Fractal energy spectrum of a polariton gas in a Fibonacci quasi-periodic potential
Dimitrii Tanese, Evgeni Gurevich, Florent Baboux, Thibaut Jacqmin, Aristide Lemaître, Elisabeth Galopin, Isabelle Sagnes, Alberto Amo, Jacqueline Bloch, Eric Akkermans
FFractal energy spectrum of a polariton gas in a Fibonacci quasi-periodic potential
D. Tanese , E. Gurevich , F. Baboux , T. Jacqmin , A. Lemaˆıtre ,E. Galopin , I. Sagnes , A. Amo , J. Bloch , E. Akkermans Laboratoire de Photonique et de Nanostructures,LPN/CNRS, Route de Nozay, 91460 Marcoussis, France and Department of Physics, Technion Israel Institute of Technology, Haifa 32000, Israel (Dated: August 15, 2018)We report on the study of a polariton gas confined in a quasi-periodic one dimensional cavity,described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space,we observe features characteristic of their fractal energy spectrum such as the opening of mini-gapsobeying the gap labeling theorem and log-periodic oscillations of the integrated density of states.These observations are accurately reproduced solving an effective 1D Schr¨odinger equation, illus-trating the potential of cavity polaritons as a quantum simulator in complex topological geometries.
PACS numbers: 71.36.+c,78.55.Cr,78.67.-n, 05.45.Df, 61.43.Hv, 71.23.Ft
Free quantum particles or waves propagating in a spa-tially varying potential present modifications of theirspectral density, which depend on the symmetry of thispotential. The richness of spectral distributions in con-strained geometries has long been recognized. The caseof a periodic potential described by means of the Blochtheorem is a significant example. The notion of spec-tral distribution has been deepened in the wake of quasi-crystals discovery and it led to a classification of energyspectra into absolutely continuous, pure point and sin-gular continuous spectral distributions [1]. The latterclass proved to be surprisingly rich and it encompasses abroad range of potentials, such as quasi-periodic poten-tials which have been thoroughly studied [2, 3].An interesting quasi-periodic potential can be designedusing a Fibonacci sequence. The corresponding singu-lar continuous energy spectrum has a fractal structure ofthe Cantor set type [4–7], and it displays self-similarity i.e. , a symmetry under a discrete scaling transformation.Denoting ρ ( ε ) the relevant density of states (DOS) in ε (either energy or frequency), a discrete scaling symmetry about a particular value ε u is expressed by the property µ ( ε u + ∆ ε ) − µ ( ε u ) = µ ( ε u + β ∆ ε ) − µ ( ε u ) α , (1)where µ ( ε ) = (cid:82) ε −∞ ρ ( ε (cid:48) ) dε (cid:48) is the integrated density ofstates (IDOS), or density measure, and α and β are scal-ing parameters which usually, depend on ε u . Defininga shifted IDOS by N ε u ( ε ) ≡ µ ( ε ) − µ ( ε u ), the generalsolution of (1) can be written as [8] N ε u ( ε ) = | ε − ε u | γ F (cid:18) ln | ε − ε u | ln β (cid:19) , (2)where γ = ln α ln β is the local ( ε u -dependent) scaling ex-ponent and F ( z ) is a periodic function of period unity,whose (non-universal) form depends on the problem athand. Generally, the exponent γ takes values betweenzero and unity, so that the density ρ ( ε ) is a singular func-tion. Such scaling properties of a fractal spectrum are expected to modify the behavior of physical quantities[8]. Recently studied examples include thermodynamicproperties of photons [9], random walks [10], quantumdiffusion of wave packets [11] and spontaneous emissiontriggered by a fractal vacuum [12]. The diffusion of awave packet in a quasi-periodic medium is predicted tobe neither diffusive, nor ballistic but to present a behav-ior characterized by non-universal exponents and a log-periodic modulation of its time dynamics. Experimen-tal demonstration of these specific properties of quasi-periodic structures is still missing as yet. We propose touse cavity polaritons to evidence such a fractal behavior.Cavity polaritons are quasi-particles arising from thestrong coupling between the optical mode of an opticalcavity and excitons confined in quantum wells [13]. Theyhave appeared recently as a promising system to real-ize quantum simulators [14, 15]. Engineering of the po-tential landscape is possible and allows implementing alarge variety of physical situations such as 1D [14, 16, 17]and 2D periodic potentials [18, 19] with the generationof gap solitons [17, 20], non-linear resonant tunnelingdevices[21], or triangular [22] and honeycomb [23, 24] lat-tices, which enables the exploration of graphene physics.Polaritons offer experimental possibilities not availablein 1D or 2D photonic quasi-crystals such as direct time-and energy-resolved measurements of the excitations inboth space and momentum domains. Thus, one can di-rectly visualize individual eigenmodes, and the dynamicsof wave packets.In this letter, we use this well-controlled system to in-vestigate both theoretically and experimentally the spec-tral properties of a polariton gas in a quasi-periodic po-tential. To do so, we have sculpted the lateral profile of aquasi-1D cavity in the shape of a Fibonacci sequence. Us-ing non resonant excitation in the low density regime, weprobe the modes both in real and reciprocal space. Weobserve a quantitative agreement between experimentsand the calculated modes and density of states. In par-ticular, we evidence features of a fractal energy spectrum, a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b namely gaps densely distributed and an integrated den-sity of states (IDOS) reflecting the existence of a discretescaling symmetry as expressed by (2).In our sample, cavity polaritons are confined withinnarrow strips (wire cavities), whose width is modu-lated quasi-periodically. These wires are fabricated pro-cessing a planar high quality factor ( Q ∼ λ/ Ga . Al . As layer surrounded by two Ga . Al . As/Ga . Al . As Bragg mirrors with 28 and40 pairs in the top/bottom mirrors respectively. 12 GaAsquantum wells of width 7 nm are inserted in the structureresulting in a 15 meV
Rabi splitting. 200 µm long wireswith lateral dimension modulated quasi-periodically aredesigned using electron beam lithography and dry etch-ing (Figs. 1(a-b)). The modulation consists in two wiresections (”letters”) A and B of same length a but dif-ferent widths w A and w B respectively (Fig.1(b)). Themodulation of the wire width induces an effective 1D po-tential for the longitudinal motion of polaritons, as dis-cussed in the sequel. The letters are arranged accordingto the Fibonacci sequence [4] using the recursion, S j ≥ = [ S j − S j − ] , and S = B, S = A, (3)where [ S j − S j − ] means concatenation of two sub-sequences S j − and S j − . The number of letters (length)of a sequence S j is given by the Fibonacci number F j ,such that F j +1 = F j + F j − . The ratio F j +1 /F j tendsto the golden mean σ = (1 + √ / (cid:39) .
62 in the limit j → ∞ , while the corresponding sequence S ∞ becomesrigorously quasi-periodic and invariant, i.e. self-similar,under the iteration transformation (3). Our sample cor-responds to S counting 233 letters with a = 0 . µm , w A = 3 . µm and w B = 1 . µm . To study the po- A A A AB A B B.... .... x V μ m a μ m (a) (b)(c) FIG. 1: (Color online) (a) Scanning electron microscopy im-age of an array of modulated wires. (b) Zoom on a particularwire, showing the shape of the A and B letters. (c) Schematicof the nominal potential corresponding to the lateral shapingof the wire cavity. lariton modes in these quasi-periodic wires, we performlow temperature (10 K) micro-photoluminescence exper-iments. Single wires are excited non-resonantly using acw monomode laser tuned typically 100 meV above thepolariton resonances. The excitation spot extends overa 80 µm -long region along the wire. The sample emis- sion is collected with a 0.65 numerical aperture objectiveand focused on the entrance slit (parallel to the wire) ofa spectrometer coupled to a CCD camera. Imaging ofthe sample surface (resp. the Fourier plane of the col-lection objective) allows studying the spectrally resolvedpolariton modes in real (resp. reciprocal) space. Excita-tion power is kept low enough to stay below condensationthreshold and obtain a nearly homogeneous populationof the lower energy polariton states.Fig.2.a displays the spatially and spectrally resolvedemission measured on a single modulated wire cavity foran exciton-photon detuning around − meV (defined asthe energy difference between the cavity mode at nor-mal incidence and the exciton resonance). Several po-lariton modes are imaged. They present complex pat-terns of bright spots distributed all over the region of thewire under investigation. To understand the nature ofthese modes and properties of their spectral density, wehave calculated the polariton eigenstates in such quasi-periodic structures. FIG. 2: (Color online) (a) Spectrally and spatially resolvedemission measured on a single modulated wire (the linear po-larization parallel to the wire is selected). Bottom of thefigure: letter sequence corresponding to a part of the whole S potential sequence. (b) Calculated polariton Fibonaccimodes as a function of energy and real space coordinate. In our model whose details are given in the supplement[25], we describe the confined photon modes using a 2Dscalar wave equation with vanishing boundary conditionson the boundary of the wire, considered as an axiallysymmetric strip where the longitudinal coordinate x ∈ [0 , L ] ( L being the length of the wire), and the transversecoordinate − w ( x )2 ≤ y ≤ w ( x )2 . Here, w ( x ) > x -dependent width of the wire (Fig.1.c), i.e. aquasi-periodic sequence of segments of width w A and w B ,as defined in (3). In the supplement [25], we show howto map this 2D problem onto a 1D Schr¨odinger equationwith the effective potential: V ( x ) = π w ( x ) + π + 312 (cid:18) w (cid:48) ( x ) w ( x ) (cid:19) . (4)The first term of V ( x ) is the usual adiabatic approxi-mation. The second term accounts for the sharpness ofthe steps. It is not perturbative, and it cannot be ne-glected (see supplement [25]). As clearly visible on Fig.1,the strip shape is not perfectly abrupt but presents somesmoothness in the width variation introduced by the ac-tual etching process. The smoothness scale is used asa fitting parameter in the calculations. The eigenfunc-tions φ q ( x ) and eigenenergies E C,q are obtained numeri-cally. To calculate the polariton modes, we consider theradiative coupling between excitons with a flat disper-sion to the photon modes we have obtained in our sim-ulations. Since the coupling is diagonal in the index q ,the resulting polariton eigenfunctions and photons havethe same spatial behavior. Fig. 2.b shows the polaritonmodes thus obtained numerically. Since experimentallywe cannot resolve states which are separated by less thanthe polariton linewidth, we have averaged the intensityover eigenmodes close in energy. Thus, what appears inFig.2.b as bright intensity spots at different energies areactually bands separated by gaps. Clearly the calcula-tion reproduces very accurately the spatial structure ofthe polariton modes observed in the experiment. This di-rect imaging of the Fibonacci modes in a quasi-periodicstructure is a clear asset offered by cavity polaritons.Probing the polariton modes in reciprocal space pro-vides also remarkable information about the eigenmodes.This is illustrated on Fig.3.a, where taking advantage ofthe one-to-one relation between angle of emission and in-plane momentum of polaritons, far field imaging of thepolariton emission is shown for the same wire as in Fig.2.A complex band structure appears with the opening ofgaps not regularly spaced unlike the case of a periodicmodulation [17]. The calculated band structure repro-duces quantitatively the measurements (Fig.3.b).In the rest of the paper, we show that despite the finitesize of the system, both in the numerics and in the exper-iments, fundamental physical properties are evidenced inthis complex band-structure which indicate the onset ofa fractal density of states. To study the spectrum andthe position of its gaps, it is convenient to rewrite thequasi-periodic potential V ( x ) in (4) under the form, V ( x ) = (cid:88) n χ ( σ − n ) u b ( x − an ) (5)valid in principle [4] for an infinitely long system namely j → ∞ in (3). u b ( x ) (which depends on w ( x )) describesthe shape of the letter B while the periodic function χ ( x )defined, within [0 , χ ( x ) = 1 for 0 < x < − σ and χ ( x ) = 0 for 2 − σ < x <
1, accounts for the quasi-periodic order. The Fourier transform of V ( x ) consists FIG. 3: (Color online) (a) Spectrally resolved far field emis-sion measured on the same wire cavity used in Fig.2; (b)Corresponding simulation. Position of the gaps labeled withtwo integers [ p, q ] is indicated with red arrows. of Bragg peaks and is given by, V ( k ) = ˜ u b ( k ) (cid:88) p,q χ q δ (cid:0) ka − π ( p + qσ − ) (cid:1) (6)with obvious notations. Since σ is irrational, each Braggpeak of the quasi-periodic potential can be uniquely la-beled with a set [ p, q ] of two integers so that the cor-responding wave number is k = Q p,q ≡ πa (cid:0) p + qσ − (cid:1) .Similarly to the Bloch theorem for a periodic modula-tion, we may expect that a series of gaps opens at eachindependent Bragg peak Q p,q . Thus, to label the gapsand to obtain the IDOS given in (2), it is tempting toconsider the quasi-periodic potential V ( x ) as a small per-turbation. Albeit not justified in the present experimen-tal case, we shall first use this assumption since it al-lows to give a more intuitive derivation of gap labeling.But the Bragg peaks being a dense set, we must be cau-tious and first approximate σ by its finite approximants σ j = F j +1 /F j as defined after (3). Then, V ( x ) in (5)becomes a periodic approximant V j +1 ( x ), built from pe-riodically repeated cells S j +1 of length a F j +1 . Thus, theproperties of the single cell S j +1 studied experimentallyare essentially those of the periodic potential V j +1 ( x ). ItsFourier transform V j +1 ( k ) is obtained replacing σ by σ j in (6). V j +1 ( k ) thus defined, is the structure factor ofa periodic structure and therefore it has a finite densityof Bragg peaks spaced by ∆ k = 2 π/ ( aF j +1 ). Pertur-bation theory in | V | (cid:28) k = Q p,q ≡ πa ( F j +1 p + F j q )hybridizes the degenerate Bloch waves at wave numbers ± Q p,q /
2. The coupling between these plane waves isbest described by a two-level Hamiltonian with diagonal, ε ≡ E Q p,q / = E − Q p,q / , and off-diagonal, V q ≡ V χ q ,matrix elements. The doubly degenerate level ε splitsinto ε ± | V q | and a gap of width 2 | V q | opens at this en- D O S b y a g u g I D O S b y a g u g b THEORY
EXPERIMENT [2mz3][z/m2] [/mz/] [z4m7][4mz6] ya5 yc5yb5 yd5
EXPERIMENT l n b y I D O S THEORY ye5
EXPERIMENT THEORY9 I n t en s i t y b y a g u g [z3m5] [2mz3][z/m2] [/mz/] [z4m7][4mz6] N o r m a li z ed bI n t eg r a t ed bI n t en s i t y b y a g u g [2mz3] [z/m2] [/mz/] [z4m7][4mz6] [2mz3] [z/m2] [/mz/] [z4m7][4mz6] EnergybymeV5/592 /594 /596 EnergybymeV5/592 /594 /596 lnb[yEnergybzE R ] z3z2z/z5 z4 z3 z2 z/ 9 [z3m5] FIG. 4: (Color online) (a) Measured total (angularly integrated) emission spectrum I ( ε ) of the quasi-periodic wire and (b)Spectrally integrated emission intensity (cid:82) εE I ( ε (cid:48) ) dε (cid:48) (where E is the lower energy state); (c-d) Calculated normalized DOS (c)and IDOS (d). (e) Display of the log-periodic oscillations of the IDOS in a log-log plot of numerical (red) and experimental(blue) IDOS in the vicinity of E (normalized to E R = (cid:126) π / (8 a m p ), with m p the polariton mass). ergy. Accordingly, there is a one-to-one correspondencebetween the Bragg peaks and the gaps generated throughthe hybridization of plane waves, so that each gap canalso be labeled with the two integers [ p, q ]. Noting that Q p,q a/ π = p + qσ − is the proportion of unperturbedeigenmodes whose energies are less than ε = E Q p,q / , theIDOS inside the [ p, q ]-gap is N ( ε = E Q p,q / ) = p + qσ − = qσ − (mod. 1) (7)for N ( ε = E Q p,q / ) normalized to unity at E Q , .While the previous result has been obtained using per-turbation theory, it happens that it has a much broaderrange of validity generally expressed by the so calledgap labeling theorem [27] formulated by Bellissard andcoworkers. This theorem provides a precise frameworkfor applicability and allows to compute values of theIDOS in the gaps of the spectrum of 1D Schr¨odingerHamiltonians with bounded potentials V ( x ). An impor-tant consequence of that theorem is the topologically sta-ble nature of the IDOS values in the gaps which extendsbeyond perturbation theory. Those specific values areobtained [27] from some prescribed linear combinationsof components of eigenvectors of the corresponding sub-stitution matrix characteristic of the quasi-periodic po-tential. For the Fibonacci sequence defined in (3), thatprescription reduces to linear combinations of 1 and σ − namely to (7). In Fig.3.a, we indicate with red arrowsthe labeling of the gaps using the set [ p, q ], demonstratingthat the positions of the gaps are accurately determinedby the positions of the Bragg peaks even for a relativelyshort Fibonacci sequence such as considered here. Thesepositions are topological quantities, namely independentof the strength of the potential. These observed spec-tral features are thus independent of the (large enough)sample size and of the realization of the potential. Thesepoints are further discussed in the supplement [25]. Onthe other hand, the energy width of the gaps depends on the heights of the Bragg peaks, i.e. on the details of thepotential u b ( x ) (and w ( x )).The peculiar structure of the emission spectrum ap-pears also clearly by considering the total emission in-tensity I ( ε ) nearly proportional to the DOS for low ex-citation powers. Fig.4.a displays peaks and deeps cor-responding respectively to bands and pseudo-gaps. Themeasured integrated intensity (cid:82) εE I ( ε (cid:48) ) dε (cid:48) (with E be-ing the lower energy state), is reported in Fig.4.b togetherwith the numerically calculated DOS and IDOS (Figs 4.c-d). Applying (7), valid in principle in the infinite limit,to the gaps [2 , − , [ − , , [1 , −
1] indicated in Figs.4.b-d,gives respectively N ( E Q p,q / ) = 0 . , . , .
38. Thesenumbers are in excellent agreement with the experiment,confirming the good homogeneity achieved in populatingthe polariton states.For the infinite system, there exists an infinite seriesof gaps at p + qσ − ∈ [0 ,
1] . Thus the energy spectrum,which is the complementary of these gaps, is singularcontinuous. It is a Cantor like set whose total widthvanishes. The high resolution available in the numericsallows to consider finer details of the IDOS as predictedby the scaling form (2). In Fig.4.e, we have plotted in alog-log scale the IDOS as a function of (properly normal-ized) energy. It is noticeable that, even for such a finitesized system, we indeed observe a power law behaviormultiplied by a log-periodic function. More interesting isthe experimental observation of these log-periodic oscil-lations, showing two periods of oscillations, which con-stitutes a direct and so far unobserved signature of thefractal character of the Fibonacci spectrum.In summary, probing the luminescence of a polaritongas laterally confined by a Fibonacci quasi-periodic po-tential, we have observed the characteristic behavior ofthe associated fractal energy spectrum: gaps densely dis-tributed, and an IDOS well described by the scaling form(2) and following the gap labeling theorem (7). We haveobtained a spectrally and spatially resolved image of thepolariton modes which is in good quantitative agreementwith theoretical and numerical results. Our results sup-port the idea that topological features of a fractal spec-trum are robust and show up quite accurately even fora relatively short structure. Those results evidence thegreat interest of cavity polaritons to study the anoma-lous time expansion of a polariton wave-packet [11], morecomplex quantum systems e.g.
2D quasi-crystals [28] andmore generally to realize quantum simulators.Acknowledgements: This work was supported bythe Israel Science Foundation Grant No.924/09, bythe ’Agence Nationale pour la Recherche’ project”Quandyde” (ANR-11-BS10-001), by the FP7 ITN”Clermont4” (235114) , by the french RENATECH net-work, the LABEX NanoSaclay and the Honeypol ERCstarting grant. [1] H.L. Cycon, R.G. Froese, W. Kitsch and B. Simon,Schr¨odinger Operators, (Springer, Berlin, 1987) and M.Reed and B. Simon, Methods of Modern MathematicalPhysics (Academic Press, California, 1980).[2] D. Damanik and A. Gorodetski, Commun. Math. Phys. , 221 (2011) and D. Damanik, M. Embree, A.Gorodetski, S. Tcheremchantsev, Commun. Math. Phys. , 499 (2008)[3] For a recent review see Z. V. Vardeny, A. Nahat and A.Agrawal, Nature Photonics 7, 177187 (2013).[4] M. Kohmoto, B. Sutherland and C. Tang, Phys. Rev.B , 1020 (1987); J.M. Luck, Phys. Rev. B , 5834(1989).[5] W. Gellermann, M. Kohmoto, B. Sutherland and P.C.Taylor , Phys. Rev. Lett. , 633 (1994).[6] M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev.Lett. , 1870 (1983) and S. Ostlund and S.Kim, PhysicaScripta , 193 (1985). For a review see E.L. Albuquerqueand M.G. Cottam, Phys. Rep. , 225 (2003); E. Maci´a,Rep. Prog. Phys. , 397 (2006).[7] M. Kohmoto, B. Sutherland and K. Iguchi, Phys. Rev.Lett. , 2436 (1987) ; D. W¨urtz, T. Schneider and M.P.Soerensen, Physica A , 343 (1988).[8] For a recent review, E. Akkermans, Contemporary Math-ematics , 1-22 (2013), arXiv:1210.6763.[9] E. Akkermans, G.V. Dunne and A. Teplyaev, Phys. Rev.Lett. , 230407 (2010).[10] E. Akkermans, O. Benichou, G. Dunne, A. Teplyaev andR. Voituriez, Phys. Rev. E , 061125 (2012).[11] I. Guarneri and G. Mantica, Phys. Rev. Lett. , 3379(1994) and S. Abe and H. Hiramoto, Phys. Rev. A ,5349 (1987).[12] E. Akkermans and E. Gurevich, Europhys. Lett. ,30009 (2013).[13] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa,Phys. Rev. Lett. , 3314 (1992)[14] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H.Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada,T. Fujisawa and Y. Yamamoto, Nature , 529 (2007)[15] I. Carusotto and C. Ciuti, Rev. Mod. Phys. , 299 (2013)[16] E. A. Cerda-M´endez, D. N. Krizhanovskii, M. Wouters,R. Bradley, K. Biermann, K. Guda, R. Hey, P. V. Santos,D. Sarkar, and M. S. Skolnick, Physical Review Letters , 116402 (2010)[17] D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A.Lemaˆıtre, E.Galopin, R. Braive, P. Senellart, I. Sagnes,G. Malpuech and J. Bloch, Nature Communication ,1749 (2013)[18] N. Y. Kim, K. Kusudo,C. Wu, N. Masumoto, A. L¨offler,S.H¨ofling, N. Kumada, L. Worschech, A.Forchel and Y.Yamamoto, Nature Physics ,681 (2011)[19] E. A. Cerda-M´endez, D. N. Krizhanovskii, K. Biermann,R. Hey, M. S. Skolnick, and P. V. Santos, Phys. Rev. B , 100301 (2012).[20] E. A. Cerda-M´endez, D. Sarkar, D. N. Krizhanovskii,S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V.Santos, Phys. Rev. Lett. , 146401 (2013)[21] H. S. Nguyen, D. Vishnevsky, C. Sturm, D. Tanese, D.Solnyshkov, E. Galopin, A. Lemaˆıtre, I. Sagnes, A. Amo,G. Malpuech, and J. Bloch Phys. Rev. Lett. , 236601(2013)[22] N. Y. Kim, K. Kusudo, A. L¨offler, S. H¨ofling, A.Forchel,and Y. Yamamoto, New Journal of Physics ,035032 (2013)[23] N. Y. Kim, A. L¨offler, S. H¨ofling, A. Forchel, and Y.Yamamoto, Phys. Rev. B , 214503 (2013)[24] T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D.Solnyshkov, G. Malpuech, E. Galopin, A. Lemaˆıtre, J.Bloch and A. Amo, arXiv:1310.8105 (2013).[25] See supplementary materials.[26] L. D. Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. John-son, A. Lagendijk, R. Righini, M. Colocci, and D. S.Wiersma, Phys. Rev. Lett. , 055501 (2003).[27] J. Bellissard, A. Bovier and J.M. Ghez, Reviews inMath. Physics, Vol. 4, No. 1, 1-37 (1992) and B. Simon,Adv. Appli. Math. , 463 (1982) and J. Bellissard, LesHouches, Springer, J.M. Luck, P. Moussa and M. Wald-schmidt Eds., (1993).[28] J-M. Gambaudo and P. Vignolo, arXiv:1309.6420 (2013). Supplementary material for ”Fractal energy spectrum of a polariton gas in aFibonacci quasi-periodic potential”
D. Tanese , E. Gurevich , F. Baboux , T. Jacqmin , A. Lemaˆıtre ,E. Galopin , I. Sagnes , A. Amo , J. Bloch , E. Akkermans Laboratoire de Photonique et de Nanostructures,LPN/CNRS, Route de Nozay, 91460 Marcoussis, France and Department of Physics, Technion Israel Institute of Technology, Haifa 32000, Israel (Dated: February 24, 2014)In this Supplementary Material, we present additional experimental data supporting our claimabout fractal properties of the Fibonacci spectrum and in particular the invariance of the IDOS inthe gaps in accordance with the gap labeling theorem (7) discussed in the Letter. We then present abrief but explicit derivation of the 1D Schr¨odinger equation with the effective potential V ( x ) givenby equation (4) of the Letter. We stress the importance of the second, often omitted, term in V ( x )and then compare the results obtained within this effective 1D approach to those obtained using afull fledged numerical calculation of the 2D polariton spectrum. EXPERIMENTAL ILLUSTRATION OF THETOPOLOGICAL INVARIANCE OF THE IDOS
In order to illustrate the topological invariance of theintegrated density of states (IDOS) measured on a po-lariton gas laterally confined by a Fibonacci potential,we describe here additional data obtained for differentsystem sizes and realizations of the potential. The re-sults are presented in a similar way as in the paper sothat they can be directly compared to Figs. 3 and 4.These new data confirm that the wave vector values atwhich the mini-gaps open in the Fibonacci spectrum, aswell as the corresponding values of the IDOS, are invari-ant quantities of topological nature, i.e. that they do notdepend on the specific shape of the quasi-periodic poten-tial felt by the polaritons nor on the size of the letters.We study two additional samples (hereafter called sam-ples 2 and 3 by contrast to sample 1 which correspondsto the one presented in the Letter). These samples havea different length a of the letters than sample 1 (whichhad a = 0 . µ m). Sample 2 has longer letters ( a = 1 . µ m), while sample 3 has shorter ones ( a = 0 . µ m). Fur-thermore, while sample 3 has letter widths identical tothose of sample 1 ( w A = 3 . µ m and w B = 1 . µ m),sample 2 has a different w B = 2 . µ m, which resultsin a smaller potential contrast between the two types ofletters. Moreover samples 2 and 3 also differ from sample1 by their total number of letters, and thus correspondto different orders of the Fibonacci sequence: S (144letters) for sample 2, S (377 letters) for sample 3, ascompared to S (233 letters) for sample 1.Figures 1 and 2 display the spectrally resolved far fieldemission, the density of states (DOS) and the IDOS mea-sured on samples 2 and 3, together with correspondingcalculations.Let us first discuss the far field emission shown in thetop panels of the two figures. As in Fig. 3 of the Letter,mini-gaps open, whose positions in momentum space canbe accurately labeled (see arrows) by means of two inte- gers [ p, q ] such that k = πa ( p + qσ − ), in accordance withthe gap labeling theorem (7). Comparing these spectrawith sample 1 allows to understand their scaling proper-ties. Since the momentum k -positions of the gaps scaleas πa , the spectrum of sample 2 appears ”compressed”in energy with respect to that of sample 1: for instance,the gap [ − ,
2] appears much closer to the bottom of theparabola. Thus, due to the finite polariton linewidth,only one mode is visible below this gap (instead of threefor sample 1 in the Letter). For sample 3 instead, thespectrum appears ”stretched” with respect to sample 1,and more modes and mini-gaps can be experimentallyidentified below this [ − ,
2] gap than in the Letter.The values of the IDOS in the mini-gaps are also in-variant topological quantities. This is illustrated in Figs.1(d) and 2(d) where the theoretical values N ( E Q p,q / ) = p + qσ − for the IDOS inside the gaps (see Eq. (7) ofthe Letter), predicted by the gap labeling theorem, areindicated with red horizontal arrows. Both numericaland experimental results reproduce well the values of theheights of the plateaus.To conclude, the overall data presented in our workprovides a solid illustration of the scaling propertiesof the gaps positions and topological invariance of theIDOS, as ensured by the gap labeling theorem (7). DERIVATION OF THE EXPRESSION OF THEEFFECTIVE POTENTIAL GIVEN BYEQUATION (4)
We now turn to the derivation and discussion of thevalidity of the 1D Schr¨odinger equation with the effectivepotential V ( x ), V ( x ) = π w ( x ) + π + 312 (cid:18) w ( x ) w ( x ) (cid:19) (1)given by Eq. (4) in the Letter. To that aim, we needto map the original 3D setup onto an effective 1D prob- [PkX8] TdH TeHTfHTcH b I n t en s i t y T a M u M H bM5kb N o r m a li z ed n t eg r a t ed n t en s i t y T a M u M H bM8bM4 D O S T a M u M H I D O S T a M u M H bbM5kEnergy4TmeVHk57k k573k578 k574 Energy4TmeVH THEORY
EXPERIMENTEXPERIMENT THEORY [kXPk][bXk]
EXPERIMENT k578k573k574k575k576k577k57k E ne r g y T m e V H TbH k4T μ m Pk H b k 8 TbH
THEORY k4T μ m Pk H bPkP8 [kXb] [PkX8][kXPk][bXk] [P8X4][3XP4] TaH [PkX8][kXPk][bXk][PkX8] [bXk] [P8X4][3XP4] [P8X4][3XP4] [kXPk] [PkX8] [bXk] [P8X4][3XP4] [kXPk] bM6bM8bbM8bM4bM6bM8 bbM8bM4bM6bM8 k57k k573k578 k574
FIG. 1: (Color online) (a) Spectrally resolved far field emis-sion measured on sample 2 (parameters given in the text)and (b) corresponding numerical results obtained with theeffective 1D model described in the Letter and in the nextsection. The positions of the gaps are labeled with two inte-gers [ p, q ] and indicated with red arrows. (c) Measured total(angular-averaged) emission spectrum I ( ε ) and (d) normal-ized integrated emission intensity R εE I ( ε ) dε (with E thelower energy state). (e) Calculated DOS smoothed for thecomparison with I ( ε ) in (c). (f) Normalized calculated IDOS. lem. As described in more detail in the Letter, polari-tonic wires are fabricated by processing a planar λ/ n its effective refractive index. Theelectromagnetic field is confined along the (vertical) z -direction using two Bragg mirrors. This confinement ismuch tighter than that in the perpendicular xy -plane.For the latter, we impose zero boundary conditions, anapproximation justified by the high contrast in refrac-tive index between dielectric and air (see, e.g., Ref. [1]).Under the above assumptions, the corresponding electro-magnetic field eigenmodes can be chosen to have eitherTE or TM polarizations. The polarization splitting islarge in an etched wire cavity, probably because of strainrelaxation. Since in the experiment we detect only onepolarization, we do not include the polarization degree offreedom in the simulation and we consider a scalar waveapproximation. Then, looking for separable solutions be-tween vertical and lateral coordinates, leads to the follow- [R3H5] [bHR3] [RMHb] [R6HMP][7HRMM] [R3H5][bHR3] [RMHb] [R6HMP][7HRMM] EdV EeVEfVEcV P krE μ m RM V RMRb P krE μ m RM V bM P I n t en s i t y r E a Y u Y V PY5MP N o r m a li z ed rI n t eg r a t ed rI n t en s i t y r E a Y u Y V PYbPY4 D O S r E a Y u Y V I D O S r E a Y u Y V r PPY5MPPYbPY4EnergyrEmeVVM594 M596M595 M597 EnergyrEmeVVM594 M596M595 M597 [R3H5] [bHR3] [RMHb] [R6HMP][7HRMM] [R3H5][bHR3] [RMHb] [R6HMP][7HRMM]
THEORY
EXPERIMENTEXPERIMENT THEORY [RMHb] [R3H5] [R6HMP][7HRMM][R4H7] [bHR3]
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M594M595 E ne r g y r E m e V V M596M597
FIG. 2: (Color online) Same as Fig. 1 for sample 3 (parame-ters in the text). ing two-dimensional (2D) stationary wave equation Eψ ( x, y ) = − ~ m ph ⊥ ψ ( x, y ) , (2)where m ph ≡ n E c /c is the effective photon mass, E c ≡ ~ cn k z is the energy associated with the fundamental modeof the λ/ ⊥ ≡ ∂ x + ∂ y is the transverseLaplacian. Since E (cid:28) E c , the total photon energy canbe expanded in E so that, ~ ω ≈ E c + E . (3)As a result of our assumed zero boundary conditions inthe xy -plane, the electromagnetic field ψ ( x, y ) vanisheson the boundary.Note that Eq.(2), with the same boundary conditions,also holds to describe the center-of-mass motion of theexcitons confined to the xy -plane by the quantum wells.Therefore, within the same approximations, the electro-magnetic field and the excitons have similar eigenmodesand energy spectrum (up to the difference in their effec-tive mass), so that the photon-exciton coupling is diago-nal in the eigenmode index. A flat exciton dispersion isused because of their relatively large mass. The finding ofthe eigenmodes of the 2D problem (2) on a strip can eas-ily be done numerically. Nevertheless, it is useful to have
3a well controlled 1D effective model providing intuitionand insight of the essential features of the problem athand. This is particularly relevant for the quasi-periodicpotential we study and its fractal polariton spectrum,since a broad range of analytical and numerical tools arespecifically available for the 1D problem, such as the gaplabeling theorem used in the Letter.To proceed further and establish the expression of the1D effective potential Eq.(1), we look for solutions ofthe wave equation Eq. (2) on a symmetric strip de-fined by its longitudinal coordinate x ∈ [0 , L ], where L is the length of the wire, and its transverse coordinate − w ( x )2 ≤ y ≤ w ( x )2 . The function w ( x ) >
0, which de-fines the x -dependent width of the wire, is assumed tobe differentiable. The sought solution can generally bewritten in the form of a Fourier series over the transversequasi-modes, ψ ( x, y ) = ∞ X n =0 ψ n ( x ) s w ( x ) cos ( k y,n ( x ) y ) , (4)where both the transverse wave vector, k y,n ( x ) = π n +1 w ( x ) ,and the expansion coefficients ψ n ( x ), are x -dependent.This solution is symmetric with respect to the middleline y = 0, and it is not coupled to the similar anti-symmetric one (note that for a non-symmetric strip, bothsolutions would participate to the expansion (4)). Weneed to consider only symmetric solutions, since they in-clude the lowest frequency branch, corresponding to thelowest transverse quasi-mode, k y, ( x ) = πw ( x ) . An infi-nite hierarchy of coupled differential equations for ψ m ( x )is obtained by substituting the expansion (4) into thewave equation (2) and subsequently integrating over y with the weight q w ( x ) cos ( k y,m ( x ) y ). Neglecting thecoupling to the higher quasi-modes, leads to the follow-ing approximate equation for the lowest quasi-mode: Eψ ( x ) = ~ m ph (cid:20) − d dx + V ( x ) (cid:21) ψ ( x ) , (5)where V ( x ) given in Eq.(1), defines the effective 1D po-tential along the strip for the lowest transverse mode.Similar results have been obtained for the study of coldatoms in optical trap waveguides [2]. The coupling of ψ ( x ) to the higher quasi-modes leads to the appearanceof additional terms in Eq.(5), involving various deriva-tives of w ( x ). For a coupling strength between quasi-modes small compared to the energy separation to thenext mode, we can neglect those additional terms. Thedetailed analysis of this conditions is, however, beyondthe scope of this supplement. Instead, we justify this ap-proximation comparing our results to the full fledged 2Dnumerics.The first term in the potential V ( x ) given in Eq.(1)is the usual adiabatic approximation, proportional to k y, ( x ), which accounts for the distribution of the ”ki-netic” energy between the transversal and the longitudi-nal degrees of freedom. For a constant w ( x ), the prob-lem is separable. It leads to uncoupled transverse modes ψ m ( x ) and the adiabatic kinetic term is the only remain-ing contribution to V ( x ). For a varying profile w ( x ) suchas the one we consider, the problem is not separable any-more, and the second term in Eq.(1) becomes relevant.This term is sensitive to the stiffness of the boundaryvariation [2]. For a smoothly varying width, w ( x ) issmall and the second term is negligible compared to thekinetic term. For a sharper step structure, like the onewe consider (see Fig. 1(b)-(c) of the Letter), the twoterms in the effective potential become comparable. Inthe limit in of sharp steps for V ( x ), the second term inEq.(1) becomes singular, namely a repulsive δ -functionsquared. In that case, higher transversal quasi-modesmust be included. COMPARISON BETWEEN THE EXACT 2DCALCULATION AND THE EFFECTIVE 1DPOTENTIAL
We wish now to show that the effective 1D modelprovides a quantitatively good description of the mea-sured polariton spectrum provided we include the sec-ond term in the potential (1) which account for thesharp boundary modulation. To that purpose, we com-pare the low energy eigenmode spectra obtained fromthe exact two-dimensional (2D) and the effective one-dimensional calculations. The 2D calculation is done us-ing, instead of (4), a complete two-dimensional Fourierexpansion, and then diagonalizing the Hamiltonian inthis two-dimensional basis. In addition, a scale is in-troduced over which we smoothen the width profile bymeans of a convolution of the binary width profile withthe Gaussian kernel, g ( x ) ∝ e − ( x/ηa ) , (6)where a is the letter length and the relative dimensionlesssmoothness scale η is used as a fitting parameter (to theexperimental data). This is justified looking at the micro-graph of the wire in Fig. 1(b) of the Letter. Obviously,there is some smoothness in the wire width variation,introduced by the etching process. Its scale, however,is hard to quantify from the direct measurement, andshould be considered as a phenomenological parameter.In order to compare the effective 1D description to thefull 2D calculation, we consider sample 2 described above( a = 1 . µ m), and plot in Fig. 3 the integrated densityof states (IDOS) for different values of the fitting param-eter η . We note that the position and the width of thegaps of the 2D spectrum are significantly less sensitiveto the parameter η than the effective 1D spectrum. Note E [meV] N o r m a li ze d I D O S η = 0 .
22D (2 modes), η = 0 . η = 0 .
12D (2 modes) , η = 0 . FIG. 3: (Color online) Comparison, for sample 2, between theresults obtained for the IDOS based on the full 2D and the1D calculations using the effective potential V ( x ) given byEq. (1). The values of smoothing parameter η are indicatedin the inset. however, that the value of the IDOS in the gaps is in-dependent of η in both cases, since it is a topologicallystable quantity. These different sensitivities to the pa-rameter η can be rather exactly compensated by increas-ing the smoothness scale in the 1D calculation relativelyto the corresponding 2D case. This is demonstrated in Fig. 3 by superimposing the two results for different setsof choices of η . On the other hand, the 1D calculationusing only the first (kinetic) term in V ( x ) does not showany specific dependence on η even for rather large valuesof the smoothing. It is thus not possible to use this ap-proximation to reproduce the 2D calculation. Moreover,the 1D potential based on the first kinetic term only inEq.(1), is unable to reproduce the gap structure of thespectrum, even qualitatively. To show this, we have plot-ted in the left panel of Fig. 4, the spectral function ofthe 2D calculation. It is compared (right panel) to the1D spectral function obtained using the first term onlyin Eq. (1). We note the discrepancy in the position ofthe gaps which cannot be handled by a proper choice of η . More important, the higher energy gaps ( e.g. the onelabeled [1 , η reproduces faithfully both the 2D and the measuredspectra. [1] A. Kuther, M. Bayer, T. Gutbrod, A. Forchel, P. A.Knipp, T. L. Reinecke, and R. Werner, Phys. Rev. B ,15744 (1998).[2] S Schwartz, M Cozzini, C Menotti, I Carusotto, P Bouyerand S Stringari, New Journal of Physics , 162 (2006). (cid:239) (cid:239) (cid:239) µ m (cid:239) ] E [ m e V ] µ m (cid:239) ] FIG. 4: (Color online) Comparison, for sample 2, between thespectral function obtained from the full 2D calculation (left)versus the 1D calculation using the effective potential V ( xx