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Nanolasers, characterized by enhanced optical localization at subwavelength scale, have emerged as promising coherent light sources for ultra-compact, high-speed and energy-efficient photonic integrated circuits. Twisted photonic crystal nanocavity, constructed by stacking two layers of photonic crystal structure with a specified rotation angle, enables strong light confinement with an ultra-small mode volume and an extremely high quality factor. The twisted angle can be randomly selected, providing the possibility of actively tuning the resonant wavelength and optical mode distribution within a nanoscale twisted cavity. Here, we demonstrate large-angle twisted single-mode photonic crystal nanolasers operating in the C-band with an exceptionally ultra-compact footprint of approximately 25 $\mu m^2$ and an ultra-small mode volume of 0.47 $(\lambda/n)^3$. The reported twisted photonic crystal nanolasers are optically pumped at room temperature with an ultra-low threshold of $\sim$ 1.25 $kW/cm^2$. Our work provides a prospective method for easily constructing robust nanolasers by twisting angles, and paves the way for achieving high-performance nanoscale coherent light sources for densely integrated photonic chips.

Density-integrated, multi-wavelength nanoscale lasers with ultra-low power consumption and ultra-compact footprints are essential for energy-efficient, fast and high-throughput data processing. Currently, on-chip multi-wavelength lasers predominantly rely on arrays of discrete large-scale conventional semiconductor lasers that are susceptible to the fabrication imperfections. Topological rainbow nanolasers, which spatially confine and emit specific topologically protected light frequencies, offer a prospective approach for achieving ultra-compact integrated multi-wavelength light sources with enhanced robustness against perturbations and defects. However, it remains a significant challenge to achieve highly localized topological rainbow trapping in nanocavities for laser emission with both high quality factors and ultra-small mode volumes. Here, we experimentally report ultra-compact topological photonic crystal rainbow nanolasers operating in the 1550 nm telecom band. Specifically, we present rainbow-like emission with uniform wavelength spacing and wavelength-scale mode volume $\sim 0.7 \left(\frac{\lambda}{n}\right)^3$ in a one-dimensional topological rainbow nanolaser, exhibiting robust lasing operation across a wide temperature range and a spectral tuning capability of approximately 70 nm. Additionally, we demonstrate an ultra-compact two-dimensional topological rainbow nanolaser in an exceptionally compact footprint of nearly 0.002 $\text{mm}^2$, featuring a broad rainbow spectra with 64 continuously tuned lasing peaks. Our work provides a promising method for realizing robust and nanoscale multi-wavelength tunable laser sources, paving the way for numerous potential applications in ultra-compact photonic chips.

Nanolasers operating at low power levels are strongly affected by intrinsic quantum noise, affecting both intensity fluctuations and laser coherence. Starting from semi-classical rate equations and making a simple hypothesis for the phase of the laser field, a simple stochastic model for the laser quantum noise is suggested. The model is shown to agree quantitatively with quantum master equations for microscopic lasers with a small number of emitters and with classical Langevin equations for macroscopic systems. In contrast, neither quantum master equations nor classical Langevin equations adequately address the mesoscopic regime. The stochastic approach is used to calculate the linewidth throughout the transition to lasing, where the linewidth changes from being dominated by the particle-like nature of photons below threshold to the wave-like nature above threshold, where it is strongly influenced by index fluctuations enhancing the linewidth., Comment: 6 pages, 4 figures. Submitted for PRL

Tailoring the losses of optical systems within the frame of non-Hermitian physics has appeared very fruitful in the last few years. In particular, the description of exceptional points (EPs) with coupled resonators have become widespread. The on-chip realization of these functionalities is highly significant for integrated nanophotonics, but requires fine control techniques of the nanodevice properties. Here, we demonstrate pump-controlled directional emission of two coupled nanolasers that distantly interact via an integrated waveguide. This coupling scheme unusually enables both frequency- and loss-couplings between two cavities, which can be advantageously exploited to reach EPs by either detuning the cavities or controlling the gain of nanolasers. The system can be readily reconfigured from bidirectional to unidirectional emission by adjusting the pump power.

The miniaturization and integration of beam steering devices have consistently been the focus of the field. Conventional methods alter the eigenmode of the optical cavity by regulating the refractive index. Due to the weak nonlinear effect of the optical system, the device must be sufficiently large to achieve sufficient light modulation. The effective method for miniaturizing beam steering devices currently in use is based on metasurfaces. However, this type of device necessitates the input of a laser source, which precludes the simultaneous generation and control of light in a single device. Here we propose a miniaturized beam steering device that employs mode selection between different bound states in the continuum (BIC) states through phase change material. The device is capable of simultaneously achieving both light generation and beam steering (33{\deg}) in a single device with a size of only 25 {\mu}m and with a low threshold of 8.9 kW cm-2. Furthermore,it is possible to achieve a significant degree of dynamic wavelength tunability, with a range extending up to 296 nm. This method achieves high-efficient regulation of light properties by dynamically controlling the system's topological charge, circumventing the problem of weak nonlinearity in traditional methods. Furthermore, the integration of phase change materials with nanolasers enables the direct alteration of lasing properties, which provides a novel idea for dynamic light control. The device process scheme based on phase change materials is straightforward, direct, and highly compatible, which will be advantageous for its intended application.

Coupled nanolasers are of growing interest for on-chip optical computation and data transmission, which requires an understanding of how lasers interact to form complex systems. The non-Hermitian interaction between two coupled resonators, when excited selectively, can lead to parity-time symmetry, the formation of exceptional points, and subsequently spectral control and increased sensitivity. These investigations have been limited to pump energies close to the lasing threshold, and large or narrow-line lasers. Here, by programmable optical excitation we study two coupled nanolasers significantly above threshold, where mode instability plays an important role. We map the mode evolution around two exceptional points, and observe lasing gaps due to reversed pump dependence which compare well with nonlinear theory. Finally, the coupling can be exploited to control the lasing threshold and wavelength, and for frequency switching around the lasing gap. Controlled and integrated nanolasers constitutes a promising platform for future highly sensitive and programmable on-chip laser sources., Comment: 8 pages, 4 figures

Nanolasers based on emerging dielectric cavities with deep sub-wavelength confinement of light offer a large light-matter coupling rate and a near-unity spontaneous emission factor, $\beta$. These features call for reconsidering the standard approach to identifying the lasing threshold. Here, we suggest a new threshold definition, taking into account the recycling process of photons when $\beta$ is large. This threshold with photon recycling reduces to the classical balance between gain and loss in the limit of macroscopic lasers, but qualitative as well as quantitative differences emerge as $\beta$ approaches unity. We analyze the evolution of the photon statistics with increasing current by utilizing a standard Langevin approach and a more fundamental stochastic simulation scheme. We show that the threshold with photon recycling consistently marks the onset of the change in the second-order intensity correlation, $g^{(2)}(0)$, toward coherent laser light, irrespective of the laser size and down to the case of a single emitter. In contrast, other threshold definitions may well predict lasing in light-emitting diodes. These results address the fundamental question of the transition to lasing all the way from the macro- to the nanoscale and provide a unified overview of the long-lasting debate on the lasing threshold.

Selective control of light is essential for optical science and technology with numerous applications. Nanophotonic waveguides and integrated couplers have been developed to achieve selective coupling and spatial control of an optical beam according to its multiple degrees of freedom. However, previous coupling devices remain passive with an inherently linear response to the power of incident light limiting their maximal optical selectivity. Here, we demonstrate nonlinear optical selectivity through selective excitation of individual single-mode nanolasers based on the spin and orbital angular momentum of light. Our designed nanolaser circuits consist of plasmonic metasurfaces and individual perovskite nanowires, enabling subwavelength focusing of angular-momentum-distinctive plasmonic fields and further selective excitation of single transverse laser modes in nanowires. The optically selected nanolaser with nonlinear increase of light emission greatly enhances the baseline optical selectivity offered by the metasurface from about 0.4 up to near unity. Our demonstrated nonlinear optical selectivity may find important applications in all-optical logic gates and nanowire networks, ultrafast optical switches, nanophotonic detectors, and on-chip optical and quantum information processing.

We detail the derivation of nanolaser models that include coherent and incoherent variables and predict the existence of a laser threshold, irrespective of cavity size and emitter number, for both single- and multi-electron systems. The growth in photon number in the lasing mode is driven by an increase in correlation between absorption and emission processes, leading to the onset of self-sustained stimulated emission (laser threshold), followed, in turn, by a correlation decrease and ending with the dominance of coherent emission. The first-order coherence $g^{(1)}$ steadily increases, as the pump grows towards the laser threshold value, and reaches unity at or beyond threshold. The transition toward coherent emission becomes increasingly sharp as the number of emitters and of the coupled electromagnetic cavity modes increase, continuously connecting, in the thermodynamic limit, the physics of nano- and macroscopic lasers at threshold. Our predictions are in remarkable agreement with experiments whose first-order coherence measurements have so far been explained only phenomenologically. A consistent evaluation of different threshold indicators provides a tool for a correct interpretation of experimental measurements at the onset of laser action., Comment: 11 pages, 5 figures

Nanolasers have great potential as both on-chip light sources and optical barcoding particles. We demonstrate ultrasmall InGaP and InGaAsP disk lasers with diameters down to 360 nm (198 nm in height) in the red spectral range. Optically pumped, room-temperature, single-mode lasing was achieved from both disk-on-pillar and isolated particles. When isolated disks were placed on gold, plasmon polariton lasing was obtained with Purcell-enhanced stimulated emission. UV lithography and plasma ashing enabled the fabrication of nanodisks on a wafer-scale, with intended random size variation. Silica-coated nanodisk particles generated stable sub-nanometer spectra from within biological cells across an 80 nm bandwidth from 635 to 715 nm., Comment: 25 pages, 5 main figures, 8 supplementary figures, 3 supplementary tables

The pursuit of compact lasers with low-thresholds has imposed strict requirements on tight light confinements with minimized radiation losses. Bound states in the continuum (BICs) have been recently demonstrated as an effective mechanism to trap light along the out-of-plane direction, paving the way to low-threshold lasers. To date, most reported BIC lasers are still bulky due to the absence of in-plane light confinement. In this work, we combine BICs and photonic band gaps to realize three-dimensional (3D) light confinements, as referred to miniaturized (mini-) BICs. Together with 3D carrier confinements provided by quantum dots (QDs) as optical gain materials, we have realized highly-compact active BIC resonators with a record-high quality ($Q$) factor up to 32500, which enables single-mode continuous wave (CW) lasing with the lowest threshold of 80 W/cm$^{2}$ among the reported BIC lasers. In addidtion, our photon statistics measurements under both CW and pulsed excitations confirm the occurence of the phase transition from spontaneous emission to stimulated emission, further suggesting that conventional criteria of input-output and linewidth are not sufficient for claiming nanoscale lasing. Our work reveal a via path towards compact BIC lasers with ultra-low power consumption and potentially boost the applications in cavity quantum electrodynamics (QEDs), nonlinear optics and integrated photonics., Comment: 12 pages,4 figures in maintext

The Siegert relation connects the first- and second-order coherence properties of light. While strictly valid only in the thermal regime and in the absence of correlations, this relation is routinely extended to the partially coherent regime in the study of high-$\beta$ nanolasers, where it aids in the identification of the lasing threshold. We re-evaluate the use of a generalized Siegert relation in different device regimes. A full two-time quantum-optical theory is derived as a reference for obtaining first- and second-order correlation functions $g^{(1)}(\tau)$ and $g^{(2)}(\tau)$ in the steady state. We find that even in the partially coherent regime the generalized Siegert relation is well suited as an approximation to $g^{(2)}(\tau)$ as long as emitter correlation are negligible, but does not apply well in the quantum regime of few-emitter nanolasers, or to devices featuring sub- and superradiant emission.

Control over the shape and polarization of the beam emitted by a laser source is important in applications such as optical communications, optical manipulation and high-resolution optical imaging. In this paper, we present the inverse design of monolithic whispering-gallery nanolasers which emit along their axial direction with a tailored laser beam shape and polarization. We design and experimentally verify three types of sub-micron cavities, each one emitting into a different laser radiation mode: an azimuthally polarized doughnut beam, a radially polarized doughnut beam and a linearly polarized gaussian-like beam. The measured output laser beams yield a field overlap with respect to the target mode of 92%, 96% and 85% for the azimuthal, radial and linearly polarized cases, respectively, thereby demonstrating the generality of the method in the design of ultra-compact lasers with tailored beams.

Since the development of micro- and nanolasers, the question of laser threshold has been subject to debate. Different definitions have been used to try and establish its occurrence, often encountering major obstacles. We examine a set of common physical definitions which we apply to measurements taken in a micro-VCSEL. Their predictions not only clearly disagree, pointing to different pump values at which the laser should cross threshold, but they also correspond to autocorrelation values which demonstrate very low field coherence. A topological analysis of the rate equations, with average spontaneous emission added to the lasing mode, clearly identifies the contradictions and explains the origin of the discrepancies. Additional considerations help understanding the failure of the approach and highlight the path towards a unique and general definition of threshold in all lasers, irrespective of their sizes. A critical scrutiny of the assumptions made in the rate equations with spontaneous emission illustrates their strength and weaknesses and better defines the bounds within which their predictions hold. We remark in the conclusions how the main results of this paper could hold for other small systems., Comment: Main article: 29 pages, 24 figures Supplementary material: 16 pages, 25 figures

Recent advancements in nanolaser design and manufacturing open up unprecedented perspectives in terms of high integration densities and ultra-low power consumption, making these devices ideal for high-performance optical computing systems. In this work we exploit the symmetry properties of the collective modes of a nanolaser array for binary image classification. The implementation is based on a 8x8 array, and relies on the activation of a collective optical mode of the array, the so-called "zero mode", under spatially modulated pump patterns. We demonstrate that a simple training strategy allows us to achieve an overall success rate of 98% in binary image recognition.

A theory of lasing in a two-dimensional array of metal nanoparticles (MNPs) covered with a thin layer of fluorescent molecules is developed from first principles. The approach is based on a rigorous account of the local field in a close vicinity of a reflective surface which provides a feedback for molecular dipole oscillations. The theory predicts the lasing threshold in such an open cavity in terms of the polar angle of laser emission, MNPs shape and the molecular layer thickness. It is demonstrated that the latter parameter plays a crucial role in the lasing condition and controls a switching from conventional lasing to lasing without inversion. This research is inspired by recent experiments in this field [N. Toropov et al, Adv. Photonics Res. {\bf 2}, 2000083 (2021)] and provides the numerical calculations carried out for the experimental conditions., Comment: 8 pages, 6 figures

Light emitters in a single-mode nanolaser interact with the same cavity field, that gives rise to polarization correlations which transform the cavity mode. Usually these correlations are ignored, however, collective phenomena can lead to the distinct sub- and superradiance, whose fully quantum description is challenging. Here, we develop a simple yet rigorous picture of radiative transitions in single-mode nanolasers that accounts for polarization correlations. We show that the collective behavior of emitters modifies the photonic density of states leading to gain-dependent Purcell enhancement of spontaneous emission. Moreover, the stimulated emission rate is dependent on both the photon number and the laser lineshape. As the laser line narrows, stimulated emission becomes stronger than predicted by Einstein's relations and the nanolaser reaches the threshold earlier. Finally, we provide concise, ready-to-use expressions for spontaneous and stimulated emission rates seamlessly describing both conventional and superradiant nanolasers., Comment: 9 pages, 4 figures

Josephson effects are commonly studied in quantum systems in which dissipation or noise can be neglected or do not play a crucial role. In contrast, here we discuss a setup where dissipative interactions do amplify a photonic Josephson current, opening a doorway to dissipation-enhanced sensitivity of quantum-optical interferometry devices. In particular, we study two coupled nanolasers subjected to phase coherent drivings and coupled by a coherent photon tunneling process. We describe this system by means of a Fokker-Planck equation and show that it exhibits an interesting non-equilibrium phase diagram as a function of the coherent coupling between nanolasers. As we increase that coupling, we find a non-equilibrium phase transition between a phase-locked and a non-phase-locked steady-state, in which phase coherence is destroyed by the photon tunneling process. In the coherent, phase-locked regime, an imbalanced photon number population appears if there is a phase difference between the nanolasers, which appears in the steady-state as a result of the competition between competing local dissipative dynamics and the Josephson photo-current. The latter is amplified for large incoherent pumping rates and it is also enchanced close to the lasing phase transition. We show that the Josephson photocurrent can be used to measure optical phase differences. In the quantum limit, the accuracy of the two nanolaser interferometer grows with the square of the photon number and, thus, it can be enhanced by increasing the rate of incoherent pumping of photons into the nanolasers., Comment: 25 pages, 8 figures. Comments are welcome

Nanolasers are considered ideal candidates for communications and data processing at chip-level thanks to their extremely reduced footprint, low thermal load and potentially outstanding modulation bandwidth, which in some case has been numerically estimated to exceed hundreds of GHz. The few experimental implementations reported to date, however, have so-far fallen very short of such predictions, whether because of technical difficulties or of overoptimistic numerical results. We propose a methodology to study the physical characteristics which determine the system's robustness and apply it to a general model, using numerical simulations of large-signal modulation. Changing the DC pump values and modulation frequencies, we further investigate the influence of intrinsic noise, considering, in addition, the role of cavity losses. Our results confirm that significant modulation bandwidths can be achieved, at the expense of large pump values, while the often targeted low bias operation is strongly noise- and bandwidth-limited. This fundamental investigation suggests that technological efforts should be oriented towards enabling large pump rates in nanolasers, whose performance promises to surpass microdevices in the same range of photon flux and input energy.

The different dynamical behaviors of the Hermite-Gaussian (HG) modes of mode-locked nanolasers based on a harmonic photonic cavity are investigated in detail using a model based on a modified Gross-Pitaevskii Equation. Such nanolasers are shown to exhibit mode-locking with a repetition rate independent of the cavity length, which is a strong asset for compactness.The differences with respect to conventional lasers are shown to originate from the peculiar gain competition between HG modes, which is investigated in details. In the presence of a saturable absorber, the different regimes, i. e. Q-switching, Q-switched mode-locking, and continuous-wave (cw) mode locking, are isolated in a phase diagram and separately described. Mode-locking is found to be robust against phase-intensity coupling and to be achievable in a scheme with spatially separated gain and absorber., Comment: to be published in Physical Review A

Based on a rate equation model for single-mode two-level lasers, two algorithms for stochastically simulating the dynamics and steady-state behaviour of micro- and nanolasers are described in detail. Both methods lead to steady-state photon numbers and statistics characteristic of lasers, but one of the algorithms is shown to be significantly more efficient. This algorithm, known as Gillespie's First Reaction Method (FRM), gives up to a thousandfold reduction in computation time compared to earlier algorithms, while also circumventing numerical issues regarding time-increment size and ordering of events. The FRM is used to examine intra-cavity photon distributions, and it is found that the numerical results follow the analytics exactly. Finally, the FRM is applied to a set of slightly altered rate equations, and it is shown that both the analytical and numerical results exhibit features that are typically associated with the presence of strong inter-emitter correlations in nanolasers.

Nanoscale light sources are being intensively investigated for their potential to enable low-energy, high-density optical communication and sensing systems. Both nano-light-emitting diodes (nanoLEDs) and nanolasers have been considered, based on advanced nanophotonic concepts such as photonic crystals and plasmonic structures, with dimensions well into the sub-micrometer domain. With decreasing dimensions, light-matter interaction becomes stronger, potentially leading to efficient and ultrafast radiative emission, both in the spontaneous and stimulated regime. These features have created wide expectations for the practical prospects of such nanoscale light sources, in particular for optical interconnects. In this article we examine the limits to the downscaling of LEDs and lasers, and ask ourselves which type of source is most suited to ultralow-power optical communications. Based on simple physical considerations on the scaling of spontaneous and stimulated emission rates for semiconductor active regions at room temperature, we analyze the speed and energy limits for nanoLEDs and nanolasers as a function of their size. The role of spontaneous emission enhancement (Purcell effect) in practical nanophotonic sources is also revisited. The main conclusion is that nanoLEDs reach a fundamental energy/speed limit for data rates exceeding a few Gb/s, whereas nanolasers with active dimensions in the range of few 100s nm may enable direct modulation rates larger than 40 Gb/s at power levels adequate for short-distance and low-energy optical interconnects., Comment: 13 pages, 8 figures

The topological lasers, which are immune to imperfections and disorders, have been recently demonstrated based on many kinds of robust edge states, being mostly at microscale. The realization of 2D on-chip topological nanolasers, having the small footprint, low threshold and high energy efficiency, is still to be explored. Here, we report on the first experimental demonstration of the topological nanolaser with high performance in 2D photonic crystal slab. Based on the generalized 2D Su-Schrieffer-Heeger model, a topological nanocavity is formed with the help of the Wannier-type 0D corner state. Laser behaviors with low threshold about 1 $\mu W$ and high spontaneous emission coupling factor of 0.25 are observed with quantum dots as the active material. Such performance is much better than that of topological edge lasers and comparable to conventional photonic crystal nanolasers. Our experimental demonstration of the low-threshold topological nanolaser will be of great significance to the development of topological nanophotonic circuitry for manipulation of photons in classical and quantum regimes., Comment: 15 pages, 4 figures

Plasmonic nanolasers have received a substantial interest for their promising applications in integrated photonics, optical sensing, and biomedical imaging. To date, a room-temperature plasmonic nanolaser, submicron in all dimensions, remains elusive in the visible regime due to high metallic losses. Here, we demonstrate single-particle lasing around 2.3 eV with full-submicron, cesium lead bromide perovskite (CsPbBr3) crystals atop polymer-coated gold substrates at room temperature. With a large number (~100) of devices in total, we systematically study the lasing action of plasmonic test and photonic control groups. The achieved smallest plasmonic laser was 0.56 micrometer x 0.58 micrometer x 0.32 micrometer in size, ten-fold smaller than that of our smallest photonic laser. Key elements to efficient plasmonic lasing are identified as enhanced optical gain by the Purcell effect, long carrier diffusivity, a large spontaneous emission factor, and a high group index. Our results shed light on three-dimensional miniaturization of plasmonic lasers., Comment: 35 pages, 4 main figures, 15 supplementary figures, 2 supplementary tables

The intensity noise of nanolasers and nanoLEDs is analyzed for conventional and quiet pumping of electrons into the active region. It is shown that nanoLEDs with Purcell-enhanced emission rates, enabled by extreme dielectric confinement, may be excellent sources of intensity-squeezed light within a bandwidth of several gigahertz. The physics of intensity squeezing is elucidated and it is shown that the parameter dependence can be explained by an effective Fano factor., Comment: 5 pages, 6 figures

A new approach for analytically solving quantum nonlinear Langevin equations is proposed and applied to calculations of spectra of superradiant lasers where collective effects play an important role. We calculate lasing spectra for arbitrary pump rates and recover well-known results such as the pump dependence of the laser linewidth across the threshold region. We predict new sideband peaks in the spectrum of superradiant lasers with large relaxation oscillations as well as new nonlinear structures in the lasing spectra for weak pump rates. Our approach sheds new light on the importance of population fluctuations in the narrowing of the laser linewidth, in the structure of the lasing spectrum, and in the transition to coherent operation., Comment: 19 pages, 10 figures

Two coupled semiconductor nanolasers exhibit a mode switching transition, theoretically characterized by limit cycle $-$or mode-beating$-$ oscillations. Their decay rate is vanishingly small in the thermodynamic limit, i.e. when the spontaneous emission noise $\beta$-factor tends to zero. We provide experimental evidence of mesoscopic limit cycles $-$with $\sim 10^3$ intracavity photons$-$ through photon statistics measurements. We first show that the order parameter quantifying the limit cycle amplitude can be reconstructed from the mode intensity statistics. As a main result we observe a maximum of the averaged amplitude at the mode switching, accounting for limit cycle oscillations. We finally relate this maximum to a dip of mode cross-correlations, reaching a minimum of $g_{ij}^{(2)}=2/3$, which we show to be a mesoscopic limit. Coupled nanolasers are thus an appealing testbed for the investigation of spontaneous breaking of time-translation symmetry in presence of strong quantum fluctuations.

We report on our initial attempt to characterize the intrinsic frequency response of metal-clad nanolasers. The probed nanolaser is optically biased and modulated, allowing the emitted signal to be detected using a high-speed photodiode at each modulation frequency. Based on this technique, the prospect of high-speed operation of nanolasers is evaluated by measuring the D-factor, which is the ratio of the resonance frequency to the square root of its output power (fR/P_out^2). Our measurements show that for nanolasers, this factor is an order of magnitude greater than that of other state-of-the-art directly modulated semiconductor lasers. The theoretical analysis, based on the rate equation model and finite element method simulations of the cavity is in full agreement with the measurement results., Comment: Optics Express

Using a fully stochastic numerical scheme, we investigate the behaviour of a nanolaser in the low-coherence regime at the transition between spontaneous emission and lasing under the influence of intensity feedback. Studying the input-output curves as well as the second order correlations for different feedback fractions, we obtain an insight on the role played by the fraction of photons reinjected into the cavity. The interpretation of the observation is strengthened through the comparison with the temporal traces of the emitted photons and with the radiofrequency power spectra. The results give insight into the physics of nanolasers as well as validate the use of the second order autocorrelation as a sufficient tool for the interpretation of the dynamics. This confirmation offers a solid basis for the reliance on autocorrelations in experiments studying the effects of feedback in nanodevices.

The ongoing miniaturization of semiconductor lasers has enabled ultra-low threshold devices and even provided a path to approach thresholdless lasing with linear input-output characteristics. Such nanoscale lasers have initiated a discourse on the origin of the physical mechanisms involved and their boundaries, such as the required photon number, the importance of optimized light confinement in a resonator and mode-density enhancement. Here, we investigate high-$\beta$ metal-clad coaxial nanolasers, which facilitate thresholdless lasing. We experimentally and theoretically investigate both the conventional lasing characteristics, as well as the photon statistics of the emitted light. While the former lacks adequate information to determine the threshold to coherent radiation, the latter reveals a finite threshold pump power. Our work clearly highlights an important and often misunderstood aspect of high-$\beta$ lasers, namely that a thresholdless laser does have a finite threshold pump power and must not be confused with a hypothetical zero threshold laser., Comment: 7 pages, 4 figures