Phase-space views into dye-microcavity thermalized and condensed photons

We have observed momentum- and position-resolved spectra and images of the photoluminescence from thermalized and condensed dye-microcavity photons. The spectra yield the dispersion relation and the potential energy landscape for the photons. From this dispersion relation, below condensation threshold, we find that the effective mass is that of a bare cavity photon, not a polariton. Above threshold, we place an upper bound on the dimensionless two-dimensional interaction strength of $\tilde{g}\lesssim {10}^{-3}$, which is compatible with existing estimates. Both photon-photon and photon-molecule interactions are weak. The temperature is found to be independent of momentum, but dependent on pump spot size, indicating that the system is ergodic but not perfectly at thermal equilibrium. Condensation always happens first in the mode with lowest potential and lowest kinetic energy, although at very high pump powers multimode condensation occurs into other modes.

Figure 1

Left: a schematic of the optics for momentum- and position-resolving spectroscopy. Double solid-dashed lines are beam splitters. Lenses are marked with double-headed arrows. Right: an example momentum-resolved spectrum. The only post-processing performed on the color image is taking the square root for display purposes. Calibration of the axes is described later in this paper. Dark lines are dirt stuck on the slit in momentum space.

Figure 2

Momentum calibration using a graticule with 800 $\mu \mathrm{m}$ pitch in the back focal plane of the imaging objective. Each line corresponds to an in-plane momentum difference of $1.8\times {10}^{-29}$ N s. Left: a momentum-space image. Right: momentum-resolved spectroscopy, with the horizontal axis being energy increasing to the left.

Figure 3

Momentum-resolved spectra $n(\mathbf{p},E)$ of photoluminescence below threshold (top) and BEC just above (middle) and far above threshold (bottom). The pump spot size was $70\mu \mathrm{m}$, optimized to produce a thermal distribution of photons at room temperature. The grey dashed lines are dispersion relations for particles of mass $7.8\times {10}^{-36}$ kg. The blue dot-dashed lines on the lower two panels are the Bogoliubov dispersion relations assuming a dimensionless two-dimensional interaction parameter $\tilde{g}={10}^{-4}$ (see text for more details).

Figure 4

Temperature vs momentum for several pump spot sizes, derived from momentum-resolved spectra. Thermalization is not complete, but larger spot sizes lead to higher temperatures. For comparison, the thermal-equilibrium cloud size is 70 $\mu \mathrm{m}$. Temperature is independent of momentum, implying that populations depend only on energy so ergodicity is respected.

Figure 5

The measured effective mass as a function of cavity cutoff wavelength obtained by parabolas to the cutoff in the the momentum-resolved spectrum. The dashed line is the bare cavity photon mass. The dashed line uses Eq. (7), and we rule out any coupling strength greater than 1 THz. The shapes of the points represent discrete manual positioning of the diffraction grating, and each set is wavelength calibrated independently.

Figure 6

Momentum-space $\mathbf{p}=(p_x,p_y)$ images, $n\left(\mathbf{p}\right)$. As pump power is increased (top to bottom), to the thermal distribution (top panel) is added a momentum-narrow condensate (middle), followed by a broadened multimode condensate (bottom panel). The calibration is accurate for $\left|\mathbf{p}\right|<4\times {10}^{-29}$ N s.

Figure 7

Cuts through the same data as Fig. 6, top two panels. The greyed out region is known to suffer from pincushion distortion, rendering the calibration inaccurate. The dashed line is a thermal distribution at room temperature. The dot-dashed line is the expected momentum distribution for the noninteracting Bose-Einstein condensate.

Figure 8

Top: The measured parabolic curvature $a$ of the position dependence of the confining potential for photons depends on the longitudinal mode number (dots). Data were all taken with cutoff ${\lambda }_0=591\pm 0.7$ nm. A fit to Eq. (9) (solid line) allows us to calibrate the magnification in the position-resolved spectra. Part of the light penetrates the cavity mirrors, to an optical depth equivalent to $L_{\mathrm{off}}$, also obtained from the fit. Bottom: an example calibrated position-resolved spectrum $n(\mathbf{r},E)$ with the fitted parabolic potential.

Figure 9

Position-resolved spectra $n(\mathbf{r},E)$ (left side) and images $n\left(\mathbf{r}\right)$ with $\mathbf{r}=(x,y)$ (right side) of photoluminescence below threshold (top) and BEC just above (middle) and far above threshold (bottom). The parabolic curve of the harmonic potential energy is clearly visible. Condensation occurs at the center, where potential energy is lowest. At higher pump powers, multimode condensation is observed.