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Two-Photon Emission from Semiconductors

We report the first experimental observations of two-photon emission from semiconductors, to the best of our knowledge, and develop a corresponding theory for the room-temperature process. Spontaneous two-photon emission is demonstrated in optically-pumped bulk GaAs and in electrically-driven GaInP/AlGaInP quantum wells. Singly-stimulated two-photon emission measurements demonstrate the theoretically predicted two-photon optical gain in semiconductors - a necessary ingredient for any realizations of future two-photon semiconductor lasers. Photon-coincidence experiment validates the simultaneity of the electrically-driven GaInP/AlGaInP two-photon emission, limited only by detector's temporal resolution.


Two-photon emission (TPE) is a process in which electron transition between
quantum levels occurs via simultaneous emission of two photons. This phenomenon is important for astrophysics and atomic physics 1, 2 , while semiconductor TPE was recently proposed as a compact source of entangled photons, essential for practical quantum information processing [3][4][5] , three orders of magnitude more efficient 6 than the existing down-conversion schemes. Two-photon absorption in semiconductors has been substantially investigated [7][8][9][10][11] ; however spontaneous semiconductor TPE has been neither observed nor fully analyzed theoretically so far.
Two-photon transition is basically much weaker than the related first-order process. Therefore observations of multi-photon spontaneous decays have so far been restricted to a few atomic transition cases, where the lowest-order transition is forbidden by selection rules 16 or suppressed by a cavity 17 , while their two-photon spectrum is continuous and centered at about half the one-photon transition frequency 1,18 .
Semiconductors can be injected with very high charge carrier densities, making even the weak second-order spontaneous processes measurable, and their TPE spectrum is expected to be determined by the photonic state density as well as by the carrier energy distribution.
Here we report the first experimental observation of spontaneous TPE from roomtemperature semiconductors, excited both optically in bulk GaAs and electrically in receiver via a lock-in amplifier. The spectrum was obtained by an Acton-Research monochromator using a 1600nm-blazed grating and ~10nm spectral resolution. The measured spectrum for the TPE was very wide as expected ( Fig. 1-a) complying with the theory described below, while the overall collected optical power was ~3nW.
In order to validate unambiguously that the observed luminescence is indeed a TPE, and to dismiss any possibility of inhomogeneously broadened emission from midgap levels, we measured singly-stimulated TPE by launching a specific wavelength into the optically pumped GaAs. As a result of this excitation, a complementary TPE wavelength peak appeared. In all the stimulated TPE experiments only the carrier density was modulated for lock-in detection, whereas the stimulating lasers were not modulated, while the observed signals were proportional to the pump power. Stimulation with 0.2mW 1630nm (0.761eV) laser resulted in a peak at 1451nm (0.854eV), while a 1600nm (0.775eV) laser yielded a peak at 1476nm (0.84eV) ( Fig.1-a). In both cases the photon energies are complementary about the center of the spontaneous emission at 0.81eV within a ~3meV error, as expected due to energy conservation. Changing the stimulation wavelength to 1570nm (0.79eV) caused the stimulating and the complementary peaks to merge approaching the degenerate case. The peaks appeared to merge for any stimulation energy closer than 30meV to the spontaneous TPE center.
Furthermore, during this excitation, the rest of the spontaneous emission spectrum was suppressed ( Fig. 1), clearly demonstrating that the wideband emission is homogeneously broadened. In TPE, the wideband homogeneous broadening stems from the very short lifetime of the intermediate virtual state, in contrast to an inhomogeneously broadened emission from any possible deep levels within the band-gap 19 .
In a third experiment, we have performed singly-stimulated TPE measurements in order to study the dependence of the complementary wavelength peak on the stimulating power. To increase the dynamic range of the measurement the pump power was increased to 180mW resulting in ~2·10 18 cm -3 carrier concentration and a 0.84eV TPE center.
Stimulation with 1310nm (0.946eV) laser yielded a complementary peak at 1691nm Hence, due to the observations described above, the measured IR luminescence is validated as TPE from bulk GaAs. These stimulated TPE experiments also demonstrate two-photon optical gain in semiconductors, which was proposed theoretically in a degenerate case for two-photon semiconductor amplifiers 12, 14 , 15 and two-photon semiconductor lasers 13 . Two-photon gain and lasing were observed previously only in discrete-level atomic systems 20, 21 exhibiting interesting nonlinear behavior. Usually twophoton lasers require seed optical power to achieve threshold. Given the nonlinearity of two-photon gain, despite the fact that one-photon oscillator strength is much larger, at high seed intensities two-photon gain can become the dominant process. Moreover, semiconductor laser complex cavities can be designed for the two-photon wavelengths only, which do not support any one-photon wavelength modes, considering the material dispersion or specially designed reflectors. Employing this effect in semiconductor materials may result in novel integrated-photonics sources allowing ultra-short pulse generation 15 , and nonlinear devices for optical switching and digital logic 22 . which was raised to above 200mA by applying an anti-reflection coating on the facets, in order to enable higher carrier population inversion, while the samples were kept at 300K by closed-loop thermoelectric cooling. Spatial distribution of the wide-band IR TPE intensity was centered near the QWs, whereas the one-photon ~1.4eV narrow-band emission was located in the GaAs cap layer region (Fig. 2). Carrier population was modulated by the injection current at 316Hz for the lock-in detection through a 500nmblazed grating monochromator. The measured spectrum for the TPE was very wide as expected and centered at 0.98eV in good agreement with the calculations (Fig. 3-a) using theoretical model described below. The overall TPE measured optical power was around 30nW, which is 5 orders of magnitude weaker than the measured one-photon emission ( Fig. 3-a inset), corresponding to TPE rates much higher than TPA at a single-photon level due to a continuum of photonic vacuum states in TPE.
The fifth experiment was aimed at observing singly-stimulated TPE in electrically-pumped GaInP/AlGaInP QWs. The stimulating laser was facet-coupled into the waveguide with about 5µm 2 mode area by a polarization maintaining lensed fiber ( Fig. 3-b inset). The measurements were taken using a 500nm-blazed grating 5nmresolution monochromator. Stimulation with 1500nm (0.826eV) laser resulted in a complementary peak at 1100nm (1.127eV), and 1450nm (0.855eV) stimulation yielded a 1117nm (1.11eV) peak ( Fig. 3-b). Both cases are complementary around the center of TPE at 0.98eV according to the theory within 4meV error. The width of the complementary peak is close to the one-photon emission spectral width, as expected (Fig  3-a inset), however the effect of stimulation is much smaller in QWs than in bulk GaAs due to the very small overlap of the optical mode with the QWs (<10 -2 ).
with q ε -photon polarization, 0 ε -vacuum permittivity, q -photon wavevector, q ωphoton radial frequency, c V -field quantization volume, q a and q a + are the field annihilation and creation operators and q n -material refractive index at q ω . Using the photon state density per unit frequency of ( ) ( ) where r m is the reduced electron mass, cv p is the bulk material momentum matrix  (Fig 1), where the transition dephasing Γ is determined mainly by the carrier decoherence time -less than 100fs for such injection levels. GaInP/AlGaInP QWs TPE spectrum was also calculated for the given current density (Fig 3-a). According our model, the ratio of spontaneous TPE versus one-photon emission is in general dependent on the injection levels because the TPE matrix element is k-dependent while one-photon emission is not (for parabolic bands). However, for the specific injection levels considered in GaInP/AlGaInP QWs, TPE is 5 orders of magnitude weaker than onephoton emission. This ratio is in good agreement with the experiments presented.
In the sixth and last experiment, the photon emission simultaneity was demonstrated by probing the coincidences in TPE from GaInP/AlGaInP QWs. In semiconductors, TPE coincidence measurements are more involved than in typical atomic systems due to the relatively low photon energies. For GaInP/AlGaInP QW TPE, at least one InGaAs-based photon counter (PC) must be used. Unlike Si PCs, InGaAsbased PCs must be operated in gated mode due to high dark-count rates, and they suffer from significant afterpulsing 30 (Fig 4). Positive integer-period number relative delay (Si output after InGaAs) resulted in coincidences rates decaying nearly exponentially with ~50µs time constant due to the InGaAs PC afterpulsing. In conclusion, we have demonstrated experimentally and analyzed theoretically TPE from semiconductors, which has not been studied before. Spontaneous and singlystimulated TPE was demonstrated in optically-pumped GaAs and in current-driven GaInP/AlGaInP QWs. Our theoretical calculations validate the experimental observations. The overall TPE is relatively strong -around 30nW in GaInP/AlGaInP QWs. Such high TPE intensities may help exploiting this fundamental physical phenomenon for practical uses in both science and engineering, while the demonstrated photon coincidences are crucial for quantum applications such as photon entanglement.