Undergraduate Research (Math 4994)
Instructor: Dr. Turker Topcu
Weekly meetings: Tue 12:00-1:15 pm, Fri 3:00-5:00 pm in MCB 558
This section of MATH 4994 is a two-credit course I created for undergraduates interested in working on computational research projects in the field of AMO physics. Please refer to the course policies sheet posted on the Canvas page for assignments and grading policy. Interested students need to contact me for a code that will allow them to enroll in the course.
Class Meetings: We will meet in person on Tuesdays between 3:00–4:15 pm in the Commons Room (MCB 455) in the Mathematics Department to go over material posted on the Canvas page of the course and discuss results obtained from simulations.
Course Content: Quantum Information Processing (QIP) using neutral atoms trapped in optical lattices is a promising platform for scalable quantum computing. In experiments, quantum gates are simulated by exploiting the Rydberg blockade mechanism, where a ground-Rydberg state transition is used to mediate conditional quantum logic.
Studying Rydberg atoms have a strong classical physics component where significant insight can be obtained from numerical simulations of Newton’s equations. Although atoms are quantum mechanical objects, Rydberg atoms are highly excited, and classical equations of motion yield surprisingly relevant insights into dynamical processes. Obtaining this insight can be otherwise difficult through a quantum mechanical approach. The numerical treatment of these problems fosters a solid foundational understanding of both numerical techniques in applied mathematics and physics. Studying such systems is accessible to undergraduate students as it mainly involves solving classical equations of motion. Therefore, this course is intended for undergraduate students with little to no prior exposure to quantum mechanics. However, previous exposure to calculus-based physics sequence is necessary.
Once the students acquire basic skills, such as numerical differentiation, integration, and phase space, they will be given more sophisticated codes to simulate the dynamics of an electron loosely bound to an atom, commonly referred to as a Rydberg or a highly excited atom. A system like this can be studied either in one- or three-dimensions, and publishable results can be obtained from applications to problems where Rydberg atoms play an important role, such as in engineering physical implementations of quantum computers using neutral atoms in quantum gate protocols. The students are expected to produce a preliminary manuscript by the end of the semester.
Syllabus & Grading: Cumulative final grade is based on four homework assignments distributed across three units.
Unit 1: In the first part of this course, the students will learn about numerical differentiation, integration and apply these to coding time-dependent dynamics of a simple harmonic oscillator by solving the classical equations of motion. The students will become familiar with basic numerical concepts and physical concepts such as phase space. Topics include:
- Numerical differentiation, integration
- Euler’s method for ODEs
- Staggered leapfrog algorithm for ODEs
- Apply these methods and produce computer code to solve the one-dimensional classical equations of motion for simple harmonic oscillator
- Classical phase space for classical harmonic oscillator (Assessment 1)
- Nonlinear oscillators (Assessment 2)
Unit 2: In the second phase of the course, the students will be introduced to the general idea of a Monte Carlo simulation:
- Classical Trajectory Monte Carlo as a general strategy for simulating classical systems
- Buffon’s needle experiment to estimate π.
- Particle in an asymmetric double-well potential (Assessment 3)
Unit 3: In the final phase of the course, the students will study the classical time-dependent dynamics of a Rydberg atom inside an optical lattice using the fourth-order Runga-Kutta algorithm. The computer codes will be provided, and we will spend some time going through the code and explaining its essential components. Topics to be covered:
- Rydberg (Ry) atoms, basic concepts. Why study Ry atoms classically?
- Physical implementations of Quantum information processing, why experimental facts of life must be taken into account when designing quantum information protocols
- Neutral atoms as a physical platform for quantum information processing,
- Rydberg blockade and the CNOT gate implementation
The students can then bifurcate into the following more specific projects:
- Atoms trapped in a non-twist optical lattice for decoherence-limited optical trapping.
- Excitation/ionization of Ry atoms in optical lattices beyond the dipole approximation. Students will learn what the dipole approximation is and why it breaks down for Ry atoms trapped in an optical lattice.