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.

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)

Lecture 3 was held on Monday February 12, 10-12 in AlbaNova C4:3059 - Café Planck (AlbaNova Main Building).

Lectures 4-12 were held 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 14-116
• Mar 18 10-12
• Mar 20 14-16
• Apr 8   10-12
• Apr 11 10-12

In the course I used some extra material, some for the first time, some already used in 2022 as a larger stand-alone course. The extra material used in 2022 is also suitable extra material for the 2023/2024 version, or for general understanding and is therefore also reproduced below. 

1. In 2022 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.

This material was not covered in the 2023/2024 course.

2.  In 2022 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.

In 2023/2024 I intended first to cover the corresponding material more superficially, as Feynman-Vernon is now part of the MSc level course, but in the end felt compelled to treat it again on the same level of detail.

4. In 2023/2024 lecture 4 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.

In 2022 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 in 2022) 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 2022) 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. 

In 2022 I further discussed later attempts along the same lines:

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

and

   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 2023/2024 I instead discussed instead of the above more recent papers along these lines lines:

   On the unraveling of open quantum dynamics
   Donvil & Muratore-Ginanneschi
   Open Systems & Information Dynamics 30(03): 2350015 (2023)
   https://arxiv.org/pdf/2309.13408.pdf

and

   Time-local unraveling of non-Markovian stochastic Schrödinger equations
   Antoine Tilloy
   Quantum 1, 29 (2017)
   https://quantum-journal.org/papers/q-2017-09-19-29/

While simpler than the earlier attempts, the treatment in the four papers listed above remains mathematically somewhat complex.

5. In 2023/2024 I discussed the Hierarchical Equation of Motion (HEOM) method, mostly after the original paper

   Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath
   Tanimura & Kubo
   J. Phys. Soc. Jpn., 58 (1): 101–114 (1989)
   https://journals.jps.jp/doi/10.1143/JPSJ.58.101

I then also discussed briefly the recent modifification where the poles of the spectral function (assumed in HEOM, leading to exponential terms in the Feynman-vernon kernels) are free adjustment parameters.

   Taming quantum noise for efficient low temperature simulations of open quantum systems
   Xu, Yan, Shi, Ankerhold & Stockburger
   Physical Review Letters 129 (23):230601 (2022)
   https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.230601

6. In 2023/2024 I discussed the reactive coordinate method as a general idea and then in more detail the recent implementation where a system interacting with a harmonic bath of general spectrum is modelled as a system interacting with a finite set of harmonic oscillators

   Pseudomode description of general open quantum system dynamics: non-perturbative master equation for the spin-boson model
   Pleasance & Petruccione
   (2021)
   https://arxiv.org/abs/2108.05755

In the above (so far unpublished) method, the spectral function is also assumed to have poles, and the Markov open quantum system dynamics (Lindblad equation) of the system and the finite set of harmonic oscillators is deduced from the location and the residues of these poles. 

7. In 2023/2024 I discussed after completing the discussion of BP chapter 12 also the paper

Two-slit diffraction with highly charged particles: Niels Bohr's consistency argument that the electromagnetic field must be quantized
   Baym & Ozawa
   PNAS 106 (9) 3035-3040

Homework exercise (2023/2024)

Delivery date: As agreed upon (turned out to be 2024-04-05)

Due date: one week later   (turned out to be 2024-04-12)

Consider the two papers Xu et al (below called A) and Pleasance & Petruccione (below called B). Both papers address the problem of the evolution of one quantum system (in the example considered one qubit) interacting with an environment of harmonic oscillators with non-trivial spectrum. In both papers the interaction is bilinear in the system and the environment, i.e. of the type A_S \otimes B_E. From Feynman-Vernon theory it is known that all the influence of the environment on the system can be expressed through the correlation function of the operator B_E. Both papers assume that this correlation function is a sum of exponentials, as follows if the spectrum of B_E (Fourier transform of the correlation function) is a meromorphic function of frequency in the lower half plane.

Paper A and B develop two different ways to compute the evolution of the quantum system in this setting, and test these on two different environments.

The exercise is to take one of the environments, analyze it with the method in the other paper, and compare. See notes below.

Note 1. The report shall contain a clear presentation of the method and the environment chosen, and a statement of the task.

Note 2. Numerical solution hinges on availability of codes and/or precise enough description of software used in the two papers. It is part of the exercise to investigate if the authors (either set of them) have given enough information and / or detail so that the above task is possible to perform in the allotted time (one week). From scratch construction or implementation of a full computational pipeline is not considered a feasible task.

Note 3. A possible outcome of the exercise is that the task is not possible. Such a conclusion will the then have to be argued in a detailed manner in the report.

-----------------------

A.
   Taming quantum noise for efficient low temperature simulations of open quantum systems
   Xu, Yan, Shi, Ankerhold & Stockburger
   Physical Review Letters 129 (23):230601 (2022)
   https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.230601

B.
   Pseudomode description of general open quantum system dynamics: non-perturbative master equation for the spin-boson model
   Pleasance & Petruccione
   (2021)
   https://arxiv.org/abs/2108.05755

 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.