## Abstract

In a Bose–Einstein condensate, bosons condense in the lowest-energy mode available and exhibit high coherence. Quantum condensation is inherently a multimode phenomenon, yet understanding of the condensation transition in the macroscopic limit is hampered by the difficulty in resolving populations of individual modes and the coherences between them. Here, we report non-equilibrium Bose–Einstein condensation of 7 ± 2 photons in a sculpted dye-filled microcavity, where the extremely small particle number and large mode spacing of the condensate allow us to measure occupancies and coherences of the individual energy levels of the bosonic field. Coherence of the individual modes is found to generally increase with increasing photon number. However, at the break-down of thermal equilibrium we observe phase transitions to a multimode condensate regime wherein coherence unexpectedly decreases with increasing population, suggesting the presence of strong intermode phase or number correlations despite the absence of a direct nonlinearity. Experiments are well-matched to a detailed non-equilibrium model. We find that microlaser and Bose–Einstein statistics each describe complementary parts of our data and are limits of our model in appropriate regimes, providing elements to inform the debate on the differences between the two concepts^{1,2}.

## Main

While Bose–Einstein condensation (BEC) is a general phenomenon, the standards of experimental evidence needed to claim BEC differ among different physical realizations. Ultracold atomic gases are very nearly closed systems for which a purely equilibrium description is often sufficient, so macroscopic occupancy of one state is considered proof of BEC. How condensation can be demonstrated with microscopic particle numbers is an open question. Quantum gases in pure states with as few as two fermions^{3} or six bosons^{4} have been created in very specific configurations, but BEC is notoriously difficult to achieve by thermalization in smooth traps^{5,6}.

For polariton condensates in microcavities it is now accepted that the build-up of coherence and population in lasing arise from stimulated emission^{7}, but in condensation the build-up is caused by bosonically stimulated scattering among the polaritons. Despite the finite number of particles, as few as about 100 in one study^{8}, the question of how a threshold for BEC (a phase transition defined only in the thermodynamic limit) can be determined has rarely been considered.

The original thermodynamic-limit, fully equilibrium Penrose–Onsager criterion for BEC is that, as system size grows towards infinity, a finite fraction of particles remain found in the lowest-energy mode^{9}. More general definitions of condensation in multimode systems have recently come into question, applicable not only to bosonic statistics out of equilibrium^{10} but also to networks, traffic jams, evolutionary game theory, population dynamics and chemical reaction kinetics^{11}. Condensation is said to have occurred when the populations of some modes of the system grow linearly with total population as the total tends to infinity, while other modes are depleted, with sub-linear growth or saturation. BEC is the special case in which the only mode to condense is the lowest-energy mode. These clear theoretical definitions are not applicable to experiments, which cannot reach infinite populations.

An operational criterion for condensation applicable to experiments would consist of an inequality that defines a parameter region of condensation. Distinctions between true or near thermal equilibrium and non-equilibrium situations call for robust criteria, which we will define here. BEC can be distinguished from general or multimode condensation in that only the lowest-energy mode is condensed, and all other modes are depleted.

In dye-microcavity photon BEC, thermalization among the particles is (uniquely for quantum fluids) completely negligible as interactions are weak^{12,13,14}. Condensation is distinguished from lasing by thermalization through multiple reabsorption and emission events for photons before they leave the microcavity^{15}. On average, two such events are sufficient to stabilize condensation in the lowest-energy mode^{16}, but more events are required to achieve complete thermalization to equilibrium at room temperature. Either a good fit to the Bose–Einstein distribution^{17}, or the robustness of the lowest-energy mode as the strongly populated mode, are considered proof of BEC.

Photon BEC has previously been reported with as few as 70 photons^{14}. In this work, we have achieved photon thermalization and condensation with just seven photons, one of the smallest such condensates ever published.

Figure 1c shows the population of each of the lowest four energy levels as a function of total population or pump power (Fig. 1d), for a single dataset. Threshold is a broad feature, characteristic of finite-sized systems. We compare the populations of all modes to a simple thermal equilibrium Bose–Einstein distribution and to a full non-equilibrium model^{18,19}, as well as a single-mode microlaser model for the lowest energy level. In the Methods section we show how the equilibrium and microlaser models can be derived as limiting cases of the full model. The thermal equilibrium model (Fig. 1c, solid lines) uses Bose–Einstein statistics so the population *n*_{i} of the *i*th excited mode is

with the degeneracy *g*_{i} = *i* + 1 for a single spin state of a two-dimensional harmonic oscillator (2DHO), the mode energy is \(\epsilon _i = i{\kern 1pt} hf\), the typical thermal energy is *k*_{B}*T* (where *k*_{B} is the Boltzmann constant and *T* the temperature) and *μ* is the chemical potential that determines *n*_{tot}. The least-squares fit returns *T* = 150 ± 20 K; this is below room temperature because the system is not quite at equilibrium.

The non-equilibrium model has three adjustable parameters: the pump-spot size, the rate of spontaneous emission into free space and the calibration of the detection system in terms of photon number (see Supplementary Information for details). Notably, the spontaneous emission rate is reduced significantly from its free-space value, as most of the emitted light is recaptured by the cavity mirrors. The populations of excited levels nearly saturate above threshold (as they would for exact equilibrium), a feature that is well described by the non-equilibrium model.

We have taken measurements through a linear polarizer aligned to maximize transmission of the condensate light, to avoid ambiguities in the role of polarization, which recent results indicate will not affect our conclusions^{20,21} (see also Supplementary Information for details).

Four suitable criteria for condensation, based on mode populations *n*_{i} and the total population *n*_{tot} = \(\mathop {\sum}\nolimits_i {\kern 1pt} n_i\), where the indices *i* run over all modes that can be measured or calculated are: (1) *n*_{0} > *n*_{tot}/2 (ref. ^{10}); (2) *n*_{tot} > \({\mathrm{lim}}_{n_{\mathrm {tot}} \to \infty }\) {*n*_{tot} − *n*_{0}} (ref. ^{22}); (3) *n*_{i} > \(k_{\mathrm {B}}T{\mathrm{/}}\epsilon\) (ref. ^{18}); and (4) *n*_{i} > \(n_{\mathrm {tot}}^{1/\alpha }\) where *α* is the dimensionality of the system (at least unity). These concepts are discussed in more detail in the Supplementary Information. Criteria (2) and (3) are defined in near-equilibrium conditions, and (1) is very strict, forcing single-mode condensation. Thus, (4) is the only criterion that is also applicable to the multimode condensation that is known to occur in photon condensates^{16,23}. It would also be useful in categorizing other non-equilibrium condensation processes, such as prethermalization^{24}.

The dimensionality of this system is *α* = 2, so criterion (4) for condensation in mode *i* becomes *n*_{i} > \(\sqrt {n_{\mathrm {tot}}}\). The mirror shape defines an effective 2DHO potential, for which the critical total particle number (2) becomes *n*_{tot} > \(\left( {\uppi ^2{\mathrm{/}}6} \right)\left( {k_{\mathrm {B}}T{\mathrm{/}}\epsilon } \right)^2\) for level spacing \(\epsilon\). In Fig. 1c the dashed line illustrates criterion (4). Criteria (2)–(4) nearly coincide and yield a threshold of *n*_{tot} = 7 ± 2 photons. Criterion (1) contradicts these, requiring not only that condensation be found but also that multimode condensation be excluded, and gives *n*_{tot} = 11 photons for condensation. Even at 7 ± 2 photons, condensation is well established.

For the conditions of Fig. 1, absorption events happen four times faster than cavity loss, so photons can exchange energy with the thermal bath of dye-solvent vibrations, and thermal equilibrium and BEC are good descriptions.

In Fig. 2 the effect of reducing the rate of thermalization through reabsorption is shown. The reabsorption rate is \(\overline n _{\mathrm{mol}}\sigma (\lambda )c^ \ast\), where \(\overline n _{{\mathrm {mol}}}\) is the effective molecular number density (see Supplementary Information for details), *c*^{*} the speed of light in the medium and *σ*(*λ*) the absorption cross-section at wavelength *λ*. The degree of thermalization is parameterized by the ratio of reabsorption to cavity loss rates: *γ* = \(\overline n _{\mathrm {mol}}\sigma (\lambda )c^ \ast {\mathrm{/}}\kappa\). We estimate (see below) that *κ* = 5 ps. Experiments (upper row) are compared with the full non-equilibrium model (lower row), with the same parameters as Fig. 1 except the pump spot size, which is set to 2.4 μm. In the left panels, the degree of thermalization is *γ* = 6.7, so the system is strongly thermalized and a condensation threshold is reached for the lowest-energy mode, and no other level. Up to 95% of photons are in this nearly pure BEC.

For weaker thermalization (centre panels, *γ* = 2.7), the lowest-energy level shows threshold, but one or more excited levels also show threshold, and the lowest-energy–level fraction peaks around 75%. Multimode condensation occurs at higher pump powers. For very weak reabsorption (right panels, *γ* = 0.15), multiple modes not including the ground state show threshold, and the system cannot even approximately be described as a BEC.

Having established that the near-thermalized photon population can be described by either Bose–Einstein statistics or a non-equilibrium model, we now apply these descriptions to the phase coherence, *g*^{(1)}. We measure *g*^{(1)} using a spectrometer on the output of a Mach–Zehnder interferometer, where we can delay one interferometer arm by some time *τ*, as in an earlier study^{23}.

For a non-dissipative thermal Bose gas below condensation threshold number, *g*^{(1)}(*τ*) decays as a Gaussian with a characteristic time of order *h*/k_{B}*T* ≃ 100 fs^{25}. It is predicted that revivals of all correlation functions will occur at intervals of the oscillation period^{26} as a consequence of uniform energy-level spacing. They are diminished by a slight anisotropy of the mirrors. Both effects, decay and revival, can be seen in Fig. 3 (top panel). Here *g*^{(1)} is inferred from the fringe visibility after summing the signals of several modes, which cover almost all of the population. The theory plotted is based on a previous study^{26} that made use of a decomposition of the photon field-annihilation operator in a basis of the trap states. Taking a density operator that describes an equilibrium distribution at room temperature with energy spacings *h* × 1.42 and 1.48 THz for the two axes, we then calculate the expectation of the field–field correlations. Fluctuations of the photon field are propagating back and forth across the trapping potential as weakly damped wavepackets.

The decay of revivals is in part due to dissipation. In Fig. 3 (bottom panel) we show the coherence for the lowest-energy level alone, without summing with other modes before inferring visibility. *g*^{(1)}(*τ*) fits well to an exponential decay with a coherence time *τ*_{c} for a variety of parameters.

By treating just one cavity mode, and assuming photon and molecule states are separable, one can reach a closed form for coherence time^{18}: see Methods. For large photon numbers \(n \gg 1\) (above threshold), the coherence time *τ*_{c} ∝ *n*, yielding the Schawlow–Townes limit. For \(n \ll 1\), *τ*_{c} is independent of *n*, given by

Coherence decays at half of the rate at which photons are removed from the mode, both by cavity loss and by reabsorption, which in turn is the thermalization rate. In Fig. 4 we see quantitative agreement between experiment and theory for most parameters. The theory has only one adjustable parameter, the cavity-loss rate, for which we find 1/*κ* = 5.2 ± 0.8 ps, in agreement with the value obtained through observations of the break-down of the BEC description, Fig. 2.

For very large photon numbers, *n* ≳ 50, coherence time decreases markedly with increasing photon number, in direct contradiction to the single-mode theory. This might be attributed to a break-down of photon–molecule separability, or to inhomogeneous coupling of multiple modes to the molecular excitations^{16}.

The first-order coherence of the light can be interpreted through two complementary physical models: as a conservative thermalized Bose gas when accounting for many energy levels, or as a driven-dissipative open quantum system when inspecting the coherence of the lowest-energy mode alone. As an open quantum system, the coherence is limited by the reabsorption of the light, which is the very mechanism that induces the coherence-enhancing BEC itself. The tension between coherence and decoherence resolves at large photon numbers by a dramatically reduced coherence time, accompanied by multimode condensation.

From this extra decoherence mechanism, we infer that there is an effective interaction that couples the quantum states of the light across the multiple cavity modes, mediated by the dye molecules. This is despite the small measured value of the direct optical nonlinearity^{17}. Additionally, in a multimode condensate, photons in one condensed mode could act as reservoirs of excitations for other modes, enhancing number fluctuations and hence decreasing phase correlations^{27}. Through this mechanism we anticipate that higher-order coherences such as intermode number correlations will lead to non-classical states of light, possibly including number squeezing. If such states can be understood and observed, they may well prove a valuable resource for quantum metrology as well as a fascinating subject of study in their own right, enabled by our microfabricated mirrors and photon thermalization techniques.

## Methods

### Experimental techniques

Our experimental setup is largely as described in our previous articles^{23,28,29}, with an open microcavity using one planar mirror and one curved mirror, where the space between is filled with fluorescent dye (Rhodamine 6G in ethylene glycol). We optically pump the dye. To prevent triplet-state population, we use 350 ns pump pulses, and as we vary the pump laser power we retain near-constant time-averaged power by also changing the repetition rate. Through incoherent emission and reabsorption and dye ro-vibrational relaxation, excitations are exchanged between dye molecules and cavity photons, and the photons can approach thermal equilibrium near room temperature^{18}. We use microfabricated mirrors to achieve large mode spacings, with spectroscopic resolution of the individual energy levels for the bosonic field: see Fig. 1a.

For these experiments, the radius of curvature (RoC) for the curved mirror was 400 μm, which we achieved by machining a superpolished substrate with focused-ion-beam (FIB) milling^{30,31,32} followed by smoothing by laser-induced heating^{33} and commercial ion-beam sputtered coating of dielectric mirrors. This curved mirror defines a 2DHO potential of frequency *f* = 1.4–1.7 THz (level spacing \(\epsilon\) = *hf*) depending on the longitudinal mode number *q* (we use 9 ≤ *q* ≤ 11). By varying the cavity length, we set the energy of the lowest level, equivalent to a cutoff wavelength *λ*_{0}, between 555 and 580 nm: see Fig. 1b. We observe cavity photoluminescence with a spectrometer of resolution 0.3 nm (equivalent to 0.3 THz), sufficient to distinguish individual energy levels.

Light leaks from both sides of the microcavity. We image from the planar-mirror side onto a monitoring spectrometer. The response of this spectrometer to intra-cavity light is calibrated by comparison to the response to a known light power, together with the transmission of the mirrors. From the microfabricated mirror side, light is sent to a beamsplitter, with some directed to a camera and most to an interferometer. The controls and data analysis techniques used with the interferometer are detailed in a previous study^{23}. For this work, one output of the interferometer goes to a camera and the other quadrature goes to a spectrometer. We use this spectrometer for inferring coherence *g*^{(1)}.

### A simplified model for dye-microcavity photons

We make use of the simplest version Kirton and Keeling model, which assumes that all dye molecules couple equally to all cavity modes^{18,34}, unlike the more recent versions that include spatial^{19} and rotational^{20} inhomogeneities. The solution of a full quantum master equation can be simplified to a system of rate equations:

where *n*_{m} is the photon number in the *m*th mode of degeneracy *g*_{m}, *p*_{e} the fraction of excited-state molecules, *N*_{mol} the total number of molecules, *Γ*_{↑} the pumping rate and *Γ*_{↓} the de-excitation rate not including emission into cavity modes (dominated by spontaneous emission into free space). The functions *A*(*δ*_{m}) and *E*(*δ*_{m}) are the absorption and emission amplitudes to and from the cavity modes at detuning *δ*_{m} from molecular resonance. {*n*_{m}} is the set of photon numbers in all modes.

#### Thermal-equilibrium limit

The Kennard–Stepanov/McCumber relation imposes that the absorption and emission spectra are related by a principle of detailed balance through rapid vibrational thermalization of the dye molecules, so that \(A{(\delta)}=E{(\delta)}^{{{\delta}/{k}_{\mathrm{B}}}{\mathrm{T}}}\), where *k*_{B}*T* is the thermal energy scale at room temperature. In the limit of low losses (*κ*, *Γ*_{↓} → 0), the steady-state solution can be written in the form

which is simply the Bose-Einstein distribution as shown in equation (1). The chemical potential is given by \({\mathrm{exp}}\left( {\mu {\mathrm{/}}k_{\mathrm {BT}}} \right)\) = \({\it{\Gamma }}_ \uparrow ^{\mathrm {tot}}{\it{/\Gamma }}_ \downarrow ^{\mathrm {tot}}\).

#### Microlaser limit

In a different limit, the model can be compared to a microlaser. First, we simplify the description to just a single cavity mode red-detuned from the molecular resonance, *δ* < 0, and free space for spontaneous emission. We treat the limit where excitation of the dye molecules by cavity light is negligible, *A* → 0 while *E* remains finite. Equations (3) and (4) become:

These equations are equivalent to standard equations of motion used to describe microlasers^{35,36}, which are derived in the approximation that molecular excited-state saturation is negligible, \(p_{\mathrm {e}}\ll 1\), an approximation that is typically valid for microcavity experiments not involving polaritons.

These equations are normally written using a parameter *β* that is the fraction of all spontaneous emission that goes into the cavity mode, which is *β* = *E*/(*E* + *Γ*_{↓}) where *E* is the emission rate evaluated at the cavity detuning. The equations admit an analytic, steady-state solution for the population *n* of the only mode:

with *ρ* = *Γ*_{↑}*N*_{mol}/*κ* being the normalized pump rate, proportional to pump laser power.

### Driven-dissipative theory of microlaser coherence

The well-established master equation for the molecules and photons among the various cavity modes is numerically and analytically difficult to solve, so we will make a single-mode approximation, as done in an earlier study^{18}. The density operator can be solved for in a basis using photon numbers and excited molecule numbers. From the number–off-diagonal elements, the first-order correlation function *g*^{(1)}(*τ*) as a function of relative time can be calculated. A time-evolution equation for those off-diagonal elements can be extracted from the full master equation, and solved through recursion of higher and lower photon numbers, assuming that the molecular state is independent of the photon state. Such an approximation is likely to be valid except in the limit of very large photon numbers, and also in the multimode condensate regimes.

In the steady state, the first-order correlation function *g*^{(1)}(*τ*) is approximately exponential as a function of the absolute value of *τ*, but not necessarily exactly (so the spectrum may deviate from Lorentzian). Nonetheless, one can define a coherence time from the decay of *g*^{(1)}. Numerical solution of the equations is intractable for experimentally appropriate molecule numbers (around 10^{6}), but assuming molecule and photon states are separable, mean-field results can be obtained.

The model is similar to equations (8) and (9), but retains the absorption coefficient *A*. One solves for the average excited-state fraction of molecules *p*_{e} and obtains:

where \({\it{\Gamma }}_ \uparrow ^{\mathrm {tot}}\) = Γ_{↑} + *A*(*δ*)\(\left\langle n \right\rangle\) and \({\it{\Gamma }}_ \downarrow ^{\mathrm {tot}}\) = *Γ*_{↓} + *E*(*δ*)(\(\left\langle n \right\rangle\) + 1), and \(\left\langle n \right\rangle\) is the mean number of photons in the mode of detuning *δ* relative to the zero-phonon line.

Then the Lorentzian linewidth *Γ*^{T}, equivalent to the inverse coherence time 1/*τ*_{c} is given by:

The photon absorption rate *A*(*δ*) in the limit of weak molecular excitation \(p_{\mathrm {e}}\ll 1\) is related directly to the measured absorption cross-section *σ*(*λ*) by *N*_{mol}*A* = *n*_{mol}*σ*(*λ*)*c*^{*}. Here *δ* = \(2\pi c\left( {\frac{1}{\lambda }_{\mathrm {ZPL}} - \frac{1}{\lambda }} \right)\) is the detuning from the molecular zero-phonon line and *n*_{mol} is the effective number density of molecules (given by the true molecular density times the fraction of the light that is in the open part of the microcavity not within the mirrors, (*q* − *q*_{0})/*q*). Thus in the limit of weak pumping and few photons we find a simple expression for the decoherence rate (inverse coherence time) that contains experimentally measurable quantities:

Decoherence is simply caused by photon loss, either by emission from the cavity or by reabsorption, which is the mechanism by which photon thermalization occurs. In the limit of large photon numbers, it can be shown that the coherence time is proportional to photon number^{18}.

The theory qualitatively matches the experimental data well for small and moderate photon numbers, with the only adjustable parameter being *κ*. Photon number calibration uses the same method and value as Fig. 2. We assume here that the dye concentration is 2.4 mM; this is slightly higher than the expected value but within uncertainty, which is perhaps a factor of 2 (ref. ^{28}). The dye is changed between datasets and the concentration is not always constant, which means that the data in the two panels of Fig. 4 show small, but statistically significant, differences.

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## Acknowledgements

We thank R. Oulton for enlightening discussions. We are grateful to the UK Engineering and Physical Sciences Research Council for supporting this work through fellowship no. EP/J017027/1 (to R.A.N.) and the Controlled Quantum Dynamics CDT EP/L016524/1 (B.T.W. and H.J.H.). D.H. thanks the DFG cluster of excellence ‘Nanosystems Initiative Munich’. L.C.F., A.A.P.T and J.M.S. acknowledge support from the Leverhulme Trust.

## Author information

### Affiliations

#### Quantum Optics and Laser Science group, Blackett Laboratory, Imperial College, London, UK

- Benjamin T. Walker
- , Henry J. Hesten
- , Florian Mintert
- & Robert A. Nyman

#### Centre for Doctoral Training in Controlled Quantum Dynamics, Imperial College, London, UK

- Benjamin T. Walker
- & Henry J. Hesten

#### Department of Materials, University of Oxford, Oxford, UK

- Lucas C. Flatten
- , Aurélien A. P. Trichet
- & Jason M. Smith

#### Physikalisches Institut, Karlsruher Institut für Technologie, Karlsruhe, Germany

- David Hunger

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### Contributions

B.T.W. carried out the experiments with assistance from R.A.N., and both analysed the data. L.C.F., A.A.P.T., J.M.S. and D.H. fabricated the mirrors and assessed their performance. H.J.H. and R.A.N. and worked out the theory with assistance from F.M. R.A.N. conceived the experiment, and wrote the manuscript with input from all authors.

### Competing interests

The authors declare no competing interests.

### Data availability

The data underlying this manuscript, source code to reproduce the figures and any further details are available from the corresponding author upon reasonable request.

### Corresponding author

Correspondence to Robert A. Nyman.

## Supplementary information

### Supplementary information

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