Observation of quantum depletion in a non-equilibrium exciton-polariton condensate

The study investigates whether quantum depletion—a beyond mean-field effect predicted by Bogoliubov theory for interacting Bose-Einstein condensates—can be directly observed in driven-dissipative exciton-polariton condensates, and how non-equilibrium conditions modify its signatures. In equilibrium BECs, interactions produce a phonon-like excitation spectrum and a finite occupation of elementary excitations at zero temperature, leading to a momentum distribution with a k^-4 tail at large k. Measuring this in atomic gases is challenging due to time-of-flight distortions. Exciton-polariton condensates offer direct optical access to energy- and momentum-resolved spectra via photoluminescence, but their finite lifetime and continuous pumping drive them out of equilibrium and can suppress or alter quantum depletion, particularly affecting the excitation spectrum (including a possible Goldstone mode modification and the presence of a negative-energy ghost branch). The purpose of the study is to generate a steady-state, single-mode, high-density polariton condensate and determine whether the ghost branch (GB), populated solely by quantum fluctuations, can be detected and analyzed to quantify depletion, interactions, and the crossover between equilibrium-like and non-equilibrium regimes as a function of exciton-photon detuning (polaritonic matter fraction).

Bogoliubov’s theory (1947) established how interactions modify BEC excitation spectra, producing two Bogoliubov branches and quantum depletion. Quantum depletion has been observed in weakly interacting ultracold atomic BECs, and accounts for reduced condensed fraction in strongly interacting liquid ⁴He. Equilibrium theory predicts a momentum occupation N(k) ~ k^-4 at large k. In polariton systems, photoluminescence grants direct access to dispersion and occupations, but their driven-dissipative nature modifies spectra (gapped or flat Goldstone modes) and can suppress quantum depletion signatures. Prior polariton experiments probed excitation spectra using resonant pump-probe schemes or intense incoherent driving, often forcing GB population via defects or resonant beams and not in spontaneous steady-state configurations. Linear Bogoliubov spectra in spontaneous condensates have been observed, but only the normal branch (NB) was visible, dominated by thermal excitations scaling as k^-2 at low k. Theory has proposed that non-equilibrium effects reduce GB visibility. Recent quantum interaction effects in polaritons were shown at the single-particle level via correlation measurements in confined systems. This work addresses the gap by seeking spontaneous, steady-state observation of GB without external forcing, enabling direct assessment of quantum depletion in non-equilibrium condensates.

Sample and cryogenics: An ultrahigh-Q GaAs-based microcavity with cavity photon lifetime >100 ps was used. The 3λ/2 cavity comprises 32/40 pair AlGaAs/AlAs DBRs and 12 GaAs/AlAs QWs (7 nm) in three groups at photon field maxima. Rabi splitting ΔΩ = 15.9 ± 0.1 meV; exciton energy Ex = 1.6062 eV; cavity photon effective mass ≈ 3.6 × 10^-5 me. Experiments were performed at 7–8 K in a continuous-flow helium cryostat. Pumping and trapping: Non-resonant CW Ti:Sapphire laser (≈719 nm, reflectivity minimum) chopped at 10 kHz with 5% duty cycle to reduce heating. An axicon-based confocal setup created a ring-shaped (annular) excitation profile, producing an incoherent exciton reservoir ring that both provided gain and generated an optical trapping potential. The condensate formed within the ring, spatially separated from the reservoir. Optical collection and filtering: Emission was collected with NA = 0.5 objective and imaged through a four-lens confocal system to access near- and far-field planes. Image filtering employed an iris in real space (blocking reservoir/barrier luminescence) and a movable razor-blade edge in momentum space to block the intense condensate signal up to k ≈ 0.55 µm^-1, revealing weaker excitation branches. Emission was dispersed by a monochromator grating and detected on a high-efficiency EMCCD. Operating regimes and detunings: By tuning the sample position along the wedge, two representative detunings were studied: Δ = +1.8 meV (excitonic) and Δ = −3.7 meV (photonic), corresponding at k=0 to Hopfield exciton fractions |X|^2 = 0.56 and 0.39, respectively. Pump power was varied to access below-threshold, multimode condensation, and a high-density single-mode Thomas-Fermi regime with homogeneous density in the trap. Spectral analysis: Energy- and momentum-resolved photoluminescence was measured. At intermediate densities, spectra resembled single-particle dispersions with thermal NB occupation characterized by an effective temperature (extracted by fits). At high density, after k-space edge filtering, both NB and GB became visible. Dispersions of NB and GB were fitted using the equilibrium Bogoliubov relation ε(k) = sqrt(E(k)[E(k) + 2gn]) with a single fit parameter µ = gn (interaction energy), where E(k) was taken from the measured lower polariton dispersion ELP(k) − ELP(0). Driven-dissipative corrections (Wouters-Carusotto model) were evaluated but neglected at the large k probed owing to long polariton lifetimes (ħΓ ≈ 3–5 µeV). Occupation extraction: Momentum-space occupation numbers N(k) were obtained by integrating the calibrated photon count rate, accounting for system collection efficiency η (calibrated with a reference laser), polariton lifetime τp(k) (from Hopfield coefficients and cavity photon lifetime), spin degeneracy, momentum-space pixel volume, duty cycle D of the chopper, and real-space filter area A. This provided NB and GB occupation distributions over k > ξ^-1, where ξ = ħ/√(2mgn) is the healing length (reported as inverse ξ values for datasets). Interaction strength extraction: From µ versus density n, g was obtained as slope, and rescaled per quantum well. Data were compiled across detunings to test g ∝ |X|^4 gx and compared with independent pulsed-regime measurements in the Thomas-Fermi limit.

- Direct observation of the ghost branch (GB) photoluminescence in a spontaneously formed, steady-state, non-resonantly pumped exciton-polariton condensate, evidencing quantum depletion due to interaction-induced quantum fluctuations. - Dispersions: NB and GB branches were resolved after momentum-space edge filtering. Fits to Bogoliubov dispersion with a single parameter µ = gn captured the renormalized spectra at both excitonic and photonic detunings. Example conditions: Δ = −3.7 meV at n ≈ 1340 µm^-2 and Δ = +1.8 meV at n ≈ 1848 µm^-2. - Interaction energy and strength: Extracted chemical potential µ scaled linearly with density (µ = gn). The polariton-polariton interaction strength per quantum well followed the expected quadratic dependence on exciton fraction: g ∝ |X|^4 gx. The inferred exciton-exciton interaction constant was gx = 13.5 ± 0.6 µeV µm^2, in excellent agreement with independent pulsed-regime measurements. - Normal branch (NB) occupations showed non-equilibrium features: at excitonic detuning and lower densities, thermal-like tails with effective temperatures Teff ≈ 10 K (Fig. 2b) and ≈30 K (guide in Fig. 5b) were observed; at highest densities, occupation shifted to excited trap states and high-k population due to inefficient relaxation and reservoir-origin high-energy polaritons, especially pronounced for photonic detuning. - Ghost branch (GB) occupations exhibited detuning-dependent scaling: at excitonic detuning (|X|^2 ≈ 0.56), NGB(k) displayed the equilibrium-like k^-4 decay at large k and increased ∝ n^2, consistent with Bogoliubov theory and Tan’s contact behavior. At photonic detuning (|X|^2 ≈ 0.39), NGB(k) deviated from k^-4 with increasing density, tending to a plateau approaching k^-1, indicative of strong non-equilibrium, reservoir-mediated fluctuations. - Additional quantitative notes: polariton lifetime >100 ps (ħΓ ~ 3–5 µeV, negligible at measured k); condensate fraction at photonic detuning capped around ρ ≈ 0.5 as pump increased; healing-length-related crossover momenta reported per dataset (e.g., ξ^-1 ≈ 0.33–0.73 µm^-1).

The detection of the GB in photoluminescence confirms quantum depletion in a driven-dissipative polariton condensate without resorting to resonant forcing or defect scattering, directly revealing beyond mean-field fluctuations. The analysis demonstrates a crossover in fluctuation behavior controlled by detuning (and hence exciton fraction): for more excitonic polaritons, the GB occupation follows equilibrium Bogoliubov predictions (k^-4 tail and n^2 scaling), implying near-equilibrium quantum fluctuations even as NB shows non-equilibrium features. For more photonic polaritons, deviations from equilibrium arise in the GB occupation, trending toward a k^-1-like behavior, consistent with theories where reservoir-condensate coupling dominates fluctuations and polariton-polariton interactions are weaker. The linear µ(n) and g(|X|)^4 scaling validate the interaction model and provide a robust measurement of interaction strengths in steady-state CW operation. Together, these results bridge equilibrium and non-equilibrium condensate physics, highlighting the significant role of reservoir dynamics and opening a path to quantify universal quantities (e.g., Tan’s contact) in polariton systems when in the equilibrium-like regime.

This work provides the first direct observation of quantum depletion in a spontaneous, steady-state exciton-polariton condensate by resolving the ghost branch of Bogoliubov excitations. By fitting excitation dispersions, the study quantifies interaction energies and confirms the expected scaling of polariton-polariton interactions with exciton fraction, yielding an exciton-exciton interaction constant gx = 13.5 ± 0.6 µeV µm^2. The momentum-space occupations reveal an equilibrium-like k^-4 GB tail at excitonic detuning and a crossover to non-equilibrium reservoir-dominated behavior at photonic detuning. These findings establish polariton condensates as a platform for probing many-body quantum fluctuations beyond mean-field and motivate theoretical advances to describe quantum depletion across the equilibrium to far-from-equilibrium crossover. Future work should develop comprehensive driven-dissipative theories of depletion, elucidate detailed population pathways of NB and GB, measure Tan’s contact in polariton condensates, and control reservoir-condensate coupling to tune fluctuation regimes.

- The analysis relied on an equilibrium Bogoliubov dispersion approximation; while justified at large k due to long lifetimes (small ħΓ), full driven-dissipative effects may alter low-k behavior not accessible here. - The experimental dynamic range necessitated momentum-space edge filtering and saturation of images; very low-k regions (k → 0) were blocked or dominated by the condensate, preventing direct access to the small-k asymptotics and precise contact extraction in this dataset. - In CW operation, incomplete depletion of the exciton reservoir raises the effective zero-point energy, precluding a direct interaction-strength measurement via condensate blueshift and requiring dispersion-based extraction. - Non-parabolicity of the polariton dispersion introduces small deviations from ideal k^-4 scaling, though estimated to be minor over the accessible k-range. - Reservoir effects are significant at photonic detuning, leading to non-thermal, non-equilibrium occupations; the precise reservoir dynamics were not directly measured and require further modeling and diagnostics. - Momentum distributions were characterized mainly for k > ξ^-1; occupations at lower momenta and trap-state details are influenced by confinement and were not the focus of the depletion analysis.