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Higherorder meanfield theory of chiral waveguide QED
by Kasper J. Kusmierek, Sahand Mahmoodian, Martin Cordier, Jakob Hinney, Arno Rauschenbeutel, Max Schemmer, Philipp Schneeweiss, Jürgen Volz, Klemens Hammerer
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Submission summary
Authors (as registered SciPost users):  Kasper Jan Kusmierek · Maximilian Schemmer 
Submission information  

Preprint Link:  https://arxiv.org/abs/2207.10439v2 (pdf) 
Date accepted:  20230109 
Date submitted:  20221206 17:02 
Submitted by:  Kusmierek, Kasper Jan 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Abstract
Waveguide QED with cold atoms provides a potent platform for the study of nonequilibrium, manybody, and opensystem quantum dynamics. Even with weak coupling and strong photon loss, the collective enhancement of lightatom interactions leads to strong correlations of photons arising in transmission, as shown in recent experiments. Here we apply an improved meanfield theory based on higherorder cumulant expansions to describe the experimentally relevant, but theoretically elusive, regime of weak coupling and strong driving of large ensembles. We determine the transmitted power, squeezing spectra and the degree of secondorder coherence, and systematically check the convergence of the results by comparing expansions that truncate cumulants of fewparticle correlations at increasing order. This reveals the important role of manybody and longrange correlations between atoms in steady state. Our approach allows to quantify the tradeoff between antibunching and output power in previously inaccessible parameter regimes. Calculated squeezing spectra show good agreement with measured data, as we present here.
Author comments upon resubmission
We thank the referee for the careful reading of the manuscript, the positive feedback, and the useful comments. Based on the listed weaknesses and requested changes, we edited the text and especially also the figures of our manuscript. In particular, we improved the style of presentation of figures and also reduced their number in order to keep a better focus on the main points of the manuscript. We hope the presentation is now clearer and the figures are easier to understand. The experimental apparatus and procedure is now described in more detail. Below we address the list of questions asked by the referee:
 Q: Why are the invidual positions of the atoms appearing nowhere in the model?
The position of the atoms comes into play as a phase e^(±ikx_i) with a positive sign for absorption and a negative one for emission of photons. Thus, for forward scattering, and using the excitation laser as a phase reference, the phases cancel and the position of the atoms drops out. The position may still come in as a time delay between the atoms. In [1] it was shown that even in the non Markovian limit the time delay between emitters can be gauged away by redefinig the times for each consecutive system. In the Markovian limit, where the time delays are negligible, the distance between emitters amounts to an overall phase that can be gauged away by redefining the spin operators [2]. For this reason the positions of the atoms does not appear in the model.
 Q: Why totally neglect backscattering / backward emission of photons; can the results be expected to be particuarly sensitive to this?
Back scattering can be largely suppressed in wave guide QED with chiral coupling. This motivates our consideration of the manybody dynamics in the ideal limit of exclusive forward scattering, realizing a cascaded quantum system. We believe it is natural and important to try to characterize and understand this limit first, before considering imperfections arising from back scattering. In the experiment, a certain level of back scattering is unavoidable. The fact the our theory compares very well to experimental data in many respects shows that the results tolerate a certain level of nonchirality. We also note that the inclusion of back scattering would complicate the theoretical model considerably, since the position of atoms would become relevant and, moreover, the cumulant expansion would become nonlinear. We also note that if the atoms are not ordered in a lattice which is commensurate with the wavelength of the input field the backscattered light will not add up coherently. The backscatteringchannel therefore plays a similar role as the other loss channels describing scattering out of the wave guide. The chiral arrangement with external loss channels is therefore a useful model for any ensemble of atoms that is not arranged into a lattice that is commensurate with the wavelength of the probe field.
 Q: Why are collective effects of the outside fibre loss/decay here completely negligible as atomic distances are of wavelength size?
Even though the distance between neighboring lattice sites d = 447 nm is on the order of λ/2 = 426 nm, we expect that collective effects of the outside fiber loss/decay can be neglected. First, collective effects are very sensitive to the distance. In particular, the freespace coupling strength between atoms only becomes comparable to the freespace decay rate once the condition d ≤ λ/2π is reached (see Eq. (4a) in NJP, 21(2), 025004, 2019). Second, and more importantly, in spite of the close proximity of the trapping sites, the mean distance between two atoms is significantly larger than the wavelength due to the fact that the majority of the trapping sites remains empty. Concretely, in our experimental realization, we load ≈ 180 atoms over a length on the order of 1 mm into the optical lattice potential along the nanofiber. These atoms are thus distributed over ≈ 2, 000 lattice sites. Hence, we operate our system in a regime where collective effects from unguided modes can be neglected.
 Q: Why are terms <sigma^x sigma^z> generally ignored in the expansion?
We know from equations (12) that moments <sigma^x_i> are zero, due to the resonant drive. Because of this, several terms in the equations of motion for <sigma^x sigma^y,z> cancel. The remaining terms describe a coupling among <sigma^x sigma^z> and <sigma^x sigma^y> only. Therefore, these quantities decouple from the dynamics and remain zero for all times. For this reason we left them out for sake of simplicity and readability. In our numerical calculations we take these terms into account and solve the complete set of equations without approximation (beyond the cumulant expansions). The numerical results confirm that the moments <sigma^x sigma^y,z> are zero on resonance.
 Q: In Fig.6: what does a "meanfield" result for g2(0) represent?
We assume the referee was referring to Fig. 8 (Fig. 6 deals with squeezing spectra). Here, as well as regarding other observables, "meanfield" means truncation of the cumulant expansion at order 1, that is, higher order moments are factorized into products of firstorder moments. For example, in equation (28), meanfield theory assumes <sigma^x_i sigma^y_j sigma^x_k sigma^y_l>=<sigma^x_i><sigma^y_j><sigma^x_k><sigma^y_l>. On the level of density matrices this assumes a tensorproduct state.
[1] Gardiner, Phys. Rev. Lett. 70, 2269, 1993 [2] Pichler et al. , PRA 91, 042116, 2015
List of changes
 Fig. 2: Column corresponding to Pin/Psat=0.01 deleted and greater fontsize
 Fig. 3: Spectra computed in meanfield theory deleted and greater fontsize
 Fig. 4: yaxes in scientific notation; legend below plots; greater fontsize
 Fig. 5: greater fontsize; line width increased
 Fig. 6: greater fontsize
 Fig. 7: Rows corresponding to <<sigma^y_i sigma^y_j>> and <<sigma^x_i sigma^x_j>> deleted; greater fontsize
 Fig. 8: Plot correspondig to TO 3 deleted; greater fontsize
 Fig. 9: Unified the xaxis of both plots; rearanged the plots in a column; greater fontsize; corrected minor mistake in legend
 Fig. 10: greater fontsize; corrected minor mistake in yaxis label
 Added Fig. 11 depicting the experimental setup
 Corrected and completed the affiliations of authors
 Added acknowledgements
 Corrected equation (26a) for <<sigma^x_i sigma^x_j>>
 Experimental setup more detailed explained
 Added references
Published as SciPost Phys. Core 6, 041 (2023)
Reports on this Submission
Anonymous Report 1 on 20221214 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2207.10439v2, delivered 20221214, doi: 10.21468/SciPost.Report.6302
Strengths
see my first report
Weaknesses
1technical quality of the figures and the graphical presentation is of sufficiently high quality now
2description of the experimental implementation is improved
3my other concerns here are largely answered satisfactorily
Report
Acceptance critera are now clearly met
Requested changes
remaining things to consider:
while the dilute nature of the atoms along the fiber will reduce nearest neighbor shifts and interactions, collective effects via far field interference of the scattered light could still be important in the large atom number limit
<sigma_i^x sigma_i^z> ~ <sigma_i^y > terms should be kept in principle
 Fig 8a: "meanfield" result for g2(0) ?
Shouldn'd this be always equal to 1 as
(a'a)^2 = (a')^2*(a)^2) in mean field ?
Author: Kasper Jan Kusmierek on 20221215 [id 3136]
(in reply to Report 1 on 20221214)We thank the referee for the positive feedback. Below we want to address the remaining questions:
We agree, that in principle collective effects may arise also due to decay to external modes. However, it is to be expected that these effects will be relatively small compared to the collective decay channels inside the waveguide on which we focused in the present work.
We agree on that point as well. Indeed, these terms are kept in our treatment. Products of Pauli operators referring to the same atom were always treated exactly by using the Pauli algebra. For example, we would write the above expression as <sigma_i^x sigma_i^z> ~ <sigma_i^y >=i<sigma_i^y ><sigma_i^y >. Factorizations always refer to tensor products of Paulis referring to different atoms.
One has to differentiate on which level the meanfield approximation is applied. Throughout the work we only applied the cumulant expansion method on the level of atomic operators. For calculating g2, we first apply inputoutput relations to express g2 in terms of atomic correlations, which gives Eq. (28) of the manuscript. ’Mean field’ then means that we factorize all two, three and fourbody correlation functions into products of singleparticle averages. We emphasize that this is different from the factorization (a†a)2 = (a†)^2 · (a)^2 on the level of field operators. E.g. mean field on the level of atomic operators still allows for antibunching arising from each atom individually.