Preliminary schedule (2023)
Lecture 1 has been scheduled on Monday February 5, 10-12 in AlbaNova C4:3059 - Café Planck (AlbaNova Main Building).
Lectures 2-12 have preliminarily been scheduled for February 7, 19, 21, 26, 29 and March 4, 6, 11, 13, 18, 20, and April 8 and 10. Dates and times of these lectures may be changed according to the schedules of the students following the course.
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 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?
