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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 10792–10800
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Build up of off-diagonal long-range order in microcavity exciton-polaritons across the parametric threshold

R. Spano, J. Cuadra, C. Lingg, D. Sanvitto, M. D. Martin, P. R. Eastham, M. van der Poel, J. M. Hvam, and L. Viña  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 10792-10800 (2013)
http://dx.doi.org/10.1364/OE.21.010792


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Abstract

We report an experimental study of the spontaneous spatial and temporal coherence of polariton condensates generated in the optical parametric oscillator configuration, below and at the parametric threshold, and as a function of condensate area. Above the threshold we obtain very long coherence times (up to 3 ns) and a spatial coherence extending over the entire condensate (40μm). The very long coherence time and its dependence on condensate area and pump power reflect the suppression of polariton-polariton interactions by an effect equivalent to motional narrowing.

© 2013 OSA

1. Introduction

Planar microcavity exciton-polaritons are composite particles generated by strong coupling between excitons and cavity photons that, below the Mott transition, follow the Bose statistics. They have been extensively investigated in the past decades for their unique properties, such as very light effective mass and the consequent high critical temperature for condensation [1

1. Physics of Semiconductor Microcavities, ed. B. Deveuad, (Wiley-VCH, Berlin, 2007).

]. It is well known that the occurrence of such a phase transition is accompanied by the onset of long range phase coherence [2

2. O. Penrose and L. Onsager, “Bose-Einstein Condensation and Liquid Helium,” Phys. Rev. 104, 576–584 (1956) [CrossRef] .

]. In fact, since the unambiguous demonstration of polariton condensation its coherence properties have been investigated [3

3. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, 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] .

], and the attention of the scientific community on the matter is high, as witnessed by the recent work by Wertz et al. on 1D ballistically expanding condensates [4

4. E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaitre, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, and J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nature Phys. 6, 860–864 (2010) [CrossRef] .

], which demonstrates an extremely large coherence. Moreover, the extension and eventually the decay of the spatial coherence can give useful information on the kind of transition the system undergoes [5

5. L. Pitaevskii and S. Stringari, Bose-Einstein condensation(Oxford University Press, Oxford, 2003).

, 6

6. F. P. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, “Spontaneous Coherence Buildup in a Polariton Laser,” Phys. Rev. Lett. 93, 016402 (2004) [CrossRef] .

]. Recently, polariton condensates have attracted intense attention for their potential application in the field of quantum information, and also some properties suitable for such applications, like Josephson oscillations and spin-switching, have been already demonstrated [7

7. K. G. Lagoudakis, B. Pietka, M. Wouters, R. André, and B. Deveaud-Plédran, “Coherent Oscillations in an Exciton-Polariton Josephson Junction,” Phys. Rev. Lett. 105, 120403 (2010) [CrossRef] [PubMed] .

, 8

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

]. Therefore, the achievement of extended spatial and temporal coherence for polariton condensate is crucial.

Polariton condensates can be created using off-resonant, resonant or under parametric optical excitation. The optical parametric oscillation (OPO) is a third order non-linear phenomenon, arising from interactions between the excitonic components of the polaritons. It takes place when two pump polaritons at the inflection point of the lower polariton branch (LPB), scatter efficiently into a signal and an idler polariton [9

9. 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, 3680–3683 (2000) [CrossRef] [PubMed] .

11

11. A. I. Tartakovskii, D. N. Krizhanovskii, and V. D. Kulakovskii, “Polariton-polariton scattering in semiconductor microcavities: Distinctive features and similarities to the three-dimensional case,” Phys. Rev. B 62, R13298–13301 (2000) [CrossRef] .

]. Spatial coherence properties have been studied theoretically for the OPO condensate by Carusotto and Ciuti [12

12. I. Carusotto and C. Ciuti, “Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies,” Phys. Rev. B 72, 125335 (2005) [CrossRef] .

]. They investigated numerically the first-order coherence function (g(1)) for a finite condensate paying special attention to its behaviour across the OPO parametric threshold (ETh) [13

13. In this work we consider two kinds of threshold, associated with tuning the pump energy and power. They are denoted EThand PThrespectively.

]. It is found that, for excitation frequencies ωp below that of the threshold, g(1) has a finite correlation length. Increasing ωp in order to approach ETh, mantaining fixed the pump angle (and therefore the wavevector), they predict the build up of macroscopic phase coherence extending over the entire condensate [12

12. I. Carusotto and C. Ciuti, “Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies,” Phys. Rev. B 72, 125335 (2005) [CrossRef] .

]. In this regime the spatial fluctuations are negligible, so the temporal coherence properties should be captured by the theory of Whittaker and Eastham [14

14. D. M. Whittaker and P. R. Eastham, “Coherence properties of the microcavity polariton condensate,” Europhys. Lett. 87, 27002 (2009) [CrossRef] .

]. In this theory the temporal coherence is limited by fluctuations in the particle number, which due to the polariton-polariton interactions imply a broadening of the emission [15

15. F. Tassone and Y. Yamamoto, “Lasing and squeezing of composite bosons in a semiconductor microcavity,” Phys. Rev. A 62, 063809 (2000) [CrossRef] .

18

18. F. P. Laussy, I. A. Shelykh, G. Malpuech, and A. Kavokin, “Effects of Bose-Einstein condensation of exciton polaritons in microcavities on the polarization of emitted light,” Phys. Rev. B 73, 035315 (2006) [CrossRef] .

]. This broadening mechanism, however, would be suppressed if the intensity fluctuations relax rapidly, in a form of motional narrowing effect [19

19. A. Berthelot, I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, and J. M. Gérard, “Unconventional motional narrowing in the optical spectrum of a semiconductor quantum dot,” Nat. Phys. 2, 759–764 (2006) [CrossRef] .

]. Thus for appropriate pump powers and condensate areas very long coherence times could be obtained.

Using a high-quality sample and a narrow-bandwidth pump laser we obtain spatially extended single-mode polariton condensates, with uniform spatial coherence extending over our entire pump spot. The temporal coherence decay of our condensates reveals the two timescales associated with the interaction-induced broadening of the condensate and the relaxation of intensity fluctuations. We show that the finite-size scaling laws describing the variation of these timescales with condensate area qualitatively agree with theory.

The main factor that limits the coherence is the quality of the cavity: the presence of defects creates a disorder potential that traps the condensate, potentially leading to multi-mode and inhomogeneous states [20

20. D. N. Krizhanovskii, K. G. Lagoudakis, M. Wouters, B. Pietka, R. A. Bradley, K. Guda, D. M. Whittaker, M. S. Skolnick, B. Deveaud-Plédran, M. Richard, R. André, and Le Si Dang, “Coexisting nonequilibrium condensates with long-range spatial coherence in semiconductor microcavities,” Phys. Rev. B 80, 045317 (2009) [CrossRef] .

]. Another, very important, detrimental effect is caused by the fluctuations of the excitation laser that hinder the attainment of the intrinsic coherence of the condensate. Initial studies [3

3. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, 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] .

, 20

20. D. N. Krizhanovskii, K. G. Lagoudakis, M. Wouters, B. Pietka, R. A. Bradley, K. Guda, D. M. Whittaker, M. S. Skolnick, B. Deveaud-Plédran, M. Richard, R. André, and Le Si Dang, “Coexisting nonequilibrium condensates with long-range spatial coherence in semiconductor microcavities,” Phys. Rev. B 80, 045317 (2009) [CrossRef] .

, 21

21. H. Deng, G. S. Solomon, R. Hey, K. H. Ploog, and Y. Yamamoto, “Spatial Coherence of a Polariton Condensate,” Phys. Rev. Lett. 99, 126403 (2007) [CrossRef] [PubMed] .

] have been performed by non-resonantly pumping the mi-crocavity, and in such cases the resulting distribution of the population at the bottom of the lower polariton branch is subjected to fluctuations due to the reservoir of particles at the bottleneck. These fluctuations, broadening the distribution of polaritons in energy and momentum space, translate, according to the Wiener-Khinchin identity [22

22. H. P. Baltes, “Coherence and the radiation laws,” Appl. Phys. 12, 221–244 (1977) [CrossRef] .

, 23

23. M. Richard, M. Wouters, and L. S. Dang, in Optical Generation and Control of Quantum Coherence in Semiconductor Nanostructures, NanoScience and Technology 146, eds. G. Slavcheva and P. Roussignol, (Springer-VerlagBerlin2010) Chap. 11 .

], into a faster decay of the temporal and spatial coherence. Although analogous broadening by the fluctuating population of pump polaritons can occur in the OPO [17

17. D. N. Krizhanovskii, D. Sanvitto, A. P. D. Love, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Dominant Effect of Polariton-Polariton Interactions on the Coherence of the Microcavity Optical Parametric Oscillator,” Phys. Rev. Lett. 97, 097402 (2006) [CrossRef] [PubMed] .

], the threshold pump density (PTh) for a resonant-gain process is much lower than that for non-resonant gain and the effect of the reservoir po-laritons is not relevant, since the reservoir is either completely empty or very weekly occupied. Thus the coherence exhibited by an OPO polariton-condensate is expected to decay over much longer times and larger distances than the one produced by non-resonant techniques. Here we realize experimentally a spatially extended, and with long temporal coherence, condensate in the OPO configuration and identify the mechanism driving the extended temporal coherence.

2. Sample and experimental setup

To avoid that the pump fluctuations conceal the genuine coherence properties of the condensate, we use a CW monomode laser with a very narrow bandwidth of 75 kHz to excite the sample, which is a high-quality λ-microcavity grown by molecular beam epitaxy, composed by 16 periods of λ/4 AlAs/Al0.15Ga0.85As on its top and 25 periods on its bottom. A single 10-nm GaAs/Al0.3Ga0.7As QW is placed at the antinode of the cavity. The Rabi splitting is 4.2 meV and we select a uniform area of the sample with a small defect density and a detuning of δ ∼ −3.8 meV [24

24. The detuning δis defined as the difference between the bare cavity and exciton energies.

]. In order to investigate the spatial coherence properties of a 2D condensate we create one with a diameter of ∼ 40μm, by pumping resonantly the LPB close to its inflection point, with a Gaussian laser beam, maintaining the temperature of the sample at 10 K. We adopt the experimental conditions to drive the sample into the OPO regime at the parametric threshold, ETh, and then we follow a similar approach to that in [12

12. I. Carusotto and C. Ciuti, “Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies,” Phys. Rev. B 72, 125335 (2005) [CrossRef] .

], i.e., we decrease the pump frequency ωp keeping fixed the angle of incidence.

3. Results

Fig. 1 Interference pattern (a/d) and corresponding coherence (b/e) of the condensate generated at/below the parametric threshold, ETh, corresponding to a pump energy E1 = 1552.574 meV/ E2 = 1552.567 meV and power density P = 15.8kW/cm2. Horizontal profiles at the center (c/f) showing the constant coherence along the entire condensate for E1, and an exponentially decaying one for E2.The red line is the Lorentzian fit.
Fig. 2 Coherence length Lc as a function of the energy distance from the threshold δEEThE. Red line is a guide to the eye.
Fig. 3 (a) Temporal coherence decay above parametric threshold for condensate area 2.7 × 103μm2 at a pump power density Pd = 10kW/cm2, the central line is a fit to Eq. (1), side bands define the confidence range in which the experimental points fall within a probability of 95%. (b) Temporal coherence decay for Pd = 5.4kW/cm2 (black squares) and 10kW/cm2 (blue dots) at A = 70μm2. Lines are fits to Eq. (1); (c) τc as a function of the condensate area A, the line is a fit to τcAx, with x = 0.41 ± 0.05; (d) τr as a function of condensate area A, the line is a fit to τrAx with x = 0.41 ± 0.17.
Fig. 4 Temporal coherence Tc as obtained from Eq. (2) as a function of the condensate area A. Red solid line is a fit to square root dependence on area A, dashed line corresponds to the best fit to a linear dependence.

4. Conclusions

In summary, we have investigated the spatial and temporal coherence properties of polariton condensates generated by parametric scattering. The spatial coherence was studied below and above the OPO condensation threshold, obtaining a very large coherence length for 2D GaAs microcavity polaritons. Measurements of the temporal coherence reveal a predominantly exponential decay caused by polariton-polariton interactions in a motional narrowing regime. Although similar to the exponential decay associated with the Schawlow-Townes linewidth, the mechanism is different, and gives a different dependence of coherence time on condensate area. By varying the area of the condensate and comparing with scaling laws, we observe that the dephasing time follows the predictions of the coherence theory of polariton condensates. Constructing condensates of large area, we are able to achieve long coherence times, which is a crucial step forward for exploiting polariton condensates in quantum and ultrafast devices.

Acknowledgments

The work was supported by the FP7 ITNs Clermont4 ( 235114) and Spin-optronics ( 237252), Spanish MEC ( MAT2011-22997), CAM ( S-2009/ESP-1503), and Science Foundation Ireland SIRG/I1592 (PRE). We thank C. Tejedor and I. Carusotto for valuable discussions.

References and links

1.

Physics of Semiconductor Microcavities, ed. B. Deveuad, (Wiley-VCH, Berlin, 2007).

2.

O. Penrose and L. Onsager, “Bose-Einstein Condensation and Liquid Helium,” Phys. Rev. 104, 576–584 (1956) [CrossRef] .

3.

J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, 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] .

4.

E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaitre, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, and J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nature Phys. 6, 860–864 (2010) [CrossRef] .

5.

L. Pitaevskii and S. Stringari, Bose-Einstein condensation(Oxford University Press, Oxford, 2003).

6.

F. P. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, “Spontaneous Coherence Buildup in a Polariton Laser,” Phys. Rev. Lett. 93, 016402 (2004) [CrossRef] .

7.

K. G. Lagoudakis, B. Pietka, M. Wouters, R. André, and B. Deveaud-Plédran, “Coherent Oscillations in an Exciton-Polariton Josephson Junction,” Phys. Rev. Lett. 105, 120403 (2010) [CrossRef] [PubMed] .

8.

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

9.

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, 3680–3683 (2000) [CrossRef] [PubMed] .

10.

J. J. Baumberg, P. G. Savvidis, R. M. Stevenson, A. I. Tartakovskii, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Parametric oscillation in a vertical microcavity: A polariton condensate or micro-optical parametric oscillation,” Phys. Rev. B 62, R16247–16250 (2000) [CrossRef] .

11.

A. I. Tartakovskii, D. N. Krizhanovskii, and V. D. Kulakovskii, “Polariton-polariton scattering in semiconductor microcavities: Distinctive features and similarities to the three-dimensional case,” Phys. Rev. B 62, R13298–13301 (2000) [CrossRef] .

12.

I. Carusotto and C. Ciuti, “Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies,” Phys. Rev. B 72, 125335 (2005) [CrossRef] .

13.

In this work we consider two kinds of threshold, associated with tuning the pump energy and power. They are denoted EThand PThrespectively.

14.

D. M. Whittaker and P. R. Eastham, “Coherence properties of the microcavity polariton condensate,” Europhys. Lett. 87, 27002 (2009) [CrossRef] .

15.

F. Tassone and Y. Yamamoto, “Lasing and squeezing of composite bosons in a semiconductor microcavity,” Phys. Rev. A 62, 063809 (2000) [CrossRef] .

16.

D. Porras and C. Tejedor, “Linewidth of a polariton laser: Theoretical analysis of self-interaction effects,” Phys. Rev. B 67, 161310(R)(2003) [CrossRef] .

17.

D. N. Krizhanovskii, D. Sanvitto, A. P. D. Love, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Dominant Effect of Polariton-Polariton Interactions on the Coherence of the Microcavity Optical Parametric Oscillator,” Phys. Rev. Lett. 97, 097402 (2006) [CrossRef] [PubMed] .

18.

F. P. Laussy, I. A. Shelykh, G. Malpuech, and A. Kavokin, “Effects of Bose-Einstein condensation of exciton polaritons in microcavities on the polarization of emitted light,” Phys. Rev. B 73, 035315 (2006) [CrossRef] .

19.

A. Berthelot, I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, and J. M. Gérard, “Unconventional motional narrowing in the optical spectrum of a semiconductor quantum dot,” Nat. Phys. 2, 759–764 (2006) [CrossRef] .

20.

D. N. Krizhanovskii, K. G. Lagoudakis, M. Wouters, B. Pietka, R. A. Bradley, K. Guda, D. M. Whittaker, M. S. Skolnick, B. Deveaud-Plédran, M. Richard, R. André, and Le Si Dang, “Coexisting nonequilibrium condensates with long-range spatial coherence in semiconductor microcavities,” Phys. Rev. B 80, 045317 (2009) [CrossRef] .

21.

H. Deng, G. S. Solomon, R. Hey, K. H. Ploog, and Y. Yamamoto, “Spatial Coherence of a Polariton Condensate,” Phys. Rev. Lett. 99, 126403 (2007) [CrossRef] [PubMed] .

22.

H. P. Baltes, “Coherence and the radiation laws,” Appl. Phys. 12, 221–244 (1977) [CrossRef] .

23.

M. Richard, M. Wouters, and L. S. Dang, in Optical Generation and Control of Quantum Coherence in Semiconductor Nanostructures, NanoScience and Technology 146, eds. G. Slavcheva and P. Roussignol, (Springer-VerlagBerlin2010) Chap. 11 .

24.

The detuning δis defined as the difference between the bare cavity and exciton energies.

25.

A. P. D. Love, D. N. Krizhanovskii, D. M. Whittaker, R. Bouchekioua, D. Sanvitto, S. Al Rizeiqi, R. Bradley, M. S. Skolnick, P. R. Eastham, R. André, and Le Si Dang, “Intrinsic Decoherence Mechanisms in the Microcavity Polariton Condensate,” Phys. Rev. Lett. 101, 067404 (2008) [CrossRef] [PubMed] .

26.

A. Baas, J.-Ph. Karr, M. Romanelli, A. Bramati, and E. Giacobino, “Quantum Degeneracy of Microcavity Polaritons,” Phys. Rev. Lett. 96, 176401 (2006) [CrossRef] [PubMed] .

27.

M. Wouters and I. Carusotto, “Goldstone mode of optical parametric oscillators in planar semiconductor microcavities in the strong-coupling regime,” Phys. Rev. A 76, 043807 (2007) [CrossRef] .

28.

P. R. Eastham and P. B. Littlewood, “Finite-size fluctuations and photon statistics near the polariton condensation transition in a single-mode microcavity,” Phys. Rev. B 73, 085306 (2006) [CrossRef] .

29.

P.R. Eastham, “Mode locking and mode competition in a nonequilibrium solid-state condensate,” Phys. Rev. B 78, 035319 (2008) [CrossRef] .

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(240.5420) Optics at surfaces : Polaritons
(020.1475) Atomic and molecular physics : Bose-Einstein condensates
(100.3175) Image processing : Interferometric imaging
(140.3945) Lasers and laser optics : Microcavities

ToC Category:
Optics at Surfaces

History
Original Manuscript: February 22, 2013
Revised Manuscript: April 15, 2013
Manuscript Accepted: April 17, 2013
Published: April 25, 2013

Citation
R. Spano, J. Cuadra, C. Lingg, D. Sanvitto, M. D. Martin, P. R. Eastham, M. van der Poel, J. M. Hvam, and L. Viña, "Build up of off-diagonal long-range order in microcavity exciton-polaritons across the parametric threshold," Opt. Express 21, 10792-10800 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10792


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References

  1. Physics of Semiconductor Microcavities, ed. B. Deveuad, (Wiley-VCH, Berlin, 2007).
  2. O. Penrose and L. Onsager, “Bose-Einstein Condensation and Liquid Helium,” Phys. Rev.104, 576–584 (1956). [CrossRef]
  3. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and Le Si Dang, “Bose-Einstein condensation of exciton polaritons,” Nature443, 409–414 (2006). [CrossRef] [PubMed]
  4. E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaitre, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, and J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nature Phys.6, 860–864 (2010). [CrossRef]
  5. L. Pitaevskii and S. Stringari, Bose-Einstein condensation(Oxford University Press, Oxford, 2003).
  6. F. P. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, “Spontaneous Coherence Buildup in a Polariton Laser,” Phys. Rev. Lett.93, 016402 (2004). [CrossRef]
  7. K. G. Lagoudakis, B. Pietka, M. Wouters, R. André, and B. Deveaud-Plédran, “Coherent Oscillations in an Exciton-Polariton Josephson Junction,” Phys. Rev. Lett.105, 120403 (2010). [CrossRef] [PubMed]
  8. A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nat. Photonics4, 361–366 (2010). [CrossRef]
  9. 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, 3680–3683 (2000). [CrossRef] [PubMed]
  10. J. J. Baumberg, P. G. Savvidis, R. M. Stevenson, A. I. Tartakovskii, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Parametric oscillation in a vertical microcavity: A polariton condensate or micro-optical parametric oscillation,” Phys. Rev. B62, R16247–16250 (2000). [CrossRef]
  11. A. I. Tartakovskii, D. N. Krizhanovskii, and V. D. Kulakovskii, “Polariton-polariton scattering in semiconductor microcavities: Distinctive features and similarities to the three-dimensional case,” Phys. Rev. B62, R13298–13301 (2000). [CrossRef]
  12. I. Carusotto and C. Ciuti, “Spontaneous microcavity-polariton coherence across the parametric threshold: Quantum Monte Carlo studies,” Phys. Rev. B72, 125335 (2005). [CrossRef]
  13. In this work we consider two kinds of threshold, associated with tuning the pump energy and power. They are denoted EThand PThrespectively.
  14. D. M. Whittaker and P. R. Eastham, “Coherence properties of the microcavity polariton condensate,” Europhys. Lett.87, 27002 (2009). [CrossRef]
  15. F. Tassone and Y. Yamamoto, “Lasing and squeezing of composite bosons in a semiconductor microcavity,” Phys. Rev. A62, 063809 (2000). [CrossRef]
  16. D. Porras and C. Tejedor, “Linewidth of a polariton laser: Theoretical analysis of self-interaction effects,” Phys. Rev. B67, 161310(R)(2003). [CrossRef]
  17. D. N. Krizhanovskii, D. Sanvitto, A. P. D. Love, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Dominant Effect of Polariton-Polariton Interactions on the Coherence of the Microcavity Optical Parametric Oscillator,” Phys. Rev. Lett.97, 097402 (2006). [CrossRef] [PubMed]
  18. F. P. Laussy, I. A. Shelykh, G. Malpuech, and A. Kavokin, “Effects of Bose-Einstein condensation of exciton polaritons in microcavities on the polarization of emitted light,” Phys. Rev. B73, 035315 (2006). [CrossRef]
  19. A. Berthelot, I. Favero, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, and J. M. Gérard, “Unconventional motional narrowing in the optical spectrum of a semiconductor quantum dot,” Nat. Phys.2, 759–764 (2006). [CrossRef]
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