Multiple-pulse lasing from an optically induced harmonic confinement in a highly photoexcited microcavity

We report the observation of macroscopic harmonic states in an optically induced confinement in a highly photoexcited semiconductor microcavity at room temperature. The spatially photomodulated refractive index changes result in the visualization of harmonic states in a micrometer-scale optical potential at quantized energies up to 4 meV even in the weak-coupling plasma limit. We characterize the time evolution of the harmonic states directly from the consequent pulse radiation and identify sequential multiple $\sim$10 ps pulse lasing with different emitting angles and frequencies.

Optical trapping of atoms is essential to realize atomic Bose-Einstein condensates [1].Similarly, the 2D spatial confinement of excitons or exciton-polaritons by inhomogeneous strains [2], natural defects and potential fluctuations [3][4][5], nanofabrication [6][7][8][9][10], and optical potentials [11][12][13][14] in a semiconductor heterostructure/microcavity is considered conducive for the formation and control of the condensates of these quasiparticles.These dynamic condensates can form a meta-stable state in a finite-momentum state [7,15] and multiple spatial modes [16].The optical visualization of macroscopic interacting quantum states in solid-state systems has been demonstrated in condensates of excitons (bound electron-hole pairs) and kindred quasiparticles.Such light-matter hybrid condensates are typically formed at cryogenic temperatures in a photoexcited density much below the Mott transition where constituent quasiparticles can be considered boson-like.Optically defined potentials allow for real-time manipulation of these light-matter fluids in 2D semiconductor chips.For example, the imaging of wavefunctions of a quantum oscillator and petal-shaped polariton condensates confined in optically defined potentials has been reported [12][13][14].In these experiments, the interacting polaritons produce stable in-plane potentials in a planar semiconductor microcavity under a continuous-wave nonresonant optical excitation; selforganized macroscopic patterns unseen for other quasiparticles in solids result.However, monitoring the dynamics of polariton condensates directly from the emissions is challenging because the system is in a steady state.
In this study, we report the optical visualization of dynamic quantized states in a highly photoexcited microcavity at room temperature.At room temperature, photoexcitation creates mainly free pairs of electrons and holes in a III-V-based semiconductor quantum well (QW) as a result of thermal ionization [17,18].In an optical potential initiated by nonresonant ps pulse excitation, macroscopically coherent and spin-polarized states emerge sequentially in time at several quantized energies.Nevertheless, the nonlinearities of the cavity-induced many-body correlated e-h state in the high-density e-h plasma enable the sizable quantized energy levels of a few meVs to be observed.In the highly photoexcited microcavity studied here, the lasing spectral blueshifts and confining potential arise from photomodulated refractive index changes enhanced by many-body effects, such as the e-h correlation [19,20].
The Fabry-Pérot microcavity sample consists of a λ GaAs cavity layer containing three sets of three In-GaAs/GaAs QWs embedded within GaAs/AlAs distributed Bragg reflectors (DBRs) (see Supplementary Materials and Methods).The sample is nonresonantly excited by a 2-ps pulse pump laser at E p = 1.58 eV at room temperature.The pump energy is about 250 meV above the QW band gap (E g ≈ 1.33 eV) and 170 meV above the cavity resonance (E c ≈ 1.40-1.41eV).The high-density e-h plasmas of a density ≈ 1−5×10 12 cm −2 per QW per pulse are formed momentarily after nonresonant pulse excitation as a result of rapid (<10 ps) energy dissipation through optical phonons.The radiative recombination rate of these e-h carriers in the reservoir is suppressed because the cavity resonance E c is detuned to ∼ 80-90 meV above the QW bandgap E g , i.e., the e1hh1 transition between the first quantized electron and (heavy-)hole states in a QW.Below the lasing threshold, the high-density e-h plasmas in the microcavity are subject to non-radiative loss with a long decay time (>500 ps) (Supplementary Fig. S4).Therefore, the chemical potential of the e-h plasmas (µ 0 ) appears to be stationary within ∼100 ps after pulse excitation.The bare cavity resonance E c of the sample studied here is close to E g , the e2hh2 transition between the second quantized electron and hole levels InGaAs/GaAs QWs.When µ 0 advances toward E c , the refractive index near E c can be greatly modified in the presence of correlated e-h pairs (cehp) formed near the Fermi edge as a result of effective coupling to the cavity light field (Supplementary Figs.S3  and S5).
The mean energy of cehp (µ) is largely determined by the chemical potential of reservoir carriers µ 0 .In general, µ increases with the photoexcited density as a result of net repulsive Coulomb and exchange interactions [21].Therefore, a quasi-stationary confining potential for  (c-d) r-space imaging spectra at P = 0.8 P th and 1.3 P th .The black dashed line represents the harmonic confining potential V (x), whereas the white lines represent the spatial probability distributions of the lowest three states of a corresponding quantum oscillator.(e-f) k-space imaging spectra.The energy splitting is ω ≈ 2 meV, consistent with the quantized energy of a quantum oscillator for a particle with mass m * = 3 × 10 −5 me, as determined by the E vs. k dispersion (doted grey line).The quantized modes spectrally blue shift about 1 meV from P = 0.8 to 1.3 P th , whereas the quantized energy splitting remains the same.The potential and spectral shifts are due to a density-dependent increase in the chemical potential of the high-density e-h plasma in the reservoir.
cehps can be established by a ring-shaped spatial distribution of photoexcited carrier density.In our experiments, we use a double-hump-shaped beam to fixate the orientation of the optically defined potential (Fig. 1a-b).
Real-space (r-space) imaging spectra provide direct visualization of the optical potential.Fig. 1c-d shows the r-space imaging spectra of a narrow cross-sectional stripe across the trap.A nearly parabolic potential well of ∼10 meV across 3 µm is revealed.Such a quasi-1D harmonic potential, V (x) = 1/2 αx 2 , is spontaneously formed under 2 ps pulse excitation.The resultant radiation appears in-between the two humps, a few micrometers away from the pump spot.The r-space intensity profiles agree with the probability distributions of the quantized states of a harmonic oscillator (harmonic states).The deviations of the actual potential from a perfectly harmonic trap result in slightly asymmetric luminescence intensity distributions.These standing wave patterns form in the selfinduced harmonic optical potential when a macroscopic coherent state emerges.At a critical density, the occupation number (n i ) in these quantized harmonic states approaches unity when the conversion from the cehps (N eh ) overcomes the decay of the harmonic states.Consequently, when a stimulated process (∝ N eh n i ) prevails, n i and the resultant radiation increase nonlinearly by a few orders of magnitude with the increasing N eh .
An even more regular pattern appears in k-space imaging spectra (Fig. 1e-f).The E vs. k dispersion measured below the threshold allows the direct measurement of the effective mass m * = 3 × 10 5 m e , where m e is the electron rest mass.Moreover, the strength of the opti- cally defined harmonic potential α can be tuned through a variation in the spatial distance between two humps of the pump beam (Fig. 2).The energy quantization ( ω) varies with α/m * , while the emission patterns evolve into the probability distributions for a particle with an effective mass m * in V (x) (Fig. 2 and Supplementary Fig. S1).Temporally, these harmonic states emerge sequentially and display distinct density-dependent dynamics.In Fig. 3a, we study the time evolutions of three harmonic states in k-space.The corresponding time-integrated imaging spectra in k-space are shown in Supplementary Fig. S2.The energy relaxation of these three states is revealed in the time-resolved spectra in Fig. 3b.At a critical photoexcited density, the high-energy E 3 state arises ∼ 25 ps after pulse excitation and lasts for ∼ 20 ps.The corresponding pump flux is defined as the threshold (P th ).With the increasing pump flux, the E 2 state emerges ∼ 25 ps after E 3 at 1.1 P th , whereas the ground E 1 state appear 50 ps after E 2 at 1.2 P th .The multiple pulse lasing is due to the decreasing chemical potential and the resultant time-dependent conversion efficiency of the cehp into the E i state (Figs.S4d, S5 and S6).
We further analyze the emission flux and energy of these harmonic states by using time-integrated spectra measured with increasing photoexcited density (Fig. 4 and Supplementary Fig. S3).Far below the threshold (P < 0.4 P th ), emission is dominated by luminescence from GaAs spacer layers.When the pump flux is increased, the emissions from the cehp states become increasingly dominant, and the E 3 state eventually lases at the threshold.The emission fluxes of all three states increase nonlinearly by more than two orders of magnitude across a threshold and then reach a plateau at a saturation density (Fig. 4a).On the other hand, the emission energy of these three states increases to a constant with the increasing pump flux (Fig. 4b).The en-ergy spacing ω only increases slightly with density.The spectral linewidths increase slightly for the E 1 state but by about a factor of 10 for the E 3 state.Next, we study the density-dependent dynamics.Fig. 4c shows the rise times and pulse durations for E 3 , E 2 , and E 1 .The product of the variances of the spectral linewidth (∆E) and the pulse duration (∆t) is found to be close to that of a transform-limited pulse: 4 and ≈ 1 for the E 3 /E 1 and E 2 states, respectively.These harmonic states are macroscopically coherent states with finite phase and intensity fluctuations induced by interactions.
We use a rate-equation model to describe the dynamic formation of quantized states in an optically defined harmonic potential (Supplementary Information).This phenomenological model reproduces qualitatively the dynamics and integrated emission flux of the harmonic states when photoexcited density is varied (Supplementary Figs.S5, S6, and S7).Non-equilibrium polariton condensates have been modeled by a modified Gross-Pitaevskii (GP) (or complex Ginzberg-Landau [cGL]) equation that accounts for the finite lifetime of polaritons [22,23].However, cGL-type equations are inapplicable for the multiple dynamic states examined in this study.Additionally, the formation of a BEC-like condensate that underpins the GP-or cGL-type equation is not necessarily justified in our room-temperature experiments.
Transverse light-field patterns and confined optical modes have been identified in nonlinear optical systems [24], vertical-cavity surface-emitting lasers (VC- SELs) [25,26], and microscale photonic structures [27].In principle, the multiple transverse mode lasing in a high-density e-h-plasma described in the present study can be modeled by a self-consistent numerical analysis with Maxwell-Bloch equations developed for conventional semiconductor lasers [28,29], provided that the strong optical nonlinearities induced by Coulomb manybody effects, such as screening, bandgap renormalization, and phase-space filling, are all considered.For example, one can consider the formation of index-guided multiple transverse modes as a result of an optically-induced re-fractive index reduction (δn c (x)).The cavity resonance shift (δE) can be estimated from δE c /E c = −δn c /n c , where n c is the effective refractive index averaged over the longitudinal cavity photon mode which spans over ∼ µm in the growth direction.Therefore, a cavity resonance shift δE ∼10 meV (Figs. 1 and 2) corresponds to |δn c /n c | ∼ 1%.Such a significant refractive index is probable with resonance-enhance optical nonlinearity [19,20] or carrier-induced change in refractive index at a high carrier density (of ∼ 10 19 cm −3 or more) [30,31].To uncover the microscopic formation mechanisms of such a sizable spatial modulation in refractive index or equivalent effective harmonic potential under pulse excitation, further characterizing the photoexcited density distribution with the use of other ultrafast spectroscopic techniques, such as a pump-probe spectroscopy, is necessary.
We identify sequential multiple ∼10 ps pulse lasing in an optically induced harmonic potential in a semiconductor microcavity at room temperature.Laser radiation emerges at the quantized states of an optically induced harmonic potential.The lasing frequency, rise time, pulse width, radiation angle, and polarization can be controlled through a variation in the photoexcited density or optical pump spot dimensions in real time.The sample has a composition structure similar to the widely used VC-SELs and can be replicated with the standard molecular beam epitaxy method.Our demonstration of a roomtemperature macroscopic quantum oscillator improve understanding of nonlinear laser dynamics and should stimulate studies on emergent ordered states near the Fermi edge of a high-density e-h plasma in a semiconductor cavity [32][33][34].

Sample.
The microcavity sample is grown on a semi-insulating (100)-GaAs substrate with the molecular beam epitaxy method.The top (bottom) DBR consists of 17 (20) pairs of GaAs(61-nm)/AlAs (78-nm) /4 layers.The central cavity layer consists of three sets of three In 0.15 Ga 0.85 As/GaAs (6 nm/12 nm) quantum wells each, positioned at the anti-nodes of the cavity light field.The structure is undoped and contains a GaAs cavity sandwiched by DBRs.The bare cavity resonance E c ⇡ 1.41 eV at room temperature.The QW bandgap (E 0 g ⇠1.33 eV) is tuned through a rapid thermal annealing process (at 1010 C-1090 C for 5-10 s), in which the InGaAs QW bandgap blueshifts because of the di↵usion of gallium ions into the MQW layers.With increasing photoexcited density, the chemical potential µ 0 of the e-h plasma can increase up to ⇠80 meV above E 0 g , approaching E c .The cavity quality factor Q is about 4000-7000, corresponding to a cavity photon lifetime of ⇠2 ps.
Optical excitation.The front surface of the sample is positioned at the focal plane of a high-numerical-aperture objective (N.A. = 0.42, 50⇥, e↵ective focal length: 4 mm).The holographic beam shaping is achieved with a reflected Fourier transform imaging system that consists of a 2D phase-only spatial light modulator (SLM), a 3⇥ telescope, a Faraday rotator, a polarizing beam splitter, and the objective.The light fields at the SLM and the sample surface form a Fourier transform pair.The 2D SLM (1920 ⇥ 1080 pixels, pixel pitch = 8 µm) enables us to generate arbitrary pump geometries with a ⇡2 µm spatial resolution at the sample surface by using computer-generated phase patterns.The polarization properties of pump and luminescence are controlled/analyzed with liquid-crystal devices.The pump flux is varied by more than two orders of magnitude with a liquid-crystal-based attenuator.
Thermal management.At a high pump flux, the steady-state incident power transmitted to the sample at the 76 MHz repetition rate of the laser would exceed 50 mW, and significant thermal heating results.Thermal heating can inhibit laser action and lead to spectrally broad redshifted luminescence.To suppress thermal heating, we temporally modulate the 2 ps 76 MHz pump laser pulse train with a duty cycle (on/o↵ ratio) < 0.5% by using a double-pass acousto-optic modulator system.The time-averaged power is limited to below 1 mW for all experiments.R-space and k-space imaging spectroscopy.We measure the angular, spectral, and temporal properties of luminescence in the reflection geometry.The angle-resolved (k-space) luminescence images and spectra are measured through a Fourier transform optical system.A removable f = 200 mm lens enables the projection of either the k-space or r-space luminescence on to the entrance plane or slit of the spectrometer.Luminescence is collected through the objective, separated from the reflected specular and scattered pump laser light with a notch filter, and then directed to an imaging spectrometer.A lasing mode with a spatial diameter ⇡8 µm is isolated for measurements through a pinhole positioned at the conjugate image plane of the microcavity sample surface.The spectral resolution is ⇡0.1 nm (150 µeV), which is determined by the dispersion of the grating (1200 grooves/mm) and the entrance slit width (100-200 µm).The spatial (angular) resolution is ⇡ 0.3 µm (6 mrad) per CCD pixel.) 0 1 P = 0.9 P th Fig. S2.K-space imaging spectra.The energy vs. in-plane momentum k || dispersion corresponding to the data set in the main paper with the increasing pump flux from 0.9 to 1.3 P th .The quantized E3, E2, and E1 states lase in sequence with the increasing pump flux.E3-state lasing commences first, followed by E2-state lasing at about 1.2 P th , and then E1-state lasing appears at 1.3 P th and dominates over the other two states at a higher pump flux.1e and Fig. 2a-c in the main text.Quantized harmonic states are formed above ⇠ 0.8P th .Below the threshold, the luminescence from all states is unpolarized.Above the threshold, the E3 state is highly circularly polarized, where as the E1 state has a diminishing circular polarization.The fourth quantized state (E4) is weakly confined.

S.3. THEORETICAL MODELING
To model the dynamic formation of a macroscopic quantum oscillator, we use a set of rate equations that consider a stimulated process and the spatial distribution of e-h carriers.We consider the temporal evolution of the reservoir distribution N R (x, t) and quantized sates n i (t).The e-h pairs photoexcited non-resonantly by a 2 ps pulse laser cool down rapidly (<5 ps) to the band edge.The cooled carriers in the reservoir N R (x, t) are subject to non-radiative loss ( nr ) and thus result in a slow decrease (⇠0.1 meV/ps) in the chemical potential µ.A fraction of the reservoir carriers [N eh (x, t) = N R (x, t)] near the Fermi edge could couple e↵ectively to the cavity light field when µ advances toward the cavity resonance E c (Fig. S5).As a result, correlated e-h pairs (cehp) are formed near the Ferm edge.The double-hump-shaped spatial distribution of e-h plasma results in the establishment of a harmonic chemical potential profile (V (x) / x 2 ) in which the standing waves of a macroscopic coherent state are formed.However, only cehps near in resonance with the quantized states of the harmonic potential can spontaneously be scattered into a specific E i state at a rate W s .Above threshold, the population of E i states increases nonlinearly as a result of spin-dependent stimulation (/ W ss N eh n i ).The consequent leakage photons from the decay of these confined cehps at a rate associated with the cavity photon decay rate c are then measured experimentally.The dynamics of the quantum oscillator of cehps can then be described by the following set of coupled rate equations: The spatial generation rate G(x, t) follows the 2 ps Gaussian temporal and double-hump-shaped beam profile (Fig. S5c and Fig. S6c).P is the pump flux of helicity ±. ⇣ is an energy dependent e↵ective formation e ciency of the cehps.We approximate ⇣ as a Lorentzian function , where µ is the energy range that cehps are formed.
⇤ is the coupling e ciency of the cehps to the E i state, where E i is the spectral linewidth of the E i state.
This model reproduces the time evolution of the quantized states with increasing photoexcited density (Fig. S6).Below the threshold, the scattering of the cehps into the quantized states occurs through a spontaneous decay process.The radiative quantum e ciency is low (< 10 4 ) because of the dominant non-radiative recombination loss.Above the threshold, the quantum e ciency increases by two to three orders of magnitude as a result of stimulation.This model reproduces qualitatively the rise times and the time-integrated emission fluxes of the multiple pulse lasing from the quantized states when the pump flux is varied (Fig. S7).
Discussion.Experimentally, the quantized states appear temporally in the order of E 3 , E 2 , and E 1 .To reproduce the observed cascade pulses sequence, it is necessary to consider the spatial carrier di↵usion and coupling (cascade) between the quantized states.In the simple theoretical mode, we use a µ(x, t) that is dependent on the initial N eh = N R to emulate the coupling between states (cascading) and spatial di↵usion of e-h carriers.A nearlyidentical stationary harmonic potential V (x) is used for all calculations when the N R increases from slight below the threshold.The population of the quantized state E i exceeds unity when the conversion e ciency (⇠) of e-h pairs into the E i state surpasses the loss (i.e., The conversion e ciency is spatially integrated over the e-h reservoir N R (x, t) when the cehp formation e ciency, ⇣(µ E c ), and the coupling e ciency of cehp to the E i state, ⌘ i (µ E i ), are considered (Fig. S5).When the pump flux is slightly below or above the threshold, the overall conversion e ciency for the E 3 state is the largest (i.e., ⇠ 3 > ⇠ 2 > ⇠ 1 ), as demonstrated in simulated (Fig. S6a and Fig. S7) and measured (Fig. S4c-d and Fig. 3) dynamics.
We note a few shortcomings of this simplified model: (1) The model does not consider the local energy shift and the evolution of potentials induced by the interactions of the cehps because the density-dependent µ(N R , N eh , n i ) is not well determined.As a result, the density-dependent energy shifts and linewidths cannot be reproduced.( 2   (b-c) Similar to (a), but for P = 1.15 and 1.5 P th .For simplicity, the spin degree of freedom is neglected in these calculations.Additionally, the radiative decay rates of these states are assumed to be identical and equal to c, resulting in an identical pulse duration.

FIG. 1 .
FIG. 1. Visualization of the harmonic states of a macroscopic quantum oscillator.(a) Intensity image of the ring-shaped pump laser beam.(b) Photoluminescence (PL) image under a pump flux of about 1.3 P th , where the threshold pump flux P th = 1.8 × 10 8 photons per pulse.The white dashed line represents the intensity peak of the pump.PL emerges at the center with a minimal overlap with the annular pump laser beam.(c-d) r-space imaging spectra at P = 0.8 P th and 1.3 P th .The black dashed line represents the harmonic confining potential V (x), whereas the white lines represent the spatial probability distributions of the lowest three states of a corresponding quantum oscillator.(e-f) k-space imaging spectra.The energy splitting is ω ≈ 2 meV, consistent with the quantized energy of a quantum oscillator for a particle with mass m * = 3 × 10 −5 me, as determined by the E vs. k dispersion (doted grey line).The quantized modes spectrally blue shift about 1 meV from P = 0.8 to 1.3 P th , whereas the quantized energy splitting remains the same.The potential and spectral shifts are due to a density-dependent increase in the chemical potential of the high-density e-h plasma in the reservoir.

FIG. 2 .
FIG. 2. Quantized states in optically controlled confining potentials.K-space imaging spectra under belowthreshold (a-b) and above-threshold (d-e) for two doublehump-shaped pump beams with peak-to-peak distances of 5 µm and 3 µm, respectively.For comparison, the k-space imaging spectra under a flat-top pump beam are shown in (c) and (f).The corresponding r-space images and spectra are shown in Supplementary Fig. S1.

2 FIG. 3 .
FIG. 3. Dynamics.(a) Time-dependent luminescence in k-space at P = 1.0, 1.1, 1.2 and 1.6 P th .The E3, E2 and E1 states appear sequentially with the increasing pump flux.The rise times decrease with the increasing pump flux for all states.(b) Time-dependent spectra in r-space.The false color represents normalized intensities.

FIG. 4 .
FIG. 4. Density dependence.(a) Temporally and spectrally integrated emission flux vs. pump flux.All three modes display non-linear increases in intensity by more than two orders of magnitude, and saturate at 1.1, 1.2 and 1.3 P th , respectively.(b) Peak energy (solid shapes) and linewidths 2∆E (error bars) vs. pump flux.These states spectrally blue shift by 1 to 4 meV.The spectral linewidths (∆E) and pulsewidths (∆t) are reciprocal with a product of ∆E × ∆t ≈ 4 ( ) for E3 and E1 (E2), which is closed to the uncertainty (Fouriertransform) limit.(c) Rise time vs. pump flux for the three states E1 (blue), E2 (red) and E3 (black).The error bar represents 2∆t.
Fig. S1.Quantized states vs. the trap size.(a-c) Optical images of pump beam profiles at the front sample surface for two double-hump-shaped and one flat-top focal spots.The false color represents the photoexcited density (cm 2 ) per quantum well per pulse at 0.8 P th for (a) and at 0.4 P th for (b).P th = 1.8 ⇥ 10 8 per pulse, as defined in the main text.The lasing thresholds are 0.8 P th for (a) and 0.4 Pth for (b).(d-f) Real-space imaging spectra measured below the threshold.The black dashed line shows the e↵ective harmonic potential, whereas the white lines represent the spatial probability distributions of the lowest few states of a corresponding quantum oscillator.The quantized energy splittings for (d) and (e) are 2.2 and 4.0 meV, respectively.The false color represents the square root of intensity.(g-h) Real-space imaging spectra under above-threshold pump.(i) The measured weave functions of the two lowest states E2 and E1 at 0.4 P th (solid blue lines) and 0.5 P th (solid red lines) as shown in (e) and (h).The ideal single-particle wave functions of the corresponding quantum simple harmonic oscillator (SHO) are represented by the dashed lines.

Fig
Fig. S3.Polarized luminescence spectra vs. pump flux.Normalized spectra integrated over k-space as a function of pump flux: Co-circularly (a) and cross-circularly (b) polarized component.Cavity-induced correlated e-h pairs are formed near above 0.4 P th , as demonstrated by the E vs. k k dispersion shown in Fig.1eand Fig.2a-cin the main text.Quantized harmonic states are formed above ⇠ 0.8P th .Below the threshold, the luminescence from all states is unpolarized.Above the threshold, the E3 state is highly circularly polarized, where as the E1 state has a diminishing circular polarization.The fourth quantized state (E4) is weakly confined.
Fig. S4.Below-threshold dynamics.(a) Time-dependent spectrally and spatially integrated luminescence at P = 0.9 P th .The luminescence decays with a > 500 ps time constant largely because of the nonradiative loss.(b) Spatial luminescence distribution integrated up to a delay of 50 ps after pulse excitation, revealing a double-hump-shaped profile.(c) Timedependent luminescence spectra at P = 0.9 P th .The false color represents the square root of the luminescence intensity.(d) Time-dependent luminescence intensity for the E3 (black), E2 (red) and E1 (blue) harmonic states.The intensity is integrated over a spectral range of 1 meV centering the energies, as indicated by the dashed lines in (c).
) The model does not consider carrier di↵usion, which a↵ects dynamics at a density far above the threshold.(3) Phase and intensity fluctuations are neglected.

Fig. S5 .
Fig.S5.Schematic diagrams for the theoretical framework.(a-b) Energy level diagrams at delays t1 and t2 (t1 < t2) after injection of e-h carriers in the reservoir.The chemical potential µ(t) at a fixed location decreases with time largely because of the nonradiative loss.The formation e ciency of the correlated e-h pairs (cehps) is approximated by a Lorentzian function centering at the cavity resonance Ec, whereas the conversion e ciency of cehps into quantized harmonic states is represented by a Gaussian function centering at quantized energies E3, E2, and E1.(c) (top panel) The initial spatial distribution of e-h carriers in the reservoir (NR(x, t = 0 + ) / Gp(x, t = 0 + )).(bottom panel) The confining potential V (x) (solid blue line), which is assumed to be stationary.The spatial µ at t1 (solid black line) and t2 (dashed black line).The cehps with µ close to the energies of Ei can scatter e↵ectively into the Ei states.
Fig.S6.Simulated dynamics of the harmonic states.(a) The time-dependent occupation number of harmonic states E3 (solid black line), E2 (solid red line), E1 (solid blue line), and the sum of the three states (dashed black line) for P = 0.8 P th .(b-c) Similar to (a), but for P = 1.15 and 1.5 P th .For simplicity, the spin degree of freedom is neglected in these calculations.Additionally, the radiative decay rates of these states are assumed to be identical and equal to c, resulting in an identical pulse duration.
Fig. S7.Simulated rise times and time-integrated flux of the harmonic states.The calculated rise times (a) and time-integrated flux (b) of the E3 (solid black line), E2 (solid red line), and E1 (solid blue line) states as a function of pump flux.