Polaritons in a nonideal periodic array of microcavities

Superlattices and Microstructures 89 (2016) 409e418

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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices

Polaritons in a nonideal periodic array of microcavities Vladimir Rumyantsev a, b, Stanislav Fedorov a, Kostyantyn Gumennyk a, *, Marina Sychanova a, Alexey Kavokin b, c, d a

Galkin Institute for Physics & Engineering, Donetsk 83114, Ukraine Mediterranean Institute of Fundamental Physics, 00047 Marino, Rome, Italy c CNR-SPIN, Viale del Politecnico 1, I-00133 Rome, Italy d Physics and Astronomy School, University of Southampton, Highfield, Southampton, SO171BJ, United Kingdom b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 25 October 2015 Received in revised form 23 November 2015 Accepted 24 November 2015 Available online 30 November 2015

The virtual crystal approximation is employed to numerically model the propagation of localized electromagnetic excitations through a two-dimensional array of coupled microcavities containing atomic clusters (quantum dots). The constructed model allows for the presence of defects (absence of cavities and/or quantum dots) at certain sites of the supercrystal. We derive the dispersion relations for polaritonic modes as functions of defect concentrations. This permits to evaluate the densities of states of polaritons and their effective masses as well as the band gaps for any prescribed values of defect concentrations. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Two-dimensional supercrystal Coupled microcavities Quantum dots Localized electromagnetic excitations Polaritonic spectrum

1. Introduction Advances in crystal optics during the past 50 years provide a solid foundation for the progress of modern photonics. Concepts developed in physics of crystalline solids prove to a large extend applicable to photonic supercrystals. Whereas the theory of impurity bands and excitons in semiconductor crystals has been constructed in 1970e1980s a similar theory for photonic crystals is yet to be developed. Fabrication of novel materials for the creation of sources of coherent radiation is presently a vast interdisciplinary area of theoretical and experimental investigations, which involve condensed matter physics, nanotechnologies, chemistry as well as information science [1,2]. In this connection some promising vistas can be opened by the so-called polaritonic crystals [3,4], which represent a particular type of photonic crystals [5] featured by a strong coupling between quantum excitations (excitons) and electromagnetic waves. An example of polaritonic structure can be provided by an array of coupled microcavities [6]. Optical modes in microcavity systems have been attracting considerable attention due to the progress in fabrication of novel optoelectronic devices [7,8]. Worthy of note are the defect-based microresonators in photonic crystals [9], which were shown to strongly interact with quantum dots [10]. Formation of quantum solitons coupled to lower dispersion branch (LDB) polaritons in a chain of microresonators was theoretically studied in Refs. [3,4]. The obtained results were found promising for the purposes of

* Corresponding author. E-mail address: [email protected] (K. Gumennyk). http://dx.doi.org/10.1016/j.spmi.2015.11.029 0749-6036/© 2015 Elsevier Ltd. All rights reserved.

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quantum information processing. Microresonator systems can also be potentially utilized for manufacturing of exceptionally accurate optical clockworks [11e13]. Another related and actively developing area is the photonics of imperfect structures. Some of our recent works are devoted to optical properties of imperfect photonic crystals [14] as well as to dispersion of exciton-like electromagnetic excitations in nonideal crystals composed of coupled microcavities [15,16]. Defects in photonic crystals provide an additional powerful tool for controlling and modeling the propagation of electromagnetic excitations. In the present paper we use the previously developed concepts of photonic structures [15,16] to treat a nonideal polaritonic crystal formed by a topologically ordered array of coupled microcavities (resonators) and a subsystem of embedded atomic clusters (quantum dots). Our main concern is the sensitivity of polaritonic spectrum to imperfections and mutual interaction of the photonic and electronic subsystems.

1.1. Theoretical background The conventional polaritonic model [3,4,15,16] of the atomic-optical interaction is only applicable to the case of ultracold atoms with frozen-out degrees of freedom (at temperatures of the order of several mK). Its validity depends on the number of atoms in individual cavities, which should be sufficiently small (N
104) so that the value of the strong coupling parameter of the atomic-optical interaction should much exceed the inverse coherence time of the atomic-optical system. Physically, the latter is the time of thermodynamic equilibration of the atomic system, which interacts with the quantized field in a polaritonic crystal. In such a case the spectral line broadening can be ignored and the quantum states of the atomic-field subsystems can be considered as pure and thermodynamically equilibrium. Polaritonic crystal can be formed by trapping of two-level atoms by an ideal coupled-resonator optical waveguide (CROW) [3] or by a nonideal array of microcavities [16]. For the sake of generality let us first examine a two-dimensional lattice of microcavities with an arbitrary number s of sublattices and then tackle an interesting specific case of a two-sublattice array. Assume that the microcavity array is comprised by resonators of s(a) types (where a is the number of sublattice) randomly distributed over a square lattice with a period d (see Fig. 1). Each resonator contains a quantum dot (a two-level assembly of atoms) pertaining to one of r(a) of types, which interacts with the resonator-localized quantized electromagnetic field. Each of tunnel-coupled microcavities possesses a single optical mode. Hamiltonian of the described system in coordinate representation writes

b þH b ; b ¼H b at þ H H int ph

(1)

Fig. 1. Schematic of a nonideal two-dimensional two-sublattice array of tunnel-coupled resonators with trapped quantum dots (the atomic subsystem). In the first sublattice (green) the atomic subsystem contains vacancies denoted by Vat. In the second sublattice (orange) a certain fraction of resonators is absent, which is viewed as the presence of vacancies (denoted by Vph) in the photonic subsystem. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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411

b at , the photonic subsystem H b b where Hamiltonians of the atomic subsystem H ph and their interaction H int equal correP P P P P b nm and H b n . Here H b ¼ b ¼ b at;n is the Hamiltonian of a b at ¼ b at;n þ 1=2 b nm , H b A G spondingly to H H V H int ph ph;n  1=2 n

n;m

n

n;m

n

b nm is the operator of Coulomb interaction between quantum dots stationary (ultracold) quantum dot at the n-th resonator, V b embedded at the n-th and m-th resonators, operator H ph;n defines the state of electromagnetic excitation localized at the n-th b nm describes an overlap between the optical fields pertaining to the n-th and m-th resonators (and resonator whereas A therefore defines the transfer probability of the corresponding electromagnetic excitation). Writing the interaction operator b n reflects the fact that the electromagnetic excitation localized at the n-th resonator b H as a sum of unitary operators G int

interacts only with the quantum dot residing at the same resonator. n and m are complex indices n ≡ (n,a), m ≡ (m,b), where the two-dimensional vectors n, m define localizations of elementary cells in the array, and a and b numerate sublattices and assume values 1,2,3…s. Let us (in accordance with [17,18]) calculate the energy spectrum of the described system by writing down its Hamiltonian b as a sum of the following second-quantization operators: H

b at ¼ H

X

þ

b b εat nf b nf b nf þ

n;f

b ¼ H ph

X n;m

b ¼ H int

  at at 1 X X X bþ bþ at  b b  b nf b mg 〈4at nf 4mg Vnm 4mh 4nl 〉 b mh b nl ; 2 nsm f ;g

1 ph b þ b εnm f nm f nm  2

X X n;f ;g m;n

h;l

X XX nsm m;n

   ph ph b b bþ b þ ph ph  f nm f mn 〈4nm 4mn Anm 4ml 4nr 〉 f ml f nr ;

(2)

l;r

  þ at ph  b  at ph b b b b ng fþ nm 〈4ng 4nm  G n 4nf 4nn 〉 b nf f nn :

at b b b Here εat , εph nm are the eigenvalues of operators H at and H ph , respectively. Wave functions 4nf , f nl characterize the states of the nf þ bþ , f b are Pauli operators, which quantum dot and electromagnetic field pertaining to the n-th microcavity. b b ,b b and f nf

nf

nm

nm

þ þ þ þ satisfy the following commutation relations b b nf b b mg þ b b nf ¼ 1, b b mg ¼ b b mg ¼ 0 for nf ¼ mg and b b mg  b b nf ¼ b mg b b nf b b nf b b nf b b mg b þ þ b bþ , f b nm ). b mg  b b nf ¼ 0 for nf s mg (analogous relations hold for f b nf b b mg b nm Below we shall assume that the density of excited states of elements in the atomic and resonator subsystems is low. This allows to simplify the expressions, which enter the energy operator (1)e(2) by approximating Pauli operators with Bose þ þ b bþ , b b b þ bþ b b bþ b b n0 z B operators: b b nf b nf b n0 b nf z B nf , f nm f n0 z J nm , f n0 f nm z J nm . As applied to a one-level model (where indices f, g, h, l in expression (2) assume values 0 and a while m, n, l, r assume values 0 and 1) such a procedure yields (within the Heitlerb and J b nm ) constituent of Hamiltonian (2): London approximation) the following expressions for the quadratic (in B nf

b HL ¼ H at

X

Dεna þ

n

X m









at  b at at  b  at at  at at 〈4at na 4m0 V nm 4m0 4na 〉  〈4n0 4m0 V nm 4m0 4n0 〉



!

bþ B b B na na þ

n;m

X X ðaÞ þ  bþ B b bþ b b B b b nm B Zuat Vnm B  V na ma ≡ na ma na B na B na þ n

b HL ¼ H ph

X n

X  at  at at 〈4at n0 4ma 4m0 4na 〉 (3)

n;m

!     X ph ph   ph ph  ph ph ph ph  b b b bþ J 〈4n1 4m0  A nm 4m0 4n1 〉  〈4n0 4m0  A nm 4m0 4n0 〉 J En1  n1 n1 m

 X ph ph  X ph þ X b nm 4ph 4ph 〉 J bþ J b b J b bþ J b  〈4n0 4m1  A Zun1 J Anm J n1 m1 ≡ n1 n1  n1 m1 m0 n1 n;m

b HL ¼ H int

X n

n

j j

(4)

n;m

j j

þ at ph b at ph at ph b at ph b b b bþ B J n1 na 〈4n0 4n1 G n 4na 4n0 〉 þ J n1 B na 〈4na 4n0 G n 4n0 4n1 〉≡

X

 þ þ  b b b b B gn J n1 na þ J n1 B na :

(5)

n

at In relations (3) and (4) appear the characteristic frequencies uph n1 , una of the atomic and photonic subsystems, which comprise the array, as well as the matrix elements of interatomic and inter-cavity interactions:

    at at  ph ph ðaÞ ph ph  b at  b  〈4at n0 4ma V nm 4m0 4na 〉≡Vnm ; 〈4n0 4m1  A nm 4m0 4n1 〉≡Anm :

(6)

In (5) it is taken into account that the wave functions of quantum dots and electromagnetic fields are real-valued and     ph  b  at ph at ph  b  at ph hence 〈4at n0 4n1  G n 4na 4n0 〉 ¼ 〈4na 4n0  G n 4n0 4n1 〉≡gn . To model a nonideal structure we allow for compositional disordering of both atomic and photonic subsystems. This ðaÞ at makes quantities uph n1 , una , Vnm ,Anm and gn configurationally dependent, whereas Hamiltonian (1) becomes not translation

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invariant. The general scheme for calculating the spectrum of quasiparticle excitations in a nonideal system with randomly distributed elements is based on finding the poles of configurationally averaged solution of the corresponding Hamiltonian [19]. The latter is translation invariant and so the corresponding spectrum of elementary excitations can be characterized by a wave vector k. In order to obtain the above solution one should inevitably adopt a certain approximation specific to the studied system. A frequently used method of calculation of quasiparticle states in disordered media is the virtual crystal approximation (VCA) [19,20]. It provides an appropriate tool for elucidation of spectrum transformations caused by changes in defect concentrations. Within the VCA the averaged Hamiltonian solution is identical to the solution of the averaged Hamiltonian. Below this approximation is used to analyze the spectrum of electromagnetic excitations as well as the optical characteristics of the studied nonideal superlattice. Similarly to the procedures described in our previous publications [15,16] let us express the configurationally dependent ðaÞ nðaÞ nðaÞ ph ph at quantities uat na ≡una , un1 ≡una , Vnm ,Anm and gn through the random quantities h ! ðh ! Þ: at; n a ph; n a rðaÞ X

uat na ¼ ph una

¼

nðaÞ¼1 sðaÞ X

nðaÞ nðaÞ uph;a hph;na

rðaÞrðbÞ X

ðaÞ

nðaÞ nðaÞ

uat;a hat;na ;

Vnamb ¼ ;

Anamb ¼

nðaÞ¼1

nðaÞmðbÞ

Vab

nðaÞ;mðbÞ¼1 sðaÞsðbÞ X

nðaÞ

mðbÞ

ðn  mÞhat;na hat;mb ; (7)

nðaÞmðbÞ nðaÞ mðbÞ Aab ðn  mÞhph;na hph;mb :

nðaÞ;mðbÞ¼1

Quantity gn reflects the configurational dependency of both subsystems

X

gn ¼

nðaÞ

mðaÞ

ganðaÞmðaÞ hat;na hph;na :

(8)

nðaÞ;mðaÞ

nðaÞ

nðaÞ

In (7) and (8) hat;na ðhph;na Þ ¼ 1 if a quantum dot of the n(a)-th type resides at the na-th site of the lattice and ¼ 0 in all other cases. Assuming that the atomic and photonic disorders are caused by independent mechanisms we arrive at the following expressions for configurationally averaged quantities:

nðaÞ nðaÞ hat;na ðhph;na Þ

rðaÞ P

〈uat na 〉 ¼

nðaÞ¼1 sðaÞ P

〈uph na 〉 ¼

nðaÞ¼1

〈gn 〉 ¼

rðaÞsðaÞ P nðaÞ;mðaÞ

nðaÞ nðaÞ

uat;a Cat;a ; nðaÞ

nðaÞ

uph;a Cph;a ;

ðaÞ

〈Vnamb 〉 ¼ 〈Anamb 〉 ¼

rðaÞrðbÞ X

nðaÞmðbÞ

Vab

nðaÞ;mðbÞ¼1 sðaÞsðbÞ X

nðaÞmðbÞ

Aab

nðaÞ mðbÞ

ðn  mÞCat;a Cat;b ; nðaÞ

mðbÞ

(9)

ðn  mÞCph;a Cph;b ;

nðaÞ;mðbÞ¼1 nðaÞ mðaÞ

ganðaÞmðaÞ Cat;a Cph;a ; nðaÞ

nðaÞ

mðbÞ

mðbÞ

where angular brackets denote the averaging procedure, Cat;a ðCph;a Þ and Cat;b ðCph;b Þ are concentrations of the n(a)-th and m (b)-th types of elements in the atomic and resonant subsystems. There must obviously hold the relations PrðaÞ nðaÞ PsðaÞ nðaÞ C ¼ 1; C ¼ 1. Configurational averaging “restores” the translation invariance of the structure, which allows nðaÞ at;a nðaÞ ph;a to characterize the eigenvalues and eigenfunctions of the resulting virtual crystal by a wave vector k ¼ (kx,ky,0). In k-representation Hamiltonian 〈H〉 has the following form

b ¼ 〈H b at 〉 þ 〈 H b 〉 þ 〈H b 〉 ; 〈 H〉 int k ph k k k

(10)

where

Xh

i þ ðaÞ b b Z〈uat na 〉dab þ Vab ðkÞ B aa ðkÞ B ba ðkÞ; a;b h i X b 〉 ¼ b bþ Z〈uph 〈H na 〉dab  Aab ðkÞ J a1 ðkÞ J b1 ðkÞ; ph k b at 〉 ¼ 〈H k

a;b

(11)

h

i X b aa ðkÞ þ J b þ ðkÞ : b 〉 ¼ b a1 ðkÞ B b þ ðkÞ B 〈gna 〉 J 〈H int k aa a1 a

b ba ðkÞ, J b na , and J b a1 ðkÞ are the Fourier components of quantities 〈V ðaÞ 〉,〈Anamb〉, B b m1 , respectively in (11) Aab(k), B namb (cf. [15,16]). Diagonalization of Hamiltonian 〈H〉k with the use of Bogolyubov-Tyablikov transformation [17] yields the expressions for the energy of polaritonic excitations in the microcavity crystal. ðaÞ Vab ðkÞ,

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413

2. Results and discussion To make our further discussion more specific let us study polaritonic excitations and the role of impurities (vacancies) for a two-sublattice model. In such a case indices a and b in expressions (11) assume only two values 1 and 2. Diagonalization of Hamiltonian 〈H〉k yields a system of linear homogeneous equations, whose solvability condition is formulated as the equality of the following determinant to zero:

  Z〈uat 〉 þ V ðaÞ ðkÞ  ZU  n1 11  ðaÞ  V21 ðkÞ   〈gn1 〉    0

ðaÞ

V12 ðkÞ ðaÞ

Z〈uat n2 〉 þ V22 ðkÞ  ZU 0 〈gn2 〉

     0 〈gn2 〉 ¼0  ph Z〈un1 〉  A11 ðkÞ  ZU A12 ðkÞ   ph A ðkÞ Z〈u 〉  A ðkÞ  ZU  〈gn1 〉

21

0

n2

(12)

22

Determinant (12) defines a fourth-order equation for the unknown dispersion relation U(k). According to (9) polaritonic ðaÞ ph frequencies U(k) as well as the parameters 〈uat na 〉, 〈una 〉, Vab , Aab and 〈gna〉 are the functions of concentrations of structural elements (cavities and quantum dots), which comprise the array. As an important particular case, let us assume that the photonic subsystem in the first sublattice is ideal while the atomic ð1Þ

ð1Þ

ð2Þ

subsystem contains defects (vacancies), i.e. Cph;1 ¼ 1, Cat;1 ≡C1at , Cat;1 ≡C1V . An opposite assumption is made for the second sublattice, namely that its photonic subsystem contains vacancies (absent resonators), while its atomic subsystem is ideal, i.e. ð1Þ

ð2Þ

ð1Þ

ph at 1 at 1 Cph;2 ≡C2ph , Cph;2 ≡C2V and Cat;2 ¼ 1. Hence, in view of (9) we obtain 〈uat n1 〉 ¼ C1 uat;1 , 〈un2 〉 ¼ uat;2 C2 (since quantum dots can ph ph 1 1 11 at 11 ph only reside in the available cavities) and 〈uph n1 〉 ¼ uph;1 ; 〈un2 〉 ¼ C2 uph;2 , 〈gn1 〉 ¼ g1 C1 ; 〈gn2 〉 ¼ g2 C2 .

Further calculations will be carried out within the nearest-neighbor approximation for a square Bravais lattice with period d(Fig. 1) [17]. Positions of cavities shall be described by a radius-vector rna ¼ rn þ ra, which makes their positions in the zeroth elementary cell (rn ¼ 0) equal to r01 ¼ 0 and r02 ¼ a in the first and second sublattices, respectively. It shall be assumed that ðaÞ a << d. Having made these assumptions we can fairly accurately write the matrix elements Aab(k) and Vab ðkÞ as

2     A11 ðkÞz2A11 ðdÞ cos kx d þ cos ky d ; A22 ðkÞz2A22 ðdÞ cos kx d þ cos ky d 1  C1V     A12 ðkÞzA12 ð0Þ 1  C1V expðik$aÞ; A21 ðkÞzA21 ð0Þ 1  C1V expðik$aÞ; 2  2     ðaÞ  ðaÞ V11 ðkÞz2V11 ðdÞ 1  C1V cos kx d þ cos ky d ; V22 ðkÞz2V22 ðdÞ 1  C2V cos kx d þ cos ky d ;       ðaÞ ðaÞ V12 ðkÞzV12 ð0Þ 1  C1V 1  C2V expðik$aÞ; V21 ðkÞzV21 ð0Þ 1  C1V 1  C2V expðik$aÞ:

(13)

Here the characteristic overlap A11(22)(d) of optical fields corresponding to different cavity modes defines the transfer probability of electromagnetic excitation between the nearest neighbors within the first (second) sublattice, while A12(21)(a) defines the corresponding transfer probability between resonators pertaining to the first (second) and the second (first) sublattices and the zeroth elementary cell. Similarly, quantities Vab describe the Coulomb interaction between quantum dots in the neighboring cavities. ðaÞ Substitution of expressions (13) for Aab(k) and Vab ðkÞ into (12) yields the dispersion law Un ðk; C1V ; C2V Þ (n ¼ 1,2,3,4) of electromagnetic excitations in the considered modeling system. We have performed their numerical evaluation for the frequencies of resonance photonic modes localized at resonators of the first and second sublattices taken to be u1ph;1 ¼ 2p  155THz, u1ph;2 ¼ 2p  245THz and the averaged frequencies of optical transitions in quantum dots being u1at;1 ¼ 2p  191THz, u1at;2 ¼ 2p  202THz. Parameters which govern an interaction between the atomic and photonic subsystems as well as an overlap of optical fields and an interaction of quantum dots in the neighboring microcavities were chosen as follows: g111 =Z ¼ 5,1013 Hz, g211 =Z ¼ 1; 5,1013 Hz, A11(d)/2Z ¼ 2$1014 Hz, A22(d)/2Z ¼ 3$1014 Hz, A12(a)/Zz A21(a)/ Z¼1$1014 Hz, V11(d)/2Z ¼ 6$1013 Hz, V22(d)/2Z ¼ 9$1013 Hz, V12(a)/Z z V21(a)/Z ¼ 3$1013 Hz. The lattice period was set equal to d ¼ 3$107m. Such a choice of parameters maximizes the polariton-induced transformation of photon bands in the crystal. It's worthwhile noting here that the adopted approach is quite general and can be applied to a large variety of photonic microcavity crystals ranging from natural opals to artificially fabricated arrays of pillar microcavities. The values of the above parameters should be adjusted appropriately in every specific case. Fig. 2 shows the numerically evaluated dispersion bands of an ideal structure (C1V ¼ C2V ¼ 0). In Fig. 3 the corresponding plots are shown for several specific realizations of a non-ideal array. Fig. 3a,b illustrate the effect of varying C1V (and unchanged C2V ) on the dispersion surfaces Un ðkx ; ky ; C1V ; C2V Þ; Fig. 3c,d demonstrate the effect of changing C2V at the constant C1V . Fig. 3e,f demonstrate the transformation of surfaces in Fig. 3d under changing g111 and g211 (and the rest of parameters kept invariable). Arrows point at the widening (narrowing) “bottle necks”, resulting from the increase of g211 by a factor of five in Fig. 3e and the decrease of g111 by a factor of five in Fig. 3f. Fig. 4a,b,c show the concentration dependencies of indirect (in the general case) band gaps

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V V V Fig. 2. Dispersion of electromagnetic excitations U1;2;3;4 ðkx ; ky ; CV 1 ; C2 Þ in an ideal array (C1 ¼ C2 ¼ 0).

V V V Fig. 3. Dispersion of electromagnetic excitations U1;2;3;4 ðkx ; ky ; CV 1 ; C2 Þ in a nonideal array where concentrations of vacancies equal to (a) C1 ¼ 0:2, C2 ¼ 0:5, (b) V V V V V 11 CV 1 ¼ 0:6, C2 ¼ 0:5, (c) C1 ¼ 0:5, C2 ¼ 0:6, (d) C1 ¼ 0:5, C2 ¼ 0:25. (e) and (f) show the transformation of the (d)-case dispersion surfaces under changing g1 11 and g11 2 (and the rest of parameters being kept invariable). In (e) and (f) arrows point at widening (narrowing) “bottle necks”, resulting from the increase of g2 by a factor of five and the decrease of g11 1 by a factor of five, respectively.

V. Rumyantsev et al. / Superlattices and Microstructures 89 (2016) 409e418

415

V Fig. 4. Concentration dependency of the band gap (a) DU1;2 ðCV 1 ; C2 Þ between the first (lowest in Figs. 2 and 3) and the second dispersion bands; (b) V Þ between the second and the third dispersion bands; (c) DU V V DU2;3 ðCV ; C 3;4 ðC1 ; C2 Þ between the third and the fourth dispersion bands. 1 2

V V V Fig. 5. Densities of states rn ðU; CV 1 ; C2 Þ; ðn ¼ 1; 2; 3; 4Þ in four dispersion bands of an ideal array (C1 ¼ C2 ¼ 0, see Fig. 2).

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V. Rumyantsev et al. / Superlattices and Microstructures 89 (2016) 409e418

Fig. 6. (a) Variation of the density of states in the upper (fourth) dispersion band U4(kx,ky) under changing vacancy concentration in the second sublattice CV 2 . (b) equifrequency lines of the dispersion surface U4(kx,ky) corresponding to the solid curve in (a); heavy equifrequency lines are responsible for peculiarities (singularities 1,2 and the jump discontinuity 3) of the density of states occurring at U's designated by dashed vertical lines in (a) and marked by corresponding numbers.

            DUn;nþ1 C1V ; C2V ≡Q min Unþ1 k; C1V ; C2V  max Un k; C1V ; C2V  min Unþ1 k; C1V ; C2V  max Un k; C1V ; C2V ; k

k

k

k

where Q(x) is the Heaviside step function. It is often important to know how the specifics of a quasiparticle spectrum is manifested in the corresponding quasiparticle densities of states rn ðU; C1V ; C2V Þ. In our case (see Ref. [21]) the functions rn ðU; C1V ; C2V Þ are given by the formula

ðnÞ

V Fig. 7. Concentration dependencies of polariton effective masses meff ðk; CV 1 ; C2 Þ; ðn ¼ 1; 2; 3; 4Þ calculated for four dispersion branches Un.

V. Rumyantsev et al. / Superlattices and Microstructures 89 (2016) 409e418

  d rn U; C1V ; C2V ¼ 2p

I Un ðkÞ¼U

dl ; jVk Un ðkÞj

417

(14)

where index n¼1,2,3,4 numerates the four dispersion branches Un ðk; C1V ; C2V Þ. Integration in (14) is carried out along various equifrequency contours falling within the first Brillouin zone. In Fig. 5 the densities of states rn(U) are numerically evaluated for the four dispersion bands of an ideal array (shown in Fig. 2). Fig. 6a shows the transformation of the density of states under changing concentration of vacancies in the second sublattice; the dashed and solid curves correspond to the upper (forth) dispersion band and concentration values C1V ¼ 0:5, C2V ¼ 0:6 and C1V ¼ 0:5, C2V ¼ 0:25 respectively. Fig. 6b illustrates how the topology of a two-dimensional dispersion surface is manifested in the corresponding density of states. Gradient VkUn(k) equals to zero at each point of contour 1 in Fig. 6b, resulting in a singularity to the right of the vertical asymptote 1 in Fig. 6a; gradient VkUn(k) turns to zero at the corner points of the square-like contour 2 in Fig. 6b, yielding singularities on both sides of asymptote 2 in Fig. 6a; the shrinking to zero contour 3 in Fig. 6b is responsible for the jump discontinuity at U corresponding to the vertical line 3 in Fig. 3a. An important property of band gap photonic structures is their capability of conducting the so-called “slow light”, which appears to be highly promising for applications in quantum information processing [22]. For example, the effective decrease in the group velocity of quasiparticles was observed in coupled waveguide optical resonators [23,24] as well as in various ðnÞ

types of layered semiconductor structures [25]. The key factor, which governs the group velocity is the effective mass meff of the so-called “dark” and “bright” polaritons, which are linear superpositions of the photonic states of photonic subsystems and of the coherent perturbations of one-level atomic subsystems. Fig. 7 shows the concentration dependencies of polariton   ðnÞ Þ1 , where n numerates dispersion branches and k can be either kx or effective masses meff ðC1V ; C2V Þ≡Zðv2 Un ðk; C1V ; C2V Þ=vk2  k¼0

ðnÞ

ky. Plots meff ðC1V ; C2V Þ demonstrate how the “slow light” can in principle be attained by an appropriate choice of concentrations C1V and C2V . 3. Conclusions We use the virtual crystal approximation to construct a sufficiently simple model of a nonideal two-dimensional microcavity array with embedded quantum dots. The model allows to analyze the polaritonic spectrum of the disordered photonic crystal. Numerical calculations show that variation of defect concentrations results in a considerable transformation of the energy structure, renormalization of polaritonic spectrum and a significant change of optical properties of the array. The presence of point defects is known to increase the effective mass of polaritons and therefore to decrease their group velocity (as compared to an ideal polaritonic crystal [3]). Here we specifically consider an important case of polaritonic crystal with vacancies in both its photonic and atomic subsystems. It should be noted that the numerical modeling of more complex objects usually requires the use of other (more sophisticated) approaches such as the one- or multi-node coherent potential method [20], the averaged T-matrix method and their various modifications. Our quasi-analytical results contribute to the understanding of physics of complex light-matter coupled structures and help modeling of polaritonic functional materials with controllable propagation of electromagnetic waves. Author contributions V.R. proposed the idea and has written the bulk of the paper. V.R., S.F. performed analytical calculations, K.G. and M.S. performed numerical simulations. V.R., S.F. and A.K. contributed to the discussion and paper writing. Acknowledgments This work was supported by the European contract FP7-PEOPLE-2013-IRSES (Grant # 612600 “LIMACONA”). References [1] W. Cai, V. Shalaev, Optical Metamaterials: Fundamentals and Applications, Springer, New York, 2010. [2] M. Razeghi, Technology of Quantum Devices, Springer, New York, 2010. [3] A.P. Alodjants, I.O. Barinov, S.M. Arakelian, Strongly localized polaritons in an array of trapped two-level atoms interacting with a light field, J. Phys. B At. Mol. Opt. Phys. 43 (2010) 095502. [4] E.S. Sedov, A.P. Alodjants, S.M. Arakelian, Y.Y. Lin, R.-K. Lee, Nonlinear properties and stabilities of polaritonic crystals beyond the low-excitationdensity limit, Phys. Rev. A 84 (2011) 013813. [5] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals. Molding the Flow of Light, Princeton University Press, Princeton, 2008. [6] K.J. Vahala, Optical microcavities, Nature 424 (2003) 839. [7] M.A. Kaliteevskii, Coupled vertical microcavities, Tech. Phys. Lett. 23 (1997) 120. [8] V.G. Golubev, A.A. Dukin, A.V. Medvedev, A.B. Pevtsov, A.V. Sel’kin, N.A. Feoktistov, Splitting of resonant optical modes in Fabry-Perot microcavities, Semiconductors 37 (2003) 832. [9] J. Vu ckovi c, M. Loncar, H. Mabuchi, A. Scherer, Design of photonic crystal microcavities for cavity QED, Phys. Rev. E 65 (2001) 016608.

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