Abstract

Optical methods, which allow the determination of the dominant channels of energy and phase relaxation, are the most universal techniques for the investigation of semiconductor quantum dots. In this paper, we employ the kinetic Pauli equation to develop the first generalized model of the pulse-induced photoluminescence from the lowest-energy eigenstates of a semiconductor quantum dot. Without specifying the shape of the excitation pulse and by assuming that the energy and phase relaxation in the quantum dot may be characterized by a set of phenomenological rates, we derive an expression for the observable photoluminescence cross section, valid for an arbitrary number of the quantum dot’s states decaying with the emission of secondary photons. Our treatment allows for thermal transitions occurring with both decrease and increase in energy between all the relevant eigenstates at room or higher temperature. We show that in the general case of N states coupled to each other through a bath, the photoluminescence kinetics from any of them is determined by the sum of N exponential functions, whose exponents are proportional to the respective decay rates. We illustrate the application of the developed model by considering the processes of resonant luminescence and thermalized luminescence from the quantum dot with two radiating eigenstates, and by assuming that the secondary emission is excited with either a Gaussian or exponential pulse. Analytic expressions describing the signals of secondary emission are analyzed, in order to elucidate experimental situations in which the relaxation constants may be reliably extracted from the photoluminescence spectra.

© 2012 OSA

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2012 (2)

K. Rivoire, S. Buckley, Y. Song, M. L. Lee, and J. Vučković, “Photoluminescence from In0.5Ga0.5As/GaP quantum dots coupled to photonic crystal cavities,” Phys. Rev. B85, 045319 (2012).
[CrossRef]

E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
[CrossRef] [PubMed]

2011 (8)

J. Gomis-Bresco, G. Muñoz-Matutano, J. Martínez-Pastor, B. Alén, L. Seravalli, P. Frigeri, G. Trevisi, and S. Franchi, “Random population model to explain the recombination dynamics in single InAs/GaAs quantum dots under selective optical pumping,” New J. Phys.13, 023022 (2011).
[CrossRef]

I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express19, 15459–15482 (2011).
[CrossRef] [PubMed]

X. M. Dou, B. Q. Sun, D. S. Jiang, H. Q. Ni, and Z. C. Niu, “Electron spin relaxation in a single InAs quantum dot measured by tunable nuclear spins,” Phys. Rev. B84, 033302 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
[CrossRef]

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photonics Rev.5, 607–633 (2011).

F. Schulze, M. Schoth, U. Woggon, A. Knorr, and C. Weber, “Ultrafast dynamics of carrier multiplication in quantum dots,” Phys. Rev. B84, 125318 (2011).
[CrossRef]

V. K. Turkov, S. Yu. Kruchinin, and A. V. Fedorov, “Intraband optical transitions in semiconductor quantum dots: Radiative electronic-excitation lifetime,” Opt. Spectrosc.110, 740–747 (2011).
[CrossRef]

2010 (4)

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photoluminescence mediated by incoherent reversible energy transport,” Phys. Rev. B81, 245303 (2010).
[CrossRef]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys.133, 104704 (2010).
[CrossRef] [PubMed]

M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc.109, 358–365 (2010).
[CrossRef]

2009 (4)

H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B80, 235319 (2009).
[CrossRef]

M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Impact ionization in InSb probed by terahertz pump—terahertz probe spectroscopy,” Phys. Rev. B79, 161201 (2009).
[CrossRef]

M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots,” J. Appl. Phys.106, 053504 (2009).
[CrossRef]

E. G. Kavousanaki, O. Roslyak, and S. Mukamel, “Probing excitons and biexcitons in coupled quantum dots by coherent two-dimensional optical spectroscopy,” Phys. Rev. B79, 155324 (2009).
[CrossRef]

2008 (3)

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Resonant energy transfer in quantum dots: Frequency-domain luminescent spectroscopy,” Phys. Rev. B78, 125311 (2008).
[CrossRef]

H.-Y. Liu, Z.-M. Meng, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and T. Yang, “Ultrafast carrier dynamics in undoped and p-doped InAs/GaAs quantum dots characterized by pump-probe reflection measurements,” J. Appl. Phys.103, 083121 (2008).
[CrossRef]

A. Pandey and P. Guyot-Sionnest, “Slow electron cooling in colloidal quantum dots,” Science322, 929–932 (2008).
[CrossRef] [PubMed]

2007 (3)

C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
[CrossRef]

A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett.90, 093104 (2007).
[CrossRef]

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
[CrossRef]

2006 (8)

S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B74, 235328 (2006).
[CrossRef]

E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
[CrossRef] [PubMed]

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. B73, 235354 (2006).
[CrossRef]

G. A. Narvaez, G. Bester, and A. Zunger, “Carrier relaxation mechanisms in self-assembled (In,Ga)As/GaAs quantum dots: Efficient P → S Auger relaxation of electrons,” Phys. Rev. B74, 075403 (2006).
[CrossRef]

J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
[CrossRef]

A. V. Fedorov and I. D. Rukhlenko, “Study of electronic dynamics of quantum dots using resonant photoluminescence technique,” Opt. Spectrosc.100, 716–723 (2006).
[CrossRef]

A. V. Fedorov and I. D. Rukhlenko, “Propagation of electric fields induced by optical phonons in semiconductor heterostructures,” Opt. Spectrosc.100, 238–244 (2006).
[CrossRef]

I. D. Rukhlenko and A. V. Fedorov, “Penetration of electric fields induced by surface phonon modes into the layers of a semiconductor heterostructure,” Opt. Spectrosc.101, 253–264 (2006).
[CrossRef]

2005 (4)

T. B. Norris, K. Kim, J. Urayama, Z. K. Wu, J. Singh, and P. K. Bhattacharya, “Density and temperature dependence of carrier dynamics in self-organized InGaAs quantum dots,” J. Phys. D: Appl. Phys.38, 2077 (2005).
[CrossRef]

P. Guyot-Sionnest, B. Wehrenberg, and D. Yu, “Intraband relaxation in cdse nanocrystals and the strong influence of the surface ligands,” J. Chem. Phys.123, 074709 (2005).
[CrossRef] [PubMed]

R. D. Schaller, J. M. Pietryga, S. V. Goupalov, M. A. Petruska, S. A. Ivanov, and V. I. Klimov, “Breaking the phonon bottleneck in semiconductor nanocrystals via multiphonon emission induced by intrinsic nonadiabatic interactions,” Phys. Rev. Lett.95, 196401 (2005).
[CrossRef] [PubMed]

P. Borri, W. Langbein, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Exciton dephasing via phonon interactions in InAs quantum dots: Dependence on quantum confinement,” Phys. Rev. B71, 115328 (2005).
[CrossRef]

2004 (4)

V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys.67, 433 (2004).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Intraband carrier relaxation in quantum dots mediated by surface plasmon-phonon excitations,” Opt. Spectrosc.97, 56–67 (2004).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon-phonon modes,” Semicond.38, 1065–1073 (2004).
[CrossRef]

M. I. Vasilevskiy, E. V. Anda, and S. S. Makler, “Electron-phonon interaction effects in semiconductor quantum dots: A nonperturbative approach,” Phys. Rev. B70, 035318 (2004).
[CrossRef]

2003 (2)

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and Y. Masumoto, “New many-body mechanism of intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Solid State Commun.128, 219–223 (2003).
[CrossRef]

A. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Phys. Rev. B68, 205318 (2003).
[CrossRef]

2002 (6)

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance secondary emission in semiconductor quantum dots,” Opt. Spectrosc.93, 52–60 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of the quasi-elastic resonant secondary emission: Semiconductor quantum dots,” Opt. Spectrosc.92, 732–738 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of thermalized luminescence in semiconductor quantum dots,” Opt. Spectrosc.93, 555–558 (2002).
[CrossRef]

M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B65, 041308 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystal–dielectric matrix systems,” Solid State Commun.122, 139–144 (2002).
[CrossRef]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Relaxation and dephasing of multiexcitons in semiconductor quantum dots,” Phys. Rev. Lett.89, 187401 (2002).
[CrossRef] [PubMed]

2001 (2)

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]

A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
[CrossRef]

1999 (1)

X.-Q. Li, H. Nakayama, and Y. Arakawa, “Phonon bottleneck in quantum dots: Role of lifetime of the confined optical phonons,” Phys. Rev. B59, 5069–5073 (1999).
[CrossRef]

1998 (1)

X.-Q. Li and Y. Arakawa, “Anharmonic decay of confined optical phonons in quantum dots,” Phys. Rev. B57, 12285–12290 (1998).
[CrossRef]

1997 (2)

T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quantum dots,” Phys. Rev. B56, R4355–R4358 (1997).
[CrossRef]

M. Grundmann and D. Bimberg, “Theory of random population for quantum dots,” Phys. Rev. B55, 9740–9745 (1997).
[CrossRef]

1996 (4)

S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence spectroscopy of single CdSe nanocrystallite quantum dots,” Phys. Rev. Lett.77, 3873–3876 (1996).
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T. Inoshita and H. Sakaki, “Electron-phonon interaction and the so-called phonon bottleneck effect in semiconductor quantum dots,” Physica B: Cond. Matt.227, 373–377 (1996).
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D. F. Schroeter, D. J. Griffiths, and P. C. Sercel, “Defect-assisted relaxation in quantum dots at low temperature,” Phys. Rev. B54, 1486–1489 (1996).
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1995 (1)

P. C. Sercel, “Multiphonon-assisted tunneling through deep levels: A rapid energy-relaxation mechanism in non-ideal quantum-dot heterostructures,” Phys. Rev. B51, 14532–14541 (1995).
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1992 (1)

T. Inoshita and H. Sakaki, “Electron relaxation in a quantum dot: Significance of multiphonon processes,” Phys. Rev. B46, 7260–7263 (1992).
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1991 (1)

H. Benisty, C. M. Sotomayor-Torrès, and C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems,” Phys. Rev. B44, 10945–10948 (1991).
[CrossRef]

1990 (1)

U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B42, 8947–8951 (1990).
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1986 (1)

J. S. Melinger and A. C. Albrecht, “Theory of time and frequency resolved resonance secondary radiation from a three-level system,” J. Chem. Phys.84, 1247–1258 (1986).
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1979 (1)

G. Nienhuis, “Time and frequency dependence of nearly resonant light scattered from collisionally perturbed atoms,” Physica C96, 391–409 (1979).
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1977 (1)

1953 (1)

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and S. V. Gaponenko, “Enhanced intraband carrier relaxation in quantum dots due to the effect of plasmon–LO-phonon density of states in doped heterostructures,” Phys. Rev. B71, 195310 (2005).

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M. Abbarchi, C. Mastrandrea, T. Kuroda, T. Mano, A. Vinattieri, K. Sakoda, and M. Gurioli, “Poissonian statistics of excitonic complexes in quantum dots,” J. Appl. Phys.106, 053504 (2009).
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J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
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Al Salman, A.

A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett.90, 093104 (2007).
[CrossRef]

Albrecht, A. C.

J. S. Melinger and A. C. Albrecht, “Theory of time and frequency resolved resonance secondary radiation from a three-level system,” J. Chem. Phys.84, 1247–1258 (1986).
[CrossRef]

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J. Gomis-Bresco, G. Muñoz-Matutano, J. Martínez-Pastor, B. Alén, L. Seravalli, P. Frigeri, G. Trevisi, and S. Franchi, “Random population model to explain the recombination dynamics in single InAs/GaAs quantum dots under selective optical pumping,” New J. Phys.13, 023022 (2011).
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A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science271, 933–937 (1996).
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M. I. Vasilevskiy, E. V. Anda, and S. S. Makler, “Electron-phonon interaction effects in semiconductor quantum dots: A nonperturbative approach,” Phys. Rev. B70, 035318 (2004).
[CrossRef]

Anderson, K. E. H.

S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B74, 235328 (2006).
[CrossRef]

Arakawa, Y.

X.-Q. Li, H. Nakayama, and Y. Arakawa, “Phonon bottleneck in quantum dots: Role of lifetime of the confined optical phonons,” Phys. Rev. B59, 5069–5073 (1999).
[CrossRef]

X.-Q. Li and Y. Arakawa, “Anharmonic decay of confined optical phonons in quantum dots,” Phys. Rev. B57, 12285–12290 (1998).
[CrossRef]

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E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
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T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
[CrossRef]

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V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: A key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys.67, 433 (2004).
[CrossRef]

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M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
[CrossRef]

Baranov, A. V.

E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
[CrossRef] [PubMed]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Degenerate pump-probe spectroscopy,” Opt. Spectrosc.109, 358–365 (2010).
[CrossRef]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photoluminescence mediated by incoherent reversible energy transport,” Phys. Rev. B81, 245303 (2010).
[CrossRef]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys.133, 104704 (2010).
[CrossRef] [PubMed]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Resonant energy transfer in quantum dots: Frequency-domain luminescent spectroscopy,” Phys. Rev. B78, 125311 (2008).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Intraband carrier relaxation in quantum dots mediated by surface plasmon-phonon excitations,” Opt. Spectrosc.97, 56–67 (2004).
[CrossRef]

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon-phonon modes,” Semicond.38, 1065–1073 (2004).
[CrossRef]

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and Y. Masumoto, “New many-body mechanism of intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Solid State Commun.128, 219–223 (2003).
[CrossRef]

A. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Phys. Rev. B68, 205318 (2003).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance secondary emission in semiconductor quantum dots,” Opt. Spectrosc.93, 52–60 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of the quasi-elastic resonant secondary emission: Semiconductor quantum dots,” Opt. Spectrosc.92, 732–738 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of thermalized luminescence in semiconductor quantum dots,” Opt. Spectrosc.93, 555–558 (2002).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystal–dielectric matrix systems,” Solid State Commun.122, 139–144 (2002).
[CrossRef]

A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
[CrossRef]

A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, and S. V. Gaponenko, “Enhanced intraband carrier relaxation in quantum dots due to the effect of plasmon–LO-phonon density of states in doped heterostructures,” Phys. Rev. B71, 195310 (2005).

A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Yu. Kruchinin, Optical Properties of Semiconductor Quantum Dots (Nauka, St. Petersburg, 2011).

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U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B42, 8947–8951 (1990).
[CrossRef]

Bawendi, M. G.

S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence spectroscopy of single CdSe nanocrystallite quantum dots,” Phys. Rev. Lett.77, 3873–3876 (1996).
[CrossRef] [PubMed]

Bayer, M.

H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B80, 235319 (2009).
[CrossRef]

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
[CrossRef]

M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B65, 041308 (2002).
[CrossRef]

Baymuratov, A. S.

Benisty, H.

H. Benisty, C. M. Sotomayor-Torrès, and C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems,” Phys. Rev. B44, 10945–10948 (1991).
[CrossRef]

Berstermann, T.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B76, 165318 (2007).
[CrossRef]

Berwick, K.

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys.133, 104704 (2010).
[CrossRef] [PubMed]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photoluminescence mediated by incoherent reversible energy transport,” Phys. Rev. B81, 245303 (2010).
[CrossRef]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Resonant energy transfer in quantum dots: Frequency-domain luminescent spectroscopy,” Phys. Rev. B78, 125311 (2008).
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G. A. Narvaez, G. Bester, and A. Zunger, “Carrier relaxation mechanisms in self-assembled (In,Ga)As/GaAs quantum dots: Efficient P → S Auger relaxation of electrons,” Phys. Rev. B74, 075403 (2006).
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T. B. Norris, K. Kim, J. Urayama, Z. K. Wu, J. Singh, and P. K. Bhattacharya, “Density and temperature dependence of carrier dynamics in self-organized InGaAs quantum dots,” J. Phys. D: Appl. Phys.38, 2077 (2005).
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P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Relaxation and dephasing of multiexcitons in semiconductor quantum dots,” Phys. Rev. Lett.89, 187401 (2002).
[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).
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M. Grundmann and D. Bimberg, “Theory of random population for quantum dots,” Phys. Rev. B55, 9740–9745 (1997).
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K. Blum, Density Matrix Theory and Applications (Springer, Berlin, 2012), 3rd ed.
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U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B42, 8947–8951 (1990).
[CrossRef]

Bonati, C.

C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
[CrossRef]

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E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
[CrossRef] [PubMed]

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P. Borri, W. Langbein, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Exciton dephasing via phonon interactions in InAs quantum dots: Dependence on quantum confinement,” Phys. Rev. B71, 115328 (2005).
[CrossRef]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Relaxation and dephasing of multiexcitons in semiconductor quantum dots,” Phys. Rev. Lett.89, 187401 (2002).
[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]

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K. Rivoire, S. Buckley, Y. Song, M. L. Lee, and J. Vučković, “Photoluminescence from In0.5Ga0.5As/GaP quantum dots coupled to photonic crystal cavities,” Phys. Rev. B85, 045319 (2012).
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C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
[CrossRef]

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M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
[CrossRef]

Chergui, M.

C. Bonati, A. Cannizzo, D. Tonti, A. Tortschanoff, F. van Mourik, and M. Chergui, “Subpicosecond near-infrared fluorescence upconversion study of relaxation processes in PbSe quantum dots,” Phys. Rev. B76, 033304 (2007).
[CrossRef]

A. Al Salman, A. Tortschanoff, M. B. Mohamed, D. Tonti, F. van Mourik, and M. Chergui, “Temperature effects on the spectral properties of colloidal CdSe nanodots, nanorods, and tetrapods,” Appl. Phys. Lett.90, 093104 (2007).
[CrossRef]

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S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B74, 235328 (2006).
[CrossRef]

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M.-R. Dachner, E. Malic, M. Richter, A. Carmele, J. Kabuss, A. Wilms, J.-E. Kim, G. Hartmann, J. Wolters, U. Bandelow, and A. Knorr, “Theory of carrier and photon dynamics in quantum dot light emitters,” Phys. Stat. Solidi (b)247, 809–828 (2010).
[CrossRef]

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H.-Y. Liu, Z.-M. Meng, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and T. Yang, “Ultrafast carrier dynamics in undoped and p-doped InAs/GaAs quantum dots characterized by pump-probe reflection measurements,” J. Appl. Phys.103, 083121 (2008).
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J. H. Davies, The Physics of Low-Dimensional Semiconductors: An Introduction (Cambridge University Press, Cambridge, 1998), 1st ed.

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A. V. Baranov, V. Davydov, A. V. Fedorov, H.-W. Ren, S. Sugou, and Y. Masumoto, “Coherent control of stress-induced InGaAs quantum dots by means of phonon-assisted resonant photoluminescence,” Physica Status Solidi (b)224, 461–464 (2001).
[CrossRef]

de Mello Donegá, C.

E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donegá, D. Vanmaekelbergh, and M. Bonn, “Direct observation of electron-to-hole energy transfer in CdSe quantum dots,” Phys. Rev. Lett.96, 057408 (2006).
[CrossRef] [PubMed]

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W. Demtröder, Laser Spectroscopy: Basic Concepts and Instrumentation (Springer, Berlin, 2002), 3rd ed.

Dias, E. A.

S. L. Sewall, R. R. Cooney, K. E. H. Anderson, E. A. Dias, and P. Kambhampati, “State-to-state exciton dynamics in semiconductor quantum dots,” Phys. Rev. B74, 235328 (2006).
[CrossRef]

Dou, X. M.

X. M. Dou, B. Q. Sun, D. S. Jiang, H. Q. Ni, and Z. C. Niu, “Electron spin relaxation in a single InAs quantum dot measured by tunable nuclear spins,” Phys. Rev. B84, 033302 (2011).
[CrossRef]

Eberly, J. H.

Ema, K.

J. Ishi-Hayase, K. Akahane, N. Yamamoto, M. Sasaki, M. Kujiraoka, and K. Ema, “Long dephasing time in self-assembled InAs quantum dots at over 1.3 μm wavelength,” Appl. Phys. Lett.88, 261907 (2006).
[CrossRef]

Empedocles, S. A.

S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence spectroscopy of single CdSe nanocrystallite quantum dots,” Phys. Rev. Lett.77, 3873–3876 (1996).
[CrossRef] [PubMed]

Fedorov, A. V.

E. V. Ushakova, A. P. Litvin, P. S. Parfenov, A. V. Fedorov, M. Artemyev, A. V. Prudnikau, I. D. Rukhlenko, and A. V. Baranov, “Anomalous size-dependent decay of low-energy luminescence from PbS quantum dots in colloidal solution,” ACS Nano6, 8913–8921 (2012).
[CrossRef] [PubMed]

I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express19, 15459–15482 (2011).
[CrossRef] [PubMed]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient intraband light absorption by quantum dots: Pump-probe spectroscopy,” Opt. Spectrosc.111, 798–807 (2011).
[CrossRef]

M. Yu. Leonov, A. V. Baranov, and A. V. Fedorov, “Transient interband light absorption by quantum dots: Non-degenerate case of pump-probe spectroscopy,” Opt. Spectrosc.110, 24–32 (2011).
[CrossRef]

V. K. Turkov, S. Yu. Kruchinin, and A. V. Fedorov, “Intraband optical transitions in semiconductor quantum dots: Radiative electronic-excitation lifetime,” Opt. Spectrosc.110, 740–747 (2011).
[CrossRef]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photoluminescence mediated by incoherent reversible energy transport,” Phys. Rev. B81, 245303 (2010).
[CrossRef]

S. Yu. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Electron-electron scattering in a double quantum dot: Effective mass approach,” J. Chem. Phys.133, 104704 (2010).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Optical (solid arrows) and relaxation (dashed arrows) transitions corresponding to the processes of (a) resonant luminescence and (b) thermalized luminescence from a semiconductor quantum dot with three energy levels. Ket vectors |0〉, |1〉, |2〉, and |3〉 denote the ground and excited states of electron–hole pairs; |n〉 is the high-energy state that does not directly contribute to the secondary emission; ωLk (ωL) and ωλk (k = 1, 2, 3) are the excitation frequencies and the frequencies of emitted photons; ζkk is the rate of transitions |k′〉 → |k〉 due to the thermal interaction with a bath. The spectral width of the excitation pulse is assumed to be much smaller than the dephasing rates of all interband optical transitions.

Fig. 2
Fig. 2

Three-dimensional plot of log[(2σF)Q(α, t)] as a function of the dimensionless parameters αt and α/σ [see Eq. (17)]. Function values are shown by labels of contour levels. Three regimes of excitation by a Gaussian pulse are clearly seen: (i) adiabatic excitation for log(α/σ) ≳ 1; (ii) instantaneous excitation for log(α/σ) ≲ −1; and (iii) pulse-sensitive excitation for |log(α/σ)| ≲ 0.5.

Fig. 3
Fig. 3

Two-dimensional spectra of (a) resonant luminescence and (b) thermalized luminescence for two relevant eigenstates of the quantum dot’s electronic subsystem. The widths of the peaks in the spectra are determined by the dephasing rates of optical transitions, while the relative peak intensities depend on the interband matrix elements [see Eqs. (5) and (7)], relaxation constants, and temperature of the system. Peak 4 vanishes in the limit of small temperatures.

Fig. 4
Fig. 4

Variation of (a) s1 and (b) log s2 with radius R of PbS quantum dot and ratio y = ζ02/ζ01 of radiative recombination rates for two relevant eigenstates of electron–hole pairs (s1 and s2 are in meV). The quantum numbers of the eigenstates are n1 = 1, l1 = 0 and n2 = 2, l2 = 0. Bound by red curves are the domains of strong thermal coupling between the eigenstates, where 2ζ12 exp[−E21/(2T)]/(γ22γ11) > 0.1. Quantum dot’s boundary is assumed to be impenetrable for both electrons and holes; ζ01 = 200 μeV, ζ12 = 150 μeV, and T = 25 meV. For other parameters, refer to the text.

Fig. 5
Fig. 5

Temporal evolution of resonant luminescence from PbS quantum dots of different sizes excited by a Gaussian pulse. Excitation and detection frequencies correspond to the peaks in Fig. 3(a): (a) ωL = ωF = ω1; (b) ωL = ωF = ω2; (c) ωL = ω2, ωF = ω1; (d) ωL = ω1, ωF = ω2. Blue and red curves indicate decay/buildup of luminescence with characteristic times 1/s1 and 1/s2, respectively. The relevant eigenstates are weakly coupled via thermal interaction with the bath for R ≲ 8 nm. Simulation parameters are γ̂1i = γ̂2i = γ̂f 1 = γ̂f 2 = 20 meV, ΓF = 10 meV, σ = 1 meV, ζ12 = 100 μeV, and ζ01 = ζ02 = 10 μeV. Parameters of the quantum dot are the same as in 4. For other parameters, refer to the text.

Fig. 6
Fig. 6

Temporal evolution of thermalized luminescence from PbS quantum dots of different sizes excited by a Gaussian pulse. Excitation and detection frequencies correspond to the peaks in Fig. 3(b): (a) ωL = ωn, ωF = ω1; (b) ωL = ωn, ωF = ω2. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. It was assumed that γ̂ni = 20 meV, γnn = 2 meV, and ζ1n = ζ2n = 500 μeV. The other parameters are the same as in Fig. 5.

Fig. 7
Fig. 7

The same as in Fig. 6 but for (a) ζ1n = 50 μeV, ζ2n = 500 μeV and (b) ζ1n = 500 μeV, ζ2n = 50 μeV. The relevant eigenstates are weakly coupled via thermal interaction for R ≲ 8 nm. The buildup and decay of luminescence from a 5-nm quantum dot in (a) allow one to determine relaxation constants γ22 and γ11, respectively.

Tables (1)

Tables Icon

Table 1 Time dependence of photoluminescence signal and experimentally measurable relaxation constants in different excitation regimes. It is assumed that experiments are conducted with either a single quantum dot or a quantum dot ensemble with a narrow size distribution, and that the quantum dot has two eigenstates decaying with the emission of secondary photons. The eigenstates are assumed to be excited independently of one another and the photoluminescence from each of them is meant to be measured separately.

Equations (80)

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H 0 = n h ¯ ω n | n n | + λ h ¯ ω λ c λ + c λ
H L ( t ) = E L n ϕ ( t ) V n , 0 ( L ) e i ω L t | n 0 | + H . c . ,
H V = λ n i g λ V 0 , n ( λ ) c λ + | 0 n | + H . c . ,
ρ n n t = γ n n ρ n n + n n ζ n n ρ n n + f n ( t ) ,
W n = ( E L h ¯ ) 2 | V n , 0 ( L ) | 2 2 γ ^ n i ( ω n ω L ) 2 + γ n i 2 ,
Σ ( ω L , ω λ , t ) = h ¯ 2 ω λ 4 π c 4 E L 2 n = 1 N ρ n n ( ω L , t ) W ˜ n ( ω λ ) ,
W ˜ n ( ω λ ) = | V 0 , n ( λ ) | 2 h ¯ 2 2 γ f n ( ω n ω λ ) 2 + γ f n 2
Σ ¯ ( ω L , ω F , t ) = 0 d τ Γ F e Γ F τ d ω λ Γ F / ( 2 π ) ( ω F ω λ ) 2 + ( Γ F / 2 ) 2 Σ ( ω L , ω λ , t τ ) = 1 π ( ω F c ) 4 n = 1 N | V 0 , n ( λ ) | 2 2 ( γ f n + Γ F / 2 ) ( ω n ω F ) 2 + ( γ f n + Γ F / 2 ) 2 R n ( t ) E L 2 ,
R n ( t ) = 0 ρ n n ( t τ ) Γ F e Γ F τ d τ
ρ 11 t = γ 11 ρ 11 + ζ 12 ρ 22 + W 1 ϕ 2 ,
ρ 22 t = γ 22 ρ 22 + ζ 21 ρ 11 + W 2 ϕ 2 .
ρ 11 ( t ) = t ϕ 2 ( t ) [ ( ρ W 1 + ϑ 12 W 2 ) e s 1 ( t t ) + ( q W 1 ϑ 12 W 2 ) e s 2 ( t t ) ] d t ,
ρ 22 ( t ) = t ϕ 2 ( t ) [ ( q W 2 + ϑ 21 W 1 ) e s 1 ( t t ) + ( p W 2 ϑ 21 W 1 ) e s 2 ( t t ) ] d t ,
p = γ 22 s 1 D , q = s 2 γ 22 D , ϑ i j = ζ i j D ,
D = ( γ 11 γ 22 ) 2 + 4 ζ 12 ζ 21 ,
s 1 = 1 2 ( γ 11 + γ 22 D ) , s 2 = 1 2 ( γ 11 + γ 22 + D ) .
R 1 ( t ) = η 12 + e 2 σ t H ( t ) + ( η 12 e 2 σ t + A 12 ( p ) e s 1 t + B 12 ( q ) e s 2 t ) H ( t ) ,
R 2 ( t ) = η 21 + e 2 σ t H ( t ) + ( η 21 e 2 σ t + A 21 ( q ) e s 1 t + B 21 ( p ) e s 2 t ) H ( t ) ,
η i j ± = ( γ i j ± 2 σ ) W i + ζ i j W j ( s 1 ± 2 σ ) ( s 2 ± 2 σ ) Γ F Γ F ± 2 σ ,
A i j ( r ) = 4 σ 4 σ 2 s 1 2 Γ F Γ F s 1 ( r W i + ϑ i j W j ) ,
B i j ( r ) = 4 σ 4 σ 2 s 2 2 Γ F Γ F s 2 ( r W i ϑ i j W j ) .
R 1 ( t ) = ξ 12 Q ( Γ F , t ) + C 12 ( p ) Q ( s 1 , t ) + D 12 ( q ) Q ( s 2 , t ) ,
R 2 ( t ) = ξ 21 Q ( Γ F , t ) + C 21 ( q ) Q ( s 1 , t ) + D 21 ( p ) Q ( s 2 , t ) ,
ξ i j = ( γ j j Γ F ) W i + ζ i j W j ( Γ F s 1 ) ( Γ F s 2 ) ,
C i j ( r ) = r W i + ϑ i j W j Γ F s 1 , D i j ( r ) = r W i ϑ i j W j Γ F s 2 ,
Q ( α , t ) = Γ F 2 σ exp ( α 2 8 ς 2 ) erfc ( α 4 ς 2 t 2 2 ς ) e α t .
ρ 11 t = γ 11 ρ 11 + ζ 12 ρ 22 + ζ 1 n ρ n n ,
ρ 22 t = γ 22 ρ 22 + ζ 21 ρ 11 + ζ 2 n ρ n n ,
ρ n n t = γ n n ρ n n + W n ϕ 2 .
ρ 11 ( t ) = t g ( t ) [ ( p w 1 + ϑ 12 w 2 ) e s 1 ( t t ) + ( q w 1 ϑ 12 w 2 ) e s 2 ( t t ) ] d t ,
ρ 22 ( t ) = t g ( t ) [ ( q w 2 + ϑ 21 w 1 ) e s 1 ( t t ) + ( p w 2 ϑ 21 w 1 ) e s 2 ( t t ) ] d t ,
g ( t ) = γ n n t ϕ 2 ( t ) e γ n n ( t t ) d t .
R 1 ( t ) = χ 12 + e 2 σ t H ( t ) + ( χ 12 e 2 σ t + E 12 ( p ) e s 1 t + F 12 ( q ) e s 2 t + G 12 e γ n n t ) H ( t ) ,
R 2 ( t ) = χ 21 + e 2 σ t H ( t ) + ( χ 21 e 2 σ t + E 21 ( q ) e s 1 t + F 21 ( p ) e s 2 t + G 21 e γ n n t ) H ( t ) ,
χ i j ± = γ n n γ n n ± 2 σ ( γ i j ± 2 σ ) w i + ζ i j w j ( s 1 ± 2 σ ) ( s 2 ± 2 σ ) Γ F Γ F ± 2 σ ,
E i j ( r ) = γ n n γ n n s 1 4 σ 4 σ 2 s 1 2 Γ F Γ F s 1 ( r w i + ϑ i j w j ) ,
F i j ( r ) = γ n n γ n n s 2 4 σ 4 σ 2 s 2 2 Γ F Γ F s 2 ( r w i ϑ i j w j ) ,
G i j = 4 σ γ n n γ n n 2 4 σ 2 ( γ n n γ j j ) w i ζ i j w j ( γ n n s 1 ) ( γ n n s 2 ) Γ F Γ F γ n n .
R 1 ( t ) = η 12 Q ( Γ F , t ) + K 12 ( p ) Q ( s 1 , t ) + L 12 ( q ) Q ( s 2 , t ) + M 12 Q ( γ n n , t ) ,
R 2 ( t ) = η 21 Q ( Γ F , t ) + K 21 ( q ) Q ( s 1 , t ) + L 21 ( p ) Q ( s 2 , t ) + M 21 Q ( γ n n , t ) ,
η i j = γ n n γ n n Γ F ( γ i j Γ F ) w i + ζ i j w j ( Γ F s 1 ) ( Γ F s 2 ) ,
K i j ( r ) = γ n n γ n n s 1 r w i + ϑ i j w j Γ F s 1 , L i j ( r ) = γ n n γ n n s 2 r w i ϑ i j w j Γ F s 2 ,
M i j = γ n n Γ F γ n n ( γ j j γ n n ) w i + ζ i j w j ( γ n n s 1 ) ( γ n n s 2 ) .
W ¯ n ( ω F ) = | V 0 , n ( λ ) | 2 h ¯ 2 2 ( γ f n + Γ F / 2 ) ( ω n ω F ) 2 + ( γ f n + Γ F / 2 ) 2 ,
Σ ¯ RL ( 0 ) ( ω L , ω F ) = Ξ ( W ¯ 1 γ 22 W 1 + ζ 12 W 2 γ 11 γ 22 ζ 12 ζ 21 + W ¯ 2 γ 11 W 2 + ζ 21 W 1 γ 11 γ 22 ζ 12 ζ 21 )
Σ ¯ TL ( 0 ) ( ω L , ω F ) = Ξ ( W ¯ 1 γ 22 ζ 1 n + ζ 12 ζ 2 n γ 11 γ 22 ζ 12 ζ 21 + W ¯ 2 γ 11 ζ 2 n + ζ 21 ζ 1 n γ 11 γ 22 ζ 12 ζ 21 ) W n γ n n
Σ ¯ RL ( 0 ) = Ξ ( W ¯ 1 W 1 γ 11 + W ¯ 2 W 2 γ 22 + W ¯ 1 ζ 12 W 2 γ 11 γ 22 )
Σ ¯ TL ( 0 ) = Ξ ( W ¯ 1 ζ 1 n γ 11 + W ¯ 2 ζ 2 n γ 22 + W ¯ 1 ζ 12 ζ 2 n γ 11 γ 22 ) W n γ n n .
ψ = ζ 12 ζ 21 γ 11 γ 22 = Σ ¯ RL ( 0 ) ( ω 1 , ω 2 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 1 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 2 ) .
χ = ζ 1 n ζ 2 n γ 22 ζ 12 = φ 1 1 φ ψ ,
φ = Σ ¯ TL ( 0 ) ( ω n , ω 1 ) Σ ¯ TL ( 0 ) ( ω n , ω 2 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 2 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 1 ) .
γ 11 ( x ) = ζ 01 + ζ 12 e x , γ 22 = ζ 02 + ζ 12 .
a = Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 2 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 1 ) = γ 11 ( x 1 ) γ 22 ζ 12 2 e x 1 γ 11 ( x 2 ) γ 22 ζ 12 2 e x 2 ,
b = Σ ¯ RL ( 0 ) ( ω 2 , ω 2 , x 1 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 1 ) Σ ¯ RL ( 0 ) ( ω 1 , ω 1 , x 2 ) Σ ¯ RL ( 0 ) ( ω 2 , ω 2 , x 2 ) = γ 11 ( x 1 ) γ 11 ( x 2 ) .
γ 11 ( x ) = ( e x + e x 1 b e x 2 b 1 ) ζ 12 ,
γ 22 = b 1 b a e x 1 a e x 2 e x 1 e x 2 ζ 12 .
Σ ¯ RL ( i ) Σ ¯ RL ( 0 ) e 2 σ | t | , Σ ¯ TL ( i ) Σ ¯ TL ( 0 ) e 2 σ | t | ,
erfc ( α 4 ς 2 t 2 2 ς ) = exp [ ( α 4 ς 2 t ) 2 8 ς 2 ] ( 2 ς α 2 π + O [ ( ς / α ) 2 ] ) ,
Σ ¯ RL ( i ) Σ ¯ RL ( 0 ) A e 2 ς 2 t 2 , Σ ¯ TL ( i ) Σ ¯ TL ( 0 ) A e 2 ς 2 t 2 .
q p 1 , s 1 γ 11 , s 2 γ 22 ,
E 21 = 3 π 2 h ¯ 2 2 μ R 2 ,
Σ ¯ RL ( ii ) Ξ σ { [ W ¯ 1 ( p W 1 + ϑ 12 W 2 ) + W ¯ 2 ( q W 2 + ϑ 21 W 1 ) ] e s 1 t + [ W ¯ 1 ( q W 1 ϑ 12 W 2 ) + W ¯ 2 ( p W 2 ϑ 21 W 1 ) ] e s 2 t } .
Σ ¯ RL ( ii ) { p e s 1 t + q e s 2 t for ω L , ω F ω 1 , q e s 1 t + p e s 2 t for ω L , ω F ω 2 , e s 1 t e s 2 t for ω L ω 1 ( 2 ) , ω F ω 2 ( 1 ) ,
s 1 1 ψ 1 / γ 11 + 1 / γ 22 s 2 γ 11 + γ 22 .
γ 11 = 1 2 ( s 1 + s 2 ( s 1 s 2 ) 2 4 ζ 12 ζ 21 ) ,
γ 22 = 1 2 ( s 1 + s 2 + ( s 1 s 2 ) 2 4 ζ 12 ζ 21 ) .
γ 11 = 1 2 ( s 1 + s 2 ( s 1 + s 2 ) 2 4 s 1 s 2 / ( 1 ψ ) ) ,
γ 22 = 1 2 ( s 1 + s 2 + ( s 1 + s 2 ) 2 4 s 1 s 2 / ( 1 ψ ) ) .
Σ ¯ TL ( ii ) Ξ σ { W ¯ 1 ( p ζ 1 n + ϑ 12 ζ 2 n ) + W ¯ 2 ( q ζ 2 n + ϑ 21 ζ 1 n ) γ n n s 1 e s 1 t + W ¯ 1 ( q ζ 1 n ϑ 12 ζ 2 n ) + W ¯ 2 ( p ζ 2 n ϑ 21 ζ 1 n ) γ n n s 2 e s 2 t + W ¯ 1 [ ( γ 22 γ n n ) ζ 1 n + ζ 12 ζ 2 n ] + W ¯ 2 [ ( γ 11 γ n n ) ζ 2 n + ζ 21 ζ 1 n ] ( γ n n s 1 ) ( γ n n s 2 ) e γ n n t } W n .
Σ ¯ TL ( ii ) { ( p μ 1 + ϑ 12 μ 2 ) e s 1 t + ( q μ 1 ϑ 12 μ 2 ) e s 2 t for ω F ω 1 , ( q μ 2 + ϑ 21 μ 1 ) e s 1 t + ( p μ 2 ϑ 21 μ 1 ) e s 2 t for ω F ω 2 ,
t m = 1 s 2 s 1 × { ln ( s 2 s 1 ϑ 12 ζ 2 n q ζ 1 n ϑ 12 ζ 2 n + p ζ 1 n ) for ω F ω 1 , ln ( s 2 s 1 ϑ 21 ζ 1 n q ζ 2 n ϑ 21 ζ 1 n + p ζ 2 n ) for ω F ω 2 .
Σ ¯ RL ( iii ) Σ ¯ RL ( 0 ) e Γ F t , Σ ¯ TL ( iii ) Σ ¯ TL ( 0 ) e Γ F t .
ρ n n t + n = 1 N a n n ρ n n = f n ( t ) ,
n = 1 N b n ρ n n t + n = 1 N ρ n n n = 1 N b n a n n = n = 1 N b n f n ( t ) .
n = 1 N b n a n n = s n b n .
a ^ n n = ( γ 11 ζ 12 e E 21 / T ζ 13 e E 31 / T ζ 1 N e E N 1 / T ζ 12 γ 22 ζ 23 e E 32 / T ζ 2 N e E N 2 / T ζ 13 ζ 23 γ 33 ζ 3 N e E N 3 / T ζ 1 N ζ 2 N ζ 3 N γ N N )
| a ^ n n | > n n | a ^ n n | .
x n = n = 1 N b n ( n ) ρ n n ,
x n t + s n x n = n = 1 N b n ( n ) f n ( t ) .
x n ( t ) = n = 1 N b n ( n ) t f n ( τ ) e s n ( t τ ) d τ .

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