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Calculation and analysis for light amplification without inversion in a semiconductor quantum well
15 February 1997
OPTICs COMMUNICATIONS Optics Communications 135 (1997) 315-320
ELSEVIER
Calculation and analysis for light amplification without inversion in a semiconductor quantum well Tie-Jun Chang b, Jin-Yue Gao a,b,Yao-Jun Qiao b, Zhi-Ren Zheng b a CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Jilin University, Changchun 130023, China
Received 16 September 1996; accepted 23 October 1996
Abstract In the presence of a coherent coupling between intersubbands of the conduction band in a semiconductor quantum well, light amplification without inversion is investigated by solving the density matrix equation and analyzed by the electron renormalization theory. The quasi-Fermi energy level is used to take the place of injection rate. This shows that the amplification of the interband transition without population inversion in the bare state can emerge on the lower frequency side of the resonant transition only. This result could be understood as population inversion in the dressed state representation.
1. Introduction The generation of atomic coherence has been attracting considerable interest in the last few years, since it gives rise to such interesting phenomena as lasing without the requirement of population inversion, electromagnetically induced transparency and enhanced nonlinearity with reduced absorption [l-8]. Lasers with a coherent system may have interesting statistical properties, such as narrower intrinsic linewidths due to reduced spontaneous emission noise. After having been successful in an atomic system, the studies on lasing or amplification without inversion in solid materials have attracted more and more attention in recent years. Wei and Manson have shown that inversionless amplification of a two-level model could be obtained in diamond [9]. Recently, the analogies between the energy levels of atomic systems and the subband energy levels of reduced dimensionality semiconductor structures have been analysed [lo- 141. The progress in semiconductor-layer growth techniques, such as MBE and MOCVD, has enabled us to make a desired fine structure such as a superlattice and quantum well. The quantum wells will play an increasingly
important role in the optoelectronic industry. But up to now, these optoelectronic devices have all been based on the intrinsic properties of quantum wells. Recently, theoretical studies show that these intrinsic properties could be modified through coherent optical processes under a resonant coupling field. Zhao and his colleagues have studied field-induced lasing without inversion, and electromagnetically induced transparency in a quantum well with the electron renormalization theory [lo- 131. They showed lasing without inversion in two-level models and electromagnetically induced transparency in two-level and three-level models in a quantum well. Up to now, all studies of lasing or amplification without inversion had to work with the two-level model in solid materials. In this paper, we focus our study on inversionless amplification with a three-level model in solid materials. We take the semiconductor quantum well as an example and study the possibility of interband light amplification without population inversion in the presence of coherent coupling between intersubbands of the conduction band in a semiconductor quantum well. The light amplification without inversion is investigated by solving the density matrix equation, and then analyzed by the
0030-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PI1 SOO30-4018(96)00668-2
316
T.-J. Chang er af./ Optics CommunicationsI35 (1997) 315-320
electron renormalization theory. In this calculation and analysis, the quasi-Fermi energy level is used to take the place of the injection rate.
2. The quantum
well structure used in the calculation
The quantum well scheme used in this paper is illustrated in Fig. 1, where both the growth direction (2) and the transverse momentum (k I) are displayed along the horizontal axis. Due to the Femi distributions, the uper subbands in the conduction band are almost empty and the lower subbands in the valence band are filled with inactive electrons. So we consider one subband in the valence band and two subbands in the conduction band only. States 1a), 1b) and 1c) represent the above subbands, respectively. la>=
I&,
0,
lb)=
I&,
kb).
Ic>= I&’
kc), (1)
where k,, k, and k, are the wave vectors of the electron in the x-y plane, and E,, , E,, and Ecz refer to the quantized z components of the first state in the valence band and the first and second states in the conduction band, respectively. The energies corresponding to 1a), I b), and 1c) are E,(k), E,(k) .and E,(k), respectively E,(k)
= -fi*ki/2mz,
+ Evl,
(2)
Ej( k) = fi*kj2/2mj* + Ej, (j=bfor/‘=cl
(3)
and j=cforJ=c2),
quire that k, = k, = k,, in other words, obey the A k = 0 selection rule. In the semiconductor quantum well laser, the electrons are injected into the conduction band in one side and the holes are injected into the valence band in the other side by an electronic current. We assume that electrons and holes in the well are in equilibrium determined by the quasi-Fermi levels EF and Eg respectively, which are related to the densities of electrons, N, and holes, P, injected into the well [15]
=gFln{l+exp[(E;
PEN=-
(4
-Evi)/KT]},
(5)
m:KT fi*wa X Ch{l
+exp[(Ec
where W is the well width, rni is the effective mass of the heavy-hole band. The possibilities of electron occupation in the subband of the conduction band and the valence band are determined by Fermi distribution functions f,[E,(k)-E,+l,f,[E,(k)-E,land f,[E,(k)-&I. The condition of population inversion is Eg - Eg > hu. In the following calculation, we assume that the inversionless condition is satisfied for all wave vectors &[E,(k)
where m:,, rn,*1and rn,** are the effective masses in the first valence barrier subband, and in the first and second conduction quantum well subbands, respectively. Intersubband coupling and direct interband optical transition re-
-E,,)/KT]},
I
---%I
>f,[E,(k)
-&I,
f,[E,(k)
-51. (6)
i.e. E; - Ez < hv, as depicted in Fig. 1. The intersubband resonant coupling field F, is added between the E,(k) and E,(k), using a strong optical field such as provided by a CO, laser. The lasing light field Fp corresponds to the interband transition between E,(k) and E,(k). The polarization of the interband light Fp and the intersubband resonant light F, are assumed parallel (TB) and perpendicular (TM) to the well plane, respectively. These polarizations are chosen to obtain the largest matrix elements for the optical absorption/gain [ 1.51.
3. The calculation matrix equation
and the results
with the density
3.1. The density matrix equation
(b) Fig. 1. Schematic energy level diagram of the semiconductor quanNnI
We&
The density matrix theory is used in this calculation. Because a direct interband optical transition obeys the Ak = 0 selection rule, for a certain wave vector k, let p’ be the one-electron density matrix for the system. The one-electron density matrix equation is expressed as a# 1 -=&H,fH,)4”]
at
-4(rpk+pkr),
(7)
T.-f. Chang et d/Optics
where Ho and H, are the unperturbed and the perturbed &rniltonians, respectively. Here Ho is a diagonal matrix in 1a), 1b) and 1c) with energies E,(k), E,(k) and E,(k), respectively. IY is given as the inverse of the relaxation time ri for each state I i), lYii = (i l(l/~~) I i>, with i = a, b, c. H, is expressed as
0 G,
H, = [
G
0
0
G,
0
1
Gp . Cl
(-Ybo
-
me
PL pEb and pt, can be interpreted as normalized population distributions. From the equations above, p& can be obtained. Eqs. (2) and (3) show that the subbands are continuous due to the continuous wave vectors, taking this into account, we obtain
(8)
+&f%(k) --%I +L[W)
--%I) kdk(15)
The Rabi frequencies are given by Gp = PboF,,/fi and G, = JL,~F,/C, where Pba = P,b and Pcb = PbC cOrrespend to the intersubband and interband transitions, respectively. In the interaction picture, the density matrix elements equations are aPkb,/af =
317
Communications 135 (1997)315-320
ispk)da
(9)
Electron state density and electron distributions with different wave vectors are considered in this expression. The linear absorption coefficient of the lasing field is propor. . tional to the imaginary part of pba 0 a
Irn(pba).
(16)
If the coefficient CLis positive, the probing light would be amplified and otherwise, it will be absorbed. In the following, the numerical results will be discussed. 3.2. Numerical results
+
13pk,,/i3t
=
(
G(d,
-
- yen
-
db)
+
iG,d,
T
(~0)
is,” - i S,k)p&
+ iG,P:b - iG,Pi,, db
=
k
(11) _
k‘
PL = Pi; 9 Pnc - Pen 1
f$i,
(12)
where 8,” = [E,(k) - ~,(k)l/fi - or, and 8,” = [E,(k) E,(k)l/fi - 0,. The relaxation rate is defined as yij = (Iii + Fjj)/2, with i, j = a, b, c. For steady state, ap:j/dt = 0 in Eqs. (9)-(11). For simplicity, we take Pab = PvlC, = (vl I ezIcl) and PbC = PL,iC2= (cl 1ez 1~2) and assume that they are constant for different wave vector k. When the quantum well with the width of W is embedded in the infinite barrier layers, we have [15] Pcb = 16eW/(3a)‘.
(13)
On the other hand, Pba can be expressed --
efi
Pba - 2Eb,
as
‘/2
Es(‘%+4 (E,+24,/3)mf
The calculation is carried out for the case of an Al,,Ga,,As-GaAs quantum well, where the width of the GaAs well is 7.9 nm. In this structure EC2 -EC, = 0.117 eV, i.e. 10.6 Pm and EC, -E,, = 1.477 eV, and the effective masses are assumed to be rrz:, = rn& = 0,0665m, and rn:, = 0.62m,, where m0 is the free-space mass of an electron. Because of the same effective masses in the conduction quantum well subband, E,(k) - E,(k) is a constant for all k, being 117 meV.
1
’
(14)
where Eg and A, are the bandgap and the spin-orbit splitting, respectively. mf is the effective mass in the conduction band. In order to get a strong coherent coupling between the intersubband, a large Rabi frequency G, (= pabFc/h) is necessary. For the case of an Al,.,Ga,.,AsGaAs quantum well, Pcb is larger than Pbo by one order. In other words, compared with the probe beam, the strongcoherent coupling is easy to obtain. This is one of the advantages of this model.
Fig. 2. Linear interband absorption coefficient, probe detuning, Sop = Eb - E, - h,, strength, Rabi frequency G,.
(Y, as a function of and the coupling field
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T.-J. Chang ef al./Oprics
Communications
0
-1.5
(mev) Fig. 3. Linear interband absorption coefficient, Q , asa function of probe detuning with different coupling fields calculated using density matrix equation, (1) G, = 0, (2) G, = 5 meV, (3) G, = 10 meV, (4) G, = 15 meV.
The probing field strength was taken as GP = 0.1 meV, much less than the intersubband relaxation rate of yrb = 5 meV and the interband relaxation rate of -yCa= yba = 2 meV. The intersubband coupling field strength ranged from 1 meV to 15 meV, which are less than any known intersubband saturation limits. In Fig. 2, we show the linear interband absorption coefficient, (Y, as a function of probing photon energy and coupling field strength, Rabi frequency G,. Fig. 3 shows the light amplification as a function of probing photon energy with different coupling field. With increasing coupling field, in the range of ho, < E,, - E,, the absorption coefficient increases in the positive direction, becoming
135 (1997) 315-320
amplified as showed in Fig. 3. In other words, the amplification only takes place at the lower energy side of the resonant frequency. The splitting between the absorption peak in the high energy side and the amplification peak is a result of the Rabi splitting in the electron energy subbands. With the increasing of the Rabi frequency of the coherent field, the amplification peak becomes higher. As indicated above, the condition of no inversion, EF - Eg < hu, which depends on the injection rate, is satisfied throughout the whole calculation. Fig. 4 shows the effect of the quasi-Fermi energy level, in other words the injection rate, on the amplification. Eg = 0 indicates that the distributions of electrons on the bottom of the lower subband in the conduction band and on the top of the valence band are the same. With increasing EC, the electron number of the first subband of the conduction band decreases, and as a result, the amplification becomes absorption at a threshold. This threshold could be changed with temperature and coupling field strength.
4. Analysis with the electronic renormalization
4.1. The electronic renormalization
theory
theory
The electronic renormalization on the band edge from a Coulomb interaction can be included through the following effective potential Schrodinger equation [ 161
F-L* d2 ---+U,,(z)+Vt.t(z)+v~~(z) 2m* dz2
1
@j”(Z)
= E,!“+j’)( z),
(l-3
where U,,,(z) is the bare quantum-well potential, V,( z> is the Hartree potential and V$( z) is the exchange-correlation potential. We consider the intersubband coherent pump field in the following. Under the rotation-wave approximation, two electron subband dispersion can be written as ,$I)( r
,+( S
k)
=
“”
-
_
2rny
-
’ [@)_@)-&,I,
2
62k2 1 k) = -2m’ ++a’-Ej+hw,],
( 19)
2
for the two coupled subbands in the conduction 1 _$I’( k) = @’ _ --Z[E!L’+Ef)+3?iwc] ;;I
band and
(20)
3
Fig. 4. Linear interband absorption coefficient, (Y , as a function of probe detuuing and the quasi-Fermi energy level I$ in the case of G, = 10 meV.
for the remaining uncoupled subband in the valence band, where w, is the frequency of the intersubband coherent
T.-J. Chang et al./
Optics Communications
319
135 (1997) 315-320
pump laser. The light-induced electronic renormalization of the electron subband dispersion is calculated as [ 171
Here, for simplicity, we assume yrt = ysr = y and kr, = psi = pba, being constant for different wave vector k,
E;*‘(k)
o(w,)
= ;(E!“(k)
+Eil)(k)
= ;(E;‘)(k)
(25)
k dk.
Using the equations above, we can obtain the linear absorption or amplification of the probe field. In the calculation, the constants in the right side of Eqs. (24) and (25) are neglected.
+[[E,1’(k)-E~1~(k)]2+4fi2G~]1’2}, if E(‘)(k) r
= ~/o+m~i(wp)
> E(‘)(k) S +E;*‘(k)
4.3. The analysis for light amplification I[
E;“(k)
if E!‘)(k) E(*)(k) S
= 1 E”‘(k) 2’ (
- I$)( k)] 2 + 4fi2G,2 “* 1 > < I$‘)( k),
(21)
+ E(‘)(k) S
- [[El’)(k)-E!‘)(k)2]2+4fi2G:ll/2}, if E:‘)(k)
> Ei’)( k)
With increasing Rabi frequency of the coherent coupling field, one of the two dressed subbands in the conduction band becomes lower as the other becomes higher. Fig. 5 shows the difference between the electron distribution functions of the lower coupled dressed subbands in the conduction band and that of the uncoupled subband in the valence band, 6fik*, G,), as a function of k2 and the Rabi frequency G, of the coupling field, 6f(k2,
=+‘(k)
G,) =f,[E;2’(k)
if E?(k) +[[E~1)(k)-E~1)(k)]2+4A2G~]1’2) if E(‘)(k) r
< E(‘)(k) S
7
6f(k*, (22)
E(‘)(k) remains unchanged, Ej*)(k) = Ej’)( k). Now, the &normalized energy states, in other words, the dressed states, can be obtained. For consistency with the calculation using the density matrix equation above, we assume hw, + E:‘) - Ef), i.e. the intersubband coupling field infinitely approaching resonance with the two subbands in the conduction band. 4.2. The absorption and amplification
-j@;*‘(k)
-EF+]
< Ei’)( k),
G,) =f,[Ei*)(k)-E,]
if I$‘)( k) > E$‘)( k).
(26) -f,[E?(k)-E;] (27)
The population between the lower dressed subband and the subband in the valence band, inversionless in the bare state, will be inverted in some range of k2 with a strong coherent coupling fields, as showed in Fig. 5. Fig. 6 shows the linear interband absorption coefficient, ~1, as a function of probing photon energy with different coupling fields. It is consistent qualitatively with the result obtained using the
of the probe light
In the following calculation, we also assume inversionless condition is satisfied fe[ E!‘)(k)
-E;]
+E;‘)(k)
that the
-E;]
>f,[E!“(k)-E,-],f,[E~“(k)-E,-1.
(23)
By using the dressed states obtained above, we can calculate the linear optical response to a weak probing field [ 151
ak(wp) = X
$
CL
- - WP E P7vW
{fe[E!*‘(k) -EF] -L[
E:*‘(k)-EF+]}iwrr
[E;2’(k)-E;2’(k)-ho,]2+y;~ {fe[ ES2)(k) -EF]
-fe[
E;*)(k)
+ [E~2)(k)-E;2)-h~p]2+~fi
--E,f]}wst
1.(24)
Fig. 5. The difference of the electron distribution between the lower coupled dressed subbands in the conduction band and that of the uncoupled subband in the valence band, 6fik*, CC>,as a function of k* and the Rabi frequency G, of the coupling field.
320
T.-J, Chnng et al./Optics
Communications
135 (1997) 315-320
the intensity of the coupling field and is exactly equal to the Rabi frequency in this model. This is the reason why the difference of the frequencies corresponding to the amplification peak and absorption peak depends on the Rabi frequency. The electron distributions in these three states are determined by quasi-Fermi levels. So with the renormalization theory, it is easy to understand that there is always a threshold of quasi-Fermi levels for getting inversionless light amplification with a certain strong coherent coupling field.
Acknowledgements
Fig. 6. Linear interband absorption coefficient, a., as a function of probe detuning with different coupling fields by using electron renormalization, (1) G, = 0, (2) G, = 5 meV, (31 G, = 10 meV, (4) G, = 15 meV.
We are pleased to acknowledge the support from the National Natural Science Foundation of China to this work.
References density matrix equation, Fig. 3. The different shapes of the amplification peak are mainly due to the inconsistent relaxation rate used in the two methods.
5. Conclusion In conclusion, light amplification without inversion with coherent coupling can be realized in a semiconductor quantum well. It shows that the population, inversionless in the bare state, will be inverted in the eigenstates of renormalization, i.e. the dressed states, with a strong coherent coupling field. The character of the amplification of the interband transition with intersubband coherent coupling is that amplification emerges on the lower energy side of the resonant frequency only. This result can be understood easily by the electron renormalization theory. In this theory, the first order energies of the two dressed states in a conduction band are the same in this model, in which the coupling field is infinitely approaching resonance with the two subbands in the conduction band. The second order energies of these two dressed states will change with the Rabi frequency of the coupling field. With increasing Rabi frequency, one of these two states goes down and the other goes up related to the first order energies. The electron population of the lower dressed state can be inverted compared to the subband of the valence band at a certain Rabi frequency of the coupling field. The probe resonant with the transition of the lower state to the subband of the valence band will be amplified. On the contrary, the electron population of the other dressed state will decrease with the Rabi frequency of the coupling field. The probe resonant with the transition of this state to the subband of the valence band will be absorbed. The split of this two dressed states depends on
[l] S.E. Harris, Phy. Rev. Lett. 62 (1989) 1033. [2] O.A. Kocharovskaya and Mandel, Phys. Rev. A 42 (1990) 523. [3] L.M. Narducci, H.M. Doss, P. Ru, M.O. Scully, S.Y. Zhu and G. Keitel, Optics Comm. 81 (1991) 379. [4] J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang and D. Jiang, Optics Comm. 93 (19921 323; H.M. Doss, L.M. Narducci, M.O. Scully and Jinyue Gao, Optics Comm. 95 ( 1993) 57; J.Y. Gao, H.Z. Zhang, H.F. Cui, X.Z. Guo, Y. Jiang, Q.W. Wang, G.X. Jin and J.S. Li, Optics Comm. 110 (1994) 590. [5] E.S. Fry, X. Li, D. Nikonov, G.G. Padmabandu, M.O. Scully, A.V. Smith, F.K. Tittel, C. Wang and S.R. Wilkinson, Phys. Rev. Len. 70 (199313235. [6] K. Hakuta, L. Marmet and B.P. Stoicheff, Phys. Rev. Lett. 66 (1991) 596. [7] M.O. Scully and Shi-Yao Zhu, Optics Comm. 87 (1992) 134. [8] Hong Yuan Ling, Phys. Rev. A 49 (1994) 2827. [9] Changjiang Wei and N. Manson, Phys. Rev. A 49 (1994) 4751. [lo] Dong S. Lee and K.J. Malloy, IEEE J. Quantum Electron. 30 (1994185. [ll] Y. Zhao, D. Huang and C. Wu, Optics Lett. 19 (1994) 816. [ 121 D. Huang, C. Wu and Y. Zhao, .I. Opt. Sot. Am. B 11 (1994) 2258. [13] Y. Zhao, D. Huang and C. Wu, Quantum Optics 6 (1994) 327. [14] D. Huang, and Y. Zhao, Phys. Rev. A 51 (199% February. [15] W. Streifer and M. Ettenberg, eds., Semiconductor diode lasers, Vol. 1 (The Institute of Electrocal and Electronics Engineers, Inc., New York, 1991). 1161 M. Asada, A. Kameyama and Y. Sumeatsu, IEEE J. Quantum Electron. QE-20 (1984) 745. [17] T. Ando, A.B. Fowler and F. Stem, Rev. Mod. Phys. 54 (1982) 437. [IS] S. Schmitt-Rink and D.S. Chemla and H. Haug, Phys. Rev. Lett. B 37 (1988) 941.
OPTICs COMMUNICATIONS Optics Communications 135 (1997) 315-320
ELSEVIER
Calculation and analysis for light amplification without inversion in a semiconductor quantum well Tie-Jun Chang b, Jin-Yue Gao a,b,Yao-Jun Qiao b, Zhi-Ren Zheng b a CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Jilin University, Changchun 130023, China
Received 16 September 1996; accepted 23 October 1996
Abstract In the presence of a coherent coupling between intersubbands of the conduction band in a semiconductor quantum well, light amplification without inversion is investigated by solving the density matrix equation and analyzed by the electron renormalization theory. The quasi-Fermi energy level is used to take the place of injection rate. This shows that the amplification of the interband transition without population inversion in the bare state can emerge on the lower frequency side of the resonant transition only. This result could be understood as population inversion in the dressed state representation.
1. Introduction The generation of atomic coherence has been attracting considerable interest in the last few years, since it gives rise to such interesting phenomena as lasing without the requirement of population inversion, electromagnetically induced transparency and enhanced nonlinearity with reduced absorption [l-8]. Lasers with a coherent system may have interesting statistical properties, such as narrower intrinsic linewidths due to reduced spontaneous emission noise. After having been successful in an atomic system, the studies on lasing or amplification without inversion in solid materials have attracted more and more attention in recent years. Wei and Manson have shown that inversionless amplification of a two-level model could be obtained in diamond [9]. Recently, the analogies between the energy levels of atomic systems and the subband energy levels of reduced dimensionality semiconductor structures have been analysed [lo- 141. The progress in semiconductor-layer growth techniques, such as MBE and MOCVD, has enabled us to make a desired fine structure such as a superlattice and quantum well. The quantum wells will play an increasingly
important role in the optoelectronic industry. But up to now, these optoelectronic devices have all been based on the intrinsic properties of quantum wells. Recently, theoretical studies show that these intrinsic properties could be modified through coherent optical processes under a resonant coupling field. Zhao and his colleagues have studied field-induced lasing without inversion, and electromagnetically induced transparency in a quantum well with the electron renormalization theory [lo- 131. They showed lasing without inversion in two-level models and electromagnetically induced transparency in two-level and three-level models in a quantum well. Up to now, all studies of lasing or amplification without inversion had to work with the two-level model in solid materials. In this paper, we focus our study on inversionless amplification with a three-level model in solid materials. We take the semiconductor quantum well as an example and study the possibility of interband light amplification without population inversion in the presence of coherent coupling between intersubbands of the conduction band in a semiconductor quantum well. The light amplification without inversion is investigated by solving the density matrix equation, and then analyzed by the
0030-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PI1 SOO30-4018(96)00668-2
316
T.-J. Chang er af./ Optics CommunicationsI35 (1997) 315-320
electron renormalization theory. In this calculation and analysis, the quasi-Fermi energy level is used to take the place of the injection rate.
2. The quantum
well structure used in the calculation
The quantum well scheme used in this paper is illustrated in Fig. 1, where both the growth direction (2) and the transverse momentum (k I) are displayed along the horizontal axis. Due to the Femi distributions, the uper subbands in the conduction band are almost empty and the lower subbands in the valence band are filled with inactive electrons. So we consider one subband in the valence band and two subbands in the conduction band only. States 1a), 1b) and 1c) represent the above subbands, respectively. la>=
I&,
0,
lb)=
I&,
kb).
Ic>= I&’
kc), (1)
where k,, k, and k, are the wave vectors of the electron in the x-y plane, and E,, , E,, and Ecz refer to the quantized z components of the first state in the valence band and the first and second states in the conduction band, respectively. The energies corresponding to 1a), I b), and 1c) are E,(k), E,(k) .and E,(k), respectively E,(k)
= -fi*ki/2mz,
+ Evl,
(2)
Ej( k) = fi*kj2/2mj* + Ej, (j=bfor/‘=cl
(3)
and j=cforJ=c2),
quire that k, = k, = k,, in other words, obey the A k = 0 selection rule. In the semiconductor quantum well laser, the electrons are injected into the conduction band in one side and the holes are injected into the valence band in the other side by an electronic current. We assume that electrons and holes in the well are in equilibrium determined by the quasi-Fermi levels EF and Eg respectively, which are related to the densities of electrons, N, and holes, P, injected into the well [15]
=gFln{l+exp[(E;
PEN=-
(4
-Evi)/KT]},
(5)
m:KT fi*wa X Ch{l
+exp[(Ec
where W is the well width, rni is the effective mass of the heavy-hole band. The possibilities of electron occupation in the subband of the conduction band and the valence band are determined by Fermi distribution functions f,[E,(k)-E,+l,f,[E,(k)-E,land f,[E,(k)-&I. The condition of population inversion is Eg - Eg > hu. In the following calculation, we assume that the inversionless condition is satisfied for all wave vectors &[E,(k)
where m:,, rn,*1and rn,** are the effective masses in the first valence barrier subband, and in the first and second conduction quantum well subbands, respectively. Intersubband coupling and direct interband optical transition re-
-E,,)/KT]},
I
---%I
>f,[E,(k)
-&I,
f,[E,(k)
-51. (6)
i.e. E; - Ez < hv, as depicted in Fig. 1. The intersubband resonant coupling field F, is added between the E,(k) and E,(k), using a strong optical field such as provided by a CO, laser. The lasing light field Fp corresponds to the interband transition between E,(k) and E,(k). The polarization of the interband light Fp and the intersubband resonant light F, are assumed parallel (TB) and perpendicular (TM) to the well plane, respectively. These polarizations are chosen to obtain the largest matrix elements for the optical absorption/gain [ 1.51.
3. The calculation matrix equation
and the results
with the density
3.1. The density matrix equation
(b) Fig. 1. Schematic energy level diagram of the semiconductor quanNnI
We&
The density matrix theory is used in this calculation. Because a direct interband optical transition obeys the Ak = 0 selection rule, for a certain wave vector k, let p’ be the one-electron density matrix for the system. The one-electron density matrix equation is expressed as a# 1 -=&H,fH,)4”]
at
-4(rpk+pkr),
(7)
T.-f. Chang et d/Optics
where Ho and H, are the unperturbed and the perturbed &rniltonians, respectively. Here Ho is a diagonal matrix in 1a), 1b) and 1c) with energies E,(k), E,(k) and E,(k), respectively. IY is given as the inverse of the relaxation time ri for each state I i), lYii = (i l(l/~~) I i>, with i = a, b, c. H, is expressed as
0 G,
H, = [
G
0
0
G,
0
1
Gp . Cl
(-Ybo
-
me
PL pEb and pt, can be interpreted as normalized population distributions. From the equations above, p& can be obtained. Eqs. (2) and (3) show that the subbands are continuous due to the continuous wave vectors, taking this into account, we obtain
(8)
+&f%(k) --%I +L[W)
--%I) kdk(15)
The Rabi frequencies are given by Gp = PboF,,/fi and G, = JL,~F,/C, where Pba = P,b and Pcb = PbC cOrrespend to the intersubband and interband transitions, respectively. In the interaction picture, the density matrix elements equations are aPkb,/af =
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Communications 135 (1997)315-320
ispk)da
(9)
Electron state density and electron distributions with different wave vectors are considered in this expression. The linear absorption coefficient of the lasing field is propor. . tional to the imaginary part of pba 0 a
Irn(pba).
(16)
If the coefficient CLis positive, the probing light would be amplified and otherwise, it will be absorbed. In the following, the numerical results will be discussed. 3.2. Numerical results
+
13pk,,/i3t
=
(
G(d,
-
- yen
-
db)
+
iG,d,
T
(~0)
is,” - i S,k)p&
+ iG,P:b - iG,Pi,, db
=
k
(11) _
k‘
PL = Pi; 9 Pnc - Pen 1
f$i,
(12)
where 8,” = [E,(k) - ~,(k)l/fi - or, and 8,” = [E,(k) E,(k)l/fi - 0,. The relaxation rate is defined as yij = (Iii + Fjj)/2, with i, j = a, b, c. For steady state, ap:j/dt = 0 in Eqs. (9)-(11). For simplicity, we take Pab = PvlC, = (vl I ezIcl) and PbC = PL,iC2= (cl 1ez 1~2) and assume that they are constant for different wave vector k. When the quantum well with the width of W is embedded in the infinite barrier layers, we have [15] Pcb = 16eW/(3a)‘.
(13)
On the other hand, Pba can be expressed --
efi
Pba - 2Eb,
as
‘/2
Es(‘%+4 (E,+24,/3)mf
The calculation is carried out for the case of an Al,,Ga,,As-GaAs quantum well, where the width of the GaAs well is 7.9 nm. In this structure EC2 -EC, = 0.117 eV, i.e. 10.6 Pm and EC, -E,, = 1.477 eV, and the effective masses are assumed to be rrz:, = rn& = 0,0665m, and rn:, = 0.62m,, where m0 is the free-space mass of an electron. Because of the same effective masses in the conduction quantum well subband, E,(k) - E,(k) is a constant for all k, being 117 meV.
1
’
(14)
where Eg and A, are the bandgap and the spin-orbit splitting, respectively. mf is the effective mass in the conduction band. In order to get a strong coherent coupling between the intersubband, a large Rabi frequency G, (= pabFc/h) is necessary. For the case of an Al,.,Ga,.,AsGaAs quantum well, Pcb is larger than Pbo by one order. In other words, compared with the probe beam, the strongcoherent coupling is easy to obtain. This is one of the advantages of this model.
Fig. 2. Linear interband absorption coefficient, probe detuning, Sop = Eb - E, - h,, strength, Rabi frequency G,.
(Y, as a function of and the coupling field
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0
-1.5
(mev) Fig. 3. Linear interband absorption coefficient, Q , asa function of probe detuning with different coupling fields calculated using density matrix equation, (1) G, = 0, (2) G, = 5 meV, (3) G, = 10 meV, (4) G, = 15 meV.
The probing field strength was taken as GP = 0.1 meV, much less than the intersubband relaxation rate of yrb = 5 meV and the interband relaxation rate of -yCa= yba = 2 meV. The intersubband coupling field strength ranged from 1 meV to 15 meV, which are less than any known intersubband saturation limits. In Fig. 2, we show the linear interband absorption coefficient, (Y, as a function of probing photon energy and coupling field strength, Rabi frequency G,. Fig. 3 shows the light amplification as a function of probing photon energy with different coupling field. With increasing coupling field, in the range of ho, < E,, - E,, the absorption coefficient increases in the positive direction, becoming
135 (1997) 315-320
amplified as showed in Fig. 3. In other words, the amplification only takes place at the lower energy side of the resonant frequency. The splitting between the absorption peak in the high energy side and the amplification peak is a result of the Rabi splitting in the electron energy subbands. With the increasing of the Rabi frequency of the coherent field, the amplification peak becomes higher. As indicated above, the condition of no inversion, EF - Eg < hu, which depends on the injection rate, is satisfied throughout the whole calculation. Fig. 4 shows the effect of the quasi-Fermi energy level, in other words the injection rate, on the amplification. Eg = 0 indicates that the distributions of electrons on the bottom of the lower subband in the conduction band and on the top of the valence band are the same. With increasing EC, the electron number of the first subband of the conduction band decreases, and as a result, the amplification becomes absorption at a threshold. This threshold could be changed with temperature and coupling field strength.
4. Analysis with the electronic renormalization
4.1. The electronic renormalization
theory
theory
The electronic renormalization on the band edge from a Coulomb interaction can be included through the following effective potential Schrodinger equation [ 161
F-L* d2 ---+U,,(z)+Vt.t(z)+v~~(z) 2m* dz2
1
@j”(Z)
= E,!“+j’)( z),
(l-3
where U,,,(z) is the bare quantum-well potential, V,( z> is the Hartree potential and V$( z) is the exchange-correlation potential. We consider the intersubband coherent pump field in the following. Under the rotation-wave approximation, two electron subband dispersion can be written as ,$I)( r
,+( S
k)
=
“”
-
_
2rny
-
’ [@)_@)-&,I,
2
62k2 1 k) = -2m’ ++a’-Ej+hw,],
( 19)
2
for the two coupled subbands in the conduction 1 _$I’( k) = @’ _ --Z[E!L’+Ef)+3?iwc] ;;I
band and
(20)
3
Fig. 4. Linear interband absorption coefficient, (Y , as a function of probe detuuing and the quasi-Fermi energy level I$ in the case of G, = 10 meV.
for the remaining uncoupled subband in the valence band, where w, is the frequency of the intersubband coherent
T.-J. Chang et al./
Optics Communications
319
135 (1997) 315-320
pump laser. The light-induced electronic renormalization of the electron subband dispersion is calculated as [ 171
Here, for simplicity, we assume yrt = ysr = y and kr, = psi = pba, being constant for different wave vector k,
E;*‘(k)
o(w,)
= ;(E!“(k)
+Eil)(k)
= ;(E;‘)(k)
(25)
k dk.
Using the equations above, we can obtain the linear absorption or amplification of the probe field. In the calculation, the constants in the right side of Eqs. (24) and (25) are neglected.
+[[E,1’(k)-E~1~(k)]2+4fi2G~]1’2}, if E(‘)(k) r
= ~/o+m~i(wp)
> E(‘)(k) S +E;*‘(k)
4.3. The analysis for light amplification I[
E;“(k)
if E!‘)(k) E(*)(k) S
= 1 E”‘(k) 2’ (
- I$)( k)] 2 + 4fi2G,2 “* 1 > < I$‘)( k),
(21)
+ E(‘)(k) S
- [[El’)(k)-E!‘)(k)2]2+4fi2G:ll/2}, if E:‘)(k)
> Ei’)( k)
With increasing Rabi frequency of the coherent coupling field, one of the two dressed subbands in the conduction band becomes lower as the other becomes higher. Fig. 5 shows the difference between the electron distribution functions of the lower coupled dressed subbands in the conduction band and that of the uncoupled subband in the valence band, 6fik*, G,), as a function of k2 and the Rabi frequency G, of the coupling field, 6f(k2,
=+‘(k)
G,) =f,[E;2’(k)
if E?(k) +[[E~1)(k)-E~1)(k)]2+4A2G~]1’2) if E(‘)(k) r
< E(‘)(k) S
7
6f(k*, (22)
E(‘)(k) remains unchanged, Ej*)(k) = Ej’)( k). Now, the &normalized energy states, in other words, the dressed states, can be obtained. For consistency with the calculation using the density matrix equation above, we assume hw, + E:‘) - Ef), i.e. the intersubband coupling field infinitely approaching resonance with the two subbands in the conduction band. 4.2. The absorption and amplification
-j@;*‘(k)
-EF+]
< Ei’)( k),
G,) =f,[Ei*)(k)-E,]
if I$‘)( k) > E$‘)( k).
(26) -f,[E?(k)-E;] (27)
The population between the lower dressed subband and the subband in the valence band, inversionless in the bare state, will be inverted in some range of k2 with a strong coherent coupling fields, as showed in Fig. 5. Fig. 6 shows the linear interband absorption coefficient, ~1, as a function of probing photon energy with different coupling fields. It is consistent qualitatively with the result obtained using the
of the probe light
In the following calculation, we also assume inversionless condition is satisfied fe[ E!‘)(k)
-E;]
+E;‘)(k)
that the
-E;]
>f,[E!“(k)-E,-],f,[E~“(k)-E,-1.
(23)
By using the dressed states obtained above, we can calculate the linear optical response to a weak probing field [ 151
ak(wp) = X
$
CL
- - WP E P7vW
{fe[E!*‘(k) -EF] -L[
E:*‘(k)-EF+]}iwrr
[E;2’(k)-E;2’(k)-ho,]2+y;~ {fe[ ES2)(k) -EF]
-fe[
E;*)(k)
+ [E~2)(k)-E;2)-h~p]2+~fi
--E,f]}wst
1.(24)
Fig. 5. The difference of the electron distribution between the lower coupled dressed subbands in the conduction band and that of the uncoupled subband in the valence band, 6fik*, CC>,as a function of k* and the Rabi frequency G, of the coupling field.
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T.-J, Chnng et al./Optics
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the intensity of the coupling field and is exactly equal to the Rabi frequency in this model. This is the reason why the difference of the frequencies corresponding to the amplification peak and absorption peak depends on the Rabi frequency. The electron distributions in these three states are determined by quasi-Fermi levels. So with the renormalization theory, it is easy to understand that there is always a threshold of quasi-Fermi levels for getting inversionless light amplification with a certain strong coherent coupling field.
Acknowledgements
Fig. 6. Linear interband absorption coefficient, a., as a function of probe detuning with different coupling fields by using electron renormalization, (1) G, = 0, (2) G, = 5 meV, (31 G, = 10 meV, (4) G, = 15 meV.
We are pleased to acknowledge the support from the National Natural Science Foundation of China to this work.
References density matrix equation, Fig. 3. The different shapes of the amplification peak are mainly due to the inconsistent relaxation rate used in the two methods.
5. Conclusion In conclusion, light amplification without inversion with coherent coupling can be realized in a semiconductor quantum well. It shows that the population, inversionless in the bare state, will be inverted in the eigenstates of renormalization, i.e. the dressed states, with a strong coherent coupling field. The character of the amplification of the interband transition with intersubband coherent coupling is that amplification emerges on the lower energy side of the resonant frequency only. This result can be understood easily by the electron renormalization theory. In this theory, the first order energies of the two dressed states in a conduction band are the same in this model, in which the coupling field is infinitely approaching resonance with the two subbands in the conduction band. The second order energies of these two dressed states will change with the Rabi frequency of the coupling field. With increasing Rabi frequency, one of these two states goes down and the other goes up related to the first order energies. The electron population of the lower dressed state can be inverted compared to the subband of the valence band at a certain Rabi frequency of the coupling field. The probe resonant with the transition of the lower state to the subband of the valence band will be amplified. On the contrary, the electron population of the other dressed state will decrease with the Rabi frequency of the coupling field. The probe resonant with the transition of this state to the subband of the valence band will be absorbed. The split of this two dressed states depends on
[l] S.E. Harris, Phy. Rev. Lett. 62 (1989) 1033. [2] O.A. Kocharovskaya and Mandel, Phys. Rev. A 42 (1990) 523. [3] L.M. Narducci, H.M. Doss, P. Ru, M.O. Scully, S.Y. Zhu and G. Keitel, Optics Comm. 81 (1991) 379. [4] J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang and D. Jiang, Optics Comm. 93 (19921 323; H.M. Doss, L.M. Narducci, M.O. Scully and Jinyue Gao, Optics Comm. 95 ( 1993) 57; J.Y. Gao, H.Z. Zhang, H.F. Cui, X.Z. Guo, Y. Jiang, Q.W. Wang, G.X. Jin and J.S. Li, Optics Comm. 110 (1994) 590. [5] E.S. Fry, X. Li, D. Nikonov, G.G. Padmabandu, M.O. Scully, A.V. Smith, F.K. Tittel, C. Wang and S.R. Wilkinson, Phys. Rev. Len. 70 (199313235. [6] K. Hakuta, L. Marmet and B.P. Stoicheff, Phys. Rev. Lett. 66 (1991) 596. [7] M.O. Scully and Shi-Yao Zhu, Optics Comm. 87 (1992) 134. [8] Hong Yuan Ling, Phys. Rev. A 49 (1994) 2827. [9] Changjiang Wei and N. Manson, Phys. Rev. A 49 (1994) 4751. [lo] Dong S. Lee and K.J. Malloy, IEEE J. Quantum Electron. 30 (1994185. [ll] Y. Zhao, D. Huang and C. Wu, Optics Lett. 19 (1994) 816. [ 121 D. Huang, C. Wu and Y. Zhao, .I. Opt. Sot. Am. B 11 (1994) 2258. [13] Y. Zhao, D. Huang and C. Wu, Quantum Optics 6 (1994) 327. [14] D. Huang, and Y. Zhao, Phys. Rev. A 51 (199% February. [15] W. Streifer and M. Ettenberg, eds., Semiconductor diode lasers, Vol. 1 (The Institute of Electrocal and Electronics Engineers, Inc., New York, 1991). 1161 M. Asada, A. Kameyama and Y. Sumeatsu, IEEE J. Quantum Electron. QE-20 (1984) 745. [17] T. Ando, A.B. Fowler and F. Stem, Rev. Mod. Phys. 54 (1982) 437. [IS] S. Schmitt-Rink and D.S. Chemla and H. Haug, Phys. Rev. Lett. B 37 (1988) 941.