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Polariton spontaneous emission superradiance and polariton-impurity states in III–V semiconductors
30 June 1997 PHYSICS LETTERS A
Physics Letters A 231 (1997) 115-122
ELSBVIER
Polariton spontaneous emission superradiance and polariton-impurity states in III-V semiconductors Mahi R. Singh, Wayne Lau Department of Physics,
University of Western Ontario, London. Canada N6A 3K7
Received 20 January 1997; revised manuscript received 2 April 1997; accepted for publication 15 April 1997 Communicated by L.J. Sham
Abstract In this paper, we study the polariton energy dispersion of two different types of two-level atoms placed within a III-V semiconductor which has a polariton gap due to photon coupling to optical phonons. We employ the spherical harmonic representation to derive the model Hamiltonian within the framework of the dipole resonance approximation. ‘ILvopolariton impurity states lying within the polariton gap are found. The spontaneous emission rate of a system with two identical two-level atoms placed within a III-V semiconductor is also studied as a function of interatomic distance when the atomic resonance frequency lies in the polariton continuous spectra. It is found that when the interatomic separation between the atoms is small, the polariton-atom system in the symmetric state can radiate a polariton with a probability that is twice that of the independent or single atom case (i.e. superradiance). On the other hand, when the polariton-atom system in the antisymmetric state, the system is prevented from radiating a polariton (i.e. s&radiance). Numerical calculations are performed for polariton-impurity states and polariton spontaneous decay rate of the polariton-atom system for GaAs semiconductor. @ 1997 Elsevier Science B.V.
1. Introduction In the past few years considerable attention has been paid to photonic band gap materials due to their unusual optical properties and potential applications [ 11. The most interesting phenomena in photonic band gap materials are the formation of photon-atom bound states and suppression of spontaneous emission from the photon-atom bound state [ 21. Recently, Rupasov and Singh have studied the quantum electrodynamics of a two-level atom placed within a frequency dispersive medium whose polariton spectrum contains a energy gap [ 31. In photonic band gap materials, the existence of the photonic band gap is due to multiple photon scattering by spatially correlated scatters, while in dispersive media such as semiconductors and dielectrics, the energy gap is caused by photon coupling to an elementary excitation (excitons, optical phonons etc.) of the media. They found that if the atomic resonance frequency lies within the gap, then the spectrum of the system contains a polar&on-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. They predicted that the spontaneous emission of the polariton-atom system is significantly suppressed due to the presence of the bound state. Rupasov and Singh 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PflSO375-9601(97)00295-8
116
M.R. Singh, W La/Physics
Letters A 231 (1997) 115-122
have also studied the quantum electrodynamics of two identical two-level atoms placed within a frequency dispersive medium [ 41. The aim of the present paper is to study the spontaneous emission of photons for III-V semiconductors doped with two two-level atoms. This phenomenon has not been studied by Rupasov and Singh, and also their theory has not been applied to the III-V semiconductors such as GaAs, InAs, GaSb. We have also derived an expression for the energy dispersion of polaritons (photons dressed by an interaction with a medium excitation) when two different types of two-level atoms are placed within the III-V semiconductors. We follow the method of Rupasov and Singh to derive the expression of the energy dispersion relation. In this method, the Hamiltonian of the system is expressed in the spherical harmonic representation [ 51. The Hamiltonian is then diagonalized exactly in the one-particle sector of the entire Hilbert space within the dipole resonance approximation. We consider that the atomic resonance frequencies of the two-level atoms lie either inside the polariton gap or the polariton continuous spectra. It is found that when the atomic resonance frequencies of both the atoms lie within the gap, the spectrum of the system contains two polariton-atom bound states with two eigenfrequencies lying within the gap. As the interatomic distance between the two atoms decreases, the energy difference between the bound states increases. This is due to the increase of the effective coupling between the atom-atom interaction as the interatomic distance decreases. The two localized states merge into one if the two atoms are taken to be identical, which agrees with the earlier finding of Rupasov and Singh [5]. To study the spontaneous decay rate of an initially excited atomic state of polariton-atom system, we consider that the atomic resonance frequencies of two atoms lie either inside the polariton gap or the polariton continuous spectra. For simplicity, we restrict ourself to the case of two identical atoms because the calculation of the two different atoms case is mathematically very complicated and difficult to get analytical results. We consider the case in which one atom is in the excited state while the other is in the ground state with no polaritons present in the system and the atomic resonance frequencies of two atoms lie in the polariton continuous spectra. It is found that when the interatomic distance between the atoms becomes very large, the spontaneous decay rate of the excited atomic state is equal to that of the single atom case. In other words, the existence of the second atom does not contribute to the spontaneous emission of the polariton. Note that when the interatomic distance becomes very large, the excited states of the system are twice degenerate. On the other hand, when the two atoms are very close to each other, the degenerate states split into a symmetric and an antisymmetric states. Therefore, for very small interatomic distances, it is found that the rate of spontaneous emission from the symmetric state is two times that of the single atom case. This means that the polariton-atom system in the symmetric state can radiate a polariton with a probability that is twice of the independent or single atom case. We call this phenomenon as spontaneous superradiance since it is related to superradiance in quantum optics [ 6-81. For the polariton-atom system in the antisymmetric state, the spontaneous emission rate is found to be zero. This means that the system is prevented from radiating a polariton from the excited state to the ground state due to the presence of another atom in the system. This phenomenon is related to subradiance in quantum optics [6-81 and we call it spontaneous subradiance. Numerical calculations are performed for polariton-impurity states and polariton spontaneous decay rate of the excited atomic state of the polariton-atom system for GaAs semiconductor.
2. Theory Let us consider a situation where two different types of two-level atoms are placed within a III-V semiconductor. The polariton spectrum of the III-V semiconductor has an energy gap due to photon coupling to the optical phonons [ 91. The excited state of a two-level atom lies either in the polariton gap or the polariton continuous spectra. Following the method of Rupasov and Singh, in the spherical harmonic representation [ 51, we obtained the following model Hamiltonian:
M.R. Singh, W La/Physics
Letters A 231 (1997) 115-I22
117
(1)
where
gjiiw= J do,Tjm(n) G(E)
= fi(fk
-
exp(fik(e)Z.n/2),
E)2/[
(&
-
E)2 + tc’],
k(E) = E(fk
- E)(&
- E)_l,
Kj accounts for relaxation processes in the III-V semiconductors, 111is the interatomic separation, and C.C.stands for the complex conjugate of the last term in the Hamiltonian. fit_ and & are maximum and minimum energy points in the lower and upper branches of the polariton spectrum, respectively. The first and second terms in the above equation represent the free atoms and polaritons, respectively, while the last term represents the coupling between atoms and polaritons. Here wi is the energy difference between the excited state and ground state of the ith atom. cjm and cTrnare the raising and lowering operators in states with the angular momentum j and z-component of angular momentum m. Tjm( n) is the spherical harmonic along the unit vector n which is parallel to the wave vector k [4], yi is the coupling constant between photons and the ith atom, and zi( E) reflects the growth of yi and the density of polariton states near the upper edge of the lower polariton branch. In obtaining Eq. ( 1) , we have used the resonance (rotating-wave) approximation [ 7,8]. The integration contour C consists of two semi-infinite intervals, C = (-00, 0~1 U[ &, 00) and .& - & is the energy gap appearing in the polariton spectrum. a: and uf = a; f iaf are the Pauli spin operators. Here an atom is regarded as a quantum system in which the ground state (singlet) has j = 0 and m = 0, while the excited state (triplet) has j = 1 and M = 0, f 1. In this paper, we only consider the transition j = 0, m = 0 +--+ j = 1, m = 0 for the one-particle excitations of the polariton-atom system. Let us now look for the solution of the Schrodinger equation (i.e. HI A) = AlA)) for one-particle eigenstates. We seek the one-particle eigenstates [3] in the form of
w;
IA)=
+g2
u+c 2’
Jg
fjm(E)CL(E)
jm c
i
1
10)~
where A is the eigenenergy and IO) is the vacuum state. The first term of Pq. (2) indicates that the first atom is in excited state while the second atom is in the ground state and the number of polaritons is zero. The second term indicates that the second atom is in an excited state while first atom is in the ground state and the number of polaritons is zero. The last term indicates the number of polaritons is one, while both atoms are in their respective ground states. With the help of E!qs. ( 1) and (2), the Schrodinger equation reduces to a set of coupled equations
(E -
A)fjm(EIA)
J
=
g
[zl(E)gl(A)[S~(E)l*+z2(E)g2(A)[~~(E)l*]
7
(3)
C
(~1 - A)gl(A)
=C
(wZ-
=C
A)gz(A)
J~zI(E)firn(~IA)[i:m(E)l*, jm c J$a(~)fjrn(sIA)[i;,(r)I*.
jm c
(4)
(5)
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M.R. Singh, W Lm/Physics
Letters A 231 (1997) 115-122
Let us first consider the case when the atomic resonance frequencies (i.e. wi), lie in the polariton gap and the polariton continuous spectra. For the eigenenergies lying in the continuous spectra, i.e., A E C, the general solution of Eq. (3) is
fjm(lIA)from Eq. (6):
Eqs. (4) and (5) r ed uce as follows after substituting the expression of [A -
w1+
[A-
w
-s(A)lg1(~)
+ &2(Mg2(M
=cl(h),
+ -C2(Mlg2(4
+ &2(Ngl(A)
=C2(A),
(7)
where Cl(A) =
-zI(A)
CFj,(A)6&(E),
G(A)
= -zz(A)
CFjm(A)S,-,(e)v
im
im
and
s
de z?(e) GE-A-i0
2i(A) =
&2(A)
=
C
Q(E) =
F:yl j’m’
s
de ZI(~)ZZ(E)&(E) e-A-i0
G
(8)
’
c
[[&,(~)l*
+ [4’$~>1*]
5j?ml(~) = Xsin(k(E)0
- UEY
cos(k(~)l>>/[k(~>Zl~.
jm
_Z12( A) describes the contribution of the effective interatomic coupling to the atoms. After some mathematical manipulation we get the atomic wave function coefficients of Eq. (7) as
(A - WI+ &(A))CI(N
atA) = g2(4
LA_
- ZntA)GtA)
WI +&(A)l[A-~2+S22(A)l
= [A _
- [&2(A)12’
(A - ~2 + Z2z(A))C,(A)
(9)
+ ZIZ(A)CI(A)
~1 +&(A)l[A-~2+S22(A)l
- [&2(A)12’
Note that when the energy of the atomic resonance state lies outside the gap, there are no eigenfrequencies lie within the gap. Let us now look for the solution of the Schriidinger equations corresponding to the atomic resonance frequencies and eigenenergies lie within the gap. A = A E [&, 0~1. In this case, the solution of Eq. (3) is now given by zlwgl(A)[S~wl* fjm(EIA)
+Z2wg2(AH5,~wl*
=
Substituting I$
(10)
E-A ( 10) into Eqs. (4) and (5)) we get
[A - WI + 21 (A) la(A)
+ &z(A)gz(A)
= 0,
[A - ~2 + Z2(A)lg2(A)
+ .%2(A)gl(A)
= 0,
(11)
where Xi and 212 are real quantities and are given by &(A)
=
I
C
de zfW ATE-A’
&2(A)
=
I
C
de z~(E)zz(E)&(E) 2?T
E-A
*
(12)
M.R. Singh, W Lou/Physics LettersA 231 (1997) 115-122
The bound-state [A-w,
eigenenergies
are the roots of the following
+~1(A)ltA-~2+~2(A)l
equation
119
which is obtained
from Eq.( 11)
-[&2(A)12=0.
(13)
This equation gives two discrete modes which should be treated as polariton-atom bound states in which the radiation and medium polarization are localized in the vicinity of the two atoms. The above expressions can be reduced to the problem of two identical atoms by putting o1 = w2 and zr (E) = z2 ( E) . In this case we get 21 (A) = 22 (A), and Rq. (7) for A E C reduces to [A - 01 + -&(A)lg*(A) &(A) &(A)
= A+
Ah(A)
= G(A)*
(14)
G(h)
= g1 (A) f g2(A),
= Cl(A)
f
C2(A).
(A) + iT& (A) is known as the self-energy,
*Q(E)]
de z2(e)[1
=P s
C
27T
E-A
’
&(A)
= 21(A)
and A* (A) and r&(A)
T&(A) = +r]z2(A)(l
*Qe(A))l.
f
212(A).
are given by
(15)
where the positive sign corresponds to the system in the symmetric state (triplet state) and the negative sign corresponds to the system in the antisymmetric state (singlet state) [4]. Similarly, Rq. ( 13) for the case A=A E [&,&I reduces to A-WI where &(A) A&(A)
+&(A)
=O,
(16)
= 2:1(A) f 212(A) = A*(A)
&Q(c)1
de z2W[l
=P I c
!iiG
+iTi(A),
E-n
’
with
T*.(A) =O.
These expressions are obtained in Ref. [4]. Now we study the spontaneous emission rate for the system in which two identical atoms are doped in a semiconductor. Note that the case for two different atoms is mathematically complicated and difficult to get analytical results for the spontaneous emission rate. First, we study the case when the resonance frequency of the two-level atoms lies in the polariton continuous spectra. The spontaneous emission rate (i.e. transition probability per unit time) is given by the imaginary part of the self-energy, r*(A), given in Eq. ( 15). The relation between the spontaneous emission rate and the imaginary part of self-energy can be found in the text books (for example see Ref. [9] ). Here we consider the spontaneous emission of a polariton from an initial state in which one atom is in an excited state while the other is in the ground state and the number of polaritons is zero, to a final state in which both atoms are in their ground states and the number of polaritons is one. The expression of the spontaneous emission rate for this process is obtained from Eq.( 15) by replacing A with wi. First let us consider the situation where the interatomic distance between the atoms is very large (i.e. I + oo) . In this case, the symmetric and antisymmetric states are degenerate and Q(wi) becomes zero. Therefore, the expression for the rate of spontaneous emission reduces to r( 01) = iyz2( WI) which is nothing but the spontaneous emission rate due to the presence of one atom in the medium [ 31. Hence, in the limit of large interatomic distances, the existence of the second atom does not contribute to the rate of spontaneous decay rate of the excited atom since the atom-atom coupling is negligible, and the two atoms behave as two independent atoms. In the opposite limit, when two atoms are very close to each other (i.e. E N 0), the symmetric and antisymmetric states are nondegenerate. When the system is in the symmetric state, the expression for the rate of spontaneous emission reduces to yz 2 ( wi ) , which is two times that of the single atom case. This means that the system in the symmetric state can radiate a polariton with a probability that is twice that of the independent or single atom case. On the other hand, when the system is in the antisymmetric state, the expression for the
M.R. Singh, U! Lau/Physics Letters A 231 (1997) 115-122
120
0.25
9.25
18.25
27.25
36.25
0.04
0.24
hTI
0.44
0.64
LTI
Fig. 1. The spontaneous emission rate as functions of the interatomic separation, hri. The upper, lower, and middle curves correspond to the superradiance [ f + ( wr ) 1, the subradiance [ r- (A) 1, and the spontaneous radiation from the single atom case, respectively. Here kr = L+/c and c is the velocity of light. Fig. 2. The energies of the impurity states as functions of the interatomic separation hrl. Here AT = e/c
and c is the velocity of light.
spontaneous emission rate becomes zero. This means that the polariton-atom system is prevented from radiating polaritons from the excited state to the ground state due to the presence of another atom in the system. The spontaneous emission rate is also derived in quantum optics [ 81 when two identical two-level atoms are placed in a cavity. The spontaneous emission from the symmetric state and antisymmetric state are called superradiance and subradiance respectively in Ref. [ 81. Therefore, in this paper, we will use the same nomenclature as that of in Ref. [ 81. Note that superradiance in the quantum optics is a collective phenomenon involving many photons. To include the effects of many photons in our theory is very complicated and it is beyond the scope of this paper. Finally, in case where the resonance frequency of the atoms lies within the gap, r( wt) is equal to zero. Hence the spontaneous emission rate is zero and the polariton-atom bound states are present within the polariton gap.
3. Results
and discussion
Using the expression r&( A = wr ) given in Eq. ( 15), we have calculated the spontaneous emission rate from an initial state in which one atom is in an excited state while the other atom is in ground state and the number of polaritons is zero, to a final state in which both atoms are in their ground states and the number of polaritons is one. We choose the excited states of atoms to lie in the continuous energy spectrum (i.e. wi /& = 1 .OS). The numerical calculations are presented in Fig. 1 for GaAs semiconductors. The parameters used in the calculations are: & = 5.5 x 1013 s-l, & = 5.1 x 1013 s-l, yt/& = yz/& = 1.19 x 10e5, and K/J& = 0.01. In the figure, the upper and lower curves correspond to the emission rate from the symmetric (r+ (wt ) ) and antisymmetric (r_ (wt ) ) states respectively, while the middle curve corresponds to the emission rate (r( wt ) ) for the single atom case. The magnitudes of r+ ( wt ) , lY_ (01) , and r( ~1) are normalized with respect to that of r( wt ) . Note from Fig. 1 that when the interatomic distance is close to zero, the spontaneous emission rate from the symmetric state (upper curve) is two times that of the single atom case (middle curve). This means that the system in the symmetric state can radiate a polariton with a probability that is twice of the independent or single atom case. On other hand, the rate of spontaneous emission from the antisymmetric state (lower curve) is equal to zero. This means that the system in the antisymmetric state is prevented from radiating a polariton. Therefore, we call the upper curve in Fig. 1 as the superradiant curve and the lower curve as the subradiant curve. Note also from the figure that as the interatomic distance increases, superradiant transition rate decreases whereas the subradiance emission rate increases with interatomic separation. In the limit of large interatomic
M.R. Singh, W. L.au/Physics
Letters A 231 (1997) 115-122
121
distance, the emission rates reduce to that of the single atom case. Therefore, the presence of the second atom does not contribute to the spontaneous emission of polariton. It is interesting to note that the behavior of the spontaneous decay rate of the excited atom is oscillatory. In the last section, we have shown that there are two discrete impurity states in III-V semiconductors doped with two different types of atoms. Using Eq. (13), we have calculated the eigenenergies of the impurity states for GaAs semiconductors as a function of interatomic separation, and the results are presented in Fig. 2. The parameters used in the calculations are: WI/& = 1.040, wz/& = 1.041, rl/& = 1.19 x 10e5, and ye/& = 1.67 x 10v5. Clearly, as the interatomic distance increases, the energy difference between the two impurity states decreases. If we set WI = w2 and yi = y2 in our calculations, then the two impurity states reduce to a single state which agrees with the earlier finding [4]. In the limit of large interatomic distances, the atom-atom interaction vanishes; consequently, the value of the lower curve approaches the energy, WI /a, while the value of the upper curve approaches the energy, w2/&. In the limit of large interatomic distance, unlike the case of two identical atoms, where the polariton-atom bound states is twice degenerate, the bound states in this case are nondegenerate. We have also calculated the eigenenergies of the impurity states as a function of interatomic distance for the two identical atom case. we also found the behavior of the eigenenergies is oscillatory. This figure is not presented here. But Fig. 2 is calculated for two different atoms which the resonance energies of the two atoms are different. In this case, we found that the oscillatory behavior disappears or is negligible. Finally, we would like to suggest a few possible experiments where the phenomena mentioned above can be observed. III-V semiconductor compounds doped with donors or acceptors can be used to study the above physical quantities provided the resonance states of donors or acceptors lie in the range of polariton spectra of these semiconductors. There are donors and acceptors in the literature [lo] whose resonance frequencies lie in the energy range where we are interested. Two semiconductor quantum dots doped in III-V semiconductors can be also used to study the above phenomena. Here a quantum dot acts as a two-level system. The required resonance frequencies of quantum dots can be obtained by changing their physical dimensions. The above phenomena can be also observed in III-V semiconductor doped with two quantum wells such as A/B/A = AlAs/GaAs/AlAs. The quantum wells act as two-level systems and their resonance frequencies can be tuned by changing the width of the B layer to meet the requirement of our theory. The experiments can be also preformed in a system consists of two quantum wells (i.e. A/B/A/B/A) m which the width of the A layers is very large compare to that of the B layers. The layer B will acts as a two-level system. In the present theory, we need three dimensional phonons to get the polariton spectrum, and the system of two quantum wells will give us three dimensional phonons. If we apply the magnetic filed along the growth direction in the last two systems, we get Landau levels. These Landau levels can act as two-level systems and their energy difference can be changed by changing the magnetic field strength to meet the requirement of our theory.
Acknowledgments The authors would like to thank Dr. V. I. Rupasov for helpful discussions and Mr. M. Casse for computational help. One of the authors (MS) is thankful to NSERC of Canada for financial support in the form of a research grant.
References I I 1 E. Yablonovitch, Phys. Rev. L-S. 58 (1987) 2059; S. John, Phys. Rev. J_ett. 58 (1987) 2486; E. Yablonovitch, T.J. Grimmer and K.M. Leung, Phys. Rev. Len. 67 ( 1991) 2295.
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MR. Singh, N! Lau/Physics
Letters A 231 (1997) 115-122
[2] Z. Zhang and S. Satpathy, Phys. Rev. Len. 65 (1990) 2650;
S. John and J. Wang, Phys. Rev. Lett. 64 ( 1990) 2418.
[ 31 V.I. Rupasov and M. Singh, Phys. L&t. A 222 (1996) 258. [4] [ 51 [ 61 [7] [ 81 [9]
V.I. Rupasov and M. Singh, Phys. Rev. A., submitted (1996). V.B. Bemstetskii et al., Quantum Electrodynamics (Pergamon, Oxford, 1982). R.H. Dicke, Phys. Rev. 93 (1954) 99. L. Allen and J.H. Eberly, Optical Resonance and lXvo-level Atoms (Wiley, New York, 1975). M. Weissbluth, Photon-Atom Interactions (Academic Press, New York, 1989). C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986); V.M. Agranovich and V.L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, Berlin, 1984). [IO] E.S. Kotekes, S. Zemon and P. Morris, Gallium Arsenide and Related Compounds 1985, Institute of Physics Conference Series, Vol 79, p. 259.
Physics Letters A 231 (1997) 115-122
ELSBVIER
Polariton spontaneous emission superradiance and polariton-impurity states in III-V semiconductors Mahi R. Singh, Wayne Lau Department of Physics,
University of Western Ontario, London. Canada N6A 3K7
Received 20 January 1997; revised manuscript received 2 April 1997; accepted for publication 15 April 1997 Communicated by L.J. Sham
Abstract In this paper, we study the polariton energy dispersion of two different types of two-level atoms placed within a III-V semiconductor which has a polariton gap due to photon coupling to optical phonons. We employ the spherical harmonic representation to derive the model Hamiltonian within the framework of the dipole resonance approximation. ‘ILvopolariton impurity states lying within the polariton gap are found. The spontaneous emission rate of a system with two identical two-level atoms placed within a III-V semiconductor is also studied as a function of interatomic distance when the atomic resonance frequency lies in the polariton continuous spectra. It is found that when the interatomic separation between the atoms is small, the polariton-atom system in the symmetric state can radiate a polariton with a probability that is twice that of the independent or single atom case (i.e. superradiance). On the other hand, when the polariton-atom system in the antisymmetric state, the system is prevented from radiating a polariton (i.e. s&radiance). Numerical calculations are performed for polariton-impurity states and polariton spontaneous decay rate of the polariton-atom system for GaAs semiconductor. @ 1997 Elsevier Science B.V.
1. Introduction In the past few years considerable attention has been paid to photonic band gap materials due to their unusual optical properties and potential applications [ 11. The most interesting phenomena in photonic band gap materials are the formation of photon-atom bound states and suppression of spontaneous emission from the photon-atom bound state [ 21. Recently, Rupasov and Singh have studied the quantum electrodynamics of a two-level atom placed within a frequency dispersive medium whose polariton spectrum contains a energy gap [ 31. In photonic band gap materials, the existence of the photonic band gap is due to multiple photon scattering by spatially correlated scatters, while in dispersive media such as semiconductors and dielectrics, the energy gap is caused by photon coupling to an elementary excitation (excitons, optical phonons etc.) of the media. They found that if the atomic resonance frequency lies within the gap, then the spectrum of the system contains a polar&on-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. They predicted that the spontaneous emission of the polariton-atom system is significantly suppressed due to the presence of the bound state. Rupasov and Singh 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PflSO375-9601(97)00295-8
116
M.R. Singh, W La/Physics
Letters A 231 (1997) 115-122
have also studied the quantum electrodynamics of two identical two-level atoms placed within a frequency dispersive medium [ 41. The aim of the present paper is to study the spontaneous emission of photons for III-V semiconductors doped with two two-level atoms. This phenomenon has not been studied by Rupasov and Singh, and also their theory has not been applied to the III-V semiconductors such as GaAs, InAs, GaSb. We have also derived an expression for the energy dispersion of polaritons (photons dressed by an interaction with a medium excitation) when two different types of two-level atoms are placed within the III-V semiconductors. We follow the method of Rupasov and Singh to derive the expression of the energy dispersion relation. In this method, the Hamiltonian of the system is expressed in the spherical harmonic representation [ 51. The Hamiltonian is then diagonalized exactly in the one-particle sector of the entire Hilbert space within the dipole resonance approximation. We consider that the atomic resonance frequencies of the two-level atoms lie either inside the polariton gap or the polariton continuous spectra. It is found that when the atomic resonance frequencies of both the atoms lie within the gap, the spectrum of the system contains two polariton-atom bound states with two eigenfrequencies lying within the gap. As the interatomic distance between the two atoms decreases, the energy difference between the bound states increases. This is due to the increase of the effective coupling between the atom-atom interaction as the interatomic distance decreases. The two localized states merge into one if the two atoms are taken to be identical, which agrees with the earlier finding of Rupasov and Singh [5]. To study the spontaneous decay rate of an initially excited atomic state of polariton-atom system, we consider that the atomic resonance frequencies of two atoms lie either inside the polariton gap or the polariton continuous spectra. For simplicity, we restrict ourself to the case of two identical atoms because the calculation of the two different atoms case is mathematically very complicated and difficult to get analytical results. We consider the case in which one atom is in the excited state while the other is in the ground state with no polaritons present in the system and the atomic resonance frequencies of two atoms lie in the polariton continuous spectra. It is found that when the interatomic distance between the atoms becomes very large, the spontaneous decay rate of the excited atomic state is equal to that of the single atom case. In other words, the existence of the second atom does not contribute to the spontaneous emission of the polariton. Note that when the interatomic distance becomes very large, the excited states of the system are twice degenerate. On the other hand, when the two atoms are very close to each other, the degenerate states split into a symmetric and an antisymmetric states. Therefore, for very small interatomic distances, it is found that the rate of spontaneous emission from the symmetric state is two times that of the single atom case. This means that the polariton-atom system in the symmetric state can radiate a polariton with a probability that is twice of the independent or single atom case. We call this phenomenon as spontaneous superradiance since it is related to superradiance in quantum optics [ 6-81. For the polariton-atom system in the antisymmetric state, the spontaneous emission rate is found to be zero. This means that the system is prevented from radiating a polariton from the excited state to the ground state due to the presence of another atom in the system. This phenomenon is related to subradiance in quantum optics [6-81 and we call it spontaneous subradiance. Numerical calculations are performed for polariton-impurity states and polariton spontaneous decay rate of the excited atomic state of the polariton-atom system for GaAs semiconductor.
2. Theory Let us consider a situation where two different types of two-level atoms are placed within a III-V semiconductor. The polariton spectrum of the III-V semiconductor has an energy gap due to photon coupling to the optical phonons [ 91. The excited state of a two-level atom lies either in the polariton gap or the polariton continuous spectra. Following the method of Rupasov and Singh, in the spherical harmonic representation [ 51, we obtained the following model Hamiltonian:
M.R. Singh, W La/Physics
Letters A 231 (1997) 115-I22
117
(1)
where
gjiiw= J do,Tjm(n) G(E)
= fi(fk
-
exp(fik(e)Z.n/2),
E)2/[
(&
-
E)2 + tc’],
k(E) = E(fk
- E)(&
- E)_l,
Kj accounts for relaxation processes in the III-V semiconductors, 111is the interatomic separation, and C.C.stands for the complex conjugate of the last term in the Hamiltonian. fit_ and & are maximum and minimum energy points in the lower and upper branches of the polariton spectrum, respectively. The first and second terms in the above equation represent the free atoms and polaritons, respectively, while the last term represents the coupling between atoms and polaritons. Here wi is the energy difference between the excited state and ground state of the ith atom. cjm and cTrnare the raising and lowering operators in states with the angular momentum j and z-component of angular momentum m. Tjm( n) is the spherical harmonic along the unit vector n which is parallel to the wave vector k [4], yi is the coupling constant between photons and the ith atom, and zi( E) reflects the growth of yi and the density of polariton states near the upper edge of the lower polariton branch. In obtaining Eq. ( 1) , we have used the resonance (rotating-wave) approximation [ 7,8]. The integration contour C consists of two semi-infinite intervals, C = (-00, 0~1 U[ &, 00) and .& - & is the energy gap appearing in the polariton spectrum. a: and uf = a; f iaf are the Pauli spin operators. Here an atom is regarded as a quantum system in which the ground state (singlet) has j = 0 and m = 0, while the excited state (triplet) has j = 1 and M = 0, f 1. In this paper, we only consider the transition j = 0, m = 0 +--+ j = 1, m = 0 for the one-particle excitations of the polariton-atom system. Let us now look for the solution of the Schrodinger equation (i.e. HI A) = AlA)) for one-particle eigenstates. We seek the one-particle eigenstates [3] in the form of
w;
IA)=
+g2
u+c 2’
Jg
fjm(E)CL(E)
jm c
i
1
10)~
where A is the eigenenergy and IO) is the vacuum state. The first term of Pq. (2) indicates that the first atom is in excited state while the second atom is in the ground state and the number of polaritons is zero. The second term indicates that the second atom is in an excited state while first atom is in the ground state and the number of polaritons is zero. The last term indicates the number of polaritons is one, while both atoms are in their respective ground states. With the help of E!qs. ( 1) and (2), the Schrodinger equation reduces to a set of coupled equations
(E -
A)fjm(EIA)
J
=
g
[zl(E)gl(A)[S~(E)l*+z2(E)g2(A)[~~(E)l*]
7
(3)
C
(~1 - A)gl(A)
=C
(wZ-
=C
A)gz(A)
J~zI(E)firn(~IA)[i:m(E)l*, jm c J$a(~)fjrn(sIA)[i;,(r)I*.
jm c
(4)
(5)
118
M.R. Singh, W Lm/Physics
Letters A 231 (1997) 115-122
Let us first consider the case when the atomic resonance frequencies (i.e. wi), lie in the polariton gap and the polariton continuous spectra. For the eigenenergies lying in the continuous spectra, i.e., A E C, the general solution of Eq. (3) is
fjm(lIA)from Eq. (6):
Eqs. (4) and (5) r ed uce as follows after substituting the expression of [A -
w1+
[A-
w
-s(A)lg1(~)
+ &2(Mg2(M
=cl(h),
+ -C2(Mlg2(4
+ &2(Ngl(A)
=C2(A),
(7)
where Cl(A) =
-zI(A)
CFj,(A)6&(E),
G(A)
= -zz(A)
CFjm(A)S,-,(e)v
im
im
and
s
de z?(e) GE-A-i0
2i(A) =
&2(A)
=
C
Q(E) =
F:yl j’m’
s
de ZI(~)ZZ(E)&(E) e-A-i0
G
(8)
’
c
[[&,(~)l*
+ [4’$~>1*]
5j?ml(~) = Xsin(k(E)0
- UEY
cos(k(~)l>>/[k(~>Zl~.
jm
_Z12( A) describes the contribution of the effective interatomic coupling to the atoms. After some mathematical manipulation we get the atomic wave function coefficients of Eq. (7) as
(A - WI+ &(A))CI(N
atA) = g2(4
LA_
- ZntA)GtA)
WI +&(A)l[A-~2+S22(A)l
= [A _
- [&2(A)12’
(A - ~2 + Z2z(A))C,(A)
(9)
+ ZIZ(A)CI(A)
~1 +&(A)l[A-~2+S22(A)l
- [&2(A)12’
Note that when the energy of the atomic resonance state lies outside the gap, there are no eigenfrequencies lie within the gap. Let us now look for the solution of the Schriidinger equations corresponding to the atomic resonance frequencies and eigenenergies lie within the gap. A = A E [&, 0~1. In this case, the solution of Eq. (3) is now given by zlwgl(A)[S~wl* fjm(EIA)
+Z2wg2(AH5,~wl*
=
Substituting I$
(10)
E-A ( 10) into Eqs. (4) and (5)) we get
[A - WI + 21 (A) la(A)
+ &z(A)gz(A)
= 0,
[A - ~2 + Z2(A)lg2(A)
+ .%2(A)gl(A)
= 0,
(11)
where Xi and 212 are real quantities and are given by &(A)
=
I
C
de zfW ATE-A’
&2(A)
=
I
C
de z~(E)zz(E)&(E) 2?T
E-A
*
(12)
M.R. Singh, W Lou/Physics LettersA 231 (1997) 115-122
The bound-state [A-w,
eigenenergies
are the roots of the following
+~1(A)ltA-~2+~2(A)l
equation
119
which is obtained
from Eq.( 11)
-[&2(A)12=0.
(13)
This equation gives two discrete modes which should be treated as polariton-atom bound states in which the radiation and medium polarization are localized in the vicinity of the two atoms. The above expressions can be reduced to the problem of two identical atoms by putting o1 = w2 and zr (E) = z2 ( E) . In this case we get 21 (A) = 22 (A), and Rq. (7) for A E C reduces to [A - 01 + -&(A)lg*(A) &(A) &(A)
= A+
Ah(A)
= G(A)*
(14)
G(h)
= g1 (A) f g2(A),
= Cl(A)
f
C2(A).
(A) + iT& (A) is known as the self-energy,
*Q(E)]
de z2(e)[1
=P s
C
27T
E-A
’
&(A)
= 21(A)
and A* (A) and r&(A)
T&(A) = +r]z2(A)(l
*Qe(A))l.
f
212(A).
are given by
(15)
where the positive sign corresponds to the system in the symmetric state (triplet state) and the negative sign corresponds to the system in the antisymmetric state (singlet state) [4]. Similarly, Rq. ( 13) for the case A=A E [&,&I reduces to A-WI where &(A) A&(A)
+&(A)
=O,
(16)
= 2:1(A) f 212(A) = A*(A)
&Q(c)1
de z2W[l
=P I c
!iiG
+iTi(A),
E-n
’
with
T*.(A) =O.
These expressions are obtained in Ref. [4]. Now we study the spontaneous emission rate for the system in which two identical atoms are doped in a semiconductor. Note that the case for two different atoms is mathematically complicated and difficult to get analytical results for the spontaneous emission rate. First, we study the case when the resonance frequency of the two-level atoms lies in the polariton continuous spectra. The spontaneous emission rate (i.e. transition probability per unit time) is given by the imaginary part of the self-energy, r*(A), given in Eq. ( 15). The relation between the spontaneous emission rate and the imaginary part of self-energy can be found in the text books (for example see Ref. [9] ). Here we consider the spontaneous emission of a polariton from an initial state in which one atom is in an excited state while the other is in the ground state and the number of polaritons is zero, to a final state in which both atoms are in their ground states and the number of polaritons is one. The expression of the spontaneous emission rate for this process is obtained from Eq.( 15) by replacing A with wi. First let us consider the situation where the interatomic distance between the atoms is very large (i.e. I + oo) . In this case, the symmetric and antisymmetric states are degenerate and Q(wi) becomes zero. Therefore, the expression for the rate of spontaneous emission reduces to r( 01) = iyz2( WI) which is nothing but the spontaneous emission rate due to the presence of one atom in the medium [ 31. Hence, in the limit of large interatomic distances, the existence of the second atom does not contribute to the rate of spontaneous decay rate of the excited atom since the atom-atom coupling is negligible, and the two atoms behave as two independent atoms. In the opposite limit, when two atoms are very close to each other (i.e. E N 0), the symmetric and antisymmetric states are nondegenerate. When the system is in the symmetric state, the expression for the rate of spontaneous emission reduces to yz 2 ( wi ) , which is two times that of the single atom case. This means that the system in the symmetric state can radiate a polariton with a probability that is twice that of the independent or single atom case. On the other hand, when the system is in the antisymmetric state, the expression for the
M.R. Singh, U! Lau/Physics Letters A 231 (1997) 115-122
120
0.25
9.25
18.25
27.25
36.25
0.04
0.24
hTI
0.44
0.64
LTI
Fig. 1. The spontaneous emission rate as functions of the interatomic separation, hri. The upper, lower, and middle curves correspond to the superradiance [ f + ( wr ) 1, the subradiance [ r- (A) 1, and the spontaneous radiation from the single atom case, respectively. Here kr = L+/c and c is the velocity of light. Fig. 2. The energies of the impurity states as functions of the interatomic separation hrl. Here AT = e/c
and c is the velocity of light.
spontaneous emission rate becomes zero. This means that the polariton-atom system is prevented from radiating polaritons from the excited state to the ground state due to the presence of another atom in the system. The spontaneous emission rate is also derived in quantum optics [ 81 when two identical two-level atoms are placed in a cavity. The spontaneous emission from the symmetric state and antisymmetric state are called superradiance and subradiance respectively in Ref. [ 81. Therefore, in this paper, we will use the same nomenclature as that of in Ref. [ 81. Note that superradiance in the quantum optics is a collective phenomenon involving many photons. To include the effects of many photons in our theory is very complicated and it is beyond the scope of this paper. Finally, in case where the resonance frequency of the atoms lies within the gap, r( wt) is equal to zero. Hence the spontaneous emission rate is zero and the polariton-atom bound states are present within the polariton gap.
3. Results
and discussion
Using the expression r&( A = wr ) given in Eq. ( 15), we have calculated the spontaneous emission rate from an initial state in which one atom is in an excited state while the other atom is in ground state and the number of polaritons is zero, to a final state in which both atoms are in their ground states and the number of polaritons is one. We choose the excited states of atoms to lie in the continuous energy spectrum (i.e. wi /& = 1 .OS). The numerical calculations are presented in Fig. 1 for GaAs semiconductors. The parameters used in the calculations are: & = 5.5 x 1013 s-l, & = 5.1 x 1013 s-l, yt/& = yz/& = 1.19 x 10e5, and K/J& = 0.01. In the figure, the upper and lower curves correspond to the emission rate from the symmetric (r+ (wt ) ) and antisymmetric (r_ (wt ) ) states respectively, while the middle curve corresponds to the emission rate (r( wt ) ) for the single atom case. The magnitudes of r+ ( wt ) , lY_ (01) , and r( ~1) are normalized with respect to that of r( wt ) . Note from Fig. 1 that when the interatomic distance is close to zero, the spontaneous emission rate from the symmetric state (upper curve) is two times that of the single atom case (middle curve). This means that the system in the symmetric state can radiate a polariton with a probability that is twice of the independent or single atom case. On other hand, the rate of spontaneous emission from the antisymmetric state (lower curve) is equal to zero. This means that the system in the antisymmetric state is prevented from radiating a polariton. Therefore, we call the upper curve in Fig. 1 as the superradiant curve and the lower curve as the subradiant curve. Note also from the figure that as the interatomic distance increases, superradiant transition rate decreases whereas the subradiance emission rate increases with interatomic separation. In the limit of large interatomic
M.R. Singh, W. L.au/Physics
Letters A 231 (1997) 115-122
121
distance, the emission rates reduce to that of the single atom case. Therefore, the presence of the second atom does not contribute to the spontaneous emission of polariton. It is interesting to note that the behavior of the spontaneous decay rate of the excited atom is oscillatory. In the last section, we have shown that there are two discrete impurity states in III-V semiconductors doped with two different types of atoms. Using Eq. (13), we have calculated the eigenenergies of the impurity states for GaAs semiconductors as a function of interatomic separation, and the results are presented in Fig. 2. The parameters used in the calculations are: WI/& = 1.040, wz/& = 1.041, rl/& = 1.19 x 10e5, and ye/& = 1.67 x 10v5. Clearly, as the interatomic distance increases, the energy difference between the two impurity states decreases. If we set WI = w2 and yi = y2 in our calculations, then the two impurity states reduce to a single state which agrees with the earlier finding [4]. In the limit of large interatomic distances, the atom-atom interaction vanishes; consequently, the value of the lower curve approaches the energy, WI /a, while the value of the upper curve approaches the energy, w2/&. In the limit of large interatomic distance, unlike the case of two identical atoms, where the polariton-atom bound states is twice degenerate, the bound states in this case are nondegenerate. We have also calculated the eigenenergies of the impurity states as a function of interatomic distance for the two identical atom case. we also found the behavior of the eigenenergies is oscillatory. This figure is not presented here. But Fig. 2 is calculated for two different atoms which the resonance energies of the two atoms are different. In this case, we found that the oscillatory behavior disappears or is negligible. Finally, we would like to suggest a few possible experiments where the phenomena mentioned above can be observed. III-V semiconductor compounds doped with donors or acceptors can be used to study the above physical quantities provided the resonance states of donors or acceptors lie in the range of polariton spectra of these semiconductors. There are donors and acceptors in the literature [lo] whose resonance frequencies lie in the energy range where we are interested. Two semiconductor quantum dots doped in III-V semiconductors can be also used to study the above phenomena. Here a quantum dot acts as a two-level system. The required resonance frequencies of quantum dots can be obtained by changing their physical dimensions. The above phenomena can be also observed in III-V semiconductor doped with two quantum wells such as A/B/A = AlAs/GaAs/AlAs. The quantum wells act as two-level systems and their resonance frequencies can be tuned by changing the width of the B layer to meet the requirement of our theory. The experiments can be also preformed in a system consists of two quantum wells (i.e. A/B/A/B/A) m which the width of the A layers is very large compare to that of the B layers. The layer B will acts as a two-level system. In the present theory, we need three dimensional phonons to get the polariton spectrum, and the system of two quantum wells will give us three dimensional phonons. If we apply the magnetic filed along the growth direction in the last two systems, we get Landau levels. These Landau levels can act as two-level systems and their energy difference can be changed by changing the magnetic field strength to meet the requirement of our theory.
Acknowledgments The authors would like to thank Dr. V. I. Rupasov for helpful discussions and Mr. M. Casse for computational help. One of the authors (MS) is thankful to NSERC of Canada for financial support in the form of a research grant.
References I I 1 E. Yablonovitch, Phys. Rev. L-S. 58 (1987) 2059; S. John, Phys. Rev. J_ett. 58 (1987) 2486; E. Yablonovitch, T.J. Grimmer and K.M. Leung, Phys. Rev. Len. 67 ( 1991) 2295.
122
MR. Singh, N! Lau/Physics
Letters A 231 (1997) 115-122
[2] Z. Zhang and S. Satpathy, Phys. Rev. Len. 65 (1990) 2650;
S. John and J. Wang, Phys. Rev. Lett. 64 ( 1990) 2418.
[ 31 V.I. Rupasov and M. Singh, Phys. L&t. A 222 (1996) 258. [4] [ 51 [ 61 [7] [ 81 [9]
V.I. Rupasov and M. Singh, Phys. Rev. A., submitted (1996). V.B. Bemstetskii et al., Quantum Electrodynamics (Pergamon, Oxford, 1982). R.H. Dicke, Phys. Rev. 93 (1954) 99. L. Allen and J.H. Eberly, Optical Resonance and lXvo-level Atoms (Wiley, New York, 1975). M. Weissbluth, Photon-Atom Interactions (Academic Press, New York, 1989). C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986); V.M. Agranovich and V.L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer, Berlin, 1984). [IO] E.S. Kotekes, S. Zemon and P. Morris, Gallium Arsenide and Related Compounds 1985, Institute of Physics Conference Series, Vol 79, p. 259.