Topological phase transition in an all-optical exciton-polariton lattice
M. Pieczarka, E. Estrecho, S. Ghosh, M. Wurdack, M. Steger, D. W. Snoke, K. West, L. N. Pfeiffer, T .C. H. Liew, A. G. Truscott, E. A. Ostrovskaya
TTopological phase transition in an all-optical exciton-polariton lattice
M. Pieczarka,
1, 2, ∗ E. Estrecho, S. Ghosh, M. Wurdack, M. Steger, † D. W. Snoke, K. West, L. N. Pfeiffer, T. C. H. Liew, A. G. Truscott, and E. A. Ostrovskaya ‡ ARC Centre of Excellence in Future Low-Energy Electronics Technologies and Nonlinear Physics Centre,Research School of Physics, The Australian National University, Canberra, ACT 2601, Australia Department of Experimental Physics, Wroc(cid:32)law University of Science and Technology,Wyb. Wyspia´nskiego 27, 50-370 Wroc(cid:32)law, Poland Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA Laser Physics Centre, Research School of Physics,The Australian National University, Canberra, ACT 2601, Australia
Topological insulators [1] are a class of electronicmaterials exhibiting robust edge states immuneto perturbations and disorder. This concepthas been successfully adapted in photonics [2–4],where topologically nontrivial waveguides [5–7]and topological lasers [8–11, 17] were developed.However, the exploration of topological proper-ties in a given photonic system is limited to afabricated sample, without the flexibility to re-configure the structure in-situ . Here, we demon-strate an all-optical realization of the orbital Su-Schrieffer-Heeger (SSH) model [18] in a micro-cavity exciton-polariton system [19–22], wherebya cavity photon is hybridized with an exciton ina GaAs quantum well. We induce a zigzag poten-tial for exciton polaritons all-optically, by shap-ing the nonresonant laser excitation, and measuredirectly the eigenspectrum and topological edgestates of a polariton lattice in a nonlinear regimeof bosonic condensation. Furthermore, taking ad-vantage of the tunability of the optically inducedlattice we modify the intersite tunneling to re-alize a topological phase transition to a trivialstate. Our results open the way to study topo-logical phase transitions on-demand in fully re-configurable hybrid photonic systems that do notrequire sophisticated sample engineering.
Microcavity exciton polaritons (polaritons therein),hybrid quasiparticles resulting from strong coupling ofexcitons and photons in a semiconductor microcavity[19–22], have emerged as a perfect platform for numer-ous applications in nonlinear and topological photonics[4, 23–25]. These interacting bosons combine a verylow effective mass inherited from cavity photons with re-pulsive interactions inherited from excitons, allowing forbosonic condensation at elevated temperatures.Taking advantage of the photonic part of a polariton,one can modify the planar microcavity by various fabrica-tion techniques and realize polariton trapping potentials,as well as a lattice of coupled traps [17, 25–27]. Addi-tionally, the TE-TM polarized modes splitting in a planar cavity results in an effective spin-orbit interaction for po-laritons enabling realizations of topological and flat-bandsystems [25, 27–29]. Nevertheless, this technological ap-proach has a major practical drawback as once the sam-ple is made, there is little or no room for modification ofits properties. This limits the applications of polariton-based photonic topological devices, where active controlis highly desirable [30–33].The solution to this issue lies in using the excitoniccomponent of the polariton for engineering the trappingpotential. Under nonresonant optical excitation abovethe semiconductor material bandgap, the pumping lasercreates a high-energy excitonic reservoir [34], which actsas a non-Hermitian potential that replenishes and repelspolaritons due to exciton-polariton interactions [26, 35,36]. The excitonic potential can therefore be shaped viaspatially structured laser excitation.In this work, we employ a spatially structured laserbeam, imaged via a microscope objective onto a planarmicrocavity sample with embedded GaAs quantum wells,see Fig. 1(a). Spatial structuring is achieved by selec-tively reflecting the laser beam from a programmed dig-ital micromirror device (DMD) (see Methods). The spa-tial distribution of the pump effectively creates a chain ofcoupled circular traps in the zigzag geometry, first pro-posed for the realization of low-dimensional topologicalsystems in plasmonics and nanophotonics [12–16]. Exci-ton polaritons trapped in this optically induced chain re-alize the orbital version of the Su-Schrieffer-Heeger (SSH)model, previously demonstrated in etched samples withcoupled micropillars [17, 27, 37].The SSH chain [18] is the simplest realization of a topo-logical insulator in one dimension (1D). Here, we focus onthe SSH model realized with the p -modes of each circulartrap, coupled as shown in Fig. 1(b,d). In contrast to lin-ear chains with alternating distances between the latticesites [8, 9, 38], the orthogonality of the p -modes makesthe SSH model valid for d A = d D = d , where d A and d D are the lattice constants in the antidiagonal ( A ) and di-agonal ( D ) directions [17]. This is because the tunnelingamplitudes in these two directions, t A and t D , are differ- a r X i v : . [ phy s i c s . op ti c s ] F e b a b c d e Laser beamDMD
Objective
Microcavity sample E ( m e V ) Eigenvalue N E ( m e V ) Eigenvalue 𝑥𝑦 FIG. 1.
Realization of an orbital SSH Hamiltonian with a nonresonant optical excitation. a , Simplified schemeof the experimental setup for creating exciton polaritons in an optically-induced trapping potential. Inset presents the spatialdistribution of the laser pump reflected from the sample. Dark areas correspond to polariton traps with the trap diameter D trap ≈ . µ m, arranged into a zigzag chain. White scale bar corresponds to 10 µ m. b,d , Sketch of the orbital SSH modelfor ( b ) topologically trivial and ( d ) nontrivial cases, realized with the trapped p -modes of different orientations A, D . c,e Eigenenergies of a tight-binding Hamiltonian model corresponding to the N = 6 chain in ( b,d ). Edge states within the gapare indicated with an arrow in panel ( e ). Inset shows the probability density distribution of the edge state. ent due to the collinear or orthogonal alignment of the p -modes with the axis linking the consecutive traps. Asa result, two configurations exist in finite length chainswith an even number of lattice sites, with the staggeringorder being trivial t D > t A for diagonal, p D , and non-trivial t D < t A for antidiagonal, p A , mode orientations.The two different configurations (phases) are presentedin Fig. 1(b,d). They are characterized by phase windingnumbers W = 0 for trivial and W = 1 for topologicalone (see Methods). The eigenenergies of a tight bindingHamiltonian model for finite sized chains investigated inthis work are presented in Fig. 1(c,e). The normal phasein Fig. 1(c) is characterised by a spectrum with a trivialbandgap. The topologically nontrivial phase, presentedin Fig. 1(e), differs from Fig. 1(c), as two eigenvaluesclose to zero energy emerge in the bandgap. The corre-sponding eigenstates are strongly localized at the edgesof the chain [see inset in Fig. 1(e)] and are an indicatorof a topological phase. In short chains, the spectra arediscrete and the edge states can have energies slightlydifferent from zero, depending on the ratio of the tun-nelling amplitudes t D /t A . Nevertheless, the topologicalproperties of the system are maintained.We investigate the optically-induced SSH chain atpump powers slightly above the exciton-polariton con-densation threshold. In this regime, the condensate occu-pies high-symmetry points of the momentum-space bandstructure (for details, see the data in Fig. S1 of the Sup-plementary Information). The narrowing of the linewidth of the exciton-polariton emission above the condensationthreshold allows us to resolve the ground and excitedbands with the corresponding bandgaps. The low-energy s -band is highly populated due to efficient energy relax-ation of polaritons towards the ground state [39]. Wealso observe nonzero occupation of excited states form-ing a higher-energy band. The corresponding position-resolved spectrum is presented in Fig. 2(a). The intensesignal from the s -band at low energies is clearly sepa-rated from the p -band by a bandgap. It indicates thatthe created potential confines the excited states and re-alizes the physics discussed in Fig. 1. More importantly,the p -band is split into two sub-bands corresponding tothe bonding (in-phase) and anti-bonding (out-of-phase)coupling between the lattice sites (in analogy to elec-tron orbitals in molecules). The gap between the two p -bands is larger than the linewidth, which allows us todirectly identify the localized states at the edges of theSSH chain. The energies of these states lie inside the p -band gap, as indicated by the arrows in Fig. 2. The spec-tral cross-sections in the bulk (i.e., in the middle) and atthe edge of the chain are presented in Figs. 2(c,d), wherethe shaded areas present the result of fitting the spec-trum with Lorentzian lines. Existence of the in-gap edgemodes confirmed by these measurements is a signatureof a topological phase of the SSH lattice [1, 17, 27, 37].We note that the occupation of all bands is not spatiallyhomogeneous, as the sample is characterized with a in-trinsic linear energy gradient due to the spatially varying -20 -10 0 10 201.59901.59921.59941.59961.59981.6000 x ( m m) E ne r g y ( e V ) I n t en s i t y ( a r b . u . ) Energy (eV) -20 -10 0 10 201.59901.59921.59941.59961.59981.6000 x ( m m) E ne r g y ( e V ) e a b I n t en s i t y ( a r b . u . ) Energy (eV) c d f g h i j experiment simulation ∇𝐸 ∇𝐸 FIG. 2.
Position-resolved spectrum and density distribution of exciton polaritons in the SSH chain. a , Exper-imentally measured spectrum along the chain with N = 6 sites, integrated over the orthogonal direction. b , Correspondingspectrum obtained by numerical simulations of the mean-field open-dissipative model, Eq. (4) in Methods. c,d , Experimentalspectrum measured at a position ( c ) in the centre of the chain, and ( d ) at the left edge of the chain. Shaded areas representthe result of fitting with Lorentzian lines. p -bands and d -band peaks are in semitransparent blue and the edge mode peakis coloured in semitransparent red. e-i , Experimental images of the exciton-polariton emission corresponding to the spatialdensity distributions taken at the energies of the ( e ) upper p -band, ( g ) edge states (middle of the bandgap) and ( i ) the lower p -band. f-j , Density distributions obtained by numerical simulations of the model Eq. (4) corresponding to the panels ( e-i ).The scale bar corresponds to 10 µ m and the direction of the energy gradient in the sample is indicated with an arrow. thickness of the cavity (see Methods), oriented antidiag-onally with respect to the x -axis in the presented data.This effect is captured in our simulations of the full open-dissipative mean-field model [see Methods, Eq. (4)], withthe results presented in Fig. 2(b), showing an excellentagreement with the experimental observations. We em-phasize that the simulations are based on experimentallydetermined parameters of the sample (see Methods). Theenergy gradient does not change the topology of the sys-tem, as the polariton states at the individual lattice sitesare hybridized, which is reflected in the opening of thebandgap (see also Fig. S1).The spatial distribution of the exciton-polariton den-sity in the chain is obtained by real-space spectral to-mography, which enables selective real-space imaging ofthe polariton emission intensity at a given energy (seeMethods). The experimental results are presented inFig. 2(e,g,i) together with the results of numerical mod-elling of Eq. (4) in Fig. 2(f,h,j). The lower p -bandshows a characteristic spatial distribution of a bondingstate, where the p -modes from neighbouring traps over-lap [Fig. 2(i,j)]. The upper p -band shows an antibond-ing character with pronounced density dips between thetraps indicative of the nodes in the probability densitydistribution [Fig. 2(e,f)]. Finally, the edge states arecomposed of antidiagonal p -mode configuration formingthe topologically nontrivial realization of the SSH model,Fig. 2(g,h).To test the robustness of the topologically protected edge states, we imprinted different chains at different po-sitions on the sample with similar detunings. The edgestate for the N = 6 chain similar to that in Fig. 2, butwith a different orientation with respect to the energygradient, is presented in Fig. 3(a) with the correspond-ing numerical simulation shown in Fig. 3(b). For this ori-entation, the diagonal p -mode configuration is the topo-logically nontrivial one. Similarly to the N = 6 case,the orbital SSH model with an odd number of sites sup-ports two edge states, however each edge state comesfrom a different p -mode configuration [17, 27, 37]. Thisis clearly seen in our experimental data for a chain with N = 5 sites, presented in Fig. 3(c), in agreement withsimulations in Fig. 3(d). These results demonstrate theinsensitivity of the all-optical realization of a topologicalSSH model to the orientation of the energy gradient andlocal disorder of the sample.To demonstrate an optically-driven topological phasetransition in our system, we modified the topology of the N = 6 chain by changing the ratio between the tunnelingamplitudes from t A /t D > t A /t D < N = 6chain. The coupling between the nearest-neighbour sitesdepends on the potential barrier amplitude as well ason the distance between the traps [40]. Therefore, weincrease the coupling t A or t D by reducing the trap sep-arations in A or D directions, while keeping all otherparameters constant. In this way, we modified the SSHHamiltonian for p A configuration. Fig. 4(a,b) presentsthe intensity distribution of the laser excitation reflected a bc d experiment simulation ∇𝐸∇𝐸 FIG. 3.
Topological edge states in different lattice re-alizations. a,c , Experimental spatial density distributionsof the edge state for ( a ) a chain of N = 6 sites with a dif-ferent orientation compared to the chain in Fig. 2 and ( c ) achain of N = 5 sites. b,d , Results of numerical simulationsof the model Eq. (4) corresponding to the cases in ( a ) and( c ). The scale bar corresponds to 10 µ m and the direction ofthe energy gradient is indicated with an arrow. from the sample for the chains of modified dimerisations.The chain maintaining the topological phase is presentedin Fig. 4(a) and the chain in the trivial phase is shown inFig. 4(b). In both cases, the lattice constant in one of thedirections was reduced by 20%. The measured real-spacespectra for these geometries are shown in Fig. 4(c,d). Oneobserves an increase of the topological and the trivialbandgaps between the p -bands in comparison to the un-modified chain: ∆ E topo = 174 ± µ eV for d D /d A = 0 . E triv = 218 ± µ eV for d D /d A = 1 .
25 in compar-ison to ∆ E topo = 151 ± µ eV for d D /d A = 1 .
0, fromFig. 2. Growth of this value for the modified chains is adirect manifestation of the control over the tunneling am-plitudes, as the bandgap in the SSH model is proportionalto | t D − t A | (full set of values is presented in Supplemen-tary Information). The geometrical modification leadingto the change of the tunneling amplitudes influences the s -band as well, which is now clearly split, and signaturesof edge modes are visible in Fig. 4(c). The asymmetry ofthe p -band gap values is caused by the energy gradient,which influences the sensitivity of t A and t D to the trapseparation parameter. Additional input to the tunnelingamplitudes comes from the intrinsic TE-TM splitting andweak birefringence of the sample [33, 41].The cavity gradient influences the occupation of thetopological edge states in the measured spectrum inFig. 4(c), with one of the edge states dominating thespectrum. On the other hand, the spectrum in Fig. 4(d)shows no signatures of the edge states, as expected fora topological phase transition to a trivial configurationwith W = 0.The flexibility of our all-optical potential allows us totune the lattice geometry parameters continuously. Thus, we performed a series of measurements to pinpoint thetopological phase transition in the chain. Direct mea-surement of the winding number W is challenging in ourexperimental configuration [43], hence we measure theintensity distribution at the mid-gap and observe the ap-pearance of the edge states as a signature of the topo-logical phase, see Fig. 4(g). The transition to a trivialstate occurs around d D /d A ≈ . − .
95, where we ob-serve an abrupt change in the intensity at the edge ofthe chain [there is a clear asymmetry in the edge modes’occupations, see inset in Fig. 4(g)]. We reproduce thisobservation by calculating the winding number in thetight-binding Hamiltonian, including the potential gra-dient and random disorder (see Methods), which softensthe transition threshold and moves it away from the point t A /t D = 1, Fig. 4(h), as observed in the experiment.To summarize, we have demonstrated an all-opticallydriven topological phase transition in a fully reconfig-urable optically-induced orbital SSH lattice created in anopen-dissipative exciton-polariton system. The transi-tion is controlled by fine-tuning the strength of tunnelingbetween the lattice sites. We emphasize that implement-ing this kind of control would typically require fabrica-tion of many different samples with different implemen-tations of lattice geometries. Moreover, we have demon-strated the robust topologically protected edge states in aregime, where exciton polaritons are condensed and non-linear (density-dependent) effects could begin to play arole [44]. Combined with the experimental control of thegain and loss (linewidth) of the trapped polariton con-densates [36] our system represents an attractive, flexi-ble platform for further studies of topological effects in anonlinear and non-Hermitian hybrid photonic system. METHODS
Sample and experiment
The sample used in the ex-periment is a high-quality GaAs-based microcavity witha long cavity photon lifetime exceeding 100 ps [42].The cavity of the length 3 λ/ . Ga . As/AlAs layer pairs. The active region is madeof 12 GaAs/AlAs quantum wells (QWs) of 7 nm nominalthickness positioned in three groups at the maxima ofthe confined photon field. The measured Rabi spliting isabout (cid:126)
Ω = 15 . ± . − .
43 meV.All results are obtained with a microcavity kept in a con-tinuous flow helium cryostat, ensuring the sample tem-perature of 7-8K.The nonresonant optical excitation was provided by acontinuous-wave (CW) Ti: Sapphire laser (M SquaredSolsTiS), tuned to the cavity reflectivity minimum abovethe QW bandgap. The Gaussian laser beam was trans- -20 -10 0 10 201.59901.59921.59941.59961.59981.6000 x ( m m) E ne r g y ( e V ) -20 -10 0 10 201.59901.59921.59941.59961.59981.6000 x ( m m) E ne r g y ( e V ) a b c de f gh E ne r g y ( e V ) x ( m m) E ne r g y ( e V ) x ( m m) experiment experimentsimulation simulation -20 -10 0 10 20 x ( m m) N o r m . i n t en s i t y ( a r b . u . ) d D / d A W t A / t D FIG. 4.
Measurement of the topological phase transition in SSH chains. a,b , Spatial distributions of the excitationintensity reflected from the sample for the modified chains in the ( a ) topological and ( b ) the trivial phase. The scale barcorresponds to 10 µ m. c,d , Experimental spectra corresponding to the cases presented in ( a,b ). e,f , Numerically computedspectra corresponding to the experimental measurements in ( c,d ). g , Intensity of the exciton-polariton edge state at the rightend of the modified chains as a function of d D /d A . Inset shows the intensity distributions measured in the middle of the p -bandgap. h , Winding number W calculated from the tight-binding Hamiltonian, including the energy gradient and disorder [seeMethods, Eqs. (1) and (3)], as a function of t A /t D . The trivial phase is coloured in blue and the topological phase is colouredin red in ( g,h ). formed and focused to a top-hat distribution by a shapinglens (Eksma Optics GTH-5-250-4) and imaged onto thedigital micromirror device (DMD). The shape of the lat-tice potential was encoded on the DMD which reflectedthe laser selectively and then imaged onto the sample viaset of lenses and a microscope objective [36]. Photolu-minescence from the sample was collected with the sameobjective in the reflection geometry and imaged with aset of confocal lenses onto the slit of the spectrometer(Princeton Instruments IsoPlane 320), equipped with a2D sensitive CCD (Andor iXon Ultra 888). The imag-ing lens in front of the spectrometer was mounted on amotorized stage, enabling the spectral tomography of theemission. The tomography was done by collecting a setof spectral images, scanning the full real-space emissionby moving the image with respect to the entrance slit. Theory.
The SSH model is described by a tight-bindingHamiltonian:ˆ H SSH = (cid:88) i (cid:16) t D ˆ a † i ˆ a (cid:48) i + t A ˆ a † i +1 ˆ a (cid:48) i + h.c. (cid:17) (1)where ˆ a i and ˆ a (cid:48) i are the annihilation operators in the i th unit cell of a lattice. The eigenfunctions of the Hamilto-nian ˆ H SSH are given by ( ± e − iϕ ( k ) , k being the wave vector. The winding number W is defined by, W = 12 π (cid:90) BZ dk, ∂ϕ ( k ) ∂k (2)corresponding to the Zak phase Z = π W , which is a 1Dlattice equivalent of the geometric Berry phase. In anideal SSH model, W = 1 for t D < t A and W = 0 for t D > t A . In our system, the lattice is perturbed by apotential gradient and disorder. To include these effects,we consider a Hamiltonian given by:ˆ H = ˆ H SSH + (cid:88) i (cid:16) V i − ˆ a † i ˆ a i + V i ˆ a (cid:48)† i ˆ a (cid:48) i (cid:17) . (3)The potential energy is given by V i = v i + v ( i − N/ v is a constant, v i is a random value represent-ing disorder in the system, and N is the number of unitcells in the lattice. For our simulations v = 0 . /N , v i is randomly distributed within the interval [ ± . t A , t D ).Within the mean-field approximation, exciton polari-tons in optically induced potentials can be described bya driven-dissipative equation for the polariton wavefunc-tion, ψ ( x, y, t ): i (cid:126) ∂ψ∂t = (cid:20) − λ (cid:126) m ∇ + iP + V ( x, y ) + α | ψ | (cid:21) ψ (4)where λ = (1 − iλ ), λ is a phenomenological pa-rameter describing the energy relaxation [45], and m is the effective mass of the polaritons. The net gain P ( x, y ) = P ( x, y ) − γ is given by the difference be-tween the pump strength (gain) P ( x, y ) and the loss γ (linewidth). The parameter α = ( α R − iα I ) repre-sents the polariton-polariton interaction α R and nonlin-ear decay α I . V ( x, y ) accounts for a real part of thepotential (the SSH lattice) induced by the repulsive in-teractions of the polaritons with an optically injected ex-citonic reservoir, as well as for the energy gradient inthe cavity. For the numerical simulations, we consider α R = α I and a transformation ψ → ψ/ √ α R to obtain anequation describing the exciton polaritons: i (cid:126) ∂ψ/∂t =[ − ( λ (cid:126) ) / (2 m ) ∇ + iP ( x, y ) + V ( x, y ) + (1 − i ) | ψ | ] ψ .The loss and effective mass parameters are obtained fromthe experimental data in [42]. Considering the long life-time of exciton polaritons in our GaAs-based sample, weuse a linewidth γ = 5 . µ eV corresponding to a lifetime120ps. We take m = 7 . × − m e , where m e is themass of an electron. Other parameters were chosen as λ = 0 .
05 meV and max( P ) = 10 . µ eV, to fit the phe-nomenology of the present experiment. Author contribution statement
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