Quantum information and optics with superconducting systems

Physics 681/682, Advanced Topics in Solid State, is a special topics course on superconducting quantum systems and quantum optics. The first part of the course introduces quantum optics, superconductivity and superconducting circuits. The second half explores quantum information and optics through contemporary experiments in the field, ending with recent developments in quantum error correction.

The course has been taught three times, in Fall 2022, Fall 2024 and Spring 2026, with a slightly different emphasis each time: superconductivity in Fall 2022, circuit quantization in Fall 2024, and open quantum systems in Spring 2026. The lecture notes below are a collation of the notes from all three offerings, organized by topic rather than by lecture date.

Module	Topic	# of lectures
1	Engineered quantum systems	2
2	Primer on Quantum mechanics	2
3	Introduction to Quantum Optics	3.5
4	Primer on Superconductivity	2
5	Quantum electrical circuits	2
6	Superconducting qubits	3
7	Decoherence in Superconducting Qubits, Open Quantum systems	2
8	Circuit Quantum Electrodynamics	2
9	Superconducting Quantum Optics	2
10	Quantum Computing with superconducting qubits	2
11	Quantum error correction	3

 

Engineered quantum systems

• Introduction
• DiVincenzo Criteria
• Overview of different platforms
  • Atoms
  • Ions
  • Photons controlled by atoms
  • Solid state defects
  • Superconducting circuits

Primer on Quantum mechanics

• Review of postulates of Quantum Mechanics
• Qubits, Spin 1/2 system, Bloch sphere, Density matrices
• Entanglement, Reduced density matrices, Entanglement entropy
• Quantum measurement

Introduction to Quantum Optics 

• Oscillators
  • Quantization of fields in the cavity and free space
  • Properties of photons, field commutation relations
  • Number states, coherent states, squeezed states
• Semi-classical light-matter interactions
• Coupling a qubit to an oscillator: Jaynes-Cummings model
• Cavity Quantum electrodynamics
• Stimulated and spontaneous emission processes

Primer on superconductivity

• Phenomenology
• BCS theory
• Josephson effect
• Single particle quantum mechanics of the superconducting phase

Quantum electrical circuits

• Quantum LC oscillator 
• Charge and phase as conjugate variables
• Circuit quantization
• Transmission lines

Superconducting qubits

• Charge qubits
• Transmon
• Flux qubits/Fluxonium
• Decoherence and Noise in Superconducting Circuits
• Protected qubits, zero pi, current mirror, Bifluxon

Open quantum systems

• Density matrix description
• The Lindblad operator and master equations
• Decay and dephasing in the driven Rabi model
• Ramsey experiments
• T1, Spin echo measurements and Dynamical decoupling

Circuit Quantum Electrodynamics

• Interaction between a superconducting qubit and a microwave resonator
• Protecting a qubit from the continuum, The Purcell effect
• Vacuum Rabi oscillations
• The dispersive limit, dispersive shift, Schrieffer-Wolf transformation
• Concept of coopoperativity
• Quantum Measurements

Quantum optics with superconducting cavities and circuits

• Photon number splitting
• Universal Quantum control of cavity states
  • Selective number dependent arbitrary phase (SNAP) gates
  • 4-wave mixing using the nonlinearity of the Josephson Junction.
  • Photon Blockade
• Coherent homodyne and heterodyne detection, coherence functions, Wigner tomography
• Generating non-classical states, Cat States, Fock states 
• Cavity gates, CNOT, Beamsplitters

Quantum computing with superconducting qubits

• Fixed frequency versus tunable architectures
• Cavities as a bus between qubits
• Parametric gates
• Cross resonance gate

Quantum error correction

• Error-correction with distributed qubits
  • Bit flip code
  • Laflamme criterion
  • Shor/Steane codes
  • Surface code
• Quantum error correction of cavity states
  • Amplitude damping codes
  • Cat codes
  • Binomial codes
  • GKP codes
• Autonomous error correction

Engineered quantum systems All notes for this module

• Course overview, introduction to engineered quantum systems. Notes
• Overview of different physical platforms: atoms, ions, photons, defect centers, superconducting qubits. DiVincenzo criteria. Notes

Primer on quantum mechanics All notes for this module

• Primer on quantum mechanics, quantum states and observables, qubits, Bloch spheres, quantum measurement, unitary evolution, canonical quantization, density matrices. Notes
• Composite systems, entanglement, oscillators, quantization of fields, commutation relations, number states. Notes

Introduction to quantum optics All notes for this module

• Coherent states, phase space representations of an oscillator, Husimi Q function, Wigner functions, cat states, squeezed states. Notes
• Phase space distributions continued, quantizing electromagnetic fields, semiclassical light-matter interactions, Rabi oscillations, Ramsey experiments. Notes
• Cavity QED, Jaynes-Cummings model, vacuum Rabi oscillations, universal control, dispersive limit of the JC model. Notes

Primer on superconductivity All notes for this module

• QND measurements using the dispersive interaction. Start of the primer on superconductivity, superconductivity phenomenology. Notes
• London’s equations, explaining the Meissner effect, penetration depth, the isotope effect and the role of electron-phonon coupling, Cooper instability, Cooper pairs. Notes
• BCS theory, wavefunction, number and phase as canonical conjugates, mean field approximation, Bogoliubov quasiparticles, gap equation. Notes
• Ginzburg-Landau theory for superfluids and superconductors, gauge symmetry, coherence length and penetration depth, Meissner effect, fluxoid quantization, Josephson effect. Notes
• Canonical conjugateness of number/charge and phase revisited, more on the Josephson effect, current-phase relation, AC Josephson effect, Ambegaokar-Baratoff relation, introduction to superconducting electrical circuits, LC oscillator. Notes

Quantum electrical circuits All notes for this module

• Lumped circuit elements, quantum LC oscillator, relation between flux and phase, zero point flux, phase, and voltage fluctuations, coupled LC oscillators. Notes
• Circuit networks, node and branch variables, SQUIDs and flux periodicity, generalization of Kirchhoff’s laws. Circuit quantization, spanning trees and closure branches, flux in a loop, method of nodes, transmission lines. Notes

Superconducting qubits and circuit QED All notes for this module

• Overview of superconducting qubits, charge qubits, inductively shunted qubits, Cooper pair box, Hamiltonian and spectrum, sensitivity to charge noise, eliminating charge noise and the transmon. Notes
• The analogy between a CPB and a quantum rotor, the Cooper pair box in the phase basis, the transmon, charge and phase fluctuations, connections with Bloch theory, the dependence of wavefunctions on ng, exponential suppression of charge noise dephasing. Notes
• More on the transmon, insensitivity to charge noise dephasing, anharmonicity and gate speed, charge matrix elements and sensitivity to decay, tunable transmon. Notes
• JC model revisited, vacuum Rabi oscillations, dispersive limit, readout, quantum non-demolition measurement using cQED. Notes
• The fluxonium circuit, junction array as a superinductor, fluxonium Hamiltonian, insensitivity to offset charges, potential as a function of flux, fluxons and plasmons, spectra, wavefunctions, and transitions, matrix elements. Notes

Decoherence and open quantum systems All notes for this module

• Fluxonium continued. Qubit relaxation and decay, Fermi’s golden rule, qubit connected to an environmental impedance, quantum limit of Johnson noise, effect of a finite temperature bath, detailed balance
• Sources of qubit decay (capacitive loss, quasiparticles) and dephasing (flux and charge noise), coherence time versus flux for fluxonium, the Purcell effect
• Open quantum systems, Markovian baths, CPTP maps, the Lindblad master equation, master equation for qubit decay, measuring decoherence with T1 and Ramsey experiments, qubit spectroscopy. Notes

Quantum computing with superconducting qubits

• Quantum computing with superconducting qubits, fixed frequency versus tunable architectures, cavities as a bus between qubits, parametric gates, cross resonance gates, randomized benchmarking, process tomography

Superconducting quantum optics

• Quantum optics with high Q cavities, control of cavity states with a superconducting qubit, photon number splitting, SNAP gates, four-wave mixing, parity measurements, Wigner tomography, cavity gates including CNOT and beamsplitter operations

Quantum error correction

• Error correction with superconducting qubits, bit flip code, Laflamme criterion, Shor and Steane codes, surface code, quantum error correction of cavity states