Dimensional Crossover in a Quantum Gas of Light

TL;DR

This work addresses how dimensionality shapes Bose-Einstein condensation in a photonic quantum gas by engineering a tunable 2D-to-1D crossover in a dye-filled microcavity using polymer nanostructures. The authors implement a parabolic trap with controllable aspect ratio $\Lambda=\omega_y/\omega_x$ to transition from isotropic 2D to quasi-1D confinement and measure caloric properties via mode-resolved spectra, extracting the chemical potential $|\mu|$ and internal energy $U$. They observe a sharp second-order phase transition in the 2D regime that gradually softens into a smooth crossover in 1D, with intermediate cases showing partial softening consistent with an effective non-integer dimension and finite-size effects. The results establish a versatile photonic platform for studying dimensional crossover phenomena and potentially exploring driven-dissipative and correlation-rich regimes in low-dimensional quantum gases.

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 (BEC) in one dimension requires stronger confinement than in two dimensions. We experimentally study the properties of a harmonically trapped photon gas undergoing Bose-Einstein condensation along the dimensional crossover from one to two dimensions. 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 an 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.

Figures (5)

• Figure 1: Dye-filled microcavity experimental setup and cavity mirror nanostructing.a, Dye-filled microcavity experimental setup. The photon gas is created by pumping the intracavity dye solution using a laser beam spatially shaped using a spatial light modulator (SLM), and focused with a 10$\times$ objective into the microcavity. The cavity consists of two plane mirrors, with a polymer structure printed on one of them to provide a potential for the photons. The cavity emission is sampled using an imaging objective and subsequently analysed either spatially or spectrally. b, The polymer structure (refractive index $n_{s}$) surrounded by dye solution (refractive index $n$) results in a potential for the trapped photon gas. c, Direct laser writing scheme, using a focused laser beam to polymerize the photoresist on top of the mirror surface.
• Figure 2: Spatial density distribution. Density distribution of photons in the quantum degenerate regime in a 1D (a, $\Lambda =22$), 2D-1D (b, $\Lambda =5$) and 2D (c, $\Lambda =1$) harmonic oscillator potentials, respectively. Microscope images of the corresponding polymer structures on the cavity mirror are shown as insets. The dashed lines indicate the position of the cuts through the center of the cloud along the horizontal and vertical axis shown in the side and upper panel. The dashed black line in the cut panels shows the contribution from thermal modes, the solid red line the contribution of the ground mode, showing the macroscopic contribution from the ground mode in all panels. For the theoretical expectations we assume a Bose-Einstein distribution of the population within the modes, with the total photon number of $N = 54$ (1D), $N = 357$ (2D-1D) and $N = 3958$ (2D), respectively. The visible deviation in the 1D case is attributed to the emission of free-space modes which are excited at the rim of the potential.
• Figure 3: Photon gas spectroscopy.a, The integrated spectrum of the cavity fluorescence for 2D ($\Lambda=1$), 2D-1D ($\Lambda=5$) and 1D ($\Lambda=22$) harmonic oscillator potential and the black curve is the expected Bose-Einstein distribution of photons at $T=300K$, evaluated at the measured position of the harmonic oscillator modes. b, Imaging the cavity emission dispersed by a grating onto a camera (raw spectra) allows one to image the shape of the first few modes. For the 2D-1D case the energy states associated with the first excited mode of the tightly confined dimension appear at around the 5th mode. The photon number $N$ in each case is chosen in the quantum degenerate regime, such that all modes are visible, i.e. the population $N_0$ in the ground mode does not significantly exceed that of the excited modes, i.e. $N_0/N \approx 0.3$, $0.18$ and $0.3$ for 2D, 2D-1D and 1D respectively. Additional spectra deep in the quantum degenerate regime for 1D are shown in the online Methods.
• Figure 4: Ground vs Excited modes population.a - c, show the population in the ground mode (blue dots) and excited modes (green diamonds), for the 1D case ($\Lambda =22$, a, the 1D-2D case ($\Lambda =5$, b) and the isotropic 2D potential in c for a varying total number of photons. The solid lines give the theory expectations assuming a Bose-Einstein distribution within the modes. Panel d, compares the theory expectations for the population in the ground mode for a 1D (red) and a 2D (blue) potential with an equal number of energy levels, with trap depth of $1.2 {k_{B}T}$ (solid lines) for finite size system and quasi-infinite system (dash-dot lines) with a depth of $10 {k_{B}T}$. One clearly observes that the effects from the finite size of the system are weaker than the effects of the dimensional crossover. For better comparison, the horizontal axis for each data set is scaled to the photon number $\tilde{N}$, with $\tilde{N}=628$, 64 and 23 photons for the 2D, the 2D-1D and the 1D harmonic oscillator potentials, see text. Error bars showing the statistical standard deviations are of the order of the marker size. Data are presented as mean values +/- SD.
• Figure 5: Caloric properties. The change from a phase transition in 2D to a crossover in 1D is visible in the chemical potential. a, Symbols represent the measured change in the absolute of chemical potential $|\mu|$ in the units of thermal energy ($k_\mathrm{B}T$), from 1D to 2D (1D, 2D-1D and 2D) harmonic oscillator potentials as function of the normalised photon number $N/\tilde{N}$, where the zero-point energy is set to zero. Solid lines are theory expectations for corresponding harmonic oscillator potentials. b, The measured internal energy per particle (photon), in units of $k_\mathrm{B}T$, as a function of the normalised photon number $N/\tilde{N}$, from top to bottom, 2D, 2D-1D and 1D harmonic oscillator potential, with their corresponding theory expectations in solid lines. Error bars show statistical standard deviations and data are presented as mean values +/- SD.