(1)

Hence, from the Schroedinger equation , we have:

--- (1-1)

--- (1-2)

(2) Defining as with a initial condition , from(1-2), we have:

--- (2-1)

Substituting this result into (1-1), we have:

where

By applying the Markovian approximation, we have:

where

--- (2-2)

(3) From (2-1), we have:

From the assumption , based on the same argument as (7.18), we have . Then, by following the same argument (7.19) and (7.20), we have:

--- (3-1)

(4) From the assumption , we have:

So, from (2-2), we have:

--- (4-1)

Defining as with an initial condition , from (4-1), we have a solution:

With input photons , by taking , we have:

Hence, from(3-1), we have:

Hence the spectrum has multiple peaks at each with height and width .

#### 7.7

where

Around the resonance frequency , suppose that the couplings are almost constant , the system is effectively described with:

where

So the system has effectively a single ground state , and the state orthogonal to becomes a dark state.

???

???

Likewise,