Abstract

The photon emission into different spatial directions of a quantum dot in a micropillar cavity is theoretically analyzed. We propose two types of photon emission statistics from a single quantum light device: (i) single photon emission into the axial, strong coupling direction and a two-photon emission into the lateral, weak coupling direction, as well as (ii) the simultaneous use of both emission directions for the temporally ordered generation of two photons within a defined time-bin constituting a heralded single photon source. Our results open up exciting perspectives for solid state based quantum light sources, which can be generalized to any quantum emitter-microcavity system featuring spatially distinct emission channels between the resonator and unconfined modes.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
  2. O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
    [Crossref] [PubMed]
  3. J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
    [Crossref] [PubMed]
  4. J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
    [Crossref]
  5. J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
    [Crossref]
  6. G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
    [Crossref]
  7. S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
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  8. D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
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    [Crossref]
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    [Crossref]
  22. A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
    [Crossref]
  23. We neglect lateral mode emissions caused by photon scattering processes at defects in the micropillar.
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    [Crossref]
  25. F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
    [Crossref]
  26. Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
    [Crossref] [PubMed]
  27. M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
    [Crossref]
  28. M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
    [Crossref]
  29. A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
    [Crossref]
  30. R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
    [Crossref]
  31. Later in the derivation of the source field expression of the g(2), the continuum modes (and thus k) in lateral direction inside a solid angle towards the detector are included.
  32. M. Kira and S. Koch, Semiconductor Quantum Optics (Cambridge University, 2011).
    [Crossref]
  33. F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
    [Crossref]
  34. C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
    [Crossref]
  35. Y. Mu and C. M. Savage, “One-atom lasers,” Phys. Rev. A 46, 5944–5954 (1992).
    [Crossref] [PubMed]
  36. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
    [Crossref] [PubMed]
  37. S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
    [Crossref]
  38. R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
    [Crossref] [PubMed]
  39. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007).
    [Crossref] [PubMed]
  40. S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express 18, 9909–9921 (2010).
    [Crossref] [PubMed]
  41. S. Schumacher, J. Förstner, A. Zrenner, M. Florian, C. Gies, P. Gartner, and F. Jahnke, “Cavity-assisted emission of polarization-entangled photons from biexcitons in quantum dots with fine-structure splitting,” Opt. Express 20, 5335–5342 (2012).
    [Crossref] [PubMed]
  42. T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
    [Crossref] [PubMed]
  43. τ = 0 can be clearly identified by a discontinuity for glat−ax(2)(−τ)=gax−lat(2)(τ). Therefore, differences in the time of flight for the two modes can be compensated by identifying this discontinuity.
  44. Similar to gax−ax(2)(0) the value of glat−ax−ax(3)(τ′) at τ′ = 0 is not exactly zero, since there are small contributions from the biexciton cascade decay despite the off-resonance of the biexciton-exciton transition to the axial cavity modes.
  45. Z. Vernon, M. Liscidini, and J. E. Sipe, “No free lunch: the trade-off between heralding rate and efficiency in microresonator-based heralded single photon sources,” Opt. Lett. 41, 788–791 (2016).
    [Crossref] [PubMed]
  46. Assuming for the exciton-photon coupling elements of the excitons a difference |Mge↑m−Mge↓m|=20μeV is probably a very extreme case and the differences are often smaller. However, we wanted to test the robustness of the effect with the extreme case.

2016 (2)

F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
[Crossref]

Z. Vernon, M. Liscidini, and J. E. Sipe, “No free lunch: the trade-off between heralding rate and efficiency in microresonator-based heralded single photon sources,” Opt. Lett. 41, 788–791 (2016).
[Crossref] [PubMed]

2015 (4)

Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
[Crossref] [PubMed]

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
[Crossref]

C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
[Crossref]

2014 (1)

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

2013 (3)

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
[Crossref] [PubMed]

W. D. Newman, C. L. Cortes, and Z. Jacob, “Enhanced and directional single-photon emission in hyperbolic metamaterials,” J. Opt. Soc. Am. B 30, 766–775 (2013).
[Crossref]

2012 (3)

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

S. Schumacher, J. Förstner, A. Zrenner, M. Florian, C. Gies, P. Gartner, and F. Jahnke, “Cavity-assisted emission of polarization-entangled photons from biexcitons in quantum dots with fine-structure splitting,” Opt. Express 20, 5335–5342 (2012).
[Crossref] [PubMed]

2011 (1)

P. K. Pathak and S. Hughes, “Coherent generation of time-bin entangled photon pairs using the biexciton cascade and cavity-assisted piecewise adiabatic passage,” Phys. Rev. B 83, 245301 (2011).
[Crossref]

2010 (3)

A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
[Crossref]

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express 18, 9909–9921 (2010).
[Crossref] [PubMed]

2009 (4)

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
[Crossref]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
[Crossref]

2008 (2)

P. Machnikowski, “Theory of two-photon processes in quantum dots: Coherent evolution and phonon-induced dephasing,” Phys. Rev. B 78, 195320 (2008).
[Crossref]

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
[Crossref]

2007 (1)

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007).
[Crossref] [PubMed]

2006 (1)

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

2005 (1)

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[Crossref] [PubMed]

2004 (5)

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004).
[Crossref] [PubMed]

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

2003 (1)

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
[Crossref]

2001 (1)

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

2000 (2)

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
[Crossref] [PubMed]

1998 (1)

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[Crossref]

1995 (2)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

1992 (1)

Y. Mu and C. M. Savage, “One-atom lasers,” Phys. Rev. A 46, 5944–5954 (1992).
[Crossref] [PubMed]

1956 (1)

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Akopian, N.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

Albert, F.

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

Arakawa, Y.

Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
[Crossref] [PubMed]

Arie, I.

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

Arnold, C.

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
[Crossref] [PubMed]

Arnold, F.

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

Aßmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Avron, J.

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

Bahgat Shehata, A.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Bakker, M. P.

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Rodt, S.

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[Crossref] [PubMed]

Scarcella, C.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Schettini, V.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
[Crossref]

Schliwa, A.

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[Crossref] [PubMed]

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
[Crossref]

Schneider, C.

F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
[Crossref]

A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
[Crossref]

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Schneider, S.

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Schumacher, S.

Seguin, R.

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[Crossref] [PubMed]

Sek, G.

J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004).
[Crossref] [PubMed]

Sellin, R. L.

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
[Crossref]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Senellart, P.

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
[Crossref] [PubMed]

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

Shahnia, S.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

Silberhorn, C.

A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
[Crossref]

Simon, C.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Sipe, J. E.

Solomon, G. S.

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

Steel, M.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

Tejedor, C.

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
[Crossref]

Tittel, W.

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

Tosi, A.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Traina, P.

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

Troiani, F.

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
[Crossref]

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Ulm, G.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

Ulrich, S. M.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

Unold, T.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

V., W. L.

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

van Exter, M. P.

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

Vernon, Z.

Vo, T.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Weihs, G.

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Weinfurter, H.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Wieck, A. D.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

Wiersig, J.

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

Woggon, U.

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

Xiong, C.

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Yamamoto, Y.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007).
[Crossref] [PubMed]

Zbinden, H.

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Zheng, S.-B.

S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
[Crossref] [PubMed]

Zrenner, A.

Zwinkels, J. C.

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

Appl. Phys. Lett. (2)

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, P. Traina, A. Della Frera, A. Tosi, A. Bahgat Shehata, C. Scarcella, A. Gulinatti, M. Ghioni, S. V. Polyakov, A. Migdall, and A. Giudice, “An extremely low-noise heralded single-photon source: A breakthrough for quantum technologies,” Appl. Phys. Lett. 101, 221112 (2012).
[Crossref]

F. Albert, C. Hopfmann, A. Eberspacher, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, and S. Reitzenstein, “Directional whispering gallery mode emission from limacon-shaped electrically pumped quantum dot micropillar lasers,” Appl. Phys. Lett. 101, 021116 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

Metrologia (1)

J. C. Zwinkels, E. Ikonen, N. P. Fox, G. Ulm, and M. L. Rastello, “Photometry, radiometry and ’the candela’: evolution in the classical and quantum world,” Metrologia 47, R15 (2010).
[Crossref]

Nat Commun. (1)

M. Collins, C. Xiong, I. Rey, T. Vo, J. He, S. Shahnia, C. Reardon, T. Krauss, M. Steel, A. Clark, and B. Eggleton, “Integrated spatial multiplexing of heralded single-photon sources,” Nat Commun. 4, 2582 (2013).
[Crossref]

Nat. Commun. (2)

O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, “Bright solid-state sources of indistinguishable single photons,” Nat. Commun. 4, 1425 (2013).
[Crossref] [PubMed]

H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, “Time-bin entangled photons from a quantum dot,” Nat. Commun. 5, 4251 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Nature (4)

J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004).
[Crossref] [PubMed]

J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

L. Peter, A. F. van Driel, I. S. N., I. Arie, O. Karin, V. Daniel, and W. L. V., “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[Crossref]

New J. Phys. (2)

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004).
[Crossref]

A. M. Brańczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded fock states using parametric down-conversion,” New J. Phys. 12, 063001 (2010).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (4)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref] [PubMed]

G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini, V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall, “Improved implementation of the alicki–van ryn nonclassicality test for a single particle using Si detectors,” Phys. Rev. A 79, 044102 (2009).
[Crossref]

C. Gies, F. Jahnke, and W. W. Chow, “Photon antibunching from few quantum dots in a cavity,” Phys. Rev. A 91, 061804 (2015).
[Crossref]

Y. Mu and C. M. Savage, “One-atom lasers,” Phys. Rev. A 46, 5944–5954 (1992).
[Crossref] [PubMed]

Phys. Rev. B (8)

S. Rodt, R. Heitz, A. Schliwa, R. L. Sellin, F. Guffarth, and D. Bimberg, “Repulsive exciton-exciton interaction in quantum dots,” Phys. Rev. B 68, 035331 (2003).
[Crossref]

M. P. Bakker, A. V. Barve, T. Ruytenberg, W. Löffler, L. A. Coldren, D. Bouwmeester, and M. P. van Exter, “Polarization degenerate solid-state cavity quantum electrodynamics,” Phys. Rev. B 91, 115319 (2015).
[Crossref]

F. Troiani and C. Tejedor, “Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering,” Phys. Rev. B 78, 155305 (2008).
[Crossref]

A. Carmele, A. Knorr, and M. Richter, “Photon statistics as a probe for exciton correlations in coupled nanostructures,” Phys. Rev. B 79, 035316 (2009).
[Crossref]

F. Hargart, M. Müller, K. Roy-Choudhury, S. L. Portalupi, C. Schneider, S. Höfling, M. Kamp, S. Hughes, and P. Michler, “Cavity-enhanced simultaneous dressing of quantum dot exciton and biexciton states,” Phys. Rev. B 93, 115308 (2016).
[Crossref]

A. Musiał, C. Hopfmann, T. Heindel, C. Gies, M. Florian, H. A. M. Leymann, A. Foerster, C. Schneider, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Correlations between axial and lateral emission of coupled quantum dot-micropillar cavities,” Phys. Rev. B 91, 205310 (2015).
[Crossref]

P. Machnikowski, “Theory of two-photon processes in quantum dots: Coherent evolution and phonon-induced dephasing,” Phys. Rev. B 78, 195320 (2008).
[Crossref]

P. K. Pathak and S. Hughes, “Coherent generation of time-bin entangled photon pairs using the biexciton cascade and cavity-assisted piecewise adiabatic passage,” Phys. Rev. B 83, 245301 (2011).
[Crossref]

Phys. Rev. Lett. (11)

I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96, 130501 (2006).
[Crossref] [PubMed]

S.-B. Zheng and G.-C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity qed,” Phys. Rev. Lett. 85, 2392–2395 (2000).
[Crossref] [PubMed]

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[Crossref]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Y. Ota, R. Ohta, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Vacuum rabi spectra of a single quantum emitter,” Phys. Rev. Lett. 114, 143603 (2015).
[Crossref] [PubMed]

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[Crossref] [PubMed]

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007).
[Crossref] [PubMed]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in ingaas quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[Crossref] [PubMed]

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical stark effect in a quantum dot: Ultrafast control of single exciton polarizations,” Phys. Rev. Lett. 92, 157401 (2004).
[Crossref] [PubMed]

Other (6)

τ = 0 can be clearly identified by a discontinuity for glat−ax(2)(−τ)=gax−lat(2)(τ). Therefore, differences in the time of flight for the two modes can be compensated by identifying this discontinuity.

Similar to gax−ax(2)(0) the value of glat−ax−ax(3)(τ′) at τ′ = 0 is not exactly zero, since there are small contributions from the biexciton cascade decay despite the off-resonance of the biexciton-exciton transition to the axial cavity modes.

Assuming for the exciton-photon coupling elements of the excitons a difference |Mge↑m−Mge↓m|=20μeV is probably a very extreme case and the differences are often smaller. However, we wanted to test the robustness of the effect with the extreme case.

Later in the derivation of the source field expression of the g(2), the continuum modes (and thus k) in lateral direction inside a solid angle towards the detector are included.

M. Kira and S. Koch, Semiconductor Quantum Optics (Cambridge University, 2011).
[Crossref]

We neglect lateral mode emissions caused by photon scattering processes at defects in the micropillar.

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Figures (6)

Fig. 1
Fig. 1 (a) QD-micropillar cavity: a single QD is embedded between two DBRs requiring an emission in axial and lateral direction. The QD directly couples to the lateral mode continuum { d k , l a t ( ) }. The axial cavity modes { b m ( ) } couple to an axial mode continuum { d k , a x ( ) }. (b) QD four-level system: ground state |g〉, two spin-degenerated exciton states |e ↑〉 and |e ↓〉, biexcitonic state |f〉 with the binding energy EB. P e σ g ( P f e σ ) denotes incoherent pumping and γ e σ g ( γ f e σ ) radiative decay.
Fig. 2
Fig. 2 Second order correlation functions for a exciton-photon coupling strength M = 10 μeV: (a) g ax ax ( 2 ) ( τ ) shows anti-correlation and g lat lat ( 2 ) ( τ ) correlation. (b) g lat ax ( 2 ) ( τ ) reaches a maximum at τmax for the emission in axial direction. The temporal width of the maximum Δτ is characterized by the time, where g lat ax ( 2 ) ( τ ) is decayed to half the difference to the uncorrelated value 1. g ax lat ( 2 ) ( τ ) shows anti-correlation.
Fig. 3
Fig. 3 (a) HSPS-proposal based on a QD-micropillar cavity system. (b) Characteristic parameters Δτ (left axis), τmax (left axis) and g lat ax ( 2 ) ( τ m a x ) (right axis) of g lat ax ( 2 ) ( τ ) for different exciton-photon coupling strength M. (c) Third order correlation function g lat ax ax ( 3 ) ( τ ) for M = 30 µeV showing anti-bunching.
Fig. 4
Fig. 4 Second order correlation functions (a) without a fine-structure splitting and for a constant exciton-photon coupling strength M = 10 µeV and (b) for a fine-structure splitting of 20 µeV and the exciton-photon coupling elements M g e m = M e f m = 30 μ e V and M g e m = M e f m = 10 μ e V. All signatures (bunching for g l a t l a t ( 2 ) ( τ ), anti-bunching for g a x a x ( 2 ) ( τ ) and the existence of a emission maximum for g lat ax ( 2 ) ( τ ) ), important for the applications, are still visible in (b).
Fig. 5
Fig. 5 Second order correlation functions for different radiative life times γ r a d 1 = 333 ps (value used in the manuscript), γ r a d 1 = 1 ns and γ r a d 1 = 2 ns: The qualitative behavior does not change with increasing radiative life time.
Fig. 6
Fig. 6 (a) Scheme of the measurement of the axial and lateral second order correlation function. (b) g mix mix ( 2 ) ( τ ) for different coefficients α and β and an exciton-photon coupling strength of 10 μeV. The superposition of axial and lateral direction generates curves between the pure axial-axial and lateral-lateral correlation functions.

Equations (18)

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g x x ( 2 ) ( τ ) = : I x ( t ) I x ( t + τ ) : I x ( t ) I x ( t ) .
H a x S R = k , m κ k b m d k , a x + h . a .
H l a t S R = σ , k ( M k d k , l a t | g e σ | + M k d k , l a t | e σ f | ) + h . a . .
H e l p h S = σ , m ( M g e σ m b m | g e σ | + M e σ f m b m | e σ f | ) + h . a . ,
H 0 S = s ω s | s s | + m ω m b m b m and H 0 R = k x ω k , x d k , x d k , x
ρ x ( t + τ ) U S ( t + τ , t ) ( μ x ϱ 0 S μ x )
t ρ = i [ H 0 S + H e l p h S , ρ ] + L d ρ L S ρ ,
L d ρ = i γ i ( L i ρ L i 1 2 ( ρ L i L i + L i L i ρ ) ) ,
g x x ( 2 ) ( τ ) = t r S ( μ x ρ x ( t + τ ) μ x ) t r S ( μ x ρ 0 μ x ) t r S ( μ x ρ 0 μ x ) ,
g lat ax ax ( 3 ) ( τ ) = : I l a t ( t ) I a x ( t + τ m a x ) I a x ( t + τ m a x + τ ) : : I l a t ( t ) I a x ( t + τ m a x ) : I a x ( t )
tr S ( μ mix ρ μ mix ) = | α | 2 tr S ( μ lat ρ μ lat ) + | β | 2 tr S ( μ ax ρ μ ax ) + α β * tr S ( μ ax ρ μ lat ) + α * β tr S ( μ lat ρ μ ax )
t ρ n 1 , n 2 , g n 1 , n 2 , g = σ γ e σ g ρ n 1 , n 2 , e σ n 1 , n 2 , e σ σ P e σ g ρ n 1 , n 2 , g n 1 , n 2 , g m = 1 , 2 [ ( i ω m ( n m n m ) + γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , g n 1 , n 2 , g + γ m n m + 1 n m + 1 ρ g g ( m , 1 , 1 ) + i σ ( M e σ g m n m ρ e σ g ) ( m , 0 , 1 ) M g e σ m n m ρ g e σ ( m , 1 , 0 ) ) ] ,
t ρ n 1 , n 2 , e σ n 1 , n 2 , e σ = ( γ e σ g + γ e σ g 2 P f e σ + P f e σ 2 ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ f e σ ρ n 1 , n 2 , f n 1 , n 2 , f δ σ , σ + P e σ g ρ n 1 , n 2 , g n 1 , n 2 , g δ σ , σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ e σ e σ ( m , 1 , 1 ) + i M g e σ m n m + 1 ρ g e σ ( m , 0 , 1 ) i M e σ f m n m ρ e σ f ( m , 1 , 0 ) i M e σ g m n m + 1 ρ e σ g ( m , 0 , 1 ) + i M f e σ m n m ρ f e σ ( m , 0 , 1 ) ] .
t ρ n 1 , n 2 , g n 1 , n 2 , e σ = ( i ( ω e σ ω g ) γ e σ g 2 σ P e σ g 2 P f e σ 2 γ p u r e ) ρ n 1 , n 2 , g n 1 , n 2 , e σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , g n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ g e σ ( m , 1 , 1 ) i M e σ f m n 1 ρ g f ( m , 1 , 0 ) i M e σ g m n 1 + 1 ρ g g ( m , 1 , 0 ) + i σ M e σ g m n 1 ρ e σ e σ ( m , 0 , 1 ) ] .
t ρ n 1 , n 2 , e σ n 1 , n 2 , e σ = ( γ e σ g + γ e σ g 2 P f e σ + P f e σ 2 ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ f e σ ρ n 1 , n 2 , f n 1 , n 2 , f δ σ , σ + P e σ g ρ n 1 , n 2 , g n 1 , n 2 , g δ σ , σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ e σ e σ ( m , 1 , 1 ) + i m d g e σ n m + 1 ρ g e σ ( m , 0 , 1 ) i m d e σ f n m ρ e σ f ( m , 1 , 0 ) i m d e σ g n m + 1 ρ e σ g ( m , 1 , 0 ) + i m d f e σ n m ρ f e σ ( m , 0 , 1 ) ]
t ρ n 1 , n 2 , f n 1 , n 2 , e σ = ( γ p u r e i ( ω e ω f ) γ e σ g 2 σ γ f e σ 2 P f e σ 2 ) ρ n 1 , n 2 , f n 1 , n 2 , e σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , f n 1 , n 2 , e σ + γ m n m + 1 n m + 1 ρ f e σ ( m , 1 , 1 ) i M e σ f m n m ρ f f ( m , 1 , 0 ) + i σ M e σ f m n m + 1 ρ e σ e σ ( m , 0 , 1 ) i M e σ g m n m + 1 ρ f g ( m , 1 , 0 ) ] .
t ρ n 1 , n 2 , g n 1 , n 2 , f = ( γ p u r e i ( ω f ω g ) σ P e σ g 2 σ γ f e σ 2 ) ρ n 1 , n 2 , g n 1 , n 2 , f + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , g n 1 , n 2 , f + γ m n m + 1 n m + 1 ρ g f ( m , 1 , 1 ) + i σ M e σ g m n m ρ e σ f ( m , 0 , 1 ) i σ M f e σ m n m + 1 ρ g e σ ( m , 1 , 0 ) ]
t ρ n 1 , n 2 , f n 1 , n 2 , f = σ γ f e σ ρ n 1 , n 2 , f n 1 , n 2 , f + σ P e σ f ρ n 1 , n 2 , e σ n 1 , n 2 , e σ + m = 1 , 2 [ ( i ω m ( n m n m ) γ m 2 ( n m + n m ) ) ρ n 1 , n 2 , f n 1 , n 2 , f + γ m n m + 1 n m + 1 ρ f f ( m , 1 , 1 ) i σ M f e σ m n m + 1 ρ f e σ ( m , 1 , 0 ) + i σ M e σ f m n m + 1 ρ e σ f ( m , 0 , 1 ) ] .

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