Abstract

Semiconductor quantum dots are inevitably coupled to the vibrational modes of their host lattice. This interaction reduces the efficiency and the indistinguishability of single-photons emitted from semiconductor quantum dots. While the adverse effects of phonons can be significantly reduced by embedding the quantum dot in a photonic cavity, phonon-induced signatures in the emitted photons cannot be completely suppressed and constitute a fundamental limit to the ultimate performance of single-photon sources based on quantum dots. In this paper, we present a self-consistent theoretical description of phonon effects in such sources and describe their influence on the figures of merit.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

3 January 2020: A typographical correction was made to the body text.


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2019 (8)

J. Liu, R. Su, Y. Wei, B. Yao, S. F. C. da Silva, Y. Yu, J. Iles-Smith, K. Srinivasan, A. Rastelli, J. Li, and X. Wang, “A solid-state source of strongly entangled photon pairs with high brightness and indistinguishability,” Nat. Nanotechnol. 14(6), 586–593 (2019).
[Crossref]

H. Wang, Y.-M. He, T.-H. Chung, H. Hu, Y. Yu, S. Chen, X. Ding, M.-C. Chen, J. Qin, X. Yang, R.-Z. Liu, Z.-C. Duan, J.-P. Li, S. Gerhardt, K. Winkler, J. Jurkat, L.-J. Wang, N. Gregersen, Y.-H. Huo, Q. Dai, S. Yu, S. Höfling, C.-Y. Lu, and J.-W. Pan, “Towards optimal single-photon sources from polarized microcavities,” Nat. Photonics 13(11), 770–775 (2019).
[Crossref]

H. Choi, D. Zhu, Y. Yoon, and D. Englund, “Cascaded cavities boost the indistinguishability of imperfect quantum emitters,” Phys. Rev. Lett. 122(18), 183602 (2019).
[Crossref]

Z.-X. Koong, D. Scerri, M. Rambach, T. S. Santana, S.-I. Park, J. D. Song, E. M. Gauger, and B. D. Gerardot, “Fundamental limits to coherent photon generation with solid-state atom-like transitions,” Phys. Rev. Lett. 123(16), 167402 (2019).
[Crossref]

A. J. Brash, J. Iles-Smith, C. L. Phillips, D. P. S. McCutcheon, J. O’Hara, E. Clarke, B. Royall, J. Mørk, M. S. Skolnick, A. M. Fox, and A. Nazir, “Light scattering from solid-state quantum emitters: Beyond the atomic picture,” Phys. Rev. Lett. 123(16), 167403 (2019).
[Crossref]

O. Černotík, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122(24), 243601 (2019).
[Crossref]

S. Lüker and D. E. Reiter, “A review on optical excitation of semiconductor quantum dots under the influence of phonons,” Semicond. Sci. Technol. 34(6), 063002 (2019).
[Crossref]

A. Carmele and S. Reitzenstein, “Non-Markovian features in semiconductor quantum optics: quantifying the role of phonons in experiment and theory,” Nanophotonics 8(5), 655–683 (2019).
[Crossref]

2018 (8)

F. Peyskens and D. Englund, “Quantum photonics model for nonclassical light generation using integrated nanoplasmonic cavity-emitter systems,” Phys. Rev. A 97(6), 063844 (2018).
[Crossref]

C. Gustin and S. Hughes, “Pulsed excitation dynamics in quantum-dot–cavity systems: Limits to optimizing the fidelity of on-demand single-photon sources,” Phys. Rev. B 98(4), 045309 (2018).
[Crossref]

P. Tighineanu, C. L. Dreessen, C. Flindt, P. Lodahl, and A. S. Sørensen, “Phonon decoherence of quantum dots in photonic structures: Broadening of the zero-phonon line and the role of dimensionality,” Phys. Rev. Lett. 120(25), 257401 (2018).
[Crossref]

B. Kambs and C. Becher, “Limitations on the indistinguishability of photons from remote solid state sources,” New J. Phys. 20(11), 115003 (2018).
[Crossref]

E. V. Denning, J. Iles-Smith, A. D. Osterkryger, N. Gregersen, and J. Mork, “Cavity-waveguide interplay in optical resonators and its role in optimal single-photon sources,” Phys. Rev. B 98(12), 121306 (2018).
[Crossref]

D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, “Nonperturbative treatment of non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 120(3), 030402 (2018).
[Crossref]

S. Gerhardt, J. Iles-Smith, D. P. S. McCutcheon, Y.-M. He, S. Unsleber, S. Betzold, N. Gregersen, J. Mørk, S. Höfling, and C. Schneider, “Intrinsic and environmental effects on the interference properties of a high-performance quantum dot single-photon source,” Phys. Rev. B 97(19), 195432 (2018).
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M. Cosacchi, M. Cygorek, F. Ungar, A. M. Barth, A. Vagov, and V. M. Axt, “Path-integral approach for nonequilibrium multitime correlation functions of open quantum systems coupled to Markovian and non-Markovian environments,” Phys. Rev. B 98(12), 125302 (2018).
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2017 (8)

G. Hornecker, A. Auffèves, and T. Grange, “Influence of phonons on solid-state cavity-QED investigated using nonequilibrium Green’s functions,” Phys. Rev. B 95(3), 035404 (2017).
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J. Iles-Smith, D. P. S. McCutcheon, J. Mørk, and A. Nazir, “Limits to coherent scattering and photon coalescence from solid-state quantum emitters,” Phys. Rev. B 95(20), 201305 (2017).
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A. Strathearn, B. W. Lovett, and P. Kirton, “Efficient real-time path integrals for non-Markovian spin-boson models,” New J. Phys. 19(9), 093009 (2017).
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T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lemaître, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118(25), 253602 (2017).
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A. Reigue, J. Iles-Smith, F. Lux, L. Monniello, M. Bernard, F. Margaillan, A. Lemaître, A. Martinez, D. P. S. Mccutcheon, J. Mørk, R. Hostein, and V. Voliotis, “Probing electron-phonon interaction through two-photon interference in resonantly driven semiconductor quantum dots,” Phys. Rev. Lett. 118(23), 233602 (2017).
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P. Senellart, G. Solomon, and A. White, “High-performance semiconductor quantum-dot single-photon sources,” Nat. Nanotechnol. 12(11), 1026–1039 (2017).
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J. Iles-Smith, D. P. S. McCutcheon, A. Nazir, and J. Mørk, “Phonon scattering inhibits simultaneous near-unity efficiency and indistinguishability in semiconductor single-photon sources,” Nat. Photonics 11(8), 521–526 (2017).
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K. A. Fischer, L. Hanschke, J. Wierzbowski, T. Simmet, C. Dory, J. J. Finley, J. Vučković, and K. Müller, “Signatures of two-photon pulses from a quantum two-level system,” Nat. Phys. 13(7), 649–654 (2017).
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2016 (7)

X. Ding, Y. He, Z.-C. Duan, N. Gregersen, M.-C. Chen, S. Unsleber, S. Maier, C. Schneider, M. Kamp, S. Höfling, C.-Y. Lu, and J.-W. Pan, “On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar,” Phys. Rev. Lett. 116(2), 020401 (2016).
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N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaître, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state,” Nat. Photonics 10(5), 340–345 (2016).
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R. N. Malein, T. S. Santana, J. M. Zajac, A. C. Dada, E. Gauger, P. M. Petroff, J. Y. Lim, J. D. Song, and B. D. Gerardot, “Screening nuclear field fluctuations in quantum dots for indistinguishable photon generation,” Phys. Rev. Lett. 116(25), 257401 (2016).
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A. Thoma, P. Schnauber, M. Gschrey, M. Seifried, J. Wolters, J.-H. Schulze, A. Strittmatter, S. Rodt, A. Carmele, A. Knorr, T. Heindel, and S. Reitzenstein, “Exploring dephasing of a solid-state quantum emitter via time- and temperature-dependent hong-ou-mandel experiments,” Phys. Rev. Lett. 116(3), 033601 (2016).
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N. Gregersen, D. P. S. McCutcheon, J. Mørk, J.-M. Gérard, and J. Claudon, “A broadband tapered nanocavity for efficient nonclassical light emission,” Opt. Express 24(18), 20904–20924 (2016).
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A. Nazir and D. P. S. McCutcheon, “Modelling exciton–phonon interactions in optically driven quantum dots,” J. Phys.: Condens. Matter 28(10), 103002 (2016).
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P. Tighineanu, R. S. Daveau, T. B. Lehmann, H. E. Beere, D. A. Ritchie, P. Lodahl, and S. Stobbe, “Single-photon superradiance from a quantum dot,” Phys. Rev. Lett. 116(16), 163604 (2016).
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2015 (5)

T. Grange, G. Hornecker, D. Hunger, J.-P. Poizat, J.-M. Gérard, P. Senellart, and A. Auffèves, “Cavity-funneled generation of indistinguishable single photons from strongly dissipative quantum emitters,” Phys. Rev. Lett. 114(19), 193601 (2015).
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K. Roy-Choudhury and S. Hughes, “Spontaneous emission from a quantum dot in a structured photonic reservoir: phonon-mediated breakdown of Fermi’s golden rule,” Optica 2(5), 434–437 (2015).
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K. Roy-Choudhury and S. Hughes, “Quantum theory of the emission spectrum from quantum dots coupled to structured photonic reservoirs and acoustic phonons,” Phys. Rev. B 92(20), 205406 (2015).
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T. Huber, A. Predojević, D. Föger, G. Solomon, and G. Weihs, “Optimal excitation conditions for indistinguishable photons from quantum dots,” New J. Phys. 17(12), 123025 (2015).
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P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87(2), 347–400 (2015).
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2014 (3)

P. Kaer and J. Mørk, “Decoherence in semiconductor cavity QED systems due to phonon couplings,” Phys. Rev. B 90(3), 035312 (2014).
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R.-C. Ge, P. T. Kristensen, J. F. Young, and S. Hughes, “Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics,” New J. Phys. 16(11), 113048 (2014).
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A. Alkauskas, B. B. Buckley, D. D. Awschalom, and C. G. V. de Walle, “First-principles theory of the luminescence lineshape for the triplet transition in diamond NV centres,” New J. Phys. 16(7), 073026 (2014).
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2013 (8)

D. P. S. McCutcheon and A. Nazir, “Model of the optical emission of a driven semiconductor quantum dot: Phonon-enhanced coherent scattering and off-resonant sideband narrowing,” Phys. Rev. Lett. 110(21), 217401 (2013).
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P. Kaer, P. Lodahl, A.-P. Jauho, and J. Mork, “Microscopic theory of indistinguishable single-photon emission from a quantum dot coupled to a cavity: The role of non-Markovian phonon-induced decoherence,” Phys. Rev. B 87(8), 081308 (2013).
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M. Glässl, A. M. Barth, and V. M. Axt, “Proposed robust and high-fidelity preparation of excitons and biexcitons in semiconductor quantum dots making active use of phonons,” Phys. Rev. Lett. 110(14), 147401 (2013).
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A. Nysteen, P. Kaer, and J. Mork, “Proposed quenching of phonon-induced processes in photoexcited quantum dots due to electron-hole asymmetries,” Phys. Rev. Lett. 110(8), 087401 (2013).
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P. Kaer, N. Gregersen, and J. Mork, “The role of phonon scattering in the indistinguishability of photons emitted from semiconductor cavity QED systems,” New J. Phys. 15(3), 035027 (2013).
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Y.-M. He, Y. He, Y.-J. Wei, D. Wu, M. Atatüre, C. Schneider, S. Höfling, M. Kamp, C.-Y. Lu, and J.-W. Pan, “On-demand semiconductor single-photon source with near-unity indistinguishability,” Nat. Nanotechnol. 8(3), 213–217 (2013).
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N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE J. Sel. Top. Quantum Electron. 19(5), 1–16 (2013).
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2012 (2)

P. Kaer, T. R. Nielsen, P. Lodahl, A.-P. Jauho, and J. Mørk, “Microscopic theory of phonon-induced effects on semiconductor quantum dot decay dynamics in cavity QED,” Phys. Rev. B 86(8), 085302 (2012).
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S. Stobbe, P. T. Kristensen, J. E. Mortensen, J. M. Hvam, J. Mørk, and P. Lodahl, “Spontaneous emission from large quantum dots in nanostructures: Exciton-photon interaction beyond the dipole approximation,” Phys. Rev. B 86(8), 085304 (2012).
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2011 (2)

A. Vagov, M. D. Croitoru, M. Glässl, V. M. Axt, and T. Kuhn, “Real-time path integrals for quantum dots: Quantum dissipative dynamics with superohmic environment coupling,” Phys. Rev. B 83(9), 094303 (2011).
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C. Roy and S. Hughes, “Phonon-dressed mollow triplet in the regime of cavity quantum electrodynamics: Excitation-induced dephasing and nonperturbative cavity feeding effects,” Phys. Rev. Lett. 106(24), 247403 (2011).
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2010 (4)

D. P. S. McCutcheon and A. Nazir, “Quantum dot Rabi rotations beyond the weak exciton–phonon coupling regime,” New J. Phys. 12(11), 113042 (2010).
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A. J. Ramsay, A. V. Gopal, E. M. Gauger, A. Nazir, B. W. Lovett, A. M. Fox, and M. S. Skolnick, “Damping of exciton Rabi rotations by acoustic phonons in optically excited InGaAs/GaAs quantum dots,” Phys. Rev. Lett. 104(1), 017402 (2010).
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A. Dousse, J. Suffczyński, A. Beveratos, O. Krebs, A. Lemaître, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466(7303), 217–220 (2010).
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S. Stobbe, T. Schlereth, S. Höfling, A. Forchel, J. M. Hvam, and P. Lodahl, “Large quantum dots with small oscillator strength,” Phys. Rev. B 82(23), 233302 (2010).
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2009 (1)

S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Post-selected indistinguishable photons from the resonance fluorescence of a single quantum dot in a microcavity,” Phys. Rev. Lett. 103(16), 167402 (2009).
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2008 (1)

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser Photonics Rev. 2(6), 514–526 (2008).
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2007 (4)

V. M. Rao and S. Hughes, “Single quantum-dot purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007).
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V. M. Rao and S. Hughes, “Single quantum dot spontaneous emission in a finite-size photonic crystal waveguide: proposal for an efficient "on chip" single photon gun,” Phys. Rev. Lett. 99(19), 193901 (2007).
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2006 (1)

N. Akopian, N. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. Gerardot, and P. Petroff, “Entangled photon pairs from semiconductor quantum dots,” Phys. Rev. Lett. 96(13), 130501 (2006).
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A. Kiraz, M. Atatüre, and A. Imamoğlu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A 69(3), 032305 (2004).
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E. A. Muljarov and R. Zimmermann, “Dephasing in quantum dots: Quadratic coupling to acoustic phonons,” Phys. Rev. Lett. 93(23), 237401 (2004).
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P. Lalanne, J. P. Hugonin, and J. M. Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84(23), 4726–4728 (2004).
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2003 (2)

J. Förstner, C. Weber, J. Danckwerts, and A. Knorr, “Phonon-assisted damping of Rabi oscillations in semiconductor quantum dots,” Phys. Rev. Lett. 91(12), 127401 (2003).
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2002 (3)

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2001 (3)

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Figures (7)

Fig. 1.
Fig. 1. a. When a quantum dot decays from its lowest-lying exciton state, $|{e}\rangle$ , to its ground state, $|{g}\rangle$ , a phonon wavepacket may be emitted or absorbed. This leads to a broad sideband (SB) in the photon emission spectrum along with the narrow zero-phonon line (ZPL), arising from relaxation not involving phonons. b. Emission spectrum showing the sideband and zero-phonon line. The emission spectrum has been calculated with a polaron master equation, described in Sec. 3. c. By introducing a cavity with narrow linewidth, the zero-phonon line can be selectively enhanced through the Purcell effect. Here, the optical spectral density of a Fabry-Pérot cavity, Eq. (5), is shown.
Fig. 2.
Fig. 2. a. Schematic of cavity model consisting of a waveguiding background medium with two mirrors. b. Artistic illustration of the micropillar single-photon source geometry. c. $J_0(\omega )$ (black) and $\Gamma _{\mathrm {R}}(\omega )$ (red) computed using the full model (points) and the single-mode model, Eq. (5) (solid lines). Both are normalized to the LDOS of a bulk material, $\Gamma _{\mathrm {bulk}}$ . The pillar features 15 (25) DBR layer pairs in the top (bottom) mirror and a diameter of 2.5 $\mu$ m, see Ref. [41] for further geometrical details. d.–g. Waveguide $\Gamma _{\mathrm {WG}}$ (green solid) and cavity (orange dotted, plotted as offset from green solid line) contributions to LDOS, where $\Delta = c/(\bar {n} L)$ is the free spectral range of the cavity. The LDOS from Eq. (5) is plotted with blue shading. Panels d-g correspond to mirror reflectivities of 0, 0.1, 0.5 and 0.8, respectively.
Fig. 3.
Fig. 3. a. Emitter–cavity coupling strength, b. cavity linewidth and c spontaneous emission rate into waveguide as a function of mirror reflectivity for a symmetric cavity, $r_1=r_2=r$ . The blue, orange and green lines correspond to $L=1\mathrm {\;\mu m},\; L=5\mathrm {\;\mu m}$ and $L=10\mathrm {\;\mu m}$ , respectively. The spontaneous emission rate into the waveguide, $\Gamma _{\mathrm {WG}}$ , is independent of the cavity length. Parameters: $\bar {n}=2,\;\hbar \Gamma _0=\mathrm {1\; \mu eV}$ .
Fig. 4.
Fig. 4. a. Two phonon-induced processes affect the optical emission properties of QDs: Emission/absorption of phonon wavepacket during exciton relaxation and virtual transitions to higher-lying excitonic states through scattering of thermal phonons. b. Emission or absorption of a phonon wavepacket leads to a broad phonon sideband in the emission spectrum. c. Virtual scattering of thermal phonons leads to broadening of the zero-phonon line due to pure dephasing. d. When the QD is placed in an optical cavity, the electromagnetic LDOS influences the shape of the spectrum by funneling emission into the cavity resonance. In the strong coupling regime, the phonon scattering leads to an asymmetry between the polariton peaks [55].
Fig. 5.
Fig. 5. a. Indistinguishability, $\mathcal {I}$ b. efficiency, $\mathcal {E}$ , and c. their product, $\mathcal {IE}$ as a function of cavity linewidth. Parameters: $\hbar \Gamma _0=1\mathrm {\;\mu eV},\; \hbar \Gamma _{\mathrm {R}}=1\mathrm {\;\mu eV},\; \hbar \gamma _0=0.5\mathrm {\; \mu eV},\hbar g=100\mathrm {\;\mu eV},\; T=4\mathrm {\;K}$ and QD size $d=10\mathrm {\; nm}$ .
Fig. 6.
Fig. 6. a. Indistinguishability, $\mathcal {I}$ b. efficiency, $\mathcal {E}$ , and c. their product, $\mathcal {IE}$ as a function of cavity linewidth, where the blue lines correspond to a temperature of $5\mathrm {\; K}$ , the brown lines to $55\mathrm {\; K}$ and the remaining lines spaced regularly in steps of $10\mathrm {\;K}$ . d. Frank-Condon factor, $B^2$ , as a function of temperature, with coloured circles indicating the temperatures of the corresponding coloured lines in panels a-c. e. Phonon-induced pure dephasing rate of the zero-phonon line as a function of temperature. As for panel e, the coloured circles correspond to the coloured lines in panels a-c. Parameters: $\hbar \Gamma _0=1\mathrm {\;\mu eV},\; \hbar \Gamma _{\mathrm {R}}=1\mathrm {\;\mu eV},\; \hbar \gamma _0=0.5\mathrm {\; \mu eV},\hbar g=100\mathrm {\;\mu eV}$ and QD size $d=10\mathrm {\; nm}$ .
Fig. 7.
Fig. 7. a. Indistinguishability, $\mathcal {I}$ b. efficiency, $\mathcal {E}$ , and c. their product, $\mathcal {IE}$ as a function of cavity linewidth, where each line corresponds to a different QD size d. Frank-Condon factor, $B^2$ , as a function of QD size, with coloured circles indicating the temperatures of the corresponding coloured lines in panels a-c. e. Phonon-induced pure dephasing rate of the zero-phonon line as a function of QD size. As for panel e, the coloured circles correspond to the coloured lines in panels a-c. Parameters: $\hbar \Gamma _0=1\mathrm {\;\mu eV},\; \hbar \Gamma _{\mathrm {R}}=1\mathrm {\;\mu eV},\; \hbar \gamma _0=0.5\mathrm {\; \mu eV},\hbar g=100\mathrm {\;\mu eV}, \; T=10\mathrm {\; K}.$

Tables (1)

Tables Icon

Table 1. Material parameters for GaAs [72,73] used for calculation of electron–phonon coupling parameters. m 0 denotes the free electron mass.

Equations (20)

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H = H E + H P + H F + H E P + H E F ,
J ( r , ω , n p ) = μ [ n p ( u μ ( r , ω ) u μ ( r , ω ) ) n p ) ] δ ( ω μ ω ) .
J ( r , ω , n p ) = π p 2 ω ϵ 0 J ( r , ω , n p ) = 2 π μ | h μ | 2 δ ( ω ω μ ) ,
J ( ω ) = J 0 ( ω ) + Γ R ( ω ) ,
J 0 ( ω ) = Γ 0 { [ 1 + r ~ 1 ( ω ) ] [ 1 + r ~ 2 ( ω ) ] 1 r ~ 1 ( ω ) r ~ 2 ( ω ) } .
V L = k g k ( b k + b k ) ,
M a , k i j = ν k 2 ϱ c s 2 V D a ψ i , a ( r ) ψ j , a ( r ) e i k r   d 3 r ,
V Q = k k f k , k ( b k + b k ) ( b k + b k ) ,
ρ ˙ ( t ) = i [ H E , ρ ( t ) ] + γ D σ σ [ ρ ( t ) ] + ( Γ W G + Γ R ) D σ [ ρ ( t ) ] + κ D a [ ρ ( t ) ] + W [ ρ ( t ) ] ,
W [ ρ ( t ) ] = g 2 { [ X , ρ ( t ) Θ X ] + [ Y , ρ ( t ) Θ Y ] + H . c . } ,
Λ X ( τ ) = 1 2 B 2 [ e φ ( τ ) + e φ ( τ ) 2 ] , Λ Y ( τ ) = 1 2 B 2 [ e φ ( τ ) e φ ( τ ) ] .
σ ( t + τ ) σ ( t ) G P ( τ ) Tr [ σ e L τ σ e L t ρ ( 0 ) ] , τ , t > 0 ,
S ( ω , ω ) = G E ( ω ) G E ( ω ) S 0 ( ω , ω )
G E ( ω ) = [ 1 + r ~ 1 ( ω ) ] t 2 1 r ~ 1 ( ω ) r ~ 2 ( ω ) ,
I = ( Γ 0 2 P ) 2 d ω d ω | S ( ω , ω ) | 2 .
S ¯ 0 z p l ( ω ) = B 2 d t d τ σ ( t + τ ) σ ( t ) e i ω τ ,
S ¯ 0 p s b ( ω ) = d t d τ ( G P ( τ ) B 2 ) σ ( t + τ ) σ ( t ) e i ω τ .
γ P = α μ ν c 4 0 ν 10 e 2 ν 2 / ν c 2 n ( ν ) ( n ( ν ) + 1 ) d ν ,
I = Γ t o t Γ t o t + γ t o t [ ( Γ W G + Γ c a v ) B 2 ( Γ W G + Γ c a v ) B 2 + 2 Γ 0 F ( 1 B 2 ) ] 2 E = ( Γ c a v + Γ W G ) B 2 + 2 Γ 0 F ( 1 B 2 ) ( Γ c a v + Γ W G ) B 2 + 2 Γ 0 F ( 1 B 2 ) + Γ R ,
I = Γ t o t Γ t o t + γ , E = Γ c a v + Γ W G Γ t o t .

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