Lectures

LECTURE 1: Second quantization formalism for harmonic oscillator

• Example: Coherent (quasiclassical) and squeezed states.
• Example: Isotropic three-dimensional harmonic oscillator.
• Example: Phonons in solids.

References

• Chapter 1.9 and 7.1 of Nazarov & Danon textbook.

LECTURE 2: Quantization of the electromagnetic field

• Example: Nonclassical light: Fock, squeezed, antibunched, and entangled states photons.
• Example: Light-matter interaction.

References

• A. Pathaka and A. Ghatak, Classical light vs. nonclassical light: Characterizations and interesting applications, J. Electromagn. Waves Appl. 32, 229 (2018). [PDF]
• K. C. Tan and H. Jeong, Nonclassical light and metrological power: An introductory review, AVS Quantum Sci. {\bf 1}, 014701 (2019). [PDF]

LECTURE 3: Second quantization formalism for bosons

• Example: Bose-Hubbard dimer.
• Example: Quantum phase transitions of Bose-Hubbard model of cold atoms in optical lattice.
• Example: Bogoliubov theory of superfluidity.
• Example: Magnons in ferromagnets and antiferromagnets.

References

• Chapters 3, 4.5 and 6 of Nazarov & Danon textbook.
• W. E. Lawrence, Algebraic identities relating first- and second-quantized operators, Am. J. Phys. 68, 167 (2000). [PDF]
• C. Timm, Lecture Notes on Theory of Magnetism
• J. M. Zhang and R. X. Dong, Exact diagonalization: The Bose–Hubbard model as an example, Eur. J. Phys. 31, 591 (2010). [PDF]

LECTURE 4: Second quantization formalism for fermions

• Example: Fermi-Hubbard dimer.
• Example: Mean-field theory of magnetism in Fermi-Hubbard model with positive U.
• Example: Mean-field (BCS) theory of superconductivity in Fermi-Hubbard model with negative U.
• Example: Hartree-Fock theory of electrons in metals.

References

• Chapters 3 and 5 of Nazarov & Danon textbook.
• C. Timm, Lecture Notes on Theory of Superconductivity
• D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, The Hubbard dimer: A density functional case study of a many-body problem, J. Phys.: Condens. Matter 29, 019501 (2017). [PDF]
• Y. Claveau, B. Arnaud and S. Di Matteo, Mean-field solution of the Hubbard model: the magnetic phase diagram, Eur. J. Phys. 35, 035023 (2014). [PDF]
• E. Erlandsen, A. Kamra, A. Brataas, and A. Sudbø, Enhancement of superconductivity mediated by antiferromagnetic squeezed magnons, Phys. Rev. B 100, 100503(R) (2019). [PDF]

LECTURE 5: Time-dependent perturbation theory: Dyson vs. Magnus series

• Example: Dyson vs. Magnus expansion for driven harmonic oscillator.

References

• Chapter 1 of Nazarov and Danon textbook.
• S. Blanes, F. Casas, J. A. Oteo, and J. Ros, A pedagogical approach to the Magnus expansion, Eur. J. Phys. 31, 907 (2010). [PDF]

LECTURE 6: Floquet theory of periodically driven quantum systems

• Example: AC Stark effect for two-level atom in classical electromagnetic wave.
• Example: Floquet topological insulators.

References

• A. Eckardt and E. Anisimovas, High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective, New J. Phys. 17, 093039 (2015). [PDF].

LECTURE 7: Dissipative quantum mechanics with application to qubits

• Example: Damped harmonic oscillator.
• Example: Qubit coupled to dissipative environment (spin-boson model).
• Example: Linblad master equation and quantum jump solution.

References

• Chapters 11 and 12 of Nazarov & Danon textbook.

LECTURE 8: Relativistic quantum mechanics

References

• Chapter 13 of Nazarov & Danon textbook.