Abstract

Optical microcavities supporting exciton–polariton quasi-particles offer one of the most powerful platforms for the investigation of the rapidly developing area of topological photonics in general, and of photonic topological insulators in particular. Energy bands of the microcavity polariton graphene are readily controlled by a magnetic field and influenced by the spin-orbit (SO) coupling effects, a combination leading to the formation of linear unidirectional edge states in polariton topological insulators as very recently predicted. In this work we depart from the linear limit of non-interacting polaritons and predict instabilities of the nonlinear topological edge states resulting in the formation of the localized topological quasi-solitons, which are exceptionally robust and immune to backscattering wave packets propagating along the graphene lattice edge. Our results provide a background for experimental studies of nonlinear polariton topological insulators and can influence other subareas of photonics and condensed matter physics, where nonlinearities and SO effects are often important and utilized for applications.

© 2016 Optical Society of America

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References

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2016 (3)

Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93, 155112 (2016).
[Crossref]

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Interacting quantum fluid in a polariton Chern insulator,” Phys. Rev. B 93, 085438 (2016).
[Crossref]

2015 (7)

A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton Z topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
[Crossref]

C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Topological polaritons and excitons in garden-variety systems,” Phys. Rev. B 91, 161413(R) (2015).
[Crossref]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaitre, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

S. Dufferwiel, F. Li, E. Cancellieri, L. Giriunas, A. A. P. Trichet, D. M. Whittaker, P. M. Walker, F. Fras, E. Clarke, J. M. Smith, M. S. Skolnick, and D. N. Krizhanovskii, “Spin textures of exciton-polaritons in a tunable microcavity with large TE-TM splitting,” Phys. Rev. Lett. 115, 246401 (2015).
[Crossref]

M. Milicevic, T. Ozawa, P. Andreakou, I. Carusotto, T. Jacqmin, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Edge states in polariton honeycomb lattices,” 2D Mater. 2, 034012 (2015).
[Crossref]

M. J. Ablowitz and Y.-P. Ma, “Strong transmission and reflection of edge modes in bounded photonic graphene,” Opt. Lett. 40, 4635–4638 (2015).
[Crossref]

P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Ultra-low-power hybrid light-matter solitons,” Nat. Commun. 6, 8317 (2015).
[Crossref]

2014 (5)

M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma, “Linear and nonlinear traveling edge waves in optical honeycomb lattices,” Phys. Rev. A 90, 023813 (2014).
[Crossref]

P. Cilibrizzi, H. Ohadi, T. Ostatnicky, A. Askitopoulos, W. Langbein, and P. Lagoudakis, “Linear wave dynamics explains observations attributed to dark solitons in polariton quantum fluids,” Phys. Rev. Lett. 113, 103901 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
[Crossref]

M. Sich, F. Fras, J. K. Chana, M. S. Skolnick, D. N. Krizhanovskii, A. V. Gorbach, R. Hartley, D. V. Skryabin, S. S. Gavrilov, E. A. Cerda-Méndez, K. Biermann, R. Hey, and P. V. Santos, “Effects of spin-dependent interactions on polarization of bright polariton solitons,” Phys. Rev. Lett. 112, 046403 (2014).
[Crossref]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. Solnyshkov, G. Malpuech, E. Galopin, A. Lematre, J. Bloch, and A. Amo, “Direct observation of Dirac cones and flat-band in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

2013 (6)

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111, 243905 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref]

M. J. Ablowitz, C. W. Curtis, and Y. Zhu, “Localized nonlinear edge states in honeycomb lattices,” Phys. Rev. A 88, 013850 (2013).
[Crossref]

D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaitre, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, “Polariton condensation in solitonic gap states in a one-dimensional periodic potential,” Nat. Commun. 4, 1749 (2013).
[Crossref]

E. A. Cerda-Mendez, D. Sarkar, D. N. Krizhanovskii, S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, “Exciton-polariton gap solitons in two-dimensional lattices,” Phys. Rev. Lett. 111, 146401 (2013).
[Crossref]

E. A. Ostrovskaya, J. Abdullaev, M. D. Fraser, A. S. Desyatnikov, and Y. S. Kivshar, “Self-localization of polariton condensates in periodic potentials,” Phys. Rev. Lett. 110, 170407 (2013).
[Crossref]

2012 (4)

F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre, V. Savona, and B. Deveaud, “Dissociation dynamics of singly charged vortices into half-quantum vortex pairs,” Nat. Commun. 3, 1309 (2012).
[Crossref]

M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez, K. Biermann, R. Hey, and P. V. Santos, “Observation of bright polariton solitons in a semiconductor microcavity,” Nat. Photonics 6, 50–55 (2012).
[Crossref]

R. O. Umucalilar and I. Carusotto, “Fractional quantum Hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref]

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2012).
[Crossref]

2011 (4)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332, 1167–1170 (2011).
[Crossref]

G. Nardin, G. Grosso, Y. Leger, B. Pietka, F. Morier-Genoud, and D. Deveaud-Pledran, “Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid,” Nat. Phys. 7, 635–641 (2011).
[Crossref]

2010 (3)

M. I. Molina and Y. S. Kivshar, “Discrete and surface solitons in photonic graphene nanoribbons,” Opt. Lett. 35, 2895–2897 (2010).
[Crossref]

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

H. Flayac, I. A. Shelykh, D. D. Solnyshkov, and G. Malpuech, “Topological stability of the half-vortices in spinor exciton-polariton condensates,” Phys. Rev. B 81, 045318 (2010).
[Crossref]

2009 (4)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nat. Phys. 5, 805–810 (2009).
[Crossref]

A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnik, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009).
[Crossref]

O. A. Egorov, D. V. Skryabin, A. V. Yulin, and F. Lederer, “Bright cavity polariton solitons,” Phys. Rev. Lett. 102, 153904 (2009).
[Crossref]

2008 (1)

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

2006 (1)

2004 (1)

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref]

Abbarchi, M.

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaitre, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. Solnyshkov, G. Malpuech, E. Galopin, A. Lematre, J. Bloch, and A. Amo, “Direct observation of Dirac cones and flat-band in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

Abdullaev, J.

E. A. Ostrovskaya, J. Abdullaev, M. D. Fraser, A. S. Desyatnikov, and Y. S. Kivshar, “Self-localization of polariton condensates in periodic potentials,” Phys. Rev. Lett. 110, 170407 (2013).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz and Y.-P. Ma, “Strong transmission and reflection of edge modes in bounded photonic graphene,” Opt. Lett. 40, 4635–4638 (2015).
[Crossref]

M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma, “Linear and nonlinear traveling edge waves in optical honeycomb lattices,” Phys. Rev. A 90, 023813 (2014).
[Crossref]

M. J. Ablowitz, C. W. Curtis, and Y. Zhu, “Localized nonlinear edge states in honeycomb lattices,” Phys. Rev. A 88, 013850 (2013).
[Crossref]

Adrados, C.

A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nat. Phys. 5, 805–810 (2009).
[Crossref]

Aitchison, J. S.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
[Crossref]

Alu, A.

Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93, 155112 (2016).
[Crossref]

Amo, A.

F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
[Crossref]

V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaitre, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

M. Milicevic, T. Ozawa, P. Andreakou, I. Carusotto, T. Jacqmin, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Edge states in polariton honeycomb lattices,” 2D Mater. 2, 034012 (2015).
[Crossref]

T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. Solnyshkov, G. Malpuech, E. Galopin, A. Lematre, J. Bloch, and A. Amo, “Direct observation of Dirac cones and flat-band in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
[Crossref]

D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaitre, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, “Polariton condensation in solitonic gap states in a one-dimensional periodic potential,” Nat. Commun. 4, 1749 (2013).
[Crossref]

A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332, 1167–1170 (2011).
[Crossref]

A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdre, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nat. Phys. 5, 805–810 (2009).
[Crossref]

A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnik, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009).
[Crossref]

Andre, R.

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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
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Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111, 243905 (2013).
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D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaitre, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, “Polariton condensation in solitonic gap states in a one-dimensional periodic potential,” Nat. Commun. 4, 1749 (2013).
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H. Flayac, I. A. Shelykh, D. D. Solnyshkov, and G. Malpuech, “Topological stability of the half-vortices in spinor exciton-polariton condensates,” Phys. Rev. B 81, 045318 (2010).
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M. Sich, F. Fras, J. K. Chana, M. S. Skolnick, D. N. Krizhanovskii, A. V. Gorbach, R. Hartley, D. V. Skryabin, S. S. Gavrilov, E. A. Cerda-Méndez, K. Biermann, R. Hey, and P. V. Santos, “Effects of spin-dependent interactions on polarization of bright polariton solitons,” Phys. Rev. Lett. 112, 046403 (2014).
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M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez, K. Biermann, R. Hey, and P. V. Santos, “Observation of bright polariton solitons in a semiconductor microcavity,” Nat. Photonics 6, 50–55 (2012).
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P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Ultra-low-power hybrid light-matter solitons,” Nat. Commun. 6, 8317 (2015).
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M. Sich, F. Fras, J. K. Chana, M. S. Skolnick, D. N. Krizhanovskii, A. V. Gorbach, R. Hartley, D. V. Skryabin, S. S. Gavrilov, E. A. Cerda-Méndez, K. Biermann, R. Hey, and P. V. Santos, “Effects of spin-dependent interactions on polarization of bright polariton solitons,” Phys. Rev. Lett. 112, 046403 (2014).
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E. A. Cerda-Mendez, D. Sarkar, D. N. Krizhanovskii, S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, “Exciton-polariton gap solitons in two-dimensional lattices,” Phys. Rev. Lett. 111, 146401 (2013).
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M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez, K. Biermann, R. Hey, and P. V. Santos, “Observation of bright polariton solitons in a semiconductor microcavity,” Nat. Photonics 6, 50–55 (2012).
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A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnik, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009).
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P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Ultra-low-power hybrid light-matter solitons,” Nat. Commun. 6, 8317 (2015).
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M. Sich, F. Fras, J. K. Chana, M. S. Skolnick, D. N. Krizhanovskii, A. V. Gorbach, R. Hartley, D. V. Skryabin, S. S. Gavrilov, E. A. Cerda-Méndez, K. Biermann, R. Hey, and P. V. Santos, “Effects of spin-dependent interactions on polarization of bright polariton solitons,” Phys. Rev. Lett. 112, 046403 (2014).
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M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez, K. Biermann, R. Hey, and P. V. Santos, “Observation of bright polariton solitons in a semiconductor microcavity,” Nat. Photonics 6, 50–55 (2012).
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T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. Solnyshkov, G. Malpuech, E. Galopin, A. Lematre, J. Bloch, and A. Amo, “Direct observation of Dirac cones and flat-band in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
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D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaitre, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, “Polariton condensation in solitonic gap states in a one-dimensional periodic potential,” Nat. Commun. 4, 1749 (2013).
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O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Interacting quantum fluid in a polariton Chern insulator,” Phys. Rev. B 93, 085438 (2016).
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H. Flayac, I. A. Shelykh, D. D. Solnyshkov, and G. Malpuech, “Topological stability of the half-vortices in spinor exciton-polariton condensates,” Phys. Rev. B 81, 045318 (2010).
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J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
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J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004).
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D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaitre, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, “Polariton condensation in solitonic gap states in a one-dimensional periodic potential,” Nat. Commun. 4, 1749 (2013).
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A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnik, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009).
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P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Ultra-low-power hybrid light-matter solitons,” Nat. Commun. 6, 8317 (2015).
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R. O. Umucalilar and I. Carusotto, “Fractional quantum Hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
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A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnik, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009).
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S. Dufferwiel, F. Li, E. Cancellieri, L. Giriunas, A. A. P. Trichet, D. M. Whittaker, P. M. Walker, F. Fras, E. Clarke, J. M. Smith, M. S. Skolnick, and D. N. Krizhanovskii, “Spin textures of exciton-polaritons in a tunable microcavity with large TE-TM splitting,” Phys. Rev. Lett. 115, 246401 (2015).
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P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Ultra-low-power hybrid light-matter solitons,” Nat. Commun. 6, 8317 (2015).
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S. Dufferwiel, F. Li, E. Cancellieri, L. Giriunas, A. A. P. Trichet, D. M. Whittaker, P. M. Walker, F. Fras, E. Clarke, J. M. Smith, M. S. Skolnick, and D. N. Krizhanovskii, “Spin textures of exciton-polaritons in a tunable microcavity with large TE-TM splitting,” Phys. Rev. Lett. 115, 246401 (2015).
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P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Ultra-low-power hybrid light-matter solitons,” Nat. Commun. 6, 8317 (2015).
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O. A. Egorov, D. V. Skryabin, A. V. Yulin, and F. Lederer, “Bright cavity polariton solitons,” Phys. Rev. Lett. 102, 153904 (2009).
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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
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M. Milicevic, T. Ozawa, P. Andreakou, I. Carusotto, T. Jacqmin, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Edge states in polariton honeycomb lattices,” 2D Mater. 2, 034012 (2015).
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D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaitre, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, “Polariton condensation in solitonic gap states in a one-dimensional periodic potential,” Nat. Commun. 4, 1749 (2013).
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A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2012).
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Nat. Photonics (2)

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
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M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gorbach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez, K. Biermann, R. Hey, and P. V. Santos, “Observation of bright polariton solitons in a semiconductor microcavity,” Nat. Photonics 6, 50–55 (2012).
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M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
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Nature (3)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
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A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lemaitre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnik, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009).
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O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Interacting quantum fluid in a polariton Chern insulator,” Phys. Rev. B 93, 085438 (2016).
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M. Sich, F. Fras, J. K. Chana, M. S. Skolnick, D. N. Krizhanovskii, A. V. Gorbach, R. Hartley, D. V. Skryabin, S. S. Gavrilov, E. A. Cerda-Méndez, K. Biermann, R. Hey, and P. V. Santos, “Effects of spin-dependent interactions on polarization of bright polariton solitons,” Phys. Rev. Lett. 112, 046403 (2014).
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E. A. Cerda-Mendez, D. Sarkar, D. N. Krizhanovskii, S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, “Exciton-polariton gap solitons in two-dimensional lattices,” Phys. Rev. Lett. 111, 146401 (2013).
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F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaitre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Türeci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016).
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S. Dufferwiel, F. Li, E. Cancellieri, L. Giriunas, A. A. P. Trichet, D. M. Whittaker, P. M. Walker, F. Fras, E. Clarke, J. M. Smith, M. S. Skolnick, and D. N. Krizhanovskii, “Spin textures of exciton-polaritons in a tunable microcavity with large TE-TM splitting,” Phys. Rev. Lett. 115, 246401 (2015).
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A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton Z topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
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Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111, 243905 (2013).
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R. O. Umucalilar and I. Carusotto, “Fractional quantum Hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
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V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaitre, H. Terças, A. Nalitov, M. Abbarchi, E. Galopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-orbit coupling for photons and polaritons in microstructures,” Phys. Rev. X 5, 011034 (2015).

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Science (1)

A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332, 1167–1170 (2011).
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Figures (9)

Fig. 1.
Fig. 1. First row shows examples of the array with zigzag and bearded edges (three periods of the structure are shown in the y direction). The second, third, and fourth rows illustrate the transformation of the edge-state profiles corresponding to the red lines (top gap) in Figs. 2(c) and 3(a) upon variation of the Bloch momentum k. The fifth row shows the edge state on the other end of the lattice. The sixth row shows the edge state corresponding to the magenta line from the bottom gap. In all cases the x[24,+24] window is shown, β=0.3, Ω=0.5. The left and right columns show |ψ+| and |ψ|, respectively.
Fig. 2.
Fig. 2. Energy–momentum diagrams ε(k) obtained for the lattice with zigzag edges at (a) β=0, (b) β=0.15, and (c) β=0.3 and for the lattice with bearded edges at (d) β=0.3. The black lines correspond to modes residing in the bulk of the lattice, while the colored lines indicate edge states. The topologically protected edge states in (b)–(d) with opposite slopes of ε(k) belong to the opposite edges of the lattice. In all cases Ω=0.5.
Fig. 3.
Fig. 3. (a) Width wx of linear edge states along the x axis and (b) the second-order dispersion coefficient 2ε/k2 for branches associated with edge states versus k at β=0.3, Ω=0.5. The red and green circles correspond to the linear modes depicted in Fig. 1. The line colors correspond to the colors used in Fig. 2(c) to denote different branches.
Fig. 4.
Fig. 4. Transformation of the nonlinear edge state bifurcating from the red branch in Fig. 2(c) at k=0.55  K with a decrease of μ. The left column shows |ψ+|, and the right column shows |ψ|. In all cases β=0.3, Ω=0.5.
Fig. 5.
Fig. 5. Norm U per period and peak amplitudes a+, a of the spin components versus μ for nonlinear edge states with (a) k=0.45  K and (b) k=0.55  K bifurcating from the red branch in Fig. 2(c) at μ=ε. (c) The X-width of the nonlinear edge state versus μ at k=0.55  K. The circles in (c) correspond to the states shown in Fig. 4. The dashed lines indicate the edge of the band for the corresponding k values. (d) The perturbation growth rate for the nonlinear edge state with μ=3.21 versus frequency of the small modulation κ. In all cases β=0.3, Ω=0.5.
Fig. 6.
Fig. 6. Instability development for the nonlinear edge state with μ=3.25, k=0.55  K stimulated by a small input noise. The first row shows the peak amplitude of the ψ component versus time, while the second row shows the array and |ψ| distributions at different moments of time corresponding to red circles. Here β=0.3, Ω=0.5.
Fig. 7.
Fig. 7. Comparison of nonlinear (top row) and linear (bottom row) evolution of the edge state with the localized soliton envelope at με=0.02, k=0.55  K. The middle row shows the peak amplitude a of the ψ component for the linear and nonlinear cases. The |ψ| distributions in the top and bottom rows correspond to red circles. Notice that the edge states in all panels in this figure move upward. To simultaneously stress the fact that the edge state moves in one direction and to provide details of its shape, we show ψ distributions within a relatively small vertical window, but we also applied an identical vertical shift for distributions at time moments t=420,540,660, where the edge state in the nonlinear medium remains nearly invariable. The distribution at t=0 was not shifted. Here β=0.3, Ω=0.5.
Fig. 8.
Fig. 8. Evolution of edge state with the localized soliton envelope at με=0.02, k=0.55  K in a rectangular array. Only |ψ| distributions are shown. In all cases β=0.3, Ω=0.5. The left plot shows the array.
Fig. 9.
Fig. 9. Passage of the soliton obtained from the edge state with the localized envelope at με=0.02, k=0.55  K through a surface defect. The soliton was allowed to move along the surface of the array over a considerable time interval before its collision with the defect. Only the ψ component is shown. Here β=0.3, Ω=0.5.

Equations (8)

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iψ±t=22m*(2x2+2y2)ψ±+β2m*(xiy)2ψ+[ε0R(x,y)±εz]ψ±+ε0(|ψ±|2+σ|ψ|2)ψ±.
iψ±t=12(2x2+2y2)ψ±+β(xiy)2ψ+R(x,y)ψ±±Ωψ±+(|ψ±|2+σ|ψ|2)ψ±,
χ±=U±2π/K+π/Kdy|ψ±|4dx,U±=π/K+π/Kdy|ψ±|2dx
μu±=12(2x2+2y2+2ikyk2)u±+R(x,y)u±+β(xi(y+ik))2u±Ωu±+(|u±|2+σ|u|2)u±.
Ψ(x,y,t)=eiεt+ikyn=0,(i)nn!(nU/kn)[nA(y,t)/yn]
LΨ=eiεt+ikyn=0,(i)nn![n(εU)/kn][nA(y,t)/yn].
iAt=iεAy+12ε2Ay2+gA|A|2,
A(y,t)=[2(εμ)/g]1/2sech{[2(εμ)/ε]1/2(y+εt)}×exp[i(με)t],

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