Recent studies of two-photon excitation of exciton-polaritons in microcavities have considered the possibility of an allowed absorption process into the -state of the excitons that participate in the polariton effect. Here we report time-resolved measurements of two-photon excitation directly into the lower polariton states, invoking the state of the excitons. Although this process is forbidden by symmetry for light at normal incidence, it is allowed at a nonzero angle of incidence due to state mixing. We examine the polarization dependence of two-photon absorption at finite both theoretically and experimentally. Previous results should be reevaluated in light of the mechanism observed here.
© 2017 Optical Society of America
The exciton-polariton is a quantum superposition of light and matter that has been studied extensively for its bosonic properties. The canonical system consists of quantum well (QW) excitons embedded in a two-dimensional microcavity (for reviews see, e.g., Refs. [1–4]). At low temperatures, this system exhibits Bose–Einstein condensation [5–7], superfluidity [8–10], and quantized vortices [11–15] and may have applications as low-threshold coherent light source and highly nonlinear optical system. Polaritons are metastable particles, as they can leak through the mirrors into external photons.
Recently, two-photon excitation of exciton-polaritons has gained attention [16–18] for its possible application in polariton lasers [16,19,20]. In general, two-photon excitation of polaritons provides another way of using the super-nonlinearities of the polaritons for optical modulation schemes. References [16,18] proposed a mechanism via absorption into the state, while Ref.  investigated this claim using a narrowband source and cast doubt on that mechanism because they saw no strong two-photon absorption at the energy of the state. Since there exist other mechanisms that could permit the conversion of dark excitons into the lower polaritons, in this paper we perform time-resolved measurements to investigate this possibility and show successful direct excitation of exciton-polaritons by two-photon absorption . We show that this is possible with an incident beam with a finite in-plane momentum due to “bright” state/“dark” state mixing and study the polarization dependence of this absorption both theoretically and experimentally.
In the GaAs-based structures we use, the lowest QW exciton states consist of the two “bright” states and two “dark” states. These states are made of the conduction electrons with and the heavy holes with angular momentum . In our samples with narrow (7 nm) QWs, the light hole exciton states are about 30 meV higher than the heavy hole states, which is greater than the upper polariton/lower polariton splitting and greater than the spectral width of the lasers we use, so the light hole states are unlikely to be excited in resonant excitation of the heavy hole states. The polaritons are formed by coupling the cavity photon to the “bright” heavy hole exciton state. The coupling of the photon and exciton states leads to two new sets of states, the upper polariton and lower polariton states, split by about 12 meV. The “dark” exciton states do not couple to photons to make polaritons.
Our exciton-polariton samples are made up of GaAs QWs with AlAs barriers, in three sets of four, placed at the antinodes of the microcavity made of two Bragg mirrors, which are made of AlAs and repeating layers. The details of these long lifetime samples are given in Ref. , and the measurement of the lifetime () is discussed in Ref. . This lifetime corresponds to a quality factor of over 300,000, compared to a quality factor of less than 10,000 for the samples used in Refs. [17,18]. The long lifetime of the samples allowed us to study ballistic propagation of the population injected by two-photon excitation. The ballistic propagation was good evidence that the injection was directly into polariton states, rather than higher-energy states that would have to thermalize/scatter into the lower polariton states.
These samples were mounted in a cryostat and held at a fixed temperature in the range of 4 K–8 K. Two-photon excitation of the samples was done using a Coherent optical parametric amplifier (OPA) system consisting of a femtosecond pulsed Ti-sapphire laser, a regenerative amplifier with a repetition rate of 250 kHz, and an OPA pumped with tunable output wavelength. Since the energy of the lower polariton in our samples was about 1.593 eV, we tuned the OPA to give a beam with half that energy (0.7965 eV). This output beam had a spectral full width at half-maximum (FWHM) of 15 meV. A pictorial representation of our experimental setup is provided in Appendix A.
We used a dichroic mirror and a 1000 nm longpass filter in the path of our pump beam in order to remove leaked signal coming from the regenerative amplifier. Also, since the photon energy of the regenerative amplifier beam is much lower than the energy of the polariton emission, it was not detected by our spectrally resolved detection system. The emission signal from the polaritons was spectrally resolved using a 0.25 m spectrometer and time resolved using a Hamamatsu streak camera. The time-averaged signal was simultaneously viewed on a Princeton CCD camera. Figure 1(a) shows a typical spatially resolved, time-averaged spectrum. The energy of the emission from the polaritons varies across the sample because there is a wedge in the cavity thickness, giving a spatial gradient to the photon energy. Figure 1(b) shows a typical time-resolved spectrum. As seen in this figure [as well as in Fig. 3(b)], there is a fast rise time of the emission, comparable to our time resolution. The fast fall time of the emission is actually an artifact due to the motion of the polaritons in the cavity gradient. Instead of remaining at the spot where they were generated, the polaritons accelerate in the direction of lower cavity photon energy. This leads to two effects that suppress their collection by our detection system. First, as they move, they can move out of the spatial field of view of the lens collecting the emission. Second, as they accelerate to higher momentum in the plane (corresponding to higher in-plane wavenumber ), their photon emission occurs at a higher angle, and therefore will not be collected by a low-NA system.
In order to ensure that we were observing two-photon excitation and not a higher-order excitation or single-photon excitation due to leaked photons in the pump beam, we did a power series measurement by varying the pump power and measured the time-averaged intensity. As seen in Fig. 2, the good fit to the power law confirms that we have observed two-photon excitation.
Figure 3 shows the results of the time-resolved measurements for various pump wavelengths. The wavelength of 1555 nm corresponds to the resonant condition of the pump photon energy, exactly half the lower polariton energy. When exciting with exactly half the resonant energy [Fig. 3(b)], we see a short (16 ps) peak. However, as we increase the pump photon energy, we see the initial peak disappear and a signal with a long rise time take its place.
Figure 4 shows the polariton intensity at constant pump power as the pump wavelength is varied. When we plot only the intensity of the initial peak, as shown in Fig. 4(a), we see that the intensity is the maximum when the pump photon energy is half the lower polariton energy and disappears at a higher pump photon energy. The FWHM of this peak is 15 meV, which is the same as the pump laser spectral FWHM. If we plot the total intensity, as shown in Fig. 4(b), we see that the intensity increases with increasing pump photon energy. We also observe a small peak while pumping at an energy corresponding to half the energy of the upper polariton energy (0.807 eV) in both cases. The apparent dip is within the uncertainty of our measurements.
For each wavelength, we measured the power dependence for the initial fast-risetime peak and the slow-risetime signal separately. Both signals had an intensity that was proportional to the square of the pump power, indicating that both cases corresponded to two-photon absorption.
In order to understand the slow-risetime signal, we measured the signal as a function of temperature. As seen in Fig. 5, when the temperature is lower, the slow-risetime signal has a greater relative weight. This is consistent with higher-energy states cooling down into the ground state of the lower polariton. At a higher temperature, these states will be scattered to higher -states, while at a lower temperature, they can cool down to states near . Since corresponds to emission normal to the cavity, and our detection system has a low NA, we observe only states with in our experiment. The short initial peak shows no change in intensity in the temperature range we studied (2.5 K to 10 K).
The picture thus arises that polaritons are created by two different mechanisms. One mechanism is direct two-photon creation of polaritons, which occurs most efficiently when the pump laser photon energy is at exactly half the lower polariton energy. The second process is two-photon absorption into excitons in higher-energy states, which then relax down into the lower polariton states with a time constant of several hundred picoseconds. These higher-energy states may be either “dark” () exciton states or states of the excitons. The short peak (due to direct creation of the lower polariton by resonant excitation) remains unaffected by temperature, since two-photon absorption cross sections do not change with temperature. However, nonresonant excitation (which causes the long peak) results in thermalization into all states and is affected by changes in temperature.
In order to explain the luminescence from two-photon absorption, other works [16,18] considered a mechanism based on absorption into the state of the exciton, which then relaxes into the lower polariton by emitting a terahertz photon, while Ref.  has found no evidence of direct two-photon pumping into the state. Time resolving the luminescence (Figs. 3 and 4) leads us to believe that we are directly exciting the polariton states. Although two-photon excitation of the states is forbidden by symmetry at (normal incidence), away from , mixing of the and states occurs.
Our pump photons are linearly polarized (along the axis). Since linear polarization can be viewed as the superposition of two opposite circular polarizations, two photons from the pump beam will couple to a net state of the excitons. Such a state exists for the light holes, corresponding to the two states and . The “bright” state polaritons are nominally defined as and , where are the cavity photon states with . The light-hole states couple to the heavy-hole states through the L–K Hamiltonian when . The term increases linearly with and and is zero for . The value of is determined by the QW confinement. While the light-hole states are 30 meV higher in energy, due to coupling between the heavy-hole and light-hole states in the L–K Hamiltonian, we are able to a access the states by exciting at nonzero .
We thus see that the lower polaritons are not purely made from heavy-hole excitons for finite in-plane ; the excitonic part of the polariton includes a “dark” exciton fraction. The “dark” exciton fraction will slightly reduce the polaritonic coupling to the cavity photons but will not lead to drastic changes of the polariton behavior. Diagonalizing the L–K Hamiltonian, setting , , , , which, as mentioned above, is about 30 meV, and , the hole eigenstates are
We further note that the eigenstates of the light holes are given by 
Calculating the optical momentum matrix element between these states and the conduction band states for the “dark” exciton, we obtain
If the plane of incidence is the plane, then and the TE polarization is along the axis. We obtain the relevant matrix elements by looking at the component, which gives us , since . To obtain the matrix element corresponding to the TM polarization, we look at the component, which gives us . In our experiment, we measure the angle , where is the angle between the normal to the sample and the incident beam and , giving us a matrix element .
The dependence of the term on and the dependence of the polarization selection rule on give us a factor of for the matrix element. The rate of two-photon absorption is proportional to the square of the matrix element, which means that we should see the intensity increase as for TM-polarized light, and we should not see any luminescence from purely TE-polarized light. To check this, we varied for incident TM- and TE-polarized light and observed that the intensity was consistent with a dependence in the TM-polarized case (Fig. 6) and no luminescence in the TE-polarized case. The spread of angles in the incoming beam gives a nonzero contribution, even at , i.e., normal incidence. When the signal was sent through a polarizer, the polaritons formed were seen to be TM polarized as well. A change in the polarization of the pump beam from linear to circular polarization caused a decrease in absorption.
Further support for this conclusion comes from experiments in which we placed the sample in a magnetic field. No change in the intensity of the polariton emission generated by two-photon excitation was seen. Since the “dark” state/“bright” state mixing is already allowed at finite in-plane , a change in the magnetic field up to did not produce a substantial increase in the mixing.
In conclusion, direct two-photon excitation of the lower polariton branch of exciton-polaritons in a microcavity is possible at a nonzero angle of incidence, without involving higher-lying or exciton states. When the pump photon energy is tuned to be resonant with those higher-lying states, we do see evidence for those states being excited, which then leads to polaritons appearing at lower energy with a long rise time. If the two-photon beam has TM polarization and is incident at an angle, we have a direct excitation of the lower polariton, while we see no excitation if the incident beam is TE polarized. Direct two-photon excitation of polaritons leads us to expect novel nonlinear effects with interaction of macroscopically occupied polariton states and light waves at half their frequency. Future work will address this.
Figure 7 shows a pictorial representation of our experimental setup. The beam from our laser is sent through longpass filters. Telescope and microscope objectives help focus the beam onto the sample. Quarter-wave plates are used to control the polarization of the beam. The beam from the sample is analyzed using a spectrometer and a streak camera.
National Science Foundation (NSF) (DMR-0819860, PHY-1205762); Gordon and Betty Moore Foundation.
The work at the University of Pittsburgh was supported by the NSF under grant PHY-1205762. The work at Princeton University was partially funded by the Gordon and Betty Moore Foundation as well as the NSF MRSEC Program through the Princeton Center for Complex Materials (DMR-0819860).
1. H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton Bose–Einstein condensation,” Rev. Mod. Phys. 82, 1489–1537 (2010). [CrossRef]
2. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013). [CrossRef]
3. A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).
4. D. Sanvitto and V. Timofeev, Exciton Polaritons in Microcavities: New Frontiers (Springer, 2012).
5. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. Keeling, F. Marchetti, M. Szymańska, R. André, J. Staehli, V. Savona, P. Littlewood, B. Deveaud, and L. Dang, “Bose–Einstein condensation of exciton polaritons,” Nature 443, 409–414 (2006). [CrossRef]
6. R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose–Einstein condensation of microcavity polaritons in a trap,” Science 316, 1007–1010 (2007). [CrossRef]
7. T. Byrnes, N. Y. Kim, and Y. Yamamoto, “Exciton-polariton condensates,” Nat. Phys. 10, 803–813 (2014). [CrossRef]
8. A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nat. Phys. 5, 805–810 (2009). [CrossRef]
9. P. Cristofolini, A. Dreismann, G. Christmann, G. Franchetti, N. G. Berloff, P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg, “Optical superfluid phase transitions and trapping of polariton condensates,” Phys. Rev. Lett. 110, 186403 (2013). [CrossRef]
10. A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. Martin, A. Lemaitre, J. Bloch, D. Krizhanovskii, M. Skolnick, C. Tejedor, and L. Vina, “Collective fluid dynamics of a polariton condensate in a semiconductor microcavity,” Nature 457, 291–295 (2009). [CrossRef]
11. G. Liu, D. Snoke, A. Daley, L. Pfeiffer, and K. West, “A new type of half-quantum circulation in a macroscopic polariton spinor ring condensate,” Proc. Natl. Acad. Sci. USA 112, 2676–2681 (2015). [CrossRef]
12. K. G. Lagoudakis, T. Ostatnický, A. V. Kavokin, Y. G. Rubo, R. André, and B. Deveaud-Plédran, “Observation of half-quantum vortices in an exciton-polariton condensate,” Science 326, 974–976 (2009). [CrossRef]
13. D. Sanvitto, F. M. Marchetti, M. H. Szymańska, G. Tosi, M. Baudisch, F. Laussy, D. Krizhanovskii, M. Skolnick, L. Marrucci, A. Lemaitre, J. Bloch, C. Tejedor, and L. Viña, “Persistent currents and quantized vortices in a polariton superfluid,” Nat. Phys. 6, 527–533 (2010). [CrossRef]
14. R. Hivet, E. Cancellieri, T. Boulier, D. Ballarini, D. Sanvitto, F. Marchetti, M. Szymanska, C. Ciuti, E. Giacobino, and A. Bramati, “Interaction-shaped vortex-antivortex lattices in polariton fluids,” Phys. Rev. B 89, 134501 (2014). [CrossRef]
15. F. Manni, T. C. H. Liew, K. G. Lagoudakis, C. Ouellet-Plamondon, R. André, V. Savona, and B. Deveaud, “Spontaneous self-ordered states of vortex-antivortex pairs in a polariton condensate,” Phys. Rev. B 88, 201303 (2013). [CrossRef]
16. A. V. Kavokin, I. A. Shelykh, T. Taylor, and M. M. Glazov, “Vertical cavity surface emitting terahertz laser,” Phys. Rev. Lett. 108, 197401 (2012). [CrossRef]
17. J. Schmutzler, M. Aßmann, T. Czerniuk, M. Kamp, C. Schneider, S. Hofling, and M. Bayer, “Nonlinear spectroscopy of exciton-polaritons in a GaAs-based microcavity,” Phys. Rev. B 90, 075103 (2014). [CrossRef]
18. G. Lemenager, F. Pisanello, J. Bloch, A. V. Kavokin, A. Amo, A. Lemaître, E. Galopin, I. Sagnes, M. De Vittorio, E. Giacobino, and A. Bramati, “Two-photon injection of polaritons in semiconductor microstructures,” Opt. Lett. 39, 307–310 (2014). [CrossRef]
19. P. Bhattacharya, B. Xiao, A. Das, S. Bhowmick, and J. Heo, “Solid state electrically injected exciton-polariton laser,” Phys. Rev. Lett. 110, 206403 (2013). [CrossRef]
20. C. Schneider, A. Rahimi-Iman, N. Y. Kim, J. Fischer, I. G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V. Kulakovskii, I. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Höfling, “An electrically pumped polariton laser,” Nature 497, 348–352 (2013). [CrossRef]
21. Preliminary results from this work were reported in August 2013; see C. Gautham and D. Snoke, “Modulation of two-photon excitation by a polariton condensate,” in Fundamental Optical Processes and Semiconductors 2013, Kodiak Island, Alaska, 12 –16 August 2013.
22. B. Nelsen, G. Liu, M. Steger, D. Snoke, R. Balili, K. West, and L. Pfeiffer, “Dissipationless flow and sharp threshold of a polariton condensate with long lifetime,” Phys. Rev. X 3, 041015 (2013). [CrossRef]
23. M. Steger, C. Gautham, D. Snoke, L. Pfeiffer, and K. West, “Slow reflection and two-photon generation of microcavity exciton-polaritons,” Optica 2, 1–5 (2015). [CrossRef]
24. S. Chaung, Physics of Photonic Devices, Wiley Series in Pure and Applied Optics (Wiley, 2009), p. 399.
25. J. K. Wuenschell, N. W. Sinclair, Z. Voros, D. W. Snoke, L. N. Pfeiffer, and K. W. West, “Darkening of interwell excitons in coupled quantum wells due to a stress-induced direct-to-indirect transistion,” Phys. Rev. B 92, 235415 (2015). [CrossRef]