Thermalization of a two-dimensional photon gas in a polymeric host matrix

We investigate thermodynamic properties of a two-dimensional photon gas confined by a dye-filled optical microcavity. A thermally equilibrated state of the photon gas is achieved by radiative coupling to a heat bath that is realized with dye molecules embedded in a polymer at room temperature. The chemical potential of the gas is freely adjustable. The optical microcavity consisting of two curved mirrors induces both a non-vanishing effective photon mass and a harmonic trapping potential for the photons. While previous experiments of our group have used liquid dye solutions, the measurements described here are based on dye molecules incorporated into a polymer host matrix. We describe studies of fluorescence properties of dye-doped polymers, and discuss the applicability of Kennard-Stepanov theory in this system. We observe a thermalized two-dimensional photon gas in the solid state based microresonator system. In the future, dye-based solid state systems hold promise for the realization of single-mode light sources in thermal equilibrium based on Bose-Einstein condensation of photons, as well as for solar energy concentrators.


Introduction
Since the first experimental realization of Bose-Einstein condensation, much effort has been expended in further, extensive examination of this macroscopic quantum state of matter [1][2][3][4]. This includes the extremely successful area of Bose-Einstein condensates realized with dilute, ultracold atomic gases [5]. More recently, condensation has also been reported with solid state quasiparticles, namely exciton-polaritons and magnons [6][7][8][9][10]. Exciton-polaritons arise from bound electron-holepairs in semiconductors, which are strongly coupled to a light field within an optical resonator.
Due to their short lifetime of typically 10 −12 s, a thermalization by polariton-polariton scattering processes is usually only observed at polariton densities above the condensation threshold [6].
Two recent works using GaAs based microcavities have reported a crossover from an excitonpolariton to a photon-like lasing regime, the latter featuring a thermal wing coexisting with the lasing mode [11,12]. Both experiments have been carried out not far from the strong coupling regime, but nevertheless would be consistent with a thermalization mechanism that differs from interparticle scattering between the material parts of the coupled matter-light states.
We here consider thermalization of a photon gas by means of repeated absorption-fluorescence processes of dye molecules embedded in a polymer within an optical microcavity in a regime of strong decoherence. Utilizing this fluorescence-induced thermalization mechanism, a Bose-Einstein condensation of pure photons in thermal equilibrium, both below and above the phase transition, could be recently observed in our group [13]. Our earlier series of experiments was based on dye molecules in liquid solution [13,14].
The used dye molecules obey the Kennard-Stepanov law, which states that the frequencydependent ratio of the absorption and fluorescence strength corresponds to a Boltzmann-function [15,16]. Such a relationship has been verified for dyes in liquid solutions, and also in the luminescence spectra of semiconductors and in vapor phase experiments [17][18][19][20]. In our previous work, a liquid dye solution was placed between the mirrors of a high-finesse optical cavity. The cavity, with a mirror spacing in the micrometer regime, establishes a low-frequency cut-off for the photons in the optical regime and a non-vanishing effective photon mass. Inside the resonator, the photon dispersion acquires particle-like (quadratic) character, which along with an effective trapping potential induced by the curved mirrors makes the system formally equivalent to a twodimensional gas of trapped, massive bosons. The photons thermalize to the temperature of the dye solution by repeated absorption and emission processes, fulfilling a detailed balance condition [21].
In this system, we have observed a Bose-Einstein condensation of photons at room temperature [13].
These findings raise the question whether such an experimental scheme is also feasible in solid state environments rather than in liquid solutions promising an enhanced practicability of the setup. The occurrence of a photon thermalization is a priori uncertain, as the fluorescence characteristics of the dye molecules can be altered by changing solvents or host materials, including condensed matter substances in particular [22,23]. In liquid solution, it is well known that for many dyes the rovibrational sublevels of both the ground and electronically excited manifold are occupied according to a Boltzmann-distribution at the temperature of the solvent [15,16,24]. This rovibrational equilibration is caused by frequent collisions between dye and solvent molecules, which occur many times (typically ∼ 1000 at room temperature) during one absorption and fluorescence cycle of a molecule [25]. In this way the equilibrated dye molecules imprint their thermalized state onto the photons simply by emission of fluorescence, as is discussed for the polymer-based system in greater detail in chapter 4. Furthermore, we emphasize that the system under observation is in the weak coupling regime due to the collision-induced decoherence, which annihilates nearly all correlations between absorbed and emitted photonic states after a single absorption-emission process, as stated by Kasha's rule [26].
In the present work, dye molecules are embedded in an amorphous polymer at room temperature, which exhibits a high transmission in the visible spectrum. In such a host material rapid decoherence and vibronic equilibration on a femtosecond timescale is achieved by dissipation of vibrational energy of the molecules into the phonon bath of the polymer [27]. We demonstrate the validity of the Kennard-Stepanov law for dye molecules in the polymer material by deriving the spectral temperature from the dye spectra. For this purpose, studies of absorption and emission spectra as well as the fluorescence quantum yield of the chromophores are carried out and compared to the spectroscopic properties of liquid dye solutions. Furthermore, we observe experimental evidence for a thermalized two-dimensional photon gas at the host temperature in the microcavity.
According to the spectral distribution of the emitted radiation, the temperature of the photon gas is found to well agree with a room temperature spectrum above the low-frequency cavity cut-off.
In addition, evidence for a thermalization of the photon gas is obtained from the observed spatial redistribution of fluorescence into the trap center, where the effective trapping potential induced by the mirror curvature exhibits a minimum value. At high average cavity photon numbers, a peak in the spatial intensity distribution is observed, which could be a hint to a Bose-Einstein-condensate.
Due to fast and irreversible photobleaching of the dye molecules a more detailed investigation is precluded. Thus, no quantitative proof of a phase transition can presently be given.
In the following, chapter 2 describes the used experimental setup, while chapter 3 presents measurements on the fluorescence properties of dye molecules embedded in polymer films. Further, chapter 4 contains measurements of the thermalized photon gas in the microcavity and discusses the fluorescence-induced thermalization process. Finally chapter 5 gives conclusions.

Experimental scheme
The used experimental setup, as shown in figure 1, consists of an optical microcavity filled with a dye-doped polymer, a pump source and both a charge-coupled device (CCD) as well as a spectrometer used for analysis of the emerging cavity light.
We ensure small photon loss rates by utilizing a high-finesse optical microcavity. It is formed by two Bragg mirrors (surface area 1 × 1 mm 2 ) with a radius of curvature of R = 1 m and a reflectivity above 99.997% in the spectral range from 500 − 590 nm, yielding a finesse of F ≃ 10 5 for the empty cavity. One mirror is mounted onto a voltage controlled piezoelectric crystal, allowing for an initial precise tuning of the mirror separation on the optical axis to the desired value of D 0 ≈ 1.64 µm. This yields a free spectral range of two adjacent longitudinal modes The used dyes are either rhodamine 6G (R6G) or perylene diimide (PDI). In liquid solvents these dyes show a fluorescence quantum yield close to unity, Φ R6G ≃ 0.95 and Φ PDI ≃ 0.97 [28,29].
We assume that the rovibrational occupation of the molecular bands is thermally equilibrated by their contact to the phonon bath of the polymer host matrix, acting as a heat bath at room temperature. For a thermalized distribution of rovibrational levels both in the ground and electronically excited state of the dye molecule, it can be shown that by absorption and emission the photon gas thermalizes to the dye temperature [21]. The thermalization process is discussed more precisely along with experimental results in chapter 4.
The fluorescence bandwidth ∆λ dye of both dyes is in the order of 80 nm and therefore comparable to the free spectral range. We find that photons in good approximation are only emitted into transversal resonator modes TEM 9mn with quantum numbers (9, m, n), where the longitudinal degree of freedom is frozen out. On the one hand, excitation of photons into energetically higher modes with q ′ = 10 is suppressed due to insufficient bandwidth of the dye.
On the other hand, we do not observe considerable emission into highly excited transversal modes TEM 8m'n' with the same energy as TEM 9mn modes. We attribute this to their higher mode volume and consequently smaller overlap with the emitting molecular dipoles, which are predominantly located in the center of the resonator (see figure 2). Hence, fluorescence into modes with q = 9 is intrinsically preferred by our system.
The cavity dispersion relation as a function of the transverse momentum in the paraxial approximation (r ≪ R, k r ≪ k z (r) = qπ/D(r)) reads where m ph =hqπn 0 /(cD 0 ) ≃ 9.5 × 10 −36 kg is the effective mass of the two-dimensional photons and D(r) the mirror separation at a distance r from the optical axis. The harmonic oscillator In our experiments, a commercially available polymer substance is applied as a host matrix for the dye molecules. The chromophores are solved in the optical adhesive NOA 61 (Norland Products), which is a colorless, liquid photopolymer curable by exposure to UV light. It offers a high optical transmission in the visible and near-infrared region. The liquid solution is purified using filters with 0.45 µm pore size to minimize scattering centers in the host matrix. After insertion of the viscous, liquid polymer the required mirror separation is adjusted by shining a He-Ne laser beam into the cavity. The reflectivity of the cavity mirrors at the He-Ne laser wavelength of 632.8 nm is around 80%, and in transmission we observe circular interference patterns with radii depending on the distance between the mirrors, as indicated in figure 1. The red laser light is deflected from the optical axis using a notch filter and imaged onto a CCD camera. After tuning the cavity length to the desired cavity cut-off by adjusting to a certain diameter of the He-Ne interference rings, the thin polymer layer is cured by irradiation of light emitting diodes at 365 nm, while the cavity length is kept at a given value by maintaining the He-Ne laser interference pattern constant by piezo tuning. An adjustment of the cavity cut-off by using dye fluorescence proved to be unfeasible as the pump radiation leads to heat deposition in the polymer due to nonradiative decays. This causes structural defects by partial curing of the initially liquid polymer, which is avoided when using off-resonant He-Ne radiation.
Insertion of the initial photon gas and compensation of losses is achieved by optically pumping for typical dye concentrations of ρ = 1 × 10 −3 M. This calculation takes into account both the finite mirror transmission T 45 • and also the effective absorption length in the dye-polymer-film D dye . The latter deviates from D 0 due to a penetration of the light field into the mirror material by q 0 = 4.68 ± 0.17 halfwaves [31].
To avoid excitation of long-lived triplet states for experiments carried out at higher pump rates (P pump ≈ 1 − 2 W), the incident light is chopped by an acousto-optical modulator into τ pulse = 500 ns pulses with a repetition rate of f rep = 250 Hz. The emitted cavity light is imaged onto a CCD chip and analyzed by a spectrometer, providing both spectral and spatial information on the trapped photon gas.

Properties of dye molecules embedded in polymer films
First we investigate the radiative properties of dye molecules incorporated into an amorphous polymer at room temperature. Therefore, the free-space absorption and fluorescence spectra of and PMMA films [22]. We find a zero-phonon line of λ 0 = 542 nm for R6G and λ 0 = 534 nm for PDI, respectively.
From the measured spectra we deduce the spectral temperature of the dye molecules as defined in Kennard-Stepanov theory [24], denote the dimensionless absorption and fluorescence profiles respectively in free space for an arbitrary ω 0 [31]. By repeated absorption and emission processes the photon gas is expected to thermalize to the spectral temperature, which usually corresponds to room temperature. However, deviations can arise if the fluorescence quantum yield is below unity [32].   [34]. A similar behavior of fluorescence quenching of R6G in PMMA matrices was observed using dual beam thermal lens spectroscopy [35].

Thermalization of the photon gas in the optical microcavity
Provided the previously discussed verification of the Kennard-Stepanov law in the dye-doped polymer, we have tested for thermalization of the photon gas in the dye-polymer-filled microcavity system in subsequent measurements. By insertion of the polymer film into a microresonator, the photonic mode density g(u n,m ) and, correspondingly, the fluorescence spectra of the dye molecules are modified, as described in chapter 2 in more detail. As a consequence, the photons occupy the transversal modes TEM 9mn associated with a fixed longitudinal wavenumber q = 9. If the mirror reflectivity is high enough to achieve reabsorption, a light-matter thermalization can be observed.
The predicted mean occupation number at a temperature T as a function of the transversal excitation energy u n,m reads wavelength. The thermal wing with its exponential decay to lower wavelengths, a characteristic signature for a Boltzmann distribution, is visible in all shown spectra, indicating the observation of a thermalized two-dimensional photon gas at room temperature. The peak at 532 nm originates from residual pump light scattered at the resonator. Although the signal-to-noise ratio is reduced for smaller cut-off wavelengths, in the spectral regime under investigation evidence for a room temperature thermalization of the two-dimensional photon gas is found.
In other measurements, we have investigated thermalization by varying the spatial position of the pump spot with respect to the center of the photon trapping potential. Figure  where the confining potential imposed by the curved mirrors exhibits a minimum value. The region where the spatial pulling into the center is observed is less than half the range as obtained in our earlier measurements carried out with liquid dye solutions [14]. This could be due to larger photon losses from a reduced dye quantum efficiency in the polymer based system.
As inferred from the afore-mentioned measurements, there is a limitation in the extent of the spatial relaxation mechanism. This limitation is related to the average number of reabsorption cyclesn re a photon undergoes before it is lost from the cavity. For clarification, we will in the following provide an explanation on the thermalization procedure and give an estimate for the number of reabsorptions: After a typical free propagation time of τ ph ≈ 20 ps in the dye-doped polymer-filled resonator, a photon is absorbed by a dye molecule. Reemission of a photon occurs after a molecular lifetime τ exc of typically a few nanoseconds [33]. The photon propagates freely again, and the absorption-reemission process is repeated. This type of thermalization constitutes a contact to a heat and particle reservoir and thus differs from a thermalization via particle-particle interactions as for cold atoms [21]. The photon gas will be in a thermal equilibrium with the bath, if the rovibrational structure of the dye molecules is equilibrated before the reemission. This is easily fulfilled within the lifetime τ exc of the dye, while the relaxation in the polymer host matrix is on subpicosecond timescales [27]. We verified this by our measurement of the spectral temperature T spec ≈ 300 K, see figure 3(b).
In principle, only a single absorption-reemission cycle is required to thermalize the state of the photon in a spatially homogeneous system. This follows from Kasha's rule, stating that the fluorescence cycle is completely uncorrelated to the state of the absorbed light [26]. However, the harmonic trapping potential constitutes an inhomogeneous system. If photons are generated far from the potential minimum, they first have to move into the center. While a spectral thermalization is achieved after a single absorption of a pump photon, a spatial relaxation towards the minimum of the trapping potential (from a displaced pump spot) requires a sufficiently high number of absorption-fluorescence cycles a photon undergoes before being lost from the system.
Such photon losses can be due to non-radiative decay of the dye molecules (quantum yield Φ < 1), finite resonator finesse (mirror reflectivity R < 1), as well as emission into modes not confined by the cavity. A measurement of the number of scattering events taking place within a lifetime of a photon inside the cavity has not yet been performed for the case of the polymer based host material. However, it is expected to be in the same order of magnitude for the solid state based setup and for the liquid dye scheme. We have estimated it based on a steady state condition by relating the intracavity power to the pumping power to ben re ≈ 3.8 ± 2.5 for the liquid solvent case, as derived in an earlier article [31].
Compared to the vast number of collisions taking place in atomic gases, this value seems to be rather small. But one has to consider that nearly all correlations between the absorbed and emitted photon, except for a necessary spatial overlap between both photon states, have vanished after a single absorption-emission cycle [26]. Therefore, the contact of the photon gas to the heat bath provides a stronger thermalization process compared to two-body collisions in atomic gases.
We can experimentally verify the thermalization to operate properly both in the spectral and spatial regime, as investigated thoroughly in the case of the liquid dye system [14], see the data shown in figures 4 and 5, respectively, for the solid state based system discussed here.
The spatial intensity distribution of the photons inside the resonator arises from a thermal average over all harmonic oscillator eigenfunctions Ψ n,m (x, y) and reads as where τ rt = 2D 0 n 0 /c ≃ 17 fs denotes the cavity round trip time for a photon. In our case (T = 300 K, µ = −9.1 × k B T ) this corresponds to a Gaussian intensity distribution with a full width at half maximum of 230 µm. In figure 6 recorded CCD-images (false color) are shown for different pump powers. At low average photon numbers (P pump = 5 mW) we observe a width of ∆ 1/2 = (260 ± 40) µm for a Gaussian photon distribution, consistent with theoretical expectations. When increasing the pump power, the shape of the spatial distribution remains qualitatively unchanged up to a value of approximately 2.5 W. At pump powers above this value, we additionally observe a bright spot of light with a width of ∆ 1/2 = (70 ± 10) µm emerging in the center of the spatial distribution ( figure 6(b)). By comparison, the diameter of the fundamental TEM 900 mode for an ideal Bose gas corresponds to ∆ ideal = 2 h ln 2/m ph Ω ≈ 12.2 µm. According to a previous observation of a Bose-Einstein condensation of photons in liquid dye solutions, the above described measurements suggest a similar behavior [13].
In the liquid-dye experiment the measured condensate width equals the diameter of the fundamental oscillator mode at moderate condensate fractions. Here we find a width increased by a factor of ∼ 6 of the concentrated photon-distribution, which could be attributed to thermo-optical lensing caused by reduced quantum efficiency (as shown for R6G, figure 3). In a more detailed analysis, the intensity-modulated index of refraction n( r) = n 0 + ∆n r , with ∆n r = n 2 I( r), can be interpreted as an effective photon-photon interaction in the Gross-Pietaevskii equation [31]. If one considers the observed peak to be a broadened TEM q00 mode, a dimensionless self-interaction constantg ≈ 2.7 × 10 −2 would be necessary to be consistent with the measured diameter. For the liquid dye scheme, we have determined a dimensionless interaction constant ofg = 7.5 × 10 −4 [13]. This is derived by comparing the measured diameter of an interacting Bose condensate in Thomas-Fermi-approximation to that of an ideal condensate, The condensate fraction is given by N 0 /N = 1 − N c /N and from figure 6 it is estimated to Together with a total photon number of N ≈ 1.7 × 10 5 , which is obtained by integrating over the intensity distributions, we derive a ground state occupation of N 0 ≈ 6 × 10 4 .
From this condensate fraction one deduces a critical photon number of N c,exp ≈ 10 5 , which is consistent with the theoretical prediction mentioned above.
However, no detailed analysis of this effect, such as a spectral study, was carried out due to immediate photobleaching of the dye molecules within τ = 1.36 s (inset of figure 6), which turns out to irreversibly inhibit further fluorescence processes within the central volume of the cavity.
Therefore, the values ofg ≃ 2.7 × 10 −2 , N ≈ 1.7 × 10 5 and N c,exp ≈ 10 5 should be taken with care, since a progressed, partial bleaching of molecules in the moment of the data acquisition cannot be precluded. The intensity distribution after a full photodegradation of dye molecules in the cavity center is given by a Gaussian distribution with a peak-intensity I(0, 0) ≃ 200 (in arbitrary units) as in figure 6(a). In that case a repeated observation of a photon condensate using the same polymer film is impossible, but the cavity mirrors can be recycled by removing the layer from the mirror surfaces.
Different from the detailed verification of the photon Bose-Einstein condensate in liquid dye solutions, the presently described experiment lacks additional investigation of the critical particle number and the spectral distribution due to the short-lived chemical stability of the dye molecules embedded in the polymer. Photobleaching also occurs in liquid dye solutions [36]. However, due to constant molecular motion in liquid solvents this effect is less pronounced than in solid state systems with localized chromophores. For the polymer-based system a multimode occupancy cannot be a precluded by the discussed measurements. However, the results are evidently resembling a non-classical distribution of photons, which deviates from Boltzmann statistics. In the future, a more detailed investigation of the observed concentration of fluorescence light to the trap center will be necessary.

Conclusion
We presented the observation of a thermalized, two-dimensional photon gas in a solid state system.
A dye-doped amorphous photopolymer is incorporated into a high-finesse optical microcavity to allow a light-matter thermalization. An external pump source is employed to adjust the chemical potential of the photon gas and to compensate for photon losses. By investigating the emitted cavity light, we find that the spatial and spectral intensity distribution of the photons correspond well to that of a harmonically confined gas of particles at 300 K. Furthermore, the observation of a thermalized photon gas is evidenced by the measurement of a fluorescence redistribution effect, which depends on the cavity cut-off.
The absorption and fluorescence profiles as well as the quantum yield of the used dye molecules Irreversible photodegradation becomes apparent as the main limiting factor for extended studies on the dye-polymer-samples at this stage. Therefore, we will prospectively study the incorporation of different dyes with high quantum yields into amorphous polymer matrices.
The application of deoxygenated PMMA hosts with excellent optical quality, combined with photostable dyes are promising candidates for the described experimental scheme [35,37,38].
Altogether, photochemically stable setups hold promise for the realization of a new type of light source, which produces coherent light in thermal equilibrium. Additionally, the reported light concentration effect could be envisaged to enhance the efficiency of solar collectors [39]. The usage of plastics rather than liquids as dye-solvents should enormously improve the practicability of such devices, including the possibility to use organic materials which can be pumped electrically. Other advantages are the mechanical stability and the protection of the mirrors due to the permanent polymer-based dye film between the cavity mirrors.  show good agreement to a Boltzmann distribution (according to equation (3)) at T = 300 K and µ = −9.1 × k B T (solid line). Residual pump light gives rise to a peak at 532 nm. In (b) and (d) the signal-to-noise ratio decreases due to an increased photon reabsorption probability within the cavity at smaller cut-off wavelengths. Upon variation of x exc the photons are spatially redistributed to the cavity center provided λ cut-off = 584 nm is in the reabsorbing regime fulfilling the thermalization condition (red squares).
For λ cut-off = 620 nm, a coincidence of fluorescence and excitation spot is observed (black dots).
This we attribute to a thermalization breakdown in the non-absorbing regime, where the photon gas does not exhibit a Boltzmann-distribution. This can also be seen in corresponding spectra,