FDD3026 Open Quantum Systems 7.5 credits

The course builds a master-level course DD2366 with the same name and same course literature (Breuer & Petruccione, BP). The master-level course covers the foundational parts / chapters in BP while the PhD level course covers the more advance parts / chapters. Relativistic aspects, in the first instance the important application to interaction with the quantized electromagnetic field, is only included in the PhD-level course. The PhD course comprises 12 lectures.

Chapters in Breuer & Petruccione are referred to as BP1, BP2, etc.

Lecture 1: Recall of the Feynman-Vernon approach and derivation of the Feynman-Vernon influence functional including the form of the kernels [extra material]

Lecture 2: Evolution of open quantum systems as PDEs and as integro-differential equations. Stochastic simulation methods. The quantum jump method. Stochastic Schrödinger equations. BP7.

Lecture 3. The hierarchical equations of motion method (HEOM) [extra material]

Lecture 4. Applications to quantum optics systems I. Continuous measurements in quantum electrodynamics. The microscopic Hamiltonian. Incomplete measurements. BP8.

Lecture 5. Applications to quantum optics systems II. Dark states. An atom evolving in interaction with the quantum electrodynamic field as an environment. BP8.

Lecture 6. Application to quantum optics III. Strong field interaction and the Floquet picture. BP8.

Lecture 7. Relativistic quantum theory on the formal level. Schwinger-Tomonaga equation. States as functionals of spacelike hypersurfaces. Foliations of space-time. The measurement of local observables. Relativistic state reduction. BP11.

Lecture 8. EPR correlations. Non-local measurements and causality. Entangled quantum probes. Quantum state verification. Quantum non-demolition verification of non-local states. BP11.

Lecture 9. Quantum teleportation. Teleportation and Bell-state measurement. A survey of experimental realization and implementations. BP11 and additional material.

Lecture 10. Density matrix theory for quantum electro-dynamics. Field equations and correlation functions. The influence functional (Feynman-Vernon functional). BP12.

Lecture 11. Vacuum-to-vacuum amplitudes. Decoherence by the emission of brehmstrahlung. The decoherence functional. Evolution of the decoherence functional for a quantum test body interacting with the quantized electromagnetic field. BP12.

Lecture 12. Decoherence of many-particle states. Limits to quatum information processing from the interactions with photons. Bp12.

1.                       After completing the course the students will be familiar with the concepts of pure states, mixed states, observables, density matrices, entanglement and von Neumann entropy of a reduced density matrix as a measure of entanglement. The students will be able to describe orally and/or in writing the Einstein­Podolski-Rosen experiment as an example of quantum correlations than can be given a classical interpretation (local hidden variables) and Bell theorem as an example of quantum correlations that cannot be given a classical interpretation (no local hidden variables). The students will be aware of the Local operations and classical communication (LOCC) paradigm, and of the quantum no cloning theorem.

2.                       After completing the course the students will be familiar with the concepts of Kraus operators, quantum Markov process, Lindblad equation, Lindblad operators and Feynman-Vernon functionals, and will know the number of parameters describing a general open quantum system evolution.

3.                       After completing the course the students will be aware of Feynman-Vernon functionals as encoding decoherence and dissipation, and be able to derive the Feynman-Vernon functional using operator techniques. The students will also be able to derive Lindblad equation from the Feynman-Vernon functional as a memory-less limit.

4.                       After completing the course the students will be aware of simulation techniques for open quantum systems without or with memory including the Lindblad equation, the quantum jump method, the stochastic Schrodinger equations and the hierachical equations of motion method (HEOM). The students will be able to compare the different methods to each other, and to decide which is more appropriate in a given application.

5.                       After completing the course the students will be able to consider in quantitative detail an atom evolving in interaction with the quantum electrodynamic field as an instance of an open quantum system interacting with an environment.

6.                       After completing the course the students will be able to discuss Einstein­Podolsky-Rosen correlations, non-local measurements and quantum causality in the language of relativistic quantum theory, and will be able to describe quantum non-demolition verification of non-local states.

7.                       After completing the course the students will understand the concept of quantum teleportation, and will be able to describe the principles of currently leading experimental realization and implementations.

8.                       After completing the course the students will understand decoherence through interaction with the quantum electrodynamic field as emission of brehmstrahlung and will be able to describe on a qualitative level the evolution of the decoherence functional for a quantum test body interacting with the quantized electromagnetic field.

No special prerequisites.

 MSc in physics or mathematics or electrical engineering MSc at KTH in engineering physics, electrical enginnering, or equivalent. Students with KTH MSc in computer science background are admitted if having corresponding mathematical background.

Lecture room

The Theory of Open Quantum Systems
  Heinz-Peter Breuer and Francesco Petruccione
  Oxford University 2007
  Published to Oxford Scholarship Online: February 2010  
  DOI:10.1093/acprof:oso/9780199213900.001.0001

If the course is discontinued, students may request to be examined during the following two academic years.

• EXA1 - Examination, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

The examination will be in three parts: (1) a standard written exam covering the same material as the master-level course (lectures 1-6); (2) correctly executed three homeworks on the material specific to the PhD-level lecture (lectures 7-18) ; (3)  an oral exam.

Passed written exam, correctly excuted homeworks, passed oral exam.

Erik Aurell

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

None