The understanding of interacting cold bosonic gases was significantly advanced by Bogoliubov's theory, which predicted the effects of interparticle interactions on Bose-Einstein Condensates (BECs). This theory describes a modified phonon-like dispersion of elementary excitations at long wavelengths, explaining the superfluidity of weakly interacting BECs. A crucial aspect is the Bogoliubov transformation of quasiparticles, where a condensate excitation is a superposition of counter-propagating single-particle states. Quantum fluctuations in the ground state lead to non-zero occupation of these excitations at zero temperature, causing quantum depletion of the condensate population. While quantum depletion has been observed in weakly interacting BECs of ultra-cold atoms and is crucial for understanding strongly interacting superfluid ⁴He, measuring it is experimentally challenging, especially in driven-dissipative exciton-polariton condensates. These condensates, formed in semiconductor microcavities, are inherently non-equilibrium systems due to the finite lifetime of the photon state, requiring constant pumping. This non-equilibrium nature is predicted to suppress quantum depletion. Exciton-polariton condensates, however, retain superfluid properties, although their features differ from equilibrium BECs. The Bogoliubov excitation spectrum features two branches: a positive (normal branch-NB) and a negative (ghost branch-GB). The GB is populated solely by quantum depletion, making it a direct probe of beyond mean-field effects. Previous attempts to measure the full excitation spectrum often involved non-spontaneous condensate creation or forced GB population, hindering observation of interaction-driven quantum effects. This study aims to overcome these limitations by creating a steady-state, high-density condensate and directly observing quantum depletion in the excitation spectra.

Bogoliubov's theory provides the foundation for understanding interacting BECs, predicting a modified phonon-like dispersion of elementary excitations and explaining superfluidity. Experimental observation of quantum depletion in ultra-cold atomic BECs has been reported, showcasing the non-zero occupation of elementary excitations even at zero temperature due to quantum fluctuations. However, these measurements face challenges due to the time-of-flight measurement affecting momentum distribution. Exciton-polariton condensates, offering direct measurement of momentum space distribution, present a different avenue. Their driven-dissipative nature, stemming from the finite lifetime of the photon state and the need for continuous pumping, introduces complexities. While maintaining superfluid properties, these non-equilibrium condensates are expected to exhibit different behavior compared to their equilibrium counterparts. Theoretical work has explored the excitation spectrum of these condensates, including the prediction of a gapped mode or flat Goldstone mode at low energies, gradually transitioning to Bogoliubov dispersion at larger wavevectors. The emergence of two branches in the Bogoliubov spectrum, a normal branch and a ghost branch, is noteworthy. The ghost branch, populated solely by quantum depletion, serves as a critical indicator of beyond-mean-field effects. Previous research has demonstrated quantum effects in polariton interactions at the single-particle level but has yet to provide a clear picture of the full excitation spectrum of spontaneously formed condensates in a steady-state. The difficulty in observing the ghost branch stems from the non-equilibrium effects that suppress its signal. Therefore, understanding the interplay between equilibrium and non-equilibrium behavior in exciton-polariton condensates is a significant challenge.

The experiment utilized a high-quality GaAs-based microcavity with a long cavity photon lifetime exceeding 100 ps. A continuous-wave (CW) Ti:Sapphire laser provided non-resonant excitation, creating a ring-shaped distribution of incoherent excitonic reservoir particles. This configuration provided both gain and a trapping potential for exciton polaritons. At high pump power, a single-mode condensate formed within the trap. Spatial separation of the pump and condensate allowed filtering of photoluminescence originating from regions with significant reservoir overlap. The experiment explored various detunings (Δ = Ec − Ex) between cavity photon and exciton energies to test equilibrium and non-equilibrium features, altering the exciton fraction of the polaritons. Two representative detunings, positive (+1.8 meV, excitonic) and negative (−3.7 meV, photonic), were examined. Momentum space photoluminescence was measured to analyze condensate excitations. High-density regimes revealed a strong ground state signal, and momentum space filtering was used to reveal weaker excitation branch signals. The excitation branches were fitted to extract condensate interaction energy (chemical potential) as a function of density, yielding polariton-polariton interaction strengths. Momentum occupation distributions (N(k)) were determined for both the normal and ghost branches, analyzed in the long wavevector range (k > 1). The Bogoliubov dispersion was fitted to the data using the condensate interaction energy (μ = gn) as a fitting parameter. The analysis considered the full polariton dispersion, going beyond a parabolic effective mass approximation. The occupation numbers of polaritons were calculated considering the collection efficiency of the experimental setup, polariton lifetime, spin degeneracy, and momentum space volume subtended by each pixel. The asymptotic behavior of the ghost branch occupation distribution, particularly the k⁻⁴ decay at large wavevectors, was used to examine the approach to the equilibrium predictions and the validity of the Bogoliubov theory. The effect of the non-parabolic dispersion on the momentum distribution was also investigated.

The study successfully created a steady-state, high-density exciton-polariton condensate in a high-quality GaAs-based microcavity. Direct observation of quantum depletion was achieved through the detection of the ghost branch in the condensate excitation spectrum. The interaction energy (chemical potential) of the condensate showed the expected linear dependence on density (μ = gn), allowing extraction of the polariton-polariton interaction strength. This strength exhibited a quadratic dependence on the exciton fraction, consistent with theoretical predictions. Analysis of momentum occupation distributions revealed a crossover behavior. At excitonic detunings, the ghost branch occupation closely followed the k⁻⁴ decay predicted by the equilibrium Bogoliubov theory, indicating near-equilibrium behavior. However, at photonic detunings, deviations from equilibrium were observed, showing a transition from k⁻⁴ to a k⁻¹ distribution. This suggests that at photonic detunings, reservoir-condensate interactions play a dominant role, while polariton-polariton interactions are less significant. The normal branch displayed non-equilibrium characteristics at both detunings, showing populations of high-k states possibly linked to inefficient energy relaxation and polariton-to-reservoir upconversion. The observed differences in ghost branch occupation between excitonic and photonic detunings highlight the non-equilibrium nature of the system and its departure from the simple Bogoliubov picture. These deviations are consistent with non-equilibrium theories suggesting the importance of reservoir fluctuations. The non-parabolic dispersion of polaritons was accounted for in the analysis, confirming that the observed deviations from k⁻⁴ were not due to this effect. The study paves the way for measurement of Tan's contact for exciton-polariton condensates, allowing investigation of universal thermodynamic properties.

The results directly demonstrate quantum fluctuations and interparticle interactions in a many-body driven-dissipative BEC. A transition from a near-equilibrium regime (excitonic detunings) to a fully non-equilibrium regime (photonic detunings) was observed, emphasizing the influence of reservoir fluctuations. This behavior extends beyond current theoretical understanding, necessitating further development of theories describing quantum depletion in the crossover from equilibrium to far-from-equilibrium conditions. The detailed mechanisms populating both the normal and ghost branches in the non-equilibrium regime require further investigation. In the near-equilibrium regime, the study opens avenues for measuring Tan's contact, providing insights into the universal thermodynamic properties of systems with contact interactions.

This study presents a direct observation of quantum depletion in a non-equilibrium exciton-polariton condensate, revealing a crossover from near-equilibrium to far-from-equilibrium behavior depending on the exciton fraction. The findings highlight the limitations of current theoretical understanding and necessitate the development of more comprehensive models for non-equilibrium condensates. Future research should focus on refining theoretical descriptions of quantum depletion in this crossover regime and investigating the detailed mechanisms populating the excitation branches. Measurement of Tan's contact is suggested as a next step, potentially revealing universal thermodynamic properties.

The study's interpretation relies on approximations, such as using the equilibrium Bogoliubov dispersion to fit non-equilibrium data. While the non-parabolic nature of the polariton dispersion was considered, other effects not accounted for in the model might introduce small uncertainties. The experimental setup might not perfectly capture all aspects of the complex many-body dynamics in the exciton-polariton condensate. Furthermore, the interpretation of the high-k populations relies on theoretical conjectures and could benefit from further validation through other methods.