Abstract
The dimensionality of a system profoundly influences its physical behaviour, leading to the emergence of different states of matter in many-body quantum systems. In lower dimensions, fluctuations increase and lead to the suppression of long-range order. For example, in bosonic gases, Bose–Einstein condensation in one dimension requires stronger confinement than in two dimensions. Here we observe the dimensional crossover from one to two dimensions in a harmonically trapped photon gas and study its properties. The photons are trapped in a dye microcavity where polymer nanostructures provide the trapping potential for the photon gas. By varying the aspect ratio of the harmonic trap, we tune from isotropic two-dimensional confinement to an anisotropic, highly elongated one-dimensional trapping potential. Along this transition, we determine the caloric properties of the photon gas and find a softening of the second-order Bose–Einstein condensation phase transition observed in two dimensions to a crossover behaviour in one dimension.
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Data availability
The data presented in this manuscript are available via Zenodo at https://doi.org/10.5281/zenodo.10571407 (ref. 44). Source data are provided with this paper.
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Acknowledgements
We acknowledge valuable discussions with E. Stein and A. Pelster. This work has been supported by the Deutsche Forschungsgemeinschaft through CRC/Transregio 185 OSCAR (project no. 277625399, all authors). We acknowledge financial support by the EU (ERC, TopoGrand, 101040409, J.Schmitt), by the DLR with funds provided by the BMWi (grant no. 50WM2240, F.V. and M.W.) and by the Cluster of Excellence ML4Q (EXC 2004/1-390534769, M.W. and J. Schmitt).
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G.v.F., J. Schulz, K.K.U. and F.V. conceived and designed the experiment. J. Schulz performed the nanofabrication. K.K.U. designed and performed the experiment. K.K.U., J. Schulz and F.V. analysed the data. J. Schmitt and M.W. contributed materials/analysis tools. All authors discussed the results. K.K.U., J. Schulz and F.V. wrote the initial paper. All authors contributed to the critical review of the paper.
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Extended data
Extended Data Fig. 1 Nanostructure surface profile.
a, AFM measurement of the surface of a printed isotropic paraboloid. b Sections through the AFM data as marked in (a) along the x axis (green) and the y axis (red) compared to a parabola with the curvature programmed for the 3D print (blue). c Difference of the programmed height profile and the measured height profile of the printed structures.
Extended Data Fig. 2 Momentum space distributions.
Exemplary momentum space distributions of the cavity fluorescence for the 2D (panel a) and the 2D-1D case (panel b) in the thermal (that is classical) and quantum degenerate regime. The red dashed circle denotes wavevectors with \({k}_{x}^{2}+{k}_{y}^{2}=2mV/{\hslash}^{2}\), and the black circle the numerical aperture of the imaging system. The side panels show distributions integrated along the kx direction, together with the corresponding expectation as a red dotted line. The broad distribution extending to regions with wavevectors which cannot be trapped by the potential (indicated by the grey shaded areas) is attributed to the emission from free-space modes not confined in the potential.
Extended Data Fig. 3 Spectum Analysis.
Exemplary spectrum of the cavity fluorescence for the 1D case. a, raw image of the spectrum on the spectrometer camera, the vertical axis is the spatial axis and the horizontal axis is both the (compressed) spatial and dispersive axis. The integration is along the vertical axis as marked by the arrow. b, the integrated, transmission-corrected spectrum, in a linear scale. Panel c shows the position of the observed modes from b as a function of the mode number. As expected for a harmonic oscillator potential, we observe a linear increase in mode energy with the mode number.
Extended Data Fig. 4 Thermal tail temperature.
Comparison of the thermal tail of the measured spectrum in the 2D case, together with the theoretical expectation, convoluted with the mode profiles for a temperature of 300K (panel a), 380 K (panel b) and 220 K (panel c), respectively. The data is the same as shown in Fig. 3a, note that only the thermal modes with energies at least h × 1THz above the ground mode are shown. The good agreement shows that our photon gas within uncertainties can be described by a room temperature distribution.
Extended Data Fig. 5 Raw spectrogram for the 2D potential.
Exemplary spectrum of the cavity fluorescence. In contrast to the main text, the raw spectrum for the 2D potential is here shown using a linear scale color map.
Extended Data Fig. 6 Alternative definition of \(\tilde{N}\).
Double log plot of the theoretically expected inner energy U/N as a function of photon number N (black points) for 1D (a), 2D-1D (b) and 2D (c). The green and red lines are fits in the linear regions before and after the curve starts changing slope. The photon number where both lines intersect is indicated by the black dashed line. This number is determined to be \(\tilde{N}=19\) (a), \(\tilde{N}=63\) (b), and \(\tilde{N}=611\) (c).
Extended Data Fig. 7 Binning effects on caloric properties for the 2D potential.
Binning effects on U/N and ∣μ∣ for the 2D potential. a–c, The mean value of the unbinned internal energy per particle U/N is plotted in grey in the upper row with corresponding theoretical expectations in the black solid curve. Binned data as mean values +/- SD with a geometric series spacing with common ratios 1.145, 1.271 and 1.381 are plotted in blue, green and red in three separate columns. d–f, Corresponding data for the chemical potential ∣μ∣ (symbols) is plotted in the lower row with corresponding theoretical expectations in the black solid curve.
Extended Data Fig. 8 Binning effects on caloric properties for the 2D-1D potential.
Binning effects on U/N and ∣μ∣ for the 2D-1D potential. a–c, The mean value of the unbinned internal energy per particle U/N is plotted in grey in the upper row with corresponding theoretical expectations in the black solid curve. Binned data {as mean values +/- SD} with a geometric series spacing with common ratios 1.189, 1.278 and 1.523 are plotted in blue, green and red in three separate columns. d–f, Corresponding data for the chemical potential ∣μ∣ (symbols) is plotted in the lower row with corresponding theoretical expectations in the black solid curve.
Extended Data Fig. 9 Binning effects on caloric properties for the 1D potential.
Binning effects on U/N and ∣μ∣ for the 1D potential. a–c, The mean value of the unbinned internal energy per particle U/N is plotted in grey in the upper row with corresponding theoretical expectations in the black solid curve. Binned data {as mean values +/- SD} with a geometric series spacing with common ratios 1.144, 1.198 and 1.318 are plotted in blue, green and red in three separate columns. d–f, Corresponding data for the chemical potential ∣μ∣ (symbols) is plotted in the lower row with corresponding theoretical expectations in the black solid curve.
Source data
Source Data Fig. 2
2D (x–y) csv data as well as tiff images for the individual panels.
Source Data Fig. 3
2D (x–y) csv data as well as tiff images for the individual panels.
Source Data Fig. 4
2D (x–y) csv data including error bars.
Source Data Fig. 5
2D (x–y) csv data including error bars.
Source Data Extended Data Fig./Table 1
2D (x–y) csv data, 3D csv data and tiff images for the individual panels.
Source Data Extended Data Fig./Table 2
2D (x–y) csv data as well as source images (tiff or pdf) for the individual panels.
Source Data Extended Data Fig./Table 3
2D (x−y) csv data of the individual panels including error bars.
Source Data Extended Data Fig./Table 4
2D (x–y) csv data of the individual panels including error bars.
Source Data Extended Data Fig./Table 6
2D (x–y) csv data of the individual panels including error bars.
Source Data Extended Data Fig./Table 7
2D (x–y) csv data of the individual panels including error bars.
Source Data Extended Data Fig./Table 8
2D (x–y) csv data of the individual panels including error bars.
Source Data Extended Data Fig./Table 9
2D (x–y) csv data of the individual panels including error bars.
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Karkihalli Umesh, K., Schulz, J., Schmitt, J. et al. Dimensional crossover in a quantum gas of light. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02641-7
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DOI: https://doi.org/10.1038/s41567-024-02641-7
