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

Superfluidity, first discovered in liquid 4He, is closely related to Bose–Einstein condensation (BEC) phenomenon. However, even at zero temperature, a fraction of the quantum liquid is excited out of the condensate into higher momentum states via interaction-induced fluctuations—the phenomenon of quantum depletion. Quantum depletion of atomic BECs in thermal equilibrium is well understood theoretically but is difficult to measure. This measurement is even more challenging in driven-dissipative exciton–polariton condensates, since their non-equilibrium nature is predicted to suppress quantum depletion. Here, we observe quantum depletion of a high-density exciton–polariton condensate by detecting the spectral branch of elementary excitations populated by this process. Analysis of this excitation branch shows that quantum depletion of exciton–polariton condensates can closely follow or strongly deviate from the equilibrium Bogoliubov theory, depending on the exciton fraction in an exciton polariton. Our results reveal beyond mean-field effects of exciton–polariton interactions and call for a deeper understanding of the relationship between equilibrium and non-equilibrium BECs.


Supplementary Note 2: Spatial shape of the condensate at large densities
The high--density, interaction--dominated regime is characterised by a smooth and nearly-homogeneous distribution of the ground state wavefunctions within the area defined by the optical trap. To verify this property, we extracted the real--space density distributions of condensed polaritons, which is reflected in the intensity distribution of the cavity photoluminescence taken along the real--space spectrum at the energy of the condensate, see

Supplementary Note 3: Condensate filtering in real space
To ensure that the collected photoluminescence (PL) originates from the condensate in the trap only and not from polaritons near the pump region, we use a real space filter as shown in Supplementary  Figure  3a. The filter is an iris placed in the intermediate real space (near--field) image plane of the optical setup, centred with respect to the condensate position. The diameter is optimized to minimise the diffraction on its edges and to filter out the signal from the barrier region. All dispersion measurements presented in the main text were performed using this filtering technique. Supplementary Figure  3b shows the real--space spectrum when we combine the real space and the momentum space edge filters. Similarly to the momentum--resolved spectra of Figure  3 of the main text, the dominant emission originates from the condensate at 1.600 meV. More importantly, one can observe that the rest of the emission that constitute the NB and GB states are spread all over the trap and do not come from the barrier region. The signal outside the real space filter is due to diffraction.

Supplementary Figure 3 (a)
Energy integrated PL of the high--density condensate showing the edge of the real space filter (dotted circle) and the outline of the excitation profile (dashed circle). (b) Real space spectrum of the high--density condensate filtered in both real space (dotted lines) and momentum space (edge filter, not shown). The signal outside the real space filter is due to diffraction on the filter edges. Image is saturated and the colour scale is logarithmic.

Supplementary Note 4: Extraction and fitting of the excitation branches
Below we provide details of the fitting procedure of the excitation spectra presented in Figure 3 of the main text. The excitation spectra in momentum space are recorded on the CCD camera, where each pixel column corresponds to a wavevector ∥ . Examples of spectral profile at different finite wavevectors are presented in Supplementary Figures 6a,b and 6d,e. The signal is dominated by the diffracted light from the condensate, which arises from the real space filtering. On the high--energy side, one can observe the photoluminescence of the normal branch (NB), whose spectral lineshape is irregular and is composed of the occupation of many confined excited states of the optically--induced trap. The lineshape of the NB is fit with a Voigt function to extract the intensity and energy from the peak area and centre, respectively, see dispersion of the GB is then fitted with the Bogoliubov spectrum ( ) (see main text, Methods), with the condensate interaction energy = is the only fitting parameter. Examples of the renormalized spectra fits are presented in Supplementary  Figures  4c  and  4f. Note that we only fit the GB dispersion due to the clarity of this data, as the GB states are populated only via quantum depletion. This is unlike the NB that has additional contributions from high--energy states coming from the top of the barrier, which results in an extracted dispersion that deviates from the Bogoliubov prediction, see Figure 3 in the main text. manifested in larger--than--expected slopes and non--zero low--density limit, Δ , of the linear dependencies of the blueshift on density, = + Δ . In the cases presented here, the slopes are: = 0.249 ± 0.003 µμeVµμm ! for the excitonic detuning and = 0.279 ± 0.004 µμeVµμm ! for the photonic detuning. These values are about two times larger than those expected to arise due to polariton--polariton interactions and the slope is larger for the photonic detuning (smaller Hopfield coefficient), which contradicts the expected behaviour of the blueshift, based on the expression of the polariton--polariton interaction energy 8--10 : where ! is the excitonic Hopfield coefficient determining the fraction of an exciton in the exciton--polariton quasiparticle, and ! = 6 ! ! ! is the exciton--exciton interaction constant 9 , with ! and ! denoting the exciton binding energy and Bohr radius, respectively. The factor of 2 comes from the fact that the condensate is linearly polarized (i.e., has an equal mixture of two spin components), and the interaction of excitons with the opposite spin is negligible. The total interaction constant for total polariton density distributed amongst all quantum wells should be divided by !" .
The correct polariton--polariton interaction constants, , were extracted from fitting the ghost branch of the Bogoliubov dispersion at various detunings and densities, using the interaction where , ! are the condensate and reservoir densities, , ! are the respective radiative decay rates, is the rate of stimulated scattering into the condensate mode, and is the rate of is a guide to the eye for the power law ∥ !! . One can observe that the power--law decay is also present in the raw data. Shaded zones represent error bars of the occupation numbers extraction taking into account fitting errors of the spectra at a given ∥ .

Supplementary Note 11: Density dependence of the contact
As described in Methods section of the main text, the direct measurement of the GB occupation in momentum space allows one to extract the value of the Tan's contact from the !" ∝ !! dependence at large --vectors (henceforth denotes the in--plane momentum ∥ ).
Assuming the local density approximation (LDA) in the middle of the trap, where the condensate density is a smooth function, the proportionality coefficient should depend quadratically on the peak density ∝ ! . Verification of this relation is presented in Supplementary Figure 10 at excitonic fraction of ! ≈ 0.56. The peak density has been extracted from real--space spectra in the middle of the trap, to avoid averaging of the density with the periphery of the condensate. We note that the contact ! measured for an atomic BEC, e.g., in Ref. 7 , is defined as the limit of density distribution in momentum space rather than the occupation number. In a two--dimensional quantum gas, it is therefore related to the quantity defined above as ! = / 2π ! , where is the area of the condensate in real space. Both ! and exhibit quadratic dependence on the peak density in the LDA.
Supplementary Figure 10 Experimentally determined values of the contact !"# as a function of the peak polariton density determined from the real space PL spectrum. Dashed line indicates a quadratic fit to the experimental data. Error bars are determined based on the fitting to experimental data and taking into account the estimation of absolute error for the polariton density value.
In certain regions of system parameters, spatially homogeneous condensates created in the CW regime were predicted to exhibit dynamical (modulational) instability, which is driven by fluctuations of the reservoir 11,12 . This instability was experimentally observed in large, quasi--1D condensates 13 . Analytical estimate obtained using the open--dissipative Gross--Pitaevskii model 11 yields the instability domain for the homogeneous pump rate (i.e. the rate of reservoir replenishing): where the threshold pump rate is defined as !" = ! / (other parameters are defined in Supplementary Note 5). The boundary of the instability region can be expressed as a function of the excitonic Hopfield coefficient: where we have assumed that the decay rate of highly excitonic reservoir particles is equal to that of excitons, and ! is the decay rate of cavity photons. For our long--lifetime system, the ratio ! ! ≈ 10 , and the corresponding instability domain is shown in Supplementary  Figure   11a When probing the elementary excitations of the condensate in a large momentum limit, it is important to understand to what extent the non--parabolicity of the lower polariton dispersion, !" ( ), which may become prominent in this limit, modifies the expressions for the energy of the Bogoliubov excitations used in the main text. We recall that the energy of the lower polariton branch, derived from the standard coupled exciton--photon model 10 , takes the form: where ! ( ) = ! 0 + ℏ ! ! 2 ! is the cavity photon dispersion, ! is the effective mass of the cavity photon, ! is the exciton energy (assumed constant), Δ = ! − ! is the exciton--photon detuning ranging from positive (excitonic) to negative (photonic), and ! = 2ℏΩ is the Rabi splitting at zero detuning.
Taking into account the polariton energy in momentum space, !" , the energy of the elementary excitations is expressed as follows: where !" = !" − !" 0 , and = . We note that the standard expression for the Bogoliubov dispersion in equilibrium (Eq. (2) in the main text, Methods) is recovered from Eq.
(3) by replacing the non--parabolic kinetic energy of the lower polariton !" with the parabolic (effective mass) approximation !" → = ℏ ! ! 2 , where is the effective mass of the lower polariton, as defined in the main text. The crossover wavevector, which is defined by the healing length as the value of ! = !! , at which the transition from the phonon to the free--particle behaviour of elementary excitations occurs, i.e. !" ! = . Just as in the case of massive bosons, the large wavevector limit is then defined as ≫ ! , or equivalently !" ≫ .
The expression for the amplitude of the elementary excitations can then be written as (4) and the occupation of the ghost branch is: where the ratio !" defines the ratio of the wavevectors !
for the case of non--parabolic LP dispersion. In the limit of large momenta, !" ≫ , one can expand Eq.
(5) in the powers of the small parameter !" ≪ 1. Omitting the higher--order terms allows us to obtain the asymptotic behaviour at large momenta: This function tends to a non--zero, constant value at very large momenta: which differs from the corresponding behaviour for a massive boson: where | | ! = 1 − ! is the photonic Hopfield coefficient (photon fraction) at = 0. However, for the range of the wavevectors and detunings probed in the experiment, the discrepancy between the asymptotic behaviour of !" and !" !" given by Eqs. (6) and (8) We note that the departure of !" !" from ∝ !! asymptote at large momenta due to the non--parabolicity of the polariton dispersion cannot explain the observed behaviour for photonic detunings, since !"

!"
⟶ !" for ! ! ! > 1 as | | ! ⟶ 1. As shown in Supplementary Figure 13, the fit to the ∝ !! asymptote should improve for photonic detunings, which is contrary to what is observed in the experiment (cf. Figure 5e in the main text, the highest density data shown in light green). As noted in Methods of the main text, this effect also does not affect the fitting of the experimentally measured ghost branch presented in Supplementary Note 4, since the actual, experimentally measured lower polariton dispersion !" was used in the fitting, rather than its parabolic (effective mass) approximation.
Supplementary Figure  12 Comparison between the asymptotic behavior of !" !" given by Eq.