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Polariton–polariton collinear interaction in semiconductor microcavities
Physica E 53 (2013) 55–58
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Polariton–polariton collinear interaction in semiconductor microcavities M.V. Lebedev n Institute of Solid State Physics RAS, Chernogolovka, Russia
H I G H L I G H T S
Collinear polariton–polariton scattering is found having no analogs in bulk polariton scattering. The k¼0 state plays a key role, being involved in a great number of different scattering processes. Direct population of the k¼ 0 state excludes the explanation in terms of BEC of microcavity polaritons.
art ic l e i nf o
a b s t r a c t
Article history: Received 15 February 2013 Received in revised form 15 April 2013 Accepted 22 April 2013 Available online 2 May 2013
It is shown that the existence of a small polariton mass near k ¼ 0 and a large exciton mass leads to a special type of polariton–polariton interaction: a collinear scattering, which has no analogs in bulk polariton scattering. This scattering process governed with energy and momentum conservation laws is very sensitive to the population of the k ¼ 0 state, which plays a main role in nonlinear relaxation of polaritons. & 2013 Elsevier B.V. All rights reserved.
Keywords: Polariton scattering Microcavity Bose–Einstein condensation
1. Introduction
2. Polariton–polariton scattering
While it is more or less clear that polariton–polariton scattering should be governed with energy and momentum conservation laws experimental evidence for this remains uncertain [1–4]. It is usually understood that the k ¼ 0 state cannot be populated directly through polariton–polariton interaction and macrooccupation of this state is the consequence of Bose condensation of the interacting polariton gas. The impossibility to explain some emission properties of microcavities in terms of energy and momentum conservation leads to explanations with strong renormalization of polariton dispersion and multistability. The analysis undertaken in this paper shows that many basic things may nevertheless be explained with peculiarities of energy and momentum conservation in the polariton–polariton scattering process leading to efficient collinear scattering of polaritons. It will be shown in what follows that the k ¼ 0 state plays a key role in nonlinear polariton relaxation being directly and effectively occupied and involved in a great number of nonlinear scattering processes.
Polariton–polariton scattering is a very efficient nonlinear process of population redistribution on the lower polariton branch. It is governed by energy and momentum conservation laws which have some peculiarities arising from light polariton mass near k ¼ 0and heavy exciton mass. If we rewrite the ordinary conservation laws for polariton–polariton interaction
n
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1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.04.015
! ! ! ! k1þ k2¼ k3þ k4 ℏω1 þ ℏω2 ¼ ℏω3 þ ℏω4 ð1Þ ! where k i are the wavevector components parallel to the crystal plane surface, in the form ! ! ! ! k1þ k2¼ k3þ k4 ℏω0 þ ε1 þ ℏω0 þ ε2 ¼ ℏω0 þ ε3 þ ℏω0 þ ε4
ð2Þ
We can see that these resemble in some sense the conservation laws of relativistic particles with the rest energy ℏω0 . We can consider interaction processes neglecting the rest energy but have to keep in mind that the number of quantum should be conserved. ! ! ! ! k1þ k2¼ k3þ k4
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M.V. Lebedev / Physica E 53 (2013) 55–58
ε1 þ ε2 ¼ ε3 þ ε4
ð3Þ
This last remark is important because among possible states we ! have in this picture is now also a state with k ¼ 0 andε ¼ 0 which could be formally added or removed from Eq. (3) without changing equations.
3. Dispersion curves algebra and collinear polariton–polariton scattering It is convenient to consider graphical representation of Eq. (3) in a form of a dispersion curves algebra (see Fig. 1). If we take the ðk1 ; ε1 Þ state as a first initial state (point A in Fig.1) and put at this point the origin of the dispersion curve of the second initial state ðk2 ; ε2 Þ, the coordinates of their sum (point C in Fig. 1) in the first coordinate system will be ðk1 þ k2 ; ε1 þ ε2 Þ representing thus the left part of Eq. (3). The existence of polariton mass due to the quadratic dispersion near k ¼ 0makes that the ðk2 ; ε2 Þ dispersion curve will always go to the right of the ðk1 ; ε1 Þ dispersion curve. If we suppose that the exciton mass is infinitely large we get that
polariton dispersion is tending to a finite energy εex with the increase of momentum. While ε1 is tending to εex with the increase of k1 , the energy of the intermediate stateε1 þ ε2 is tending to εex þ ε2 〉εex . This means that there exist such a finite momentum k2 that the dispersion curve of the second initial state intersects with that one of the first (see Fig. 2). In other words conservation laws ! (Eq. (3)) may be satisfied with k 4 ¼ 0 and ε4 ¼ 0. This fact of the dispersion curves intersection enables nontrivial collinear in ksolutions of the conservation laws which find no analogs in bulk polariton interactions. The dispersion of the low energy polariton branch depends critically on detuning δ ¼ Ec −Eex (where Eex is the exciton energy and Ec the cavity mode energy fork ¼ 0) and can be represented ! with a class of monotonic curves having a minimum at k ¼ 0 and ! tending asymptotically to Eex at large k (see Fig. 3). These curves have obviously an inflection point and we suppose that they have a single inflection point only, because the existence of more than one inflection point demands very specific properties of exciton– photon interaction. Let us analyze possible collinear solutions for this class of dispersion curves. Collinear intersection of dispersion curves considered above means that for every state εðk1 Þ one can find such a wavevector k2 that εðk1 Þ þ εðk2 Þ ¼ εðk1 þ k2 Þ
Fig. 1. The dispersion curves algebra.
ð4Þ
Let us analyze the behavior of the intersection point C: ðk1 þ k2 ; ε1 þ ε2 Þ with the increase of ε1 . The sum k1 þ k2 obviously tends to infinity when k1 -∞ and ε1 þ ε2 -εex . Eq. (4) is permutation symmetric against the initial polariton states εðk1 Þ and εðk2 Þ: one can reach the same final state C on the dispersion curve starting from point A:ðk1 ; ε1 Þ or point B:ðk2 ; ε2 Þ of the initial dispersion curve—this changes only the order of terms in the left hand side of Eq. (4). This permutation symmetry leads to the symmetry of possible solutions of Eqs. (3) or (4). To show this let us consider all possible final states of the polariton–polariton interaction process with the intermediate state C (Fig. 2). We can rewrite Eq. (3) in the following form: ! ! ! ! k 1 þ k 2− k 3 ¼ k 4 ε1 þ ε2 −ε3 ¼ ε4
Fig. 2. Collinear intersection of dispersion curves.
ð5Þ
This means that we can subtract from the sum ðk1 þ k2 ; ε1 þ ε2 Þ the state ðk3 ; ε3 Þ plotting its dispersion curve in the inversed coordinate system ð−k; −εÞ with the origin at the point ðk1 þ k2 ; ε1 þ ε2 Þ and finding the intersections of this curve with the dispersion curve of the first initial state (see Fig. 4). The intersection points of the inversed dispersion curve (shown with dash) with the initial dispersion curve (solid line) will be the points A and B ( for steep dispersion curves in addition D and F, see
Fig. 3. Polariton dispersion curves for different detunings.
M.V. Lebedev / Physica E 53 (2013) 55–58
57
Fig. 6. The magic point for possible dispersion curves with different detunings shown. in Fig. 3 (see text).
Fig. 4. Graphical representation of Eq. (5) for sloping dispersion curves.
Fig. 5. The magic point of the low polariton dispersion curve. Fig. 7. Graphical representation of Eq. (5) for steep dispersion curves.
Fig. 7) but also the point 0. Point A corresponds to a trivial case when polaritons do not interact and keep their initial momentums and energies. Point B means that the particles exchange their momentums and energies, this case is physically indistinguishable from A. But the interaction corresponding to point 0 is nontrivial. It means that two collinearly moving polaritons may interact producing a high energy polariton at point C and a k ¼ 0polariton. This interaction is obviously reversible. A high energy polariton may interact with a k ¼ 0polariton producing two collinearly moving in the same direction polaritons with appropriate energies and momentums. The dispersion curves are monotonic functions of momentum. Energy of the intermediate state belongs to the polariton dispersion curve that is why ε1 þ ε2 ≤ εex , and the growth of εðk1 Þ will lead to the decrease of εðk2 Þ. Points A and B in Fig. 4 will tend to each other with the increase of εðk1 Þ and at some momentum which we will call kmag they will coincide (see Fig. 5). Eq. (4) gives in this case: 2εðkmag Þ ¼ εð2kmag Þ
ð6Þ
2kmag will be the minimum value of the intermediate state momentum k1 þ k2 while dðk1 þ k2 Þ dk1 jε1 ¼ εðkmag Þ ¼ j dε1 dε1 ε1 ¼ εðkmag Þ þ
dk2 dk dk j j ¼ − j ¼0 dε ε ¼ εðkmag Þ dε ε ¼ εðkmag Þ dε1 ε1 ¼ εðkmag Þ
ð7Þ
The existence of a magic point (Eq. (6)) can be demonstrated in a more direct way if we consider a sum of two identical polariton states εðk1 Þ þ εðk1 Þ ¼ 2εðk1 Þ
ð8Þ
The energy of this intermediate state will tend to 2εex with the increase of k1 . This means that the point C with coordinatesð2k1 ; 2ε1 Þ representing the intermediate state will reach at some kmag the polariton dispersion curve and Eq. (6) will be fulfilled.
The curve OAC in Fig. 5 built of the part OA of the initial polariton dispersion curve shown with solid line and the part AC of the inversed polariton dispersion curve shown with a dash is inversely symmetric against point A. Point A is thus the inflection point of this curve. We can get the properties of the magic point for our class of dispersion curves by continuous deformation of the upper part AC of the initial polariton dispersion curve shown with solid line in Fig. 5. This is shown in Fig. 6. For curves that go to the right of the dashed curve AC point A will remain the inflection point, because the derivative dε=dk reaches its maximum value at point A. But the inflection point of curves going to the left from the dashed curve AC does not coincide with the magic point. These curves intersect the dashed AC curve in some additional point D which gives a new possible collinear scattering point of polaritons. One can see that the process of collinear scattering is always possible at momentums of the intermediate state k≥2kmag . This is because the polariton dispersion curve has a zero slope at k ¼ 0, while for high kvalues it always has a small but finite slope. The considered scattering process is one dimensional and the one dimensional density of states has two singularities: atk ¼ 0, due to the polariton mass and at k ¼ ∞, due to a heavy exciton mass. This means that polaritons with large kcan be effectively scattered to the low kstates of the dispersion curve if the k ¼ 0state is sufficiently high populated. If we consider successive scattering processes we find that every polariton state below the energy of the magic point has its counterpart above this energy and the whole low polariton dispersion curve can redistribute its population with this mechanism. The nontrivial collinear polariton– polariton interaction is very effective which was found when studying the stimulated polariton–polariton scattering [1,2]. This scattering is in fact a special case of polariton–polariton collinear interaction with k1 ¼ k2 and ε1 ¼ ε2 (that is at a magic point). We have in summary only two points of collinear scattering A and B for smooth dispersion curves and sometimes two additional points D and F for steep dispersion curves (see Fig. 7). But a finite polariton damping Γ which was neglected in our consideration can play a significant role for dispersion curves nearly inversely
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M.V. Lebedev / Physica E 53 (2013) 55–58
symmetric against the magic point. If we assume that polaritons have a finite damping we find immediately that all the right hand side and left hand side curves which differ from the dashed AC curve by not more than Γ will produce an infinite number of possible collinear scattering variants in the vicinity of the magic point. The magic point is usually recognized experimentally as a point where efficient scattering to the k ¼ 0state and k ¼ 2kp state under monochromatic pumping of the k ¼ kp state takes place. This means that at some detunings and dampings the magic point will be at the same time the inflection point of the low polariton branch, at some parameters the inflection point does not coincide with the magic one and in some cases ( at large enough Γand nearly symmetric dispersion curves) the magic point is not evidently recognized experimentally because the collinear scattering to the k ¼ 0state is possible with a monochromatic excitation not in a one point of the dispersion curve only but in some region around the magic point. 4. Can polariton–polariton interaction be regarded as a particle–particle one? The collinear polariton–polariton scattering resembles in some sense the problem of possible free space photon instability and collinear spontaneous photon decay put forward by Halpern [6] and discussed later by Bronstein [7]. In our case, by contrast, collinear polariton instability is not against a spontaneous decay but against induced by the k ¼ 0state scattering process. This means, that the k ¼ 0state, if empty, does not affect polariton relaxation, but the population of this state has significant influence on polariton dynamics. Graphical analysis of the dispersion curves intersections reveals some interesting properties of polariton–polariton interactions in semiconductor microcavities. We can state that polariton interaction cannot be interpreted as particle–particle interaction in a common sense, because the collision process when two flying in the same direction particles exchange their energy and momentum so that one of them stops moving and the other carries all the energy and momentum is for massive particles forbidden. This means that the analogy with an interacting Bose gas is incorrect and critical temperature for Bose condensation of polaritons estimated on the basis of light polariton effective mass: mp ∝10−4 me (me —free electron mass) has nothing to do with Bose properties of the polariton gas if optical pumping generates polaritons with high momentum values. Our analysis shows that parametric interaction of waves is a more adequate language for polariton–polariton interaction. The bosonic nature of this parametric interaction leads to stimulation effects just as in a usual optical parametric oscillator. In a general case we have to consider not only a collinear scattering but a two dimensional scattering of polaritons involving the k ¼ 0state. Polariton dispersion in this case will be represented with three dimensional surfaces in a 3D space ðkx ; ky ; εÞ, but the “dispersion curves algebra” remains essentially the same. To find the intersection curves of different dispersion surfaces one has to make exact calculations, but the topology of this intersection curves can be seen without calculations. This topology is shown schematically in Fig. 8, where the projection of the intersection curve to the kx ; ky plane is represented for various values of the ! ! ! total momentum k tot ¼ k 1 þ k 2 . The intersection curves are
Fig. 8. Collinear and non collinear polariton–polariton interaction including the k ¼ 0state for different total momentum (see text).
! inversely symmetric against the point ðð k tot =2Þ; ðεtot =2ÞÞ which leads to the inversion symmetry of their projections against the ! point k tot =2. At high ktot 4 4 2kmag the projections look like two circles, at ktot ¼ 2kmag this circles touch and form a 8-shaped curve and at ktot o2kmag collinear scattering disappears together with the middle point of the 8-shaped curve and the projection looks like a deformed ellipse. It is essential that the k ¼ 0state is nevertheless involved in the scattering process making polariton relaxation with rather different momentums depending on the population of this state. This makes nonlinear polariton relaxation processes in the kx ; ky plane dependent on each other, because all of them are sensitive to the population of this common state. At not to high densities polaritons can be regarded as interacting bosons. This means that cross sections of all kinds of polariton scattering processes should be proportional to the population of the final polariton state. Stimulation can make polariton relaxation nontrivial and build up some coherence effects [8]. The k ¼ 0state is involved in a great number of scattering processes and can be effectively populated with linear (resonant Rayleigh scattering of polarization waves) [5] and nonlinear mechanisms. This state is not the lowest in energy one, because a small longitudinal–transverse splitting shifts the energy minimum to k≠0[5]. Our present consideration of polariton relaxation was carried out without taking into account this splitting in order to simplify the analysis. Including of longitudinal–transverse splitting of polariton states may explain some experimental findings which should otherwise be attributed to renormalization of polariton dispersion curve [4,8].
5. Conclusions To sum up one can say that polariton relaxation at high pumping power is a nonlinear wave interaction process populating directly the k ¼ 0state and looks very different from the process of Bose-particles interactions leading to Bose–Einstein condensation of a nonideal Bose gas. The author would like to thank S.M. Dickman for his interest and helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8]
R.M. Stevenson, et al., Physical Review Letters 85 (2000) 3680. J.J. Baumberg, et al., Physical Review B 62 (2000) R16247. P.G Savvidis, et al., Physical Review B 64 (2001) 075311. R. Butte, et al., Physical Review B 68 (2003) 115325. M.V. Lebedev, A.A Demenev, Physica E 44 (2012) 1510. O. Halpern, Physical Review 44 (1934) 885. M.P. Bronstein, Soviet Physics JETP 7 (3) (1937) 335. D.N. Krizhanovskii, et al., Physical Review B 77 (2008) 115336.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Polariton–polariton collinear interaction in semiconductor microcavities M.V. Lebedev n Institute of Solid State Physics RAS, Chernogolovka, Russia
H I G H L I G H T S
Collinear polariton–polariton scattering is found having no analogs in bulk polariton scattering. The k¼0 state plays a key role, being involved in a great number of different scattering processes. Direct population of the k¼ 0 state excludes the explanation in terms of BEC of microcavity polaritons.
art ic l e i nf o
a b s t r a c t
Article history: Received 15 February 2013 Received in revised form 15 April 2013 Accepted 22 April 2013 Available online 2 May 2013
It is shown that the existence of a small polariton mass near k ¼ 0 and a large exciton mass leads to a special type of polariton–polariton interaction: a collinear scattering, which has no analogs in bulk polariton scattering. This scattering process governed with energy and momentum conservation laws is very sensitive to the population of the k ¼ 0 state, which plays a main role in nonlinear relaxation of polaritons. & 2013 Elsevier B.V. All rights reserved.
Keywords: Polariton scattering Microcavity Bose–Einstein condensation
1. Introduction
2. Polariton–polariton scattering
While it is more or less clear that polariton–polariton scattering should be governed with energy and momentum conservation laws experimental evidence for this remains uncertain [1–4]. It is usually understood that the k ¼ 0 state cannot be populated directly through polariton–polariton interaction and macrooccupation of this state is the consequence of Bose condensation of the interacting polariton gas. The impossibility to explain some emission properties of microcavities in terms of energy and momentum conservation leads to explanations with strong renormalization of polariton dispersion and multistability. The analysis undertaken in this paper shows that many basic things may nevertheless be explained with peculiarities of energy and momentum conservation in the polariton–polariton scattering process leading to efficient collinear scattering of polaritons. It will be shown in what follows that the k ¼ 0 state plays a key role in nonlinear polariton relaxation being directly and effectively occupied and involved in a great number of nonlinear scattering processes.
Polariton–polariton scattering is a very efficient nonlinear process of population redistribution on the lower polariton branch. It is governed by energy and momentum conservation laws which have some peculiarities arising from light polariton mass near k ¼ 0and heavy exciton mass. If we rewrite the ordinary conservation laws for polariton–polariton interaction
n
Tel.: +7 903 539 3097; fax: +7 496 522 8160. E-mail address: [email protected]
1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.04.015
! ! ! ! k1þ k2¼ k3þ k4 ℏω1 þ ℏω2 ¼ ℏω3 þ ℏω4 ð1Þ ! where k i are the wavevector components parallel to the crystal plane surface, in the form ! ! ! ! k1þ k2¼ k3þ k4 ℏω0 þ ε1 þ ℏω0 þ ε2 ¼ ℏω0 þ ε3 þ ℏω0 þ ε4
ð2Þ
We can see that these resemble in some sense the conservation laws of relativistic particles with the rest energy ℏω0 . We can consider interaction processes neglecting the rest energy but have to keep in mind that the number of quantum should be conserved. ! ! ! ! k1þ k2¼ k3þ k4
56
M.V. Lebedev / Physica E 53 (2013) 55–58
ε1 þ ε2 ¼ ε3 þ ε4
ð3Þ
This last remark is important because among possible states we ! have in this picture is now also a state with k ¼ 0 andε ¼ 0 which could be formally added or removed from Eq. (3) without changing equations.
3. Dispersion curves algebra and collinear polariton–polariton scattering It is convenient to consider graphical representation of Eq. (3) in a form of a dispersion curves algebra (see Fig. 1). If we take the ðk1 ; ε1 Þ state as a first initial state (point A in Fig.1) and put at this point the origin of the dispersion curve of the second initial state ðk2 ; ε2 Þ, the coordinates of their sum (point C in Fig. 1) in the first coordinate system will be ðk1 þ k2 ; ε1 þ ε2 Þ representing thus the left part of Eq. (3). The existence of polariton mass due to the quadratic dispersion near k ¼ 0makes that the ðk2 ; ε2 Þ dispersion curve will always go to the right of the ðk1 ; ε1 Þ dispersion curve. If we suppose that the exciton mass is infinitely large we get that
polariton dispersion is tending to a finite energy εex with the increase of momentum. While ε1 is tending to εex with the increase of k1 , the energy of the intermediate stateε1 þ ε2 is tending to εex þ ε2 〉εex . This means that there exist such a finite momentum k2 that the dispersion curve of the second initial state intersects with that one of the first (see Fig. 2). In other words conservation laws ! (Eq. (3)) may be satisfied with k 4 ¼ 0 and ε4 ¼ 0. This fact of the dispersion curves intersection enables nontrivial collinear in ksolutions of the conservation laws which find no analogs in bulk polariton interactions. The dispersion of the low energy polariton branch depends critically on detuning δ ¼ Ec −Eex (where Eex is the exciton energy and Ec the cavity mode energy fork ¼ 0) and can be represented ! with a class of monotonic curves having a minimum at k ¼ 0 and ! tending asymptotically to Eex at large k (see Fig. 3). These curves have obviously an inflection point and we suppose that they have a single inflection point only, because the existence of more than one inflection point demands very specific properties of exciton– photon interaction. Let us analyze possible collinear solutions for this class of dispersion curves. Collinear intersection of dispersion curves considered above means that for every state εðk1 Þ one can find such a wavevector k2 that εðk1 Þ þ εðk2 Þ ¼ εðk1 þ k2 Þ
Fig. 1. The dispersion curves algebra.
ð4Þ
Let us analyze the behavior of the intersection point C: ðk1 þ k2 ; ε1 þ ε2 Þ with the increase of ε1 . The sum k1 þ k2 obviously tends to infinity when k1 -∞ and ε1 þ ε2 -εex . Eq. (4) is permutation symmetric against the initial polariton states εðk1 Þ and εðk2 Þ: one can reach the same final state C on the dispersion curve starting from point A:ðk1 ; ε1 Þ or point B:ðk2 ; ε2 Þ of the initial dispersion curve—this changes only the order of terms in the left hand side of Eq. (4). This permutation symmetry leads to the symmetry of possible solutions of Eqs. (3) or (4). To show this let us consider all possible final states of the polariton–polariton interaction process with the intermediate state C (Fig. 2). We can rewrite Eq. (3) in the following form: ! ! ! ! k 1 þ k 2− k 3 ¼ k 4 ε1 þ ε2 −ε3 ¼ ε4
Fig. 2. Collinear intersection of dispersion curves.
ð5Þ
This means that we can subtract from the sum ðk1 þ k2 ; ε1 þ ε2 Þ the state ðk3 ; ε3 Þ plotting its dispersion curve in the inversed coordinate system ð−k; −εÞ with the origin at the point ðk1 þ k2 ; ε1 þ ε2 Þ and finding the intersections of this curve with the dispersion curve of the first initial state (see Fig. 4). The intersection points of the inversed dispersion curve (shown with dash) with the initial dispersion curve (solid line) will be the points A and B ( for steep dispersion curves in addition D and F, see
Fig. 3. Polariton dispersion curves for different detunings.
M.V. Lebedev / Physica E 53 (2013) 55–58
57
Fig. 6. The magic point for possible dispersion curves with different detunings shown. in Fig. 3 (see text).
Fig. 4. Graphical representation of Eq. (5) for sloping dispersion curves.
Fig. 5. The magic point of the low polariton dispersion curve. Fig. 7. Graphical representation of Eq. (5) for steep dispersion curves.
Fig. 7) but also the point 0. Point A corresponds to a trivial case when polaritons do not interact and keep their initial momentums and energies. Point B means that the particles exchange their momentums and energies, this case is physically indistinguishable from A. But the interaction corresponding to point 0 is nontrivial. It means that two collinearly moving polaritons may interact producing a high energy polariton at point C and a k ¼ 0polariton. This interaction is obviously reversible. A high energy polariton may interact with a k ¼ 0polariton producing two collinearly moving in the same direction polaritons with appropriate energies and momentums. The dispersion curves are monotonic functions of momentum. Energy of the intermediate state belongs to the polariton dispersion curve that is why ε1 þ ε2 ≤ εex , and the growth of εðk1 Þ will lead to the decrease of εðk2 Þ. Points A and B in Fig. 4 will tend to each other with the increase of εðk1 Þ and at some momentum which we will call kmag they will coincide (see Fig. 5). Eq. (4) gives in this case: 2εðkmag Þ ¼ εð2kmag Þ
ð6Þ
2kmag will be the minimum value of the intermediate state momentum k1 þ k2 while dðk1 þ k2 Þ dk1 jε1 ¼ εðkmag Þ ¼ j dε1 dε1 ε1 ¼ εðkmag Þ þ
dk2 dk dk j j ¼ − j ¼0 dε ε ¼ εðkmag Þ dε ε ¼ εðkmag Þ dε1 ε1 ¼ εðkmag Þ
ð7Þ
The existence of a magic point (Eq. (6)) can be demonstrated in a more direct way if we consider a sum of two identical polariton states εðk1 Þ þ εðk1 Þ ¼ 2εðk1 Þ
ð8Þ
The energy of this intermediate state will tend to 2εex with the increase of k1 . This means that the point C with coordinatesð2k1 ; 2ε1 Þ representing the intermediate state will reach at some kmag the polariton dispersion curve and Eq. (6) will be fulfilled.
The curve OAC in Fig. 5 built of the part OA of the initial polariton dispersion curve shown with solid line and the part AC of the inversed polariton dispersion curve shown with a dash is inversely symmetric against point A. Point A is thus the inflection point of this curve. We can get the properties of the magic point for our class of dispersion curves by continuous deformation of the upper part AC of the initial polariton dispersion curve shown with solid line in Fig. 5. This is shown in Fig. 6. For curves that go to the right of the dashed curve AC point A will remain the inflection point, because the derivative dε=dk reaches its maximum value at point A. But the inflection point of curves going to the left from the dashed curve AC does not coincide with the magic point. These curves intersect the dashed AC curve in some additional point D which gives a new possible collinear scattering point of polaritons. One can see that the process of collinear scattering is always possible at momentums of the intermediate state k≥2kmag . This is because the polariton dispersion curve has a zero slope at k ¼ 0, while for high kvalues it always has a small but finite slope. The considered scattering process is one dimensional and the one dimensional density of states has two singularities: atk ¼ 0, due to the polariton mass and at k ¼ ∞, due to a heavy exciton mass. This means that polaritons with large kcan be effectively scattered to the low kstates of the dispersion curve if the k ¼ 0state is sufficiently high populated. If we consider successive scattering processes we find that every polariton state below the energy of the magic point has its counterpart above this energy and the whole low polariton dispersion curve can redistribute its population with this mechanism. The nontrivial collinear polariton– polariton interaction is very effective which was found when studying the stimulated polariton–polariton scattering [1,2]. This scattering is in fact a special case of polariton–polariton collinear interaction with k1 ¼ k2 and ε1 ¼ ε2 (that is at a magic point). We have in summary only two points of collinear scattering A and B for smooth dispersion curves and sometimes two additional points D and F for steep dispersion curves (see Fig. 7). But a finite polariton damping Γ which was neglected in our consideration can play a significant role for dispersion curves nearly inversely
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symmetric against the magic point. If we assume that polaritons have a finite damping we find immediately that all the right hand side and left hand side curves which differ from the dashed AC curve by not more than Γ will produce an infinite number of possible collinear scattering variants in the vicinity of the magic point. The magic point is usually recognized experimentally as a point where efficient scattering to the k ¼ 0state and k ¼ 2kp state under monochromatic pumping of the k ¼ kp state takes place. This means that at some detunings and dampings the magic point will be at the same time the inflection point of the low polariton branch, at some parameters the inflection point does not coincide with the magic one and in some cases ( at large enough Γand nearly symmetric dispersion curves) the magic point is not evidently recognized experimentally because the collinear scattering to the k ¼ 0state is possible with a monochromatic excitation not in a one point of the dispersion curve only but in some region around the magic point. 4. Can polariton–polariton interaction be regarded as a particle–particle one? The collinear polariton–polariton scattering resembles in some sense the problem of possible free space photon instability and collinear spontaneous photon decay put forward by Halpern [6] and discussed later by Bronstein [7]. In our case, by contrast, collinear polariton instability is not against a spontaneous decay but against induced by the k ¼ 0state scattering process. This means, that the k ¼ 0state, if empty, does not affect polariton relaxation, but the population of this state has significant influence on polariton dynamics. Graphical analysis of the dispersion curves intersections reveals some interesting properties of polariton–polariton interactions in semiconductor microcavities. We can state that polariton interaction cannot be interpreted as particle–particle interaction in a common sense, because the collision process when two flying in the same direction particles exchange their energy and momentum so that one of them stops moving and the other carries all the energy and momentum is for massive particles forbidden. This means that the analogy with an interacting Bose gas is incorrect and critical temperature for Bose condensation of polaritons estimated on the basis of light polariton effective mass: mp ∝10−4 me (me —free electron mass) has nothing to do with Bose properties of the polariton gas if optical pumping generates polaritons with high momentum values. Our analysis shows that parametric interaction of waves is a more adequate language for polariton–polariton interaction. The bosonic nature of this parametric interaction leads to stimulation effects just as in a usual optical parametric oscillator. In a general case we have to consider not only a collinear scattering but a two dimensional scattering of polaritons involving the k ¼ 0state. Polariton dispersion in this case will be represented with three dimensional surfaces in a 3D space ðkx ; ky ; εÞ, but the “dispersion curves algebra” remains essentially the same. To find the intersection curves of different dispersion surfaces one has to make exact calculations, but the topology of this intersection curves can be seen without calculations. This topology is shown schematically in Fig. 8, where the projection of the intersection curve to the kx ; ky plane is represented for various values of the ! ! ! total momentum k tot ¼ k 1 þ k 2 . The intersection curves are
Fig. 8. Collinear and non collinear polariton–polariton interaction including the k ¼ 0state for different total momentum (see text).
! inversely symmetric against the point ðð k tot =2Þ; ðεtot =2ÞÞ which leads to the inversion symmetry of their projections against the ! point k tot =2. At high ktot 4 4 2kmag the projections look like two circles, at ktot ¼ 2kmag this circles touch and form a 8-shaped curve and at ktot o2kmag collinear scattering disappears together with the middle point of the 8-shaped curve and the projection looks like a deformed ellipse. It is essential that the k ¼ 0state is nevertheless involved in the scattering process making polariton relaxation with rather different momentums depending on the population of this state. This makes nonlinear polariton relaxation processes in the kx ; ky plane dependent on each other, because all of them are sensitive to the population of this common state. At not to high densities polaritons can be regarded as interacting bosons. This means that cross sections of all kinds of polariton scattering processes should be proportional to the population of the final polariton state. Stimulation can make polariton relaxation nontrivial and build up some coherence effects [8]. The k ¼ 0state is involved in a great number of scattering processes and can be effectively populated with linear (resonant Rayleigh scattering of polarization waves) [5] and nonlinear mechanisms. This state is not the lowest in energy one, because a small longitudinal–transverse splitting shifts the energy minimum to k≠0[5]. Our present consideration of polariton relaxation was carried out without taking into account this splitting in order to simplify the analysis. Including of longitudinal–transverse splitting of polariton states may explain some experimental findings which should otherwise be attributed to renormalization of polariton dispersion curve [4,8].
5. Conclusions To sum up one can say that polariton relaxation at high pumping power is a nonlinear wave interaction process populating directly the k ¼ 0state and looks very different from the process of Bose-particles interactions leading to Bose–Einstein condensation of a nonideal Bose gas. The author would like to thank S.M. Dickman for his interest and helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8]
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