Abstract

We studied optical response of microcavity non-equilibrium exciton-polariton Bose-Einstein condensate with saturable nonlinearity under simultaneous resonant and non-resonant pumping. We demonstrated the emergence of multistabile behavior due to the saturation of the excitonic absorption. Stable periodic Rabi-type oscillations of the excitonic and photonic condensate components in the regime of the stationary pump and their transition to the chaotic dynamics through the cascade of Hopf bifurcations by tuning of the electrical pump are revealed.

© 2016 Optical Society of America

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2016 (2)

I. Yu. Chestnov, S. S. Demirchyan, A.P. Alodjants, Yu.G. Rubo, and A.V. Kavokin, “Permanent Rabi oscillations in coupled exciton-photon systems with PT -symmetry,” Sci. Rep. 6, 19551 (2016).
[Crossref] [PubMed]

V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
[Crossref] [PubMed]

2015 (3)

M. Amthor, T.C.H. Liew, C. Metzger, S. Brodbeck, L. Worschech, M. Kamp, I. A. Shelykh, A. V. Kavokin, C. Schneider, and S. Hoefling, “Optical bistability in electrically driven polariton condensates,” Phys. Rev. B 91, 081404 (2015).
[Crossref]

J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2015).
[Crossref]

N. Takemura, M. D. Anderson, S. Trebaol, S. Biswas, D. Y. Oberli, M. T. Portella-Oberli, and B. Deveaud, “Dephasing effects on coherent exciton-polaritons and the breakdown of the strong coupling regime,” Phys. Rev. B 92, 235305 (2015).
[Crossref]

2014 (3)

M. De Giorgi, M. Ballarini, P. Cazzato, G. Deligeorgis, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, G. Gigli, F.P. Laussy, and D. Sanvitto, “Relaxation Oscillations in the Formation of a Polariton Condensate,” Phys. Rev. Lett. 112, 113602 (2014).
[Crossref] [PubMed]

S. S. Demirchyan, Y. Chestnov, A. P. Alodjants, M. M. Glazov, and A. V. Kavokin, “Qubits Based on Polariton Rabi Oscillators,” Phys. Rev. Lett. 112, 196403 (2014).
[Crossref] [PubMed]

T. C. H. Liew, Y. G. Rubo, and A. V. Kavokin, “Exciton-polariton oscillations in real space,” Phys. Rev. B 90, 245309 (2014).
[Crossref]

2013 (5)

M. Abbarchi, A. Amo, V. G. Sala, D. D. Solnyshkov, H. Flayac, L. Ferrier, I. Sagnes, E. Galopin, A. Lemaitre, G. Malpuech, and J. Bloch, “Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons,” Nat. Phys. 9, 275–279 (2013).
[Crossref]

C. Schneider, A. Rahimi-Iman, N. YoungKim, J. Fischer, I.G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V.D. Kulakovskii, I.A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Hoefling, “Print Share/bookmark An electrically pumped polariton laser,” Nature 497, 348–352 (2013).
[Crossref]

T. Espinosa-Ortega and T. C. H. Liew, “Complete architecture of integrated photonic circuits based on and and not logic gates of exciton polaritons in semiconductor microcavities,” Phys. Rev. B 87, 195305 (2013).
[Crossref]

D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdr, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, “All-optical polariton transistor,” Nat. Commun. 4, 1778 (2013).
[Crossref] [PubMed]

I. Carusotto and C. Ciuti, “Quantum Fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

2012 (3)

M. Steger, C. Gautham, B. Nelsen, D. Snoke, L. Pfeiffer, and K. West, “Single-wavelength, all-optical switching based on exciton-polaritons,” Appl. Phys. Lett. 101, 131104 (2012).
[Crossref]

M. De Giorgi, D. Ballarini, E. Cancellieri, F.M. Marchetti, M.H. Szymanska, C. Tejedor, R. Cingolani, E. Giacobino, A. Bramati, G. Gigli, and D. Sanvitto, “Control and ultrafast dynamics of a two-fluid polariton switch,” Phys. Rev. Lett. 109, 266407 (2012).
[Crossref]

H. Eleucha and A. Prasad, “Chaos and regularity in semiconductor microcavities,” Phys. Lett. A 376, 1970 (2012).
[Crossref]

2011 (1)

C. Adrados, T. C. H. Liew, A. Amo, M. D. Martin, D. Sanvitto, C. Anton, E. Giacobino, A. Kavokin, A. Bramati, and L. Vina, “Motion of Spin Polariton Bullets in Semiconductor Microcavities,” Phys. Rev. Lett. 107, 146402 (2011).
[Crossref] [PubMed]

2010 (5)

A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photonics 4, 361–366 (2010).
[Crossref]

T. C. H. Liew, A. V. Kavokin, T. Ostatnicky, M. Kaliteevski, I. A. Shelykh, and R. A. Abram, “Exciton-polariton integrated circuits,” Phys. Rev. B 82, 033302 (2010).
[Crossref]

T. K. Paraiso, M. Wouters, Y. Leger, F. Morier-Genoud, and B. Deveaud-Pledran, “Multistability of a coherent spin ensemble in a semiconductor microcavity,” Nature Materials 9, 655–660 (2010).
[Crossref] [PubMed]

O. A. Egorov, D. V. Skryabin, and F. Lederer, “Polariton solitons due to saturation of the exciton-photon coupling,” Phys. Rev. B 82, 165326 (2010)
[Crossref]

H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton bose-einstein condensation,” Rev. Mod. Phys. 82, 1489 (2010).
[Crossref]

2009 (2)

D. D. Solnyshkov, R. Johne, I. A. Shelykh, and G. Malpuech, “Chaotic Josephson oscillations of exciton-polaritons and their applications,” Phys. Rev. B 80, 235303 (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,” Nature Physics 5, 805810 (2009).
[Crossref]

2008 (2)

T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical Circuits Based on Polariton Neurons in Semiconductor Microcavities,” Phys. Rev. Lett. 101, 016402 (2008).
[Crossref] [PubMed]

D. Bajoni, E. Semenova, A. Lemaitre, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, and J. Bloch, “Optical Bistability in a GaAs-Based Polariton Diode,” Phys. Rev. Lett. 101, 266402 (2008).
[Crossref] [PubMed]

2007 (1)

N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization Multistability of Cavity Polaritons,” Phys. Rev. Lett. 98, 236401 (2007).
[Crossref] [PubMed]

2006 (2)

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lematre, J. Bloch, Ph. Roussignolm, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature (London) 440, 904–907 (2006).
[Crossref]

J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
[Crossref] [PubMed]

2004 (1)

M. Gurioli, L. Cavigli, G. Khitrova, and H.M. Gibbs, “Bistable optical response in quantum well semiconductor microcavity,” Semicond. Sci. Technol. 19, S345 (2004).
[Crossref]

2003 (1)

C. Ciuti, P. Schwendimann, and A. Quattropani, “Theory of polariton parametric interactions in semiconductor microcavities,” Semicond. Sci. Technol. 18, S279 (2003).
[Crossref]

2000 (3)

G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, and P. Schwendimann, “Excitonic Bloch equations for a two-dimensional system of interacting excitons,” Phys. Rev. B 61, 13856 (2000).
[Crossref]

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Angle-Resonant Stimulated Polariton Amplifier,” Phys. Rev. Lett. 84, 1547 (2000).
[Crossref] [PubMed]

R. M. Stevenson, V. N. Astratov, M. S. Skolnick, D. M. Whittaker, M. Emam-Ismail, A. I. Tartakovskii, P. G. Savvidis, J. J. Baumberg, and J. S. Roberts, “Continuous Wave Observation of Massive Polariton Redistribution by Stimulated Scattering in Semiconductor Microcavities,” Phys. Rev. Lett. 85, 3680 (2000).
[Crossref] [PubMed]

1996 (1)

A. Tredicucci, Y. Chen, V. Pellegrini, M. Berger, and F. Bassani, “Optical bistability of semiconductor microcavities in the strong-coupling regime,” Phys. Rev. A 54, 3493 (1996).
[Crossref] [PubMed]

1993 (1)

K. M. Cuomo and A. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65–68 (1993).
[Crossref] [PubMed]

1988 (1)

R. Zimmermann, “Nonlinear Optics and the Mott Transition in Semiconductors,” Physica Status Solidi (b) 146, 371 (1988).
[Crossref]

1983 (1)

M. L. Steyn-Ross and C. W. Gardiner, “Quantum theory of excitonic optical bistability,” Phys. Rev. A 27, 310 (1983).
[Crossref]

Abbarchi, M.

M. Abbarchi, A. Amo, V. G. Sala, D. D. Solnyshkov, H. Flayac, L. Ferrier, I. Sagnes, E. Galopin, A. Lemaitre, G. Malpuech, and J. Bloch, “Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons,” Nat. Phys. 9, 275–279 (2013).
[Crossref]

Abram, R. A.

T. C. H. Liew, A. V. Kavokin, T. Ostatnicky, M. Kaliteevski, I. A. Shelykh, and R. A. Abram, “Exciton-polariton integrated circuits,” Phys. Rev. B 82, 033302 (2010).
[Crossref]

Adrados, C.

C. Adrados, T. C. H. Liew, A. Amo, M. D. Martin, D. Sanvitto, C. Anton, E. Giacobino, A. Kavokin, A. Bramati, and L. Vina, “Motion of Spin Polariton Bullets in Semiconductor Microcavities,” Phys. Rev. Lett. 107, 146402 (2011).
[Crossref] [PubMed]

A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photonics 4, 361–366 (2010).
[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,” Nature Physics 5, 805810 (2009).
[Crossref]

Afanasiev, M.M.

V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
[Crossref] [PubMed]

Alodjants, A. P.

S. S. Demirchyan, Y. Chestnov, A. P. Alodjants, M. M. Glazov, and A. V. Kavokin, “Qubits Based on Polariton Rabi Oscillators,” Phys. Rev. Lett. 112, 196403 (2014).
[Crossref] [PubMed]

Alodjants, A.P.

I. Yu. Chestnov, S. S. Demirchyan, A.P. Alodjants, Yu.G. Rubo, and A.V. Kavokin, “Permanent Rabi oscillations in coupled exciton-photon systems with PT -symmetry,” Sci. Rep. 6, 19551 (2016).
[Crossref] [PubMed]

Amo, A.

M. Abbarchi, A. Amo, V. G. Sala, D. D. Solnyshkov, H. Flayac, L. Ferrier, I. Sagnes, E. Galopin, A. Lemaitre, G. Malpuech, and J. Bloch, “Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons,” Nat. Phys. 9, 275–279 (2013).
[Crossref]

C. Adrados, T. C. H. Liew, A. Amo, M. D. Martin, D. Sanvitto, C. Anton, E. Giacobino, A. Kavokin, A. Bramati, and L. Vina, “Motion of Spin Polariton Bullets in Semiconductor Microcavities,” Phys. Rev. Lett. 107, 146402 (2011).
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D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdr, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, “All-optical polariton transistor,” Nat. Commun. 4, 1778 (2013).
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C. Adrados, T. C. H. Liew, A. Amo, M. D. Martin, D. Sanvitto, C. Anton, E. Giacobino, A. Kavokin, A. Bramati, and L. Vina, “Motion of Spin Polariton Bullets in Semiconductor Microcavities,” Phys. Rev. Lett. 107, 146402 (2011).
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A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photonics 4, 361–366 (2010).
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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,” Nature Physics 5, 805810 (2009).
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M. Gurioli, L. Cavigli, G. Khitrova, and H.M. Gibbs, “Bistable optical response in quantum well semiconductor microcavity,” Semicond. Sci. Technol. 19, S345 (2004).
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M. De Giorgi, M. Ballarini, P. Cazzato, G. Deligeorgis, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, G. Gigli, F.P. Laussy, and D. Sanvitto, “Relaxation Oscillations in the Formation of a Polariton Condensate,” Phys. Rev. Lett. 112, 113602 (2014).
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M. De Giorgi, D. Ballarini, E. Cancellieri, F.M. Marchetti, M.H. Szymanska, C. Tejedor, R. Cingolani, E. Giacobino, A. Bramati, G. Gigli, and D. Sanvitto, “Control and ultrafast dynamics of a two-fluid polariton switch,” Phys. Rev. Lett. 109, 266407 (2012).
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N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization Multistability of Cavity Polaritons,” Phys. Rev. Lett. 98, 236401 (2007).
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S. S. Gavrilov, V. D. Kulakovskii, S. G. Tikhodeev, and N. A. Gippius, “Spin oscillations and chaos in polariton condensates,” arXiv:1511.07284.

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S. S. Demirchyan, Y. Chestnov, A. P. Alodjants, M. M. Glazov, and A. V. Kavokin, “Qubits Based on Polariton Rabi Oscillators,” Phys. Rev. Lett. 112, 196403 (2014).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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M. De Giorgi, M. Ballarini, P. Cazzato, G. Deligeorgis, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, G. Gigli, F.P. Laussy, and D. Sanvitto, “Relaxation Oscillations in the Formation of a Polariton Condensate,” Phys. Rev. Lett. 112, 113602 (2014).
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M. Amthor, T.C.H. Liew, C. Metzger, S. Brodbeck, L. Worschech, M. Kamp, I. A. Shelykh, A. V. Kavokin, C. Schneider, and S. Hoefling, “Optical bistability in electrically driven polariton condensates,” Phys. Rev. B 91, 081404 (2015).
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C. Schneider, A. Rahimi-Iman, N. YoungKim, J. Fischer, I.G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V.D. Kulakovskii, I.A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Hoefling, “Print Share/bookmark An electrically pumped polariton laser,” Nature 497, 348–352 (2013).
[Crossref]

Houdr, R.

D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdr, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, “All-optical polariton transistor,” Nat. Commun. 4, 1778 (2013).
[Crossref] [PubMed]

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A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photonics 4, 361–366 (2010).
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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,” Nature Physics 5, 805810 (2009).
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J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2015).
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J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2015).
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J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
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D. D. Solnyshkov, R. Johne, I. A. Shelykh, and G. Malpuech, “Chaotic Josephson oscillations of exciton-polaritons and their applications,” Phys. Rev. B 80, 235303 (2009).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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M. Amthor, T.C.H. Liew, C. Metzger, S. Brodbeck, L. Worschech, M. Kamp, I. A. Shelykh, A. V. Kavokin, C. Schneider, and S. Hoefling, “Optical bistability in electrically driven polariton condensates,” Phys. Rev. B 91, 081404 (2015).
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C. Schneider, A. Rahimi-Iman, N. YoungKim, J. Fischer, I.G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V.D. Kulakovskii, I.A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Hoefling, “Print Share/bookmark An electrically pumped polariton laser,” Nature 497, 348–352 (2013).
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Kasprzak, J.

J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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C. Adrados, T. C. H. Liew, A. Amo, M. D. Martin, D. Sanvitto, C. Anton, E. Giacobino, A. Kavokin, A. Bramati, and L. Vina, “Motion of Spin Polariton Bullets in Semiconductor Microcavities,” Phys. Rev. Lett. 107, 146402 (2011).
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A. Kavokin, J. Baumberg, G. Malpuech, and F. Laussy, Microcavities (Clarendon, 2006).

Kavokin, A. V.

M. Amthor, T.C.H. Liew, C. Metzger, S. Brodbeck, L. Worschech, M. Kamp, I. A. Shelykh, A. V. Kavokin, C. Schneider, and S. Hoefling, “Optical bistability in electrically driven polariton condensates,” Phys. Rev. B 91, 081404 (2015).
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S. S. Demirchyan, Y. Chestnov, A. P. Alodjants, M. M. Glazov, and A. V. Kavokin, “Qubits Based on Polariton Rabi Oscillators,” Phys. Rev. Lett. 112, 196403 (2014).
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T. C. H. Liew, Y. G. Rubo, and A. V. Kavokin, “Exciton-polariton oscillations in real space,” Phys. Rev. B 90, 245309 (2014).
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A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photonics 4, 361–366 (2010).
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T. C. H. Liew, A. V. Kavokin, T. Ostatnicky, M. Kaliteevski, I. A. Shelykh, and R. A. Abram, “Exciton-polariton integrated circuits,” Phys. Rev. B 82, 033302 (2010).
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T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical Circuits Based on Polariton Neurons in Semiconductor Microcavities,” Phys. Rev. Lett. 101, 016402 (2008).
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N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization Multistability of Cavity Polaritons,” Phys. Rev. Lett. 98, 236401 (2007).
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I. Yu. Chestnov, S. S. Demirchyan, A.P. Alodjants, Yu.G. Rubo, and A.V. Kavokin, “Permanent Rabi oscillations in coupled exciton-photon systems with PT -symmetry,” Sci. Rep. 6, 19551 (2016).
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J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
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M. Gurioli, L. Cavigli, G. Khitrova, and H.M. Gibbs, “Bistable optical response in quantum well semiconductor microcavity,” Semicond. Sci. Technol. 19, S345 (2004).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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S. S. Gavrilov, V. D. Kulakovskii, S. G. Tikhodeev, and N. A. Gippius, “Spin oscillations and chaos in polariton condensates,” arXiv:1511.07284.

Kulakovskii, V.D.

C. Schneider, A. Rahimi-Iman, N. YoungKim, J. Fischer, I.G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V.D. Kulakovskii, I.A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Hoefling, “Print Share/bookmark An electrically pumped polariton laser,” Nature 497, 348–352 (2013).
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J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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A. Kavokin, J. Baumberg, G. Malpuech, and F. Laussy, Microcavities (Clarendon, 2006).

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M. De Giorgi, M. Ballarini, P. Cazzato, G. Deligeorgis, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, G. Gigli, F.P. Laussy, and D. Sanvitto, “Relaxation Oscillations in the Formation of a Polariton Condensate,” Phys. Rev. Lett. 112, 113602 (2014).
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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,” Nature Physics 5, 805810 (2009).
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D. Bajoni, E. Semenova, A. Lemaitre, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, and J. Bloch, “Optical Bistability in a GaAs-Based Polariton Diode,” Phys. Rev. Lett. 101, 266402 (2008).
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C. Schneider, A. Rahimi-Iman, N. YoungKim, J. Fischer, I.G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V.D. Kulakovskii, I.A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Hoefling, “Print Share/bookmark An electrically pumped polariton laser,” Nature 497, 348–352 (2013).
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J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2015).
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T. C. H. Liew, Y. G. Rubo, and A. V. Kavokin, “Exciton-polariton oscillations in real space,” Phys. Rev. B 90, 245309 (2014).
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C. Adrados, T. C. H. Liew, A. Amo, M. D. Martin, D. Sanvitto, C. Anton, E. Giacobino, A. Kavokin, A. Bramati, and L. Vina, “Motion of Spin Polariton Bullets in Semiconductor Microcavities,” Phys. Rev. Lett. 107, 146402 (2011).
[Crossref] [PubMed]

A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nature Photonics 4, 361–366 (2010).
[Crossref]

T. C. H. Liew, A. V. Kavokin, T. Ostatnicky, M. Kaliteevski, I. A. Shelykh, and R. A. Abram, “Exciton-polariton integrated circuits,” Phys. Rev. B 82, 033302 (2010).
[Crossref]

T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, “Optical Circuits Based on Polariton Neurons in Semiconductor Microcavities,” Phys. Rev. Lett. 101, 016402 (2008).
[Crossref] [PubMed]

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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
[Crossref] [PubMed]

M. Amthor, T.C.H. Liew, C. Metzger, S. Brodbeck, L. Worschech, M. Kamp, I. A. Shelykh, A. V. Kavokin, C. Schneider, and S. Hoefling, “Optical bistability in electrically driven polariton condensates,” Phys. Rev. B 91, 081404 (2015).
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J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
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V.P. Kochereshko, M.V. Durnev, L. Besombes, H. Mariette, V.F. Sapega, A. Askitopoulos, I.G. Savenko, T.C.H. Liew, I.A. Shelykh, A.V. Platonov, S.I. Tsintzos, Z. Hatzopoulos, P.G. Savvidis, V.K. Kalevich, M.M. Afanasiev, V.A. Lukoshkin, C. Schneider, M. Amthor, C. Metzger, M. Kamp, S. Hoefling, P. Lagoudakis, and A. Kavokin, “Lasing in Bose-Fermi mixtures,” Scientific Reports 6, 20091 (2016).
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M. Abbarchi, A. Amo, V. G. Sala, D. D. Solnyshkov, H. Flayac, L. Ferrier, I. Sagnes, E. Galopin, A. Lemaitre, G. Malpuech, and J. Bloch, “Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons,” Nat. Phys. 9, 275–279 (2013).
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D. D. Solnyshkov, R. Johne, I. A. Shelykh, and G. Malpuech, “Chaotic Josephson oscillations of exciton-polaritons and their applications,” Phys. Rev. B 80, 235303 (2009).
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N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization Multistability of Cavity Polaritons,” Phys. Rev. Lett. 98, 236401 (2007).
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A. Kavokin, J. Baumberg, G. Malpuech, and F. Laussy, Microcavities (Clarendon, 2006).

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J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006).
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N. Takemura, M. D. Anderson, S. Trebaol, S. Biswas, D. Y. Oberli, M. T. Portella-Oberli, and B. Deveaud, “Dephasing effects on coherent exciton-polaritons and the breakdown of the strong coupling regime,” Phys. Rev. B 92, 235305 (2015).
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T. K. Paraiso, M. Wouters, Y. Leger, F. Morier-Genoud, and B. Deveaud-Pledran, “Multistability of a coherent spin ensemble in a semiconductor microcavity,” Nature Materials 9, 655–660 (2010).
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[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Sketch of the system under consideration: a polaritonic microcavity is pumped simultaneously with an electrical pump PX and optical pump PC.
Fig. 2
Fig. 2 (a) Exciton amplitude vs the optical pump power for the absent electrical pumping at the pumping detuning equal to −9 meV (green line), −2 meV (red line) and 5 meV (black line); all other parameters can be found in the text. (b) Exciton amplitude vs the optical pump power at δC = δX = 9meV at PX = 0 (black line), 1.12 (blue line) and 1.2 (green line. (c) Exciton amplitude vs the electrical pump power at δC = δX = −9meV at PC = 0.2 (green line), 0.5 (black line), and 0.02 (blue line). In figures (a–c) solid lines correspond to the saturable case λ = 1 and dotted to λ = 0. (d) Three dimensional plot of exciton amplitude vs electrical and optical pump for λ = 1 and δC = δX = 9 meV.
Fig. 3
Fig. 3 Spectrum (left column and phase space (|ϕ(t), χ(t)|) (middle column) and (Reϕ(t),Imχ(t)) (right column) trajectories of the system at δC = −9 meV and PC = 0.5, PX = 0.55 (a,b,c), PC = 0.5, PX = 0.27 (d,e,f), PC = 0.2, PX = 0.09 (g,h,i). In all figures solid black lines correspond to saturable case λ = 1, and dashed green to λ = 0. In figure (b) red point corresponds to the stationary solution for the saturable case.
Fig. 4
Fig. 4 Time dependance the of the exciton and photon field at the same parameters as in Fig. 3(a) for λ = 1.

Equations (9)

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= h ¯ ω C ϕ ϕ + h ¯ ω X χ χ + g 2 | χ | 4 + h ¯ Ω R ( 1 λ a B 2 | χ | 2 ) ( ϕ χ + χ ϕ ) ,
i ϕ ˜ ˙ = ( δ ˜ C i γ ˜ C ) ϕ ˜ + ( 1 λ | χ ˜ | 2 ) χ ˜ + P ˜ C , i χ ˜ ˙ = ( δ ˜ X i γ ˜ X ) χ ˜ + ( 1 λ | χ ˜ | 2 ) ϕ ˜ 2 λ χ ˜ Re ( ϕ ˜ * χ ˜ ) + g ˜ | χ ˜ | 2 χ ˜ + i P ˜ X χ ˜ ( 1 λ | χ ˜ | 2 ) ,
ϕ = e i φ 0 δ C 2 + γ C 2 ( [ 1 λ | χ | 2 ] χ + P C ) ,
δ X X + g X 3 ( 1 3 λ X 2 ) P C cos ( φ φ 0 ) ( X 4 λ X 3 + 3 λ X 5 ) cos φ 0 = 0 ,
sin ( φ φ 0 ) = X P C [ ( 1 λ X 2 ) sin φ 0 P X α 0 ] ,
φ = φ 0 ± sin 1 [ X P C ( 1 λ X 2 ) sin φ 0 ] + π 2 ( 1 1 )
3 λ X 5 ( g δ + 4 λ ) X 3 ( δ 2 1 ) X ± ( 1 3 λ X 2 ) P C = 0 .
g δ X 2 + ( δ 2 1 ) = 0 ,
3 X 4 ( g δ + 4 ) X 2 ( δ 2 1 ) = 0 .

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