# Study notes on "Quantum Information and Quantum Optics with Superconducting Circuits"

#### Derivation of the Lindblad master equation (2.26)

I found the following paper is useful to understand the general derivation of the Lindblad master equation.

It derives the master equation in two different ways. One is starting from the von Neumann equation for the density matrix (the common way of many physicists?):

The other is starting from the most general expression of time evolution of composite quantum system (CPT-maps derived from Choi-Kraus representation theorem):

I like the second one as it's based on the pure formalism and requires only assumptions of the quantum information theory! As I found some typos and unclear calculations in the paper, I created my own errata:

#### Gauge invariance of defined in (3.16)

As the gauge invariance is not explained explicitly in the book, I got confused why defined in (3.16) is called "gauge-invariant phase." I refreshed my memory on the gauge field theory, and summarized a rationale behind this definition:

#### Derivation of the gauge-independent relation between the phase and the electric field (3.22)

In the book, before deriving the relationship (3.22), it says "We have to complete our derivation to regard more general conditions!" on p.27. However, the assumptions used in the following derivation are not that clear to me. So I tried to write down all the details of the calculations:

As a result, I found that the following assumptions are required to get (3.22).

• The charge density is constant and uniform (independent of ).
• The currents are uniform (independent of ).

#### Derivation of the first and second Josephson relation (3.40) - (3.43)

On p.31, it says "As a boundary condition, we impose a current intensity flowing though the junction." So initially, I misunderstood that (3.37) can be derived from this boundary condition. After struggling with a lengthy calculation:

I finally realized that:

• The coefficients are decided by the usual(?) boundary condition, the continuity of the (gauge-invariant) wavefunction.
• The boundary condition of the current intensity imposes a relationship between the global phases and of wavefunctions outside the insulator.

Also I got a slightly different result for from (3.37) in the book (though it doesn't change the following discussions). My result is:

--- (3.37)

#### Typo in chapter 4

p.57 The critical value is written as . The last is not necessary.

p.58 (4.56)

#### Detailed derivation of the Input-Output theory

I found the discussion in Appendix B.3 is rather complicated. So I reconstructed the detailed delibation.

#### Typo in chapter 5

p.65 equation after (5.2)

( is required.)

(5.46) should be:

p.89 (5.62) should be

p.92 The Bose-Einstein distribution should be:

--- (5.72)

p.93 (Exercise 5.8) -> (Exercise 5.9) (Right before the equation (5.74))

p.104 The model Hamiltonian of Exercise 5.8 should be:

#### Convention of the matrix representation for the two level system.

It's not explicitly defined, but I assume the convention is:

where

#### Typo in chapter 6

p.115 (6.19) should be:

p.127 (6.45)

The characteristic number of Mathieu function should be . Also the variable in is not explicitly explained. In this context, it's the same as the characteristic number .

To normalize as , (6.45) should be:

p.127 (6.47)

The index function (6.47) doesn't work outside . It also generates the wrong order . The formula in Koch et al. (2007) works for all , but still generates the wrong order . I believe the right one is:

where

See this for results.

p.132 Figure 6.6, the x label should be

p.133 (6.54)

p.136 (6.59) should be:

p.136 (6.60) should be:

p.150 Solution of 6.4 (4) should be:

p.152 Problem 6.10 (1)

p.153 Problem 6.12 (3) .

p.154 Problem 6.15

p.155 Problem 6.16 (3) In the limit

#### Derivation of Input-Output relations for waveguide-QED

The equations in "7.2.2 Input-Output Relations" have several typos. I reconstructed the detailed derivation here.

#### Typo in chapter 7

p.166 The definition of (after (7.15)) should be:

p.166 (7.16) should be:

p.166 The definition of (before (7.17)) should be:

p.166 (7.17) should be:

p.169 (7.24) should be:

p.170 (7.25) should be:

p.170 (7.26) The pre-factor should be instead of

p.189 (7.57) should be:

p.194 Problem 7.1 Show how (7.14) can be... (Not (7.13))

p.194 Problem 7.2 (7.64) should be:

#### Typo in chapter 8

p.212 (8.10) should be:

p.216 The second equation of (8.13) should be:

p.216 (8.14) should be:

p.216 The middle part of (8.17) should be:

p.220 The first equation of (8.24) should be:

Note that in this section, the depolarization parameter stands for the probability of not-decaying. So it's the opposite (i.e ) of the depolarization parameter in Table 8.2.