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.

Preliminary schedule (2023 / 2024)

Lecture 1 was held on Monday February 5, 10-12 in AlbaNova C4:3059 - Café Planck (AlbaNova Main Building).

Lecture 2 was held on Wednesday February 7 10-12 in AlbaNova A4:3001 (AlbaNova Main Building)

Lectures 3-11 have been scheduled in AlbaNova A4:3001 (AlbaNova Main Building) at the following dates and times:

• Feb 19 10-12
• Feb 21 10-12
• Feb 28 10-12
• Mar 4 10-12
• Mar 6 10-12
• Mar 11 10-12
• Mar 13 10-12
• Mar 18 10-12
• Mar 20 10-12

Lecture 12 will be scheduled later.

An extra lecture has been scheduled on Monday February 12, 10-12 in AlbaNova C4:3059 - Café Planck (AlbaNova Main Building).

The course was given in 2022 as a larger stand-alone course. Some extra material used in 2022 is also suitable extra material for the 2023/2024 version, and is therefore reproduced below. The list will be revised after the course will have finished. Note that the references to lectures in the below pertain to numbering of 2022, and not 2023. 

1. In Lecture 3 I used material taken from

   Bengtsson & Zyczkowski
   Geometry of Quantum states (First ed.)
   Cambridge University Press (2016)
   Section 10:3

Similar material can also be found in the slightly earlier paper by the same authors

   On Duality between Quantum Maps and Quantum States
   Karol Życzkowski & Ingemar Bengtsson
   Open Systems & Information Dynamics volume 11, pages3–42 (2004)
   https://link.springer.com/article/10.1023/B:OPSY.0000024753.05661.c2

Most of the statements are in Breuer & Petruccione, Section 2.4.3 "Representation theorem for quantum operations", but in more compressed form.

2.  In Lectures 4 and 5 I used the presentation in the original paper

  R.P Feynman & F.L. Vernon Jr
  The theory of a general quantum system interacting with a linear dissipative system
  Annals of Physics, vol 24, pages 118-173 (1963)
  https://www.sciencedirect.com/science/article/pii/000349166390068X

For a system interacting with a bath of harmonic oscillators the general form of the influence functional depends on two kernels, these days most often written k_i and k_r; to derive them I used a method presented in

   Erik Aurell, Ryochi Kawai & Ketan Goyal
   An operator derivation of the Feynman–Vernon theory, with applications to the generating function of bath energy changes and to an-harmonic baths
   J. Phys. A: Math. Theor. 53 275303 (2020)
   https://iopscience.iop.org/article/10.1088/1751-8121/ab9274/meta

This method is essentially the same as the one used in Breuer & Petruccione chapter 12, but in simpler non-relativistic setting.

4. In lectures 10 and 11 I discussed the stochastic wave equation as a way to simulate open quantum systems without memory, following BP chapters 6 and 7. I then also covered "stochastic Liouville equation" after

    Stochastic Liouvillian algorithm to simulate dissipative quantum dynamics with arbitrary precision
    J. Chem. Phys. 110, 4983 (1999); https://doi.org/10.1063/1.478396
    Jürgen T. Stockburger and C. H. Mak

It is a method with one real random driving field F. The density matrix by solving a Markov equation for each F, and then averaging over F. 

The limitation of this method is that it requires the real Feynman-Vernon action term (kernel k_i) to be Markov. It is therefore a method well adapted to the Caldeira-Leggett model with Ohmic spectrum and arbitrary temperature because while in this case Feynman-Vernon kernel k_r has memory, k_i does not. But it does not work for problems where also kernel k_i has memory.

Then I discussed attempts to use a similar method to also eliminate k_i using a random complex field Z.

The first idea (covered in some detail) can be found in

   Linear quantum state diffusion for non-Markovian open quantum systems
   Walter Strunz
   Phys Lett A (1996)
   https://www.sciencedirect.com/science/article/pii/S0375960196008055

and then slightly later (material covered more superficially) in

   The non-Markovian stochastic Schrödinger equation for open systems
   Lajos Diosi and Walter Strunz
   Phys Lett A (1997)
   https://www.sciencedirect.com/science/article/pii/S0375960197007172

There is a major technical problem of both those papers, that the complex field Z only eliminates the cross-terms between the forward and backward path in the Feynman-vernon action. The forward-forward and backward-backward terms remain, and this means that the eventual stochastic wave equation has memory.

The two above papers also use an unnormalized propagator, hence they are not properly stochastic wave function techniques. They further have a somewhat complicated structure where

\partial_t \psi (t, [Z]) = [.....simple terms...] + q \int \alpha(t,s) \delta \psi (t, [Z])/\delta \psi (s, [Z])

where the last term is quite hard to evaluate. 

C.
I then discussed on a superficial level a slightly later development which addresses the last two problems of above.

   Non-Markovian quantum state diffusion
   Diosi, Gisin & Strunz
   Physical Review A (1998)
   https://journals.aps.org/pra/abstract/10.1103/PhysRevA.58.1699

   Open-system Dynamics with Non-Markovian Quantum Trajectories
   Strunz, Diosi & Gisin
   Phys Rev Lett (1999)
   https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.1801

In these papers the authors write the dynamics for the normalized propagator which leads to a stochastic wave equation with the characteristic non-linearity. Note that it is for the kind of stochastic wave equation with small jumps called "wave diffusion equation" (see BP). The authors further substitute \psi (t, [Z])/\delta \psi (s, [Z]) with O(t,s,[Z])  \psi (t, [Z]) where O is some operator. It renders the equations to solve much simpler -- but how to do it seems to be more an art than a science.

Homework exercise 1 (2022)

Delivery date: December 5

Due date: December 22, at 24.00

Consider the Lindblad equation for the Caldeira-Leggett model in the high-temperature limit as derived in the lecture. This equation is as in Breuer & Petruccione eq 3.411 with a non-leading correction of the same form as Breuer & Petruccione eq 3.414, but with a constant 1/6, instead of 1/8.

(a) derive the equation satisfied by the Wigner function, defined as in Breuer & Petruccione eq 2.81.

(b) compare to the classical Fokker-Planck equation.

(c) discuss the non-leading term in the time evolution of the Wigner function proportional to [p,[p, \prho]] (\rho is the density matrix).

Does this term have a classical interpretation? If so, what? Are there any settings where it can have observable effects?

 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.

The course is given immediately after an MSC course DD2366 with the same name.

The MSc course is not a mandatory prerequisite for the PhD level course, but it is recommended to review the material covered in the MSc before taking the course.

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

Lecture room

Students at KTH with a permanent disability can get support during studies from Funka:

Funka - compensatory support for students with disabilities

P, F

• 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.

• 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.