Physics 581 Spring 2023

Open Quantum Systems

University of New Mexico

Department of Physics and Astronomy

Instructor: Prof. Ivan H. Deutsch

Lectures: Tuesday and Thursday 9:30am-10:45am, P&A Room 1140

Teaching Assistants: Andrew Forbes & S. Vikas Buchemmavari

Office Hours: TBA

Synopsis: All quantum systems interact with their environment, be it through unwanted interactions with a noisy bath of thermal and quantum fluctuations, or because we want to observed the system through a quantum measurement. This class will cover the foundations of open quantum systems and its relationship to classical nonequilibrium statistical physics. New features arise in quantum mechanics -- entanglement and negative probabilities -- with profound implications. With a foundation in open quantum systems, we will address fundamental processes in quantum mechanics: dissipation, decoherence, and measurement.

The Theory of Open Quantum Systems, by H.-P. Breuer and F. Petruccione
Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, by C. Gardiner and P. Zoller
Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences, by C. Gardiner

Grading:

* Problem Sets (5-8 assignments) 75%
* Final Project 25%
*Problem sets will be available on the web, about every other week. Generally assignments will be due Thursdays in the TA mailbox.

Tentative Syllabus

I. Quantum Mechanical Building Blocks

   - Closed quantum systems: states and evolution
   - Two level systems: qubits
   - Infinite dimension systems: bosons
   - Multipartite systems: entanglement

II. Foundation theory of open quantum systems

   - Completely positive maps and quantum channels
   - Kraus representation
   - Quantum channels for qubits
   - Measurment model

III. Lindblad Master Equation - formal theory

   - Markov differential maps - Lindblad form
   - Born-Markov approximation and the derivation of the Master Equation
   - Examples for qubits: amplitude damping, dephasing, depolarizating channels
   - Damped simple harmonic oscillator

IV. Langevin equations

   - Stochastics processes
   - Brownian motion
   - Langevin equation
   - Heisenberg-Langevin

V. Fokker-Planck equation

   - Classical statistical physics theory
   - Damped harmonic oscillator
   - Decoherence and the emergence of the classical world

VI. Quantum Trajectories

   - Quantum trajectories — the measurement model
   - Quantum Monte-Carlo wavefunctions
   - Different unravellings of master equation
   - Stochastic Schrödinger equation and continuous measurement

**Lectures**

Lecture notes in pdf. Podcasts for Lectures: Microsoft Stream
Jan. 17	Introduction to Open Quantum Systems: Dissipation, Irreversibility, Decoherence, Measurement	Lecture #1
Jan. 19	Quantum Foundations: Density operators, Measurements qubits, Bloch sphere	Lecture #2
Jan. 24	Qubit Dynamics: Rabi Oscillations	Lecture #3
Jan. 26	Qubits : Phenomenological Damping - Bloch equations	Lecture #4
Jan. 31	Continuous Variable Quantum Mechanics: Simple harmonic oscillator, coherent states, boson algebra	Lecture #5
Feb.2	Introduction to Phase Space Representations	Lecture #6
Feb. 7	Continuation
Feb. 9	Quasiprobability functions Wigner (W), Husumi (Q), and Glauber (P)
Feb. 14	Dynamics in Phase Space
Feb. 16	Tensor product structure and entanglement Schmidt decomposition	Lecture #7
Feb. 21	Intro to open quantum system dynamics: Quantum operations, CP maps, Kraus Representation	Lecture #8
Feb. 23	Quantum Channels for qubits
Feb. 28	Irreversible bipartite system-reservoir interaction. Markov approximation - Lindblad Master Equation	Lecture #9
Mar. 2	Derivation of the Lindblad Master Equation Born-Markov approximation
Mar. 7	Examples of Master Equation Evolution: Damped two-level atom	Lecture #10
Mar. 9	Damped Simple Harmonic Oscillator
Mar. 13-17	Spring Break
Mar. 21	No Class
Mar. 23	Damped Simple Harmonic Oscillator in Phase Space	Lecture #11
Mar. 28	Decoherence and the Emergence of the Classical World
Mar. 30	Continuation
Apr. 4	Brownian Motion and Heisenberg-Langevin Equation	Lecture #12
Apr. 6	No lecture
Apr. 11	Continuation
Apr. 13	Classical stochastic processes and the master equation	Lecture #13
Aprl. 18	Wiener processes and Ito stochastic differential equations
Apr. 20	Continuation
Apr. 25	Quantum Trajectories I Measurement model	Lecture #14 Molmer 1 Molmer 2
Apr. 27	Quantum Trajectories II Quantum Monte-Carlo Wave Function Algorithm	Lecture #15 Molmer 3 Molmer 4
May 2	Quantum Trajectories III Different Unravelings of the Master Equation	Lecture #16
May 4	The Stochastic Schrodinger Equation. Quantum State Diffusion	Lecture #17
May 9	Continuous Measurement	Lecture #18

Problem Sets

Problem Set #1

Problem Set #2

Problem Set #3

Problem Set #4

Questions

Problem Set #5

Questions

Final Project

Questions