1. INTRODUCTION

The transition from excitons at low excitation to an electron-hole plasma in highly excited semiconductors have been extensively studied in solid-state physics [1–3]. In a two-dimensional (2D) system, the exciton state disappears and undergoes a Mott transition with increasing density as a result of both the Coulomb-potential screening and the phase-space filling [4]. A high-density system provides two advantages: (1) a condensate can form at a higher energy and temperature scale, and (2) disorder effects can be largely suppressed as a result of the screening of disorders and the occupation of localized states. Moreover, supperradiance (SR) [5] can occur at carrier densities significantly above the Mott density, such as in the e-h plasma phase [3]. Such cooperative radiative recombination of ensemble e-h pairs in semiconductors [6] can be considered as a manifestation of spontaneous macroscopic coherence in a solid-state system [7–10]. Correlation in an e-h system can also result in other effects, such as Mahan excitons near the Fermi edge [11–13] and exciton insulators [14,15].

Coupled electron-hole-photon systems in semiconductor microcavities have been a platform for studies of many-body effects and cooperative phenomena in solid states. A polariton condensate forms in the limit of weak scattering and low density, whereas conventional lasing occurs mostly at a e-h pairing-breaking regime where the density is high. In a high-density regime, a BCS-like state has been theoretically predicted to emerge in the limit of weak scattering at low temperatures. In the highly photoexcited microcavity [16,17] studied here, the nonlinearly photomodulated refractive index [18–20] results in a spatially modulated effective cavity resonance shift. Such a photocarrier-induced cavity resonance detuning is utilized to create an additional lateral optical confinement using a spatially modulated laser beam for the excitation. Under nonresonant 2-ps pulse excitation with excess energy of more than 150 meV, sequential multiple-pulse lasing commences at quantized energy levels in the optically induced harmonic confinement. Such multiple-pulse lasing is distinct from the one in conventional semiconductor lasers, where many-body effects caused by e-h pairing are often negligible. The transient multiple-pulse lasing presented here is attributed to the cooperative radiation from the correlated e-h pairs (cehp) in a high-density plasma. These correlated e-h pairs exhibit time- , density- and energy-dependent coupling efficiency to the discrete traverse modes in the harmonic lateral confinement induced by photocarriers. The dynamics of multiple lasing demonstrates sequential radiations from high- to low-energy states, which is analogous to SR [7–10] or superfluorescence [21–24] in semiconductor heterostructures. We note that, unlike condensates of exciton-polaritons, SR occurs in the e-h plasma phase [3] where the nonequilibrium carrier density is significantly above the excitonic Mott density.

2. RESULTS

The Fabry–Perot microcavity sample consists of a $\lambda$ GaAs cavity layer containing three sets of three InGaAs/GaAs quantum wells (QWs) embedded within GaAs/AlAs distributed Bragg reflectors (DBRs) (see also Supplement 1). The sample is nonresonantly excited by a 2-ps pulse pump laser at $E_p=1.58\mathrm{eV}$ at room temperature. The pump energy is about 250 meV above the QW bandgap ($E_g^{\prime }\approx 1.33\mathrm{eV}$) and 170 meV above the cavity resonance ($E_c\approx 1.40-1.41\mathrm{eV}$). Nonresonant photoexcitation creates dominantly free pairs of electrons and holes in a III–V-based semiconductor QW owing to the thermal ionization at high temperatures [25,26]. The high-density e-h plasmas of a density $\approx 1-5\times {10}^{12}{\mathrm{cm}}^{-2}$ per QW per pulse are formed momentarily after pulse excitation as a result of rapid ($<10\mathrm{ps}$) 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 $\sim 70-80$ meV above the QW bandgap $E_g^{\prime }$, i.e., the $e_{1}h{h}_1$ 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 nonradiative loss with a long decay time ($>500\mathrm{ps}$) (Supplement 1, Fig. S1). Therefore, the chemical potential of the e-h plasmas (${\mu }_0$) appears to be stationary within $\sim 100\mathrm{ps}$ after pulse excitation. The bare cavity resonance $E_c$ of the sample studied here is close to $E_{g^{\prime \prime }}$, the $e_{2}h{h}_2$ transition between the second quantized electron and hole levels of InGaAs/GaAs QWs.

At a high photoexcitation density, the average refractive index near $E_c$ can be considerably modified. This density-dependent refractive index of DBRs and QWs results in a sizable blueshift of the effective cavity resonance ($E_c^{\prime }$). In general, $E_c^{\prime }$ increases with the photoexcited density and thus can result in a finite number of transverse modes in optically induced confinement. Such an index-guiding confinement can be seen as an effective potential $V$ for cavity photons (Supplement 1, Section 3). The confinement is time dependent because of the transient density distribution of photoexcited carriers. Nonetheless, the confining potential is quasi-stationary within $\sim 100\mathrm{ps}$, where multiple-pulse lasing occurs as a result of $\sim \mathrm{ns}$ lifetimes of nonradiative carriers in the cavity spacer and DBR layers. Therefore, the quasi-stationary confining potential $V(r)$ can be established by a ring-shaped spatial distribution of the photoexcited carrier density. In our experiments, we use a double-hump-shaped beam to fixate the orientation of the optically defined potential [Figs. 1(a)–1(b)].

 

Fig. 1. Visualization of the macroscopic harmonic states. (a) Intensity image of the ring-shaped pump laser beam. (b) Photoluminescence (PL) image under a pump flux of about $1.3P_{\mathrm{th}}$, where the threshold pump flux $P_{\mathrm{th}}=1.8\times {10}^8$ 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 $P=0.8P_{\mathrm{th}}$ and $1.3P_{\mathrm{th}}$. 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 harmonic oscillator. (e)–(f) $k$-space imaging spectra. The energy splitting is $\hslash \omega \approx 2\mathrm{meV}$, consistent with the quantized energy of a quantum oscillator for cavity photons with an effective mass $m_c^{*}=3\times {10}^{-5}{\mathrm{m}}_e$, as determined by the $E$ versus $k_{\Vert }$ dispersion (dotted gray line). The quantized modes spectrally blue-shift about 1 meV from $P=0.8$ to $1.3P_{\mathrm{th}}$, 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.

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Real-space ($r$-space) imaging spectra provide direct visualization of the optical potential. Figs. 1(c)–1(d) shows the $r$-space imaging spectra of a narrow cross-sectional stripe across the trap. A nearly parabolic potential well of $\sim 10\mathrm{meV}$ across 3 μm is revealed. Such a quasi-one-dimensional harmonic potential, $V(x)=1/2\alpha x^2$, 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 asymmetrical luminescence intensity distributions. These standing-wave patterns form in the self-induced harmonic optical potential when a macroscopic coherent state emerges. An even more regular pattern appears in $k$-space imaging spectra [Figs. 1(e) and 1(f)]. The $E$ versus $k_{\Vert }$ dispersion measured below the threshold allows the direct measurement of the effective mass of cavity photons $m_c^{*}=3\times {10}^{-5}{\mathrm{m}}_e$, where ${\mathrm{m}}_e$ is the electron rest mass. Moreover, the strength of the optically defined harmonic potential $\alpha$ can be tuned through a variation in the spatial distance between two humps of the pump beam (Fig. 2). The energy quantization ($\hslash \omega$) varies with $\sqrt{\alpha /m_c^{*}}$, while the emission patterns evolve into the probability distributions for a particle with an effective mass $m_c^{*}$ in $V(x)$ (Fig. 2 and Supplement 1, Fig. S2).

 

Fig. 2. Quantized states in optically controlled confining potentials. $K$-space imaging spectra below-threshold (a)–(b) and above-threshold (d)–(e) for two double-hump-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 Supplement 1, Fig. S2.

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Under highly photoexcited conditions, optical resonance modes are modulated due to the index-dependent lateral potential. In addition, the correlation between the excited e-h pairs in QWs is much enhanced in such a high-density regime with the assistance of the common cavity light field. These correlated e-h pairs (cehps) provide the gain for the multiple-pulse lasing from the transverse optical modes. Note that the energy of cehps (μ) differs from the effective cavity resonance, $E_c^{\prime }$, and varies with the time-dependent density of the e-h plasma.

Temporally, these harmonic states emerge sequentially and display distinct density-dependent dynamics. In Fig. 3(a), we study the time evolutions of three harmonic states in $k$-space. The corresponding time-integrated imaging spectra in $k$-space are shown in Supplement 1, Fig. S3. The energy relaxation of these three states is revealed in the time-resolved spectra in Fig. 3(b). At a critical photoexcited density, the high-energy $E_3$ state arises $\sim 25\mathrm{ps}$ after pulse excitation and lasts for $\sim 20\mathrm{ps}$. The corresponding pump flux is defined as the threshold ($P_{\mathrm{th}}$). With the increasing pump flux, the $E_2$ state emerges $\sim 25\mathrm{ps}$ after $E_3$ at $1.1P_{\mathrm{th}}$, whereas the ground $E_1$ state appear 50 ps after $E_2$ at $1.2P_{\mathrm{th}}$. In the optically induced harmonic confinement studied here, the energy of cehp μ decreases with time as a result of the decay of the reservoir carriers. However, the conversion from the reservoir to the specific confined $E_i$ state can be efficient only when μ is resonant with $E_i$. Such a temporal decrease in μ results in a time- and energy-dependent conversion efficiency for these confined harmonic states (Supplement 1, Fig. S1) and consequent multiple-pulse lasing above the critical density threshold.

 

Fig. 3. Dynamics. (a) Time-dependent luminescence in $k$-space at $P=1.0$, 1.1, 1.2, and $1.6P_{\mathrm{th}}$. The $E_3$, $E_2$, and $E_1$ 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.

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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 Supplement 1, Fig. S4). Far below the threshold ($P<0.4P_{\mathrm{th}}$), the emission is dominated by luminescence from the 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. 4(a)]. On the other hand, the emission energy of these three states increases to a constant with the increasing pump flux [Fig. 4(b)]. The energy spacing $\hslash \omega$ only increases slightly with the 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. Figure 4(c) shows the rise times and pulse durations for $E_3$, $E_2$, and $E_1$. The product of the variances of the spectral linewidth ($\mathrm{\Delta }E$) and the pulse duration ($\mathrm{\Delta }t$) is found to be close to that of a transform-limited pulse: $\gtrsim 4\hslash$ and $\approx 1\hslash$ 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.

 

Fig. 4. Density dependence. (a) Temporally and spectrally integrated emission flux versus the pump flux. All three modes display nonlinear increases in intensity by more than two orders of magnitude and saturate at 1.1, 1.2, and $1.3P_{\mathrm{th}}$, respectively. (b) Peak energy (solid shapes) and linewidths $2\mathrm{\Delta }E$ (error bars) versus pump flux. These states spectrally blue-shift by 1 to 4 meV. The spectral linewidths ($\mathrm{\Delta }E$) and pulsewidths ($\mathrm{\Delta }t$) are reciprocal with a product of $\mathrm{\Delta }E\times \mathrm{\Delta }t\approx 4\hslash$ ($\hslash$) for $E_3$ and $E_1$ ($E_2$), which is closed to the uncertainty (Fourier-transform) limit. (c) Rise time versus pump flux for the three states $E_1$ (blue), $E_2$ (red), and $E_3$ (black). The error bar represents $2\mathrm{\Delta }t$.

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In Fig. 5, we study the polarization properties of the quantized harmonic states when the microcavity is excited by nonreasonant circularly polarized light. Polarized time-integrated polarized PL spectra are shown for $P=1.3P_{\mathrm{th}}$, where the lowest three states display nearly equal integrated intensities. The $E_3$ state is highly circularly polarized, with a degree of circular polarization ${\overline{\rho }}_c>\mathrm{0.8.}$ ${\overline{\rho }}_c$ for $E_2$ decreases to about 0.5, whereas $E_1$ has a diminishing ${\overline{\rho }}_c<0.05$. Temporally, the radiation from $E_3$ reaches its peak within 10 ps after pulse excitation [Fig. 4(c)]. Under a circularly polarized pump, the sub-10-ps carrier cooling results in a sizable spin imbalance despite a $\sim 10\mathrm{ps}$ electron-spin relaxation time. The conversion of cehp to the $E_3$ state is dominated by a spin-dependent stimulation process, which results in a highly circularly polarized radiation from $E_3$. Radiation from the low-energy $E_1$ state has a vanishing ${\overline{\rho }}_c$ because of the lack of spin imbalance at a time delay of 50 ps or more after the injection of the spin-polarized carriers.

 

Fig. 5. Polarization. Polarized radiation spectra at $k_{\Vert }=0$ for $P=1.3P_{\mathrm{th}}$ under a circularly polarized (${\sigma }^{+}$) pump. Black and red curves represent the co-circular ($I^{+}$) and cross-circular ($I^{-}$) components, respectively. The magnitudes of the time- and spectrally integrated degrees of circular polarization [${\overline{\rho }}_c\equiv (I^{+}-I^{-})/(I^{+}+I^{-})$] for the lowest three states are also indicated.

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3. DISCUSSION

In a highly photoexcited microcavity, the radiative behavior of a degenerate e-h plasmas gas is revealed. Due to the intense excitation and large detuning of the cavity resonance and bandgap energy, high-density e-h carriers with a chemical potential energy $\sim 80\mathrm{meV}$ above the bandgap are accumulated at room temperature. The luminescence of a high-density plasma is influenced by the surrounding light fields, which are modulated by both an optically induced trap in the transverse plane and structural confinement in the longitudinal direction. Unlike the simultaneous emissions from multiple optical modes in semiconductor lasers, the multiple-pulse lasing presented here commences sequentially from high- to low-energy modes. Such transient multiple-pulse lasing cannot be explained by the conventional mode-competition laser theory. First, the nearly energy-independent density of states in QWs and the decreasing filling factor with the increasing energy are expected to result in a higher gain at a lower energy. Second, the loss rate for a high-energy optical mode is expected to be higher than that for a low-energy mode, as revealed by the spectral linewidth of discrete lasing modes (Figs. 1 and 3). Therefore, lasing at a low-energy mode would dominate over any high-energy mode considering the gain/loss, consistent with the conventional mode-competition laser theory [27–33]. On the contrary, we observe lasing at a high-energy state followed by low-energy states in a highly photoexicited microcavity where cehps play a role.

In a highly photoexcited microcavity, cooperative optical effects can emerge when the quasi-Fermi level of e-h plasmas ${\mu }_0$ advances toward the cavity resonance, $E_c$. The phase correlation among e-h pairs can become prevalent due to the long-range Coulomb interaction and the corporative coupling effect through cavity light field. Similarly, nonequilibrium collective quantum states near the Fermi edge in a highly excited system were studied recently at cryogenic temperatures, such as Fermi-edge superfluorescence in a degenerate plasma gas [22] and Fermi-edge polaritons (or Mahan excitons) in a microcavity containing 2D electron gas [34]. Here, our experimental results support the hypothesis that a fraction of e-h pairs with energies near the optical cavity resonance can develop a phase correlation enhanced by the cavity light field. These correlated e-h pairs can form a many-body state with an effective chemical potential μ below the quasi-Fermi level of the e-h plasma (Supplement 1, Fig. S5). The critical transition density of such a many-body state or cehp is as large as ${10}^{12}{\mathrm{cm}}^{-2}$. In general, cehp is unstable at room temperature because of the small binding energy of a few milli-electronvolts (meV). However, when the energy level of cehps (μ) is far below the quasi-Fermi level of the e-h plasma, the thermal ionization of cehps can be suppressed [8]. The energy μ is largely determined by the dynamical density of photoexcited carriers in QWs and decreases with time as a result of the dissipation of carriers in QWs. The observed multiple-pulse lasing is attributed to the transient coupling of cehps to the transverse optical modes.

We use a rate-equation model to describe the dynamic formation of quantized states in an optically defined harmonic confinement (Supplement 1, Section 4). This phenomenological model reproduces qualitatively the dynamics and integrated emission flux of the harmonic states when the photoexcited density is varied (Supplement 1, Figs. S5–S7). Nonequilibrium 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 [35,36]. 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 [37], vertical-cavity surface-emitting lasers (VCSELs) [38,39], and microscale photonic structures [40]. 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 [41,42], provided that the phase correlation and strong optical nonlinearities induced by Coulomb many-body effects, such as screening, bandgap renormalization, and phase-space filling, are all considered. To uncover the microscopic formation mechanisms of such a sizable spatial modulation in the refractive index or the equivalent effective harmonic confinement 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.

4. CONCLUSION

In this study, we demonstrate a room-temperature dynamic multiple-pulse lasing of optically induced macroscopic harmonic states in high-density e-h plasmas created in a semiconductor microcavity. In an optical potential initiated by nonresonant ps pulse excitation, macroscopically coherent and spin-polarized states emerge sequentially in time at several quantized energies when these correlated e-h pairs undergo a spin-dependent stimulated process. The small effective mass ($\sim {10}^{-5}{\mathrm{m}}_e$) and unprecedented effective interactions (nonlinearities) of the constituent light–matter quasi-particle enable the sizable quantized energy levels of a few meV to be observed.

We identify sequential multiple $\sim 10\mathrm{ps}$ pulse lasing in an optically induced harmonic confinement in a semiconductor microcavity at room temperature. Laser radiation emerges at the quantized states of an optically induced harmonic confinement. The lasing frequency, rise time, pulse width, and radiation angle 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 those used for studies of polariton condensates/lasers and the widely used VCSELs. In the highly photoexcited microcavity studied here, harmonic confinement can be optically induced and controlled as a result of photomodulated refractive index changes, even in the weak-coupling plasma limit. Spatial modulations or confinements for cavity polaritons have been implemented by other static structure modifications such as (1) using quantum well structures with a spatially varying well width or cavity length, (2) adjusting the semiconductor gap of the host material locally by applying strain or local electric fields, and (3) modulating the cavity resonance by external electrodes [43]. The possibility of changing and modulating the pump flux and intensity, and thus the cavity resonance, in situ, enables a set of experiments in one sample.

The room-temperature macroscopic harmonic states in optically induced harmonic confinement presented here improve our understanding of nonlinear laser dynamics and related studies on emergent ordered states near the Fermi edge of a high-density e-h plasma in a semiconductor cavity [44–48]. The photocarrier-induced refractive index changes as well as e-h pairing and associated many-body interactions can be harnessed for the development of all-optical control of laser radiation energy and spatial and angular emission patterns.