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. 
 

On this page:


 

General Information

 


"Recommended" Texts (none required)

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

Problem Set #2

  • Questions

Problem Set #3

  • Questions
Problem Set #4
  • Questions
Problem Set #5
  • Questions