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	Bosons: Simple harmonic oscillator Fock states, coherent states, Boson algebra
Feb.2	Introduction to Phase Space Representations
Feb. 7	Quasiprobability functions Wigner (W), Husumi (Q), and Glauber (P)
Feb. 9	Dynamics in Phase Space
Feb. 14	Tensor product structure and entanglement Schmidt decomposition
Feb. 16	Intro to open quantum system dynamics: Quantum operations, CP maps, Kraus Representation
Feb. 21	Quantum Channels for qubits
Feb. 23	Irreversible bipartite system-reservoir interaction. Markov approximation - Lindblad Master Equation
Feb. 28	Derivation of the Lindblad Master Equation Born-Markov approximation
Mar. 2	Continuation
Mar. 7	Examples of Master Equation Evolution: Damped two-level atom
Mar. 9	Damped Simple Harmonic Oscillator
Mar. 13-17	Spring Break
Mar. 21	Stochastic Processes
Mar. 23	Brownian Motion and Langevin Equation
Mar. 28	Heisenberg -Langevin Equations
Mar. 30	Fokker-Planck Equation
Apr. 4	Damped Simple Harmonic Oscillator in Phase Space
Apr. 6	Continuation
Apr. 11	The Truncated Wigner Approximation Decoherence and the Emergence of the Classical World
Apr. 13	Quantum Trajectories I Measurement theory
Aprl. 18	Quantum Trajectories II Quantum Monte-Carlo Wave Function Algorithm
Apr. 20	Quantum Trajectories III Different Unravelings of the Master Equation
Apr. 25	The Stochastic Schrodinger Equation. Quantum State Diffusion
Apr. 27	QND measurement and and the Stochastic Schrodinger Equation
May 2	Slack
May 4	Slack

Problem Sets

Problem Set #1

Questions

Problem Set #2

Questions

Problem Set #3

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Problem Set #4

Questions

Problem Set #5

Questions