# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 6) : Part1

#### 6.1

Unitary operation for a rotation around axis is:

An operator to swap x and y is:

An operator to swap x and z is:

#### 6.2

The general solution to the linear interaction case is given by (5.10) as:

Note that this is valid not only for , but also for , given the boundary condition at as (annihilation operator in the Schroedinger picture).

Then, if you start from , the expectation value of the number operator is:

Hence if you start from , you can never reach (even approximately) that satisfies in the future () or in the past (). In other words, is not reachable from .

To consider the effect of nonlinearity, write down a Schroedinger equation in the interaction picture. First, without the nonlinear term:

The interaction Hamiltonian is:

where I used the relationship:

By redefining the coupling factor as , we have:

For ,

From the Schroedinger equation , we have:

You can see that each state couples with and . The coupling to is slightly stronger than the coupling to . Hence if you start from , the state diffuses into higher energy states, and cannot converge into a single energy state.

Now let's add the nonlinear interaction .

The last term can be included in the base Hamiltonian by shifting the frequency from to . Then the interaction Hamiltonian becomes:

And the corresponding equations of motion are:

You see that each state has a self-coupling term with the strength . So, if U is sufficiently large, the higher energy states decouples from other states, such as:

So, as a first order approximation, we have:

Assuming that g is a constant, this can be explicitly solved with the boundary condition .

When , it is achievable to get .

Note: The condition to ignore the higher energy states conflicts with the last assumption . To avoid it, we may need higher order nonlinearity...?

#### 6.3

The convention for the matrix representation:

(1) In the interaction picture,

Hence,

Likewise,

(2) Again, in the interaction picture,

(3) In the Schroedinger picture,

Hence,

#### 6.4

(1) In general,

Hence,

--- (1)

In the zero temperature limit , you get (6.73).

(2) Basic definitions and relationships:

Note that these relationships hold at any time in the Heisenberg picture since:

Now, from (1),

--- (2)

--- (3)

Similarly,

--- (4)

In the zero temperature limit , from (2)(3)(4),

--- (2)'

--- (3)'

--- (4)'

(3) From the expression:

The elements of Bloch vector are given by:

Hence,

Likewise,

So, by setting , we have:

(3) Since

Hence, from (2) and (3),

By defining , we have:

#### 6.5

Using the unit system ,

The contribution from the dephasing term to is:

Hence, the solution of is the same as 6.4.

The contribution from the dephasing term to is:

Hence, the equation for in 6.4 is modified as:

where

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#### 6.7

For simplicity, we use the truncated model. At the symmetry point , the Hamiltonian becomes:

For , we have:

Hence, from the Schroedinger equation ,

With the initial condition , we have:

So, the probability of finding the state is:

#### 6.8

Since , the energy gap is .

The additional energy for a Cooper pair is:

From , we have .

In general, the Hamiltonian becomes:

where

#### 6.9

(4) The standard Hamiltonian for the LC resonator is:

Impedance

Natural frequency

The transmon Hamiltonian is:

Hence, the corresponding inductance is:

Using the relationships , the corresponding impedance is:

(1)

(2)

(3) The quartic term is:

Hence the first order contribution to the diagonal elements is: