Dynamically Tuned Arrays of Polariton Parametric Oscillators
Alexander S. Kuznetsov, Galbadrakh Dagvadorj, Klaus Biermann, Marzena Szymanska, Paulo V. Santos
DDynamically Tuned Arrays of Polariton Parametric Oscillators
Alexander S. Kuznetsov, ∗ Galbadrakh Dagvadorj,
2, 3
KlausBiermann, Marzena Szymanska, and Paulo V. Santos † Paul-Drude-Institut f¨ur Festk¨orperelektronik, Leibniz-Institut imForschungsverbund Berlin e. V., Hausvogteiplatz 5-7, 10117 Berlin, Germany Department of Physics and Astronomy, University College London,Gower Street London WC1E 6BT, United Kingdom Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: March 4, 2020)Optical parametric oscillations (OPOs) – a non-linear process involving the coherent coupling ofan optically excited two particle pump state to a signal and an idler states with different energies– is a relevant mechanism for optical amplification as well as for the generation of correlated pho-tons. OPOs require states with well-defined symmetries and energies: the fine-tuning of materialproperties and structural dimensions to create these states remains a challenge for the realizationof scalable OPO-based functionalities in semiconductor nanostructures. Here, we demonstrate apathway towards this goal based on the control of confined microcavity exciton-polaritons modu-lated by the spatially and time varying dynamical potentials produced by a surface acoustic waves(SAW). The exciton-polariton are confined in µ m-sized intra-cavity traps fabricated by structuringa planar semiconductor microcavity during the epitaxial growth process. OPOs in these structuresbenefit from the enhanced non-linearities of confined systems. We show that SAW fields inducestate-dependent and time-varying energy shifts, which enable the energy alignment of the confinedlevels with the appropriate symmetry for OPO triggering. Furthermore, the dynamic acoustic tun-ing, which is fully described by a theoretical model for the modulation of the confined polaritonsby the acoustic field, compensates for fluctuations in symmetry and dimensions of the confinementpotential thus enabling a variety of dynamic OPO regimes. The robustness of the acoustic tuningis demonstrated by the synchronous excitation of an array of confined OPOs using a single acousticbeam, thus opening the way for the realization of scalable non-linear on-chip systems. I. INTRODUCTION
Microcavity exciton-polaritons (polaritons) are light-matter quasi-particles resulting from the strong couplingbetween photons confined in a semiconductor microcav-ity (MC) with excitons in a quantum well (QW) embed-ded in the MC spacer [1]. Polaritons inherit the low ef-fective mass from their photonic component, which givesthem spatial coherence lengths of several µ m. The po-lariton properties can thus be modified by confinementwithin µ m dimensions as compared to the nm dimen-sions normally required to induce quantum shifts in elec-tronic systems. In addition, the excitonic component ofpolaritons gives rise to inter-polariton interactions and,thus, non-linearities much stronger than between pho-tons. Finally, polaritons are composite bosons and canform Bose-Einstein-like condensates (BECs) at tempera-tures several orders of magnitude higher than cold atoms[2].The mixed light-matter nature of polaritons brings therich physics of correlated systems to an all-semiconductorplatform [3, 4]. It was early recognized that the stronginter-polariton interactions and the peculiar shape of thepolariton energy dispersion enable stimulated parametricamplification with very large gain [5, 6] as well as optical ∗ [email protected] † [email protected] parametric oscillations (OPOs) [6–8]. Here, two pump( p ) polaritons resonantly excited at the inflection point ofthe lower polariton dispersion can scatter into signal ( s )and idler ( i ) states while conserving energy and momen-tum [9–12]. The OPO process is a convenient approachfor direct excitation of polariton condensates by stimu-lated scattering to the s states at the bottom of the dis-persion [7]. This excitation scheme avoids the formationof a high-density reservoir of excitons with high in-planemomentum, which normally occurs for non-resonant op-tical injection.The OPO process provides a pathway for the efficientgeneration of correlated and entangled photons. Differ-ent approaches have been proposed to enhance the effi-ciency of the process including the engineering of thepolariton density of states by using multiple cavities [13]and spatial confinement [14]. Confinement creates a dis-crete spectrum of polariton levels. The latter can act aspump, signal, and idler states, provided that the symme-try and energy spacing required for OPO are satisfied.This approach profits from the high density of polari-tons that can be excited in confined potentials, whichenhances the non-linear interactions required for OPOformation. In addition, since OPO properties are con-trolled by the dimensions of µ m-sized polariton traps, afurther advantage of confinement is the scalability arisingfrom the combination of multiple OPO structures on thesame polariton MC. The design of confined OPO levelswith equally spaced pump, idler, and signal states withthe appropriate symmetries to enable mutual non-linear a r X i v : . [ c ond - m a t . o t h e r] M a r interactions remains, however, a challenging task. Thestudies in Ref. [14] show that this requirement can besatisfied by the confined states of square pillars etched ina (Al,Ga)As polariton MC. However, even a smallest de-viation in the potential shape results in non-equidistantenergy spectrum, thus preventing OPO excitation.In this work, we demonstrate a pathway for the effi-cient generation of confined polariton states with the ap-propriate symmetry and energy alignment for OPOs viathe dynamic energy tuning by a surface acoustic wave(SAW). The studies are carried out in intra-cavity po-lariton traps defined in the spacer layer of an (Al,Ga)AsMC fabricated by molecular beam epitaxy (MBE), cf.Fig. 1. We show that the spatially dependent SAW fieldsinduce state-dependent energy shifts of the confined lev-els, which enable the alignment of levels with the appro-priate symmetry for OPO triggering. Spatially resolvedwave function maps of OPO states prove that the signaland idler states must have same parity, in agreement withthe predictions of a model developed to account for non-linear interactions between confined levels in the contactapproximation regime [3]. Time-resolved investigationsprove the dynamic character of the acoustic OPO trig-gering at multiples of the SAW frequency. Finally, theacoustic modulation enables robust OPO triggering overa wide range of excitation conditions. In particular, wedemonstrate the synchronous tuning of an array of con-fined OPOs using a single SAW beam, thus proving thefeasibility of scalable OPO systems. II. EXPERIMENTAL DETAILS
Confined polariton states employed here originate from µ m-sized intra-cavity traps created within the spacerlayer of an (Al,Ga)As MC. The traps were producedby etching the spacer layer of the MC between growthsteps by molecular beam epitaxy (MBE). The samplewas grown on a GaAs (001) substrate having the struc-ture schematically illustrated in Fig. 1(a). During theMBE growth run, the lower distributed Bragg reflector(DBR) and the MC spacer region containing three pairsof 15 nm-thick GaAs QWs centered at the anti-nodes ofthe optical field were deposited and then terminated bya 120 nm-thick Al . Ga . As layer. The sample wasthen removed from the MBE chamber and patterned bymeans of photolithography and wet chemical etching toform 12 nm-high and a few µ m-wide mesas with differentshapes. For the final growth step, the sample was rein-serted in the MBE chamber for the deposition of the up-per DBR. The lower and upper DBRs consist of 58.7 nmand 65.8 nm-thick pairs of Al x Ga − x As/Al x Ga − x Aswith different Al compositions x = 0 .
15 and x = 0 . E b = 5 . FIG. 1.
Intra-cavity polariton traps. (a) Polariton mi-crocavity with intra-cavity traps defined by structuring thespacer region during the growth process. (b) Atomic force mi-crograph showing the surface relief of a nominally 4 × µ m trap. (c) Surface acoustic wave (SAW) resonator containingan intra-cavity polariton trap. an energy barrier E b can then be formed by enclosing a µ m-sized non-etched region by etched areas. Due to theconformal nature of the MBE growth, the lateral dimen-sions of the traps can be estimated by measuring the sur-face relief of the MC, as illustrated in the Fig. 1(b) for asquare trap with nominal lateral dimensions of 4 × µ m .The anisotropic MBE growth yields traps with differentprofiles along the trap sides oriented along the x = [¯110]and y = [¯1¯10] surface directions. A detailed analysis ofAFM profiles from the traps presented in Secs. SM1 ofthe Supplementary Material (SM) shows that the con-finement potential is mirror-symmetric with respect tovertical planes x = 0 and y = 0 [cf. Fig. 1(b)] but withdifferent profiles along the two directions.The spectroscopic studies were performed at a temper-ature between 6 and 10 K in an optical cryostat withradio-frequency (rf) feedthroughs for the excitation ofSAWs. We have measured the photoluminescence (PL)of small square traps (dimensions ≤ µ m) placed withina SAW delay line formed by two single-finger interdig-ital transducers (IDTs). The latter were designed forlaunching SAWs along the x || [¯110] surface direction witha wavelength of 8 µ m (corresponding to an acoustic fre-quency f SAW = 383 .
66 MHz at 10 K). The delay lineforms an acoustic resonator with a quality factor of 4700[15]. Care was taken to match the location of the trapswith the anti-nodes of the SAW strain field. Time-resolved studies of the OPO dynamics were carried outby detecting the PL with a streak camera synchronizedwith the rf-signal used to excite the SAW resonator.
FIG. 2.
OPO in a polariton trap with red-shifted pump . Photoluminescence maps of the squared wave functions | Ψ ij | ( i, j = 1 , ,
3) of confined polariton levels in a 4 × µ m intra-cavity trap projected on the (a) y = 0 and (b) x = 0 planesunder the low-density, non-resonant optical excitation. (c)-(d) Corresponding spatially resolved PL maps recorded under quasi-resonant excitation of the Ψ confined level in the (c) absence and (d) presence of a standing SAW along x [cf. Fig. 1(c)]. (e)Calculated energy evolution of the confined levels Ψ m x m y with the SAW phase, φ SAW : the labeled ones (thick red curves) arelevels of the OPO set O = { Ψ , Ψ , Ψ / Ψ } . The dash-dotted black vertical lines designate SAW phase for which thelevels correspond to the unmodulated ones. The green horizontal line marks the pump energy while the dashed vertical bluelines show the phases of equidistant separation between the levels within O . (f)-(h) Calculated | Ψ m x m y ( x, y ) | projections onthe x - y -plane corresponding to the φ SAW = 0 .
415 (2 π ). III. RESULTSA. Confined polaritons in intra-cavity traps
Figure 2(a) and 2(b) compare PL spectral maps of asquare trap with nominal dimensions of 4 × µ m pro-jected on the x = 0 and y = 0 planes, respectively. Thesemaps were recorded under low-density non-resonant opti-cal excitation conditions by collecting the PL with spatialresolution along two perpendicular directions, thus yield-ing the projection of the squared wave functions | Ψ ij | ( i, j = 1 , , . . . ) of the confined polariton levels on the x = 0 and y = 0 axes, respectively. The i and j indicesdenote the number of lobes of | Ψ ij | along the x and y directions, respectively.The polariton states in the intra-cavity traps can beapproximated by those of a rectangular one with infinitebarriers and dimensions ‘ x ∼ ‘ y along the x and y di-rections, respectively, which can be classified by indices( m x , m y ), m i = 1 , , . . . and i = x, y ) according to: E m x m y = E LP + ~ m p "(cid:18) m x − l x (cid:19) + (cid:18) m y − l y (cid:19) . (1)Here ~ is Planck’s constant, m p the reduced polaritonmass, and E LP the lower polariton energy in the absenceof lateral confinement. The corresponding wave functions can be written as:Ψ m x m y ( x, y ) = √ ( k x k y ) π × (2)[ cos (cid:0) m x k x x + ( m x − π (cid:1) × cos (cid:0) m y k y y + ( m y − π (cid:1) ] , with k i = π/‘ i . Note that the wave function projectionsin Figs. 2(a) and 2(b) do not show modes with nodes onthe y or x axis, respectively. As a consequence, some ofthe modes are either very weak or simply do not appearin the maps (one example is the Ψ mode displayed inFig. SM1(i)). B. OPO in intra-cavity traps
According to Eq. (1), the confined levels with the set { Ψ signal , Ψ pump , Ψ idler } = O = { Ψ , Ψ or Ψ , Ψ } are equidistant in energy for l x = l y , thus satisfying theenergy requirements for OPOs. The lowest three con-fined levels in Fig. 2(a), however, do not follow this be-havior. In addition, the levels Ψ and Ψ are not de-generate, cf. Fig. 2(a & b). These discrepancies arisefrom the deviation of the confinement potential from aperfect square shape due to the anisotropic effects duringthe MBE overgrowth (cf. Sec. SM1 for details) [16]. Theimpact of the anisotropic shape on the confined statescan be reproduced by assuming l x = l y in Eq. (1), whichlifts the degeneracy of the ( m x , m y ) and ( m y , m x ) stateswith ( m x = m y ) .The different inter-level spacings and the symmetry ofthe states (cf. Sec. IV B) within O prevent OPO exci-tation by optically pumping of the Ψ level. An ex-perimental implementation is illustrated in the PL mapof Fig. 2(c). Here, the trap was illuminated by a con-tinuous wave (cw) laser beam slightly red-shifted withrespect to the Ψ emission in Fig. 2(a)-(b), but withinits linewidth. The angle of incidence of the laser (of 10 ◦ )was chosen to match the emission peak in momentumspace of the Ψ state. In addition, this configurationhelps to avoid the specular reflection of the pump. Dueto focusing, the excitation beam has an angular spread of ± ◦ . The spatially resolved PL was collected for anglesbetween 0 ◦ and ± ◦ with respect to the sample surface.The map shows a slight increase of the Rayleigh scatter-ing due to pump interactions with the lobes of the Ψ state superimposed on a background of stray light fromthe pump laser.When the SAW is turned on, the intra-cavity trapbecomes subjected to an effective modulation potential V SAW given by [15]: V SAW ( x, t ) = V SAW , cos ( k SAW x ) cos ( φ SAW ) . (3)Here, k SAW = 2 π/λ
SAW and φ SAW = 2 πf SAW t denotethe wave vector and phase of the standing SAW field, re-spectively. Under the acoustic modulation, a signal-idlerpair appears at energies equidistant to the pump energy,thus signalizing OPO triggering. A detailed analysis ofthe dependence of the OPO excitation on the frequencyof the rf-drive applied to the IDT (cf. Sec. SM4) revealsthat OPO states only appear for rf-frequencies matchingthe modes of the acoustic resonator, thus unambiguouslyproving that OPO triggering is induced by the SAW field.The PL map of Fig. 2(d) also yields information aboutthe symmetry of the confined states participating in theOPO process. In fact, the pump state in this figure hastwo lobes along x , thus indicating that it corresponds tothe Ψ (rather than the closely lying Ψ state). Theidler state, in contrast, has a single lobe along x locatedin-between the two lobes of the pump state. This emis-sion pattern does not correspond to the one expected forthe Ψ state but rather to a superposition of the stateΨ and Ψ . As will be shown in Sec. IV B, this statered-shifts under the acoustic modulation to satisfy theOPO energy matching condition within the set of lev-els O = { Ψ , Ψ , Ψ / Ψ } displayed in the rightmostpanels of Fig. 2.Interestingly, the OPO level configuration is not uniqueand can be changed by varying the pump energy. Fig. 3displays PL maps recorded by setting the pump laser en-ergy in resonance to the Ψ level (in contrast to the red-shift pumping employed in Fig. 2). The real-space andmomentum-space (angle-resolved) maps were recorded inthe direction orthogonal to the SAW-propagation and op-tical excitation. While the OPO-pump state is still Ψ ,the idler has single emission maximum corresponding tothe Ψ state. The excited OPO thus corresponds to O = { Ψ , Ψ , Ψ } . We will show in Secs. IV B andIV C that this OPO mode set is in full agreement with anumerical model for the OPO based on the numerical so- FIG. 3.
OPO with resonant pump . Acoustically drivenOPO detected by (a) spatially and (b) momentum resolvedPL. The PL maps were recorded along the direction y perpen-dicular to the SAW propagation axis x . The optical pumpis in resonance to the Ψ level, cf. Fig. 2(a), and triggers atype O = { Ψ , Ψ , Ψ } OPO. lution of the Gross-Pitaevskii equation for this particularpump excitation energy.
C. OPO switching dynamics
The dynamic character of the acoustic modulationwas investigated by analyzing the time dependent PLfrom the 4 × µ m intracavity trap using a streak cam-era. Figure 4(a) displays the PL response of polari-ton condensates in the first and second confined levelsrecorded under a SAW. This spectrum was acquired un-der non-resonant excitation at 1.54 eV with an opticalpower equal to twice the condensation threshold ( P th )[2]. The right panel compares the time-integrated PLspectra recorded in the absence of a SAW under weak op-tical excitation (10 − P th ) and in the condensation regime(2 P th ). The large energy blue-shift of the condensate en-ergies with respect to the ones measured at low excita-tion are attributed to the polariton interactions with theexcitonic reservoir produced by the non-resonant excita-tion. As discussed in detail in Ref. 15, the modulationby the SAW field leads to a sinusoidal dependence of theemission energy of the confined states.The left panel in Fig. 4(b) shows the time-resolvedemission of an OPO excited in same trap by tuning thepump laser energy to the one from the second confinedlevel at low excitation conditions (right panel). Thesignal blue-shift in the OPO configuration is negligiblecompared to the one under the non-resonant excitation[cf. Fig. 4(a)] due to the absence of an excitonic reservoir.The emission from the signal and idler OPO states only FIG. 4.
Time-resolved OPO triggering . Time-resolvedPL maps of the nominally 4 × µ m trap recorded under a f SAW = 383 .
69 MHz ( T SAW = 2 . P th , where P th is the threshold power for the condensation.The right panel displays the spectral dependence of the time-integrated emission in the absence of a SAW for optical ex-citation powers of 10 − P th (black) and 2 P th (red). (b) OPOexcitation by pumping the second confined level under thesame acoustic excitation as in (a). The right panel displaysthe integrated PL in the absence of a SAW for optical exci-tation powers of 10 − P th . The phases in (a) and (b) werenot synchronized and were aligned manually using numericalsimulations of Sec. SM1. appears during the restricted range of SAW phases forwhich these states are equidistant to the pump. The timedependence confirms the dynamic nature of the acous-tic tuning. The turn-on and turn-off times of the OPO(taken as the time delay for the intensity of the signalstate to change by an order of magnitude) is far belowthe temporal resolution of the present measurements ofapprox. 100 ps.The OPO triggering dynamics depends on the SAW amplitude and phase, as well as on the pump energy.Figure 5 illustrates the different dynamic regimes thatcan be induced by varying the pump energy. At lowpump energies, the OPO is normally excited only overa small range of SAW phases. Depending on the SAWamplitude and pump energy, the OPO can be triggeredonce in a SAW cycle, as in Fig. 5(a & b), or twice in in aSAW cycle, as shown in Fig. 5(c). For high pump ener-gies, the OPO can remain triggered over a range of SAWphases – the latter enables us to follow the dynamic en-ergy modulation of the signal state induced by the SAWfield, as illustrated in Figs. 5(a) and 5(b). Note thatin all cases, the signal energy for the OPO triggering isalways slightly lower than the one for the OPO turnoff,thus showing a hysteretic dependence on the SAW phase.This behavior is attributed to the dynamic energy shiftsof the OPO states arising from polariton-polariton inter-actions. Once the OPO is triggered, the SAW-inducedenergy shifts of the OPO levels can be counteracted bychanges in the polariton density induced by the stimu-lated scattering to the signal and idler states. In this way,it becomes possible to fulfill the OPO energy conserva-tion requirement for a range of SAW phases and energiesof the pump state. FIG. 5.
OPO at different pump conditions . Time-resolved PL maps of an acoustically triggered OPO in a nomi-nally 4 × µ m trap excited by pump energies (a) 1531.5 meV,(b) 1530.3 meV, and (c) 1529.7 meV. Note that in (c) theOPO triggers twice in a SAW period ( T SAW = 2 .
61 ns).
FIG. 6.
Tuned arrays of acoustically tuned OPOs . (a) Optical micrograph of an array of nominally 3 × µ m intra-cavitytraps with a pitch of 9 µ m. The dashed rectangle designates the region probed by PL. (b)-(c) Spectral PL map of the arrayrecorded (a) under weak non-resonant excitation in the absence of SAW, and (b) resonant excitation in the presence of SAW. D. OPO arrays
We now demonstrate that the acoustic triggering isvery robust against fluctuations in trap size and energies,thus making it possible to synchronously trigger OPOsin an array of traps. The studies were carried out in asquare array of 3 × µ m traps with a pitch of 9 µ mschematically depicted in Fig. 6(a). Figure 6(b) displaysa PL map, recorded under weak non-resonant excitationand absence of SAW by collecting the PL within the areaindicated by the dashed square in the Fig. 6(a). The largespatial separation between the traps prevents tunnel cou-pling between them. The map thus reveals a series of con-fined states with almost identical energy spectrum for alltraps. OPO experiment conditions are identical to thesingle 4 × µ m trap discussed above. Figure 6(c) showsthe corresponding PL map obtained by pumping the ar-ray slightly below the Ψ level in the presence of theacoustic field. OPO is activated in all intra-cavity trapswith correlated triggering times determined by the SAWphase at the trap location. Similar to the single trap [cf.Fig. 2(c)], no OPO excitation is observed in the absenceof a SAW. The variations of the signal and idler energieson trap position arises from the Gaussian shape of theexciting laser beam, which populates the traps with dif-ferent polariton densities. These fluctuations, however,do not prevent OPO triggering in all lattice sites underthe acoustic modulation. IV. DISCUSSIONSA. Acoustic modulation of confined levels
The state dependent acoustic energy tuning mainly re-lies on the SAW strain field. The latter modulates the excitonic component of polariton via the deformation po-tential mechanism as well as the photonic component dueto modulation of the thicknesses and refractive indices ofthe MC layers [17, 18]. These two modulation mecha-nisms add in phase for the present sample and lead to theeffective standing polariton potential given by Eq. (3).The symmetry and time evolution of the polariton lev-els under a standing acoustic fields can be understood byusing perturbation theory to calculate the impact of thedynamical potential V SAW [cf. Eq. (3)] on the confinedlevels given by Eq. (1). The acoustically induced energyshifts of the Ψ m x m y can be expressed as (see Sec. SM3for details):∆ E m x m y = h Ψ m x m y | φ SAW | Ψ m x m y i (4) ≈ V SAW , (cid:18) π − m x (cid:19) (cid:18) l x λ SAW (cid:19) cos ( φ SAW ) E The acoustic modulation thus introduces a spatial dis-tortion of the confinement potential, which is dictatedby the instantaneous amplitude V SAW , cos ( φ SAW ) of thestanding field as well as by the ratio between the trap di-mensions and the acoustic wavelength. The correspond-ing energy shifts ∆ E m x m y are independent of the modeindex m y but reduce with increasing m x . This behav-ior arises from the fact that the m y lobes of the wavefunction are centered on the SAW anti-nodes, thus expe-riencing the full strain-induced energy modulation. The m x lobes, in contrast, are oriented along the SAW prop-agation direction and, thus, probe different phases of thestanding SAW field. These state-dependent shifts enablethe dynamic energy tuning for OPO triggering over awide range of trap geometries (see Sec. SM3). In partic-ular, the OPO states O = { Ψ , Ψ , Ψ } in a perfectsquare trap become equidistant in energy by selecting theSAW amplitude and phase to satisfy: V SAW , cos ( φ SAW ) = 23 (cid:18) λ SAW ‘ (cid:19) (5)In order to determine the nature of the OPO states, wefirst examine the impact of the SAW on the energy ofthe states given by Eq. (3). Figure 2(e) displays the en-ergy evolution of the confined polariton levels with theSAW phase, φ SAW . The calculations were carried outusing a numerical approach that takes into account themeasured spatial profile of the traps (see Sec. SM1 fordetails), but neglects polariton-interactions. A more re-alistic model taking into account interactions will be pre-sented in Sec. IV C. The vertical dash-dotted black linesmark the nodes of the SAW strain field, where the po-lariton states are identical to the ones of an unperturbedtrap. The states within set O = { Ψ , Ψ or Ψ , Ψ } are approximately equidistant at this phase, but, for sym-metry reasons presented in Sec. IV B, do not interact.The vertical dashed blue lines indicate the phases forwhich the OPO energy matching requirement becomessatisfied for the set O = { Ψ , Ψ , Ψ / Ψ } . We at-tribute the PL features in Fig. 2(d) to an OPO involv-ing these states. This assignment is supported by acomparison of the PL maps with their calculated wavesquared function | Ψ m x m y | at the matching SAW phasedisplayed in Figs. 2(f)-(h). The pump state has thusa predominantly Ψ character with two lobes along x .The idler state results from the SAW-induced red-shiftof the Ψ unperturbed state, which mixes with the Ψ state. The idler state, thus acquires the symmetry shownin Fig. 2(f).When the pump energy is in resonance or blue-shiftedwith respect to the Ψ state, the OPO assumes the O configuration (cf. Fig. 3) with an Ψ idler state. B. Symmetry of the OPO states
The previous sections have shown that acoustic tun-ing enables the excitation of OPO in intra-cavity trapsover a wide range optical and acoustical excitation con-ditions. One interesting question is why it is not pos-sible to trigger an OPO in the configuration O = { Ψ , Ψ (or Ψ ) , Ψ } , which has equally separatedstates for a perfect square potential. In fact, we showin the SM (Sec. SM3 A) that deviations from a squareshape can also be corrected by the acoustic field. Therequired field amplitudes are in this case much smallerthan those given by Eq. (5) for the energy matching ofthe O states.The inability to excite an O OPO arises from sym-metry requirements of the non-linear process respon-sible for OPO triggering, a critical process to initiateparametric oscillations. OPO triggering initiates whenfluctuations in population leads to the occupation ofthe pump state by two polaritons. The latter createsa non-adiabatic and non-linear potential that couples the initial two-polariton state { Ψ p Ψ p } to the final state { Ψ s Ψ i } consisting of particles in a superposition of sig-nal and idler states. In the contact approximation forpolariton-polariton interactions,[19, 20] the perturbedtwo-polariton state represented by { Ψ p Ψ p } can be ex-pressed as: { Ψ p Ψ p } ≈ { Ψ p Ψ p } + h{ Ψ s Ψ i }| δV p |{ Ψ p Ψ p }i E p − E s ) { Ψ s Ψ i } + . . . (6)The coupling Hamiltonian δV p can be expressed as δV p = Z g (cid:2) Ψ i Ψ s Ψ (cid:3) dxdy, (7)where g denotes the effective polariton-polariton cou-pling strength. The non-linear coupling δV p thus enablesthe scattering of pump polaritons to the idler and sig-nal states required to trigger the stimulated scatteringleading to parametric oscillations and amplification.Equation (7) has important consequences for poten-tials with mirror symmetry, such as the ones created bythe intra-cavity traps studied here. The confined statesΨ ij in these potentials have a well-defined parity. Since | Ψ p | in Eq. 7 is always an even function, a non-vanishingcoupling δV p requires idler and signal states with sameparity, i.e., states with indices m i ( i = x, y ) differing byan even number. This condition is satisfied for the sets O and O but not for O , in full agreement with theexperimental results.The previously mentioned symmetry requirements re-main valid under a standing SAW field, as long as thetraps are centered at a field anti-node, since in this casethe acoustic perturbation does not affect the mirror sym-metry of the trap potential. If, however, the SAW fieldhas a traveling component (or if the trap is displaced fromthe anti-nodes of a standing field), it will mix states withdifferent parities and enable other OPO configurations. C. Driven-dissipative simulations of the OPOdynamics
We present in this section a theoretical analysis basedon the numerical solution of the driven-dissipative Gross-Pitaevskii equation (GPE) for the lower-polariton field(Ψ) in an intra-cavity trap subjected to an optical field( F p ( r , t ) with r = ( x, y )) as well as to the acoustic mod-ulation potential given by Eq. (3). In contrast to thesingle-particle calculations presented above, the GPE so-lutions implicitly account for the polariton non-linearityleading to OPO formation. In atomic units, the GPE canbe expressed as: i∂ t Ψ( r , t ) = F p ( r , t ) + ( ω LP ( − i ∇ ) − iκ LP (8)+ g LP | Ψ( r , t ) | + V trap ( r ) + V SAW ( r , t ) ) Ψ( r , t ) . FIG. 7.
Theoretical analysis of acoustically tuned OPOs . Numerical simulations carried out by solving the driven-dissipative Gross-Pitaevskii equation (GPE) for polaritons with an 4 × µ m intra-cavity trap. (a) and (f) Momentum-spacespectra of the trap under non-resonant excitation and in the absence of SAW along x and y directions, respectively. (b) and (g)Momentum-space spectra of the trap under resonant excitation into Ψ state in the absence of SAW along x and y directions,respectively. (c) and (h) Similar to (b) and (g) but for the SAW phase of 0.3 in units of 2 π . (d) Evolution of the energy of theOPO levels over one SAW cycle. (e) Energy difference (∆ E = E + E − E ) between the Ψ ij levels over once SAW period.The vertical bars designate two SAW phases for which the spectrum of the trap is equidistant (∆ E = 0). (i-k) Spatial profilesof the squared wavefunctions of the signal, pump and idler states at SAW phase of 0 . π ). Here, ω LP , κ LP and g LP are lower-polariton dispersion,decay rate and polariton-polariton interaction, respec-tively. The dimensions of the intra-cavity trap were de-termined from AFM height maps, see Table SM1. Theparameters for the trap potential and the SAW modu-lation amplitude were the same as in the experiments.More details of the calculations procedure are summa-rized in Sec. SM6.Figures 7(a) and 7(f) display the wave-function projec-tions of the polariton states (in momentum space) calcu-lated for low power, non-resonant optical excitation ofthe trap. The central panels [Figs. 7(b) and 7(g)] showthe corresponding maps under polariton injection intothe Ψ state, which blue-shifts due to the polariton-polariton interactions. In agreement with the experi-mental results of in the absence of a SAW, no OPO isobserved under these conditions. In contrast, once theSAW potential is added, the signal and idler states ap-pear in the simulated spectra [cf. Figs. 7(c) and 7(h)].We emphasize here the excellent agreement between the calculated wave function projection of Fig. 7(h) and theexperimental momentum-resolved PL map of Fig. 3(b).The wave functions of the OPO states are illustratedin Figs. 7(i)-(k) correspond to the ones expected for anOPO involving the state set O = { Ψ , Ψ , Ψ } . Thedependence of their energy on the SAW phase, which issummarized in Figs. 7(d) and 7(e), shows that the OPOenergy matching condition becomes satisfied twice in aSAW cycle for φ SAW = 0 . D. Idler-signal intensity ratios
The OPO process yields pairs of signal and idler po-laritons: the same applies for the simulation of theFig. 7(c,h) that predicts almost identical amplitudes forthe signal and idler states. The PL yield from these statesdepends on how the polaritons decay to photons and maydiffer considerably due to differences in the scatteringrates, Hopfield coefficients, emission pattern, and photonre-absorption. As a result, the emission from idler statesis normally much weaker than the signal one. The latteris a main drawback for applications as sources of corre-lated photons, which ideally require comparable intensityratios.The ratio r OPO between the integrated emission inten-sity of the signal and idler in the present studies coversa wide range extending from 20 to 500. For comparison,the OPOs based on confined polariton states reportedin Ref. [14] have r OPO ratios ranging from approx. 5 to100, depending on the excitation intensity, which com-pare with the range from 5 to 10 predicted by theoreticalstudies presented in the same manuscript. The same ra-tio increases to 10 to 10 in polariton OPOs based ontripple microcavities [13].The high r OPO inferred from Figs. 2-6 is partially dueto an inefficient collection of the idler emission. In partic-ular, the emission in the PL maps will appear very weakif the idler state has a spatially-extended wave functionor, as discussed in Sec. III B, an emission node along thecollection axis. We estimate that the limited collectionof the idler emission, which can be eliminated by a fullmeasurement of the wave functions, induces an increasesof the measured r OPO by a factor between 3 and 5.Another mechanism leading to a large r OPO arises fromthe higher photonic content of the signal states in com-parison to the idler states in the present sample. Theenergy difference between signal and idler states is lessthan half of the Rabi splitting, so that the large r OPO ratios can not be solely attributed to differences in theHopfield coefficients. Finally, the large r OPO may alsoarise from decay paths from the pump to the low lyingsignal states (e.g., by a thermal process). Figure 2(d)shows, however, that the signal state only emits underOPO excitation, thus proving the absence of a parallelexcitation path. We suggest that the weak photon yieldof the idler results from the strong dephasing arising fromthe coupling with closely lying energy levels. The acous-tic field may play a role in this process: in particular,the SAWs employed here also carry piezoelectric fields,which interact strongly and can efficiently mix electronicstates. Future acoustic modulation studies using non-piezoelectric SAWs [21] will help to clarify this issue.
V. SUMMARY AND OUTLOOK
We have demonstrated an efficient and versatile ap-proach for the dynamic control of the scattering pathwaysof confined polariton condensates based on the modula-tion by spatially and time-varying potentials producedby SAWs. A unique feature arising from the spatialdependence of the SAW field is the ability to dynami- cally control the energy of individual polariton states ina confined potential. Here, the SAW is applied to tunethe energy states of confined exciton-polariton conden-sates to enable OPOs. We demonstrated that the acous-tic OPO requires not only the matching of the energyenergy-level separation, but also signal, pump, and idlerstates with the appropriate symmetry. The experimentalstudies have been complemented by a theoretical frame-work, which accounts for the required symmetry of theconfined states and also provides a quantitative determi-nation of the energy tuning parameters. Finally, we havepresented experimental results confirming the dynamiccharacter and the robust nature of the acoustic tuning,which enables OPOs under a wide range of excitationconditions.A natural future step will be the exploitation of acous-tically tuned OPOs for the generation of entangled pho-tons from a single trap and arrays. We anticipate thatone of the challenges will be to control the mismatch inemission intensity between the signal and idler states.The theoretical framework together with the ability todevelop polariton confinement potentials with the appro-priate symmetry are an excellent starting point to reachthis goal.The ability to synchronously tune several OPOs is onefurther advantage of the dynamic acoustic tuning. Thisfunctionality is demonstrated by the excitation of an ar-ray of confined OPOs using a single acoustic beam. Themodulation of the individual polariton traps at the arraynodes appropriately tunes the confined levels and coun-teracts unavoidable energy fluctuations. Furthermore,the OPO emission from the array sites is correlated bythe SAW phase. The time jitter of the emission dependson the fluctuations in the trap properties and can beminimized by increasing the SAW amplitude. The pho-ton pairs are emitted not only at well-defined locationswithin the array but also at well-determined times, a fea-ture which can enhance the fidelity of such a source ofcorrelated photons.As a final remark, we point out that strain fields inter-act with a wide variety of excitations in solid state sys-tems. The dynamical acoustic tuning reported can thusbe applied to a wide variety of systems, thus providingthe robustness in operation required for the realizationof scalable on-chip systems.
Acknowledgements:
We thank M. Ramsteiner and S.Krishnamurthy for discussions and for a critical reviewof the manuscript. We also acknowledge the techni-cal support from R. Baumann, S. Rauwerdink, and A.Tahraoui in the sample fabrication process. We acknowl-edge financial support from the German DFG (grant359162958) and from the QuantERA grant Interpol (EU-BMBF (Germany) grant nr. 13N14783). [1] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa,Observation of the coupled exciton-photon mode split- ting in a semiconductor quantum microcavity, Phys. Rev. Lett. , 3314 (1992).[2] J. Kasprzak, M. Richard, S. Kundermann, A. Baas,P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H.Szyma´nska, R. Andr´e, J. L. Staehli, V. Savona, P. B. Lit-tlewood, B. Deveaud, and L. S. Dang, Bose-Einstein con-densation of exciton polaritons, Nature , 409 (2006).[3] I. Carusotto and C. Ciuti, Quantum fluids of light, Rev.Mod. Phys. , 299 (2013).[4] D. Sanvitto and S. Kena-Cohen, The road towards po-laritonic devices, Nat Mater , 1061 (2016).[5] P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. S.Skolnick, D. M. Whittaker, and J. S. Roberts, Angle-resonant stimulated polariton amplifier, Phys. Rev. Lett. , 1547 (2000).[6] M. Saba, C. Ciuti, J. Bloch, V. Thierry-Mieg, R. Andr´e,S. S. Dang, S. Kundermann, A. Mura, G. Bongiovanni,J. L. Staehli, and B. Deveaud, High-temperature ultra-fast polariton parametric amplification in semiconductormicrocavities, Nature , 731 (2001).[7] J. J. Baumberg, P. G. Savvidis, R. M. Stevenson, A. I.Tartakovskii, M. S. Skolnick, D. M. Whittaker, and J. S.Roberts, Parametric oscillation in a vertical microcav-ity: A polariton condensate or micro-optical parametricoscillation, Phys. Rev. B , R16247 (2000).[8] W. Langbein, Spontaneous parametric scattering of mi-crocavity polaritons in momentum space, Phys Rev B ,205301 (2004).[9] C. Ciuti, Branch-entangled polariton pairs in planar mi-crocavities and photonic wires, Phys. Rev. B , 245304(2004).[10] S. Savasta, O. D. Stefano, V. Savona, and W. Lang-bein, Quantum complementarity of microcavity polari-tons, Phys. Rev. Lett. , 246401 (2005).[11] M. Romanelli, C. Leyder, J. P. Karr, E. Giacobino, andA. Bramati, Four wave mixing oscillation in a semicon-ductor microcavity: Generation of two correlated polari-ton populations, Phys. Rev. Lett. , 106401 (2007).[12] S. Portolan, O. D. Stefano, S. Savasta, and V. Savona,Emergent entanglement of microcavity polariton pairs,Journal of Physics: Conference Series , 012033(2010).[13] C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti,A. Lemaˆıtre, J. Bloch, P. Roussignol1, and C. Delalande,Parametric oscillation in vertical triple microcavities, Na- ture , 904 (2006).[14] L. Ferrier, S. Pigeon, E. Wertz, M. Bamba, P. Senellart,I. Sagnes, A. Lemaˆıtre, C. Ciuti, and J. Bloch, Polaritonparametric oscillation in a single micropillar cavity, Appl.Phys. Lett. , 031105 (2010).[15] A. S. Kuznetsov, K. Biermann, and P. V. Santos, Dy-namic acousto-optical control of confined polariton con-densates: From single traps to coupled lattices, Phys.Rev. Research , 023030 (2019).[16] A. S. Kuznetsov, P. L. J. Helgers, K. Biermann, andP. V. Santos, Quantum confinement of exciton-polaritonsin structured (Al,Ga)As microcavity, Phys. Rev. B ,195309 (2018).[17] T. Sogawa, P. V. Santos, S. K. Zhang, S. Eshlaghi, A. D.Wieck, and K. H. Ploog, Dynamic band structure modu-lation of quantum wells by surface acoustic waves, Phys.Rev. B , 121307(R) (2001).[18] M. M. de Lima, Jr. and P. V. Santos, Modulation ofphotonic structures by surface acoustic waves, Rep. Prog.Phys. , 1639 (2005).[19] C. Ciuti, P. Schwendimann, and A. Quattropani, Para-metric luminescence of microcavity polaritons, Phys.Rev. B , 041303 (2001).[20] I. Carusotto and C. Ciuti, Spontaneous microcavity-polariton coherence across the parametric threshold:Quantum Monte Carlo studies, Phys. Rev. B , 125335(2005).[21] J. Rudolph, R. Hey, and P. V. Santos, Long-range excitontransport by dynamic strain fields in a GaAs quantumwell, Phys. Rev. Lett. , 047602 (2007).[22] V. P. LaBella, D. W. Bullock, Z. Ding, C. Emery, W. G.Harter, and P. M. Thibado, Monte carlo derived diffu-sion parameters for Ga on the GaAs(001)- (2x4) sur-face: A molecular beam epitaxy scanning tunneling mi-croscopy study, J. Vac. Sci. Technol., A , 1526 (2000),http://dx.doi.org/10.1116/1.582379.[23] M. M. de Lima, Jr., M. van der Poel, P. V. Santos,and J. M. Hvam, Phonon-induced polariton superlattices,Phys. Rev. Lett. , 045501 (2006).[24] G. R. Dennis, J. J. Hope, and M. T. Johnsson, Xmds2:Fast, scalable simulation of coupled stochastic partial dif-ferential equations, Computer Physics Communications , 201 (2013). upplementary Material:Dynamically Tuned Arrays of Polariton Parametric Oscillators Alexander S. Kuznetsov, ∗ Galbadrakh Dagvadorj,
2, 3
KlausBiermann, Marzena Szymanska, and Paulo V. Santos † Paul-Drude-Institut f¨ur Festk¨orperelektronik, Leibniz-Institut imForschungsverbund Berlin e. V., Hausvogteiplatz 5-7, 10117 Berlin, Germany Department of Physics and Astronomy, University College London,Gower Street London WC1E 6BT, United Kingdom Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: March 4, 2020)
This document provides supplementary information to the experiments and data analysis presented in the maintext. We start with studies carried out to determine the potential profile (Sec. SM1) of the intra-cavity traps as wellas the energy levels of confined polaritons in these traps (Sec. SM2). The modulation of the confined states by a SAWis analyzed in Sec. SM3. The next two sections summarizes results on the dependence of the OPO properties on thefrequency of the acoustic excitation (Sec. SM4) and on the energy of the pumping laser beam (Sec. SM5).
SM1. STRUCTURE OF THE INTRA-CAVITY POLARITON TRAPS
The MBE overgrowth of the structured microcavity (MC) spacer results in a relief of the sample surface, whichreflects the structural dimensions of the polariton intra-cavity traps embedded in the spacer [ ? ]. The thick linesin Figs. SM1(a) and SM1(b) show the surface relief of the final MC induced by an intra-cavity trap measured byscanning an atomic force microscopy (AFM) along the x = [¯110] and y = [¯1¯10] crystallographic axes, respectively.The mesa with a nominal height of 12 and lateral dimensions of 4 × µ m etched in the MC spacer gives rise to asurface relief with a height of δz = 15 nm after the MBE overgrowth. While the height profiles δz ( r i ) ( r i = x, y )along the y axis closely reproduce the dimension of the mesa etched in the MC spacer, the profile shapes along x aresignificantly smoother. This behavior is attributed to the anisotropic dynamics of the MBE overgrowth process [ ? ? ]. Following the procedure delineated in Ref. [ ? ], we obtained the spatial dependence of the changes in trap height δz ( r i ) by fitting the AFM profiles along the two directions to the following expression: δz ( r i ) = δz (cid:20) Erfc (cid:18) r i − w i / √ w i (cid:19) + Erfc (cid:18) r i + w i / √ w i (cid:19)(cid:21) . (SM1)Here, Erfc ( r ) is the complementary error function and w i the effective width of the lateral interface between non-etched and etched regions. The thin blue lines superimposed on the AFM curves of Figs. SM1(a) and SM1(b) displayfits of Eq. (SM1) to the experimental data, which yield the structural parameters for the intra-cavity trap summarizedin Table SM1. TABLE SM1. Structural parameters and simulations parameters for intra-cavity traps.Parameter ValueLateral mesa dimensions, l x and l y . / . µ mLateral interface widths, w x and w y . / . µ mPotential barrier, E b m p × − m Rabi energy, ~ Ω R m is free electron mass. ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . o t h e r] M a r FIG. SM1. (a)-(b) (thick grey lines) Surface relief measured by scanning an atomic force microscopy (AFM) tip over a surfacearea on the MC containing an intra-cavity trap with nominal dimensions of 4 × µ m along the x = [¯110] and y = [¯1¯10]crystallographic axes, respectively. The thin blue lines are fits with Eq. SM1 yielding the parameters listed in Table SM1.(c) Comparison of the spatially resolved PL map with the squared wave functions projections | Ψ m x m y ( x, y ) | calculated forthe first confined states ( m x , m y = 1 , , , φ SAW of the SAW potential inEq. 3 of the main text. (f)-(k) Comparison between the squared polariton wave functions in the absence (upper panels) andpresence of the SAW potential (lower panels) corresponding to panels (d) and (e) respectively. Note that the state for the phase φ SAW = 0 .
415 in (k) is a superposition of the Ψ and Ψ states. SM2. ENERGY LEVELS OF INTRA-CAVITY TRAPS
The change δz in the MC spacer thickness results in an energy difference (barrier) of approx. E b = 6 meV betweenthe lower polariton states in the non-etched and etched regions of the MC. The energy levels and wave functionsfor polaritons confined in intra-cavity traps were determined by numerically solving the two-dimensional Schrdingerequation for a polariton confinement potential with depth E b and the lateral shape for δz given by Eq. (SM1) andTable (SM1). As described in details in Ref. [ ? ], the calculations were carried out in Fourier space for a periodicsuper-cell containing the intra-cavity trap surrounded by potential barriers. The latter were chosen to be wide enoughto avoid interactions between neighboring traps.The solid lines in Fig. SM1(c) displays the squared wave functions | Ψ m x m y (0 , y ) | calculated for the first confinedstates ( m x , m y = 1 , ,
3) of the the intra-cavity trap. The dashed line shows the polariton confinement potential alongthe x direction. The calculations reproduce well the measured energy of the confined states [cf. Fig. SM1(b)] andyield spatial extent for their wave functions very similar to the ones obtained from the PL maps. SM3. ACOUSTIC MODULATION
Different mechanisms mediating the modulation of polariton levels by SAW are reviewed in Ref. ? . Here, wewill make the approximation that the standing SAW field along x induces a modulation of the polariton levels givenby Eq. (3) of the main text with the field anti-nodes coinciding with the center of the trap at x = 0. Due torelatively low SAW frequency (384 MHz), one can safely assume that the polariton states adiabatically follow thepotential variations. The impact of the standing SAW field on the polariton levels was studied by adding to theconfinement potential the acoustic potential given by Eq. (3). Figures SM1(d) and SM1(e) compare profiles for thewave functions | Ψ m x m y ( x, | for phases φ SAW = 0 . π rad, when the standing SAW potential vanishes, with the onefor φ SAW = 0 . π rad and a SAW potential amplitude V SAW , = 2 meV. The latter yields the energy matching forOPO excitation. A. Acoustically induced energy tunning
The symmetry and time evolution of the polariton levels displayed in Fig. SM1 (as well as in Fig. 2(e) of the maintext) can be understood by treating the acoustic potential of Eq. (3) as a perturbation of the confined states of therectangular potential given by Eq. (3). For small ratios ‘ x /λ SAW and ‘ y /λ SAW , the energy changes ∆ E m x m y of theconfined states Ψ m x m y can then we written as:∆ E m x m y = h Ψ m x m y | V SAW | Ψ m x m y i ≈ V SAW , (cid:18) π − m x (cid:19) (cid:18) l x λ SAW (cid:19) cos ( φ SAW ) . (SM2)The approximation in Eq. (SM2) applies up to second order terms in the ratios ‘ x /λ SAW .The acoustically induced shift given by Eq. (SM2) enables tuning the energy levels of the trap by adjusting theSAW amplitude V SAW , to satisfy the OPO energy matching requirement E idler − E pump = E pump − E signal . (SM3)Here the subscripts denote the OPO states. By properly selecting V SAW , and λ SAW , it is possible to satisfy the OPOenergy matching criterion over a wide range of trap geometries and OPO configurations. Two examples are describedbelow: • States O = { Ψ signal , Ψ pump , Ψ idler } = { Ψ , Ψ or Ψ , Ψ } in a rectangular trap: these states are energeti-cally equidistant for a perfectly square trap with square size ‘ , but not in a rectangular with sides ‘ x = ‘ y . If l x = ‘ and ‘ y = (1 + r ‘ ) ‘ , it can be shown using Eq. (SM2) that the levels become equidistant if: V SAW , cos ( φ SAW ) = 4 r ‘ (cid:18) λ SAW ‘ (cid:19) E , with E = (1 − r ‘ ) π ~ m p ‘ . (SM4)Here, m p is the polariton mass and E the zero-motion energy in the trap (corresponding to the quantum shiftof the lowest confined level). We note, however, that this set of states miss the symmetry requirements fornon-linear coupling conditions required for OPO excitation, as discussed in Sec. IV B of the main text. • States O = { Ψ signal , Ψ pump , Ψ idler } = { Ψ , Ψ , Ψ } in a square trap: if the trap size is ‘ , the states becomesequidistant in energy by selecting V SAW , cos ( φ SAW ) = 23 (cid:18) λ SAW ‘ (cid:19) E , with E = π ~ m p ‘ . (SM5) SM4. RADIO-FREQUENCY DEPENDENCE OF OPO EXCITATION
The delay line in Fig. 1(c) forms an acoustic resonator due to the acoustic reflections at the single-finger IDTs,which act as acoustic Bragg reflectors. The properties of the resonator were accessed by measuring the radio-frequency(rf) scattering parameters using a vector network analyser. Figure SM2(a) displays the spectral dependence of the | s | parameter, which corresponds to the ratio between the reflected and incident rf-power applied to one of theIDTs. The sharp dips in the spectrum correspond to the reduction of the reflected rf power due to the excitation ofacoustic modes of the resonator. From the width of these dips one can extract the quality factor of the resonator.The sharpest of the dips at f SAW = 383 . Q = 4700. FIG. SM2. (a) Radio-frequency (rf) power reflection coefficient | s | for the delay line in Fig. 1. (b) rf dependence of the PLfrom a 4 × µ m intracavity traps. Note that OPOs are only excited at the resonance frequencies of the acoustic resonator,which correspond to the dips of the spectrum in (a). The map of Fig. SM2 show in a color scale the spectral dependence of the PL spectrum of the trap on the nominalrf frequency applied to the delay line for a fixed rf level. Note the OPO signal and idler levels only appear at theresonance frequencies of the acoustic resonator, thus unambiguously proving that OPO triggering is due to the acousticfield. In addition, the figure also shows that the energy separation between the pump and signal, while remainingequal the one between idler and pump, changes with frequency following the changes in amplitude of the SAW strainfield.
SM5. OPO DEPENDENCE ON PUMPING ENERGY
The dependence of the OPO properties on the pump intensity is summarized in Fig. SM3. In this series of plots, wemonitored the PL emitted by a 4 × µ m intra-cavity traps under a fixed amplitude of the SAW field while increasingthe optical excitation power from 10 to 1000 mW (focused into a spot of approx. 50 µ m in diameter). The OPOtriggers at approx. 30 mW (corresponding to a flux of 1.5 kW/cm ). The intensity of the signal and idler statesincreases with excitation power up to approx. 100 mW and saturates for higher powers. The saturation is attributedto the saturation of pump injection as the pump state blue-shifts due to polariton-polariton interactions. FIG. SM3. Dependence of the photoluminescence spectrum of a 4 × µ m intra-cavity trap for a fixed acoustic excitation onthe optical pump level. Only the signal and ideler energy ranges are shown. The observation of OPO over a relatively wide range of acoustic (cf. Fig. SM2) and optical (cf. Fig. SM3) excitationconditions indicates that once triggered, non-linearities induced by the polariton interactions permit the OPO over awide range of excitation conditions.
SM6. NUMERICAL SIMULATION OF THE ACOUSTICALLY DRIVEN OPO
We simulate the polariton dynamics by solving Eq. 8 in the main text with the XMDS2 software framework [ ? ]using an adaptive step-size fourth and fifth order embedded Runge-Kutta algorithm. In order to allow for the OPOregime, we have used the exact LP dispersion rather than approximating it by a quadratic form, which is only accuratefor the low energy polariton modes and suitable only for non-resonant excitations. The LP dispersion and externalcoherent pump are given by: ω LP ( − i ∇ ) = ω X + 12 − ∇ m C + δ − s(cid:18) − ∇ m C + δ (cid:19) + Ω , (SM6)and F p ( r , t ) = f p e i ( k p · r − ω p t ) . (SM7)We choose the system parameters to be close to current experiments: m C = 2 . × − m e : the effective mass of the photons due to the microcavity, κ LP = 0 . g LP = 0 .
005 meV µ m : the polariton-polariton interaction strength,Ω R = 7 . δ = ω C (0) − ω X (0) = − . f p to zero and the polariton decay rate κ LP to a very small negative value with a Gaussiannoise initial condition. To determine the numerical pump state Ψ21