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Polariton condensation into vortex states in the synthetic magnetic field of a strained honeycomb lattice

by C. Lledó, I. Carusotto, and M. H. Szymańska

This is not the current version.

Submission summary

As Contributors: Cristóbal Lledó
Preprint link: scipost_202104_00031v2
Date submitted: 2021-09-16 23:35
Submitted by: Lledó, Cristóbal
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


Photonic materials are a rapidly growing platform for studying condensed matter physics with light, where the exquisite control capability is allowing us to learn about the relation between microscopic dynamics and macroscopic properties. One of the most interesting aspects of condensed matter is the interplay between interactions and the effect of an external magnetic field or rotation, responsible for a plethora of rich phenomena---Hall physics and quantized vortex arrays. At first sight, however, these effects for photons seem vetoed: they do not interact with each other and they are immune to magnetic fields and rotations. Yet in specially devised structures these effects can be engineered. Here, we propose the use of a synthetic magnetic field induced by strain in a honeycomb lattice of resonators to create a non-equilibrium Bose-Einstein condensate of light-matter particles (polaritons) in a rotating state, without the actual need for external rotation nor reciprocity-breaking elements. We show that thanks to the competition between interactions, dissipation and a suitably designed incoherent pump, the condensate spontaneously becomes chiral by selecting a single Dirac valley of the honeycomb lattice, occupying the lowest Landau level and forming a vortex array. Our results offer a new platform where to study the exciting physics of arrays of quantized vortices with light and pave the way to explore the transition from a vortex-dominated phase to the photonic analogue of the fractional quantum Hall regime.

Current status:
Has been resubmitted

Author comments upon resubmission

Dear Editor,

Please find submitted a new version of our manuscript titled “Polariton condensation into vortex states in the synthetic magnetic field of a strained honeycomb lattice”, for publication in SciPost Physics. We have effected anew a series of amendments in the main text.

Reply to the Referee:

We thank the reviewer for his comments and his positive assessment. We have addressed his comments in the revised version of the manuscript.

  1. The choice in the direction of the strain is what breaks the symmetry between A and B sites and thus determines the A or B sublattice polarization of the n=0 Landau level (see [29-32] and Appendix A). For positive (negative) strain parameter, $\tau>0$ ($\tau<0$), the $n=0$ wavefunction is localized in the B (A) sublattice. This is mentioned in the paragraph preceding Eq.(3), ``... completely localized in the B sublattice (it would be in the $A$ sublattice if $\tau$ was negative)''. In the revised text, we have added reference to a few works where the interested reader can find a discussion of this fact. A summary of the analytical derivation is available in Appendix A.

  2. This is an interesting point. At the end of Sec.2.1, we have added a sentence explaining why we expect that the mode selection mechanism based on spatial overlap with the pump should largely exceed all other mode selection mechanisms.

  3. Indeed, as the referee points out, what we have in mind is the homogeneity of the condensate density in real space. Among the available states in which condensation can occur (the degenerate $n=0$ Landau level states), it is favourable to choose a linear combination in a single valley to reduce the interaction energy. When only positive or negative angular momenta states are available, like in rotating cold atoms, the energy is minimized forming a triangular array of vortices or antivortices, which the result of Abrikosov. But what happens if both positive and negative angular momenta are available like in our case? A linear combination of states with both chiralities would entail strong inhomogeneities in real space due to the destructive interference. To give an analytical example, let us imagine for a moment that we are dealing with a continuous space like in rotating BECs. Forming an equally-weighted linear combination of $m=1$ and $m=-1$ single-particle states, would increase, compared to a state with just $m=1$ or $-1$, the interaction energy ($\propto \int |\psi(r)|^4 d^2 r$) by a factor of $\left(\int\limits_0^{2\pi}4\cos^4(\phi)d\phi\right)/\left(\int\limits_0^{2\pi}d\phi\right) = 3/2$. Thus a single orientation is favoured.

We have modified our previous sentence, highlighted by the referee, by ``On the contrary, interactions play instead a key role in the case of wider incoherent pump profiles, where they make condensation in a single Dirac valley---as opposed to in both valleys simultaneously---more likely. In standard BECs at thermal equilibrium, the interaction energy is minimized by choosing among the single-particle ground states the one which has the most homogenous density in real space. Here, a similar mechanism hinders the condensate from occupying both valleys at the same time, which would entail large density inhomogeneities due to destructive interference of states with opposite angular momenta.''

Regarding the the two suggested papers, they are clearly relevant and we now cite them. We have now clarified in the manuscript, though, that we are not dealing with transient phenomena like Kibble-Zurek physics, which is beyond the scope of our work.

  1. We have refined the tuning of the sentences in the outlook modifying the last paragraph: ``When applied to strongly-interacting photonic lattices...''.

In view of the above modifications and qualifications, we would be grateful if you could consider our manuscript for publication in SciPost Physics.

Yours sincerely,

C. Lledó, I. Carusotto, M. H. Szymaǹska

List of changes

1) Added references [44, 45, 47, 50].

2) In the last paragraph of the Introduction, we changed the sentence " used in the recent works [27, 40–43]..." for " used in the recent experiments [27,40–43] and theoretical works [31,44,45]..."

3) In the paragraph following Eq. (3), we changed "...We will take advantage of this to induce polariton condensation into the n = 0 Landau level." for an extended version "...We will take advantage of this to induce polariton condensation into the n = 0 Landau level [47]: given the full sublattice polarization of this state, we expect that the mode selection mechanism based on spatial overlap with the pumped sublattice will largely exceed other mechanisms that may be active in polariton systems, such as energy relaxation, the exciton-photon fraction of modes, and the amount of localization of the wave function at the pillars."

4) We have corrected a typo at the beginning of Sec. 2.3 : "$P_B >~ \gamma$" has been replaced for $P_B \gtrsim \gamma$.

5) At the end of Sec. 2.2 we have included the new paragraph "The results that we are going to show in the following of the work refer to the long-time limit of the system evolution. A study of the additional features that may occur during the switch-on transient, such as Kibble-Zurek vortex nucleation phenomena [45] go beyond the scope of this work."

6) We have have erased the words "...the y-reflection symmetry is explicitly broken..." from the first paragraph of the Discussion section, as it seemed to obscure the proposal.

7) We have rewritten the last paragraph of the Outlook.

8) We have added a "Data and Code Availability" section before the Acknowledgements.

Submission & Refereeing History

Resubmission scipost_202104_00031v3 on 6 January 2022

Reports on this Submission

Anonymous Report 2 on 2021-12-19 (Invited Report)


1. interesting effect that combines interaction and strain-induced synthetic field to produce an effectively rotating polariton condensate.
2. The manuscript is well written and contains clear results. It is likely to stimulate experiments.


interaction effects are small and cannot bring the system to the even more interesting fractional Hall regime


The manuscript combines a series of insights to propose an alternative system for observing strongly rotating bosons. It is clear in its presentation and objectives, and I believe it could be published with minor edits.
1. It would be helpful to have an explanation of the pump-noise correlation below eq 3. Also, the sign of gamma in eq 3 and in the pump-noise expression is opposite. Is one of them a typo?
2. I couldn't find a discussion of the experimentally achievable interaction strength. I suggest such a discussion be added.
3. I suggest adding citations to other works producing a synthetic rotation in atomic systems:
and so on. I would encourage the authors to cite more broadly also key works on topological polaritonics.

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

Report 1 by Guillaume Malpuech on 2021-11-9 (Invited Report)


I would like first to apologize for the delay in my reply which is partly due to technical problems between scipost emails and the french university network Renater...

I thank the authors for their reply and I think the manuscript can be published without further changes.

I have still a few remarks but they don’t request actions from the authors.

Regarding the question 2 about mode selection, the overlap with the pump is reduced by a factor 2 for states being equally on A and B with respect to states B polarized. This type of factor 2 can easily be reached by the other mechanisms I was mentioning. So it is our theoretician right not to take them into account, but in a real specific case, they would need to be evaluated.

Regarding point 3, I now better understand the point. On the other hand, I would not expect the m=1 and m=-1 state being spatially at the same position. I understand also that the authors really want to discuss an equilibrium steady state and not the consequence of some transient effect and it is perfectly fine to justify that taking these effects into account is beyond the scope of the present work. On the other hand, this transient necessarily occurs when the condensate forms (also in the numerical simulation). Traces of this transient may indeed survive in the steady state. This is especially true in this lattice configuration where vortices cannot move easily once formed.

Best regards,
Guillaume MALPUECH

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
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  • grammar: -

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