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

by Aashish A. Clerk

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Submission summary

Authors (as registered SciPost users): Aashish Clerk
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-02-28
Date submitted: 2022-01-10 17:53
Submitted by: Clerk, Aashish
Submitted to: SciPost Physics Lecture Notes
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


These lecture notes from the 2019 Les Houches Summer School on Quantum Information Machines provides a pedagogical introduction to the theory of non-reciprocal quantum interactions and devices. The goal is connect various approaches and concepts, including Hamiltonians encoding synthetic gauge fields, scattering descriptions, quantum master equations, and non-Hermitian Hamiltonians. The importance of having both non-trivial synthetic gauge fields and dissipation for obtaining non-reciprocal interactions is stressed. Connections to broader topics such as quantum reservoir engineering and the quantum theory of continuous-measurement based feedforward are also discussed.

Published as SciPost Phys. Lect. Notes 44 (2022)

Reports on this Submission

Anonymous Report 1 on 2022-2-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2201.00894v1, delivered 2022-02-08, doi: 10.21468/SciPost.Report.4352


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.


The journal's acceptance criteria are met. I believe the community would benefit from these Lecture notes and thus recommend its publication.

Requested changes

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.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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