- Email: [email protected]
Polariton-atom bound state in a dispersive medium
4 November 1996 PHYSICS
ELSFNIER
LETTERS
A
Physics Letters A 222 (1996) 258-262
Polariton-atom
bound state in a dispersive medium
Valery I. Rupasov ‘, M. Singh 2 Centrefor Chemical Physics and Department
of Physics, University of Western Ontario, London, Onz., Canada N6A SK7
Received 13 May 1996; accepted for publication 2 August 1996 Communicated by L.J. Sham
Abstract The quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. If the atomic resonance frequency lies near the gap, the spectrum of the system is shown to contain a novel polariton-atom bound state with an eigenfrequency lying within the gap. The radiation and medium polarization of the bound state are localized in the vicinity of the atom. A signi~c~t suppression of spontaneous emission due to the bound state is predicted.
The prediction [1,2] and the first experimental observation [3] of beg-dimensional dielectric structures, which exhibit a complete photonic band gap (PBG) for all directions of electromagnetic propagation, have created a great interest in a number of fundamental phenomena of classical and quantum electrodynamics in these materials. It has been shown that the existence of PI3G materials gives rise to such interesting phenomena as the suppression of spontaneous emission [4], the formation of strongly localized states of light [3], photon-atom bound states [S], and the localization of superradiance near a photonic band gap f61. In ~ificial PBG materials, a suppression of the photon density of states over a narrow frequency range results from multiple photon scattering by
’ On leave from the Landau Institute for Theoretical Physics and the Institute of Spectroscopy, Russian Academy of Sciences. E-mail: v~p~ov~juli~.uwo.ca. ’ E-mail: [email protected]. 03759601/96/$12.00 Copyright PII SO375-9601(96)00626-3
spatially correlated scatterers. But it is well known that a frequency gap for propagating ele~~oma~etic modes exists also in many natural dielectrics and semiconductors. In contrast to PBG materials, gaps in these media are caused by photon couplings to elementary excitations (excitons, optical phonons, etc.) of the media [‘7,8]. The “normal” electromagnetic modes in frequency dispersive media (DM) are determined by the Maxwell equations with a frequency-dependent dielectric permeability and are treated as “photons in a medium” (or “polaritons”, i.e. photons dressed by an interaction with a medium excitation). Their spectrum consists of two branches of allowed states separated by a gap in which propagating polariton modes are completely forbidden_ Therefore, the intriguing question arises whether quantum optical phenomena predicted for PBG materials are observed in dispersive media. In this paper we report some first results of qu~~rn elec~~yn~ics of ~l~itons interacting with a single two-level atom whose resonance fre-
0 1996 Elsevier Science B.V. All rights reserved.
V.I. Rupasou, M. Singh/Physics Letters A 222 (39963 X8-262
quency lies near a polaritonic gap of DM. Within the framework of the conventional dipole resonance (rotating wave) approximation [9], we diagondize exactly the Hamiltonian of the field + medium + atom system in the one-particle sector of the entire Hilbert space. We show that the system’s spectrum contains both branches corresponding to propagating polariton modes scattered on the atom and a single discrete mode whose eigenfrequency lies within the Platonic gap. This novel discrete mode can be treated as a polariton-atom bound state in which the atomic excitation is dressed by the radiation field and the medium polarization, which are both localized in the vicinity of the atom. To clarify the role of the bound state in quantum optical phenomena in DM, we solve the problem of the spontaneous decay of an initially excited atomic state and show that the existence of the bound state leads to a significant suppression of the spontaneous emission process. The Hamiltonian of the field + medium + atom system has the form
+e61dre(t’)*E(r)--d~(0),
(1)
259
as a quantum two-level system [9] with the transition frequency 0 ,2. Due to the spherical symmetry of the problem, the three-dimensional Hamiltonian of the system can be reduced to a one-dimensional form [lo]. Let us choose the system of coordinates with the origin at the point of the atom’s position and expand field and medium variables in terms of spherical harmonic vectors [I 11,
where the Bose operators cs; (bgi) describe a photon la medium excitation) of an electric fru = e) or a magnetic (ff = m> type with a frequency w (or the modulus of a wave vector k = w), an angular momentum j and projection thereof m. The Fourier component of the functions in Eq. (2) are. given by ‘/2
where the first three terms represent the Hamiltonians of the atom, the radiation and the medium, while the fourth and fifth terms represent the medium-field and atom-field couplings, respectively. The operators E and H describe the radiation field. The operators Q(r) and P(r) are the operators of the displacement and the momentum of the medium, respectively. The medium is treated here as a continuous set of charge harmonic oscillators, each with frequency fz, charge e, and mass mo. Here n is the density of the number of oscillators, and d = de(cr+ -I-CT-) is the atomic dipole operator, where d is the matrix element ‘of the dipole operator, e is the unit vector along the z-axis, and CT’ = (CTx, IT y, CT“1; uf=crxfitry are the spin operators. For the sake of simplicity, we do not account here for a possible degeneration of atomic levels, and we treat the atom
6(k-
w)~~‘(n),
112 8(k-w)~f)(n), 1 I
S(k-
o)Y.++(n) J”’
’
, !
where it is a unit vector in the directiontof k and Y!“) are the spherical harmonic vectors [Ill. In the e&tro-dipole approximation only the electro-dipole harmonic of the radiation field (cr =.e, j = 1) is coupled to the atom. The choice of the quantum number m = 0 is determined by our choice of the polarization of the atomic transition (d 11z-axis). Therefore, putting Eq. (2) and the analogous expression for the magnetic field H into Eq. (1) and omitting .a11 the
V.I. Rupasov, M. Sir&/
260
Physics Letters A 222 (1996/ 258-262
higher h~oni~s (j >, 2) uncoupled to the atom, we derive the following model Hamiltonian, H=w,,(tT’++) =dk + o 271. kc+(k)c(k) I
+L’b+(k)b(k)
t
+m[c+(k)b(k)
+b+(k)c(k)]
-~~[c(k)crC+cr-c+(k)]),
(3)
where A== ne2n/mof2, y(k) = $k3d2. Here the terms c”b+ (cb), which create (~~hilate) both field and medium excitations simultaneously, as well as the analogous terms (c’ a+ and ccf- > in the atom-field coupling operator, are omitted in accordance with the Heitler-London [S] and the resonance 191approximations. The field-medium part of the H~iltoni~ is diagonalized in terms of polariton operators p,(k) corresponding to the lower (a = - > and upper (a = +) polariton branches, respectively [ 121. For our purposes, it is more convenient to introduce the energy variable E, k = ~(0 - E)/(G - A - E), and the polariton operators P(E) on the “energy scale”, I+
contour C consists of two intervals: -W < E < (Jz A) and f2 < E < ~0.In accordance with the resonance approximation, the integration in the lower limit is extended to - ~0, and we set E = o,~ in the expression for -y(e). The “particle” number operator of the system,
-e_(k)
/&AaEE,(k)I
commutes with the model Hamiltonian. Therefore, all the model eigenstates can be classified with respect to the number of “particles”, or eigenvalues of the operator N. In this paper we study one-particle eigenstates, which can be written in the form
where the vacuum state of the system 10) is defined by g- IO) =p(e) IO) = 0. Then, the Schrijdinger equation (H - A) 1A) = 0 takes the form @-WWV
= fizGM$ (6)
I’* P,(k),
where ~~(k)=$(L?+k>+ \1(O-k)2+4kAlare the polariton spectra. Then, the model Hamiltonian finally takes the form
co
H=#~~(rr~+i)+IC~~~+(r)P(E)
where the first and second terms are the Hamiltonians of the atom and the polariton field, respectively, while the third one describes their couphng. In Eq. (4) y = $wf2 d2 is the inverse lifetime of the excited atomic state in empty space. The atomic form-factor z(E)=(~-E>~/[(~~-A-~E)~+~c~I reflects the growth of y(k) and the density of polariton states near the upper edge of the lower polariton branch. Here we introduced some constant u to account for relaxation processes in the medium. The integration
is the self-energy and P stands for the principal value of the integral, The states of the continuous spectrum are orthonormal to each other, ( p I A) = 2&C fi A), and describe propagating polariton modes scattering on the atom. Let us now look for solutions of Eq. (6) with the eigenenergy lying within the gap, A E (0 - A, ii!). Then, the first term in Eq. (7) for the polar&on wave function fee 1A) does not contribute to either the
V.I. Rupasov, M. Singh/Physics
0.01 -0.01 2
go.03 w" -0.05
/
-0.07
1.005
1.055
5
%
1.155
1.105 ?I./%
Fig. 1. The self-energy of the bound state &h(h) for SiC. Parameters used in the computer calculations am [13]: y/o, = 3 X 10m6, wr = 0 - A = 14.9x lOi s-‘, wL = 0 = 17.9X lOI s-l; K/+ = 0.01 for the lower and K/C+ = 0.015 for the upper curve, respectively. The frequency gap A = 3 X 1Ol3 s- ’ is extended from h/o, = 1 to A/+= 1.2.
one-particle state or the equation for the atomic wave function g(h), and should be omitted. One then finds the following equation for g(A),
Letters A 222 (1996) 258-262
261
is determined from the normalization condition (A 1A> = 1. The discrete mode is orthogonal to the continuous spectrum, ( p 1A> = 0, and corresponds to the ~l~ton-atom bound state in which the atomic excitation is dressed by the polariton field localized in the vicinity of the atom. Next we study the problem of spontaneous emission assuming that at an initial moment of time t = 0 the atom is in the excited state, while the polariton field is in the vacuum state, iin) = CT+IO). The initial state is one-particle, and hence, is represented as a linear su~~sition of all the one-particle eigenstates of the atom-polariton system,
(11) where A(h)=(hlin)=g*(h)andB(A)=(Alin) = gi (A). Then, the time evolution of the initial state is given by IP(t))
=B( A)e-“‘i
A> f &EA(A)e-‘^‘I
A).
(12)
FT.+ (81 has a nontrivial solution gd( A> + 0, provided that the eigenenergy A is a root of the equation A-
“12
+&(A)
=0,,
A=(fl-A,
i’2).
As t -+ CQthe contribution of the atomic state in the second term vanishes due to spontaneous emission, and one obtains the probability of finding the atom in the excited state,
(13)
P=
(9)
The expression for ,ZJ A) is evaluated numerically
(see Fig. I). The first derivative of 2&j, d &,/dh, is obviously positive, and hence C,(A) is a monotonically increasing unction. Therefore IQ. (9) has only the solution which is given by the intersection of the curve XJh) with the straight line w12- X. Thus, we find the following expression for the discrete eigenstate of the system,
Here fd( E I .A) = fizC E>/I E - ~0, while
0.501
0.801
1.101 w/w,
1.401
-l/2
Fig. 2. The spectral density of the radiation atom with y/w, = 3X 10e6 and o,~,/o~ Sic. Here K/C+ = 0.01.
(see Eq. (14)) for the = 1.1 placed within
V.I. Rupasov, M. Singh/Physics
262
The spectral density of the radiation is given by
G(w)=
YZ2(
@>
Iw_
WI2 +zy6J)]”
+ [yz+4/2j2’
WEC,
(14)
where Z’(o) is the real part of the self-energy. The spectrum consists of two asymmetric lines separated by the polaritonic gap and is shown in Fig. 2. In empty space, G(w) inverts into the standard Wigner-Weisskopf expression for a natural radiation line /9,111. V. Rupasov is grateful to the Centre for Chemicd Physics at the University of Western Ontario for hospitality and support. M. Singh is thankful to NSERC of Canada for financial support in the form of a research grant. References [f] E. Yabionovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486.
Letters A 22.2 (1996) 258-262
I31 E. Yablonovitch, T.J. Grimmer and K.M. Leung, Phys. Rev. Lett. 67 (1991) 2295. 141Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65 (1990) 2650. I51 S. John and J. Wang, Phys. Rev. Lett. 64 (1990) 2418; Phys. Rev. B 43 (1991) 12772. Kl S. John and T. Quang, Phys. Rev. Lett. 74 (1995) 3419. M L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media (Pergamon, Oxford, 1984). @I V.M. Agranovich and V.L. Ginzburg, Crystal optics with spatial dispersion, and excitons (Springer, Berlin, 1984). t91 L. Allen and J.H. Eberly, Optical resonauce and two-level atoms (Wiley, New York, 1975). Adv. Phys. 32 ml A.M. Tsvelick and P.B. Wiegm~, (1983) 453; N. Andrei, K. Furuya and J.H. Lowenstein, Rev. Mod. Phys. 55 (1983) 331; V.I. Rupasov and V.I. Yudson, Zh. Eksp. Teor. Fiz. 87 (1984) 1617 [Sov. Phys. JETP 60 (1984) 9271. [Ill V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics (Pergamon, Oxford, 1982). 1121 AS. Davydov, Theory of molecular excitons (Plenum, New York, 1971). If31 C. Kittel, Int~uction to solid state physics (Wiley, New York, 1.986); R. Eisberg and R. Resnick, Quantum physics (Wiley, New York, 1985).
ELSFNIER
LETTERS
A
Physics Letters A 222 (1996) 258-262
Polariton-atom
bound state in a dispersive medium
Valery I. Rupasov ‘, M. Singh 2 Centrefor Chemical Physics and Department
of Physics, University of Western Ontario, London, Onz., Canada N6A SK7
Received 13 May 1996; accepted for publication 2 August 1996 Communicated by L.J. Sham
Abstract The quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. If the atomic resonance frequency lies near the gap, the spectrum of the system is shown to contain a novel polariton-atom bound state with an eigenfrequency lying within the gap. The radiation and medium polarization of the bound state are localized in the vicinity of the atom. A signi~c~t suppression of spontaneous emission due to the bound state is predicted.
The prediction [1,2] and the first experimental observation [3] of beg-dimensional dielectric structures, which exhibit a complete photonic band gap (PBG) for all directions of electromagnetic propagation, have created a great interest in a number of fundamental phenomena of classical and quantum electrodynamics in these materials. It has been shown that the existence of PI3G materials gives rise to such interesting phenomena as the suppression of spontaneous emission [4], the formation of strongly localized states of light [3], photon-atom bound states [S], and the localization of superradiance near a photonic band gap f61. In ~ificial PBG materials, a suppression of the photon density of states over a narrow frequency range results from multiple photon scattering by
’ On leave from the Landau Institute for Theoretical Physics and the Institute of Spectroscopy, Russian Academy of Sciences. E-mail: v~p~ov~juli~.uwo.ca. ’ E-mail: [email protected]. 03759601/96/$12.00 Copyright PII SO375-9601(96)00626-3
spatially correlated scatterers. But it is well known that a frequency gap for propagating ele~~oma~etic modes exists also in many natural dielectrics and semiconductors. In contrast to PBG materials, gaps in these media are caused by photon couplings to elementary excitations (excitons, optical phonons, etc.) of the media [‘7,8]. The “normal” electromagnetic modes in frequency dispersive media (DM) are determined by the Maxwell equations with a frequency-dependent dielectric permeability and are treated as “photons in a medium” (or “polaritons”, i.e. photons dressed by an interaction with a medium excitation). Their spectrum consists of two branches of allowed states separated by a gap in which propagating polariton modes are completely forbidden_ Therefore, the intriguing question arises whether quantum optical phenomena predicted for PBG materials are observed in dispersive media. In this paper we report some first results of qu~~rn elec~~yn~ics of ~l~itons interacting with a single two-level atom whose resonance fre-
0 1996 Elsevier Science B.V. All rights reserved.
V.I. Rupasou, M. Singh/Physics Letters A 222 (39963 X8-262
quency lies near a polaritonic gap of DM. Within the framework of the conventional dipole resonance (rotating wave) approximation [9], we diagondize exactly the Hamiltonian of the field + medium + atom system in the one-particle sector of the entire Hilbert space. We show that the system’s spectrum contains both branches corresponding to propagating polariton modes scattered on the atom and a single discrete mode whose eigenfrequency lies within the Platonic gap. This novel discrete mode can be treated as a polariton-atom bound state in which the atomic excitation is dressed by the radiation field and the medium polarization, which are both localized in the vicinity of the atom. To clarify the role of the bound state in quantum optical phenomena in DM, we solve the problem of the spontaneous decay of an initially excited atomic state and show that the existence of the bound state leads to a significant suppression of the spontaneous emission process. The Hamiltonian of the field + medium + atom system has the form
+e61dre(t’)*E(r)--d~(0),
(1)
259
as a quantum two-level system [9] with the transition frequency 0 ,2. Due to the spherical symmetry of the problem, the three-dimensional Hamiltonian of the system can be reduced to a one-dimensional form [lo]. Let us choose the system of coordinates with the origin at the point of the atom’s position and expand field and medium variables in terms of spherical harmonic vectors [I 11,
where the Bose operators cs; (bgi) describe a photon la medium excitation) of an electric fru = e) or a magnetic (ff = m> type with a frequency w (or the modulus of a wave vector k = w), an angular momentum j and projection thereof m. The Fourier component of the functions in Eq. (2) are. given by ‘/2
where the first three terms represent the Hamiltonians of the atom, the radiation and the medium, while the fourth and fifth terms represent the medium-field and atom-field couplings, respectively. The operators E and H describe the radiation field. The operators Q(r) and P(r) are the operators of the displacement and the momentum of the medium, respectively. The medium is treated here as a continuous set of charge harmonic oscillators, each with frequency fz, charge e, and mass mo. Here n is the density of the number of oscillators, and d = de(cr+ -I-CT-) is the atomic dipole operator, where d is the matrix element ‘of the dipole operator, e is the unit vector along the z-axis, and CT’ = (CTx, IT y, CT“1; uf=crxfitry are the spin operators. For the sake of simplicity, we do not account here for a possible degeneration of atomic levels, and we treat the atom
6(k-
w)~~‘(n),
112 8(k-w)~f)(n), 1 I
S(k-
o)Y.++(n) J”’
’
, !
where it is a unit vector in the directiontof k and Y!“) are the spherical harmonic vectors [Ill. In the e&tro-dipole approximation only the electro-dipole harmonic of the radiation field (cr =.e, j = 1) is coupled to the atom. The choice of the quantum number m = 0 is determined by our choice of the polarization of the atomic transition (d 11z-axis). Therefore, putting Eq. (2) and the analogous expression for the magnetic field H into Eq. (1) and omitting .a11 the
V.I. Rupasov, M. Sir&/
260
Physics Letters A 222 (1996/ 258-262
higher h~oni~s (j >, 2) uncoupled to the atom, we derive the following model Hamiltonian, H=w,,(tT’++) =dk + o 271. kc+(k)c(k) I
+L’b+(k)b(k)
t
+m[c+(k)b(k)
+b+(k)c(k)]
-~~[c(k)crC+cr-c+(k)]),
(3)
where A== ne2n/mof2, y(k) = $k3d2. Here the terms c”b+ (cb), which create (~~hilate) both field and medium excitations simultaneously, as well as the analogous terms (c’ a+ and ccf- > in the atom-field coupling operator, are omitted in accordance with the Heitler-London [S] and the resonance 191approximations. The field-medium part of the H~iltoni~ is diagonalized in terms of polariton operators p,(k) corresponding to the lower (a = - > and upper (a = +) polariton branches, respectively [ 121. For our purposes, it is more convenient to introduce the energy variable E, k = ~(0 - E)/(G - A - E), and the polariton operators P(E) on the “energy scale”, I+
contour C consists of two intervals: -W < E < (Jz A) and f2 < E < ~0.In accordance with the resonance approximation, the integration in the lower limit is extended to - ~0, and we set E = o,~ in the expression for -y(e). The “particle” number operator of the system,
-e_(k)
/&AaEE,(k)I
commutes with the model Hamiltonian. Therefore, all the model eigenstates can be classified with respect to the number of “particles”, or eigenvalues of the operator N. In this paper we study one-particle eigenstates, which can be written in the form
where the vacuum state of the system 10) is defined by g- IO) =p(e) IO) = 0. Then, the Schrijdinger equation (H - A) 1A) = 0 takes the form @-WWV
= fizGM$ (6)
I’* P,(k),
where ~~(k)=$(L?+k>+ \1(O-k)2+4kAlare the polariton spectra. Then, the model Hamiltonian finally takes the form
co
H=#~~(rr~+i)+IC~~~+(r)P(E)
where the first and second terms are the Hamiltonians of the atom and the polariton field, respectively, while the third one describes their couphng. In Eq. (4) y = $wf2 d2 is the inverse lifetime of the excited atomic state in empty space. The atomic form-factor z(E)=(~-E>~/[(~~-A-~E)~+~c~I reflects the growth of y(k) and the density of polariton states near the upper edge of the lower polariton branch. Here we introduced some constant u to account for relaxation processes in the medium. The integration
is the self-energy and P stands for the principal value of the integral, The states of the continuous spectrum are orthonormal to each other, ( p I A) = 2&C fi A), and describe propagating polariton modes scattering on the atom. Let us now look for solutions of Eq. (6) with the eigenenergy lying within the gap, A E (0 - A, ii!). Then, the first term in Eq. (7) for the polar&on wave function fee 1A) does not contribute to either the
V.I. Rupasov, M. Singh/Physics
0.01 -0.01 2
go.03 w" -0.05
/
-0.07
1.005
1.055
5
%
1.155
1.105 ?I./%
Fig. 1. The self-energy of the bound state &h(h) for SiC. Parameters used in the computer calculations am [13]: y/o, = 3 X 10m6, wr = 0 - A = 14.9x lOi s-‘, wL = 0 = 17.9X lOI s-l; K/+ = 0.01 for the lower and K/C+ = 0.015 for the upper curve, respectively. The frequency gap A = 3 X 1Ol3 s- ’ is extended from h/o, = 1 to A/+= 1.2.
one-particle state or the equation for the atomic wave function g(h), and should be omitted. One then finds the following equation for g(A),
Letters A 222 (1996) 258-262
261
is determined from the normalization condition (A 1A> = 1. The discrete mode is orthogonal to the continuous spectrum, ( p 1A> = 0, and corresponds to the ~l~ton-atom bound state in which the atomic excitation is dressed by the polariton field localized in the vicinity of the atom. Next we study the problem of spontaneous emission assuming that at an initial moment of time t = 0 the atom is in the excited state, while the polariton field is in the vacuum state, iin) = CT+IO). The initial state is one-particle, and hence, is represented as a linear su~~sition of all the one-particle eigenstates of the atom-polariton system,
(11) where A(h)=(hlin)=g*(h)andB(A)=(Alin) = gi (A). Then, the time evolution of the initial state is given by IP(t))
=B( A)e-“‘i
A> f &EA(A)e-‘^‘I
A).
(12)
FT.+ (81 has a nontrivial solution gd( A> + 0, provided that the eigenenergy A is a root of the equation A-
“12
+&(A)
=0,,
A=(fl-A,
i’2).
As t -+ CQthe contribution of the atomic state in the second term vanishes due to spontaneous emission, and one obtains the probability of finding the atom in the excited state,
(13)
P=
(9)
The expression for ,ZJ A) is evaluated numerically
(see Fig. I). The first derivative of 2&j, d &,/dh, is obviously positive, and hence C,(A) is a monotonically increasing unction. Therefore IQ. (9) has only the solution which is given by the intersection of the curve XJh) with the straight line w12- X. Thus, we find the following expression for the discrete eigenstate of the system,
Here fd( E I .A) = fizC E>/I E - ~0, while
0.501
0.801
1.101 w/w,
1.401
-l/2
Fig. 2. The spectral density of the radiation atom with y/w, = 3X 10e6 and o,~,/o~ Sic. Here K/C+ = 0.01.
(see Eq. (14)) for the = 1.1 placed within
V.I. Rupasov, M. Singh/Physics
262
The spectral density of the radiation is given by
G(w)=
YZ2(
@>
Iw_
WI2 +zy6J)]”
+ [yz+4/2j2’
WEC,
(14)
where Z’(o) is the real part of the self-energy. The spectrum consists of two asymmetric lines separated by the polaritonic gap and is shown in Fig. 2. In empty space, G(w) inverts into the standard Wigner-Weisskopf expression for a natural radiation line /9,111. V. Rupasov is grateful to the Centre for Chemicd Physics at the University of Western Ontario for hospitality and support. M. Singh is thankful to NSERC of Canada for financial support in the form of a research grant. References [f] E. Yabionovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486.
Letters A 22.2 (1996) 258-262
I31 E. Yablonovitch, T.J. Grimmer and K.M. Leung, Phys. Rev. Lett. 67 (1991) 2295. 141Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65 (1990) 2650. I51 S. John and J. Wang, Phys. Rev. Lett. 64 (1990) 2418; Phys. Rev. B 43 (1991) 12772. Kl S. John and T. Quang, Phys. Rev. Lett. 74 (1995) 3419. M L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media (Pergamon, Oxford, 1984). @I V.M. Agranovich and V.L. Ginzburg, Crystal optics with spatial dispersion, and excitons (Springer, Berlin, 1984). t91 L. Allen and J.H. Eberly, Optical resonauce and two-level atoms (Wiley, New York, 1975). Adv. Phys. 32 ml A.M. Tsvelick and P.B. Wiegm~, (1983) 453; N. Andrei, K. Furuya and J.H. Lowenstein, Rev. Mod. Phys. 55 (1983) 331; V.I. Rupasov and V.I. Yudson, Zh. Eksp. Teor. Fiz. 87 (1984) 1617 [Sov. Phys. JETP 60 (1984) 9271. [Ill V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics (Pergamon, Oxford, 1982). 1121 AS. Davydov, Theory of molecular excitons (Plenum, New York, 1971). If31 C. Kittel, Int~uction to solid state physics (Wiley, New York, 1.986); R. Eisberg and R. Resnick, Quantum physics (Wiley, New York, 1985).