Abstract
Bose–Einstein condensation^{1}, the macroscopic accumulation of bosonic particles in the energetic ground state below a critical temperature, has been demonstrated in several physical systems^{2,3,4,5,6,7,8}. The perhaps best known example of a bosonic gas, blackbody radiation^{9}, however exhibits no Bose–Einstein condensation at low temperatures^{10}. Instead of collectively occupying the lowest energy mode, the photons disappear in the cavity walls when the temperature is lowered—corresponding to a vanishing chemical potential. Here we report on evidence for a thermalized twodimensional photon gas with a freely adjustable chemical potential. Our experiment is based on a dyefilled optical microresonator, acting as a ‘white wall’ box for photons. Thermalization is achieved in a photonnumberconserving way by photon scattering off the dye molecules, and the cavity mirrors provide both an effective photon mass and a confining potential—key prerequisites for the Bose–Einstein condensation of photons. As a striking example of the unusual system properties, we demonstrate a yet unobserved light concentration effect into the centre of the confining potential, an effect with prospects for increasing the efficiency of diffuse solar light collection^{11}.
Main
Following the achievement of atomic Bose–Einstein condensation^{2,3,4}, we have witnessed interest in light sources where a macroscopically populated photon mode is not the consequence of laserlike amplification, but rather results from a thermal equilibrium phase transition. Work in this direction includes the proposal of a superfluid phase transition of photons in a nonlinear cavity^{12,13,14} and, albeit in the strongcoupling regime, the demonstration of a quasiequilibrium phase transition of exciton–polariton quasiparticles to ‘half matter, half light’ condensates^{6,7,8}. In the weakcoupling regime (as in our case), optical cavities have been used to achieve a modified spontaneous emission of atoms and molecules^{15,16,17}.
The main idea of our experiment is to study thermalization of a photon gas, to a heat bath near room temperature (dye molecules), in a system with reduced spatial dimensionality and an energy spectrum restricted to values far above the thermal energy. The photons are trapped in a curvedmirror optical microcavity, and repeatedly scatter off dye molecules. The longitudinal confinement (along the cavity axis) introduces a large frequency spacing between adjacent longitudinal modes and modifies spontaneous emission coupling such that basically only photons of longitudinal mode number q=7 (Fig. 1a) are observed to populate the cavity. By this, an effective low frequency cutoff ω_{cutoff} (the eigenfrequency for the corresponding TEM_{00} transverse mode) is introduced, with ℏω_{cutoff}∼2.1 eV, much larger than the thermal energy k_{B}T (∼1/40 eV at room temperature). The two remaining transverse modal degrees of freedom of light thermalize to the (internal rovibrational) temperature of the dye solution, and the photon frequencies will be distributed by an amount ∼k_{B}T/ℏ above the cavity cutoff. We expect significant population for the TEM_{nm} modes with high transversal quantum numbers n and m (of high eigenfrequency) at high temperature, whereas the population concentrates to the lower transverse modes when the system is cold. Equilibrium is reached as photons are absorbed and emitted by dye molecules many times, with the interplay between fluorescence and absorption leading to a thermal population of cavity modes, making the photon gas equilibrate at the temperature of the dye solution (see the Methods section). During the course of thermalization (in an idealized experiment) only transverse mode quantum numbers are varied, and—different from in experiments observing blackbody radiation^{9,18}—no photons are destroyed or created on average. This becomes clear by noting that the dye molecules have quantized electronic excitation levels with a transition energy near or above the cutoff, and thermal excitation is suppressed by a factor of order exp(−ℏω_{cutoff}/k_{B}T)∼10^{−36}.
The thermodynamics of the system can readily be obtained by a mapping onto the twodimensional ideal Bose gas: for paraxial propagation and a fixed longitudinal excitation number, photons can be treated as nonrelativistic massive particles with mass m_{ph}=ℏω_{cutoff}/c^{2} (see the Methods section), moving in the transverse plane of the resonator^{12}. Owing to the curvature of the mirrors, the photons are moreover under harmonic confinement (see the Methods section). When the photons are in thermal equilibrium and their average number is conserved, we expect that this trapped twodimensional ideal Bose gas will undergo a phase transition to a Bose–Einstein condensate at sufficiently low temperature and high density^{19,20}. A notable further consequence of the thermalization of the trapped photon gas is an accumulation in the trap centre, where the photon trapping potential is minimum. This behaviour is evident from the analogy to a trapped gas of material particles.
Our experimental setup shown in Fig. 1b is based on an optical resonator consisting of two highreflecting curved optical mirrors spaced 3.5 optical wavelengths apart, filled with dye solution (see the Methods section). The short distance between the cavity mirrors yields a free spectral range (∼7×10^{13} Hz) much larger than the thermal energy k_{B}T in frequency units (∼6×10^{12} Hz at room temperature), and comparable to the spectral width of the dye emission. The mean transversal excitation number (per axis) of the twodimensional photon gas is k_{B}T/ℏΩ∼160, where Ω/2π (∼4×10^{10} Hz) denotes the spacing of transversal cavity modes; that is, the spacing is so small that the transverse motion is quasicontinuous. Other than in a perfect ‘photon box’^{21} we expect losses from coupling to optical modes not confined in the cavity, nonradiative decay and mirror losses. To compensate for the loss rate, the dye is pumped with an external laser beam. In a simple model, the pumping fills a reservoir of electronic excitations in the dye that can exchange particles with the photon gas. Thus, the photon gas is seen as an open system, in the sense of a grandcanonical ensemble. The pumping maintains a steady state in which the average photon number N_{ph} will be proportional to the number of electronic excitations N_{exc} and is determined by N_{ph}/N_{exc}=τ_{ph}/τ_{exc}, where τ_{ph} and τ_{exc} denote the lifetimes of photons and electronic excitations, respectively. Despite losses and pumping we expect the photon gas to be well described by an equilibrium distribution when the thermalization is sufficiently fast; that is, a photon scatters several times off a molecule before being lost. Our present experiment is carried out at an optical pumping beam intensity of ∼10 W cm^{−2}, an estimated three orders of magnitude below the onset of a photon Bose–Einstein condensate and four orders of magnitude below saturation, so that we do not expect collective photonic effects to be relevant.
Experimental data for the spectral distribution of light transmitted by one of the cavity mirrors are shown in Fig. 2a for two different temperatures of the resonator. The connected circles represent experimental data, and the solid lines are theoretical curves for a fully thermalized gas at the corresponding temperature. The expected photon number in the cavity n_{T,μ}(u) at given transversal energy u=ɛ−ℏω_{cutoff} is the product of the energy degeneracy g(u)=2(u/ℏΩ+1) (Fig. 1a) and the Bose–Einstein distribution factor:
The chemical potential μ is determined by measuring the output power P_{out}=50±5 nW, which corresponds to a photon number of N_{ph}=60±10 in the cavity, and solving for μ. For the two cases in Fig. 2a we obtain μ/k_{B}T=−6.76±0.17 (T=300 K) and μ/k_{B}T=−7.16±0.17 (T=365 K) respectively. As a result of μ≪−k_{B}T, the term −1 in the denominator of equation (1) can be neglected and the distribution becomes Boltzmannlike, with the term e^{μ/k}_{B}T acting as a prefactor to the spectral distribution. A Bose–Einstein condensate would be expected for (at which μ→0). The shapes of the shown experimental spectra are in good agreement with theoretical expectations over a wide spectral range (a visible deviation near 532 nm stems from residual pump light). Notably, the enhancement of the shortwavelength wing for the higher temperature data by almost an order of magnitude is predicted accurately. We interpret these results as one line of evidence for the twodimensional photon gas to be in thermal equilibrium with the dye solution. Evidence is also obtained from investigating the spatial distribution of light emitted by the resonator. A typical snapshot is shown in Fig. 2b, in which a shift from the yellow spectral regime for light near the axis towards the green (that is, higher photon energy) for the radiation emitted offaxis is clearly visible. This is due to the higher energy of cavity resonances with a large transversal momentum and correspondingly large mode diameter. From the image in Fig. 2b, the spatial intensity distribution shown in Fig. 2c was extracted, which is in good agreement with a thermal average over spatial modes. In other words, the thermal distribution of the photon spectrum is also reflected by the spatial distribution.
In further measurements, we have varied the cutoff wavelength λ_{cutoff}=2πc/ω_{cutoff} by piezo tuning of the distance between cavity mirrors. A series of corresponding spectra is shown in Fig. 3a. Reasonable agreement with a thermal distribution is obtained for the two spectra shown at the top with cutoff wavelengths near 570 and 590 nm respectively, whereas the two spectra with longer cutoff wavelengths shown in the bottom seem only partly thermalized. We attribute this to weak dye reabsorption in this spectral regime, preventing repeated photon scattering off the dye molecules (also the reflectivity of cavity mirrors lessens at wavelengths above 600 nm). This illustrates the importance of both emission and reabsorption of scattered photons for thermalization, in agreement with expectations.
Finally, thermalization was investigated by varying the spatial position of the external pumping beam with respect to the centre of the trapping region. Figure 3b shows the position of the maximum of the emitted fluorescence x_{max} versus the transverse position of the pump focus x_{exc} for two different values of the cutoff wavelength. For the data recorded with a cutoff wavelength near 580 nm (squares fitted with a solid line) the fluorescence essentially freezes to the position of the trapping centre, when the excitation spot is closer than approximately 60 μm distance. This effect is not observed when tuning the cutoff wavelength to 620 nm (circles fitted with a dashed line), where weak reabsorption prevents multiple scattering events and thus an efficient thermalization (Fig. 3a). The demonstrated position locking effect is a direct consequence of the thermalization, leading to a photon accumulation at the trap centre, where the confining potential imposed by the curved mirrors exhibits a minimum value. We attribute the observed breakdown of thermalization for data points with x_{exc}>60 μm to the finite quantum efficiency of the dye (; ref. 22), and our mirrors providing only a onedimensional photonic bandgap, limiting the number of possible photon scattering processes. The relaxation also for initial states further apart from equilibrium could be improved with mirrors providing a full threedimensional photonic bandgap^{23}. Technical applications of the observed directed diffusion effect could include the collection of diffuse solar light to a central spot^{14}, with prospects for enhancing the optical phase space density. When a Bose–Einstein condensate of photons can be reached, perspectives include coherent ultraviolet sources^{24}.
Methods
Experimental setup.
Our optical resonator consists of two highreflecting dielectric mirrors with spherical curvature (R=1 m). One of the mirrors is cut to 1 mm×1 mm surface size to allow for a distance between mirrors in the micrometre regime (D_{0}≅1.46 μm) despite the mirrors’ curvature. The measured reflectivity of the cavity mirrors is ∼99.9985% in the relevant wavelength regime (520–590 nm). The resonator is filled with a drop of dye (rhodamine 6G or perylene diimide respectively) dissolved in an organic solvent and is pumped with a laser beam near 532 nm wavelength inclined at 45° to the cavity axis, exploiting a reflectivity minimum.
Photon dispersion in cavity.
The photon energy as a function of transversal (k_{r}) and longitudinal (k_{z}) wavenumber in the paraxial approximation (k_{z}≫k_{r}) is , with c as the speed of light in the medium. Within the resonator, we have k_{z}(r)=qπ/D(r) with as the mirror separation at distance r from the axis. For r≪R and a fixed longitudinal mode number q one finds
We arrive at the dispersion of a particle with mass m_{ph}=ℏk_{z}(0)/c=ℏω_{cutoff}/c^{2} subject to a harmonic oscillator potential of trapping frequency in two spatial dimensions.
Principle of radiation field thermalization.
For photons absorbed and emitted by dye molecules many times, the state of the light field can be regarded as following a random walk in configuration space. For a transition between two configurations, P and Q (that is, sets of mode occupation numbers), induced by the absorption of a photon in mode i and the emission into mode j, the transition rate is expected to follow R(P→Q)∝α_{T}(ω_{i})f_{T}(ω_{j}) with α_{T}(ω) and f_{T}(ω) as (temperature dependent) relative absorption and fluorescence strength coefficients. This random walk in configuration space will lead to a thermal equilibrium population of photon modes, if the condition of detailed balance^{25} is fulfilled:
Dye spectra are known to typically fulfil (2) within good accuracy, which itself is a consequence of a thermal equilibrium of rovibrational excitations within the ground and excited electronic molecular levels respectively^{26,27,28}.
For a simple model of the thermalization process in the dye, consider a twolevel system with energy splitting ℏω_{0} subject to further rovibrational sublevel structures of ground and excited state^{28}. Absorption α_{T}(ω) and fluorescence strength f_{T}(ω) at frequency ω can be expanded by integrating over the contributions from individual energetic sublevels, yielding a ratio
where e and e′ denote the energies of sublevels of ground and excited electronic state respectively, p(e′) the population of sublevels in the excited electronic state, g and g′ the density of states and A and B are the Einstein coefficients. The Einstein coefficients at sublevels with energies e and e′ matching ℏω+e=ℏω_{0}+e′ are connected by the A–B relation g′(e′)A(e′,ω) de′=(2ℏω^{3}/πc^{2})g(e)B(e,ω) de. When thermalization in the electronically excited molecular level is sufficiently fast, p(e′)∝e^{−e′/k}_{B}T =e^{−(e+ℏ(ω−ω}_{0}))/k_{B}T and one immediately finds f_{T}(ω)/α_{T}(ω)∝ω^{3}e^{−ℏ(ω−ω}_{0})/k_{B}T . With this, the detailed balance condition (2) relating absorption and fluorescence at two different frequencies is fulfilled for ℏω≫k_{B}T.
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Acknowledgements
We thank F. Schelle for experimental contributions during the early phase of this project. Financial support from the Deutsche Forschungsgemeinschaft within the focused research unit FOR557 is acknowledged.
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J.K. and M.W. contributed to the experimental idea; J.K. carried out the experiments. All authors analysed the experimental data and discussed the results.
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Klaers, J., Vewinger, F. & Weitz, M. Thermalization of a twodimensional photonic gas in a ‘white wall’ photon box. Nature Phys 6, 512–515 (2010). https://doi.org/10.1038/nphys1680
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DOI: https://doi.org/10.1038/nphys1680
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