# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 4)

4.1 For the thermal state : For the pure state : 4.2 By mathematical induction, you can prove: Hence: 4.4 Current conservation: Branch currents: Hence: Without the register: With the register: 4.5 4.6 This is very rough apporximation. I'm …

# Derivation of the first and second Josephson relation

In the following discussion, we assume that the quantum states are quasi-static. We solve the time-independent Schrödinger equation supposing that external potentials are independent of time . When we change them slowly enough, the corresp…

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

The effective wavefunction and the charge current are given as: --- (3.4) ---(3.13)The wavefunction follows the Schrödinger equation: --- (3.5)Without losing the generality, we can take the Coulomb gauge: --- (1)Now, we assume that the cha…

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

Quantum Information and Quantum Optics with Superconducting Circuits作者:García Ripoll, Juan JoséCambridge University PressAmazon Derivation of the Lindblad master equation (2.26) I found the following paper is useful to understand the gen…

# Gauge Invariance of superconducting circuits wavefunction model

Effective wavefunction describing the flow of superconducting elections (Cooper pairs): Schrödinger equation for : --- (3.5)I will show that this model is invariant under the gauge transformation: [Proof](3.5) is equivalent to --- (1)A sim…

# (Informal) Errata of "A short introduction to the Lindblad Master Equation"

arxiv.orgp.8 Equation (24) p.10 Equation (30) p.11 Equation (31) p.13 Derivation of (45) from (44)First, in (44), we change the integral variable from (without any approximation) to get: Then, we assume that the kernel in the integration i…