Abstract

A detailed understanding of the population and coherence dynamics in optically driven individual emitters in solids and their signatures in ultrafast nonlinear optical signals is of prime importance for their applications in future quantum and optical technologies. In a combined experimental and theoretical study on exciton complexes in single semiconductor quantum dots, we reveal a detailed picture of the dynamics by employing three-beam polarization-resolved four-wave mixing (FWM) micro-spectroscopy. The oscillatory dynamics of the FWM signals in the exciton-biexciton system is governed by the fine-structure splitting and the biexciton binding energy. There is an excellent quantitative agreement between the measurement and analytical description. The analysis of the excitation conditions exhibits a dependence of the dynamics on the specific choice of polarization configuration, pulse areas, and temporal ordering of driving fields. The interplay between the transitions in the four-level exciton system leads to the rich evolution of the coherence and population. Using two-dimensional FWM spectroscopy, we elucidate the exciton-biexciton coupling and identify neutral and charged exciton complexes in a single quantum dot. Our investigations thus clearly reveal that FWM spectroscopy is a powerful tool to characterize the spectral and dynamical properties of single quantum structures.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  40. P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. J. van der Zant, S. Michaelis de Vasconcellos, and R. Bratschitsch, “Single-photon emission from localized excitons in an atomically thin semiconductor,” Optica 2, 347–352 (2015).
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  41. M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotech. 10, 503–506 (2015).
    [Crossref]

2016 (1)

F. Fras, Q. Mermillod, G. Nogues, C. Hoarau, C. Schneider, M. Kamp, S. Höfling, W. Langbein, and J. Kasprzak, “Multi-wave coherent control of a solid state single emitter,” Nat. Photonics 10, 155–158 (2016).
[Crossref]

2015 (3)

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
[Crossref]

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotech. 10, 503–506 (2015).
[Crossref]

P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. J. van der Zant, S. Michaelis de Vasconcellos, and R. Bratschitsch, “Single-photon emission from localized excitons in an atomically thin semiconductor,” Optica 2, 347–352 (2015).
[Crossref]

2014 (3)

S. Maier, P. Gold, A. Forchel, N. Gregersen, J. Mørk, S. Höfling, C. Schneider, and M. Kamp, “Bright single photon source based on self-aligned quantum dot-cavity systems,” Opt. Express 22, 8136–8142 (2014).
[Crossref]

Y. Kodriano, E. R. Schmidgall, Y. Benny, and D. Gershoni, “Optical control of single excitons in semiconductor quantum dots,” Semicond. Sci. Technol. 29, 053001 (2014).
[Crossref]

D. E. Reiter, T. Kuhn, M. Glässl, and V. M. Axt, “The role of phonons for exciton and biexciton generation in an optically driven quantum dot,” J. Phys. Condens. Matter 26, 423203 (2014).
[Crossref]

2013 (4)

L. Monniello, C. Tonin, R. Hostein, A. Lemaitre, A. Martinez, V. Voliotis, and R. Grousson, “Excitation-induced dephasing in a resonantly driven InAs/GaAs quantum dot,” Phys. Rev. Lett. 111, 026403 (2013).
[Crossref]

A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. Warburton, “Charge noise and spin noise in a semiconductor quantum device,” Nat. Phys. 9, 570–575 (2013).
[Crossref]

J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
[Crossref]

G. Moody, R. Singh, H. Li, I. A. Akimov, M. Bayer, D. Reuter, A. D. Wieck, and S. T. Cundiff, “Fifth-order nonlinear optical response of excitonic states in an InAs quantum dot ensemble measured with two-dimensional spectroscopy,” Phys. Rev. B 87, 045313 (2013).
[Crossref]

2012 (3)

C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots,” Phys. Rev. B 85, 155303 (2012).
[Crossref]

X. Dai, M. Richter, H. Li, A. D. Bristow, C. Falvo, S. Mukamel, and S. T. Cundiff, “Two-dimensional double-quantum spectra reveal collective resonances in an atomic vapor,” Phys. Rev. Lett. 108, 193201 (2012).
[Crossref]

S. T. Cundiff, “Optical two-dimensional Fourier transform spectroscopy of semiconductor nanostructures,” J. Opt. Soc. Am. B 29, A69–A81 (2012).
[Crossref]

2011 (1)

I. Aharonovich, S. Castelletto, D. A. Simpson, C. H. Su, A. D. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
[Crossref]

2010 (4)

J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2010).
[Crossref]

F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
[Crossref]

W. Langbein, “Coherent optical spectroscopy of semiconductor nano structures,” Riv. Nuovo Cimento 33, 255–312 (2010).

A. J. Ramsay, “A review of the coherent optical control of the exciton and spin states of semiconductor quantum dots,” Semicond. Sci. Technol. 25, 103001 (2010).
[Crossref]

2008 (1)

J. Kasprzak and W. Langbein, “Vectorial four-wave mixing field dynamics from individual excitonic transitions,” Phys. Rev. B 78, 041103 (2008).
[Crossref]

2007 (1)

A. Krügel, A. Vagov, V. M. Axt, and T. Kuhn, “Monitoring the buildup of the quantum dot polaron: Pump-probe and four-wave mixing spectra from excitons and biexcitons in semiconductor quantum dots,” Phys. Rev. B 76, 195302 (2007).
[Crossref]

2006 (6)

B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time-and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B 73, 235354 (2006).
[Crossref]

T. Voss, I. Rückmann, J. Gutowski, V. M. Axt, and T. Kuhn, “Coherent control of the exciton and exciton-biexciton transitions in the generation of nonlinear wave-mixing signals in a semiconductor quantum well,” Phys. Rev. B 73, 115311 (2006).
[Crossref]

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96, 067401 (2006).
[Crossref]

R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006).
[Crossref]

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]

W. Langbein and B. Patton, “Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems,” Opt. Lett. 31, 1151–1153 (2006).
[Crossref]

2005 (5)

B. Patton, U. Woggon, and W. Langbein, “Coherent control and polarization readout of individual excitonic states,” Phys. Rev. Lett. 95, 266401 (2005).
[Crossref]

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]

R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
[Crossref]

V. M. Axt, T. Kuhn, A. Vagov, and F. M. Peeters, “Phonon-induced pure dephasing in exciton-biexciton quantum dot systems driven by ultrafast laser pulse sequences,” Phys. Rev. B 72, 125309 (2005).
[Crossref]

I. A. Akimov, K. V. Kavokin, A. Hundt, and F. Henneberger, “Electron-hole exchange interaction in a negatively charged quantum dot,” Phys. Rev. B 71, 075326 (2005).
[Crossref]

2004 (1)

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Control of fine-structure splitting and biexciton binding in InxGa1−x As quantum dots by annealing,” Phys. Rev. B 69, 161301 (2004).
[Crossref]

2003 (1)

B. Urbaszek, R. J. Warburton, K. Karrai, B. D. Gerardot, P. M. Petroff, and J. M. Garcia, “Fine structure of highly charged excitons in semiconductor quantum dots,” Phys. Rev. Lett. 90, 247403 (2003).
[Crossref]

2002 (1)

A. Vagov, V. M. Axt, and T. Kuhn, “Electron-phonon dynamics in optically excited quantum dots: Exact solution for multiple ultrashort laser pulses,” Phys. Rev. B 66, 165312 (2002).
[Crossref]

1999 (3)

G. Bacher, R. Weigand, J. Seufert, V. D. Kulakovskii, N. A. Gippius, A. Forchel, K. Leonardi, and D. Hommel, “Biexciton versus exciton lifetime in a single semiconductor quantum dot,” Phys. Rev. Lett. 83, 4417–4420 (1999).
[Crossref]

F. Gindele, K. Hild, W. Langbein, and U. Woggon, “Phonon interaction of single excitons and biexcitons,” Phys. Rev. B 60, R2157 (1999).
[Crossref]

W. Langbein, H. Gislason, and J. M. Hvam, “Transient four-wave mixing in t-shaped GaAs quantum wires,” Phys. Rev. B 60, 16667–16674 (1999).
[Crossref]

1998 (1)

N. H. Bonadeo, J. Erland, D. Gammon, D. Park, D. S. Katzer, and D. G. Steel, “Coherent optical control of the quantum state of a single quantum dot,” Science 282, 1473–1476 (1998).
[Crossref]

Aharonovich, I.

I. Aharonovich, S. Castelletto, D. A. Simpson, C. H. Su, A. D. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
[Crossref]

Akimov, I. A.

G. Moody, R. Singh, H. Li, I. A. Akimov, M. Bayer, D. Reuter, A. D. Wieck, and S. T. Cundiff, “Fifth-order nonlinear optical response of excitonic states in an InAs quantum dot ensemble measured with two-dimensional spectroscopy,” Phys. Rev. B 87, 045313 (2013).
[Crossref]

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96, 067401 (2006).
[Crossref]

I. A. Akimov, K. V. Kavokin, A. Hundt, and F. Henneberger, “Electron-hole exchange interaction in a negatively charged quantum dot,” Phys. Rev. B 71, 075326 (2005).
[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]

Andrews, J. T.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96, 067401 (2006).
[Crossref]

Arora, A.

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotech. 10, 503–506 (2015).
[Crossref]

Atkinson, P.

R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006).
[Crossref]

R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
[Crossref]

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]

Axt, V. M.

D. E. Reiter, T. Kuhn, M. Glässl, and V. M. Axt, “The role of phonons for exciton and biexciton generation in an optically driven quantum dot,” J. Phys. Condens. Matter 26, 423203 (2014).
[Crossref]

A. Krügel, A. Vagov, V. M. Axt, and T. Kuhn, “Monitoring the buildup of the quantum dot polaron: Pump-probe and four-wave mixing spectra from excitons and biexcitons in semiconductor quantum dots,” Phys. Rev. B 76, 195302 (2007).
[Crossref]

T. Voss, I. Rückmann, J. Gutowski, V. M. Axt, and T. Kuhn, “Coherent control of the exciton and exciton-biexciton transitions in the generation of nonlinear wave-mixing signals in a semiconductor quantum well,” Phys. Rev. B 73, 115311 (2006).
[Crossref]

V. M. Axt, T. Kuhn, A. Vagov, and F. M. Peeters, “Phonon-induced pure dephasing in exciton-biexciton quantum dot systems driven by ultrafast laser pulse sequences,” Phys. Rev. B 72, 125309 (2005).
[Crossref]

A. Vagov, V. M. Axt, and T. Kuhn, “Electron-phonon dynamics in optically excited quantum dots: Exact solution for multiple ultrashort laser pulses,” Phys. Rev. B 66, 165312 (2002).
[Crossref]

Bacher, G.

G. Bacher, R. Weigand, J. Seufert, V. D. Kulakovskii, N. A. Gippius, A. Forchel, K. Leonardi, and D. Hommel, “Biexciton versus exciton lifetime in a single semiconductor quantum dot,” Phys. Rev. Lett. 83, 4417–4420 (1999).
[Crossref]

Bayer, M.

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S. Maier, P. Gold, A. Forchel, N. Gregersen, J. Mørk, S. Höfling, C. Schneider, and M. Kamp, “Bright single photon source based on self-aligned quantum dot-cavity systems,” Opt. Express 22, 8136–8142 (2014).
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G. Bacher, R. Weigand, J. Seufert, V. D. Kulakovskii, N. A. Gippius, A. Forchel, K. Leonardi, and D. Hommel, “Biexciton versus exciton lifetime in a single semiconductor quantum dot,” Phys. Rev. Lett. 83, 4417–4420 (1999).
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[Crossref]

C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots,” Phys. Rev. B 85, 155303 (2012).
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W. Langbein, H. Gislason, and J. M. Hvam, “Transient four-wave mixing in t-shaped GaAs quantum wires,” Phys. Rev. B 60, 16667–16674 (1999).
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F. Fras, Q. Mermillod, G. Nogues, C. Hoarau, C. Schneider, M. Kamp, S. Höfling, W. Langbein, and J. Kasprzak, “Multi-wave coherent control of a solid state single emitter,” Nat. Photonics 10, 155–158 (2016).
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F. Fras, Q. Mermillod, G. Nogues, C. Hoarau, C. Schneider, M. Kamp, S. Höfling, W. Langbein, and J. Kasprzak, “Multi-wave coherent control of a solid state single emitter,” Nat. Photonics 10, 155–158 (2016).
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J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2010).
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J. Kasprzak and W. Langbein, “Vectorial four-wave mixing field dynamics from individual excitonic transitions,” Phys. Rev. B 78, 041103 (2008).
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Q. Mermillod, T. Jakubczyk, V. Delmonte, A. Delga, E. Peinke, J.-M. Gérard, J. Claudon, and J. Kasprzak, “Harvesting, coupling and control of single exciton coherences enabled by photonic trumpets,” submitted (2016).

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N. H. Bonadeo, J. Erland, D. Gammon, D. Park, D. S. Katzer, and D. G. Steel, “Coherent optical control of the quantum state of a single quantum dot,” Science 282, 1473–1476 (1998).
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I. A. Akimov, K. V. Kavokin, A. Hundt, and F. Henneberger, “Electron-hole exchange interaction in a negatively charged quantum dot,” Phys. Rev. B 71, 075326 (2005).
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Kodriano, Y.

Y. Kodriano, E. R. Schmidgall, Y. Benny, and D. Gershoni, “Optical control of single excitons in semiconductor quantum dots,” Semicond. Sci. Technol. 29, 053001 (2014).
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M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotech. 10, 503–506 (2015).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. Warburton, “Charge noise and spin noise in a semiconductor quantum device,” Nat. Phys. 9, 570–575 (2013).
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D. E. Reiter, T. Kuhn, M. Glässl, and V. M. Axt, “The role of phonons for exciton and biexciton generation in an optically driven quantum dot,” J. Phys. Condens. Matter 26, 423203 (2014).
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A. Krügel, A. Vagov, V. M. Axt, and T. Kuhn, “Monitoring the buildup of the quantum dot polaron: Pump-probe and four-wave mixing spectra from excitons and biexcitons in semiconductor quantum dots,” Phys. Rev. B 76, 195302 (2007).
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T. Voss, I. Rückmann, J. Gutowski, V. M. Axt, and T. Kuhn, “Coherent control of the exciton and exciton-biexciton transitions in the generation of nonlinear wave-mixing signals in a semiconductor quantum well,” Phys. Rev. B 73, 115311 (2006).
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J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
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W. Langbein, “Coherent optical spectroscopy of semiconductor nano structures,” Riv. Nuovo Cimento 33, 255–312 (2010).

J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2010).
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J. Kasprzak and W. Langbein, “Vectorial four-wave mixing field dynamics from individual excitonic transitions,” Phys. Rev. B 78, 041103 (2008).
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B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time-and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B 73, 235354 (2006).
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W. Langbein and B. Patton, “Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems,” Opt. Lett. 31, 1151–1153 (2006).
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B. Patton, U. Woggon, and W. Langbein, “Coherent control and polarization readout of individual excitonic states,” Phys. Rev. Lett. 95, 266401 (2005).
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Maingault, L.

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C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots,” Phys. Rev. B 85, 155303 (2012).
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L. Monniello, C. Tonin, R. Hostein, A. Lemaitre, A. Martinez, V. Voliotis, and R. Grousson, “Excitation-induced dephasing in a resonantly driven InAs/GaAs quantum dot,” Phys. Rev. Lett. 111, 026403 (2013).
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G. Moody, R. Singh, H. Li, I. A. Akimov, M. Bayer, D. Reuter, A. D. Wieck, and S. T. Cundiff, “Fifth-order nonlinear optical response of excitonic states in an InAs quantum dot ensemble measured with two-dimensional spectroscopy,” Phys. Rev. B 87, 045313 (2013).
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F. Fras, Q. Mermillod, G. Nogues, C. Hoarau, C. Schneider, M. Kamp, S. Höfling, W. Langbein, and J. Kasprzak, “Multi-wave coherent control of a solid state single emitter,” Nat. Photonics 10, 155–158 (2016).
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N. H. Bonadeo, J. Erland, D. Gammon, D. Park, D. S. Katzer, and D. G. Steel, “Coherent optical control of the quantum state of a single quantum dot,” Science 282, 1473–1476 (1998).
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J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2010).
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W. Langbein and B. Patton, “Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems,” Opt. Lett. 31, 1151–1153 (2006).
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B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time-and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B 73, 235354 (2006).
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B. Patton, U. Woggon, and W. Langbein, “Coherent control and polarization readout of individual excitonic states,” Phys. Rev. Lett. 95, 266401 (2005).
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V. M. Axt, T. Kuhn, A. Vagov, and F. M. Peeters, “Phonon-induced pure dephasing in exciton-biexciton quantum dot systems driven by ultrafast laser pulse sequences,” Phys. Rev. B 72, 125309 (2005).
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Petroff, P. M.

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).
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B. Urbaszek, R. J. Warburton, K. Karrai, B. D. Gerardot, P. M. Petroff, and J. M. Garcia, “Fine structure of highly charged excitons in semiconductor quantum dots,” Phys. Rev. Lett. 90, 247403 (2003).
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J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. Warburton, “Charge noise and spin noise in a semiconductor quantum device,” Nat. Phys. 9, 570–575 (2013).
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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).
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J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
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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).
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I. Aharonovich, S. Castelletto, D. A. Simpson, C. H. Su, A. D. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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G. Moody, R. Singh, H. Li, I. A. Akimov, M. Bayer, D. Reuter, A. D. Wieck, and S. T. Cundiff, “Fifth-order nonlinear optical response of excitonic states in an InAs quantum dot ensemble measured with two-dimensional spectroscopy,” Phys. Rev. B 87, 045313 (2013).
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A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. Warburton, “Charge noise and spin noise in a semiconductor quantum device,” Nat. Phys. 9, 570–575 (2013).
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W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Control of fine-structure splitting and biexciton binding in InxGa1−x As quantum dots by annealing,” Phys. Rev. B 69, 161301 (2004).
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R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006).
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R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
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M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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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).
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J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics 5, 57–63 (2010).
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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).
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Y. Kodriano, E. R. Schmidgall, Y. Benny, and D. Gershoni, “Optical control of single excitons in semiconductor quantum dots,” Semicond. Sci. Technol. 29, 053001 (2014).
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J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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Schmidt, R.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. J. van der Zant, S. Michaelis de Vasconcellos, and R. Bratschitsch, “Single-photon emission from localized excitons in an atomically thin semiconductor,” Optica 2, 347–352 (2015).
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M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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Schneider, C.

F. Fras, Q. Mermillod, G. Nogues, C. Hoarau, C. Schneider, M. Kamp, S. Höfling, W. Langbein, and J. Kasprzak, “Multi-wave coherent control of a solid state single emitter,” Nat. Photonics 10, 155–158 (2016).
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S. Maier, P. Gold, A. Forchel, N. Gregersen, J. Mørk, S. Höfling, C. Schneider, and M. Kamp, “Bright single photon source based on self-aligned quantum dot-cavity systems,” Opt. Express 22, 8136–8142 (2014).
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Schneider, R.

Schulze, J.-H.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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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).
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Seifried, M.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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G. Bacher, R. Weigand, J. Seufert, V. D. Kulakovskii, N. A. Gippius, A. Forchel, K. Leonardi, and D. Hommel, “Biexciton versus exciton lifetime in a single semiconductor quantum dot,” Phys. Rev. Lett. 83, 4417–4420 (1999).
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Shields, A. J.

R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006).
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R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
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Simpson, D. A.

I. Aharonovich, S. Castelletto, D. A. Simpson, C. H. Su, A. D. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
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Singh, R.

G. Moody, R. Singh, H. Li, I. A. Akimov, M. Bayer, D. Reuter, A. D. Wieck, and S. T. Cundiff, “Fifth-order nonlinear optical response of excitonic states in an InAs quantum dot ensemble measured with two-dimensional spectroscopy,” Phys. Rev. B 87, 045313 (2013).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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Skolnick, M. S.

R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
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Stavarache, V.

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Control of fine-structure splitting and biexciton binding in InxGa1−x As quantum dots by annealing,” Phys. Rev. B 69, 161301 (2004).
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N. H. Bonadeo, J. Erland, D. Gammon, D. Park, D. S. Katzer, and D. G. Steel, “Coherent optical control of the quantum state of a single quantum dot,” Science 282, 1473–1476 (1998).
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Steele, G. A.

Stevenson, R. M.

R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006).
[Crossref]

R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
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Strittmatter, A.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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Su, C. H.

I. Aharonovich, S. Castelletto, D. A. Simpson, C. H. Su, A. D. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
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Tartakovskii, A. I.

R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
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Thoma, A.

M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, A. Strittmatter, S. Rodt, and S. Reitzenstein, “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron-beam lithography,” Nat. Commun. 6, 7662 (2015).
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Tonin, C.

L. Monniello, C. Tonin, R. Hostein, A. Lemaitre, A. Martinez, V. Voliotis, and R. Grousson, “Excitation-induced dephasing in a resonantly driven InAs/GaAs quantum dot,” Phys. Rev. Lett. 111, 026403 (2013).
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C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots,” Phys. Rev. B 85, 155303 (2012).
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Tonndorf, P.

Urbaszek, B.

B. Urbaszek, R. J. Warburton, K. Karrai, B. D. Gerardot, P. M. Petroff, and J. M. Garcia, “Fine structure of highly charged excitons in semiconductor quantum dots,” Phys. Rev. Lett. 90, 247403 (2003).
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A. Krügel, A. Vagov, V. M. Axt, and T. Kuhn, “Monitoring the buildup of the quantum dot polaron: Pump-probe and four-wave mixing spectra from excitons and biexcitons in semiconductor quantum dots,” Phys. Rev. B 76, 195302 (2007).
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V. M. Axt, T. Kuhn, A. Vagov, and F. M. Peeters, “Phonon-induced pure dephasing in exciton-biexciton quantum dot systems driven by ultrafast laser pulse sequences,” Phys. Rev. B 72, 125309 (2005).
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C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots,” Phys. Rev. B 85, 155303 (2012).
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J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
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Woggon, U.

B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time-and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B 73, 235354 (2006).
[Crossref]

B. Patton, U. Woggon, and W. Langbein, “Coherent control and polarization readout of individual excitonic states,” Phys. Rev. Lett. 95, 266401 (2005).
[Crossref]

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Control of fine-structure splitting and biexciton binding in InxGa1−x As quantum dots by annealing,” Phys. Rev. B 69, 161301 (2004).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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F. Ding, R. Singh, J. D. Plumhof, T. Zander, V. Křápek, Y. H. Chen, M. Benyoucef, V. Zwiller, K. Dörr, G. Bester, A. Rastelli, and O. G. Schmidt, “Tuning the exciton binding energies in single self-assembled InGaAs/GaAs quantum dots by piezoelectric-induced biaxial stress,” Phys. Rev. Lett. 104, 067405 (2010).
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Nat. Commun. (1)

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Nat. Nanotech. (1)

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotech. 10, 503–506 (2015).
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Nat. Phys. (1)

A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. Warburton, “Charge noise and spin noise in a semiconductor quantum device,” Nat. Phys. 9, 570–575 (2013).
[Crossref]

Nature (1)

R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006).
[Crossref]

New J. Phys. (1)

J. Kasprzak, S. Portolan, A. Rastelli, L. Wang, J. D. Plumhof, O. G. Schmidt, and W. Langbein, “Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system,” New J. Phys. 15, 055006 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

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T. Voss, I. Rückmann, J. Gutowski, V. M. Axt, and T. Kuhn, “Coherent control of the exciton and exciton-biexciton transitions in the generation of nonlinear wave-mixing signals in a semiconductor quantum well,” Phys. Rev. B 73, 115311 (2006).
[Crossref]

G. Moody, R. Singh, H. Li, I. A. Akimov, M. Bayer, D. Reuter, A. D. Wieck, and S. T. Cundiff, “Fifth-order nonlinear optical response of excitonic states in an InAs quantum dot ensemble measured with two-dimensional spectroscopy,” Phys. Rev. B 87, 045313 (2013).
[Crossref]

C. Tonin, R. Hostein, V. Voliotis, R. Grousson, A. Lemaitre, and A. Martinez, “Polarization properties of excitonic qubits in single self-assembled quantum dots,” Phys. Rev. B 85, 155303 (2012).
[Crossref]

F. Gindele, K. Hild, W. Langbein, and U. Woggon, “Phonon interaction of single excitons and biexcitons,” Phys. Rev. B 60, R2157 (1999).
[Crossref]

W. Langbein, H. Gislason, and J. M. Hvam, “Transient four-wave mixing in t-shaped GaAs quantum wires,” Phys. Rev. B 60, 16667–16674 (1999).
[Crossref]

R. J. Young, R. M. Stevenson, A. J. Shields, P. Atkinson, K. Cooper, D. A. Ritchie, K. M. Groom, A. I. Tartakovskii, and M. S. Skolnick, “Inversion of exciton level splitting in quantum dots,” Phys. Rev. B 72, 113305 (2005).
[Crossref]

A. Krügel, A. Vagov, V. M. Axt, and T. Kuhn, “Monitoring the buildup of the quantum dot polaron: Pump-probe and four-wave mixing spectra from excitons and biexcitons in semiconductor quantum dots,” Phys. Rev. B 76, 195302 (2007).
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[Crossref]

A. Vagov, V. M. Axt, and T. Kuhn, “Electron-phonon dynamics in optically excited quantum dots: Exact solution for multiple ultrashort laser pulses,” Phys. Rev. B 66, 165312 (2002).
[Crossref]

V. M. Axt, T. Kuhn, A. Vagov, and F. M. Peeters, “Phonon-induced pure dephasing in exciton-biexciton quantum dot systems driven by ultrafast laser pulse sequences,” Phys. Rev. B 72, 125309 (2005).
[Crossref]

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Control of fine-structure splitting and biexciton binding in InxGa1−x As quantum dots by annealing,” Phys. Rev. B 69, 161301 (2004).
[Crossref]

B. Patton, W. Langbein, U. Woggon, L. Maingault, and H. Mariette, “Time-and spectrally-resolved four-wave mixing in single CdTe/ZnTe quantum dots,” Phys. Rev. B 73, 235354 (2006).
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[Crossref]

Phys. Rev. Lett. (9)

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Supplementary Material (1)

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Figures (9)

Fig. 1.
Fig. 1. Schematic picture. Sketches of the QD system for (a) circularly polarized excitation and (b) linearly polarized excitation. (c) Cartoon of the FWM experiment and the QD orientation.
Fig. 2.
Fig. 2. Quantum beats. (a) Two-pulse FWM spectrum of a single QD for co-linear polarization. Inset: Zoom-in on the exciton line for co-circular polarization, revealing FSS. (b) FWM signal as function of excitation intensity P 1 . (c) FWM signal as function of time delay τ 12 for co-circularly polarized excitation. (d) Zoom-in of (c) around τ 12 = 0 . (e) and (f) FWM signal as function of τ 12 for linear polarization with α = 0 ° . Green dots and blue squares: experimental data; gray circles: indication of noise level; red lines: theoretical calculations.
Fig. 3.
Fig. 3. Coherence dynamics for co-linear excitation. (a), (b) Measurements and (c), (d) calculations of the coherence dynamics as functions of delay τ 12 and polarization angle α for co-linear excitation for (a), (c) the GXY transition and (b), (d) the XYB transition.
Fig. 4.
Fig. 4. Coherence dynamics for cross-linear excitation. Two-pulse FWM amplitude of the coherence dynamics of GXY and XYB transitions for cross-polarized excitation with α 2 = α 1 + 90 ° and (a)  α 1 = 0 ° , (b)  α 1 = 22.5 ° , and (c)  α 1 = 45 ° . Green dots and blue squares: experimental data of GXY and XYB, respectively. The background level is indicated by dark gray circles. Theoretical predictions are given by red lines.
Fig. 5.
Fig. 5. Population dynamics. Three-pulse FWM amplitude displaying population dynamics of GXY and XYB for different polarization states of E 1,2 , 3 (a)  ( , , ) , (b)  ( , , ) , (c)  ( / , / , / ) , and (d)  ( / , / , ) . Green dots and blue squares: experimental data of GXY and XYB, respectively; gray circles: indication of noise level; red lines: theoretical calculations.
Fig. 6.
Fig. 6. Charged exciton complexes. Pictographic representation of the (a) negative trion system and (b) the charged biexciton system with the excited trion states.
Fig. 7.
Fig. 7. FWM of a charge-fluctuating QD. (a) Two-pulse FWM spectrum of a QD shown with (b) the corresponding dynamical behavior in τ 12 . (c) FWM signal of the coherence dynamics for the GX, XB, and G X transitions.
Fig. 8.
Fig. 8. 2D FWM spectroscopy map, revealing coherent couplings between exciton complexes generated by a charge-fluctuating QD.
Fig. 9.
Fig. 9. Pulse area dependence of 2D maps. 2D FWM spectroscopy map of the exciton-biexciton system for an excitation with a pulse area of (a), (c) θ 1 = 0.1 π and (b), (d) θ 1 = 0.45 π . Upper row: experimental data, lower row: theoretical calculations. The inset in (c) shows the strength of the ratio ζ of biexciton XB and the correlation peak X B ¯ as function of pulse area θ 1 ; orange dots: experiment; and blue curve: theory.

Equations (8)

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S G σ exp ( τ 12 T 2 ) cos ( δ τ 12 ) + 1 .
S G X exp ( τ 12 T 2 ) 4.36 + cos ( Δ τ 12 ) ,
S X B exp ( τ 12 T 2 ) 1.15 + cos ( Δ τ 12 ) .
S G X , X B τ < 0 e β B τ 12 .
S G X Y e γ τ 23 0.29 e 2 γ τ 23 ,
S X Y B e γ τ 23 + 0.76 e 2 γ τ 23 .
S G σ = S G X Y 6 cos ( δ τ 23 ) e ( γ + β X Y ) τ 23 + 9 e 2 γ τ 23 + e 2 β X Y τ 23 ,
S σ B = S X Y B 2 cos ( δ τ 23 ) e ( γ + β X Y ) τ 23 e 2 γ τ 23 e 2 β X Y τ 23 .

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