Slow reflection and two-photon generation of microcavity exciton-polaritons

We resonantly inject polaritons into a microcavity and track them in time and space as they feel a force due to the cavity gradient. This is an example of"slow reflection,"as the polaritons, which can be viewed as renormalized photons, slow down to zero velocity and then move back in the opposite direction. These measurements accurately measure the lifetime of the polaritons in our samples, which is 180 $\pm$ 10 ps, corresponding to a cavity leakage time of 135 ps and a cavity $Q$ of 320,000. Such long-lived polaritons propagate millimeters in these wedge-shaped microcavities. Additionally, we generate polaritons by two-photon excitation directly into the polariton states, allowing the possibility of modulation of the two-photon absorption by a polariton condensate.


I. INTRODUCTION
Since the initial observation of exciton-polaritons in a strongly coupled microcavity in 1992 1 , a wide range of quantum many-body effects have been observed in polariton fluids such as Bose-Einstein Condensation 2,3 , and superfluidity exhibiting quantized vortices 4 and solitons 5 . Most of these results have been interpreted in terms of nonequilibrium Bose gas theory, because the thermalization of the polaritons has been limited by their short cavity lifetime, on the order of 10 ps, compared to a thermalization time of the order of 1 ps.
Our recent results 6,7 have indicated that we can now produce structures which allow much longer lifetime, of the order of 200 ps. Here we report on accurate measurements of this lifetime using a unique method in which we inject polariton pulses at finite momentum into a microcavity and track their motion in time and space. This allows us to observe "slow reflection," in which renormalized light slows down to zero velocity, turns around, and goes back the other way. In addition to providing a measure of the lifetime, the long-distance propagation seen here allows the possibility of beam-like polariton-interaction experiments and all-optical switching methods over long distances.
As the technology of microcavity polaritons is now well established, much attention has turned to increasing the lifetime of the polaritons, to allow better thermalization and to allow propagation over longer distances. The lifetime of polaritons is a function of the intrinsic photon lifetime of the cavity and the fraction of photon in the polariton states. As amply discussed elsewhere 8 , a polariton state |P k is a superposition of an exciton state |e k and a photon state |γ k , where α k and β k are the k-dependent Hopfield coefficients. The ± signs indicate that there are two superpositions, known as the upper and lower polaritons; in the experiments reported here we focus entirely on the lower polariton branch. At resonance, α k = β k = 1/ √ 2, while far from resonance the polariton can be nearly fully photon-like or exciton-like. This implies that the k-dependent lifetime τ k of the polaritons is given by For polaritons in our GaAs-based samples, the rate of nonradiative recombination τ nonrad is negligible, so the lifetime is essentially entirely determined by the photon fraction and the cavity lifetime. In early polariton experiments 2,3,9,10 , the cavity lifetime was on the order of 1 ps while the polariton lifetime was at most 10-15 ps, even well into excitonic detunings. These measurements confirm the earlier estimates of the lifetime but considerably reduce the uncertainty.

II. ONE-PHOTON RESONANT INJECTION
The sample was arranged such that the gradient was aligned with the streak camera time slit, and then polaritons were injected at a large angle such that they moved directly against the gradient. The experimental setup is shown in Figure 1. We used an objective with a wide field of view in addition to a large numerical aperture. A resonantly injected picosecond pulse of polaritons was tracked as it entered the field of view, turned around and traveled away. This occurs because the sample has a cavity thickness variation that leads to an energy gradient of the polariton. In simple terms, one can think of the motion of the polaritons as governed by energy conservation with the following Hamiltonian, which is just the same as that of a massive object moving in a potential gradient: Here m eff is the effective mass of the lower polariton branch that we observe, which depends weakly on k, and is equal approximately to 5×10 −5 times the vacuum electron mass in these experiments. The force F is given by the gradient in space of the k = 0 cavity resonance energy, and is approximately equal to 10.5 meV/mm for the section of the microcavity studied here. We will refer to "uphill" as moving toward higher cavity resonance energy (narrower cavity width) and "downhill" as moving to lower energy (wider cavity width).
This experimental setup utilizes the fact that the polaritons in these high-Q samples flow over a great spatial distance and change in-plane momentum rapidly. The lifetime of shorterlived polaritons is more difficult to directly observe by streak camera measurements due to the overlap of any emission with the injecting laser. Upon resonantly injecting polaritons, the created population is in the same state as the exciting laser. The initial polariton population therefore will have the same characteristics as the exciting laser and cannot be separated from it. Observing any other state (for example by looking at cross-polarized emission) will inherently measure the scattering time of the polaritons to enter that state.
In this experiment, we rely on the fact that polaritons will flow ballistically from the point of injection to the point of detection in order to separate the observed luminescence from the reflected laser. To the extent that this motion is ballistic, integrating the population over the observed spatial region will directly yield the population decay of the polaritons.
Unlike the case of observing luminescence from a different energy or polarization state than the initial population, this method directly follows the decay of a single population rather than relying on an average over many k-states.
The momentum of the injected polaritons is controlled by the angle of the laser which generates them. The angle of incidence used here was ∼ 42 • , corresponding to an initial wavevector of 5.5 × 10 −4 cm −1 . After propagating uphill for over two millimeters, the polari-  To measure the lifetime, the bright jet of polaritons was time-resolved using a Hamamatsu streak camera. To facilitate this, the sample was initially installed such that the gradient was aligned with the horizontal time slit on the streak camera. This enabled us to track a single jet of polaritons while they propagate against the gradient, turn around, and travel backwards, as shown in Figure 3(a). The vertical distance axis in this figure corresponds to the horizontal x-axis in Figure 2. The trajectory of the polaritons is easily seen in the data, which in this region is well described by a parabolic fit, as expected for the Hamiltonian (3), which is equivalent to that of a ball moving with a constant force due to gravity. Indeed, these data directly demonstrate the in-plane velocity and acceleration of the polaritons during their trajectory. One should note that this region of observation is already more than a millimeter and nearly 200 ps from the injection point, indicating that these polaritons are propagating farther and persisting longer than those in earlier samples, even without confinement in 1D structures, such as used in Ref. 12 .
A simple analysis of this data yields the polariton lifetime after summing in the spatial dimension, as shown in Figure 3  135 ps, which corresponds to a Q-factor of over 320,000.
It should be noted that this lifetime measurement may still be an underestimation of the lifetime. Close inspection of Figure 2 reveals that individual jets of polaritons are still spreading out from the central jet. A population with some spread in initial momenta perpendicular to the cavity gradient must spread out horizontally while propagating uphill.
The fraction of polaritons that move out of our field of view will lead to an underestimation of the lifetime. This error can be compensated for by using a narrower time slit to cut out adjacent jets at early times; however, narrow slit widths can result in errors that will either underestimate or overestimate the polariton lifetime if the entirety of the main jet is not aligned with the time slit. In this experiment, data was collected over a range of slit widths from 50 to 300 µm with consistent results.

III. TWO-PHOTON RESONANT INJECTION
x-distance (mm)  In another set of experiments, we generated polaritons via a 200-fs pump pulse tuned to one-half the energy of the polaritons. Such experiments will be presented in more depth elsewhere, but we can see here that the propagation of long-lived polaritons can be used to probe the dynamics of two-photon generated polaritons. We conducted two experiments.
First, we recreated geometry of the one-photon experiments discussed above by injecting polaritons uphill and observing the turn-around point, as shown in Figure 4(a). Second, we injected polaritons at normal incidence, directly at the point of observation, as shown in Figure 4(b). In Fig. 4(b) we present data from a different setup employing two-photon generation of polaritons. In this case, we pump at normal incidence directly through the microscope objective used to image the luminescence. Since the pump laser has wavelength far from the polariton wavelength, there is no difficulty with scattered laser light. Since the entire NA of the objective was used, the angle of incidence of the pump light ranged over ± 20 • even though the intensity was maximum at 0 • and the laser was spectrally tuned to the k = 0 state. The data indicate that polaritons created directly via two-photon absorption are peaked at an initial wavevector uphill, with no k = 0 polaritons created initially. The long-lived population stationary at x = 0 is substrate luminescence excited by two-photon absorption. Figure 4(c) shows the same data as Figure 4(b), except that the image is centered to eliminate the clipping at x < 0 in (b). The majority of the injected polaritons have finite k, even though the pump light was centered at k = 0. This supports the view that that two-photon absorption of the polaritons, which should be forbidden at k = 0 due to the selection rules, becomes allowed at finite k, due to valence-band mixing with the higher-lying light-hole states.
These results show clearly that two-photon resonant generation of polaritons is possible.
One can expect very strong nonlinear effects from microcavity polaritons due to the strong interaction of the cavity mode with the quantum well exciton 13,14 . We point out that there is no comparable one-photon excitation experiment-such an experiment will not work because the exciting laser will be reflected directly back into the imaging system.

IV. CONCLUSIONS
Polaritons can be viewed as "renormalized photons," especially in the region of the cavity where the the detuning makes the polaritons mostly photon-like. As mentioned above, the behavior we have seen here can thus be viewed as a type of "slow light," or "slow reflection," in which the photons decelerate from  well as exciton energies. The polariton exhibits a Rabi coupling of 6 meV at 5 K; the cavity mode gradient is 13 meV/mm and the exciton gradient is 1.5 meV/mm.

METHODS
The sample was held in a cold-finger cryostat at 5 K for all experiments.
Emission was collected using a N.A.=0.42 microscope objective. A preliminary imaging lens permitted spatial filtering of the real space image data, and a subsequent iris in the Fourier image plane permitted filtering of the emission angle. Secondary lenses could be exchanged to image either the real-space or angle-resolved emission. Luminescence was imaged through a spectrometer onto either a standard CCD or onto a Hamamatsu streak camera.

ONE-PHOTON INJECTION
In the resonant injection experiment, polaritons were resonantly injected at λ = 778 nm with a picosecond laser far on the photonic side of the sample. The injected state had a detuning of approximately -2.6 meV and corresponded to an external angle of roughly 42 • (k = 5.5 × 10 −4 cm −1 ). The sample and pump laser were arranged such that the polaritons were moving anti-parallel to the cavity gradient, which was aligned with the time slit. The angle of incidence was larger than the collection angle of the optics, so the reflected beam was not collected. Additionally, the pump spot was spatially outside the field of view such that scattered light was not collected. At a distance of approximately 2 mm from the injection point, emission entered the collection range of the optics. At the turn-around point, the polaritons are more photonic with a detuning of -7.4 meV which corresponds to a photon fraction of 75%.

TWO-PHOTON INJECTION
For the two-photon injection experiment, a 200-fs pulse generated by a Coherent OPA system was tuned to one-half the energy of the desired polariton transition. For the 40 • injection case, the laser was tuned to excite at λ = 778 nm. For the 0 • injection case, the laser was tuned to excite at λ = 780 nm.

ANALYSIS
The x-distance from the injection point for the 40 • injection cases was estimated as follows: the x-distance in figs. 2,3(a), and 4(a) was determined from the fit of the polariton x vs t trajectory presented in fig. 3(a). Extrapolation of this fit back to time t = 0 as determined by locating scattered laser light determines the initial position of excitation. This initial excitation position is consistent with the sample parameters and injection conditions. This method assumes that the acceleration of the polaritons is strictly constant from creation to turn around. Variation in the acceleration due to a non-constant energy gradient in addition to the changing mass of the polariton implies uncertainty on the overall offset of this axis, but the spatial magnification was measured directly.