# Circuit Quantum Electrodynamics (Circuit QED)

The above Hamiltonian can be rewritten as:

By coupling these two circuit together using a capacitor, we get the circuit electrodynamics. Circuit electrodynamics is the circuit equivalent of cavity QED.

And the system can be described using dispersive Jaynes Cummings Hamiltonian:

Once you coupled these systems together, the effective resonant frequency of the readout resonator is dependent on the state of the qubit. If the qubit is pointing down or in zero state, the readout resonator will have one frequency, and if the qubit is pointing up or in one state, that frequency will shift.

The physical circuit looks like the following:

The middle section is a CPW style resonator with open boundaries.

The purple section is a flux tunable qubit and its diagram is illustrated in the right lower section.

These two sections are capatively coupled.

If a tone is sent and the qubit is at one state, a large amount of the signal will go through.

If the qubit is at another state, not a lot of signal will pass through.

That is:

**Superconducting Microwave Resonator** is useful in:

— reading out quibt states

— noise filter (isolating quibts from the environment)

— mutli-qubit quantum bus (In IBM, this linear resonator is used to couple mutlti-qubit together forming the cross resonance interaction. )

If we rearrange the Hamiltonian, we get:

# Resonator Photons Will Shift the Qubit Frequency

We observe that the qubit frequency depends on** the number of photons inside the resonator (how?)**.

The x axis is the qubit frequency being probed. If there was zero photons inside the resonator, you get the first frequency peak, etc.

This number resolved state or number resolved setup for the qubit, the non-linear element can be used to have coherent control of the linear cavity, i.e., turn the linear oscillator into a qubit.

For example, using selective number-dependent arbitrary phase (SNAP) gates,

we can get coherent control of the photons inside the linear cavity.

The black section on the right of the horizontal component is a transmon qubit coupled to a 3D ** cavity**, whose frequency is set by the pin at the bottom. Using this device, you can set up a harmonic oscillator and have coherent control over the cavity.

*These are called cat qubits.*

Source:

[1] Qiskit Global Summer School 2021-QGSS21 Lecture 9.1