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Four-wave mixing theory in a cavity-polariton system
Solid State Communications, Vol. 108, No. 5, pp. 289–293, 1998 䉷 1998 Elsevier Science Ltd. All rights reserved 0038–1098/98 $ - see front matter
Pergamon
PII: S0038–1098(98)00382-2
FOUR-WAVE MIXING THEORY IN A CAVITY-POLARITON SYSTEM Hidekatsu Suzuura, a ,† Yu.P. Svirko b and Makoto Kuwata-Gonokami a , b ,* a
Department of Applied Physics, Graduate School of Engineering, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan b Cooperative Excitation Project ERATO, Japan Science and Technology Corporation (JST), D8 K.S.P. Building, 3-2-1 Sakato, Takatsu-ku, Kanagawa 213-0012, Japan (Received 2 July 1998; accepted 3 August 1998 by T. Tsuzuki) We present a theory of frequency degenerate four-wave mixing in semiconductor microcavities in the strong coupling regime. The intensity and polarization state of the signal wave are obtained from the probability of the two-polariton scattering on the anharmonic potential arising from the fermionic character of the exciton constituencies. We show that polarization sensitive four-wave mixing measurements enable one to discriminate the contributions to the excitonic anharmonicity due to the repulsive and attractive interactions between excitons and the phase space filling effect. 䉷 1998 Elsevier Science Ltd. All rights reserved
Nonlinear optical effects near a semiconductor band edge may be described within a boson formalism [1–4]. This approach often simplifies the conventional analysis based on the semiconductor Bloch equations [5] or ¹ when accounting for the higher-order correlations ¹ equations of motions for the high-order density matrices [6, 7]. Within the bosonic framework the optical nonlinearity of the electron–hole system arises from the fermionic character of the exciton constituencies that gives rise to the anharmonicity of the excitons themselves and their coupling with photons [8]. This makes the optical nonlinearity at the semiconductor band edge to be a figure of merit for many-particle correlations allowing one to bridge qualitatively and quantitatively the optical and electronic properties of the electron–hole system. The weakly interacting boson (WIB) model accounts for boson anharmonicities due to four-carriers correlations which give rise to the third-order optical nonlinearity and may be investigated by four-wave mixing spectroscopy. In particular, frequency degenerate polarization-sensitive four-wave mixing in the time [9] and frequency-domain [10]
* To whom communication should be addressed. E-mail: [email protected] † Present address: Institute for Solide State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo.
enable us to discriminate the relative contributions of the four-particle interactions which have different symmetries. The electron–hole and boson approaches in studying of a two-dimensional electron–hole system have been recently linked on the microscopic level [11]. This allows us to employ the boson approach to study many-particle correlations in semiconductor quantum well microcavities. In the high-Q cavity exciton–photon interaction leads to normal mode coupling. In this regime the cavity polariton states replace the exciton and cavity resonances in linear reflection and transmission spectra and the cavity polaritons become the system eigenmodes [12]. Correspondingly, the nonlinear optical response of the microcavity may be consistently described in terms of the eigenmodes anharmonicity. In this report we develop a theory of the third-order optical response of the semiconductor microcavity. Specifically, we obtain the intensity and polarization state of frequency degenerate four-wave mixing (DFWM) signal by analyzing twopolariton scattering by the anharmonic potential arising from the fermionic character of the exciton constituencies. By using the polariton scattering approach, we develop a qualitative picture of the DFWM in the microcavity and obtain the dependence of the amplitude and polarization state of the DFWM signal on the different mechanisms of the many-particles interactions. We show that the cavity-polariton scattering concept enables us to explain
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FOUR-WAVE MIXING THEORY IN A CAVITY-POLARITON SYSTEM
the results of recent polarization-sensitive DFWM measurements in high-Q cavities. The harmonic part of the Hamiltonian for the semiconductor quantum well microcavity is conventionally presented as the following: X Hx ¼ ½Éqc a†kj akj þ Éqe b†kj bkj þ gða†kj bkj þ b†kj akj Þÿ kj
(1) a†kj
ðb†kj Þ
stands for the creation operators of where the photon (exciton) with wave vector k and the projection of the angular momentum Jz ¼ þ1ðj ¼ þ1Þ and Jz ¼ ¹ 1ðj ¼ ¹Þ; q c and q e are the frequencies of the cavity photon and exciton respectively and g is the energy of the harmonic exciton–photon coupling. Within the WIB model framework the anharmonic part of the excitonic Hamiltonian [8, 2] may be presented in the following form [11]: X b†k1 j b†k2 j bk3 j bk1 þk2 ¹ k3 j Hxx ¼ 12ðU ¹ U⬘Þ ¹ U⬘
X
k1 ;k2 ;k3 ;j
k1 ;k2 ;k3
b†k1 þ bk2 þ b†k3 ¹ bk1 ¹ k2 þk3 ¹ :
ð2Þ
Here U accounts for the repulsion between 1s excitons with the same angular momentum while U⬘ arises from the renormalization of the higher exciton states [11]. One may observe from equation (2) that the higher states are responsible for an attraction between the 1s excitons with opposite j and for the decreasing of the repulsion between the 1s excitons with the same j. Therefore, the measurement of U and U⬘ in the same experiment allow us to estimate the role of higher states in the third-order optical response. The anharmonicity of the exciton–photon coupling manifests itself as a decrease of the exciton–photon coupling with increasing exciton density (the phase space filling effect (PSF) [8]). Within the WIB model the PSF Hamiltonian, which accounts for density correction to the exciton–photon coupling may be written as follows [8]: X ðb†k1 bk2 j a†k3 j bk1 ¹ k2 þk3 j HPSF ¼ ¹ gn k1 ;k2 ;k3 ;j
þ
b†k1 j ak2 j b†k3 j bk1 ¹ k2 þk3 j Þ
ð3Þ
equations (1)–(3) give the system Hamiltonian. Its diagonalization allows us to obtain the probability of fourth-order scattering that conventionally gives the intensity of the four-wave mixing. Here, it is important to note that the parameters U, U⬘ and n are introduced for excitonic nonlinearity independent of cavity structure. Hamiltonian H xx determines the four-carrier correlation effects and, correspondingly, U and U⬘ do not depend on the frequency of the incident light. Hamiltonian H PSF
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accounts for a decrease of the exciton–photon coupling with increasing exciton density (the phase space filling effect (PSF)). Being a measure of the PSF effect phenomenological parameter n determines a decrease of the probability of the transition from the one- to two-exciton state. This parameter also accumulates the microscopic properties of the exciton system and does not depend on the frequency of the incident light wave. From now on we will consider degenerate four-wave mixing in the self-pumped phase conjugated geometry [10] when frequencies of the all interacting waves are the same. The pump forms a standing wave within the cavity and, when the test wave is directed onto the sample in the same direction as the pump, the phase conjugated signal wave emerges [10]. The amplitude of the signal is proportional to the product of the amplitudes of the counter-propagating pump waves in the cavity and the test wave. The total Hamiltonian of the system Htot ¼ Hx þ Hxx þ HPSF allows to obtain the probability of such a four-photon process and, therefore, the intensity of the scattered wave as the following [13]: X h0jas j f ih f jat jmihmja† jnihnja† j0i 2 p1 p2 (4) IðqÞ ⬀ f ;m;n ðEf ¹ ÉqÞðEm ¹ 2ÉqÞðEn ¹ ÉqÞ Here h0j is the vacuum state, h f j; hmj and hnj are the eigenmodes of the total Hamiltonian H tot with energies E f, E m and E n respectively; a†s , a†t and a†p1;p2 are the photon creation operators for signal, test and pump, respectively and subscripts represent the wave vector and polarization state of the correspondent waves. It has been shown that numerical diagonalization of the relevant two-particle states yields I(q) for the five polarization configurations from the experiment [10] at U : U⬘ : ng ¼ 17 : 15 : 5. This observation confirmed that the WIB model is an effective tool to describe the exciton–exciton correlations in the low density giving a consistent picture of the third-order optical response. To calculate the DFWM spectra from equation (4) (see Fig. 1) we introduce the dephasing rates G e and G c for the exciton and cavity, respectively, by substituting qe ⇒ qe þ iGe and qc ⇒ qc þ iGc into equation (1). In the strong coupling regime the light field in the cavity has pronounced polaritonic character. This implies that the harmonic exciton-photon interaction should be taken into account in the calculation of the optical response non-perturbatively. Such a treatment describes the linear optical response in terms of cavity polaritons. The cavity eigenmodes may be conventionally obtained by the diagonalization of the Hamiltonian (1): X † bkj Þ (5) Hx ¼ ðÉqa a†kj akj þ Éqb bkj k;j
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291
terms of the scattering of the cavity polaritons by the anharmonic potential arising from the exciton–exciton interaction and PSF effect. The anharmonic part of the total Hamiltonian H NL arises from two-particle interactions and affects only two-particle eigenstates. Therefore, only two-polariton scattering processes will contribute to the DFWM signal. Its intensity is given by the probability of two-polariton scattering by the anharmonic potential H NL. In particular, the intensity of the DFWM signal at the frequency of the upper and lower polariton modes are given by the following relations: Iðqa Þ ⬀ jh0jas at HNL a†p1 a†p2 j0ij2 cos8 v Iðqb Þ ⬀ jh0jbs bt HNL b†p1 b†p2 j0ij2 sin8 v: Fig. 1. FD-DFWM spectra at zero detuning for different polarization configurations. ‘‘X’’ and ‘‘Y’’ correspond to the orthohonal linear polarizations and j⫾ correspond to right- and left circular polarizations. where the akj ¼ bkj sin v þ akj cos v and bkj ¼ bkj cos v ¹ akj sin v are so-called upper- and lower cavity polariton modes with eigenfrequencies qa;b ¼ qe ⫾ g cot v, respectively. Here we have introduced p v ¼ tan ¹ 1 ðd þ 1 þ d2 Þ where d ¼ ðqc ¹ qe Þ=2g is the reduced detuning. Apparently, the cavity polariton will show weak anharmonic behavior due to the exciton–exciton interactions and the PSF effect. The anharmonic part of the total Hamiltonian, i.e. HNL ¼ Hxx þ HPSF is presumed to be a small correction to H x, that is g Ⰷ U, U⬘, gn. This justifies the descriptions of the DFWM process in
ð6Þ
Here subscripts ‘‘s’’, ‘‘t’’, ‘‘p1’’ and ‘‘p2’’ label cavity polaritons associated with the photons from the signal test and pump beams. The factors cos 8v and sin 8v emerge due to the photon mode weight in the upper and lower polariton respectively. Equations (6) give the amplitudes of the maximums in the experimental DFWM spectra [10]. The analysis of equations (6) allows us to obtain the relationships between U, U⬘ and n without numerical simulation. Indeed, the inspection of equations (2) and (3) and definitions akj and bkj shows that ðU ¹ U⬘Þ- and U⬘-proportional terms in the signal intensities are of the same sign for the both modes. In contrast, the signs of the n-proportional contributions to the I(q a) and I(q b) will be different because the PSF Hamiltonian (3) includes one photon operator. From Table 1 one may find that this holds for all polarization configurations
Table 1. Dependence of the relative intensities of the DFWM signals at the upper and lower polariton resonances as given by equations (7) for different polarization configurations (see Fig. 1) Lower mode (b) A
j 14U þ tan vgnj2
B
j 12U⬘ ¹ 14U ¹ tan vgnj2
C
j 12U ¹ 12U⬘ þ 2 tan vgnj2
D
j 14U ¹ 14U⬘ þ tan vgnj2
E
j 14U⬘j2
Upper mode (a) 2 1 1 gn 4U ¹ tan v 2 1 1 1 gn 2U⬘ ¹ 4U þ tan v 2 2 1 1 gn 2U ¹ 2U⬘ ¹ tan v 2 1 1 1 gn 4U ¹ 4U⬘ ¹ tan v j 14U⬘j2
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FOUR-WAVE MIXING THEORY IN A CAVITY-POLARITON SYSTEM
from the experiment [10]. The intensity of the lower (upper) mode approximately vanishes when pump and probe have parallel (perturbation) linear polarizations (see Fig. 1). Therefore, using Table 1 one may conclude, that jU⬘=2 ¹ U=4 ¹ ngj ⬇ jU=4 ¹ ngj ⬇ 0, that is, U⬘⬇ U ⬇ 4ng. That is, the interactions between 1s excitons and the PSF effect give contributions of the same order into the Hamiltonian H tot and should be accounted for simultaneously even in the low-excitation limit (compare with [14]). One may readily find from Tab., that for all polarization configurations from Fig. 1 the ratio I(q a)/I(q b) is determined by two parameters. Therefore, the polariton scattering picture implies the following constraints on the intensities of the DFWM signal at the frequencies of the upper and lower modes: IA ðqa;b Þ þ IB ðqa;b Þ IC ðqa;b Þ IA ðqa Þ ¹ IB ðqa Þ ¼ ¼ p ID ðqa;b Þ þ IE ðqa;b Þ 2ID ðqa;b Þ ID ðqa ÞIE ðqa Þ IB ðqb Þ ¹ IA ðqb Þ ¼ p ¼ 2: ID ðqb ÞIE ðqb Þ
ð7Þ
These relationships hold well for the results of the experiment [10] giving the same ratio between the measures of the exciton–exciton interactions and PSF effect: U : U⬘ : ng ¼ 17 : 15 : 5. Figure 2 shows the experimental dependence of the first ratio from (7) on detuning. The cavity polariton scattering picture of the nonlinear optical response in the microcavity may be employed while the polariton modes are well defined. In particular, it may be used to study the exciton dephasing dependence of the nonlinear optical response of the microcavity. In order to prove this we examined the DFWM spectra measured under irradiation of the sample with a He–Ne laser beam which increases the exciton damping. These experimental data presented on
Fig. 2. Dependence of the first ratio from equation (7) on detuning. Experimental result.
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the Fig. 3(a). The broken and solid lines correspond to the DFWM signal with and without the cw pump, respectively. The increasing of the exciton damping manifests itself as a decrease in the signal intensity and broadening of the normal modes. As follows from Fig. 3, the upper and lower modes are equally affected by the incoherent processes at zero detuning. At considerable detunings, the closer the mode frequency to the exciton resonance, the stronger it is affected by the incoherent processes. These experimental findings are well explained within the cavity polariton scattering framework. Indeed, the cavity polariton modes originate from the exciton–photon hybridization which is maximum at zero detuning. When qe ¼ qc both cavity polaritons have exciton and photon modes with equal weight, therefore, they are equally affected by the damping. At negative detunings, the upper polariton mode is mainly exciton-like, while the lower one is photon-like [see Fig. 3(a)] and vice versa for positive detunings. This behavior is well reproduced by the calculation of the DFWM spectra using the cavity polariton scattering model as shown in Fig. 3(b). All these results make it clear that the scattering via excitonic nonlinearities effectively explains DFWM spectra of cavity polariton systems. It is easy to apply this model to non-degenerate cases. Because there are only two relevant modes, that is, upper and lower polariton modes in a high-Q cavity, two-particle scattering between upper and lower modes represents nondegenerate FWM. Scattering amplitude habjHNL jabi are calculated in the same way for DFWM. As mentioned
Fig. 3. DFWM spectra in the presence of incoherent excitation; (a) experimental spectra with (broken line) and without (solid line) He–Ne laser irradiation; (b) calculated spectra using the cavity polariton scattering approach for Gc ¼ 0:4g, Ge ¼ 0:4g (broken line) and Ge ¼ 0:6g (solid line).
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FOUR-WAVE MIXING THEORY IN A CAVITY-POLARITON SYSTEM
before, we determined the ratio of the parameters U, U⬘ and n in H NL by the spectra without detuning and we assumed that they have purely excitonic origin apart from cavity structure. In fact, this model well agrees to experiment in detuning dependence of the DFWM signal for the upper and lower model as shown in our previous paper. Figures 2 and 3 also support those parameters for finite detuning case. However, the cavity polariton scattering picture may only be used to describe the optical response of high-Q systems, where the eigenmode separation is large enough in comparison to the linewidth. At the same time, the applicability of the WIB model itself, which based on the Hamiltonian H tot, is apparently wider. In particular, it also describes the exciton–photon system with strong coupling in a low-Q cavity where cavity polaritons are suppressed [9]. In such a case the description of the third-order optical response may be obtained on the basis of an effective six-level scheme which corresponds to the WIB Hamiltonian. In this semiclassical treatment the exciton–exciton interactions are taken into account by the splitting of the two particle state by the energy U. Specifically, the eigenvalues of the Hamiltonian H tot that corresponds to pairs of the 1s excitons with opposite and the same angular momenta are 2Éqe ¹ U⬘ and 2Éqe þ U ¹ U⬘, respectively. The PSF effect is accounted for by decreasing the dipole moment of the transition from one-exciton state to a two-pair state of excitons with the same angular momentum by a fraction n. This approach also allows us to take into consideration incoherent effects by introducing an excitation induced dephasing correction to the homogeneous exciton linewidth due to exciton– exciton scattering [9]. Importantly, that on the semiclassical level the non-perturbative account for the strong harmonic exciton–photon coupling means the replacement of the external electric field with a local one [15, 16]. This field non-perturbatively accounts for the cavity effects or ¹ in terms of the WIB model ¹ for the harmonic exciton– photon interaction. We have shown recently, that such semiclassical treatment within the WIB model gives the same result for a high-Q cavity as the polaritonscattering approach and also consistently describes the results of our recent experiments on DFWM in low-Q microcavities [9]. In conclusion, we have shown that third-order nonlinear processes in the strong coupling regime are
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consistently explained in terms of scattering of cavity polaritons by the anharmonic potential arising from the exciton–exciton interaction and the PSF effect. The parameters of the anharmonic potential can be obtained from polarization-sensitive four-wave mixing measurements. Acknowledgements—This work was partially supported by a grant-in-aid for scientific research in the priority area of Mutual Quantum Manipulation of Radiation Field and Matter from the Ministry of Education, Science and Culture of Japan and the Core Research for Evolution Science and Technology (CREST) by the Japan Science and Technology Corporation (JST). REFERENCES 1. Usui, T., Progr. Theor. Phys., 23, 1990, 787. 2. Ivanov, A.L., Huag, H. and Keldysh, L.V., Phys. Rep., 296, 1998, 237 and references therein. 3. Victor, K., Axt, V.B., Bartels, G., Stahl, A., Bott, K. and Thomas, P., Z. Phys., B99, 1996, 197. 4. Saiki, T., Kuwata-Gonokami, M., Matesusue, T. and Sakaki, H., Phys. Rev., B49, 1994, 7817. 5. Haug, H. and Koch, S.W., Quantum theory of Optical and Electronical Properties of Semiconductors. World Scientific, 1994. 6. Axt, V.B. and Stahl, A., Z. Phys., B93, 1994, 195. 7. Axt, V.M. and Mukamel, S., Rev. Mod. Phys., 70, 1998, 145. 8. Hanamura, E., J. Phys. Soc. Japan, 37, 1974, 1545; 1553. 9. Shirane, M., Ramkumar, C., Svirko, Yu.P., Suzuura, H., Inouye, S., Shimano, R., Smeya, T., Sakaki, H. and Kuawata-Gonokam, M., Phys. Rev., B58(12), 1998. 10. Kuwata-Gonokami, M., Inouye, S., Suzuura, H., Shirane, M., Shimano, Ryo, Someya, T. and Sakaki, H., Phys. Rev. Lett., 79, 1997, 1341. 11. Inoue, J., Brandes, T. and Shimizu, A., J. Phys. Soc. Jap., 67(10), 1998. 12. Norris, T.B., Rhee, J.-K., Sung, C.-Y., Arakawa, Y., Nishioka, M. and Weisbuch, C., Phys. Rev., B59, 1994, 14 663. 13. Loudon, R., Quantum Theory of Light. Clarendon Press, 1979. 14. Lindberg, M., Hu, Y.M., Binder, R. and Koch, S.W., Phys. Rev., B, 1994, 18060. 15. Ishihara, H. and Cho, K., Phys. Rev., B48, 1993, 7960. 16. Shimano, R., Inouye, S., Kuwata-Gonokami, M., Nakamura, T., Yamanishi, M. and Ogura, I., Jpn. J. Appl. Phys., 34, 1995, L817.
Pergamon
PII: S0038–1098(98)00382-2
FOUR-WAVE MIXING THEORY IN A CAVITY-POLARITON SYSTEM Hidekatsu Suzuura, a ,† Yu.P. Svirko b and Makoto Kuwata-Gonokami a , b ,* a
Department of Applied Physics, Graduate School of Engineering, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan b Cooperative Excitation Project ERATO, Japan Science and Technology Corporation (JST), D8 K.S.P. Building, 3-2-1 Sakato, Takatsu-ku, Kanagawa 213-0012, Japan (Received 2 July 1998; accepted 3 August 1998 by T. Tsuzuki) We present a theory of frequency degenerate four-wave mixing in semiconductor microcavities in the strong coupling regime. The intensity and polarization state of the signal wave are obtained from the probability of the two-polariton scattering on the anharmonic potential arising from the fermionic character of the exciton constituencies. We show that polarization sensitive four-wave mixing measurements enable one to discriminate the contributions to the excitonic anharmonicity due to the repulsive and attractive interactions between excitons and the phase space filling effect. 䉷 1998 Elsevier Science Ltd. All rights reserved
Nonlinear optical effects near a semiconductor band edge may be described within a boson formalism [1–4]. This approach often simplifies the conventional analysis based on the semiconductor Bloch equations [5] or ¹ when accounting for the higher-order correlations ¹ equations of motions for the high-order density matrices [6, 7]. Within the bosonic framework the optical nonlinearity of the electron–hole system arises from the fermionic character of the exciton constituencies that gives rise to the anharmonicity of the excitons themselves and their coupling with photons [8]. This makes the optical nonlinearity at the semiconductor band edge to be a figure of merit for many-particle correlations allowing one to bridge qualitatively and quantitatively the optical and electronic properties of the electron–hole system. The weakly interacting boson (WIB) model accounts for boson anharmonicities due to four-carriers correlations which give rise to the third-order optical nonlinearity and may be investigated by four-wave mixing spectroscopy. In particular, frequency degenerate polarization-sensitive four-wave mixing in the time [9] and frequency-domain [10]
* To whom communication should be addressed. E-mail: [email protected] † Present address: Institute for Solide State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo.
enable us to discriminate the relative contributions of the four-particle interactions which have different symmetries. The electron–hole and boson approaches in studying of a two-dimensional electron–hole system have been recently linked on the microscopic level [11]. This allows us to employ the boson approach to study many-particle correlations in semiconductor quantum well microcavities. In the high-Q cavity exciton–photon interaction leads to normal mode coupling. In this regime the cavity polariton states replace the exciton and cavity resonances in linear reflection and transmission spectra and the cavity polaritons become the system eigenmodes [12]. Correspondingly, the nonlinear optical response of the microcavity may be consistently described in terms of the eigenmodes anharmonicity. In this report we develop a theory of the third-order optical response of the semiconductor microcavity. Specifically, we obtain the intensity and polarization state of frequency degenerate four-wave mixing (DFWM) signal by analyzing twopolariton scattering by the anharmonic potential arising from the fermionic character of the exciton constituencies. By using the polariton scattering approach, we develop a qualitative picture of the DFWM in the microcavity and obtain the dependence of the amplitude and polarization state of the DFWM signal on the different mechanisms of the many-particles interactions. We show that the cavity-polariton scattering concept enables us to explain
289
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FOUR-WAVE MIXING THEORY IN A CAVITY-POLARITON SYSTEM
the results of recent polarization-sensitive DFWM measurements in high-Q cavities. The harmonic part of the Hamiltonian for the semiconductor quantum well microcavity is conventionally presented as the following: X Hx ¼ ½Éqc a†kj akj þ Éqe b†kj bkj þ gða†kj bkj þ b†kj akj Þÿ kj
(1) a†kj
ðb†kj Þ
stands for the creation operators of where the photon (exciton) with wave vector k and the projection of the angular momentum Jz ¼ þ1ðj ¼ þ1Þ and Jz ¼ ¹ 1ðj ¼ ¹Þ; q c and q e are the frequencies of the cavity photon and exciton respectively and g is the energy of the harmonic exciton–photon coupling. Within the WIB model framework the anharmonic part of the excitonic Hamiltonian [8, 2] may be presented in the following form [11]: X b†k1 j b†k2 j bk3 j bk1 þk2 ¹ k3 j Hxx ¼ 12ðU ¹ U⬘Þ ¹ U⬘
X
k1 ;k2 ;k3 ;j
k1 ;k2 ;k3
b†k1 þ bk2 þ b†k3 ¹ bk1 ¹ k2 þk3 ¹ :
ð2Þ
Here U accounts for the repulsion between 1s excitons with the same angular momentum while U⬘ arises from the renormalization of the higher exciton states [11]. One may observe from equation (2) that the higher states are responsible for an attraction between the 1s excitons with opposite j and for the decreasing of the repulsion between the 1s excitons with the same j. Therefore, the measurement of U and U⬘ in the same experiment allow us to estimate the role of higher states in the third-order optical response. The anharmonicity of the exciton–photon coupling manifests itself as a decrease of the exciton–photon coupling with increasing exciton density (the phase space filling effect (PSF) [8]). Within the WIB model the PSF Hamiltonian, which accounts for density correction to the exciton–photon coupling may be written as follows [8]: X ðb†k1 bk2 j a†k3 j bk1 ¹ k2 þk3 j HPSF ¼ ¹ gn k1 ;k2 ;k3 ;j
þ
b†k1 j ak2 j b†k3 j bk1 ¹ k2 þk3 j Þ
ð3Þ
equations (1)–(3) give the system Hamiltonian. Its diagonalization allows us to obtain the probability of fourth-order scattering that conventionally gives the intensity of the four-wave mixing. Here, it is important to note that the parameters U, U⬘ and n are introduced for excitonic nonlinearity independent of cavity structure. Hamiltonian H xx determines the four-carrier correlation effects and, correspondingly, U and U⬘ do not depend on the frequency of the incident light. Hamiltonian H PSF
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accounts for a decrease of the exciton–photon coupling with increasing exciton density (the phase space filling effect (PSF)). Being a measure of the PSF effect phenomenological parameter n determines a decrease of the probability of the transition from the one- to two-exciton state. This parameter also accumulates the microscopic properties of the exciton system and does not depend on the frequency of the incident light wave. From now on we will consider degenerate four-wave mixing in the self-pumped phase conjugated geometry [10] when frequencies of the all interacting waves are the same. The pump forms a standing wave within the cavity and, when the test wave is directed onto the sample in the same direction as the pump, the phase conjugated signal wave emerges [10]. The amplitude of the signal is proportional to the product of the amplitudes of the counter-propagating pump waves in the cavity and the test wave. The total Hamiltonian of the system Htot ¼ Hx þ Hxx þ HPSF allows to obtain the probability of such a four-photon process and, therefore, the intensity of the scattered wave as the following [13]: X h0jas j f ih f jat jmihmja† jnihnja† j0i 2 p1 p2 (4) IðqÞ ⬀ f ;m;n ðEf ¹ ÉqÞðEm ¹ 2ÉqÞðEn ¹ ÉqÞ Here h0j is the vacuum state, h f j; hmj and hnj are the eigenmodes of the total Hamiltonian H tot with energies E f, E m and E n respectively; a†s , a†t and a†p1;p2 are the photon creation operators for signal, test and pump, respectively and subscripts represent the wave vector and polarization state of the correspondent waves. It has been shown that numerical diagonalization of the relevant two-particle states yields I(q) for the five polarization configurations from the experiment [10] at U : U⬘ : ng ¼ 17 : 15 : 5. This observation confirmed that the WIB model is an effective tool to describe the exciton–exciton correlations in the low density giving a consistent picture of the third-order optical response. To calculate the DFWM spectra from equation (4) (see Fig. 1) we introduce the dephasing rates G e and G c for the exciton and cavity, respectively, by substituting qe ⇒ qe þ iGe and qc ⇒ qc þ iGc into equation (1). In the strong coupling regime the light field in the cavity has pronounced polaritonic character. This implies that the harmonic exciton-photon interaction should be taken into account in the calculation of the optical response non-perturbatively. Such a treatment describes the linear optical response in terms of cavity polaritons. The cavity eigenmodes may be conventionally obtained by the diagonalization of the Hamiltonian (1): X † bkj Þ (5) Hx ¼ ðÉqa a†kj akj þ Éqb bkj k;j
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terms of the scattering of the cavity polaritons by the anharmonic potential arising from the exciton–exciton interaction and PSF effect. The anharmonic part of the total Hamiltonian H NL arises from two-particle interactions and affects only two-particle eigenstates. Therefore, only two-polariton scattering processes will contribute to the DFWM signal. Its intensity is given by the probability of two-polariton scattering by the anharmonic potential H NL. In particular, the intensity of the DFWM signal at the frequency of the upper and lower polariton modes are given by the following relations: Iðqa Þ ⬀ jh0jas at HNL a†p1 a†p2 j0ij2 cos8 v Iðqb Þ ⬀ jh0jbs bt HNL b†p1 b†p2 j0ij2 sin8 v: Fig. 1. FD-DFWM spectra at zero detuning for different polarization configurations. ‘‘X’’ and ‘‘Y’’ correspond to the orthohonal linear polarizations and j⫾ correspond to right- and left circular polarizations. where the akj ¼ bkj sin v þ akj cos v and bkj ¼ bkj cos v ¹ akj sin v are so-called upper- and lower cavity polariton modes with eigenfrequencies qa;b ¼ qe ⫾ g cot v, respectively. Here we have introduced p v ¼ tan ¹ 1 ðd þ 1 þ d2 Þ where d ¼ ðqc ¹ qe Þ=2g is the reduced detuning. Apparently, the cavity polariton will show weak anharmonic behavior due to the exciton–exciton interactions and the PSF effect. The anharmonic part of the total Hamiltonian, i.e. HNL ¼ Hxx þ HPSF is presumed to be a small correction to H x, that is g Ⰷ U, U⬘, gn. This justifies the descriptions of the DFWM process in
ð6Þ
Here subscripts ‘‘s’’, ‘‘t’’, ‘‘p1’’ and ‘‘p2’’ label cavity polaritons associated with the photons from the signal test and pump beams. The factors cos 8v and sin 8v emerge due to the photon mode weight in the upper and lower polariton respectively. Equations (6) give the amplitudes of the maximums in the experimental DFWM spectra [10]. The analysis of equations (6) allows us to obtain the relationships between U, U⬘ and n without numerical simulation. Indeed, the inspection of equations (2) and (3) and definitions akj and bkj shows that ðU ¹ U⬘Þ- and U⬘-proportional terms in the signal intensities are of the same sign for the both modes. In contrast, the signs of the n-proportional contributions to the I(q a) and I(q b) will be different because the PSF Hamiltonian (3) includes one photon operator. From Table 1 one may find that this holds for all polarization configurations
Table 1. Dependence of the relative intensities of the DFWM signals at the upper and lower polariton resonances as given by equations (7) for different polarization configurations (see Fig. 1) Lower mode (b) A
j 14U þ tan vgnj2
B
j 12U⬘ ¹ 14U ¹ tan vgnj2
C
j 12U ¹ 12U⬘ þ 2 tan vgnj2
D
j 14U ¹ 14U⬘ þ tan vgnj2
E
j 14U⬘j2
Upper mode (a) 2 1 1 gn 4U ¹ tan v 2 1 1 1 gn 2U⬘ ¹ 4U þ tan v 2 2 1 1 gn 2U ¹ 2U⬘ ¹ tan v 2 1 1 1 gn 4U ¹ 4U⬘ ¹ tan v j 14U⬘j2
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from the experiment [10]. The intensity of the lower (upper) mode approximately vanishes when pump and probe have parallel (perturbation) linear polarizations (see Fig. 1). Therefore, using Table 1 one may conclude, that jU⬘=2 ¹ U=4 ¹ ngj ⬇ jU=4 ¹ ngj ⬇ 0, that is, U⬘⬇ U ⬇ 4ng. That is, the interactions between 1s excitons and the PSF effect give contributions of the same order into the Hamiltonian H tot and should be accounted for simultaneously even in the low-excitation limit (compare with [14]). One may readily find from Tab., that for all polarization configurations from Fig. 1 the ratio I(q a)/I(q b) is determined by two parameters. Therefore, the polariton scattering picture implies the following constraints on the intensities of the DFWM signal at the frequencies of the upper and lower modes: IA ðqa;b Þ þ IB ðqa;b Þ IC ðqa;b Þ IA ðqa Þ ¹ IB ðqa Þ ¼ ¼ p ID ðqa;b Þ þ IE ðqa;b Þ 2ID ðqa;b Þ ID ðqa ÞIE ðqa Þ IB ðqb Þ ¹ IA ðqb Þ ¼ p ¼ 2: ID ðqb ÞIE ðqb Þ
ð7Þ
These relationships hold well for the results of the experiment [10] giving the same ratio between the measures of the exciton–exciton interactions and PSF effect: U : U⬘ : ng ¼ 17 : 15 : 5. Figure 2 shows the experimental dependence of the first ratio from (7) on detuning. The cavity polariton scattering picture of the nonlinear optical response in the microcavity may be employed while the polariton modes are well defined. In particular, it may be used to study the exciton dephasing dependence of the nonlinear optical response of the microcavity. In order to prove this we examined the DFWM spectra measured under irradiation of the sample with a He–Ne laser beam which increases the exciton damping. These experimental data presented on
Fig. 2. Dependence of the first ratio from equation (7) on detuning. Experimental result.
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the Fig. 3(a). The broken and solid lines correspond to the DFWM signal with and without the cw pump, respectively. The increasing of the exciton damping manifests itself as a decrease in the signal intensity and broadening of the normal modes. As follows from Fig. 3, the upper and lower modes are equally affected by the incoherent processes at zero detuning. At considerable detunings, the closer the mode frequency to the exciton resonance, the stronger it is affected by the incoherent processes. These experimental findings are well explained within the cavity polariton scattering framework. Indeed, the cavity polariton modes originate from the exciton–photon hybridization which is maximum at zero detuning. When qe ¼ qc both cavity polaritons have exciton and photon modes with equal weight, therefore, they are equally affected by the damping. At negative detunings, the upper polariton mode is mainly exciton-like, while the lower one is photon-like [see Fig. 3(a)] and vice versa for positive detunings. This behavior is well reproduced by the calculation of the DFWM spectra using the cavity polariton scattering model as shown in Fig. 3(b). All these results make it clear that the scattering via excitonic nonlinearities effectively explains DFWM spectra of cavity polariton systems. It is easy to apply this model to non-degenerate cases. Because there are only two relevant modes, that is, upper and lower polariton modes in a high-Q cavity, two-particle scattering between upper and lower modes represents nondegenerate FWM. Scattering amplitude habjHNL jabi are calculated in the same way for DFWM. As mentioned
Fig. 3. DFWM spectra in the presence of incoherent excitation; (a) experimental spectra with (broken line) and without (solid line) He–Ne laser irradiation; (b) calculated spectra using the cavity polariton scattering approach for Gc ¼ 0:4g, Ge ¼ 0:4g (broken line) and Ge ¼ 0:6g (solid line).
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before, we determined the ratio of the parameters U, U⬘ and n in H NL by the spectra without detuning and we assumed that they have purely excitonic origin apart from cavity structure. In fact, this model well agrees to experiment in detuning dependence of the DFWM signal for the upper and lower model as shown in our previous paper. Figures 2 and 3 also support those parameters for finite detuning case. However, the cavity polariton scattering picture may only be used to describe the optical response of high-Q systems, where the eigenmode separation is large enough in comparison to the linewidth. At the same time, the applicability of the WIB model itself, which based on the Hamiltonian H tot, is apparently wider. In particular, it also describes the exciton–photon system with strong coupling in a low-Q cavity where cavity polaritons are suppressed [9]. In such a case the description of the third-order optical response may be obtained on the basis of an effective six-level scheme which corresponds to the WIB Hamiltonian. In this semiclassical treatment the exciton–exciton interactions are taken into account by the splitting of the two particle state by the energy U. Specifically, the eigenvalues of the Hamiltonian H tot that corresponds to pairs of the 1s excitons with opposite and the same angular momenta are 2Éqe ¹ U⬘ and 2Éqe þ U ¹ U⬘, respectively. The PSF effect is accounted for by decreasing the dipole moment of the transition from one-exciton state to a two-pair state of excitons with the same angular momentum by a fraction n. This approach also allows us to take into consideration incoherent effects by introducing an excitation induced dephasing correction to the homogeneous exciton linewidth due to exciton– exciton scattering [9]. Importantly, that on the semiclassical level the non-perturbative account for the strong harmonic exciton–photon coupling means the replacement of the external electric field with a local one [15, 16]. This field non-perturbatively accounts for the cavity effects or ¹ in terms of the WIB model ¹ for the harmonic exciton– photon interaction. We have shown recently, that such semiclassical treatment within the WIB model gives the same result for a high-Q cavity as the polaritonscattering approach and also consistently describes the results of our recent experiments on DFWM in low-Q microcavities [9]. In conclusion, we have shown that third-order nonlinear processes in the strong coupling regime are
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consistently explained in terms of scattering of cavity polaritons by the anharmonic potential arising from the exciton–exciton interaction and the PSF effect. The parameters of the anharmonic potential can be obtained from polarization-sensitive four-wave mixing measurements. Acknowledgements—This work was partially supported by a grant-in-aid for scientific research in the priority area of Mutual Quantum Manipulation of Radiation Field and Matter from the Ministry of Education, Science and Culture of Japan and the Core Research for Evolution Science and Technology (CREST) by the Japan Science and Technology Corporation (JST). REFERENCES 1. Usui, T., Progr. Theor. Phys., 23, 1990, 787. 2. Ivanov, A.L., Huag, H. and Keldysh, L.V., Phys. Rep., 296, 1998, 237 and references therein. 3. Victor, K., Axt, V.B., Bartels, G., Stahl, A., Bott, K. and Thomas, P., Z. Phys., B99, 1996, 197. 4. Saiki, T., Kuwata-Gonokami, M., Matesusue, T. and Sakaki, H., Phys. Rev., B49, 1994, 7817. 5. Haug, H. and Koch, S.W., Quantum theory of Optical and Electronical Properties of Semiconductors. World Scientific, 1994. 6. Axt, V.B. and Stahl, A., Z. Phys., B93, 1994, 195. 7. Axt, V.M. and Mukamel, S., Rev. Mod. Phys., 70, 1998, 145. 8. Hanamura, E., J. Phys. Soc. Japan, 37, 1974, 1545; 1553. 9. Shirane, M., Ramkumar, C., Svirko, Yu.P., Suzuura, H., Inouye, S., Shimano, R., Smeya, T., Sakaki, H. and Kuawata-Gonokam, M., Phys. Rev., B58(12), 1998. 10. Kuwata-Gonokami, M., Inouye, S., Suzuura, H., Shirane, M., Shimano, Ryo, Someya, T. and Sakaki, H., Phys. Rev. Lett., 79, 1997, 1341. 11. Inoue, J., Brandes, T. and Shimizu, A., J. Phys. Soc. Jap., 67(10), 1998. 12. Norris, T.B., Rhee, J.-K., Sung, C.-Y., Arakawa, Y., Nishioka, M. and Weisbuch, C., Phys. Rev., B59, 1994, 14 663. 13. Loudon, R., Quantum Theory of Light. Clarendon Press, 1979. 14. Lindberg, M., Hu, Y.M., Binder, R. and Koch, S.W., Phys. Rev., B, 1994, 18060. 15. Ishihara, H. and Cho, K., Phys. Rev., B48, 1993, 7960. 16. Shimano, R., Inouye, S., Kuwata-Gonokami, M., Nakamura, T., Yamanishi, M. and Ogura, I., Jpn. J. Appl. Phys., 34, 1995, L817.