An introduction to Markovian open quantum systems

I think this is a very nice and useful introduction to the field of open systems for the beginning graduate student. I particularly like the inclusion of a good number of practice problems. I have a few small points where I think the clarity could be improved:

1. On page 12, there is a discussion of the eigenvalues of the Lindbladian having nonpositive real part: “If the dynamics is well behaved at long times—as is true in most physical situations—all nonzero eigenvalues in the summation must have nonpositive real parts.” It could perhaps be useful to point out that this is always true for finite dimensional system (i.e. from the contraction-mapping theorem). What constitutes well-behaved dynamics is also a bit vague – one could reasonably define this as the Lindbladian being negative semi-definite, in which case this is a tautology. It could also be useful to point out there are very simple, physically motivated models in infinite dimensional Hilbert spaces (bosonic modes) where this fails, such as a simple parametric amplifier.



2. Also on page 12, when discussing the spectrum of a spin model vs a damped Harmonic oscillator, it is written: “The wedge shape in Fig. 1(b) is characteristic of a classical fixed point, whereas the parabolic shapes in Fig. 1(c) indicate damped oscillations”. I believe either of these could reasonably be called decaying oscillations. The wedge shape in 1(b) is characteristic of a non-interacting system, where there is a mode structure.

3. In section 4.1, and especially at the top of page 14 when discussing the delta function at the boundary of the integral, the text plays a bit fast and loose with the stochastic calculus. I appreciate the point of these lecture notes are not suitable for a rigorous derivation, but perhaps a reference to a proper derivation using the Ito calculus would help point the curious student in the right direction for how to put this on a more rigorous footing.

4. On page 14, colored noise is introduced without defining it. Should mention that this is any noise source with a non-flat spectral density (and perhaps some experimentally motivated examples, like 1/f noise).

5. On page 29, the text reads: “In many-body systems, generic states, including eigenstates of an interacting Hamiltonian, have correlations between far-off sites.” It is important to qualify that this is only eigenstates in the bulk of the spectrum. The reader might infer this to mean many-body ground states, which of course can only have long range correlations if the Hamiltonian is gapless.

6. On page 43, the text reads: “One can formulate a broader class of continuous phase transitions in terms of a spontaneous weak-symmetry breaking, where the symmetry-broken phase is gapless”. It is important to be careful with language here. From a quantum optics perspective, having two degenerate zero modes might be defined as a gapless spectrum, but from a more traditional condensed matter perspective, a spectrum is only gapless if there is a continuum of modes around the ground state energy, whereas just having multiple (but a finite number of) steady states separated by a finite gap would just be a ground state degeneracy. This is especially important when discussing phases of matter, which are typically only well defined for gapped Liouvillians/Hamiltonians (in the condensed matter sense), and the phase transition point is gapless. Some clarifying remarks might be useful to avoid confusion.

7. In the discussion of XXZ subject to boundary dissipation, one should also add the reference: Prosen, PRL 107, 137201 (2011). This gives exact, analytic results in agreement with Ref. [108] in the text.

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The paper meets all the criteria of SciPost Physics Lecture Notes and therefore I recommend publishing the paper.

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