Long-range ballistic motion and coherent flow of long-lifetime polaritons

Exciton polaritons can be created in semiconductor microcavities. These quasiparticles act as weakly interacting bosons with very light mass, of the order of ${10}^{-4}$ times the vacuum electron mass. Many experiments have shown effects which can be viewed as due to a Bose-Einstein condensate, or quasicondensate, of these particles. The lifetime of the particles in most of those experiments has been of the order of a few picoseconds, leading to significant nonequilibrium effects. By increasing the cavity quality, we have made samples with longer polariton lifetimes. With a photon lifetime on the order of 100–200 ps, polaritons in these structures can not only come closer to reaching true thermal equilibrium, a desired feature for many researchers working in this field, but they can also travel much longer distances. We observe the polaritons to ballistically travel on the order of 1 mm, and at higher densities we see transport of a coherent condensate, or quasicondensate, over comparable distances. In this paper we report a quantitative analysis of the flow of the polaritons both in a low-density, classical regime, and in the coherent regime at higher density. Our analysis gives us a measure of the intrinsic lifetime for photon decay from the microcavity and a measure of the strength of interactions of the polaritons.

Figure 1

Intensity of the PL emitted from the LP, as a function of energy $E$ and position $x$, recorded with an imaging spectrometer. The intensity data are presented on a log scale to highlight motion. These data are taken at a $k_{\parallel }=0$ polariton detuning of $-21$ meV with a pump power of 500 $\mu$W at 705 nm focused to a 15-$\mu$m-diameter spot size ($0.28\times {10}^6$ mW/cm${}^2$). (a) Hot-carrier luminescence seen through the reflectivity minima of the DBR stop band. The size of this spot indicates the size of the pump spot and the exciton cloud. (b) Lower polariton PL, spatially resolved but only collected near $k_{\left|\right|}=0$. The bright spot is the point of creation of the polaritons; the PL at further distances gives the $k_{\left|\right|}=0$ energy of polaritons which have moved to that point on the sample. (c) The same data taken with a larger NA, i.e., a larger range of $k_{\parallel }$. The polaritons flow outward from the creation spot to fill all space within our field of view. The polaritons are generated over a broad range of $k_{\parallel }$ at the pump spot and ballistically travel outward at constant energy. The sharp cutoff in energy on the downhill side indicates that the polaritons do not scatter once they are spatially distant from the excitation spot. The horizontal/angled cutoff at high energy is the accepted NA of the microscope objective. The cutoffs at $\pm 0.2$ mm are due to clipping in the optics and spectrometer.

Figure 2

Intensity of the PL emitted from the LP, as a function of energy $E$ and in-plane momentum $k_{\parallel }$, recorded using an imaging spectrometer focused on the far-field emission (Fourier plane). These data were taken under the same pumping conditions as Fig. 1. (a) Spatial filtering is applied to an intermediate image to isolate the dispersion relation of the LP at the pump spot. Due to nonresonant excitation, polaritons are observed filling the momentum states. (b) With no spatial filtering, the excitation spot polaritons are smeared in the downhill ($-k_{\parallel }$) direction. Polaritons at $k_{\parallel }=0$ correspond to the polaritons observed in Fig. 1. Again we observe an energy minimum coinciding with the vertex of the pump spot dispersion curve, as the polaritons scatter very little after leaving the creation region.

Figure 3

Time-resolved $k_{\parallel }\simeq 0$ PL from the lower polartion at three sample distances from the pump spot. These data were collected following a 2-ps, 2-mW pump laser with wavelength of 725 nm focused to a 50-$\mu$m-diameter pump spot ($0.1\times {10}^6$ mW/cm${}^2$ or 1.3 $\mu$J/cm${}^2$ per pulse) where the $k_{\parallel }=0$ polariton detuning was $-15$ meV. Blue lines are intensity data of photoluminescence from propagating polaritons. Each frame is taken at a different distance from the pump spot. Black lines indicate the emission of the hot carriers above the stop band which occurs very soon after the picosecond pump. Red lines are the Gaussian-exponential decay convolution fits to the data with the parameters given above each frame. $t_0$ is the central time of the Gaussian following the hot PL, $\sigma$ is the standard deviation, and $\tau$ is the exponential decay time. Note that $t_0$ is an indicator of the travel time—we know that this value must include both the time of flight as well as the time to cool down from hot carriers to the lower polariton. As an aid to the reader, the unconvolved Gaussian is presented as the dashed green line to see how the $t_0$ parameter compares to the peak of the intensity data. The convolution with a decay pushes the peak of the fit to significantly later time than the Gaussian fit alone.

Figure 4

Time of flight for polaritons with different initial momenta to reach $k_{\parallel }=0$ for the same pumping conditions as Fig. 3. Black crosses with error bars: the time of flight as determined from the time-resolved data. Blue solid line: fit of the data assuming a constant mass and constant gradient of potential energy (i.e., constant force) felt by the polariton. This model clearly fails to describe the later time arrivals. Green curve: fit of the data assuming a constant force on the polaritons but allowing for the full dispersion relation $E\left(k\right)$ of the polaritons, which has effective mass that changes at higher $k_{\parallel }$. Accounting for this changing mass improves the fit only slightly. Red curve: fit calculated by numerically propagating $x\left(t\right)$ and $k\left(t\right)$ according to the full semiclassical Hamiltonian of the LP.

Figure 5

Lifetime of the polaritons based on the normalized intensity vs a time of flight. The time values for the red crosses (data points) are the time-of-flight data presented in Fig. 4. The intensity values of these data points are the intensity detected at the point and time of measurement, normalized by the intensity at the same polariton energy taken from $k$-resolved data under the same conditions as Fig. 2 except that the pump spot detuning was the same as the time-resolved conditions. Since the emission at each spatial point corresponds to a single initial $k_{\parallel }$ state at the pump spot, this ratio gives the loss during the spatial propagation due to radiative emission and other scattering processes. The solid black line is a fit of a single exponential decay with lifetime of 200 ps.

Figure 6

Lifetime measurement based on the steady state $k$-resolved PL intensity data. (a) The same data as presented in Fig. 2, with a highlight showing a selected detuning to generate an intensity profile. (b) The intensity profile for the selected detuning. Dashed line: fit to a single exponential decay in time. Note that the polaritons travel uphill and come back down. We therefore restrict the fits to times before the polaritons have returned back to the same place, which corresponds to $k_{\parallel }$ equal but opposite the initial $k_{\parallel }$. The polaritons moving downhill from the generation point are ignored due to noise in the data and the fact that they are observed for a short period of time which renders the fits unreliable. The time calibration in this plot is generated using the $k\left(t\right)$ prediction at each energy based on the initial conditions and applying Hamilton's method, as used for the fit of Fig. 4.

Figure 7

Panels (a)–(d) show $k$-resolved PL from the polariton population at pump powers of 0.25, 21, 30, and 35 mW, respectively (pump densities of $0.14\times {10}^6$, $12\times {10}^6$, $17\times {10}^6$, and $20\times {10}^6$ mW/cm${}^2$). These data were collected using a pump laser with wavelength of 705 nm focused to a 15-$\mu$m-diameter pump spot where the LP detuning was $-8$ meV. Panels (e)–(h) show $k_{\parallel }\sim 0$ real-space-resolved emission at the same densities. Note that at the lowest density [(a) and (e)], all of $k$ space is occupied at the emission spot and the polaritons roll uphill and downhill as discussed above. However, as the pump power increases and renormalization occurs at the pump spot, a larger occupation builds up in the $k_{\parallel }=0$ state on top of the potential-energy hill at the pump spot. The high occupation of a single state is seen as a monoenergetic line in $k$ space and two spots in the low-NA, real-space data, corresponding to the excitation spot and the turnaround point 200 $\mu$m away. In real space only two spots are observed because the polaritons in between, as well as those traveling downhill, are outside the angle of emission being imaged.

Figure 8

Interference measurements conducted by overlapping PL from the pump spot with time-delayed PL from the turnaround point in the medium density regime. These data were collected using a 34-mW pump laser with wavelength of 705 nm focused to a 25-$\mu$m-diameter pump spot ($7\times {10}^6$ mW/cm${}^2$) where the LP detuning was $-4.5$ meV. Panel (a) shows the real-space luminescence from the individual points and a sample interference pattern. Panel (b) plots the visibility of the fringes as a function of delay time. The fact that the greatest visibility is seen at 140 ps makes perfect sense as this is the propagation time for the polaritons to travel 200 $\mu$m from the pump spot to the turnaround point. The high scatter and overall low visibility of the fringes is primarily due to the instability in the pump laser, which leads to instability of the blueshift peak on which the polariton quasicondensate is formed and therefore causes the both the condensate energy and turnaround point to fluctuate.

Figure 9

Comparison of simulated state evolution with observed real-space intensity. The data were collected under the same conditions as Fig. 8. Note that three simulated lifetimes are presented for comparison.