The curious case of the inverted oscillator: On the dynamics and symmetries of quadratic potentials in the lowest Landau level

Scattering and interferometry in the lowest Landau level serve as important probes of quantum Hall matter. In this work, we focus on the non-commutative nature of the lowest Landau level and its invariance under area preserving transformations, and the physical implications of these properties on such probes. Borrowing from quantum optics literature, we model quantum point contacts and other and aspects of these probes as quadratic potentials, and study the dynamics they generate in the lowest Landau level, particularly in coherent states that obey the anyonic boundary condition. We study such dynamics in two ways - in terms of evolution of coherent state distributions over the lowest Landau level, as well as in terms of trajectories in an effective phase space obtained from these coherent states. These resulting trajectories offer interesting insights that serve to distinguish between Abelian anyons of differing statistical phase factors, as well as the emergence of an effective "statistical force" in the semi-classical limit. We then focus particularly on the properties of a specific quadratic potential - the inverted harmonic oscillator, as a model for quantum point contacts. In the lowest Landau level, the transmission amplitude of the point contact has a "thermal" form, and shares crucial parallels with the Hawking-Unruh effect in uniformly accelerated spacetimes, like those associated with black holes. We show these parallels to be a result of their shared isomorphic symmetries unique to the physics of (2+1) dimensional spaces. Consequently, these connections lead to the possibility of realizing other relativistic phenomena like Wigner rotation, and time-decaying modes with quantized decay rates among the scattered states in the lowest Landau level. We then demonstrate possible physical avenues where these modes can be detected. Lastly, we take a brief detour to discuss non-Abelian anyons and their exchange properties. Specifically, we study the nucleation of Majorana bound states in two dimensional topological Josephson junctions through the application of magnetic flux in the junction area. We locate the bound states in the geometry of our system and then construct trijunction and four-point crossroad junctions as an extension of these ideas. We then provide two proof-of-concept braiding schemes to exchange the Majorana bound states and design readout mechanisms that employ a series of projective measurements using quantum dots as charge or parity detectors. We finally end with a brief summary and outlook on the work presented here.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Subramanyan, Varsha

Contributors dc:contributor

• Vishveshwara, Smitha
• Fradkin, Eduardo
• Eckstein, James
• Holder, Gilbert

• Quantum Hall Effect
• Anyons
• Rindler Physics
• Quantum Point Contacts
• Squeezing
• Quantum Optics

Statement dc:rights

• Copyright 2023 Varsha Subramanyan

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/121399

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121399