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This site has been supported by the NSF-CAREER award grant DMR-1147430, the NSF-RUI award grant DMR-1603418, the BSF-NSF grant DMR-1936006. Its purpose is to make available tutorials and computer codes developed with students and postdocs.
**) GROUP MEETINGS PRESENTATIONS (Summer 2024)
1) Devesh Karthik, undergraduate physics student (June/02/2024)
A brief introduction to the Lindblad Master Equation
The von Neumann Equation describes the time evolution of a density matrix from an initial state in closed systems. Unfortunately, true closed systems do not accurately model the evolution of states in an open quantum system, which are vital to development of quantum technologies. In this presentation, we introduce the function of the Lindblad Master Equation, which presents a model to study the time evolution of open quantum systems. We will then consider the Lindblad equation as applied to the Kerr Parametric Oscillator, and the effects of varying control parameters on the decay of the Survival Probability.
YOUTUBE
https://youtu.be/CqsK2jGknj8
SLIDES on Lindblad master equation
TUTORIAL on Lindblad master equation
2) Edson Signor, PhD physics student (June/06/2024)
Chaology of quantum maps applied to the Standard map
This tutorial aims to give a basic introduction to the concept of classical, and especially, quantum maps. Due to their simplicity and easy manipulation, quantum maps have been a great resource to investigate some intrinsic questions in quantum chaos. Here, we present the map quantization in the torus and apply it in spectral analysis as level spacing distribution. Furthermore, we bring the Husimi distributions to illustrate the “quantum” phase space of the quantum maps eigenstates. The calculations are applied to the quantum standard map but can be easily employed on any other map.
YOUTUBE
https://youtu.be/3il5rdzHs_0
TUTORIAL on quantum maps
3) Jorge Chávez-Carlos, postdoc (June/10/2024)
Dicke model: Quantum phase transitions and semiclassical techniques
The Dicke Hamiltonian is a quantum optical model that describes the interaction between a single-mode bosonic field and a collection of N two-level atoms. This model presents a zero-temperature quantum phase transition from a normal to a superradiant phase. It happens in the thermodynamic limit (as N approaches infinity) and has a well-defined classical analogy. The model also presents excited-state quantum phase transitions (ESQPTs) at different excitation energies. The classical Dicke Hamiltonian has two degrees of freedom, with energy being the only conserved quantity. As a consequence, it represents a nonintegrable system and chaos can develop. These characteristics—nonintegrability and chaos—are mirrored in the quantum domain. In this presentation, we introduce the quantum Dicke model and its properties, and after obtaining the classical Hamiltonian through the semiclassical limit, we explain how to obtain the Lyapunov exponent.
YOUTUBE
https://youtu.be/Thopsxc4mPE
Presentation
Code: Lyapunov exponent
4) David Zarate-Herrada, PhD student (Jun/13/2024)
Disordered spin-1/2 models
In this talk, I will introduce the one-dimensional Heisenberg spin-1/2 model with disorder. I explain the level statistics and structures of the eigenstates for different strengths of disorder. The analysis of the structure of the eigenstates is done in a basis that tells us about the degree of localization in real space. At the end, I will offer an analytical description of a system with two coupled spin-1/2 particles and explain how it converges to the Heisenberg model.
https://youtu.be/2Q1kkRM4Wro
Presentation
5) Miguel Prado, postdoc, and Edson Signor, PhD student (Jun/27/2024)
Leaking Classical and Quantum Systems
This presentation is about the classical and quantum standard map with a leak at different positions of the phase space. Finite Lyapunov exponents are computed in the classical domain and the Wehrl entropy is computed for the quantum states after the Schur decomposition. The correspondence between the classical and quantum results is discussed.
https://youtu.be/oQEtma08asc
Slides
6) Talk by Pedro Bento, PhD student (Oct/07/2024)
Krylov Complexity
https://youtu.be/e9rzXV5LTq4
**) Floquet systems, quasienergies, quantum chaos (file) (2022-2023)
Look also at the excellent tutorial by Holthaus.
J. Phys. B: At. Mol. Opt. Phys. 49, 013001 (2016).
It is available in the arXiv here.
**) Some of the codes used in the 2nd International Summer School on Advanced Quantum Mechanics, Sep/02-11, 2021, Prague.
Slides about STATIC properties
Slides about DYNAMICS
Fortran code for GOE matrices
Figures for the results for the GOE code
Fortran code for spin-1/2 models
Figures for the results of the spin-1/2 model with a single-defect
Figures for the results of the spin-1/2 model with couplings between 2nd neighbors
**) Notes and codes developed for the International Summer School on Exact and Numerical Methods for Low-Dimensional Quantum Structures that took place at the Izmir Institute of Technology, Turkey, from August 23 to August 31, 2014. The tutorial teaches how to exactly diagonalize one-dimensional spin-1/2 models. In hands of the eigenvalues and eigenstates, we: (i) analyze signatures of quantum phase transition, localization, and quantum chaos; (ii) investigate the dynamics of the system by studying the survival probability and the evolution of various few-body observables; (iii) compare the infinite-time averages of observables with thermal averages and identify conditions that can lead to the thermalization of isolated quantum systems. Computer programs in Mathematica and Fortran 90 are provided.
EXERCISES (Fortran Codes)
Fortran_Exercise01
Fortran_Exercise02
Fortran_Exercise03
Fortran_Exercise04
Fortran_Exercise06
Fortran_Exercise07
Fortran_Exercise09
Fortran_Exercise12
Fortran_Exercise16
Fortran_Exercise17
Fortran_Exercise20
EXERCISES (Mathematica Codes)
Mathematica_Exercise01
Mathematica_Exercise02
Mathematica_Exercise03
Mathematica_Exercise04
Mathematica_Exercise05
Mathematica_Exercise06
Mathematica_Exercise07
Mathematica_Exercise09
Mathematica_Exercise11
**) An introduction to the spectrum, symmetries, and dynamics of spin-1/2 Heisenberg chains (paper)
American Journal of Physics 81, 450 (2013)
Codes developed with undergraduate students Kira Joel and Davida Kollmar. They can be used to:
(i) Diagonalize the Hamiltonian matrix.
(ii) Find the density of states and Inverse Participation Ratio for all eigenstates.
(iii) Study the time evolution of different initial states.
(iv) Analyze the effects of the symmetries of the system.
**) Quantum chaos: an introduction via chains of interacting spins-1/2 (paper)
American Journal of Physics 80, 246 (2012)
Codes developed with undergraduate student Aviva Gubin.
(i) Density of states, level spacing distribution and NPC for spin-1/2 chain.
(spin-1/2 chain Mathematica code)
(ii) Density of states, level spacing distribution, and NPC for Gaussian Orthogonal Ensembles.
(GOE Mathematica code)
(iii) Suggestions for exercises.
(Extra Exercises)