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 14-116
- Mar 18 10-12
- Mar 20 14-16
Lecture 10-122 will be scheduled later.
An extra lecture was scheduled on Monday February 12, 10-12 in AlbaNova C4:3059 - Café Planck (AlbaNova Main Building).
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.
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 I covered the corresponding material more superficially, as Feynman-Vernon is now part of the MSc level course.
4. In 2023 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 I instead discussed instead 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 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 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.
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?
