Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance.

Abstract

We develop a low-temperature theory of quasi-elastic secondary emission from a semiconductor quantum dot, the electronic subsystem of which is resonant with the confined longitudinal-optical (LO) phonon modes. Our theory employs a generalized model for renormalization of the quantum dot's energy spectrum, which is induced by the polar electron-phonon interaction. The model takes into account the degeneration of electronic states and allows for several LO-phonon modes to be involved in the vibrational resonance. We give solutions to three fundamental problems of energy-spectrum renormalization--arising if one, two, or three LO-phonon modes resonantly couple a pair of electronic states--and discuss the most general problem of this kind that admits an analytical solution. With these results, we solve the generalized master equation for the reduced density matrix, in order to derive an expression for the differential cross section of secondary emission from a single quantum dot. The obtained expression is then analyzed to establish the basics of optical spectroscopy for measuring fundamental parameters of the quantum dot's polaron-like states.

Full text

Page 1

Theory of quasi-elastic secondary
emission from a quantum dot in the
regime of vibrational resonance
Ivan D. Rukhlenko,1,∗Anatoly V. Fedorov,2Anvar S. Baymuratov,2
and Malin Premaratne1
1Advanced Computing and Simulation Laboratory (AχL), Department of Electrical and
Computer Systems Engineering, Monash University, Clayton, VIC 3800, Australia
2Center of Information Optical Technologies, Saint Petersburg State University of Information
Technologies, Mechanics and Optics, St. Petersburg 197101, Russia
∗ivan.rukhlenko@monash.edu
Abstract: We develop a low-temperature theory of quasi-elastic secondary
emission from a semiconductor quantum dot, the electronic subsystem
of which is resonant with the confined longitudinal-optical (LO) phonon
modes. Our theory employs a generalized model for renormalization of the
quantum dot’s energy spectrum, which is induced by the polar electron-
phonon interaction. The model takes into account the degeneration of
electronic states and allows for several LO-phonon modes to be involved in
the vibrational resonance. We give solutions to three fundamental problems
of energy-spectrum renormalization—arising if one, two, or three LO-
phonon modes resonantly couple a pair of electronic states—and discuss
the most general problem of this kind that admits an analytical solution.
With these results, we solve the generalized master equation for the reduced
density matrix, in order to derive an expression for the differential cross
section of secondary emission from a single quantum dot. The obtained ex-
pression is then analyzed to establish the basics of optical spectroscopy for
measuring fundamental parameters of the quantum dot’s polaron-like states.
© 2011 Optical Society of America
OCIS
(250.5590) Quantum -well, -wire, and -dot devices; (290.5870) Scattering, Rayleigh;
(300.6280) Spectroscopy, fluorescence and luminescence; (300.3700) Linewidth.
codes:
(160.4236) Nanomaterials; (300.6470)Spectroscopy,semiconductors;
References and links
1. A. V. Fedorov, I. D. Rukhlenko, A. V. Baranov, and S. Y. Kruchinin, Optical Properties of Semiconductor Quan-
tum Dots (Nauka, 2011).
2. T.Takagahara,“Electron-phononinteractions insemiconductor quantumdots,”inSemiconductorQuantumDots:
Physics, Spectroscopy and Applications, Y. Masumoto and T. Takagahara, eds. (Springer, 2002).
3. 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).
4. E.A.Muljarovand R.Zimmermann,“ExcitondephasinginquantumdotsduetoLO-phonon coupling:anexactly
solvable model,” Phys. Rev. Lett. 98, 187401 (2007).
5. K. Kojima and A. Tomita, “Influence of pure dephasing by phonons on exciton-photon interfaces: quantum
microscopic theory,” Phys. Rev. B 73, 195312 (2006).
6. 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).
7. 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).
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15459

Page 2

8. 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).
9. 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).
10. S. Sanguinetti, E. Poliani, M. Bonfanti, M. Guzzi, E. Grilli, M. Gurioli, and N. Koguchi, “Electron-phonon
interaction in individual strain-free GaAs/Al0.3Ga0.7As quantum dots,” Phys. Rev. B 73, 125342 (2006).
11. D. Valerini, A. Cret´ ı, M. Lomascolo, L. Manna, R. Cingolani, and M. Anni, “Temperature dependence of the pho-
toluminescence properties of colloidal CdSe/ZnS core/shell quantum dots embedded in a polystyrene matrix,”
Phys. Rev. B 71, 235409 (2005).
12. A. V. Fedorov and A. V. Baranov, “Exciton-vibrational interaction of the Fr¨ ohlich type in quasi-zero-size sys-
tems,” J. Exp. Theor. Phys. 83, 610–618 (1996).
13. 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, 1645–1648 (1995).
14. I. D. Rukhlenko and A. V. Fedorov, “Propagation of electric fields induced by optical phonons in semiconductor
heterostructures,” Opt. Spectrosc. 100, 238–244 (2006).
15. 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).
16. 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. B
71, 195310 (2005).
17. A. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots
embedded in doped heterostructures,” Phys. Rev. B 68, 205318 (2003).
18. 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).
19. 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).
20. 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).
21. S. Hameau, J. N. Isaia, Y. Guldner, E. Deleporte, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, and J. M. G´ erard,
“Far-infrared magnetospectroscopy of polaron states in self-assembled InAs/GaAs quantum dots,” Phys. Rev. B
65, 085316 (2002).
22. 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).
23. S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaˆ ıtre, and J. M. G´ erard, “Strong
electron-phonon coupling regime in quantum dots: evidence for everlasting resonant polarons,” Phys. Rev. Lett.
83, 4152–4155 (1999).
24. P. Palinginis, S. Tavenner, M. Lonergan, and H. Wang, “Spectral hole burning and zero phonon linewidth in
semiconductor nanocrystals,” Phys. Rev. B 67, 201307 (2003).
25. E. A. Chekhovich, A. B. Krysa, M. S. Skolnick, and A. I. Tartakovskii, “Direct measurement of the hole-nuclear
spin interaction in single InP/GaInP quantum dots using photoluminescence spectroscopy,” Phys. Rev. Lett. 106,
027402 (2011).
26. P. Fallahi, S. T. Yılmaz, and A. Imamo˘ glu, “Measurement of a heavy-hole hyperfine interaction in InGaAs quan-
tum dots using resonance fluorescence,” Phys. Rev. Lett. 105, 257402 (2010).
27. 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).
28. O. Verzelen, R. Ferreira, and G. Bastard, “Excitonic polarons in semiconductor quantum dots,” Phys. Rev. Lett.
88, 146803 (2002).
29. T. Stauber, R. Zimmermann, and H. Castella, “Electron-phonon interaction in quantum dots: a solvable model,”
Phys. Rev. B 62, 7336–7343 (2000).
30. T. Inoshita and H. Sakaki, “Density of states and phonon-induced relaxation of electrons in semiconductor quan-
tum dots,” Phys. Rev. B 56, R4355–R4358 (1997).
31. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Exciton-phonon coupling in semiconductor quantum dots: resonant
Raman scattering,” Phys. Rev. B 56, 7491–7502 (1997).
32. A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum
dots,” Phys. Rev. B 54, 8627–8632 (1996).
33. O. Madelung, M. Schultz, and H. Weiss, eds., Semiconductors. Physics of Group IV Elements and III-V Com-
pounds, Landolt-B¨ ornstein, New Series, Group III, Vol. 17, Pt. a (Springer-Verlag, 1982).
34. A. I. Anselm, Introduction to Semiconductor Theory (Prentice-Hall, 1978).
35. A. S. Davydov, Theory of Solid State (Nauka, 1976).
36. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Company,
1968).
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15460

Page 3

37. 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).
38. A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance sec-
ondary emission in semiconductor quantum dots,” Opt. Spectrosc. 93, 52–60 (2002).
39. K. Blum, Density Matrix Theory and Its Applications (Plenum Press, 1981).
40. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
41. E. A. Zibik, T. Grange, B. A. Carpenter, R. Ferreira, G. Bastard, N. Q. Vinh, P. J. Phillips, M. J. Steer, M. Hop-
kinson, J. W. Cockburn, M. S. Skolnick, and L. R. Wilson, “Intersublevel polaron dephasing in self-assembled
quantum dots,” Phys. Rev. B 77, 041307 (2008).
42. A. V. Fedorov, A. V. Baranov, and Y. Masumoto, “Acoustic phonon problem in nanocrystaldielectric matrix
systems,” Solid State Commun. 122, 139–144 (2002).
43. A. V. Fedorov and S. Y. Kruchinin, “Acoustic phonons in a quantum dot-matrix system: hole-burning spec-
troscopy,” Opt. Spectrosc. 97, 394–402 (2004).
44. I.D.Rukhlenko,D.Handapangoda,M.Premaratne,A.V.Fedorov,A.V.Baranov,andC.Jagadish,“Spontaneous
emission of guided polaritons by quantum dot coupled to metallic nanowire: beyond the dipole approximation,”
Opt. Express 17, 17570–17581 (2009).
45. A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, T. S. Perova, and K. Berwick, “Quantum dot energy relaxation
mediated by plasmon emission in doped covalent semiconductor heterostructures,” Phys. Rev. B 76, 045332
(2007).
46. S. Y. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Perova, and K. Berwick, “Double quantum dot photolumi-
nescence mediated by incoherent reversible energy transport,” Phys. Rev. B 81, 245303 (2010).
47. D. Gammon, N. H. Bonadeo, G. Chen, J. Erland, and D. G. Steel, “Optically probing and controlling single
quantum dots,” Physica E (Amsterdam) 9, 99–105 (2001).
48. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in
In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308 (2002).
1. Introduction
The coupling between electrons and phonons is one of the most influential interactions occur-
ring inside semiconductor quantum dots, which results in the scattering of electrons, holes, and
excitons [1,2]. In the photoluminescence experiments performed on quantum dots, this cou-
pling is often manifested in enhanced dephasing rates of spectroscopic transitions [3–6] and
reduced lifetimes of electronic excitations [7–11]. Electron-phonon interaction may also cause
a significant modification of the quantum dot’s energy spectrum, known to be especially promi-
nent in the regime of vibrational resonance [12,13]. This regime is realized where the energy
of a phonon coincides with the energy spacing between a pair of states of the quantum dot’s
electronic subsystem. Renormalization of the electron-phonon energy spectrum in this situa-
tion leads to the formation of hybrid energy states characterizing a new class of polaron-like
excitations, which are similar to the polarons in a bulk semiconductor.
Although quantum dots are generally affected by a multitude of phonon-related elemen-
tary excitations—residing both inside them and in various distant parts of the host heterostruc-
tures [14–17]—the renormalizations of the quantum dot’s energy spectra, that have been ex-
perimentally observed so far, predominantly resulted from polar interaction with longitudinal-
optical (LO) phonons [13,18–23]. Despite a relatively high strength of polar interaction com-
pared to other types of electron-phonon coupling [1, 2], the energy splitting of polaron-like
states induced by the vibrational resonance is typically of the order of few millielectronvolts.
Such a fine structure of the quantum dot’s energy spectra can be experimentally resolved only
at cryogenic temperatures, when both dephasing rates of optical transitions and spectral widths
of energy levels are relatively small. Another challenge that hampers observation of singular-
ities in the quantum dots’ optical spectra in the regime of vibrational resonance stems from
the relatively broad size distribution of quantum dots, which is present even in the most finely
fabricated samples. Quantum dots of different sizes emit light of different wavelengths leading
to the inhomogeneous broadening of spectroscopic transitions. This obstacle can be avoided
by applying size-selective optical methods, such as persistent spectral hole burning in an in-
homogeneously broadened light absorption profile [20,24], one- and two-photon excited pho-
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15461

Page 4

toluminescence [13,22], or a single-quantum-dot spectroscopy [25,26]. Using these methods
for studying the effect of vibrational resonance requires an adequate theoretical treatment of
the underlying optical phenomena, taking into account specific features characteristic to the
quantum dot’s polaron-like spectrum.
Several physical theories of polaron-like excitations in semiconductor quantum dots were
developed by different research groups [13,22,23,27–30], and subsequently employed for the
interpretation of experimental data. For example, the first theory proposed in Ref. [13] helped
to explain the shape of the excitonic luminescence spectra obtained under the resonant size-
selective excitation of CuCl-nanocrystals. The model of Ref. [23] was applied for the analy-
sis of data on intraband transitions obtained with far-infrared magnetospectroscopy for self-
assembled quantum dots made of InAs [18,19,21,23]. Another model, reported in Ref. [22],
proved useful in interpreting the effect of renormalization for the lowest excitonic states in
CuCl-quantum dots embedded into NaCl-matrix, which were studied with the spectroscopy of
secondary emission under two-photon resonance excitation. Unlike theoretical treatment of the
vibrational resonance itself, the mathematical description of its manifestation in optical spectra
is developed rather poorly, and the information these spectra are capable of providing is not
perfectly understood. To the best of our knowledge, the only technique that has been placed
on solid theoretical ground up to date is the persistent spectral hole burning in an inhomoge-
neously broadened light absorption profile [27]. Thus, a fundamental theory of quantum dot
optical response allowing for the renormalization of the electron-LO-phonon energy spectrum
is still in demand.
In this work, we present a comprehensive theoretical study of quasi-elastic secondary emis-
sion from a single quantum dot that exhibits a vibrational resonance. The paper is organized
as follows. In Section 2, we outline the model for renormalization of the quantum dot’s energy
spectrum due to the resonant polar interaction with LO phonons. This model generalizes our
previous results [22, 27], by extending them to degenerate electronic levels and an arbitrary
number of LO-phonon modes involved in the vibrational resonance. Section 3 is devoted to the
solution of the generalized master equation for the reduced density matrix, and calculation of
the differential cross section of the quasi-elastic secondary emission at low temperatures. In
this section, we also analyze the effect of spectral filtration of secondary emission due to the
finite bandwidth of the photon detector, and present a general expression for the intensity of
secondary emission in the case of vibrational resonance between 3-fold and 9-fold degener-
ate electronic states. In Section 4, we illustrate the potential of optical spectroscopy based on
probing polaron-like states, by the example of a spherical quantum dot in the regime of strong
confinement. Specifically, we consider two different scenarios in which vibrational resonance
occurs between either a pair of lowest degenerate or a pair of lowest nondegenerate electronic
states. In Section 5, we summarize our results and conclude the paper.
2. Theory of vibrational resonance in a semiconductor quantum dot
We start our theoretical treatment of vibrational resonance arising in a quantum dot due to the
polar electron-phonon interaction, by introducing the Hamiltonian formalism for the physical
system being of interest.
2.1.
The physical system to be analyzed is composed of a semiconductor quantum dot, classical
excitation light, and vacuum (quantum electromagnetic) radiation field. It is convenient to take
the total Hamiltonian of this system to be of the form
Hamiltonian formalism
H(t) = H0+Hint(t),
(1)
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15462

Page 5

where
H0=∑
p
Epa+
pap+∑
q
¯ hΩqb+
qbq+∑
λ
¯ hωλc+
λcλ
(2)
gives the contribution from noninteracting electron-hole pairs, LO phonons, and photons, while
Hint(t) = He,ph+He,L(t)+He,λ
(3)
describes the interaction of electron-hole pairs (e) with phonons (ph), excitation light (L),
and radiation field (λ). The notations on the right-hand side of Eq. (2) are as follows: a+
(ap), b+
phonons, and photons, respectively; Ep= Eg+Epe+Ephis the energy of electron-hole pair
state |p⟩ ≡ |pe;ph⟩, which depends on the quantum numbers of electron (pe) and hole (ph),
and the band gap, Egof the bulk semiconductor; ¯ hΩqand ¯ hωλare the energies of LO-phonon
mode q and photon mode λ.
For simplicity, we restrict our analysis to the quantum dot in the regime of strong confine-
ment, and describe its electronic subsystem using the two-band approximation [1,2]. Although
polar electron-phonon interaction in quantum dots is relatively strong, it does not excite elec-
trons from the valence (v) band to the conduction (c) band, but induces only intraband tran-
sitions between the different states of electron-hole pairs. This fact allows electron-phonon
Hamiltonian to be represented as
∑
p1,p2̸=0
where the second summation does not extend over the quantum numbers p1= p2= 0, repre-
senting vacuum of electron-hole pairs, and the matrix element of polar electron-phonon inter-
action is given by the expression
(
in which −e is the charge of the electron; φ(q)
electric potential induced by the LO-phonon mode q; δpe2,pe1and δph2,ph1are the products of
Kronecker deltas for the quantum numbers of electron and hole. In writing Eq. (5) in the form
shown, we assumed that the spatial confinement provided by a quantum dot for electrons, holes,
and phonons is approximated suitably enough by an infinitely deep potential well.
An explicit form of the matrix elements in Eq. (5) depends on the specific shape of the
quantum dot [1,2]. For a spherical quantum dot of radius R, which will be used to illustrate our
results in Section 4, they can be expressed compactly as [31]
√
p
q(bq), and c+
λ(cλ) are the creation (annihilation) operators of electron-hole pairs, LO
He,ph=∑
q
(
V(q)
p2,p1a+
p2ap1bq+H.c.
)
,
(4)
V(q)
p2,p1= −e
φ(q)
pe2,pe1δph2,ph1−φ(q)
pe2,pe1and φ(q)
ph1,ph2δpe2,pe1
)
,
(5)
ph1,ph2are the matrix elements of the
φ(q)
pe2,pe1= φ(q)
ph2,ph1=
4(2lq+1)(2lp1+1)¯ hΩq
(2lp2+1)ε∗R
Inqlq
np2lp2,np1lp1Clp20
lq0,lp10Clp2mp2
lqmq,lp1mp1,
(6)
where ε∗=(1/ε0−1/ε∞)−1, ε0and ε∞are the low- and high-frequency dielectric permittivities
of the bulk semiconductor,
∫1
and jl(z) is the spherical Bessel function of the first kind, with ξnlbeing its nth zero [i.e.,
jl(ξnl) = 0]. Clebsch-Gordan coefficients Clp2mp2
lection rules for electron-electron and hole-hole transitions. The quantities na, la, and ma
Inqlq
np2lp2,np1lp1=
0
dx
x2jlq(ξnqlqx) jlp2(ξnp2lp2x) jlp1(ξnp1lp1x)
ξnqlqjlq+1(ξnqlq) jlp2+1(ξnp2lp2) jlp1+1(ξnp1lp1),
lqmq,lp1mp1appearing in this expression set the se-
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15463

Page 6

(a = q,p1,p2) denote the principal quantum number, angular momentum, and its projection
for LO phonon (a = q), electron (a = pe1,pe2), or hole (a = ph1,ph2).
The second and third terms in Eq. (3) account for the two mechanisms of interband transi-
tions: (i) generation of electron-hole pairs through absorption of excitation light by the quantum
dot’s electronic subsystem; and (ii) recombination of electron-hole pairs with emission of pho-
tons due to their interaction with the vacuum radiation field. The Hamiltonians describing these
mechanisms can be represented in the forms [31,32]
(
He,λ=∑
p̸=0
He,L(t) =∑
p̸=0
(
V(L)
p,0(t)a+
p+H.c.
)
,
λ∑
i¯ hgλV(λ)
p,0a+
pcλ+H.c.
)
,
where V(L)
gλ=√2πωλ/(ε∞¯ hV),V is the normalization volume, and
p,0(t) = V(L)
p,0ϕ(t)exp(−iωLt), ϕ(t) is the complex envelope of classical optical field,
V(η)
p,0= −e⟨uc|r·ˆ eη|uv⟩δpe,ph
(7)
is the matrix element of the electric dipole moment operator −er, calculated on Bloch functions
ucand uvat the Brillouin zone center, for excitation light (η =L) or emitted photon (η =λ) of
frequency ωηand polarization vector ˆ eη. Kronecker’s delta following the matrix element shows
that the dipole-allowed generation and recombination of electron-hole pairs occur only if the
quantum numbers of electron and hole are the same. In what follows, we shall consider only
quantum dots made of semiconductors with either Tdand Ohsymmetry. In this case, matrix
element in Eq. (7) is expressed through the Kane’s parameter, P of bulk material as [33,34]
⟨uc|r·ˆ eη|uv⟩ ≡√2Zcv=√2P/Eg.
2.2. Formation of polaron-like states
In the case of no interaction occurring between electrons and phonons inside a quantum dot,
a number of the total system “electrons plus phonons” energy states may be degenerate. This
happensinthesituationofvibrationalresonance,whereenergyofaparticularLO-phononmode
is close to the energy spacing between a pair of electron-hole states, i.e., when Ep2−Ep1≈
¯ hΩq. The degeneracy is removed by the polar electron-phonon interaction, which results in the
splitting of degenerate levels into two or more polaron-like states with different energies [35].
The energies and wave functions of the polaron-like states can be found by diagonalizing
Hamiltonian in Eq. (1) with respect to the electron-phonon interaction. We begin this procedure
by employing the unitary transformation
[
U = exp
−∑
p,q
(
Φ(q)
p,pbq−H.c.
)
a+
pap
]
with Φ(q)
the linear part of the electron-phonon interaction in the transformed Hamiltonian, yields [27]:
p,p=V(q)
p,p/(¯ hΩq) to eliminate the diagonal part of electron-hole coupling. Keeping only
˜ H(t) =U+H(t)U =˜ H0+˜ He,ph+˜ He,L(t)+˜ He,λ,
(8)
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15464

Page 7

where
˜ H0=∑
p
˜Epa+
pap+∑
q
¯ hΩqb+
qbq+∑
λ
(
¯ hωλc+
λcλ,
˜Ep= Ep−∑
q
¯ hΩq
??Φ(q)
p,p
??2,
(9a)
˜ He,ph=∑
{
{
q
∑
p1̸=p2̸=0
[
[
V(q)
p2,p1a+
p2ap1bq+H.c.
)
,
(9b)
˜ He,L(t) =∑
p̸=0
V(L)
p,0(t)
1+∑
q
(
(
Φ(q)
p,pbq−H.c.
)]
)]
a+
p+H.c.
}
,
(9c)
˜ He,λ=∑
λ∑
p̸=0
i¯ hgλV(λ)
p,0
1+∑
q
Φ(q)
p,pbq−H.c.
a+
pcλ+H.c.
}
.
(9d)
It is seen from Eq. (9a) that the vibrational resonance occurs in the event that the energy of
one or several LO phonons approximately coincide with the energy spacing of electron-hole
states shifted due to the electron-phonon interaction, i.e.,˜Ep2−˜Ep1≈ ¯ hΩq. To complete the
diagonalization procedure of the Hamiltonian˜ H(t), we exclude from Eq. (9b) electron-phonon
interaction coupling the states being in vibrational resonance. This can be done with an appro-
priately constructed unitary operator S, via the transformation
ˆ H(t) = S+˜ H(t)S.
(10)
The eigenstates | ˆ ψ⟩ of the resulting Hamiltonian ˆ H(t) are related to the eigenstates | ˜ ψ⟩ of
the Hamiltonian˜ H(t) through the same operator S via | ˆ ψ⟩ = S+| ˜ ψ⟩. The matrix representing
operator S is found by solving a standard eigenvalue problem, ˆ H| ˆ ψ⟩ = E| ˆ ψ⟩; in the general
case that the resonant condition is simultaneously satisfied for k LO-phonon modes, it requires
solving the (k+1)-degree algebraic equation.
2.3.
Let us now consider those eigenvalue problems associated with Hamiltonian˜ H that admit ana-
lytical solutions. The simplest problems of this type arise where two nondegenerate states |˜Ep2⟩
and |˜Ep1⟩ are coupled via one (q1), two (q1,q2), or three (q1,q2,q3) LO-phonon modes. Us-
ing Eqs. (9a) and (9b) and assuming that the temperature of the system is low enough for the
phonon modes to be unoccupied, we first draw the matrices that represent the electron-phonon
part of the Hamiltonian (9) in these three cases,
(
p2,p1




V(q3)∗
p2,p1
0
Vibrational resonance involving one, two, and three phonon modes
˜ H(1)
e,ph=
˜Ep2
V(q1)∗
V(q1)
p2,p1
˜Ep1+ ¯ hΩq1
V(q1)
p2,p1
˜Ep1+ ¯ hΩq1
0
V(q1)
p2,p1
˜Ep1+ ¯ hΩq1
0
)
,
(11a)
˜ H(2)
e,ph=

˜Ep2
V(q1)∗
p2,p1
V(q2)∗
p2,p1
V(q2)
p2,p1
0
˜Ep1+ ¯ hΩq2
V(q2)
p2,p1
0
˜Ep1+ ¯ hΩq2
0

,
p2,p1
0
0

(11b)
˜ H(3)
e,ph=



˜Ep2
V(q1)∗
p2,p1
V(q2)∗
p2,p1
V(q3)
˜Ep1+ ¯ hΩq3




.
(11c)
The diagonal elements of matrix˜ H(1)
|˜Ep1;1q1⟩ = |˜Ep1⟩|1q1⟩, where kets |0q1⟩ and |1q1⟩ stand for the vacuum of the phonon modes
e,phgive the energies of states |˜Ep2;0q1⟩ = |˜Ep2⟩|0q1⟩ and
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15465

Page 8

E
E
(k)
i
~
~
~
|E ;1
1 p
|E ;0
2 p
1
2
p
LO
E
(a)
hΩ
1
−
E
(1)
+
E
(1)
(c)
δ
−
E
(k)
+
(k)
(d)
δ /2
k
δ /2
k
1 p
E
(b)
=
q
1 q
~
Fig. 1. Formation of polaron-like states from [(a), (b)] a pair of states˜Ep2and˜Ep1coupled
via LO phonon. (c) States E(1)
+
−
˜Ep2and˜Ep1via a single phonon mode. (d) States E(k)
originate from two nondegenerate states˜Ep2and˜Ep1coupled via k phonon modes, or from
either k-fold degenerate state˜Ep2and nondegenerate state˜Ep1or nondegenerate state˜Ep2
and k-fold degenerate state˜Ep1coupled via a single phonon mode.
and E(1)
arise due to the coupling of nondegenerate states
+, E(k)
i
(i = 3,...,k+1), and E(k)
−
and one LO phonon in the mode q1[see Figs. 1(a) and 1(b)]. Being perturbed by electron-
phonon interaction, these states change their energies and wave functions according to the re-
lations
E(1)
± =1
|˜Ep1;1q1⟩,
=V(q1)
2
(˜Ep2+˜Ep1+ ¯ hΩq1±δ1
|E(1)
),
(12a)
|E(1)
where c(1)
expressions
+⟩ = c(1)
1|˜Ep2;0q1⟩+c(1)∗
= 2V(q1)
3
−⟩ = c(1)
2|˜Ep2;0q1⟩−c(1)∗
4
c(1)
1|˜Ep1;1q1⟩, (12b)
3
p2,p1/∆1, c(1)
4
p2,p1/V1, and the rest of parameters are given by the generic
c(k)
1=
˜Ep2−˜Ep1− ¯ hΩqk+δk
∆k
√(˜Ep2−˜Ep1− ¯ hΩqk+δk
and c(k)
2
do not contain LO phonons. The Hamiltonian in Eq. (11a) is reduced to the diagonal form
(
,
c(k)
2=2Vk
∆k
,
δk=
√(˜Ep2−˜Ep1− ¯ hΩqk
Vk=
∑
i=1
)2+4V2
.
k,
(13a)
∆k=
)2+4V2
k,
(
k
???V(qi)
p2,p1
???
2)1/2
(13b)
Here, c(k)
1
give the probability amplitudes that the polaron-like states |E(1)
+⟩ and |E(1)
−⟩
ˆ H(1)
e,ph=
E(1)
+
0
0
E(1)
−
)
by the matrix
S1=
(
c(1)
1
c(1)∗
3
c(1)
2
−c(1)∗
4
c(1)
1
)
.
We see that the electron-phonon interaction completely removes the degeneracy of states
|˜Ep2;0q1⟩ and |˜Ep1;1q1⟩. The formation of new energy-shifted levels of the polaron-like states
|E(1)
Closed expressions for energies and wave functions of the polaron-like states corresponding
to matrices in Eqs. (11b) and (11c) can also be found. Generally, these expressions are rather
+⟩ and |E(1)
+⟩ is illustrated in Figs. 1(a)–1(c).
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15466

Page 9

cumbersome, because their derivation amounts to solving cubic and quartic equations [36]. We,
therefore, restrict ourselves to a specific situation of dispersionless phonon modes, in which
simplified solutions to both eigenvalue problems can be obtained.
Assuming that the energies of phonons in the modes q1and q2are equal to the energy,
¯ hΩLOof the bulk LO phonon at the Brillouin zone center, we find the polaron-like states of the
Hamiltonian˜ H(2)
e,phto be
E(2)
± =1
|E(2)
2
(˜Ep2+˜Ep1+ ¯ hΩLO±δ2
),
E(2)
3
=˜Ep1+ ¯ hΩLO,
(14a)
+⟩ = c(2)
1|˜Ep2;0q⟩+c(2)∗
|E(2)
−⟩ = c(2)
and c(2)
2
= V(q1)
5
˜ H(2)
6
|˜Ep1;1q1⟩+c(2)∗
4|˜Ep1;1q2⟩,
1|˜Ep1;1q1⟩−c(2)∗
3
|˜Ep1;1q2⟩,
(14b)
3⟩ = −c(2)
2|˜Ep2;0q⟩−c(2)∗
are given by Eq. (13) after the replacement Ωqk→ ΩLO, c(2)
p2,p1/V2, c(2)
6
e,phto the diagonal formˆ H(2)


Figure 1(d) shows how the energy level that was initially triply degenerate splits into three
nondegenerate levels given in Eq. (14a).
Finally, by setting the energies of all phonon modes in Eq. (11c) alike, ¯ hΩq1= ¯ hΩq2=
¯ hΩq3= ΩLO, we may represent the polaron-like states as
(˜Ep2+˜Ep1+ ¯ hΩLO±δ3
|E(3)
|E(3)
|E(3)
|E(3)
where c(2)
c(3)
6
78
V(q2)
is realized by the matrix


c(3)∗
4
0
It is seen that electron-phonon interaction is unable to completely remove the degeneracy of
the four states in the present case. This is a consequence of the initial triple degeneracy of the
LO-phonon mode.
5|˜Ep1;1q1⟩+c(2)
c(2)
(14c)
|E(2)
4
5
c(2)
1|˜Ep1;1q2⟩,
(14d)
where c(2)
c(2)
4
13
= 2V(q2)
p2,p1/∆2,
= V(q2)
p2,p1/V2, c(2)
e,ph= diag(E(2)

= 2V(q1)
+,E(2)
p2,p1/∆2, and q = (q1,q2). The matrix that puts
3,E(2)
−
c(2)
2
−c(2)
c(2)
4
5
)is
S2=
c(2)
1
c(2)∗
6
c(2)∗
3
0
5
−c(2)
−c(2)
1c(2)∗
1c(2)∗
4

.

E(3)
± =1
+⟩ = c(3)
2
1|˜Ep2;0q⟩+c(3)∗
3⟩ = −c(3)
4⟩ = −c(3)
2|˜Ep2;0q⟩−c(3)∗
1and c(2)
= V(q1)
= V(q3)
p2,p1/V3, and q=(q1,q2,q3). The diagonalization˜ H(3)
),
E(3)
3
= E(3)
4
=˜Ep1+ ¯ hΩLO,
(15a)
9
|˜Ep1;1q1⟩+c(3)∗
5|˜Ep1;1q1⟩+c(3)
|˜Ep1;1q1⟩−c(3)
10c(3)
3
|˜Ep1;1q2⟩+c(3)∗
6|˜Ep1;1q2⟩,
|˜Ep1;1q2⟩+c(3)
11c(3)
4
|˜Ep1;1q3⟩,
(15b)
(15c)
7c(3)∗
6
7c(3)∗
5
8|˜Ep1;1q3⟩,
c(3)
(15d)
−⟩ = c(3)
1|˜Ep1;1q1⟩−c(3)∗
3=2V(q2)
p2,p1/V3, c(3)
1|˜Ep1;1q2⟩−c(3)∗
p2,p1/∆3, c(3)
= V2/V3, c(3)
9
e,ph→ˆ H(3)
71|˜Ep1;1q3⟩,
p2,p1/∆3, c(3)
p2,p1/∆3, c(3)
e,ph=diag(E(3)
c(3)
2
−c(3)
−c(3)
−c(3)
(15e)
2are given by Eq. (13), c(3)
p2,p1/V2, c(3)
4=2V(q3)
= 2V(q1)
5=V(q2)
p2,p1/V3, c(3)
+,E(3)
p2,p1/V2,
11=
4,E(3)
10= V(q1)
3,E(3)
−
)
S3=



c(3)
1
c(3)∗
9
c(3)∗
3
00
−c(3)
c(3)
6
5
−c(3)
−c(3)
7c(3)∗
7c(3)∗
c(3)
8
6
1c(3)∗
1c(3)∗
1c(3)∗
10
5
11
7




.
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15467

Page 10

By looking at Eqs. (12a), (14a), and (15a), it may be noted that they admit a straightfor-
ward generalization to the situation in which an arbitrary number of degenerate phonon modes
(q1,q2,...,qk)areresonantwiththequantumdot’selectronicsubsystem.Theresultingenergies
of the polaron-like states are of the form
E(k)
± =1
2
(˜Ep2+˜Ep1+ ¯ hΩLO±δk
),
E(k)
3
= E(k)
4
= ... = E(k)
k+1=˜Ep1+ ¯ hΩLO.
(16)
Thus, whenever the k degenerate LO-phonon modes couple a pair of quantum-dot electronic
states, it results in the formation of two nondegenerate and one (k−1)-degenerate polaron-like
states.
2.4.
We saw in the previous two subsections that vibrational resonance modifies the interaction
of electrons and holes residing in a quantum dot, with excitation light and emitted photons.
This fact is described mathematically by the transformations of the Hamiltonians˜ He,Land˜ He,λ
given in Eqs. (8) through (10). In order to evaluate the efficiency of low-temperature secondary
emission in the presence of vibrational resonance with one, two, or three LO-phonon modes,
the following matrix elements of the transformed Hamiltoniansˆ He,η(η = L,λ) are required:
Polaron-photon interaction
ˆ H(1)
e,η=



0
0
0
0
c(1)
c(1)
1V(η)
2V(η)
0
c(2)
p2,0
p2,0
c(1)
1V(η)∗
p2,0
0
0
0
c(2)
p2,0
0
0
0
0
c(3)
p2,0
c(1)
2V(η)∗
p2,0

,
p2,0

(17a)
ˆ H(2)
e,η=





0
0
0
0
0
0
0
1V(η)
0
c(2)
0
c(3)
2V(η)
p2,0
1V(η)∗
c(2)
2V(η)∗
p2,0




,
p2,0
(17b)
ˆ H(3)
e,η=







0
0
0
0
0
0
0
0
0
0
0
0
0
0
1V(η)
0
0
c(3)
0
2V(η)
p2,0
1V(η)∗
c(3)
2V(η)∗
p2,0







.
(17c)
Here the rows from top to bottom, and columns from left to right, correspond to the states
|E(k)
η = λ; |0(k)⟩ denotes the vacuum of polaron-like excitations, while |0λ⟩ and |1λ⟩ stand for
zero and one photons in the mode λ. Equations (17) are obtained with the same S-matrices
that were used to diagonalize Hamiltonians ˜ H(1)
approximation, direct generation or recombination of electron-hole pairs can occur only for
states |E(k)
ph2).
By analogy, the matrices representing transformed electron-photon Hamiltonians for an ar-
+⟩|0λ⟩,...,|E(k)
−⟩|0λ⟩, |0(k)⟩|0λ⟩ for η = L, and |E(k)
+⟩|0λ⟩,...,|E(k)
−⟩|0λ⟩, |0(k)⟩|1λ⟩ for
e,ph, ˜ H(2)
e,ph, and ˜ H(3)
e,ph. They show that in dipole
+⟩ and |E(k)
−⟩, provided the quantum numbers of electron and hole coincide (pe2=
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15468

Page 11

bitrary number of phonon modes can be obtained,
ˆ H(k)
e,η=











0
0
...
0
0
0
0
...
0
0
0
···
···
0
0
...
0
0
0
0
0
...
0
0
c(k)
1V(η)
0
...
0
c(k)
0
p2,0
···
···
···
2V(η)
p2,0
c(k)
1V(η)∗
p2,0
c(k)
2V(η)∗
p2,0











.
(18)
Notice that these matrices are Hermitian.
2.5.
Since the quantum dot usually has a large number of electronic states and confined LO-phonon
modes, multiple vibrational resonances may occur between them. Suppose that N vibrational
resonances simultaneously occur for the different electron-hole states of a single quantum dot,
such that the first pair of states is resonant to k1phonon modes, the second pair is resonant
to k2modes, and so on. By combining interaction matrices ˜ H(kj)
resonances, we can build an electron-phonon Hamiltonian of the entire system,



This quasi-diagonal block Hamiltonian allows for various types of degeneration of the electron-
hole pairs’ states and LO-phonon modes. Its diagonalization is realized with an S-matrix of a
similar block structure,


00
Analytically solvable eigenvalue problems
e,phdescribing the individual
˜
He,ph=




˜ H(k1)
e,ph
0
...
0
0
···
···
0
0
...
˜ H(k2)
e,ph
...
0
···
˜ H(kN)
e,ph







.
S =


˜

Sk1
0
...
0
···
···
0
0
...
Sk2
...
···
SkN




.
Diagonalization of the Hamiltonian
special case of any N vibrational resonances involving no more than three phonon modes, i.e.,
if max(k1,k2,...,kN) ≤ 3. It becomes particularly simple if, in addition, all modes are of the
same frequency. In this case, renormalization of electron-hole spectra is described by Eqs. (12)–
(16). The matrix H(k)
Eq. (18); its last row can be written as
He,phwith matrix S can be performed analytically in the
e,ηof the polaron-photon interaction is self-adjoint and similar in form to
(
c(k1)
1
V(η)∗
p2,0,0,...,0
? ?? ?
k1−1
,c(k1)
2
V(η)∗
p2,0,c(k2)
1
V(η)∗
p2,0,0,...,0
? ?? ?
k2−1
,c(k2)
2
V(η)∗
p2,0,...
...,c(kN)
1
V(η)∗
p2,0,0,...,0
? ?? ?
kN−1
,c(kN)
2
V(η)∗
p2,0,0
)
.
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15469

Page 12

3.Quantum dot secondary emission
Let us next mathematically describe the process of secondary emission from a single quantum
dot exhibiting vibrational resonance with confined LO-phonon modes. We assume weak optical
excitation of the system—to ensure that the interband transitions are not saturated—and restrict
ourselves, as before, to the case of low temperatures (T ≪ ¯ hΩq), in order to avoid turning the
phonon subsystem out of its state of thermodynamic equilibrium. In this situation, it is conve-
nient to describe the phenomenon of secondary emission within the density matrix formalism.
The dynamics of spectroscopic transitions in the considered quantum system is then governed
by the generalized master equation for the reduced density matrix, ρ(t) [37–40]
∂ρµν(t)
∂t
=1
i¯ h
[ˆ H(t),ρ(t)]
µν−γµνρµν(t)+δµν ∑
ν′̸=ν
ζνν′ρν′ν′(t),
(19)
whereˆ H(t) is the transformed Hamiltonian given in Eq. (10), γµν= γνµ= (γµµ+γνν)/2+ ˆ γµν
for µ ̸= ν is the coherence relaxation rate, γµµ= τ−1
lifetime) of state µ, and ˆ γµν= ˆ γνµ is the pure dephasing rate. The last term on the right-
hand side of the master equation accounts for the transitions |ν′⟩ → |ν⟩ due to the thermal
interaction with a bath through the relaxation parameters ζνν′. We allow for this interaction in
our subsequent discussion of quasi-elastic secondary emission, but neglect it in Section 4.
µ
is the population relaxation rate (inverse
3.1.
In order to calculate the intensity of secondary emission for the case in which vibrational res-
onance of the k phonon modes occurs between the nondegenerate states of the electron-hole
pairs, we take the four eigenstates of the Hamiltonian˜ H0,
Secondary emission in the case of resonance with nondegenerate electronic states
|1⟩ = |0(k)⟩|0q⟩|0λ⟩, |2⟩ = |E(k)
These eigenstates form the minimal sufficient basis for our system, since the rest of the polaron-
like states given in Eq. (16) are not involved in the direct dipole-allowed optical transitions. The
absence of the degeneration with reference to the spherical quantum dot adopted in our model,
implies that the angular momenta of the electrons and holes must be zero, le1=lh1=le2=lh2=
0.
Through use of Eq. (18), it is easy to show that the emission of photons of frequency ωλdue
to the annihilation of the polaron-like excitations is characterized by the rate [37,38]
(
where c(k)
12
are to be found from the master equation.
We now solve Eq. (19) for the case of stationary excitation, ϕ(t) = EL, using the method
of perturbation theory and assuming the inverse lifetime of the ground state |1⟩ to be zero.
The solution we obtain leads to the following expression for the differential cross section of
secondary emission per unit solid angle, dΘ and unit frequency interval, dωλ:
(
n=2
(c(k)
12 13
n=2
+⟩|0q⟩|0λ⟩, |3⟩ = |E(k)
−⟩|0q⟩|0λ⟩, |4⟩ = |0(k)⟩|0q⟩|1λ⟩.
W =1
i¯ h
[ˆ H(t),ρ(t)]
and c(k)
44= −gλ
c(k)
1V(λ)∗
p2,0ρ24(t)+c(k)
2V(λ)∗
p2,0ρ34(t)+c.c.
)
,
are given in Eq. (13), while the density matrix elements ρ24(t) and ρ34(t)
d2σ
dΘdωλ
=Vε3/2
4(πc)3IL
∞ ¯ hω3
λ
W = δpe2,ph2C(ωλ)
3
∑
2ˆ γ1n
γnn
(c(k)
(c(k)
)2
n−1
Ln+γ2
)4
1n
????
∆2
2γ1n
λn+γ2
∆2
1n
+2ζ32γ12
γ22γ33
1c(k)
∆2
2
)2
L2+γ2
2γ13
λ3+γ2
∆2
+
????
3
∑
n−1
∆Ln−iγ1n
2
γ0
(ωL−ωλ)2+(γ0/2)2
)
,
(20)
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15470

Page 13

whereIL=√ε∞cE2
c is the speed of light in a vacuum, and γ0is the inverse photon lifetime. The quantities ∆ηn=
ω(k)
states Σ(k)
The first, second, and third terms in the parenthesis of Eq. (20) describe the process of lu-
minescence, whereas the fourth term corresponds to the resonant quasi-elastic scattering. For
the given frequency detunings ∆ηn, the relative importance of these terms is set by: (i) pure
dephasing rates ˆ γ12and ˆ γ13; (ii) lifetimes, γ−1
and |3⟩; (iii) coherence relaxation rates, γ12= γ22/2+ ˆ γ12and γ13= γ33/2+ ˆ γ13, for optical
transitions with generation and recombination of the electron-hole pairs; (iv) relaxation rate
ζ32; and (v) photon lifetime. The values of these parameters depend on many factors, which
makes their precise determination a challenging experimental task that has not still been ac-
complished [41]. For example, γ12and γ13substantially increase if the transitions between
states |E(k)
tinuous energy spectrum [42,43]. The quantity ζ32is determined by the relaxation processes
with the emission of local excitations of the quantum dot, and elementary excitations of the
environment [16,17,44,45]. These processes are sensitive to the design and fabrication quality
of the sample, as well as excitation conditions. Although in most practical situations the pop-
ulation relaxation rates satisfy the inequality γ0≪ γ22, γ33, their absolute values also vary in a
broad range. It is reasonable, therefore, to treat parameters appearing in Eq. (20) as adjustable
phenomenological constants, which should be determined (separately for each individual case)
from the experimental data.
L/(2π)istheintensityoftheexcitationlight,C(ωλ)=4(Zcvωλe)4/(πc4¯ h2),
n −ωη denote detunings from the frequencies, ω(k)
2= E(k)
n
= Σ(k)
n /¯ h of the polaron-like energy
+ and Σ(k)
3= E(k)
−.
22and γ−1
33, of the electron-hole pairs in states |2⟩
+⟩ and |E(k)
−⟩ are intensified through the emission of acoustic phonons with a con-
3.2.
The spectrum expressed according to Eq. (20) cannot be directly compared with experimental
data, since the treatment of the previous subsection implicitly assumed that the photon detection
system possesses infinite frequency resolution. We can obtain expression for the secondary
emission spectrum registered by a real photon detector, by convoluting Eq. (20) with a filter
function gF(ωF−ωλ) centered at frequency ωF. Following Ref. [46], we take the filtering
properties of the detector to be identical to those of a Fabry–Perot interferometer characterized
by the function
gF(ωF−ωλ) =1
π
with ΓFbeing the detector’s bandpass. Using this expression for gFand taking into account
that ΓF≫ γ0in the majority of practical instances, we obtain after the convolution
⟨
+2ζ32γ12
γ22γ33
∆2
12
????
where ∆Fn= ω(k)
For experimental measurement of the coherence relaxation rates γ12and γ13, the bandpass is
Spectral filtration of secondary emission
ΓF/2
(ωF−ωλ)2+(ΓF/2)2,
d2σ
dΘdωλ
⟩
= δpe2,ph2C(ωF)
(
3
∑
n=2
2ˆ γ1n
γnn
(c(k)
(c(k)
)2
n−1
Ln+γ2
)4
1n
∆2
2(γ1n+ΓF/2)
Fn+(γ1n+ΓF/2)2
∆2
1c(k)
L2+γ2
2
2(γ13+ΓF/2)
F3+(γ13+ΓF/2)2
(c(k)
∆2
+
3
∑
n=2
n−1
)2
∆Ln−iγ1n
????
2
ΓF
(ωL−ωF)2+(ΓF/2)2
)
,
(21)
n −ωF.
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15471

Page 14

usually chosen to satisfy the inequalities
ΓF≪ γ12,γ13,
(22)
in which case Eq. (21) acquires the form
⟨
d2σ
dΘdωλ
⟩
= δpe2,ph2C(ωF)
(
3
∑
n=2
2ˆ γ1n
γnn
(c(k)
????
n−1
Ln+γ2
)4
1n
∆2
2γ1n
Fn+γ2
(c(k)
∆2
1n
????
+2ζ32γ12
γ22γ33
(c(k)
1c(k)
∆2
2
)2
12
L2+γ2
2γ13
F3+γ2
∆2
13
+
3
∑
n=2
n−1
)2
∆Ln−iγ1n
2
ΓF
(ωL−ωF)2+(ΓF/2)2
)
.
(23)
This expression is a function of the excitation frequency ωLand detection frequency ωF. The
possibility to vary one of them while keeping the other constant results in two types of spec-
troscopic experiments. If ωLis fixed and ωFis varied, then an ordinary spectrum of secondary
emission is obtained. On the other hand, in the situation where ωFis fixed and ωLis altered, an
excitation spectrum of secondary emission is recorded.
Comparison of luminescence and scattering contributions to the total signal of secondary
emission given in Eqs. (20), (21), and (23) reveals several important differences between their
spectra. First, if the frequency of excitation is out of resonance with either of the electronic
transitions, then the position of the scattering maximum coincides with ωL, whereas the lu-
minescence intensity peaks at frequencies ωF= ω(k)
Eq. (22) be satisfied, the scattering linewidth is narrower than the linewidths of the lumines-
cence. Third, scattering strongly masks luminescence, as the peak intensity of the scattering
signal is by far greater than the peak luminescence intensity. The domination of scattering
over luminescence in the spectra of secondary emission is due to the factors 2ˆ γ12/γ22≪ 1 and
2ˆ γ13/γ33≪ 1 entering the luminescence terms, and because of ΓF≪ γ12,γ13.
For the same reasons, scattering prevails in the excitation spectra of the secondary emission.
The excitation spectrum of scattering consists of two peaks at ωL= ω(k)
determined by the coherence relaxation rates. This fact allows parameters γ12and γ13to be
calculated through fitting the experimentally measured excitation spectra with Eq. (23).
±≡ E(k)
±/¯ h. Second, should conditions in
±, whose widths are
3.3.
The above approach allows the secondary emission to be treated analytically in the event that
the degenerate states (with nonzero angular momenta) of the electron-hole pairs are involved in
the vibrational resonance. In principle, this can be done for the situation discussed in Subsec-
tion2.5;however,suchageneraltreatmentresultsinrathercumbersomemathematicalformulas
not suitable for this paper. The simplest problem that can be solved exactly is the case of vi-
brational resonance between the 9-fold degenerate state |p2⟩ = |ne1eme;nh1hmh⟩ and either of
the 3-fold degenerate states |p1⟩ = |ne1eme;nh0h0h⟩ or |ne0e0e;nh1hmh⟩ (which correspond to
resonance in the valence or conduction band, respectively). Clearly, the 9-fold degeneracy of
the upper state is the lowest possible degeneracy for optically excitable states with equal quan-
tum numbers of electrons and holes [see Eq. (7)]. Eigenvectors for the state |p2⟩ are obtained
as a direct product of the three degenerate eigenstates for electrons and the three degenerate
eigenstates for holes,


Secondary emission in the case of resonance with degenerate electronic states
{|ne1eme;nh1hmh⟩} =

|ne,1e,−1e⟩
|ne,1e, 0e⟩
|ne,1e,+1e⟩


⊗



|nh,1h,−1h⟩
|nh,1h, 0h⟩
|nh,1h,+1h⟩


.
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15472

Page 15

For the two lower states, the eigenvectors are
|nh,0h,0h⟩⊗





|ne,1e,−1e⟩
|ne,1e, 0e⟩
|ne,1e,+1e⟩




and
|ne,0e,0e⟩⊗

|nh,1h,−1h⟩
|nh,1h, 0h⟩
|nh,1h,+1h⟩

.
Employing the results of Section 2 for the above pair of degenerate states, we solve Eq. (19)
with a new minimal basis for the Hamiltonian˜ H0based on the renormalized electron-phonon
spectrum. The obtained density matrix elements are then introduced into the expression similar
to Eq. (20) to find that the differential cross section of the secondary emission is given by
(
n=2
7
∑
r=3n=2
????
In deriving this expression, we took into account six optically allowed polaron-like states of
energies
⟨
d2σ
dΘdωλ
⟩
= δpe2,ph2C(ωF)
7
∑
2ˆ γ1n
γnn
d4
n
∆2
Ln+γ2
d2
r
∆2
1n
2(γ1n+ΓF/2)
Fn+(γ1n+ΓF/2)2
2(γ1r+ΓF/2)
Fr+(γ1r+ΓF/2)2
d2
n
∆Ln−iγ1n
∆2
+
r−1
∑
2ζrnγ1n
γnnγrr
nd2
Ln+γ2
1n
∆2
+
7
∑
n=2
????
2
ΓF
(ωL−ωF)2+(ΓF/2)2
)
.
(24)
Σ(k)
2=1
=1
2
(Ep2+˜Ep1+ ¯ hΩLO+δk
(Ep2+˜E′
(Ep2+˜E′′
p1are the energies of the electron-hole pairs shifted due to the interaction
with k, k′, and k′′LO-phonon modes according to Eqs. (5) and (9a);
√
χ = Ep2−˜Ep1− ¯ hΩLO,
and the parameters Vkare given by Eq. (13b). It was also taken into account that the diagonal
part of the electron-phonon interaction does not change the energy of the state |p2⟩, which
resulted in˜E′′
renormalization of the electron-phonon spectrum, given by
),
Σ(k)
5=1
=1
2
(Ep2+˜Ep1+ ¯ hΩLO−δk
(Ep2+˜E′
=1
2
),
Σ(k′)
3
2
p1+ ¯ hΩLO+δk′),
Σ(k′)
6
2
p1+ ¯ hΩLO−δk′),
(Ep2+˜E′′
Σ(k′′)
4
=1
2
p1+ ¯ hΩLO+δk′′),
Σ(k′′)
7
p1+ ¯ hΩLO−δk′′),
where˜Ep1,˜E′p1, and˜E′′
δk=
χ2+4V2
k,
δk′ =
χ′= Ep2−˜E′
√
χ′2+4V2
k′,
δk′′ =
√
χ′′2+4V2
k′′,
p1− ¯ hΩLO,
p1− ¯ hΩLO,
χ′′= Ep2−˜E′′
p2=˜E′p2=˜Ep2= Ep2. The quantities dnin Eq. (24) are the constants describing
d2= (χ +δk)/∆k,
d5= 2Vk/∆k,
d3= (χ′+δk′)/∆k′,
d6= 2Vk′/∆k′,
d4= (χ′′+δk′′)/∆k′′,
d7= 2Vk′′/∆k′′,
where
∆k=
√(χ +δk
)2+4V2
k,
∆k′ =
√(χ′+δk′)2+4V2
k′,
∆k′′ =
√(χ′′+δk′′)2+4V2
k′′.
#146499 - $15.00 USD
(C) 2011 OSA
Received 26 Apr 2011; revised 24 Jun 2011; accepted 11 Jul 2011; published 28 Jul 2011
1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 15473