COURSE DESCRIPTION:
Understanding, predicting, and controlling the behavior of many-body quantum systems far from equilibrium is one of the central challenges of modern physics. Rapid experimental advances in controlling the coherent dynamics of complex quantum systems have transformed this field into a vibrant interface between theory and experiment, with far-reaching scientific and technological implications. The subject bridges fields as diverse as condensed matter physics, atomic and molecular physics, high energy physics, and quantum information science. This hands-on course offers an introduction to this active research area. The lectures alternate between the presentation of new concepts and practical coding exercises.

COURSE OBJECTIVES:
Students will learn how to develop computational tools to analyze the energy spectrum, eigenstate structure, and real-time dynamics of many-body quantum systems of current experimental relevance, including platforms based on cold atoms, trapped ions, nuclear magnetic resonance, and digital quantum computers. They will gain the ability to distinguish integrable from chaotic models, identify quantum critical points, and characterize phenomena such as thermalization and many-body localization. By the end of the course, students will be able to read and critically assess research papers in many-body quantum dynamics, reproduce key results numerically, and present their findings to the group.

 

YouTube Videos:
*) Jordan-Wigner transformation
In this video, we explain how to map spin-1/2 operators onto spinless fermions using the Jordan-Wigner transformation, and how this mapping helps distinguish interacting from noninteracting terms in the Hamiltonian.
https://youtu.be/-tp_EHlsoWw

*) Holstein-Primakoff transformation
In this video, we explain how spin-1/2 operators can be mapped onto hard-core boson operators using the Holstein-Primakoff transformation. Unlike the Jordan-Wigner transformation, however, this mapping does not make the distinction between interacting and noninteracting terms in the Hamiltonian transparent, since the hard-core constraint remains implicit in the bosonic representation.
https://youtu.be/1L19bygLboc

*) Role of basis in many-body quantum systems
The choice of basis depends on the physical question being addressed. When analyzing the structure of eigenstates in interacting many-body quantum systems, particularly to understand the effects of interactions, it is often useful to express the states in the basis of eigenstates of the noninteracting part of the Hamiltonian (i.e., the part describing noninteracting particles or quasiparticles). In this representation, the interaction terms spread the eigenstate over the noninteracting basis providing direct insight into delocalization in Hilbert space, quantum chaos, and thermalization.
https://youtu.be/TRj7nVgtN_U