Abstract

We present a theory of phonon-assisted photoluminescence from a semiconductor quantum dot (QD) whose electron and phonon subsystems are resonantly coupled via the polar electron–phonon interaction. We show that the resonance-induced renormalization of the QD energy spectrum, leading to the formation of the polaron-like states, can be performed exactly in terms of the arbitrarily degenerate states of electron–hole pairs and the phonon modes of equal energies. Using the model of QDs with finite potential barriers for electron and holes leads to new selection rules of interband optical transitions and the three-particle interaction describing simultaneous absorption and/or emission of a photon and a phonon. We also derive a simple expression for the differential cross section of the stationary, low-temperature photoluminescence, which allows the fundamental parameters of the polaron-like excitations to be readily extracted from the frequency-resolved experimental spectra. In particular, the energies of the excitations and the coherence relaxation rates of the optical transitions resulting in their generation and recombination are shown to be directly given by the positions and widths of the photoluminescence peaks. The developed theory complements the existing experimental techniques of studying the phonon-assisted photoluminescence from individual nanocrystals.

© 2014 Optical Society of America

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2013 (4)

S. Wageh, A. M. El-Nahas, A. A. Higazy, and M. A. M. Mahmoud, “Preparation and characterization of a novel system of CdS nanoparticles embedded in borophosphate glass matrix,” J. Alloys Comp. 555, 161–168 (2013).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, and A. V. Fedorov, “Engineering band structure in nanoscale quantum-dot supercrystals,” Opt. Lett. 38, 2259–2261 (2013).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, M. V. Artemyev, A. O. Orlova, and A. V. Fedorov, “Anisotropy of optical transitions in ordered ensemble of CdSe quantum rods,” Opt. Lett. 38, 3426–3428 (2013).
[Crossref] [PubMed]

2012 (5)

A. S. Baimuratov, V. K. Turkov, I. D. Rukhlenko, and A. V. Fedorov, “Shape-induced anisotropy of intraband luminescence from a semiconductor nanocrystal,” Opt. Lett. 37, 4645–4647 (2012).
[Crossref] [PubMed]

G. H. Shih, C. G. Allen, and B. G. Potter, “Interfacial effects on the optical behavior of Ge:ITO and Ge:NO nanocomposite films,” Nanotechnol. 23, 075203 (2012).
[Crossref]

N. Tschirner, H. Lange, A. Schliwa, A. Biermann, C. Thomsen, K. Lambert, R. Gomes, and Z. Hens, “Interfacial alloying in CdSe/CdS heteronanocrystals: A Raman spectroscopy analysis,” Chem. Mater. 24, 311–318 (2012).
[Crossref]

T. S. Kim, Y.-H. Kil, H. D. Yang, J.-H. Yang, W.-K. Hong, S. Kang, T. S. Jeong, and K.-H. Shim, “Growth and characterization of Si1−xGex QDs on Si/Si0.8Ge0.2 layer,” Electron. Mater. Lett. 8, 559–563 (2012).
[Crossref]

K. Rezgui, S. Aloulou, J. Rihani, and M. Oueslati, “Competition between strain and confinement effects on the crystalline quality of InAs/GaAs (001) quantum dots probed by Raman spectroscopy,” J. Raman Spectrosc. 43, 1964–1968 (2012).
[Crossref]

2011 (3)

E. S. F. Neto, S. W. da Silva, P. C. Morais, M. I. Vasilevskiy, M. A. P. da Silva, and N. O. Dantas, “Resonant Raman scattering in CdSx Se1−x nanocrystals: Effects of phonon confinement, composition, and elastic strain,” J. Raman Spectrosc. 42, 1660–1669 (2011).
[Crossref]

K. Raulin, S. Turrell, B. Capoen, C. Kinowski, V. T. T. Tran, M. Bouazaoui, and O. Cristini, “Raman characterization of localized CdS nanostructures synthesized by UV irradiation in sol-gel silica matrices,” J. Ram. Spectrosc. 42, 1366–1372 (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. Express 19, 15461–15482 (2011).
[Crossref]

2010 (6)

V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and p-doped InAs/GaAs quantum dots-in-a-well,” Phys. Rev. B 82, 195314 (2010).
[Crossref]

S. Dhara, A. K. Arora, S. Bera, and J. Ghatak, “Multiphonon probe in ZnS quantum dots,” J. Raman Spectrosc. 41, 1102–1105 (2010).
[Crossref]

S. K. Gupta, R. Desai, P. K. Jha, S. Sahoo, and D. Kirin, “Titanium dioxide synthesized using titanium chloride: Size effect study using Raman spectroscopy and photoluminescence,” J. Ram. Spectrosc. 41, 350–355 (2010).

G. Yu, Q. Liang, Y. Jia, and J. Dong, “Phonon sidebands of photoluminescence in single wall carbon nanotubes,” J. Appl. Phys. 107, 024314 (2010).
[Crossref]

S. Sohal, Y. Alivov, Z. Fan, and M. Holtz, “Role of phonons in the optical properties of magnetron sputtered ZnO studied by resonance Raman and photoluminescence,” J. Appl. Phys. 108, 053507 (2010).
[Crossref]

S. Y. 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. B 81, 245303 (2010).
[Crossref]

2009 (1)

C. H. Ahn, S. K. Mohanta, N. E. Lee, and H. K. Cho, “Enhanced exciton–phonon interactions in photoluminescence of ZnO nanopencils,” Appl. Phys. Lett. 94, 261904 (2009).
[Crossref]

2008 (1)

S. K. Tripathy, G. Xu, X. Mu, Y. J. Ding, M. Jamil, R. A. Arif, N. Tansu, and J. B. Khurgin, “Phonon–assisted ultraviolet anti-stokes photoluminescence from GaN film grown on Si (111) substrate,” Appl. Phys. Lett. 93, 201107 (2008).
[Crossref]

2007 (3)

A. Cros, N. Garro, A. Cantarero, J. Coraux, H. Renevier, and B. Daudin, “Raman scattering as a tool for the evaluation of strain in GaN/AlN quantum dots: The effect of capping,” Phys. Rev. B 76, 165403 (2007).
[Crossref]

E. A. Muljarov and R. Zimmermann, “Exciton dephasing in quantum dots due to LO-phonon coupling: An exactly solvable model,” Phys. Rev. Lett. 98, 187401 (2007).
[Crossref] [PubMed]

R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B 75, 245311 (2007).
[Crossref]

2006 (13)

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]

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton–phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
[Crossref]

S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron–phonon interaction in individual strain-free GaAs/Al0.3Ga0.7 As quantum dots,” Phys. Rev. B 73, 125342 (2006).
[Crossref]

K. Kojima and A. Tomita, “Influence of pure dephasing by phonons on exciton–photon interfaces: Quantum microscopic theory,” Phys. Rev. B 73, 195312 (2006).
[Crossref]

A. Cros, N. Garro, J. M. Llorens, A. García-Cristóbal, A. Cantarero, N. Gogneau, E. Monroy, and B. Daudin, “Raman study and theoretical calculations of strain in GaN quantum dot multilayers,” Phys. Rev. B 73, 115313 (2006).
[Crossref]

A. V. Baranov, A. V. Fedorov, T. S. Perova, R. A. Moore, V. Yam, D. Bouchier, V. Thanh, and K. Berwick, “Analysis of strain and intermixing in single-layer Ge/Si quantum dots using polarized Raman spectroscopy,” Phys. Rev. B 73, 075322 (2006).
[Crossref]

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

I. D. Rukhlenko and A. V. Fedorov, “Resonant photoluminescence of quantum dots: Dynamics of electronic subsystem,” Bull. Russ. Acad. Sci. 70, 126 (2006).

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]

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

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

S. Y. Kruchinin and A. V. Fedorov, “Renormalization of the energy spectrum of quantum dots under vibrational resonance conditions: Persistent hole burning spectroscopy,” Opt. Spectrosc. 100, 41–48 (2006).
[Crossref]

R. P. Miranda, M. I. Vasilevskiy, and C. Trallero-Giner, “Nonperturbative approach to the calculation of multi-phonon Raman scattering in semiconductor quantum dots: Polaron effect,” Phys. Rev. B 74, 115317 (2006).
[Crossref]

2005 (2)

V. Preisler, R. Ferreira, S. Hameau, L. A. de Vaulchier, Y. Guldner, M. L. Sadowski, and A. Lemaitre, “Hole–LO-phonon interaction in InAs/GaAs quantum dots,” Phys. Rev. B 72, 115309 (2005).
[Crossref]

D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B 71, 235409 (2005).
[Crossref]

2004 (5)

A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B 70, 201305 (2004).
[Crossref]

V. V. Ursaki, I. M. Tiginyanu, V. V. Zalamai, V. M. Masalov, E. N. Samarov, G. A. Emelchenko, and F. Briones, “Photoluminescence and resonant Raman scattering from ZnO-opal structures,” J. Appl. Phys. 96, 1001–1006 (2004).
[Crossref]

R. W. Meulenberg, T. Jennings, and G. F. Strouse, “Compressive and tensile stress in colloidal CdSe semiconductor quantum dots,” Phys. Rev. B 70, 235311 (2004).
[Crossref]

A. V. Baranov, T. S. Perova, R. A. Moore, S. Solosin, A. V. Fedorov, V. Yam, V. Thanh, and D. Bouchier, “Polarized Raman spectroscopy of multilayer Ge/Si(001) quantum dot heterostructures,” J. Appl. Phys. 95, 2857–2863 (2004).
[Crossref]

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon–phonon modes,” Semiconductors 38, 1101–1109 (2004).
[Crossref]

2003 (2)

J. Zhao, A. Kanno, M. Ikezawa, and Y. Masumoto, “Longitudinal optical phonons in the excited state of CuBr quantum dots,” Phys. Rev. B 68, 113305 (2003).
[Crossref]

A. V. Baranov, Y. P. Rakovich, J. F. Donegan, T. S. Perova, R. A. Moore, D. V. Talapin, A. L. Rogach, Y. Masumoto, and I. Nabiev, “Effect of ZnS shell thickness on the phonon spectra in CdSe quantum dots,” Phys. Rev. B 68, 165306 (2003).
[Crossref]

2002 (3)

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gerard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B 65, 085316 (2002).
[Crossref]

E. P. Pokatilov, S. N. Klimin, V. M. Fomin, J. T. Devreese, and F. W. Wise, “Multiphonon Raman scattering in semiconductor nanocrystals: Importance of nonadiabatic transitions,” Phys. Rev. B 65, 075316 (2002).
[Crossref]

O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett. 88, 146803 (2002).
[Crossref] [PubMed]

2001 (1)

A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors 35, 1390–1397 (2001).
[Crossref]

2000 (1)

T. Stauber, R. Zimmermann, and H. Castella, “Electron–phonon interaction in quantum dots: A solvable model,” Phys. Rev. B 62, 7336 (2000).
[Crossref]

1999 (1)

S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaitre, and J. M. Gerard, “Strong electron–phonon coupling regime in quantum dots: Evidence for everlasting resonant polarons,” Phys. Rev. Lett. 83, 4152 (1999).
[Crossref]

1998 (1)

V. M. Fomin, V. N. Gladilin, J. T. Devreese, E. P. Pokatilov, S. N. Balaban, and S. N. Klimin, “Photoluminescence of spherical quantum dots,” Phys. Rev. B 57, 2415–2425 (1998).
[Crossref]

1997 (2)

T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quantum dots,” Phys. Rev. B 56, R4355 (1997).
[Crossref]

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton–phonon coupling in semiconductor quantum dots: Resonant Raman scattering,” Phys. Rev. B 56, 7491 (1997).
[Crossref]

1996 (1)

A. V. Fedorov and A. V. Baranov, “Exciton–vibrational interaction of the frohlich type in quasi-zero-size systems,” J. Exp. Theor. Phys. 83, 610–618 (1996).

1995 (2)

T. Itoh, M. Nishijima, A. I. Ekimov, C. Gourdon, A. L. Efros, and M. Rosen, “Polaron and exciton–phonon complexes in CuCl nanocrystals,” Phys. Rev. Lett. 74, 1645 (1995).
[Crossref] [PubMed]

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[Crossref]

1989 (1)

N. Mori and T. Ando, “Electron–optical-phonon interaction in single and double heterostructures,” Phys. Rev. B 40, 6175–6188 (1989).
[Crossref]

1974 (1)

I. B. Levinson and E. I. Rashba, “Threshold phenomena and bound states in the polaron problem,” Physics-Uspekhi 16, 892–912 (1974).
[Crossref]

1973 (1)

E. Evans and D. L. Mills, “Theory of inelastic scattering of slow electrons by long-wavelength surface optical phonons,” Phys. Rev. B 8, 4126–4139 (1973).

Ahn, C. H.

C. H. Ahn, S. K. Mohanta, N. E. Lee, and H. K. Cho, “Enhanced exciton–phonon interactions in photoluminescence of ZnO nanopencils,” Appl. Phys. Lett. 94, 261904 (2009).
[Crossref]

Alivov, Y.

S. Sohal, Y. Alivov, Z. Fan, and M. Holtz, “Role of phonons in the optical properties of magnetron sputtered ZnO studied by resonance Raman and photoluminescence,” J. Appl. Phys. 108, 053507 (2010).
[Crossref]

Allen, C. G.

G. H. Shih, C. G. Allen, and B. G. Potter, “Interfacial effects on the optical behavior of Ge:ITO and Ge:NO nanocomposite films,” Nanotechnol. 23, 075203 (2012).
[Crossref]

Aloulou, S.

K. Rezgui, S. Aloulou, J. Rihani, and M. Oueslati, “Competition between strain and confinement effects on the crystalline quality of InAs/GaAs (001) quantum dots probed by Raman spectroscopy,” J. Raman Spectrosc. 43, 1964–1968 (2012).
[Crossref]

Anderson, K. E. H.

R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B 75, 245311 (2007).
[Crossref]

Ando, T.

N. Mori and T. Ando, “Electron–optical-phonon interaction in single and double heterostructures,” Phys. Rev. B 40, 6175–6188 (1989).
[Crossref]

Anni, M.

D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B 71, 235409 (2005).
[Crossref]

Arif, R. A.

S. K. Tripathy, G. Xu, X. Mu, Y. J. Ding, M. Jamil, R. A. Arif, N. Tansu, and J. B. Khurgin, “Phonon–assisted ultraviolet anti-stokes photoluminescence from GaN film grown on Si (111) substrate,” Appl. Phys. Lett. 93, 201107 (2008).
[Crossref]

Arora, A. K.

S. Dhara, A. K. Arora, S. Bera, and J. Ghatak, “Multiphonon probe in ZnS quantum dots,” J. Raman Spectrosc. 41, 1102–1105 (2010).
[Crossref]

Artemyev, M. V.

Axt, V. M.

A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B 70, 201305 (2004).
[Crossref]

Baimuratov, A. S.

Bakin, A.

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Balaban, S. N.

V. M. Fomin, V. N. Gladilin, J. T. Devreese, E. P. Pokatilov, S. N. Balaban, and S. N. Klimin, “Photoluminescence of spherical quantum dots,” Phys. Rev. B 57, 2415–2425 (1998).
[Crossref]

Baranov, A. V.

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, M. V. Artemyev, A. O. Orlova, and A. V. Fedorov, “Anisotropy of optical transitions in ordered ensemble of CdSe quantum rods,” Opt. Lett. 38, 3426–3428 (2013).
[Crossref] [PubMed]

S. Y. 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. B 81, 245303 (2010).
[Crossref]

A. V. Baranov, A. V. Fedorov, T. S. Perova, R. A. Moore, V. Yam, D. Bouchier, V. Thanh, and K. Berwick, “Analysis of strain and intermixing in single-layer Ge/Si quantum dots using polarized Raman spectroscopy,” Phys. Rev. B 73, 075322 (2006).
[Crossref]

A. V. Baranov, T. S. Perova, R. A. Moore, S. Solosin, A. V. Fedorov, V. Yam, V. Thanh, and D. Bouchier, “Polarized Raman spectroscopy of multilayer Ge/Si(001) quantum dot heterostructures,” J. Appl. Phys. 95, 2857–2863 (2004).
[Crossref]

A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon–phonon modes,” Semiconductors 38, 1101–1109 (2004).
[Crossref]

A. V. Baranov, Y. P. Rakovich, J. F. Donegan, T. S. Perova, R. A. Moore, D. V. Talapin, A. L. Rogach, Y. Masumoto, and I. Nabiev, “Effect of ZnS shell thickness on the phonon spectra in CdSe quantum dots,” Phys. Rev. B 68, 165306 (2003).
[Crossref]

A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors 35, 1390–1397 (2001).
[Crossref]

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton–phonon coupling in semiconductor quantum dots: Resonant Raman scattering,” Phys. Rev. B 56, 7491 (1997).
[Crossref]

A. V. Fedorov and A. V. Baranov, “Exciton–vibrational interaction of the frohlich type in quasi-zero-size systems,” J. Exp. Theor. Phys. 83, 610–618 (1996).

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

Bastard, G.

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gerard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B 65, 085316 (2002).
[Crossref]

O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett. 88, 146803 (2002).
[Crossref] [PubMed]

S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaitre, and J. M. Gerard, “Strong electron–phonon coupling regime in quantum dots: Evidence for everlasting resonant polarons,” Phys. Rev. Lett. 83, 4152 (1999).
[Crossref]

Baymuratov, A. S.

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. Express 19, 15461–15482 (2011).
[Crossref]

Bekeny, C.

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Bera, S.

S. Dhara, A. K. Arora, S. Bera, and J. Ghatak, “Multiphonon probe in ZnS quantum dots,” J. Raman Spectrosc. 41, 1102–1105 (2010).
[Crossref]

Berwick, K.

S. Y. 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. B 81, 245303 (2010).
[Crossref]

A. V. Baranov, A. V. Fedorov, T. S. Perova, R. A. Moore, V. Yam, D. Bouchier, V. Thanh, and K. Berwick, “Analysis of strain and intermixing in single-layer Ge/Si quantum dots using polarized Raman spectroscopy,” Phys. Rev. B 73, 075322 (2006).
[Crossref]

Biermann, A.

N. Tschirner, H. Lange, A. Schliwa, A. Biermann, C. Thomsen, K. Lambert, R. Gomes, and Z. Hens, “Interfacial alloying in CdSe/CdS heteronanocrystals: A Raman spectroscopy analysis,” Chem. Mater. 24, 311–318 (2012).
[Crossref]

Blum, K.

K. Blum, Density Matrix Theory and Applications (Springer, Berlin, 2012).
[Crossref]

Bonfanti, M.

S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron–phonon interaction in individual strain-free GaAs/Al0.3Ga0.7 As quantum dots,” Phys. Rev. B 73, 125342 (2006).
[Crossref]

Borner, S.

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Borri, P.

V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and p-doped InAs/GaAs quantum dots-in-a-well,” Phys. Rev. B 82, 195314 (2010).
[Crossref]

A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B 70, 201305 (2004).
[Crossref]

Bouazaoui, M.

K. Raulin, S. Turrell, B. Capoen, C. Kinowski, V. T. T. Tran, M. Bouazaoui, and O. Cristini, “Raman characterization of localized CdS nanostructures synthesized by UV irradiation in sol-gel silica matrices,” J. Ram. Spectrosc. 42, 1366–1372 (2011).
[Crossref]

Bouchier, D.

A. V. Baranov, A. V. Fedorov, T. S. Perova, R. A. Moore, V. Yam, D. Bouchier, V. Thanh, and K. Berwick, “Analysis of strain and intermixing in single-layer Ge/Si quantum dots using polarized Raman spectroscopy,” Phys. Rev. B 73, 075322 (2006).
[Crossref]

A. V. Baranov, T. S. Perova, R. A. Moore, S. Solosin, A. V. Fedorov, V. Yam, V. Thanh, and D. Bouchier, “Polarized Raman spectroscopy of multilayer Ge/Si(001) quantum dot heterostructures,” J. Appl. Phys. 95, 2857–2863 (2004).
[Crossref]

Briones, F.

V. V. Ursaki, I. M. Tiginyanu, V. V. Zalamai, V. M. Masalov, E. N. Samarov, G. A. Emelchenko, and F. Briones, “Photoluminescence and resonant Raman scattering from ZnO-opal structures,” J. Appl. Phys. 96, 1001–1006 (2004).
[Crossref]

Cantarero, A.

A. Cros, N. Garro, A. Cantarero, J. Coraux, H. Renevier, and B. Daudin, “Raman scattering as a tool for the evaluation of strain in GaN/AlN quantum dots: The effect of capping,” Phys. Rev. B 76, 165403 (2007).
[Crossref]

A. Cros, N. Garro, J. M. Llorens, A. García-Cristóbal, A. Cantarero, N. Gogneau, E. Monroy, and B. Daudin, “Raman study and theoretical calculations of strain in GaN quantum dot multilayers,” Phys. Rev. B 73, 115313 (2006).
[Crossref]

Capoen, B.

K. Raulin, S. Turrell, B. Capoen, C. Kinowski, V. T. T. Tran, M. Bouazaoui, and O. Cristini, “Raman characterization of localized CdS nanostructures synthesized by UV irradiation in sol-gel silica matrices,” J. Ram. Spectrosc. 42, 1366–1372 (2011).
[Crossref]

Cardona, M.

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[Crossref]

Carpenter, B. A.

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

Castella, H.

T. Stauber, R. Zimmermann, and H. Castella, “Electron–phonon interaction in quantum dots: A solvable model,” Phys. Rev. B 62, 7336 (2000).
[Crossref]

Cesari, V.

V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and p-doped InAs/GaAs quantum dots-in-a-well,” Phys. Rev. B 82, 195314 (2010).
[Crossref]

Chamberlain, M. P.

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[Crossref]

Cho, H. K.

C. H. Ahn, S. K. Mohanta, N. E. Lee, and H. K. Cho, “Enhanced exciton–phonon interactions in photoluminescence of ZnO nanopencils,” Appl. Phys. Lett. 94, 261904 (2009).
[Crossref]

Cingolani, R.

D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B 71, 235409 (2005).
[Crossref]

Cockburn, J. W.

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

Cooney, R. R.

R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B 75, 245311 (2007).
[Crossref]

Coraux, J.

A. Cros, N. Garro, A. Cantarero, J. Coraux, H. Renevier, and B. Daudin, “Raman scattering as a tool for the evaluation of strain in GaN/AlN quantum dots: The effect of capping,” Phys. Rev. B 76, 165403 (2007).
[Crossref]

Cretí, A.

D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B 71, 235409 (2005).
[Crossref]

Cristini, O.

K. Raulin, S. Turrell, B. Capoen, C. Kinowski, V. T. T. Tran, M. Bouazaoui, and O. Cristini, “Raman characterization of localized CdS nanostructures synthesized by UV irradiation in sol-gel silica matrices,” J. Ram. Spectrosc. 42, 1366–1372 (2011).
[Crossref]

Cros, A.

A. Cros, N. Garro, A. Cantarero, J. Coraux, H. Renevier, and B. Daudin, “Raman scattering as a tool for the evaluation of strain in GaN/AlN quantum dots: The effect of capping,” Phys. Rev. B 76, 165403 (2007).
[Crossref]

A. Cros, N. Garro, J. M. Llorens, A. García-Cristóbal, A. Cantarero, N. Gogneau, E. Monroy, and B. Daudin, “Raman study and theoretical calculations of strain in GaN quantum dot multilayers,” Phys. Rev. B 73, 115313 (2006).
[Crossref]

da Silva, M. A. P.

E. S. F. Neto, S. W. da Silva, P. C. Morais, M. I. Vasilevskiy, M. A. P. da Silva, and N. O. Dantas, “Resonant Raman scattering in CdSx Se1−x nanocrystals: Effects of phonon confinement, composition, and elastic strain,” J. Raman Spectrosc. 42, 1660–1669 (2011).
[Crossref]

da Silva, S. W.

E. S. F. Neto, S. W. da Silva, P. C. Morais, M. I. Vasilevskiy, M. A. P. da Silva, and N. O. Dantas, “Resonant Raman scattering in CdSx Se1−x nanocrystals: Effects of phonon confinement, composition, and elastic strain,” J. Raman Spectrosc. 42, 1660–1669 (2011).
[Crossref]

Dantas, N. O.

E. S. F. Neto, S. W. da Silva, P. C. Morais, M. I. Vasilevskiy, M. A. P. da Silva, and N. O. Dantas, “Resonant Raman scattering in CdSx Se1−x nanocrystals: Effects of phonon confinement, composition, and elastic strain,” J. Raman Spectrosc. 42, 1660–1669 (2011).
[Crossref]

Daudin, B.

A. Cros, N. Garro, A. Cantarero, J. Coraux, H. Renevier, and B. Daudin, “Raman scattering as a tool for the evaluation of strain in GaN/AlN quantum dots: The effect of capping,” Phys. Rev. B 76, 165403 (2007).
[Crossref]

A. Cros, N. Garro, J. M. Llorens, A. García-Cristóbal, A. Cantarero, N. Gogneau, E. Monroy, and B. Daudin, “Raman study and theoretical calculations of strain in GaN quantum dot multilayers,” Phys. Rev. B 73, 115313 (2006).
[Crossref]

de Vaulchier, L. A.

V. Preisler, R. Ferreira, S. Hameau, L. A. de Vaulchier, Y. Guldner, M. L. Sadowski, and A. Lemaitre, “Hole–LO-phonon interaction in InAs/GaAs quantum dots,” Phys. Rev. B 72, 115309 (2005).
[Crossref]

Deleporte, E.

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gerard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B 65, 085316 (2002).
[Crossref]

Desai, R.

S. K. Gupta, R. Desai, P. K. Jha, S. Sahoo, and D. Kirin, “Titanium dioxide synthesized using titanium chloride: Size effect study using Raman spectroscopy and photoluminescence,” J. Ram. Spectrosc. 41, 350–355 (2010).

Devreese, J. T.

E. P. Pokatilov, S. N. Klimin, V. M. Fomin, J. T. Devreese, and F. W. Wise, “Multiphonon Raman scattering in semiconductor nanocrystals: Importance of nonadiabatic transitions,” Phys. Rev. B 65, 075316 (2002).
[Crossref]

V. M. Fomin, V. N. Gladilin, J. T. Devreese, E. P. Pokatilov, S. N. Balaban, and S. N. Klimin, “Photoluminescence of spherical quantum dots,” Phys. Rev. B 57, 2415–2425 (1998).
[Crossref]

Dhara, S.

S. Dhara, A. K. Arora, S. Bera, and J. Ghatak, “Multiphonon probe in ZnS quantum dots,” J. Raman Spectrosc. 41, 1102–1105 (2010).
[Crossref]

Dias, E. A.

R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B 75, 245311 (2007).
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E. P. Pokatilov, S. N. Klimin, V. M. Fomin, J. T. Devreese, and F. W. Wise, “Multiphonon Raman scattering in semiconductor nanocrystals: Importance of nonadiabatic transitions,” Phys. Rev. B 65, 075316 (2002).
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D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B 71, 235409 (2005).
[Crossref]

Vasilevskiy, M. I.

E. S. F. Neto, S. W. da Silva, P. C. Morais, M. I. Vasilevskiy, M. A. P. da Silva, and N. O. Dantas, “Resonant Raman scattering in CdSx Se1−x nanocrystals: Effects of phonon confinement, composition, and elastic strain,” J. Raman Spectrosc. 42, 1660–1669 (2011).
[Crossref]

R. P. Miranda, M. I. Vasilevskiy, and C. Trallero-Giner, “Nonperturbative approach to the calculation of multi-phonon Raman scattering in semiconductor quantum dots: Polaron effect,” Phys. Rev. B 74, 115317 (2006).
[Crossref]

Verzelen, O.

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gerard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B 65, 085316 (2002).
[Crossref]

O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett. 88, 146803 (2002).
[Crossref] [PubMed]

S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaitre, and J. M. Gerard, “Strong electron–phonon coupling regime in quantum dots: Evidence for everlasting resonant polarons,” Phys. Rev. Lett. 83, 4152 (1999).
[Crossref]

Voss, T.

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Waag, A.

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Wageh, S.

S. Wageh, A. M. El-Nahas, A. A. Higazy, and M. A. M. Mahmoud, “Preparation and characterization of a novel system of CdS nanoparticles embedded in borophosphate glass matrix,” J. Alloys Comp. 555, 161–168 (2013).
[Crossref]

Whittaker, D. M.

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

Wilson, L. R.

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

Wischmeier, L.

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Wise, F. W.

E. P. Pokatilov, S. N. Klimin, V. M. Fomin, J. T. Devreese, and F. W. Wise, “Multiphonon Raman scattering in semiconductor nanocrystals: Importance of nonadiabatic transitions,” Phys. Rev. B 65, 075316 (2002).
[Crossref]

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]

A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B 70, 201305 (2004).
[Crossref]

Xu, G.

S. K. Tripathy, G. Xu, X. Mu, Y. J. Ding, M. Jamil, R. A. Arif, N. Tansu, and J. B. Khurgin, “Phonon–assisted ultraviolet anti-stokes photoluminescence from GaN film grown on Si (111) substrate,” Appl. Phys. Lett. 93, 201107 (2008).
[Crossref]

Yam, V.

A. V. Baranov, A. V. Fedorov, T. S. Perova, R. A. Moore, V. Yam, D. Bouchier, V. Thanh, and K. Berwick, “Analysis of strain and intermixing in single-layer Ge/Si quantum dots using polarized Raman spectroscopy,” Phys. Rev. B 73, 075322 (2006).
[Crossref]

A. V. Baranov, T. S. Perova, R. A. Moore, S. Solosin, A. V. Fedorov, V. Yam, V. Thanh, and D. Bouchier, “Polarized Raman spectroscopy of multilayer Ge/Si(001) quantum dot heterostructures,” J. Appl. Phys. 95, 2857–2863 (2004).
[Crossref]

Yang, H. D.

T. S. Kim, Y.-H. Kil, H. D. Yang, J.-H. Yang, W.-K. Hong, S. Kang, T. S. Jeong, and K.-H. Shim, “Growth and characterization of Si1−xGex QDs on Si/Si0.8Ge0.2 layer,” Electron. Mater. Lett. 8, 559–563 (2012).
[Crossref]

Yang, J.-H.

T. S. Kim, Y.-H. Kil, H. D. Yang, J.-H. Yang, W.-K. Hong, S. Kang, T. S. Jeong, and K.-H. Shim, “Growth and characterization of Si1−xGex QDs on Si/Si0.8Ge0.2 layer,” Electron. Mater. Lett. 8, 559–563 (2012).
[Crossref]

Yu, G.

G. Yu, Q. Liang, Y. Jia, and J. Dong, “Phonon sidebands of photoluminescence in single wall carbon nanotubes,” J. Appl. Phys. 107, 024314 (2010).
[Crossref]

Zalamai, V. V.

V. V. Ursaki, I. M. Tiginyanu, V. V. Zalamai, V. M. Masalov, E. N. Samarov, G. A. Emelchenko, and F. Briones, “Photoluminescence and resonant Raman scattering from ZnO-opal structures,” J. Appl. Phys. 96, 1001–1006 (2004).
[Crossref]

Zeman, J.

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gerard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B 65, 085316 (2002).
[Crossref]

S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaitre, and J. M. Gerard, “Strong electron–phonon coupling regime in quantum dots: Evidence for everlasting resonant polarons,” Phys. Rev. Lett. 83, 4152 (1999).
[Crossref]

Zhao, J.

J. Zhao, A. Kanno, M. Ikezawa, and Y. Masumoto, “Longitudinal optical phonons in the excited state of CuBr quantum dots,” Phys. Rev. B 68, 113305 (2003).
[Crossref]

Zibik, E. A.

B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

Zimmermann, R.

E. A. Muljarov and R. Zimmermann, “Exciton dephasing in quantum dots due to LO-phonon coupling: An exactly solvable model,” Phys. Rev. Lett. 98, 187401 (2007).
[Crossref] [PubMed]

T. Stauber, R. Zimmermann, and H. Castella, “Electron–phonon interaction in quantum dots: A solvable model,” Phys. Rev. B 62, 7336 (2000).
[Crossref]

Appl. Phys. Lett. (3)

C. H. Ahn, S. K. Mohanta, N. E. Lee, and H. K. Cho, “Enhanced exciton–phonon interactions in photoluminescence of ZnO nanopencils,” Appl. Phys. Lett. 94, 261904 (2009).
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S. K. Tripathy, G. Xu, X. Mu, Y. J. Ding, M. Jamil, R. A. Arif, N. Tansu, and J. B. Khurgin, “Phonon–assisted ultraviolet anti-stokes photoluminescence from GaN film grown on Si (111) substrate,” Appl. Phys. Lett. 93, 201107 (2008).
[Crossref]

T. Voss, C. Bekeny, L. Wischmeier, H. Gafsi, S. Borner, W. Schade, A. C. Mofor, A. Bakin, and A. Waag, “Influence of exciton–phonon coupling on the energy position of the near-band-edge photoluminescence of ZnO nanowires,” Appl. Phys. Lett. 89, 182107 (2006).
[Crossref]

Bull. Russ. Acad. Sci. (1)

I. D. Rukhlenko and A. V. Fedorov, “Resonant photoluminescence of quantum dots: Dynamics of electronic subsystem,” Bull. Russ. Acad. Sci. 70, 126 (2006).

Chem. Mater. (1)

N. Tschirner, H. Lange, A. Schliwa, A. Biermann, C. Thomsen, K. Lambert, R. Gomes, and Z. Hens, “Interfacial alloying in CdSe/CdS heteronanocrystals: A Raman spectroscopy analysis,” Chem. Mater. 24, 311–318 (2012).
[Crossref]

Electron. Mater. Lett. (1)

T. S. Kim, Y.-H. Kil, H. D. Yang, J.-H. Yang, W.-K. Hong, S. Kang, T. S. Jeong, and K.-H. Shim, “Growth and characterization of Si1−xGex QDs on Si/Si0.8Ge0.2 layer,” Electron. Mater. Lett. 8, 559–563 (2012).
[Crossref]

J. Alloys Comp. (1)

S. Wageh, A. M. El-Nahas, A. A. Higazy, and M. A. M. Mahmoud, “Preparation and characterization of a novel system of CdS nanoparticles embedded in borophosphate glass matrix,” J. Alloys Comp. 555, 161–168 (2013).
[Crossref]

J. Appl. Phys. (4)

A. V. Baranov, T. S. Perova, R. A. Moore, S. Solosin, A. V. Fedorov, V. Yam, V. Thanh, and D. Bouchier, “Polarized Raman spectroscopy of multilayer Ge/Si(001) quantum dot heterostructures,” J. Appl. Phys. 95, 2857–2863 (2004).
[Crossref]

G. Yu, Q. Liang, Y. Jia, and J. Dong, “Phonon sidebands of photoluminescence in single wall carbon nanotubes,” J. Appl. Phys. 107, 024314 (2010).
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S. Sohal, Y. Alivov, Z. Fan, and M. Holtz, “Role of phonons in the optical properties of magnetron sputtered ZnO studied by resonance Raman and photoluminescence,” J. Appl. Phys. 108, 053507 (2010).
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V. V. Ursaki, I. M. Tiginyanu, V. V. Zalamai, V. M. Masalov, E. N. Samarov, G. A. Emelchenko, and F. Briones, “Photoluminescence and resonant Raman scattering from ZnO-opal structures,” J. Appl. Phys. 96, 1001–1006 (2004).
[Crossref]

J. Chem. Phys. (1)

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton–phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
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J. Ram. Spectrosc. (2)

S. K. Gupta, R. Desai, P. K. Jha, S. Sahoo, and D. Kirin, “Titanium dioxide synthesized using titanium chloride: Size effect study using Raman spectroscopy and photoluminescence,” J. Ram. Spectrosc. 41, 350–355 (2010).

K. Raulin, S. Turrell, B. Capoen, C. Kinowski, V. T. T. Tran, M. Bouazaoui, and O. Cristini, “Raman characterization of localized CdS nanostructures synthesized by UV irradiation in sol-gel silica matrices,” J. Ram. Spectrosc. 42, 1366–1372 (2011).
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S. Dhara, A. K. Arora, S. Bera, and J. Ghatak, “Multiphonon probe in ZnS quantum dots,” J. Raman Spectrosc. 41, 1102–1105 (2010).
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K. Rezgui, S. Aloulou, J. Rihani, and M. Oueslati, “Competition between strain and confinement effects on the crystalline quality of InAs/GaAs (001) quantum dots probed by Raman spectroscopy,” J. Raman Spectrosc. 43, 1964–1968 (2012).
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E. S. F. Neto, S. W. da Silva, P. C. Morais, M. I. Vasilevskiy, M. A. P. da Silva, and N. O. Dantas, “Resonant Raman scattering in CdSx Se1−x nanocrystals: Effects of phonon confinement, composition, and elastic strain,” J. Raman Spectrosc. 42, 1660–1669 (2011).
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Nanotechnol. (1)

G. H. Shih, C. G. Allen, and B. G. Potter, “Interfacial effects on the optical behavior of Ge:ITO and Ge:NO nanocomposite films,” Nanotechnol. 23, 075203 (2012).
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Opt. Express (1)

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. Express 19, 15461–15482 (2011).
[Crossref]

Opt. Lett. (3)

Opt. Spectrosc. (3)

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]

I. D. Rukhlenko and A. V. Fedorov, “Propagation of electric fields induced by optical phonons in semiconductor heterostructures,” Opt. Spectrosc. 100, 238–244 (2006).
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S. Y. Kruchinin and A. V. Fedorov, “Renormalization of the energy spectrum of quantum dots under vibrational resonance conditions: Persistent hole burning spectroscopy,” Opt. Spectrosc. 100, 41–48 (2006).
[Crossref]

Phys. Rev. B (26)

R. P. Miranda, M. I. Vasilevskiy, and C. Trallero-Giner, “Nonperturbative approach to the calculation of multi-phonon Raman scattering in semiconductor quantum dots: Polaron effect,” Phys. Rev. B 74, 115317 (2006).
[Crossref]

E. P. Pokatilov, S. N. Klimin, V. M. Fomin, J. T. Devreese, and F. W. Wise, “Multiphonon Raman scattering in semiconductor nanocrystals: Importance of nonadiabatic transitions,” Phys. Rev. B 65, 075316 (2002).
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T. Stauber, R. Zimmermann, and H. Castella, “Electron–phonon interaction in quantum dots: A solvable model,” Phys. Rev. B 62, 7336 (2000).
[Crossref]

V. M. Fomin, V. N. Gladilin, J. T. Devreese, E. P. Pokatilov, S. N. Balaban, and S. N. Klimin, “Photoluminescence of spherical quantum dots,” Phys. Rev. B 57, 2415–2425 (1998).
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B. A. Carpenter, E. A. Zibik, M. L. Sadowski, L. R. Wilson, D. M. Whittaker, J. W. Cockburn, M. S. Skolnick, M. Potemski, M. J. Steer, and M. Hopkinson, “Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots,” Phys. Rev. B 74, 161302 (2006).
[Crossref]

V. Preisler, R. Ferreira, S. Hameau, L. A. de Vaulchier, Y. Guldner, M. L. Sadowski, and A. Lemaitre, “Hole–LO-phonon interaction in InAs/GaAs quantum dots,” Phys. Rev. B 72, 115309 (2005).
[Crossref]

J. Zhao, A. Kanno, M. Ikezawa, and Y. Masumoto, “Longitudinal optical phonons in the excited state of CuBr quantum dots,” Phys. Rev. B 68, 113305 (2003).
[Crossref]

S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. Gerard, “Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B 65, 085316 (2002).
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A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton–phonon coupling in semiconductor quantum dots: Resonant Raman scattering,” Phys. Rev. B 56, 7491 (1997).
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M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
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E. Evans and D. L. Mills, “Theory of inelastic scattering of slow electrons by long-wavelength surface optical phonons,” Phys. Rev. B 8, 4126–4139 (1973).

N. Mori and T. Ando, “Electron–optical-phonon interaction in single and double heterostructures,” Phys. Rev. B 40, 6175–6188 (1989).
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A. V. Baranov, Y. P. Rakovich, J. F. Donegan, T. S. Perova, R. A. Moore, D. V. Talapin, A. L. Rogach, Y. Masumoto, and I. Nabiev, “Effect of ZnS shell thickness on the phonon spectra in CdSe quantum dots,” Phys. Rev. B 68, 165306 (2003).
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S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron–phonon interaction in individual strain-free GaAs/Al0.3Ga0.7 As quantum dots,” Phys. Rev. B 73, 125342 (2006).
[Crossref]

D. Valerini, A. Cretí, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the photoluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,” Phys. Rev. B 71, 235409 (2005).
[Crossref]

V. Cesari, W. Langbein, and P. Borri, “Dephasing of excitons and multiexcitons in undoped and p-doped InAs/GaAs quantum dots-in-a-well,” Phys. Rev. B 82, 195314 (2010).
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K. Kojima and A. Tomita, “Influence of pure dephasing by phonons on exciton–photon interfaces: Quantum microscopic theory,” Phys. Rev. B 73, 195312 (2006).
[Crossref]

A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri, and U. Woggon, “Nonmonotonous temperature dependence of the initial decoherence in quantum dots,” Phys. Rev. B 70, 201305 (2004).
[Crossref]

R. R. Cooney, S. L. Sewall, E. A. Dias, D. M. Sagar, K. E. H. Anderson, and P. Kambhampati, “Unified picture of electron and hole relaxation pathways in semiconductor quantum dots,” Phys. Rev. B 75, 245311 (2007).
[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).
[Crossref]

A. Cros, N. Garro, A. Cantarero, J. Coraux, H. Renevier, and B. Daudin, “Raman scattering as a tool for the evaluation of strain in GaN/AlN quantum dots: The effect of capping,” Phys. Rev. B 76, 165403 (2007).
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A. Cros, N. Garro, J. M. Llorens, A. García-Cristóbal, A. Cantarero, N. Gogneau, E. Monroy, and B. Daudin, “Raman study and theoretical calculations of strain in GaN quantum dot multilayers,” Phys. Rev. B 73, 115313 (2006).
[Crossref]

A. V. Baranov, A. V. Fedorov, T. S. Perova, R. A. Moore, V. Yam, D. Bouchier, V. Thanh, and K. Berwick, “Analysis of strain and intermixing in single-layer Ge/Si quantum dots using polarized Raman spectroscopy,” Phys. Rev. B 73, 075322 (2006).
[Crossref]

R. W. Meulenberg, T. Jennings, and G. F. Strouse, “Compressive and tensile stress in colloidal CdSe semiconductor quantum dots,” Phys. Rev. B 70, 235311 (2004).
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S. Y. 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. B 81, 245303 (2010).
[Crossref]

Phys. Rev. Lett. (4)

E. A. Muljarov and R. Zimmermann, “Exciton dephasing in quantum dots due to LO-phonon coupling: An exactly solvable model,” Phys. Rev. Lett. 98, 187401 (2007).
[Crossref] [PubMed]

S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaitre, and J. M. Gerard, “Strong electron–phonon coupling regime in quantum dots: Evidence for everlasting resonant polarons,” Phys. Rev. Lett. 83, 4152 (1999).
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T. Itoh, M. Nishijima, A. I. Ekimov, C. Gourdon, A. L. Efros, and M. Rosen, “Polaron and exciton–phonon complexes in CuCl nanocrystals,” Phys. Rev. Lett. 74, 1645 (1995).
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O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett. 88, 146803 (2002).
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Sci. Rep. (1)

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
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A. V. Fedorov, A. V. Baranov, A. Itoh, and Y. Masumoto, “Renormalization of energy spectrum of quantum dots under vibrational resonance conditions,” Semiconductors 35, 1390–1397 (2001).
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A. V. Fedorov and A. V. Baranov, “Relaxation of charge carriers in quantum dots with the involvement of plasmon–phonon modes,” Semiconductors 38, 1101–1109 (2004).
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O. Madelung, M. Schultz, and H. Weiss, eds., “Semiconductors. Physics of Group IV Elements and III–V Compounds,” Landolt-Börnstein, New Series, Group III, vol. 17, Pt. a (Springer-Verlag, Berlin, 1982).

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A. Rogach, ed., Semiconductor Nanocrystal Quantum Dots: Synthesis, Assembly, Spectroscopy and Applications (Springer, New York, 2008).
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Y. Masumoto and T. Takagahara, eds., Semiconductor Quantum Dots: Physics, Spectroscopy and Applications (Springer, New York, 2002).
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Figures (5)

Fig. 1
Fig. 1

Formation of polaron-like states from a pair of nondegenerate electron–hole states coupled via one, two, three or four degenerate LO phonon modes of energy h̄Ω. The electron–phonon interaction transforms the degenerate state of the QD into two, three, four or five nondegenerate polaron-like states.

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Fig. 2
Fig. 2

Feynman diagrams describing [(a) and (b)] creation and [(c) and (d)] annihilation of a polaron-like quasiparticle of energy Ẽp with the simultaneous (a) absorption of a photon, h̄ωL, and a phonon, h̄Ω, (b) absorption of a photon and emission of a phonon, (c) emission of a photon, h̄ωλ, and absorption of a phonon, and (d) emission of a photon and a phonon.

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Fig. 3
Fig. 3

Matrix elements and overlap integrals as functions of potential barrier heights for electrons and holes in a spherical InAs QD with R = 9.14 nm. The material parameters are: ε0 = 15.15, ε∞ = 12.25, Eg = 418 meV, h̄Ω = 29.5 meV, me = 0.0219m0, and mh = 0.43m0 [64] (m0 is the free-electron mass). For other parameter refer to the text.

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Fig. 4
Fig. 4

Photoluminescence spectra of InAs QD with a pair of nondegenerate electronic states coupled through two phonon modes. The excitation frequencies are: h ¯ ω 2 ( 2 ) = 530.6   meV, h ¯ ω 3 ( 2 ) = 533.7   meV, and h ¯ ω 1 ( 2 ) = 536.8   meV; V(2) = 3.1 meV. For material parameters refer to the text.

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Fig. 5
Fig. 5

Excitation photoluminescence spectra of InAs QD with a pair of nondegenerate electronic states coupled through two phonon modes. The detection frequencies are shown near the spectra. The parameter values and resonant frequencies ω j ( 2 ) (j = 1, 2, 3) are the same as in Fig. 4.

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Equations (55)

Equations on this page are rendered with MathJax. Learn more.

H e , ph = ∑ p 1 , p 2 ∑ q V p 2 ; p 1 ( q ) a p 2 + a p 1 b q + H . c . ,
H e , L = ∑ p V p ; 0 ( L ) a p + + H . c .
H e , r = ∑ p , λ V p ; 0 ( λ ) a p + c λ + H . c . ,
F p = 2 δ l e , l h δ m e , m h ∫ 0 ∞ R n e , l e R n h , l h x 2 d x ,
φ p k 2 , p k 1 ( q ) = 4 ( 2 l q + 1 ) ( 2 l k 1 + 1 ) h ¯ Ω ( 2 l k 2 + 1 ) ε * R C l q 0 , l k 1 0 l k 2 0 C l q m q , l k 1 m k 1 l k 2 m k 2 ℐ n k 2 l k 2 , n k 1 l k 1 n q l q ,
ℐ n k 2 l k 2 , n k 1 l k 1 n q l q = ∫ 0 ∞ ℛ n q l q R n k 1 l k 1 R n k 2 l k 2 x 2 d x ,
R n k l k = 1 A n k l k × { κ l k ( α n k l k s ) j l k ( α n k l k d x ) , x ≤ 1 j l k ( α n k l k d ) κ l k ( α n k l k s x ) , x > 1 ,
A n k l k = κ l k + 1 ( α n k l k s ) κ l k − 1 ( α n k l k s ) j l k 2 ( α n k l k d ) − j l k + 1 ( α n k l k d ) j l k − 1 ( α n k l k d ) κ l k 2 ( α n k l k s ) ,
α n k l k d = ( R / h ¯ ) 2 m k d E n k l k ,
α n k l k s = ( R / h ¯ ) 2 m k s ( V k − E n k l k ) ;
m k s α n k l k d j l k ′ ( α n k l k d ) j l k ( α n k l k d ) = m k d α n k l k s κ l k ′ ( α n k l k s ) κ l k ( α n k l k s ) ,
ℛ n q l q = 1 ξ n q l q j l q + 1 ( ξ n q l q ) × { j l q ( ξ n q l q x ) , x ≤ 1 0 , x > 1 ,
F p ≡ F l e m e , l h m h n e n h = 2 δ l e , l h δ m e , m h A n e l e A n h l h ( κ l e ( α n e l e s ) κ l h ( α n h l h s ) ∫ 0 1 j l e ( α n e l e d x ) j l h ( α n h l h d x ) x 2 d x + j l e ( α n e l e d ) j l h ( α n h l h d ) ∫ 1 ∞ κ l e ( α n e l e s x ) κ l h ( α n h l h s x ) x 2 d x )
ℐ n k 2 l k 2 , n k 1 l k 1 n q l q = κ l k 1 ( α n k 1 l k 1 s ) κ l k 2 ( α n k 2 l k 2 s ) ξ n q l q j l q + 1 ( ξ n q l q ) A n k 1 l k 1 A n k 2 l k 2                                                                                                                                                                                         × ∫ 0 1 j l q ( ξ n q l q x ) j l k 1 ( α n k 1 l k 1 d x ) j l k 2 ( α n k 2 l k 2 d x ) x 2 d x .
H ˜ = U + H U ,
U = exp ( ∑ p , q ( Φ p ; p ( q ) b q − H . c . ) a p + a p )
E ˜ p = E p − ∑ q h ¯ Ω | Φ p ; p ( q ) | 2
H ˜ e , ph = ∑ p 1 ≠ p 2 ∑ q V p 2 ; p 1 ( q ) a p 2 + a p 1 b q + H . c . ,
H ˜ e , L = ∑ p V p ; 0 ( L ) ( 1 + ∑ q ( Φ p ; p ( q ) b q − H . c . ) ) a p + + H . c . ,
H ˜ e , r = ∑ p , λ V p ; 0 ( λ ) ( 1 + ∑ q ( Φ p ; p ( q ) b q − H . c . ) ) a p + c λ + H . c .
H ^ e , L ( k ) = ( 0 0 ⋯ 0 H 1 ( L , k ) 0 0 ⋯ 0 H 2 ( L , k ) ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 0 H k + 1 ( L , k ) H 1 ( L , k ) * H 2 ( L , k ) * ⋯ H k + 1 ( L , k ) * 0 ) ,
H 1 ( L , k ) = V p 2 ; 0 ( L ) S 1 ; 1 ( k ) + V p 1 ; 0 ( L ) ∑ ν = 1 k Φ p 1 ; p 1 ( q ν ) S 1 + ν ; 1 ( k ) ,
H 2 ( L , k ) = V p 2 ; 0 ( L ) S 1 ; 2 ( k ) + V p 1 ; 0 ( L ) ∑ ν = 1 k Φ p 1 ; p 1 ( q ν ) S 1 + ν ; 2 ( k ) ,
H n ( L , k ) = V p 1 ; 0 ( L ) ∑ ν = 1 k Φ p 1 ; p 1 ( q ν ) S 1 + ν ; n ( k )
H ^ e , r ( k ) = ( 0 0 ⋯ 0 H 1 , q 1 ( r , k ) H 1 , q 2 ( r , k ) ⋯ H 1 , q k ( r , k ) 0 0 ⋯ 0 H 2 , q 1 ( r , k ) H 2 , q 2 ( r , k ) ⋯ H 2 , q k ( r , k ) ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 0 H k + 1 , q 1 ( r , k ) H k + 1 , q 2 ( r , k ) ⋯ H k + 1 , q k ( r , k ) H 1 , q 1 ( r , k ) * H 2 , q 1 ( r , k ) * ⋯ H k + 1 , q 1 ( r , k ) * 0 0 ⋯ 0 H 1 , q 2 ( r , k ) * H 2 , q 1 ( r , k ) * ⋯ H k + 1 , q 2 ( r , k ) * 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ H 1 , q k ( r , k ) * H 2 , q k ( r , k ) * ⋯ H k + 1 , q k ( r , k ) * 0 0 ⋯ 0 ) ,
H 1 , q ( r , k ) = V p 1 ; 0 ( λ ) S 1 + q ; 1 ( k ) + V p 2 ; 0 ( λ ) Φ p 2 ; p 2 ( q ) S 1 ; 1 ( k ) ,
H 2 , q ( r , k ) = V p 1 ; 0 ( λ ) S 1 + q ; 2 ( k ) + V p 2 ; 0 ( λ ) Φ p 2 ; p 2 ( q ) S 1 ; 2 ( k ) ,
H n , q ( r , k ) = V p 1 ; 0 ( λ ) S 1 + q ; n ( k )
| i 〉 = | 0 〉 | 0 q 〉 | 0 λ 〉 ,
| 1 〉 = | ℰ 1 ( k ) 〉 | 0 q 〉 | 0 λ 〉 , … , | k + 1 〉 = | ℰ k + 1 ( k ) 〉 | 0 q 〉 | 0 λ 〉 ,
| f 1 〉 = | 0 〉 | 1 q 1 〉 | 1 λ 〉 , … , | f k 〉 = | 0 〉 | 1 q k 〉 | 1 λ 〉 .
∂ ρ μ ν ( t ) ∂ t = 1 i h ¯ [ H ^ ( t ) , ρ ( t ) ] μ ν − γ μ ν ρ μ ν ( t ) + δ μ ν ∑ ν ′ ≠ ν ζ ν ν ′ ρ ν ′ ν ′ ( t ) ,
d 2 σ d Θ d ω λ = C ( ω λ ) ∑ η = 1 k ( ∑ μ = 1 k + 1 2 γ i μ γ μ μ 2 γ ^ i μ + γ ph Δ L μ 2 + γ i μ 2 | d μ ( L , k ) | 2 | d μ , q η ( λ , k ) | 2 Δ λ μ 2 + γ f μ 2                                                                                                                                                 + ∑ ν = 2 k + 1 ∑ μ = 1 , ( μ ≠ 2 ) ν − 1 2 ζ ν μ γ i ν γ μ μ γ ν ν 2 γ ν f Δ L μ 2 + γ i μ 2 | d μ ( L , k ) | 2 | d ν , q η ( λ , k ) | 2 Δ λ ν 2 + γ f ν 2 ) ,
d n ( L , k ) = F p 2 S 1 ; n ( k ) + F p 1 ∑ ν = 1 k Φ p 1 ; p 1 ( q ν ) S 1 + ν ; n ( k ) ,
d n , q ( λ , k ) = F p 1 S 1 + q ; n ( k ) + F p 2 Φ p 2 ; p 2 ( q ) S 1 ; n ( k ) .
ℋ ( l 1 , l 2 ) = ( H ( l 1 , l 2 , l 2 ) 0 ⋯ 0 0 0 H ( l 1 , l 2 , l 2 − 1 ) ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ H ( l 1 , l 2 , 1 − l 2 ) 0 0 0 ⋯ 0 H ( l 1 , l 2 , − l 2 ) ) ,
𝒮 ( l 1 , l 2 ) = ( S ( l 1 , l 2 , l 2 ) 0 ⋯ 0 0 0 S ( l 1 , l 2 , l 2 − 1 ) ⋯ 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ S ( l 1 , l 2 , 1 − l 2 ) 0 0 0 ⋯ 0 S ( l 1 , l 2 , − l 2 ) ) .
k ( l 1 , l 2 , m 2 ) = ∑ ν = 1 1 + min ( l 1 , l 2 ) min ( 2 l 1 + 1 , 2 l q ν + 1 , 2 l 1 + 2 l q ν − l 2 − | m 2 | + 1 ) .
H ( l 1 , l 2 , m 2 ) ≡ H ˜ e , ph ( k ) = ( E ˜ p 2 V p 2 ; p 1 ( q 1 ) V p 2 ; p 1 ( q 2 ) ⋯ V p 2 ; p 1 ( q k ) V p 2 ; p 1 ( q 1 ) * E ˜ p 1 + h ¯ Ω 0 ⋯ 0 V p 2 ; p 1 ( q 2 ) * 0 E ˜ p 1 + h ¯ Ω ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ V p 2 ; p 1 ( q k ) * 0 0 ⋯ E ˜ p 1 + h ¯ Ω ) .
S ( l 1 , l 2 , m 2 ) ≡ S ( k ) = ( S 1 ; 1 ( k ) S 1 ; 2 ( k ) 0 0 ⋯ 0 0 S 2 ; 1 ( k ) S 2 ; 2 ( k ) S 2 ; 3 ( k ) S 2 ; 4 ( k ) ⋯ S 2 ; k ( k ) S 2 ; k + 1 ( k ) S 3 ; 1 ( k ) S 3 ; 2 ( k ) S 3 ; 3 ( k ) S 3 ; 4 ( k ) ⋯ S 3 ; k ( k ) S 3 ; k + 1 ( k ) S 4 ; 1 ( k ) S 4 ; 2 ( k ) 0 S 4 ; 4 ( k ) ⋯ S 4 ; k ( k ) S 4 ; k + 1 ( k ) ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ S k ; 1 ( k ) S k ; 2 ( k ) 0 0 ⋯ S k ; k ( k ) S k ; k + 1 ( k ) S k + 1 ; 1 ( k ) S k + 1 ; 2 ( k ) 0 0 ⋯ 0 S k + 1 ; k + 1 ( k ) ) ,
S 1 ; 1 ( k ) = ( α + δ ( k ) ) χ ( k ) / β ( k , + ) ,                   S 1 ; 2 ( k ) = ( α − δ ( k ) ) χ ( k ) / β ( k , − ) ,
S n ; 1 ( k ) = 2 V p 2 ; p 1 ( q n − 1 ) * χ ( k ) / β ( k , + ) ,                         S n ; 2 ( k ) = 2 V p 2 ; p 1 ( q n − 1 ) * χ ( k ) / β ( k , − )
S n ; m ( k ) = − V p 2 ; p 1 ( q m − 1 ) V p 2 ; p 1 ( q n − 1 ) * / ( V ( m − 1 ) V ( m − 2 ) )
S n ; n ( k ) = V ( n − 2 ) / V ( n − 1 )
β ( k , ± ) = [ ( α ± δ ( k ) ) 2 + 4 ( V ( k ) ) 2 ] 1 / 2 ,                   χ ( k ) = | V p 2 ; p 1 ( q k ) | / V p 2 ; p 1 ( q k ) * ,
δ ( k ) = [ α 2 + 4 ( V ( k ) ) 2 ] 1 / 2 ,                               V ( k ) = ( ∑ ν = 1 k | V p 2 ; p 1 ( q ν ) | 2 ) 1 / 2 .
ℰ 1 , 2 ( k ) = 1 2 ( E ˜ p 1 + E ˜ p 2 + h ¯ Ω ± δ ( k ) ) ,
ℰ 3 ( k ) = ℰ 4 ( k ) = … = ℰ k + 1 ( k ) = E ˜ p 1 + h ¯ Ω
| ψ ^ n 〉 = ∑ m = 1 k + 1 S m ; n ( k ) * | ψ ˜ m 〉 .
ℋ ( 0 , 1 ) = ( E ˜ p 2 ( + 1 ) V p 2 ; p 1 ( 1 , 1 , − 1 ) 0 0 0 0 V p 2 ; p 1 ( 1 , 1 , − 1 ) * E ˜ p 1 + h ¯ Ω 0 0 0 0 0 0 E ˜ p 2 ( 0 ) V p 2 ; p 1 ( 1 , 1 , 0 ) 0 0 0 0 V p 2 ; p 1 ( 1 , 1 , 0 ) * E ˜ p 1 + h ¯ Ω 0 0 0 0 0 0 E ˜ p 2 ( − 1 ) V p 2 ; p 1 ( 1 , 1 , + 1 ) 0 0 0 0 V p 2 ; p 1 ( 1 , 1 , + 1 ) * E ˜ p 1 + h ¯ Ω )
𝒮 ( 0 , 1 ) = ( S ( 0 , 1 , + 1 ) 0 0 0 S ( 0 , 1 , 0 ) 0 0 0 S ( 0 , 1 , − 1 ) ) ,
ℋ ( 1 , 0 ) = ( E ˜ p 2 V p 2 ; p 1 ( 1 , 1 , + 1 ) V p 2 ; p 1 ( 1 , 1 , 0 ) V p 2 ; p 1 ( 1 , 1 , − 1 ) V p 2 ; p 1 ( 1 , 1 , + 1 ) * E ˜ p 1 ( + 1 ) + h ¯ Ω 0 0 V p 2 ; p 1 ( 1 , 1 , 0 ) * 0 E ˜ p 1 ( 0 ) + h ¯ Ω 0 V p 2 ; p 1 ( 1 , 1 , − 1 ) * 0 0 E ˜ p 1 ( − 1 ) + h ¯ Ω )
H ( 1 , 2 , ± 2 ) = ( E ˜ p 2 ( ± 2 ) V p 2 ; p 1 ( 1 , 3 , ∓ 3 ) V p 2 ; p 1 ( 1 , 3 , ∓ 2 ) V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) V p 2 ; p 1 ( 1 , 3 , ∓ 3 ) * E ˜ p 1 ( ∓ 1 ) + h ¯ Ω 0 0 0 V p 2 ; p 1 ( 1 , 3 , ∓ 2 ) * 0 E ˜ p 1 ( 0 ) + h ¯ Ω 0 0 V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) * 0 0 E ˜ p 1 ( ± 1 ) + h ¯ Ω 0 V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) * 0 0 0 E ˜ p 1 ( ± 1 ) + h ¯ Ω ) ,
H ( 1 , 2 , ± 1 ) = = ( E ˜ p 2 ( ± 1 ) V p 2 ; p 1 ( 1 , 3 , ∓ 2 ) V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) V p 2 ; p 1 ( 1 , 3 , 0 ) V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) V p 2 ; p 1 ( 1 , 1 , 0 ) V p 2 ; p 1 ( 1 , 3 , ∓ 2 ) * E ˜ p 1 ( ∓ 1 ) + h ¯ Ω 0 0 0 0 V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) * 0 E ˜ p 1 ( 0 ) + h ¯ Ω 0 0 0 V p 2 ; p 1 ( 1 , 3 , 0 ) * 0 0 E ˜ p 1 ( ± 1 ) + h ¯ Ω 0 0 V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) * 0 0 0 E ˜ p 1 ( 0 ) + h ¯ Ω 0 V p 2 ; p 1 ( 1 , 1 , 0 ) * 0 0 0 0 E ˜ p 1 ( ± 1 ) + h ¯ Ω )
H ( 1 , 2 , 0 ) = = ( E ˜ p 2 ( 0 ) V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) V p 2 ; p 1 ( 1 , 3 , 0 ) V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) V p 2 ; p 1 ( 1 , 1 , ∓ 0 ) V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) V p 2 ; p 1 ( 1 , 3 , ∓ 1 ) * E ˜ p 1 ( ∓ 1 ) + h ¯ Ω 0 0 0 0 0 V p 2 ; p 1 ( 1 , 3 , 0 ) * 0 E ˜ p 1 ( 0 ) + h ¯ Ω 0 0 0 0 V p 2 ; p 1 ( 1 , 3 , ± 1 ) * 0 0 E ˜ p 1 ( ± 1 ) + h ¯ Ω 0 0 0 V p 2 ; p 1 ( 1 , 1 , ∓ 1 ) * 0 0 0 E ˜ p 1 ( ∓ 1 ) + h ¯ Ω 0 0 V p 2 ; p 1 ( 1 , 1 , 0 ) * 0 0 0 0 E ˜ p 1 ( 0 ) + h ¯ Ω 0 V p 2 ; p 1 ( 1 , 1 , ± 1 ) * 0 0 0 0 0 E ˜ p 1 ( ± 1 ) + h ¯ Ω ) .

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