Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics

Round 1

Reviewer 1 Report

It is well known that the master equations used by the authors are correct in the high temperature limit [U. Weiss, Quantum Dissipative Systems, World Scientific, 2012]. On the other hand, quantum effects are important at low temperature, where the Markov master equation becomes nonlinear [R. Tsekov, A nonlinear master equation for open quantum systems, FNL 20 (2021) 2130004]. The authors must investigate the temperature range of validity of their approach. Additionally, it is not clear what happens near the absolute zero and the described engine could probably violate the Third law of thermodynamics.

Author Response

Thank you for the feedback on our paper and the suggestion. The aim of our paper, in line with the focus of the special issue, is to review certain approaches to calculating heat currents, and to explore how they can be used to compute laser cooling spectra. Laser cooling is only possible at temperatures where there is a non-vanishing phonon occupation at the transition energy, which is why we consider such temperatures when drawing our conclusions about the accuracy of methods for calculating cooling spectra. The behaviour at very low temperatures is not relevant to our conclusions. Nonetheless we do accept the general point that it would be useful to consider temperature in more detail. We have therefore revised the paper to add a figure and accompanying discussion, looking at the temperature dependence of the cooling power at resonance. This figure confirms that the Bloch-Redfield equation (which is not strictly speaking a Markov master equation) makes reasonable predictions over the temperature regime relevant to laser cooling, although it does become unphysical at very low temperatures, as the referee suspected.

Reviewer 2 Report

I have some questions and remarks to the manuscript.
1. In sec.2.1 the authors formulate the problem and write two equations but incorrectly de ne
states, the initial (unperturbed) Hamiltonian and the main interactions. Really, any operator
can be represented as a matrix when the bas set is chosen. The authors write about eigen
states of the Hamiltonian of the subsystem, but write its non-diagonal matrix. As is well
known, in the base of eigen states, any operator should be diagonal. Eq.(1) has a sense
when the interaction is de ned, but it is not de ned anywhere in the manuscript at all.
In addition, in Eq.(1) \)" lost, but I think it could be written in the appropriate place.
Respectively, the Rabi frequency could be de ned when the interaction is given.
2. In Eq.(5) commutator i[HSB(t); (0)] is omitted. The authors should comment on this.
3. When the problem is formulated the authors should emphasize that they use an e ective
hamiltonian of the open system (three-level impurity atom).
4. The used model of the environment is very primitive (one-dimensional system) and should
have more clear explanation.
5. The proposed model can describe kinetic processes in the open subsystem interacting with
phonons (harmonic oscillators) only, but it doesn't take into account any other interactions,
including interactions with a laser beam (electromagnetic  eld). Kinetic equations (7), (10)
and (14) take into account relaxation e ects of the  rst order of a perturbation theory, but
e ects under examination have the second order. In addition, the interaction with laser
electromagnetic  eld is not described anywhere, but the authors examine a laser-cooling
e ect.
6. The second terms (imaginary terms) in equations (7) and (14) describe energy shifts in an
open subsystem, but the authors do not discuss them.
7. When equations (5)-(9) are written down, a reference to K. Bloom's book \Theory and
Applications of the Density Matrix", publishing house \Plenum Press", 1981, or some other
books is appropriate.

Comments for author File: Comments.pdf

Author Response

We thank the reviewer for their questions and remarks on our manuscript, which we respond to as follows:

1. "In sec 2.1 the authors formulate the problem but incorrectly define states, the initial Hamiltoniaan and the main interactions ... the operator should be diagonal". 

The definitions here are correct. The off-diagonal terms in the Hamiltonian are the laser driving, which causes transitions between the electronic eigenstates. This is discussed in various books and reviews, for example, Allen and Eberly, "Optical resonance and two-level atoms", sec 2.5. We have added a comment and reference to the text to clarify this.

1. In Eq. (1) ")" is lost. 

We have checked this equation and it appears correct in our version.

2. "In Eq. (5) commutator is omitted"

This equation is in the interaction picture and so there is no commutator term.

3. "When the problem is formulated the authors should emphasise that they use an effective hamiltonian of the open system".

We do not understand this request, or how the reviewer understands the term 'effective hamiltonian'. We state, in the text, that "The Hamiltonian for the system is ..". 

4. "The model used for the environment is very primitive (one-dimensional) and should have a more clear explanation".

The model of the environment is a standard bath of harmonic oscillators, discussed in texts such as Ref. 24. It is not one-dimensional and, by using the appropriate spectral density, can describe many different physical systems, including the three-dimensional phonons and photons important in our problem. Note that the quantity k in Eq. is not a wavevector but merely an index labelling the bath modes. We have added some further details of this in the manuscript, and changed the label from 'k' to 'r' to avoid confusion.

5. "The proposed model ... doesn't take into account interactions with a laser beam. Kinetic equations 7, 10 and 14 take into account relaxation effects of the first order of perturbation theory .. but effects under examination have second order".

To take the first point, the interaction with the laser beam is the off-diagonal term in the Hamiltonian, Eq. (2), and we have clarified this as noted in our response to point 1 above. To address the second point, Eqs. (7), (10), and (14) are second order in perturbation theory. Note that they involve the spectral density J, which contains the square of the coupling strength g.

6. "The second terms in equations 7 and 14 describe energy shifts in an open subsystem, but the authors do not discuss them".

We address this by stating in the manuscript that "the quantities B_{ij} are associated with energy shifts". This general connection is discussed in textbooks, such as the Ref. [24] to which we refer. We do not think any further discussion is needed.

7. "When equations 5-9 are written down, a reference to K. Bloom's book or some other book is appropriate".

We provide a reference to the book by Breuer and Petruccione, Ref 24, at the start of the paragraph where we introduce and discuss those equations : '...we recall the standard procedure for deriving a Bloch-Redfield master equation [24,41,42]...'.

Round 2

Reviewer 2 Report

The authors corrected the secondary remarks, but, they didn't answer on principal questions
and remarks. Namely, the main question conerns the formulation of the Hamiltonian. They need
to start from the initial equation, e.g.
bH
= bH
1 + bH
2 + bH
3 + bV12 + bV13;
where bH
1 is the system unperturbed Hamiltonian, bH
2 is the thermal bath Hamiltonian and bH
3
is the laser beam Hamiltonian. Operators bV12 and bV13 are interactions of the system with the
thermal bath and laser  eld respectively.
Interaction bV12 allways exists, but interaction bV13 exists when a laser  eld is applied only. In
this case, Eq.(2) is incorrect, even if we do not go into the details of the allocation of a three-level
system.
All other my principal questions are the consequence of this  rst question.
If the authors will not give explanation of these questions their interesting work could be
considered as a solution of a pure mathematical task.
In conclusion, the authors should read carefully that in my remark \In Eq.(5) commutator
i[HSB(t); (0)] is omitted" the density matrix is taken in the initial moment of time. It follows
from the preliminary transformation of the Liouville-von Neumann equation. See cited books.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

The authors took into account some main remarks and the manuscript could be published in the present form.