Introduction to quantum non-reciprocal interactions: from non-Hermitian Hamiltonians to quantum master equations and quantum feedforward schemes

1- it is a nice pedagogical introduction to the theory of non-reciprocal quantum devices. 2- it introduces different theory and formalisms, like Hamiltonians with synthetic gauge fields, non-Hermitian effective Hamiltonians, or quantum master equations and gives useful connection between them.

1- For better clarity, the notation for the hopping strength « t » should be distinguished from the notation for time « t ». For example, « J » could be used. 2- In abstract, page 1, missing « to » in « The goal is connect […] ». 3- In introduction, page 2, missing « system » in « […] description of an open quantum […] ». 4- In section 2, page 4, «our Hamiltonian described » should be replaced by «our Hamiltonian describes », in Eqs 4 and 5, the notation q for the charge should be introduced and « the the phase » should be replaced by « the phase». 5- In page 5, in Eq 10, adding the indice j to phi (or removing the sum over j) should make the computation step clearer to understand. 6- In page 6, « is highly non-resonant can be safely dropped » should be replaced by « is highly non-resonant and can be safely dropped ». 7- In page 7, a «t» is missing in the cos() of Eq 25. 8- In page 8, « with eaching hopping phase » should be replaced by «with each hopping phase » . 9- In section 3, the notation for the Green functions should be harmonized as they are sometimes written with ( ) and sometimes with [ ]. 10- In page 12, in the sentence « on site j if we drive the system at frequency ω on site j », the second «on site j» should be «on site j’». 11- In page 12, « as well as the the tendency » should be « as well as the tendency » . 12- In page 13, « simple intuitive way of understand » should be replaced by « simple intuitive way to understand ». 13- In page 14, « the bracketed factor is the clockwise hopping » should be replaced by « the bracketed factor is the counter-clockwise hopping ». 14- In Fig 4, the one-way arrow representing the particle that starts at 1 then returns to 1 via some arbitrary trajectory is a little misleading. As it is arbitrary trajectory, a two-way arrow should be a better representation. 15- In page 15, « back and form » should be replaced by « back and forth » and « see that the this sum » by « see that this sum » . 16- In page 17, the sentence « The second term on the RHS corresponds to processes 1 and 2, where as the last term corresponds to process 3 » appears unclear and RHS is not introduced. It should be reformulated as for example, « The order of the terms in the right-hand-side (RHS) corresponds respectively to the process 1 to 3 ». 17- In page 18, «  Achieving non-reciprocity required » should be replaced by « Achieving non-reciprocity requires ». 18- In section 4, page 19, EOM is undefined and moreover it is not used afterwards. 19- In page 22, « Heisenerg-Langevin » should be replaced by « Heisenberg-Langevin » and « along with the correspond noise » by « along with the corresponding noise ». 20- In section 5, page 22, « where the the third » should be replaced by « where the third » and « In the result effective model » by « In the resulting effective model ». 21- Figure 6 is not cited in the main text. 22- In page 24, « some puirely local » should be replaced by « some purely local ». 23- In page 25, «  in the previous, » should be replaced by «  in the previous section, ». 24- In the title of section 6, « measuerment » should be replaced by « measurement ». 25- In page 27, « analogous fo Eq » should be replaced by « analogous to Eq ». 26- In the sentence « drive system 2 (via a forcing operator F2 (see Fig.8) », a parenthesis and a final dot are missing. 27- In page 28, the equation « dW^2_t =dt» should have an overline on the squared Wiener increment. 28- « paramteric two photon » should be replaced by « parametric two photon ». 29- In conclusion, page 32, « and th effective » should be replaced by « and the effective ». 30- In Acknowledgements, the guidelines regarding funding information should be removed. 31- In the title of appendix A, it would be clearer to replace « t » by « the hopping strength t ». 32- In the bibliography, a few references are missing a link.