Tailoring the potential landscape of room-temperature single-mode whispering gallery polariton condensate

A key point of exciton polaritons is real-time controllability of potential energy and its landscape due to the hybrid nature of excitons and photons. Although wide-bandgap semiconductors allow us to generate room-temperature polaritons, unintentional localizations of two-dimensional cavities caused by disorder (dephasing and potential fluctuation) still hinder the establishment of ballistic extensions of polariton condensates. This ballistic extension accompanies spatial coherence, an essential factor for phase transitions as well as any quantum controls. Here we propose a room-temperature polariton system with ultralow disorder capable of one-dimensional ballistic propagation. Selectively grown GaN wire dramatically reduces disorder in both the exciton perspective via dislocation bending and the photon perspective through crystallographically defined hexagonal cavities. This high-quality wire on a substrate forms triangular whispering gallery modes and allows us to demonstrate the room-temperature single-mode one-dimensional polariton condensate with ballistic propagation. This ballistic propagation is manipulated by active real-time control of the potential gradient. The correlation between propagation distances deduced from real and momentum space provides strong evidence of ballistic propagations in an ultralow-disordered one-dimensional system. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


INTRODUCTION
Semiconductor microcavities can allow us to enter the hybrid nature of exciton cavity polaritons (polaritons) [1], which comply with Bose-Einstein statistics [2,3]. A crucial benefit of the polariton system is real-time controllability of polariton potential via optical manipulation [4,5] or acoustic waves [6]. This potential engineering can make challenging photonic devices feasible, such as polariton spin switches [7], transistors [8], phase modulators [9], and resonant tunneling diodes [10]. The potential engineering can be conducted by the manipulation of two variables, polariton potential energy and potential landscape [11][12][13][14]. Remarkably, this potential landscape can yield ballistic extension in the polariton condensate, which accompanies the robust coherence of the polariton condensate due to spatial decoupling [12,15] between polaritons and the exciton reservoir. This macroscopic quantum coherence is an essential factor for phase transition research, such as the Berezinskii-Kosterlitz-Thouless phase transition [15] and quantum vortex [16], as well as the next generation of the polaritonic quantum fluid circuit.
Since semiconductor cavities with ultralow disorder are essential for the control of the polariton potential landscape, state-ofthe-art two-dimensional cavities based on III-arsenide materials have been mainly used, even though cryogenic temperatures are required. In order to overcome the temperature limitation, wide-bandgap semiconductors, such as perovskites, ZnO, and III-nitride systems, are promising candidates for generating room-temperature polaritons due to their large exciton (X) binding energy and high oscillator strength. In the case of perovskite materials, this ballistic transport of polariton condensate with high velocity was recently reported in a dielectric distributed Bragg reflector (DBR) based on a one-dimensional structure at room temperature [17]. Also, a one-dimensional system based on ZnO obtains the temporal and spatial information of the ballistic transports for multimode polaritons [18]. However, fabrication of a III-nitride-based planar microcavity for high-quality DBR is still challenging due to the low refractive index contrast and the large lattice misfit between AlN and GaN. Nevertheless, a conventional two-dimensional room temperature polariton condensate based on III-nitride semiconductors was previously reported using diverse structures, such as bulk [19] or multiple quantum wells [20,21] in a hybrid DBR as well as nanowires [22,23] or a membrane [24,25] embedded in an oxide-based DBR. These two-dimensional microcavity systems still confronted unintentional localizations [19][20][21][22][23][24][25] in the lateral direction blurring out the potential landscape. These unintentional localizations are caused by both the excitonic disorder from dislocations due to lattice mismatch and the photonic disorder from the imperfection of fabrication. On the other hand, one-dimensional structures (i.e., wire structures) with high-quality X and cavity photons (CPs), analogous to Tonks-Girardeau gas [26,27] in a one-dimensional ultracold atom, are unexplored systems for potential landscape manipulation, for which we can expect enhanced directionality as a result of the reduced dimensionality [28].
Here, we propose a single-mode, triangular whispering gallery (tri-WG) polariton condensate in a GaN hexagonal wire on a substrate as the room-temperature polariton platform with ultralow disorder for optically tailoring the polariton potential landscape. Excitonic disorder can be free from dephasing because of dislocation bending [29] to the bottom part of the surface and strain relaxation [30] along the c axis. Moreover, photonic disorder is dramatically reduced due to the crystallographically defined hexagonal facets, which generate the spontaneous formation of triangular whispering gallery modes (tri-WGMs) without any fabrication. The tri-WGMs can robustly keep the quality factor (Q factor) and CP energy from being influenced by disorder due to a lower number of reflection faces and greater distance from the edges. Thus, tri-WG polaritons resulting from the strong coupling between three-dimensional X and one-dimensional tri-WG photons in the wire can be generated in the paved potential system with ultralow disorder. We first demonstrate the onedimensional ballistic propagation of the single-mode tri-WG polariton condensate at room temperature. This ballistic propagation can be manipulated by active control of the potential gradient and form the state by spatially decoupling from the exciton reservoir. The good consistency between characterizations of propagation deduced from real and momentum space provides strong evidence of ballistic propagations due to an ultralowdisordered one-dimensional system.

A. Coupled Oscillator Hamiltonian
To calculate our results, we use a coupled oscillator Hamiltonian, 0 Here, Ω 1 and Ω 2 are the Rabi splitting frequencies for exciton energies X i and X j (X i X A , X j X B for TE and X i X B , X j X C for TM, respectively). For rapid calculation, we use a plane-wave model to estimate tri-WGMs inside the wire.
is the ground CP energy of the tri-WGM, where n is the refractive index of the cavity (uniaxial birefringent index n o and n e for TE and TM, respectively), N is the photonic mode number, R is the radius of a hexagonal cavity, h the Planck constant, and c is the speed of light. The factor β is equal to n and 1∕n for TE and TM modes, respectively. We assumed ℏΩ Rabi is independent of positions due to the negligible variation of the spatial overlap between photons and excitons for the micro-sized diameters. The parameters we used in the calculation correspond to X A 3.429 eV, X B 3.434 eV, X C 3.450 eV, n 0 3.013, n e 3.020, and R 830 nm.

B. Tail Fitting of an Airy Function
To consider the disorder effect via the correlation between real space and momentum space, we use the Gross-Pitaevskii equation for the polariton condensate as follows: Rn R r − γ p V dis r ψr: Here, E 0 is the minimum energy of the lower polariton branch, m is the effective mass of the polaritons, g is the polariton-polariton interaction coefficient, g X is the polariton-reservoir interaction coefficient, and γ p is the polariton loss rate, which is the inverse of the polariton lifetime.
To consider the disorder effect via the correlation between real space and momentum space, we use the regime far outside the pump area devised by Wouters et al. [31]. In this region, the density of the polariton condensate is too small to feel the repulsive interaction by the polaritons (ℏgjψzj 2 ∼ 0). Also, it is far from the pump area, so we can ignore the exciton reservoir related imaginary terms and polariton-reservoir interaction term (ℏg X n R z ∼ 0). V dis is the disorder potential term, which is linear potential (V dis ∼ αz) due to the slightly tapered geometry of our wire structure along the z axis. Here, it is worth noticing that the conventional method (∂ 2 E∕∂k 2 ) of extraction of the polariton effective mass approximation might be distorted due to the linear potential. Thus, we can simplify the Gross-Pitaevskii equation by considering our one-dimensional structure as follows: where τ p corresponds to the polariton lifetime. Also, ω c and ω o are the energies of the finite momentum of the polariton state and ground state of polaritons, respectively. The corresponding solution can be expressed as Here, ΔE and k c are determined by measured angle-resolved photoluminescence, and α (∼2.7 meV∕μm) is extracted from the line scan data. The only fitting parameters are υ g × τ p and c. Thus, we can extract the decay length via tail fitting of the spatial profile [at the shaded area on the left and right sides in Fig. 5(b)]. If severe disorder exists, this simple modeling would not show very good agreement between real space and momentum space.  Fig. 1(b). Surprisingly, this growth method provides excellent regularity of the hexagonal shape, diameter, and height with low surface roughness, as seen in the SEM. For the photonic aspect, this method can ensure a regular hexagonal shape determined by the aspect Research Article ratio of the mask opening shape [32]. This regular hexagon can be crystallographically sustained along the c axis without any stacking faults so that the photonic energy fluctuation is negligible (0.69 meV; see Supplement 1). This six-fold rotation symmetry is crucial for the spontaneous generation of WGMs, as shown in the high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image for the cross section of the wire in Fig. 1(c). Only a tri-WGM [33] with an upward direction can dwell a long time, as shown in the inset of Fig. 1(d), due to leakage of the interface between the substrate and the wire. This tri-WGM can rigidly protect the Q factor and the resonance energy of the cavity from external influences because of the far distance of the electric fields from the edges and a lower number of reflection faces. To confirm this, we conducted a finite-difference time-domain (FDTD) simulation with variation of the distance of an artificial link (dust) with a 200 nm diameter, as shown in Fig. 1(d). We considered three directions, namely, 0°f or the reflected surface, 30°for the edge, and 60°for the nonreflected surface. For 0°, the Q factor gradually decreased by as much as 64% and the CP energy showed a 3 meV redshift when the defect was attached to the reflected surface. Interestingly, there were no distinct effects on either the Q factor or the CP energy for 30°and 60°depending on the distance of the defect. This result indicates that tri-WGMs have stronger photonic tolerance than hexagonal WGMs, which is beneficial for the observation of robust polariton effects. For the excitonic aspect, the intensity fluctuation (i.e., dephasing) is dramatically suppressed by both dislocation bending [29] to the bottom part of the surface and the strain relaxation [30] along the c axis due to the lateral overgrowth mode. Consequently, most of the wire offers high-quality X, except for the bottom part acting as a sacrificial layer. In the panchromatic CL image of the wire, there was no measurable dark spot compared to the film structure. The intensity fluctuation of the normalized profile for the wire (0.015) was more than one order smaller than the film (0.23). Therefore, this real one-dimensional cavity [34] with ultralow disorder yields robust room-temperature tri-WG polaritons capable of engineering the potential energy and landscape, as shown in Fig. 1(f ).

ROOM-TEMPERATURE POLARITON CONDENSATE
To identify our one-dimensional cavity in the strong-coupling regime, we measured the angle-resolved micro-photoluminescence (ARPL) at room temperature. Figure 2(a) shows two different orientations of the polarization-resolved measurements. Electric fields of the TE modes (TM modes) is perpendicular (parallel) to the c axis. Figure 2(b) shows the polarization-resolved ARPL data obtained for detection angle θ z for the TE (left side) and TM (right side) modes, respectively. Within given polarization, several lower polariton branches appeared, but upper polaritons were invisible due to absorption of the X continuum under the condition where Rabi splitting energy (ℏΩ Rabi ) was greater than X binding energy [35]. Multiple discrete states are revealed for detection angle θ y due to two-dimensional optical confinement (see Supplement 1). Their dispersion aspects clearly indicate the strong coupling as follows. (i) The curvature of the highernumber modes (i.e., closer to the X energy) shows dispersion with lower curvature [36], while dispersions at lower energy have  Fig. 2(b)].
To investigate the cavity energy detuning effect, we characterized the position-dependent micro-photoluminescence (μPL) along the c axis of the same wire, as shown in Fig. 2(c). To calculate our results using the coupled oscillator Hamiltonian, we considered the variation of the radius along the c axis. The difference between the radius of the bottom and the top is 10 nm, confirmed by SEM. Here, the only fitting parameters were the ℏΩ Rabi values, whereas other parameters were used as determined by ARPL. The white dashed-dotted lines and red dashed lines in Fig. 2(c) represent the calculated dispersions of photonic tri-WGMs and polaritonic tri-WGMs, respectively. The results are clear evidence of the strong-coupling regime for our structure with ℏΩ Rabi values of 120 7 meV and 117 10 meV for TE and TM, respectively. The value of the Rabi splitting energy is similar to other reported WG polaritons [38] for a single GaN wire system (115 10 meV). This large ℏΩ Rabi value, which is six times larger compared to those of conventional planar cavities [35], is induced by the large spatial overlap between X and CP.
To explore the polariton condensate, we investigated the behavior of the integrated-emission intensity as a function of the pump power (P) for the lower-polariton with mode number 35 (LP35) branch, which only shows nonlinearity, as seen in Fig. 2(d) (bottom). Nonlinearity in the emission was clearly observed at a power density of 1.5 W∕cm 2 (pulse energy ∼150 μJ∕cm 2 ), the first threshold power (P th ), due to the stimulated polariton-polariton scattering. Here, the measured nonlinearity was not photonic lasing but polariton condensate, since the calculated maximum carrier density (∼1 × 10 17 cm −3 ) at the threshold was 2 orders of magnitude smaller than the X Mott density (∼1 × 10 19 cm −3 ) of GaN [39]. The nonlinearity of the emission led to the linewidth narrowing of the emission from 8 to 1.19 meV. After the linewidth minimum, linewidth broadening occurred up to 4 meV. Furthermore, a clear blueshift of the peak (8 meV) was observed because of the repulsive interactions of the polariton-polariton and polariton-exciton reservoir, as shown in Fig. 2(d) (top). Additionally, with a further increase in the power density over 5.8 W∕cm 2 (pulse energy ∼580 μJ∕cm 2 ) corresponding to the second threshold power (P th2 ), a new peak appeared at a higher energy with a narrower linewidth compared to the polariton condensate, indicating the process of the transition to the weak-coupling regime, as shown in Fig. 2(e) (see Supplement 1). Polaritons can reach the singlemode state due to large mode spacing caused by a small cavity size and relax down to the ground state through strong scattering mediated by a large ℏΩ Rabi . This large ℏΩ Rabi can produce gradual dispersion with a large excitonic fraction, which leads to efficient scattering of polaritons with acoustic phonons [40], compared to the same energy of the ground state with a small ℏΩ Rabi and more negatively detuned polariton branch (see Supplement 1). Moreover, a large ℏΩ Rabi can help to decouple with a longitudinal optical phonon bath [41], so it can have a large X fraction [42] for enhanced polariton-polariton scattering.
To identify the photonic mode, the momentum space image was measured for observation of the interference pattern [43] generated by each corner of the hexagonal cross section, as shown in Fig. 3(a). Figure 3(b) shows the measured data at 2.0 P th (right) and the simulated data by the FDTD method (left) in momentum-pace. The peculiar interference pattern along θ y is attributed to the multiple-slit interference of the single-mode tri-WG polariton condensate (see Supplement 1), which yields the spatial coherence. This leakage, which is from the four corners [four red circles in Fig. 3(a)] due to the strong scattering, is analogous to the point-like sources going through the slits. The result of the measured data is in excellent agreement with the simulated data, which is the tri-WGM. Thus, we can confidently identify the photonic mode generated in the hexagonal cavity on the substrate. As a result, a room-temperature tri-WG polariton condensate with a single mode is clearly observed on the one-dimensional polariton condensate based on III-nitride semiconductors.

TAILORING THE POTENTIAL LANDSCAPE
To explore the low disorder of the wire structure and manipulate the potential landscape, we illuminated two different beam sizes (σ) for the same wire (see Supplement 1). Figures 4(a)  Here, the polariton condensate obtains a finite momentum on the polariton branch due to the abrupt gradient of the exciton reservoir. The emission at θ z ∼ 0 in the momentum space, corresponding to the pump area in the real space, exhibits very blueshifted energy due to the repulsive interaction with the exciton reservoir generated by the pump area. The nanofocused laser beam (σ 200 nm) generates an exciton reservoir with an abrupt landscape in real space, which supplies polaritons to the condensate and also creates a Gaussian-shaped potential landscape for polaritons in real space due to the repulsive interactions between the polariton and the exciton reservoir. Polariton condensate initially occurs at the apex of this tailored potential peak, and the potential energy is transformed to kinetic energy due to the abrupt landscape of the exciton reservoir, so that the polariton condensate obtains a finite momentum [14,17]. This extended polariton condensate has never been explored in conventional III-nitride two-dimensional microcavity systems (to our knowledge) due to the unintentional lateral localization caused by the disorder.

CORRELATION BETWEEN MOMENTUM AND REAL SPACE
To explore the one-dimensional ballistic propagation of the polariton condensate, we characterized the space-resolved photoluminescence (SRPL). Figure 5(a) (top) shows a panchromatic real-space image of the wire at P 3.0P th . The main leakage of the tri-WGMs comes from each corner of the hexagonal cavity. Here, the polariton energy increases along the z axis, since the  Research Article radius of the top parts of the wire is smaller and the consequent CP photon energy becomes higher than those of the bottom parts. Therefore, with respect to the pump area, the left and right sides are called the downhill potential (V low ) and uphill potential (V high ), respectively. The results of the SRPL [ Fig. 5(a), bottom] show the polariton condensate with quasi-monochromatic energy spread along the z axis due to the finite momentum induced by the potential landscape [12]. We extracted the profiles of the extended polariton condensate along the z axis in relation to P, as shown in Fig. 5(b). The graphs in the bottom part represent the profiles of the pump and exciton reservoirs at 3450 meV. As P increases, two distinct peaks appear on the left and right sides due to the stronger repulsive interaction than in the lower P case.
To characterize the correlation between polaritons in momentum space and real space, we extracted the peak energy (E p ) and wave vector (k z ) from the ARPL in relation to P, as indicated by the colored triangles (left) and circles (right) in Fig. 5(c). As P increases, E p and consequent k z increase, completely complying with the calculated LP35 branch [solid line in Fig. 5(c)]. To understand the relative mixing fraction between the exciton (jX k z j) and photon (jCk z j) of the tri-WG polaritons, we calculated the Hopfield coefficients of the LP35 branch, as shown in Fig. 5(d). As jk z j increases, the values of jX k z j and jCk z j increase and decrease, respectively. Here, jX k z j and jCk z j are affected by k z due to the variation of the CP energy. The group velocity of the extended polariton condensate is deduced by υ g 1∕ℏ∂E p ∕∂k z , as shown in Fig. 5(e) (left axis). The polariton lifetime (τ p ) is defined as a mixed fraction of the CP lifetime (τ c ) and X lifetime (τ x ), which is τ −1 p k z jCk z j 2 ∕τ c jX k z j 2 ∕τ x . Here, τ c is determined from the lowest-energy LP branch (∼97% photonic-like branch; see Supplement 1) by Fourier transform. The value of τ c is 0.092 ps (Q factor ∼450) for LP33. Here, τ c is negligibly changed on the mode number and wave vector, so we consider it as a constant. τ x (100 ps) is the X lifetime measured at room temperature. Consequently, the value of τ p is shown in Fig. 5(e) (right axis). Here, τ p can be tailored via the X fraction in the variable of k z , which is directly controlled by P and σ.
Finally, Fig. 5(f ) shows the deduced results of υ g × τ p based on ARPL data [Figs. 4(f )-4(h)] as a solid line as a function of k z . To define υ g × τ p based on the results of SRPL, we apply the region far outside the pump area [31] for analytical estimation. In this region, polaritons are negligibly influenced by the repulsive potential of the exciton reservoirs and polariton-polariton interaction due to spatial separation from the pump area and the low density of the polaritons, respectively. Considering the tapered (d) Corresponding Hopfield coefficient for exciton fraction (orange) and photon fraction (green) from the calculated dispersion, (e) polariton velocity (blue) and polariton lifetime (red) deduced by the LP35 branch, (f) solid line shoes υ g × τ p deduced from ARPL, and colored diamonds and squares show fitted data extracted from SRPL by the tail fitting of far outside approximation. geometry of our structure and the region, the Gross-Pitaevskii equation can be reduced to a one-dimensional Airy equation (see Method B in Section 2) with enhanced directionality due to reduced dimensionality. We carried out tail fittings for spatial profiles as shown in Fig. 5(b) (see Supplement 1). The fitted results of υ g × τ p based on the results of SRPL are represented in Fig. 5(f ) by the colored diamonds (left) and squares (right). Here, the fitted results of υ g × τ p based on the SRPL data show excellent agreement with the results of υ g × τ p deduced from the ARPL data without any consideration of the disorder term. This excellent agreement between characterizations of real space and momentum space indicates strong evidence of ballistic propagation, which is the polariton transport with negligible scattering caused by the disorder, due to the ultralow-disorder onedimensional property of selectively grown wire structures. This ballistic propagation with macroscopic quantum coherence has been observed in the quasi-wire [12] and planar structure [13] based on a GaAs DBR structure at cryogenic temperature. Also, the spatiotemporal evolution of this ballistic propagation of multimode condensate [18] is recently observed in a ZnO wire structure at room temperature. A DBR-based ZnO polariton [44] can obtain ballistic propagation at room temperature, but there is no consistency between real space and momentum space. We focused on this excellent correlation between real space and momentum space for the ballistic propagation of single-mode condensate from the selectively grown GaN wire structure. Furthermore, we can manipulate υ g and τ p due to the unique features of the polariton system, which exhibits finite momentum and energy within the polariton branch depending on the pump power. Consequently, we can actively control the ballistic propagation of one-dimensional polariton condensate at room temperature. Our proposed platform, a single GaN wire on a substrate having ultralow disorder, opens up the avenue for room-temperature macroscopic quantum coherence with robustness in real one-dimensional structures.

CONCLUSION
In this work, we exploit single-mode tri-WG polaritons as an ultralow-disorder system to form and manipulate one-dimensional polariton condensate at room temperature. Selectively grown wires can offer (i) high-quality excitons, (ii) high-quality photons, and (iii) tri-WGMs protecting τ p from external influences. Nonresonant and nano focused laser illumination induces an abrupt polariton potential gradient via exciton reservoirs, so the polariton condensate acquires finite momentum and energy on the polariton branch. As a result, we can optically manipulate the ballistic propagation of the polariton condensate and consequently bring about the extended polariton condensate being spatially separated from the exciton-reservoir. We anticipate that this ballistic extension will yield a new room-temperature platform for demonstrations of fundamental phenomena analogous to one-dimensional atomic physics. Our platform with wide-bandgap semiconductors offers a breakthrough for nonlinear devices in versatile nanophotonic (polaritonic) applications capable of room-temperature operation.