Defect Proliferation at the Quasicondensate Crossover of Two-Dimensional Dipolar Excitons Trapped in Coupled GaAs Quantum Wells

We study ultracold dipolar excitons confined in a $10\mu \mathrm{m}$ trap of a double GaAs quantum well. Based on the local density approximation, we unveil for the first time the equation of state of excitons. Specifically, in this regime and below a critical temperature of about 1 K, we show that for a local density $n\sim (2-3)\times {10}^{10}{\mathrm{cm}}^{-2}$ a coherent quasicondensate phase forms in the inner region of the trap, encircled by a more dilute and normal component in the outer rim. Remarkably, this spatial arrangement correlates directly with the concentration of defects in the exciton density, which is strongly decreased in the quasicondensed region, consistent with a superfluid phase. Thus, our observations point towards a Berezinskii-Kosterlitz-Thouless crossover for two-dimensional excitons.

Figure 1

Equation of state of the trapped gas. (a) Sketch of the four nondegenerate exciton spin states, bright ($\pm 1$) and dark ($\pm 2$). Below a critical temperature of about 1 K, excitonic condensation leads to a macroscopic occupation of dark states coherently coupled to a lower population of bright excitons radiating the analyzed photoluminescence (wavy red lines). (b),(c) Photoluminescence resolved spectrally and spatially along the vertical axis of the electrostatic trap at $T_b=340\mathrm{mK}$. (b) 150 ns after the loading laser pulse the dispersion of the photoluminescence energy (dotted line) reflects the profile of the exciton density $n$, which is around $2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ at the trap center. (c) 350 ns after termination of the loading laser pulse the density at the trap center is about ${10}^9{\mathrm{cm}}^{-2}$ and we observe the energy profile of the trapping potential, well reproduced by a parabolic line shape (dashed line). (d) Phase space density $D=n{\lambda }_T^2$ as a function of the scaled chemical potential $\beta \mu =\mu /k_B{T}_b$. Experimental data are obtained by superposing density profiles measured at 3.5 K (gray), 2.3 K (red), 1.3 K (pink), and 0.33 K (blue), up to a maximum density $2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ at the trap center. Variations expected from Monte Carlo calculations are shown for $\tilde{g}=6$ and 7 (dotted and solid lines, respectively). The inset presents an enlargement of the dilute regime where we note a characteristic curvature of $D(\beta \mu )$.

Figure 2

Spatial coherence at $T_b=340\mathrm{mK}$. (a) Exciton phase space density $D$ measured across the trap at 340 mK (blue), 1.3 K (green), and 2.5 K (red), and for a central density of $2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$. Dashed lines represent the profiles expected by Monte Carlo calculations at each bath temperature. (b) The spatial coherence of the quasicondensate is assessed by splitting the photoluminescence in two equal parts, recombined after a lateral shift of $2\mu \mathrm{m}$ has been introduced. The interference visibility $V$ is measured by the amplitude modulation along the vertical direction, as shown by the right-hand panel for a profile taken in the inner part of the trap. (c) Variation of $V$ as a function of the distance to the trap center, for a statistical average of 90 experiments, where $n(||\mathbf{r}||=0)\sim 2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ at $T_b=340\mathrm{mK}$, i.e., for the same experimental conditions as in Fig. 1. The minimum contrast of 11% is given by the signal-to-noise ratio of our measurements, without any background subtraction.

Figure 3

Photoluminescence defects at $T_b=340\mathrm{mK}$. (a) Cartography of photoluminescence defects measured for 60 experiments where $n(||\mathbf{r}||=0)\sim 2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$, i.e., for the same experimental conditions as for Figs. 1 and 2. (b) Defect density $\mathcal{P}$ as a function of $n(||\mathbf{r}||=0)$, in the inner region of the trap ($||\mathbf{r}||\le 1.5\mu \mathrm{m}$) in red and in the outer region of the trap ($1.5\le ||\mathbf{r}||\le 3\mu \mathrm{m}$) in black. The blue area underlines the density range where quantum spatial coherence is resolved. (c) Spatial profile of the defect density for $n(||\mathbf{r}||=0)\sim 2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ (red), $1.8(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ (black), and $3.5(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ (blue). (d) Variation of the spatial interference contrast as a function of the defect density for $n(||\mathbf{r}||=0)\sim 2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$.

Figure 4

Density fluctuations vs bath temperature. (a) Cartography of photoluminescence defects measured for 60 experiments where $n(||\mathbf{r}||=0)\sim 2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ at 2.3 K. (b) Defect density at $T_b=2.3\mathrm{K}$ averaged in the inner region of the trap ($||\mathbf{r}||\le 1.5\mu \mathrm{m}$) in red and in the outer region of the trap ($1.5\le ||\mathbf{r}||\le 3\mu \mathrm{m}$) in black, as a function of the central density $n(||\mathbf{r}||=0$). (c) Defect density $\mathcal{P}$ resolved across the trap at $T_b=2.3\mathrm{K}$, for $n(||\mathbf{r}||=0)\sim 2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ (red), $1.5(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ (black), and $3.5(3)\times {10}^{10}{\mathrm{cm}}^{-2}$ (blue). Measurements correspond to an average of 60 experiments at every density. (d) Variation of the defect density averaged in the inner region of the trap ($||\mathbf{r}||\le 1.5\mu \mathrm{m}$) as a function of the bath temperature. For every measurement the central exciton density is kept at $2.7(3)\times {10}^{10}{\mathrm{cm}}^{-2}$.