Learning the Fuzzy Phases of Small Photonic Condensates

Phase transitions, being the ultimate manifestation of collective behavior, are typically features of many-particle systems only. Here, we describe the experimental observation of collective behavior in small photonic condensates made up of only a few photons. Moreover, a wide range of both equilibrium and nonequilibrium regimes, including Bose-Einstein condensation or laserlike emission are identified. However, the small photon number and the presence of large relative fluctuations places major difficulties in identifying different phases and phase transitions. We overcome this limitation by employing unsupervised learning and fuzzy clustering algorithms to systematically construct the fuzzy phase diagram of our small photonic condensate. Our results thus demonstrate the rich and complex phase structure of even small collections of photons, making them an ideal platform to investigate equilibrium and nonequilibrium physics at the few particle level.

Figure 1

Small photonic condensates: (a) Schematic of our multimode dye-filled microcavity. The small radius of curvature induces a tight harmonic potential, whose eigenmodes are depicted in colors. The resulting large mode spacing highly suppresses the thermal excitation of highly excited modes. Because of transverse radial symmetry, nondegenerate photonic modes are labeled by a single quantum number $m$. Their spatially dependent profiles and the resulting mode-mode competition can bring the system into complex nonequilibrium states exhibiting collective behavior even at low photon numbers. (b) Illustration of the effect of the system size, quantified by the parameters ${\beta }_m$, on the criticality of the lasing phase transition, in the limit of negligible mode-mode coupling. Small systems, ${\beta }_m\to 1$, are shown to exhibit broad transitions, while pure criticality is recovered in thermodynamic limit, ${\beta }_m^{-1}\to \infty$.

Figure 2

Occupation numbers: (a)–(c) Absolute and relative occupation numbers of the first four photonic modes, for different thermalization regimes. The latter is experimentally tuned by sweeping the cavity cutoff wavelength ${\lambda }_0$. Each set of occupation numbers defines a configuration. The dashed vertical lines mark transitions between different representative phases, which are labeled as in the text. (d)–(f) Results from the nonequilibrium model of photonic condensation.

Figure 3

Fuzzy phase diagram. (a) Total photon number as a function of the pump power and thermalization coefficient. (b) Projection of the full ten-dimensional feature space onto the two-dimensional plane spanned by the relative occupation of the ground and the first excited state, after the fuzzy clustering procedure being applied. Different representative phases are shown by different colors, with a transparency level proportional to the respective membership entropy. (c) Estimation of the number of phases by minimizing the average membership entropy, whose distribution is approximated by bootstrapping the feature space. The resulting standard deviation is depicted by the dashed gray area. This ensures the stability of this procedure. (d) Representative phase diagram. (e) Membership entropy. The fuzzy phase diagram in (f) is constructed from the representative phase diagram in (d) by assigning a transparency level proportional to the membership entropy in (e). In this way, regions of larger phase ambiguity becoming white, thus visually gauging the degree of representativeness of the representative phase diagram. The dashed vertical lines indicate the traces depicted in Fig. 2.

Figure 4

Fuzzy and sharp photonic condensates. (b) Fuzzy phase diagram obtained from the nonequilibrium model of photonic condensation, with parameters that match the experiment. As described earlier, the model slightly deviates from the experiment at weak thermalization conditions, where it predicts the condensation of the third excited cavity mode, the L3 phase. At intermediate thermalization rates, we also infer the presence of a multimode (MM) phase, characterized by the condensation of both the ground and the first excited state. This is, however, a relatively narrow and fuzzy phase which is not resolved in the experiment. The theoretical model is also used to generate configurations for both smaller (a) and larger (c) photonic condensates, quantified by the fraction of spontaneous emission into the ground state. The thermodynamic limit is approached from left to right.