Quantum 101 — Primer, Synthesis, Reflections

Primer-style LaTeX summary of quantum physics and quantum optics, with focus on selected topics in superconducting circuits, by Felix Wagner.

Pedagogical style

Each topic follows a four-step structure:

1. Phenomenon — describe what is observed and why it matters, in words.
2. Statement — state the result formally (theorem / proposition / postulate / definition).
3. Derivation — work through the math.
4. Examples / use cases — where it shows up (need not be fully worked out; a pointer to the application is fine).

Theorem-like environments for these are predefined in preamble.tex: phenomenon, theorem, proposition, definition, postulate, example, application, remark.

Repository structure

quantum101/
├── main.tex              # master document, includes all chapters
├── preamble.tex          # packages, theorem environments, macros
├── bibliography.bib      # BibTeX references
├── Makefile              # `make` to build, `make watch` for live preview
├── chapters/             # one .tex file per chapter
└── figures/              # figures, TikZ snippets, image assets

Building

make            # full build (pdflatex + bibtex + 2x pdflatex)
make watch      # latexmk continuous build (recommended while writing)
make clean      # remove aux files

Requires a TeX Live distribution with the physics, tikz/quantikz, siunitx, cleveref, and hyperref packages.

Online preview

Every push to this repository triggers a GitHub Actions workflow (.github/workflows/build-pdf.yml) that compiles main.tex and publishes the resulting PDF to GitHub Pages, embedded in index.html.

One-time setup (repository owner only):

1. Open Settings → Pages on the GitHub repo.
2. Under Build and deployment → Source, choose GitHub Actions.
3. Push any change (or re-run the latest workflow). Once the deploy job succeeds, the PDF is available at https://<user>.github.io/quantum101/ (with a "Download PDF" link), and the raw file at https://<user>.github.io/quantum101/main.pdf.

Build status and the deploy URL are visible under the Actions tab.

---

TODO — Topics to cover

This is a draft list of suggested topics; review and prune/extend as needed. Each [ ] becomes a section or chapter.

Part I — Quantum mechanics foundations

• Postulates of quantum mechanics (states, observables, measurement, time evolution)
• Hilbert spaces and Dirac (bra-ket) notation
• Operators: Hermitian, unitary, projectors; spectral theorem
• Commutators, uncertainty principle
• Schrödinger vs. Heisenberg vs. interaction picture
• Quantum harmonic oscillator and ladder operators
• Two-level systems: Pauli matrices, Bloch sphere, Rabi oscillations
• Density matrices, pure vs. mixed states, partial trace
• Composite systems, tensor products, entanglement
• Time-dependent perturbation theory and Fermi's golden rule
• Rotating-wave approximation (RWA)

Part II — Open quantum systems

• System–bath models, Born–Markov approximation
• Lindblad master equation
• Relaxation and dephasing: $T_1$, $T_2$, $T_2^*$
• Input–output theory
• Quantum Zeno effect (measurement-induced freezing of dynamics)
• Stochastic master equation / quantum trajectories (optional)

Part III — Quantum optics

• Quantization of the electromagnetic field
• Fock states and photon-number statistics
• Coherent states and their (over)completeness
• Squeezed states, displacement and squeeze operators
• Beam splitters and interferometry
• Atom–field interaction, dipole approximation
• Jaynes–Cummings model: dressed states, vacuum Rabi splitting
• Dispersive regime and dispersive shift $\chi$
• AC Stark shift (photon-number-dependent qubit frequency shift) and measurement-induced dephasing
• Cavity QED basics (strong, weak, bad-cavity limits)
• Purcell effect

Part IV — Superconducting circuits

• DiVincenzo criteria for a quantum computer
• Superconductivity primer: BCS ground state, Bogoliubov quasiparticles
• Lumped-element circuit quantization (node flux / charge)
• LC oscillator as a quantum harmonic oscillator
• Josephson junctions: current–phase and voltage–phase relations, Josephson energy
• Josephson effects: DC effect, AC effect, inverse AC effect (Shapiro steps)
• Cooper pair box and charge qubit
• Transmon qubit: anharmonicity, charge-noise insensitivity
• Flux qubit and fluxonium (overview)
• Capacitive and inductive coupling between elements
• Qubit–resonator coupling and dispersive readout
• Drive Hamiltonians and single-qubit gates
• Parametric / mixing processes: three- and four-wave mixing in Josephson nonlinearities (e.g., two resonator modes + drive + qubit as a four-wave-mixing example)
• Two-qubit gates: cross-resonance, iSWAP, CZ (overview)
• Decoherence channels in cQED hardware (incl. quasiparticle poisoning)
• Parametric amplifiers and quantum-limited readout (optional)

Experimental basics (qubit calibration)

• Resonator spectroscopy ($S_{21}$ vs. drive frequency; bare vs. dressed cavity)
• Two-tone (drive) spectroscopy of the qubit
• Rabi experiment: drive amplitude → $\pi$-pulse calibration
• Ramsey experiment: detuning and $T_2^*$ measurement
• Echo / CPMG sequences for $T_2$ (optional)
• $T_1$ measurement (optional)

Part V — Quantum information processing

A general primer that progresses from the language of quantum computing into quantum error correction. Topics in order:

• Quantum gates and the universal gate set; gate notation and diagrammatic conventions for circuit diagrams
• Quantum teleportation, fully worked
• Magic states and stabiliser-vs-non-stabiliser dichotomy
• Notes on quantum cryptography (BB84, E91) and quantum communication (no-cloning, dense coding)
• Why QEC: noise channels, no-cloning, the digitization theorem
• Repetition codes: bit-flip and phase-flip
• Shor and Steane codes (overview)
• Stabiliser formalism
• Surface code (overview)
• Bosonic codes in cQED: cat / GKP (overview, optional)

Part VI — Quantum sensing

Five chapters total:

• Experimental search for dark matter (chapters/09_dark_matter_search.tex, drafted). Detector-agnostic walk-through of both detection channels: fermion-like DM via elastic recoil and bosonic DM via absorption. Pipeline: model parameters → predicted in-detector spectrum → measured spectrum → statistical limit on the model parameter (Poisson + profile likelihood). Cross-references to the four mechanism chapters below.
• Phonon-to-quasiparticle conversion (chapters/10_phonon_quasiparticle.tex, placeholder). Cooper-pair breaking by an athermal phonon above $2\Delta$, read out by a TES or KID. Recoil channel.
• Infrared absorption in the Josephson junction (chapters/11_ir_absorption_junction.tex, placeholder). Direct IR/THz photon absorption in a junction; sensitive to dark photons with $m_{A'}\geq 2\Delta/c^2$. Absorption channel.
• Phonon-qubit coupling via piezoelectricity (chapters/12_phonon_qubit_piezoelectric.tex, placeholder). Single-phonon detection on AlN/GaAs substrates. Recoil channel.
• Microwave absorption in the transmon (chapters/13_microwave_absorption.tex, placeholder). Single-microwave-photon haloscope front end. Absorption channel.

Appendices (optional)

• Useful Gaussian integrals and operator identities (BCH, Hadamard lemma)
• Notation and conventions
• Bibliography / suggested reading