Quantum information and optics with superconducting systems

01:750:681 Advanced Solid State I is a special topics course that will introduce superconducting quantum systems/optics. The first part of the course will provide introduce Quantum optics, Superconductivity, and Superconducting circuits. The latter half of the course will explore quantum information and optics through contemporary experiments in the field, ending with recent developments in quantum error correction.

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

 

Lecture	Dates	Topic	Notes	Module
1	Sep. 6	Course Overview, Introduction to Engineered Quantum Systems	Lec 1 notes	Module 1
2	Sep. 9	Overview of different physical platforms – Atoms, Ions, Photons, Defect Center, Superconducting qubits, DiVincenzo Criteria	Lec2 notes	Module 1
3	Sep. 13	Primer on Quantum Mechanics, Quantum states and observables, Qubits, Bloch Spheres, Quantum Measurement, Unitary evolution, Canonical Quantization, Density Matrices	Lec3 notes	Module 2
4	Sep. 16	Composite Systems, Entanglement, Oscillators, Quantization of fields, Commutation relations, Number States	Lec4 Notes	Module 2
5	Sep. 20	Coherent States, Phase space representations of an oscillator, Husimi Q function, Wigner functions, Cat states, Squeezed States	Lec5 Notes	Module 3
6	Sep. 23	Phase space distributions continued,  Quantizing electromagnetic fields, Semiclassical light-matter interactions, Rabi oscillations, Ramsey experiments	Lec 6 Notes	Module 3
7	Sep. 27	Cavity QED, Jaynes Cummings model, Vacuum Rabi oscillations, Universal control, Dispersive limit of JC model	Lec 7 Notes	Module 3
8	Sep. 30	QND measurements using the dispersive interaction, Start of Primer on Superconductivity, Superconductivity Phenomenology	Lec 8 Notes	Module 4
9	Oct. 4	London’s equations, Explaining the Meissner effect, Penetration depth, The isotope effect and role of electron-phonon coupling, Cooper instability, Cooper pairs	Lec 9 Notes	Module 4
10	Oct. 7	BCS theory,  Wavefunction, Number and phase as canonical conjugates, Mean field approximation, Bogoliubov Quasiparticles, Gap equation	Lec10 Notes	Module 4
11	Oct. 10	Ginzburg Landau theory for superfluids and superconductors, Gauge symmetry, Coherence length and penetration depth, Meissner effect,  Fluxoid quantization, Josephson effect	Lec 11 notes	Module 4
12	Oct. 14	Canonical conjugateness of Number/Charge and Phase revisited, More on the Josephson effect, Current Phase relation, AC Josephson effect, Ambegaonkar Baratoff Relation, Introduction to superconducting electrical circuits, LC Oscillator	Lec 12 notes	Module 4
13	Oct. 21	Lumped circuit elements, Quantum LC Oscillator, Relation between flux and phase, Zero point flux, phase, and voltage fluctuations, Coupled LC oscillators	Lec 13 notes	Module 5
14	Oct. 28	Circuit networks, Node and Branch variables,  SQUIDs and flux periodicity, Generalization of Kirchoff’s laws		Module 5
15	Nov. 1	Circuit quantization, Spanning trees and closure branches, Flux in a loop, Method of nodes,  Transmission lines	Lec 14 and 15 notes	Module 5
16	Nov 4	Overview of superconducting qubits, Charge qubits, Inductively shunted qubits, Cooper pair box, Hamiltonian and spectrum, Sensitivity to charge noise, Eliminating charge noise – the transmon	Lec 16 notes	Module 6
17	Nov. 8	The analogy between a CPB and a quantum Rotor, The Cooper pair box in the phase basis, The transmon, Charge & Phase fluctuations, Connections with Bloch theory, the dependence of wavefunctions on ng, exponential suppression of charge noise dephasing	Lec 17 notes	Module 6
18	Nov. 11	More on the transmon, insensitivity to charge noise dephasing, Anharmonicity and gate speed, Charge matrix elements, and sensitivity to decay, Tunable transmon	Lec 18 notes	Module 6
19	Nov. 15	JC model revisited, Vacuum Rabi oscillations, Dispersive limit, Readout, Quantum Non-Demolition Measurement using cQED	Lec 19 Notes	Module 6
20	Nov. 17	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	Lec 20 Notes	Module 6
21	Nov. 18	Fluxonium contd. Qubit relaxation/decay, Fermi’s golden rule, Qubit connected to an environmental impedance, Quantum limit of Johnson noise, Effect of a finite temperature bath, Detailed balance.		Module 7
22	Nov. 22	Sources of qubit decay (capacitive loss, quasiparticles) and dephasing (flux and charge noise), Coherence time versus flux for fluxonium, The Purcell effect		Module 7
23	Nov. 29	Open quantum systems, Markovian baths, CPTP maps, The Lindblad Master equation, master equation for qubit decay, measuring decoherence: T1 & Ramsey expts, qubit spectroscopy	Lec 23 Notes	Module 7
24	Dec. 1	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
25	Dec 2	Quantum optics with High Q cavities,  Control of cavity states with a superconducting qubit, Photon number splitting, SNAP gates, 4-wave mixing, Parity measurements, Wigner tomography, Cavity gates – CNOT, Beamsplitter operations,
26	Dec. 6	Error-correction with superconducting qubits, Bit flip code, Laflamme criterion, Shor/Steane codes, Surface code, Quantum error correction of cavity states
27	Dec. 9	Course Presentations
28	Dec. 13	Course Presentations