Lectures

LECTURE 1: Second quantization formalism for harmonic oscillator

• Example: Coherent (quasiclassical) 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 electromagnetic field

• Example: Nonclassical light--Fock, squeezed, antibunched, and entangled states of photons.
• Example: Cavity quantum electrodynamics--Jaynes-Cummings model of two-level atom interacting with one mode of electromagnetic field.

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. 1, 014701 (2019). [PDF]
• J. Larson, Dynamics of the Jaynes–Cummings and Rabi models: Old wine in new bottles, Phys. Scr. 76, 146 (2007). [PDF]
• E. Munguía-González, S. Rego, and J. K. Freericks, Making squeezed-coherent states concrete by determining their wavefunction, Am. J. Phys. 89, 885 (2021). [PDF] NOTE: It nicely explains purpose of AJP articles on the top of textbook material: "Why is this not routinely discussed in textbooks? Well, we often work with squeezed-coherent states in terms of their operators and in this case, the Gaussian form of the wavefunction is hidden (discussed in Sec. II B). As we will see, it is not so easy to extract it from this operator form, hence it often is not covered in textbooks."

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 evolution of nondissipative (closed) quantum many-body systems: Perturbation and Floquet theory

• Example: Dyson vs. Magnus perturbation series with application to driven harmonic oscillator.
• Example: AC Stark effect for two-level atom in classical electromagnetic wave.
• Example: Floquet topological insulators.

References

• Chapter 1 of Nazarov and Danon textbook.
• 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].
• 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: Time evolution of dissipative (open) quantum many-body systems

• 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.
• D. Manzano, A short introduction to the Lindblad master equation, AIP Advances 10, 025106 (2020). [PDF]

LECTURE 7: Relativistic quantum mechanics

References

• Chapter 13 of Nazarov & Danon textbook.