This is a year of teaching. To children, to high-school teachers, to master students, etc.

In these days I’m teaching a four-hour course to Ph.D. candidates of the physics department on Open Quantum Systems. The idea is that of starting from the unitary dynamics of a “universe”, and resolving it into “system + environment” to find a dynamical map for the system by tracing out the environmental degrees of freedom in the limit where the interaction between system and environment is weak. The material covers the so-called “microscopic” derivation of the Lindblad equation, then focuses on the case where the environment is thermal radiation, and then further focuses on the cases where the system is either a spin or a harmonic oscillator. I briefly discuss entropy vs. entropy production, the second law of thermodynamics, and irreversibility.

I had real fun preparing this course. As I dealt with a fairly new subject, I found the same vibe as back when I was a university student, always looking for further references, scrutinizing all possible details, ending up reading four times as much material as was needed to prepare an exam. Which was one of the (many) reasons why I eventually graduated two years later (while at the time I was a bit discouraged by this delay, it eventually turned out to be one of the best “choices” in my life).

The starting reference for this basic course is the remarkable book by Breuer and Petruccione:

H. P. Bruer, F. Petruccione, *The Theory of Open Quantum Systems* (Oxford University Press)

This is perfect to have a physicist’s overview, but recently I found this other short book

A. Rivas, S. F. Huelga,* Open Quantum Systems* (SpringerBriefs)

that does a great job in resuming a lot of otherwise quite hostile math-phys literature on the subject, and sheds light on some obscure passages. In particular, in the physicist’s derivation, several different approximations are made: beside the weak coupling, there’s Born’s assumption that the system and the environment are uncorrelated at all times (which makes no sense, given that what we want to study is precisely how the environment affects the system…), the Markov property, and the so-called “rotating wave” approximation. This is probably due to a stratification of arguments and techniques that have piled up in the history of the derivation of this and similar master equations. However, the treatment given by Rivas and Huelga in terms of Nakajima-Zwanzig operators shows that the Born-Markov approximation and, to some degree that I’m not able to appreciate, even the rotating-wave approximation, all follow from the weak interaction hypothesis alone! Thus the derivation is much neater.

As I want to learn more about all these things, as final exam I will propose to the students a half-day workshop where each one of them will present a paper or a book chapter picked from the following list (or, if they don’t like, they might just choose a relevant and interesting paper of their own).

Breuer & Petruccione Secs. 3.2.1 & 3.2.2

Rivas and Huelga Ch. 4.2 & Th. 3.1.2, Th. 3.2.1

These book chapters explain how to derive the Lindblad equation from the Markov property alone, and not following a microscopic derivation. This is standard material, and should be covered with some detail.

V. V. Albert and L. Jiang, *Symmetries and conserved quantities in Lindblad master equation*, PRA 89, 022118 (2014).

The main difference between the Lindblad equation and its classical analogue, the master equation, is that the former may have multiple nontrivial steady states, or even oscillate between coherent steady states, while multiple steady states for the classical master equation are trivially due to disconnected sub-spaces, in the case of reversible rates, or absorbing basins of attraction in the case where transitions are not reversible. Allegedly this paper discusses multiple steady states and their relations to conservation laws and symmetries of the Lindbladian.

D. Taj and F. Rossi, *Completely positive Markovian quantum dynamics in the weak-coupling limit*, PRA 78, 052113 (2008)

One of the limitations of the microscopic derivations outlined above is that the system must have continuous spectrum, and the system discrete. While the first condition seems to be unavoidable (all of thermodynamics is based on the assumption that we can throw stuff away and we will never see the consequences of it…), this paper rigorously addresses the case where the system has continuous spectrum, by analyzing the separation of time scales between the system’s evolution and the time of Poincaré recurrencies in the environment.

A. Rivas, A. D. K. Plato, S. F. Huelga, *Markovian master equations: a critical study*, New J. Phys. 12, 113032 (2010).

This is the material that sustains the book by Rivas and Huelga, but it also includes a couple of more in-depth examples, the two coupled harmonic oscillators and the driven harmonic oscillator, that are worth studying in detail.

F. Carollo, J. P. Garrahan, I. Lesanovsky and C. Pérez-Espigares, *Making rare events typical in Markovian open quantum systems*, arXiv:1711.10951.

J. P. Garrahan and I. Lesanovsky, *Thermodynamics of Quantum Jump Trajectories*, PRL 104, 160601 (2010)

P. Zoller, M. Marte, and D. F. Walls, *Quantum jumps in atomic systems*, PRA 35, 198 (1987).

This triptyque can be run up-down or down-up. For those who know about the thermodynamic approach to open systems, and in particular a bit of large deviation theory (or even better, they know about rare-events algorithms) (that is, my own group members) it is best to start from the first and only refer to the other two papers to complete the information. For all others, the third paper contains a great nontrivial three-level example and some speculations about “quantum jump trajectories”, and they might just take a look at the first two.

Philip Pearle, *Simple Derivation of the Lindblad Equation, *Eur. J. Phys 33, 805 (2012).

As per title, this paper proposes a simple derivation of the Lindblad equation, of the sort based on some general principles like many derivations of quantum mechanics from reasonable axioms. I’m at the same time compelled but skeptical about this paper, so this JC should really become a critical analysis.

M. B. Plenio and P. L. Knight, *The quantum-jump approach to dissipative dynamics in quantum optics*, Rev. Mod. Phys. 70, 101 (1998).

This is a long review, dealing with the subject of “quantum jump trajectories”. It connects here and there with the Lindblad equation, but not in a rigorous way. Possibly Carlmichael’s work (cited) gives further insights. I still have a problem making sense of all this jumping, it looks way too classical (the point is, jumping from where to where? As the diagonal basis of the density matrix evolves with time, then also the preferred “locations” whereto jump change…).

R. Alicki, *The Markov master equations and the Fermi golden rule*, Int. J. Theor. Phys. 16, 351 (1977).

As per the title, this paper proposes a derivation of Fermi’s golden rule. The JC should explain in what sense this derivation is more rigorous than the usual derivations.

Breuer and Petruccione Sec. 3.5 (given 2.4.6 and other material)

Derivation of the Lindblad equation from an indirect measurement process.

Breuer & Petuccione 3.4.6

Damped harmonic oscillator (in some detail).

Furthermore, I have a couple of questions of my own I would like to inspect:

– For the “microscopically derived” Lindblad equation, there is one preferred basis where the dynamics separates into populations and coherences. This implies that, in that basis, if one starts from a purely statistical state (a diagonal density matrix), then such state only evolves by the Pauli master equation. The question is, for generic Lindblad generators not following from a microscopic derivation, does there exist a preferential basis such that, if the initial state is prepared as a purely statistical state in that basis, then they evolve by a classical master equation, that is, they do not develop correlations at later times?

– An analysis by Kossakowsky states that, for any family of orthogonal complete projections {P_i}, i = 1,…,n, where n is the dimension of the system’s Hilbert space, the quantities w_ij = tr(P_i L P_j) are the rates of a classical Markov jump process. However, such process might not be representative of the quantum evolution, as the classical evolution depends on the choice of family of projections. The idea is to study the statistics of classical Markov jump processes, randomizing over the family. How to randomize? One possibility is the following. Since for some unitary matrix U, P_i = U I_i U^-1, where I_i has only the i-th diagonal entry equal to 1, and all others vanish, we could take the unitary U from the GUE (Gaussian Unitary Ensemble).

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