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1 This is a specimen tite, C.V. Radhakrishnan a,,1, K. Bazargan a,b,2, S. Pepping c,,1,3 a River Vaey Technoogies, SJP Buiding, Cotton His, Trivandrum, Keraa, India b River Vaey Technoogies, 9, Browns Court, Kennford, Exeter, United Kingdom c Centra Appication Management, Esevier, Radarweg 29, 1043 NX Amsterdam, Netherands Abstract In this work we demonstrate the formation of a new type of poariton on the interface between a cuprous oxide sab and a poystyrene micro-sphere paced on the sab. The evanescent fied of the resonant whispering gaery mode (WGM) of the micro sphere has a substantia gradient, and therefore effectivey coupes with the quadrupoe 1S excitons in cuprous oxide. This evanescent poariton has a ong ife-time, which is determined ony by its excitonic and WGM component. The poariton ower branch has a we pronounced minimum. This suggests that this excitation is ocaized and can be utiized for possibe BEC. The spatia coherence of the poariton can be improved by assembing the micro-spheres into a inear chain. Key words: quadrupoe exciton, poariton, WGM, BEC PACS: y, Lk, c 1. Introduction Athough quadrupoe excitons (QE) in cuprous oxide crystas are good candidates for BEC due to their narrow ine-width and ong ife-time there are some factors impeding BEC Kavouakis and Baym (1996); Rosyak and Birman (2007). One of these factors is that due to the sma but non negigibe couping to the photon bath, one must consider BEC of the corresponding mixed ight-matter states caed poaritons Frohich et a. (2005). The photon-ike part of the poariton has a arge group veocity and tends to escape from the crysta. Thus, the tempora coherence of the condensate is effectivey broken E et a. (1998); Snoke (2002). One proposed soution to This document is a coaborative effort. The second tite footnote which is a onger onger than the first one and with an intention to fi in up more than one ine whie formatting. Corresponding author Principa corresponding author Emai addresses: cvr@river-vaey.com (C.V. Radhakrishnan), kaveh@river-vaey.com (K. Bazargan) URL: (S. Pepping) 1 This is the specimen author footnote. 2 Another author footnote, but a itte more onger. 3 Yet another author footnote. Indeed, you can have any number of author footnotes. Preprint submitted to Esevier February 14, 2008

2 this issue is to pace the crysta into a panar micro-cavity Kasprzak et a. (2006). But even stateof-the-art panar micro-cavities can hod the ight no onger than 10 µs. Besides, formation of the poaritons in the panar cuprous oxide micro-cavity is not effective due to quadrupoe origin of the excitons. Therefore in this work we propose to prevent the poariton escaping by trapping it into a whispering gaery mode (WGM) 4 of a poystyrene micro-sphere (PMS). We deveop a mode which demonstrates formation of a strongy ocaized poariton-ike quasi-partice. This quasi-partice is formed by the resonant interaction between the WGM in PMS and QE in the adjacent ayer of cuprous oxide. The QE interacts with the gradient of the WGM evanescent fied. There are few experiments concerned with resonant interaction of the WGM and dipoe aowed exciton (DE) Xudong Fan (1999); Fan et a. (1999). But the DE has some disadvantages compared to QE when it comes to interaction with the WGM. First, the evanescent ight has sma intensity. Therefore it is not effective for the dipoe aowed couping. But it has a arge gradient, so it can effectivey coupe through a quadrupoe part. Second, the DE has short ife time compared to the QE and therefore is not suitabe for BEC. Third, the kinetic energy of the DE is comparabe with the interaction energy. Hence the ocaization is effectivey impeded. 2. Evanescent vs. conventiona quadrupoe ight-matter couping Assume that a singe PMS of radius r 0 µm is paced at a sma 5 distance δr 0 r 0 from the cuprous oxide crysta (ɛ Cu2O = 6.5). There are severa methods to observe WGM-QE interaction. One of them is to mount a prism (or a fiber) on the top of PMS Xudong Fan (1999). But any surface nearby perturbs spherica symmetry and therefore reduces the Q factor and the ife-time of the WGM. But the positions of the corresponding Mie resonances are not affected. Therefore in this paper we adopt a sighty different picture. Namey the scattering of the buk poariton in cuprous oxide by the PMS. If one of the Mie resonances is in resonance with the QE one can expect formation of the new type of poariton. Assume some density of quadrupoe 1S excitons ([QE]), ω 1S = 2.05 ev, λ 1S = 2π/ω 1S = 6096 Å) has been created by an externa aser puse. The corresponding poaritons move in the crysta as the poariton and can be trapped by the PMS due to WGM-QE resonant interaction. The WGM evanescent fied penetration depth into the cuprous oxide adjacent crysta is much arger than the QE radius: λ 1S /2π (ɛ Cu2O 1) 1/2 = 414 Å a B = 4.6 Å Hence, the ight-matter interaction can be considered semi-cassicay. For the same reason we consider ony buk poaritons. However the theory may be expanded to incude the surface poaritons aso. In the ate case the evanescent fied of such a surface poariton coud pay an essentia roe in the interaction with the WGM, comparabe with WGM-QE couping. 4 WGM occur at particuar resonant waveengths of ight for a given dieectric sphere size. At these waveengths, the ight undergoes tota interna refection at the sphere surface and becomes trapped within the partice for timescaes of the order of ns. 5 comparing to the evanescent fied penetration depth 2

3 For resonance couping with a WGM its size parameter shoud be determined by the resonant wave vector in the cuprous oxide k 0 = m 1. For exampe, if one takes a poystyrene (refractive index ɛ 2 = 1.59) sphere of radius r 0 = 10.7 µm then k 0 r 0 = This size parameter corresponds to the 39TE1 resonance Miyazaki and Jimba (2000). The photon part of the poariton trapped inside the PMS moves as it woud move in a microcavity of the effective moda voume V 4πr 3 0 /3. Consequenty, it can escape through the evanescent fied. This evanescent fied essentiay has a quantum origin and is due to tunneing through the potentia caused by dieectric mismatch on the PMS surface. Therefore, we define the evanescent poariton (EP) as an evanescent ight - QE coherent superposition. Beow we compare the evanescent quadrupoe poariton and conventiona buk quadrupoe poariton in cuprous oxide. For simpicity et us consider the incident poariton wave vector running aong the interface (z direction). The poarization of the poariton is taken aong the x direction. Therefore, in the system of coordinates centered at the sphere, the photon part of the incident poariton can be written as Bohren and Huffman (1983): E i = E 0 i ( + 1) (M 1 in 1 ), (1) where M 1 and N 1 are vector spherica harmonics corresponding to TE- and TM- poarized modes of anguar momentum ; the z component of the anguar momentum is m = 1; E 0 is the ampitude of the eectric fied. The scattered fied is given as: E s = E 0 i ( + 1) (ia 1N 1 b 1 M 1 ), (2) here a 1 and b 1 are scattering Mie coefficients (See the Appendix). Taking into account that both WGM and QE have narrow ine-width, and the energy separation between different WGM is much bigger then the interaction energy we adopt a singe mode picture Xudong Fan (1999). Keeping ony the resonant term the ast expression yieds: E s = E 0 i 0.05b 1,39 M 1,39, (3) To cacuate the interaction of the pane wave (conventiona poariton) (1) and WGM (evanescent poariton) with cuprous oxide one has to change to the cuprous oxide centered system of coordinate (See Fig.2) Whie in the system of the coordinate, centered at the cuprous oxide, the pane wave is sti given by the expression (1), the scattered fied has to be changed according to the vector spherica harmonic addition theorem Stein (1961): M 1,39 = A m 1,39 (r 0 + δr) M m + B m 1,39 (r 0 + δr) N m (4) Here A m 1,39 and Bm 1,39 are the transationa coefficients. Their expicit expression can be found, for instance, in Fuer (1991); Miyazaki and Jimba (2000) and are expicity isted in the Appendix. The buk (incident) and evanescent poaritons in cuprous oxide are formed through the quadrupoe part of the ight-matter interaction: M 1,39 = A m 1,39 (r 0 + δr) M m + B m 1,39 (r 0 + δr) N m (5) Here e, m are the eectron charge and mass; p is the eectron momentum. For the quadrupoe 1S transition in cuprous oxide the energy of interaction can be written as: A n dx F n(x) = B n C n G n (x, y) dx dy A i=0 n + B n A n x + B n y + G n(x, y) (6) A n x + B n y 3

4 Here we introduced the initia state of the system, which transforms as irreducibe representation 1 Γ + 1 of the cubic centered group O h. The fina state is the ortho-exciton state which transforms as 3 Γ + 5,xz in Cartesian system or as 3 Γ + 5;1,2 in the corresponding spherica basis. Hence, using (1, 3, 4, 6), one can deduce that the the couping of the spherica harmonic compared to the pane wave ( g 1,2 = 124 µev) is resonanty enhanced: g 1,39 g 1,2 = i0.06b 1,39 (kr 0 ) A 1,2 1,39 (r 0 + δr) (7) Here we utiized the fact that B 1,2 1,39 A1,2 1,39. Whie the resonant enhancement is provided by the b 1,39 Mie coefficient here, the transationa coefficient reduces the effect. That is why if one tries to coupe the evanescent ight to the dipoe transition the effect is much weaker as A 0,1 1,39 A1,2 1,39. The resuting exciton - evanescent ight couping is shown in Fig.1 Both dipoe and quadrupoe Figure 1: The evanescent ight - 1S quadrupoe couping (g 1, ) scaed to the buk exciton-photon couping (g 1,2 ). The size parameter kr 0 is denoted as x and the PMS is paced directy on the cuprous oxide sampe (δr = 0, See aso Fig.2). couping rate in the actua combined semiconductor-microsphere system is smaer then that in case of conventiona poariton. This is attributed to the fact that the couping occurs in a sma region of the evanescent tai penetrating into cuprous oxide, athough the couping grows with mode number, because the gradient of the evanescent fied increases. Note that QE reaizes strong couping regime g 1,39 > γ whie DE demonstrates weak regime ony Xudong Fan (1999). The property of the scaabe couping factor can be utiized in practica appications such as non-inear optics and is the subject of our future work. 3. Resuts and discussion In this section et us utiize the above cacuated WGM-QE interaction to obtain the evanescent poariton (EP) dispersion in the framework of the couped osciator mode that has been widey used for describing couped atom-photon or exciton-photon modes in microcavity systems Carmichae (1986). Near the resonance between WGM and the quadrupoe exciton ω 1 ω 1S the EP branches are given by the eigenvaues of the foowing Hamitonian: H/ = ω 1 a xa x + ω 1S b xb x + g 1 (x) ( a k b x + a x b x), (8) 4

5 here a x, b x are annihiation operators for ight and the exciton, respectivey. We aso negected kinetic energy of the QE due to smaness of the resonant wave vector and big mass of the QE. Therefore, considering that both the exciton and WGM of a singe sphere are ocaized, the dispersion is reduced to: ω = ω 1S ± g 1 / (9) The above expression shows the formation of the doubet at resonance (both states are exacty haf-qe, haf-wgm). Reca that for DE-WGM weak couping ony WGM pattern shifted by the couping has been observed Xudong Fan (1999). The excitons are trapped in the minimum of the ower branch thus popuating the strongy ocaized states. Physicay this means that the resonant couping with ocaized WGM does not et QE escape by means of its kinetic energy. The dispersion above is simiar to the quadrupoe-dipoe hybrid in the organic-inorganic hetero-structures Rosyak and Birman (2007). In the ater case, the excited organic moecues create an evanescent fied penetrating into the cuprous oxide. Now et us consider possibe appication of the evanescent poariton to BEC. The probem of the conventiona poariton escaping from the crysta mentioned in the introduction no onger exists for the ocaized states of the evanescent poariton. The inewidth of the mixed state is expected to be in the first approximation the sum of the exciton and photon inewidth weighted by the exciton and photon component of the poariton. At resonance, the inewidth of the two eigenstates is simpy given by γ = ( ) γ 1S + γ m, /2. Where γ 1S and γ m, are the QE and WGM inewidth. Even having taken into account decrease of the Q factor due to PMS contact with the cuprous oxide sampe γ 1S γ 1,39 for the ortho-exciton. Hence, γ is defined by the QE inewidth. For the Cu 2 O para-exciton the inewidth of the poariton is given by the WGM inewidth. The para-excitons can acquire some osciator strength provided the PMS exerts a oca stress upon the cuprous oxide sampe. The appied stress changes the crysta symmetry, so that usuay opticay inactive para-excitons may coupe to the WGM. Note that due to spin-orbit interaction the para-excitons are 12 µev beow the ortho-exciton. Therefore for given radius of the PMS the resonant interaction occurs with different WGM for ortho- and para- cases. The evanescent poariton provided by a singe sphere gives the time coherence necessary for the observabe BEC of the quadrupoe exciton. But the spatia coherence is imited to a sma region near the sphere. To improve the spatia coherence one has to sacrifice the tempora coherence sighty by deocaizing the corresponding WGM. It can be done by using an array of spheres aigned aong the z direction and separated by the distance δr 0 (See Fig.2). Recent experimenta Hara et a. (2005) and theoretica Deych and Rosyak (2006) studies have shown that the WGM can trave aong the chain as heavy photons. Therefore the WGM acquires the spatia dispersion, and the evanescent quadrupoe poariton has the form (See Fig.3): (ω1,k ) 2ω = ω 1,k + ω 1S ± ω 2 1S + 4 g 1 / 2 ω 1,k = ω 1S + 2 ( g 1 1 / ) cos(x x 1 + π/2) (10) Here g 1 1 = ω 1S b 1 A 1 1 (δr 1) is the nearest-neighbor inter-sphere couping parameter. When the couping between spheres dominates (δr δr 0 ) the minimum of the ower poariton branch disappears. Consequenty, for possibe BEC of the evanescent poariton one has to keep the desired baance between spatia and tempora coherence by adjusting experimenta parameters δr and δr 0. 5

6 Figure 2: Schematic of formation of the evanescent poariton on inear chain of PMS. The actua dispersion is determined by the ratio of two couping parameters such as exciton-wgm couping and WGM-WGM couping between the microspheres. Figure 3: Dispersion of the evanescent poariton 39TE1. The dashed ine (1) corresponds to the dispersion of the chain of spheres touching each other (δr 0 = 0). The thin soid ine (3) stands for upper and ower branches of a singe sphere dispersion (δr 0 δr = 0). The thick soid curve (2) is the case of inear chain of the spheres in contact with the cuprous oxide (δr 0 = δr = 0). 6

7 Both, the energy of the 1S quadrupoe exciton and the WGM depend on the temperature. Therefore one can use a standard temperature scan to revea the evanescent poariton dispersion Peter et a. (2005). In summary, we note that there is some simiarity between BEC of akai atoms trapped by the aser fied Leggett (2001) and the ong iving QE ocaized by the resonant WGM. The theory deveoped above is appicabe aso for void cavities, spherica impurities and metaic dropets in buk cuprous oxide crysta. 4. Appendix In the appendix we ist expicit expression for the Mie scattering coefficient: a m = b m = n 2 j m (nx) [ x j m (x) ] j m (x) [ nx j m (nx) ] n 2 j m (nx) [ xh (1) m (x)] h (1) m (x) [ nx j m (nx) ] j m (nx) [ x j m (x) ] n 2 j m (x) [ nx j m (nx) ] j m (nx) [ xh (1) m (x)] n2 h (1) m (x) [ nx j m (nx) ] Here n = ɛ 2 is the refractive index of the spheres; x = kr 0 is the size parameter; j m, h m are the spherica Besse and Hanke of the first kind functions respectivey. In the case of 1 the cacuation of the transationa coefficients can be significanty simpified with the hep of the so-caed maximum term approximation Miyazaki and Jimba (2000). A 2 ( 1) +1 + π ( + 1) ( 1) ( ) ( + 1) +1 h(1) (ηx) 1 + ( 1) x i j B i A Here η defined as η = r 0 + δr /r 0 1 is a dimensioness distance between the centers of the spheres. References G. Kavouakis and G. Baym, Phys. Rev. B 53, 7227 (1996). O. Rosyak and J. Birman, arxiv:cond-mat/ , PRB to be pubished (2007). D. Frohich, G. Dasbach, G. B. Hogerstha, M. Bayer, R. Kiebera, D. Sutera, and H. Stozb, Soid State Communications 134, 139 (2005). C. E, A. L. Ivanov, and H. Haug, Phys. Rev. B 57, 9663 (1998). D. Snoke, Science 298, 1368 (2002). J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. Keeing, F. Marchetti, M. Szymanska, R. Andre, J. Staehi, et a., Nature 443, 409 (2006). H. W. Xudong Fan, Scott Lacey, Optics Letters 24, 771 (1999). X. Fan, S. Lacey, and H. Wang, Opt. Lett 24, 771 (1999). H. Miyazaki and Y. Jimba, Phys. Rev. B 62, 7976 (2000). C. Bohren and D. Huffman, Absorption and scattering of ight by sma partices (Wiey New York, 1983). S. Stein, Q. app. Math 19, 15 (1961). 7

8 K. Fuer, App. Opt 30, 4716 (1991). H. J. Carmichae, Phys. Rev. A 33, 3262 (1986). Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, Physica Review Letters 94, (2005). L. Deych and A. Rosyak, Physica Review E 73, (2006). E. Peter, P. Seneart, D. Martrou, A. Lemaître, J. Hours, J. Gérard, and J. Boch, Physica Review Letters 95, (2005). A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). 8