Algebraic order and the Berezinskii-Kosterlitz-Thouless transition in an exciton-polariton gas

We observe quasi-long-range coherence in a two-dimensional condensate of exciton-polaritons. Our measurements confirm that the spatial correlation algebraically decays with a slow power law, whose exponent quantitatively behaves as predicted by the Berezinskii-Kosterlitz-Thouless theory. The exciton-polaritons are created by nonresonant optical pumping of a microcavity sample with embedded GaAs quantum wells at liquid helium temperature. Michelson interference is used to measure the coherence of the photons emitted by decaying exciton-polaritons.

Figure 1

Pump-power-dependent density distribution. The irregular density profile at very high pump powers prevents us from extracting the exponent $a_{\mathrm{p}}$ in the very high-power regime (measured at $\delta \approx -1\mathrm{meV}$).

Figure 2

Blueshift of exciton-polariton energy. (a) Pump-power-dependent dispersion (measured at $\delta \approx -1\mathrm{meV}$). Around 6 mW, the condensation phase appears, and above this pump power, nearly all exciton-polaritons are in a ground state with zero momentum and one specific energy. A blueshift towards higher energies due to repulsive interaction between the exciton-polaritons is observed above the condensation threshold. (b) Normalized signal close to zero momentum at blue detuning of $\delta \approx -1\mathrm{meV}$. The measurement of the exciton-polariton density, described earlier, indicates that at the threshold of $p_{\mathrm{pump}}\approx 6\mathrm{mW}$, the total exciton-polariton density is $n\approx 2\mu {\mathrm{m}}^{-2}$, which corresponds to an exciton-polariton density per quantum well of $n_{\mathrm{per}\mathrm{QW}}\approx 0.5\mu {\mathrm{m}}^{-2}$. This small value confirms that we observe condensation rather than vertical cavity surface-emitting laser (VCSEL) lasing, because the latter requires significantly higher exciton-polariton densities of approximately $3\times {10}^3\mu {\mathrm{m}}^{-2}$ per quantum well to create a population inversion. (c) The same at red detuning of $\delta \approx 4\mathrm{meV}$.

Figure 3

Michelson interference. (a) Michelson interferometer setup to measure the phase and the fringe visibility. (b) Measured interferogram (with intensity in arbitrary units) for one specific path-length difference $L$, measured at a pump power above the condensation threshold. (Same measurement as in the left column of Fig. 4) (c) The intensity of each pixel behaves like a sine function of the form $I_{\mathrm{measured}}\left(\Delta L\right)=B+Asin\left(w\Delta L-{\phi }_0\right)$ if plotted as a function of the change $\Delta L$ of the path-length difference $L$. (d) Same as (b), but at a low pump power below the condensation threshold, where no spatial coherence exists over long distances. (See also Fig. 9 for the lack of coherence at low pump powers.)

Figure 4

Measured phase and visibility (measured at detuning parameter $\delta \approx -1\mathrm{meV}$). Left column: Above the condensation threshold (at $p_{\mathrm{pump}}=15\mathrm{mW}$). (a) Phase map ${\phi }_0$ produced by an ensemble of pixels. (b) Visibility map. The flip axis is shown as a line at $x=0$, and the region of interest used for further evaluation is highlighted by a rectangle. (c) The visibility within the region of interest as a function of the directed distance $x$ to the flip axis. In region II, a power-law fit with exponent $a_{\mathrm{p}}=0.082$ is shown. The regions I–III are selected manually. (Insert: The same in a log-log plot, where the power-law decay appears as a straight line.) Right column: The same characteristics as those below the condensation threshold (at $p_{\mathrm{pump}}=2\mathrm{mW}$). Absence of quasi-long-range order is clearly seen in (d)–(f), and the fitted line in (f) is a Gaussian decay function.

Figure 5

Simulated visibility. (a) Assumed position-dependent superfluid fraction as a function of radial position $r$. (b) Simulated visibility assuming $g^{\left(1\right)}=\left(n_{\mathrm{s}}/n\right){\left(\left|x\right|/0.02\mu \mathrm{m}\right)}^{-0.1}$. (c) Simulated visibility at $y\approx 0$. The simulated fast decay in region III shows that the fast decay in Fig. 4 can be explained by a decreasing superfluid fraction.

Figure 6

Transition from power law to fast decay of visibility. The dots show the cutoff points where the power law ends. They seem to form a circle that corresponds to the rotational symmetry of the Gaussian pump beam.

Figure 7

Fitting various decay functions to the visibility data. (a) The used decay functions are a power law of the form $f_{\mathrm{p}}\left(x\right)=c_{\mathrm{p}}{\left|x/\mu \mathrm{m}\right|}^{-a_{\mathrm{p}}}$, exponential function $f_{\mathrm{e}}\left(x\right)=c_{\mathrm{e}}exp\left(-a_{\mathrm{e}}\left|x/\mu \mathrm{m}\right|\right)$, and Gaussian function $f_{\mathrm{g}}\left(x\right)=c_{\mathrm{g}}exp\left[-a_{\mathrm{g}}{\left(x/\mu \mathrm{m}\right)}^2\right]$. The fits have been performed by minimizing the RMS deviation $D_i=\sqrt{\underset{n}{\Sigma }{\left[f_i\left(x_n\right)-V_n\right]}^2/N}$ between the fitted functions $f_i$ (with $i\in \left\{\mathrm{p},\mathrm{e},\mathrm{g}\right\}$) and the $N$ measured positions and visibilities $\left(x_n;V_n\right)$ with $x_{min}<\left|x_n\right|<x_{max}$, where the limits of the intermediate range $x_{min}$ and $x_{max}$ (marked by black vertical lines) have been chosen manually. We use the same fit for the $x<0$ and the $x>0$ regions. The visibility shown here has been measured at $\delta \approx -1\mathrm{meV}$ and $p_{\mathrm{pump}}=19\mathrm{mW}$. (b) The relative RMS deviations $R_i=\frac{D_i}{D_{\mathrm{p}}}$ measured at $\delta \approx -1\mathrm{meV}$. Above the condensation threshold, we always observed $D_{\mathrm{p}}<D_{\mathrm{e}}<D_{\mathrm{g}}$, which means that the power-law decay gives the best match and the Gaussian fit gives the worst match. (c) The pump power dependence of the factors $c_{\mathrm{p}}$ determined by this fit at $\delta \approx -1\mathrm{meV}$ and at $\delta \approx 4\mathrm{meV}$. The dashed line is introduced to guide the eye.

Figure 8

Pump-power dependence. Upper row: Pump-power dependence of the exponent $a_{\mathrm{p}}$ for two detuning values. The symbols show the measured exponents as determined by a power-law fit as shown in Fig. 4. The gray horizontal line at 0.25 shows the theoretically predicted exponent at the threshold. The black continuous line shows the estimated inverse superfluid phase-space density $1/\left(n_{\mathrm{s}}{\lambda }_T^2\right)$, and as predicted by the theory, it matches the measured exponents $a_{\mathrm{p}}$. Filled symbols correspond to the $x<0$ region. Lower row: Pump-power-dependent peak intensity, as determined by an energy and momentum resolved measurement. The threshold pump powers defined by the critical exponent $a_{\mathrm{p}}=0.25$ (upper traces) are indicated by the black arrows.

Figure 9

Phase-space density, visibility, and power-law exponent. As in Fig. 8, filled symbols correspond to the $x<0$ region, and different kinds of symbols correspond to different detunings. (a) The visibility at $x=\pm 2\mu \mathrm{m}$ versus the inverse phase-space density $1/\left(n{\lambda }_T^2\right)$. The threshold is upper bounded by $1/\left(n_{\mathrm{s}}{\lambda }_T^2\right)=1/4$ and lower bounded [34] by $1/ln\left[380{\hslash }^2/\left(m_{\mathrm{eff}}{g}_{\mathrm{interaction}}\right)\right]=1/10$, which are shown by the shaded area. (b) Measured exponents $a_{\mathrm{p}}$ as a function of estimated inverse superfluid phase-space density $1/\left(n_{\mathrm{s}}{\lambda }_T^2\right)$. The black line shows the predicted $a_{\mathrm{p}}^{\mathrm{calculated}}=1/\left(n_{\mathrm{s}}{\lambda }_T^2\right)$, and the region above the expected threshold of 0.25 is shaded in gray. Although we did not change the temperature during the measurement, the abscissa can be interpreted as the dimensionless temperature (in units of $2\pi m_{\mathrm{eff}}^{-1}{k}_{\mathrm{B}}^{-1}{n}_{\mathrm{s}}{\hslash }^2$), and the BKT phase transition occurs at the dimensionless temperature of 0.25 in these units.

Figure 10

Determination of the exciton-polariton temperature. (a) Position $x$ and energy $E$ resolved intensity. (b) Measured population (black) with Boltzmann (straight line) and Bose-Einstein distributions for 25 K. (c) Effective temperatures determined by fitting a thermal distribution to the population (performed for different pump powers in the case of a detuning of $\delta \approx -1\mathrm{meV}$).

Figure 11

Measured visibility at a detuning of −1 meV. One can see the existence of quasi-long-range order above the condensation threshold but only Gaussian decay below the condensation threshold.

Figure 12

Measured visibility at a detuning of 4 meV.

Figure 13

Simulated power law. (a) Phase $\Theta$ in one run of the simulation. (b) and (c) Visibility averaged over 1000 runs of the simulation. (d) The same visibility shown as a log-log plot, where the power law appears as a straight line.