PHYSICAL REVIEW B 83, 155304 (2011)
Collisional damping of dipole oscillations in a trapped polariton gas T. K. Para¨ıso,1,* R. Cerna,2 M. Wouters,2 Y. L´eger,1 B. Pietka,1 F. Morier-Genoud,1 M. T. Portella-Oberli,1 and B. Deveaud-Pl´edran1 1
Laboratory of Quantum Optoelectronics, EPFL, CH-1015, Lausanne, Switzerland 2 Institute of Theoretical Physics, EPFL, CH-1015, Lausanne, Switzerland (Received 30 August 2010; revised manuscript received 4 November 2010; published 6 April 2011) We study the relaxation dynamics of a trapped polariton gas in the nonlinear regime. We excite the three lowest energy states of the system and observe the time evolution of the polariton density in the momentum space. At a low excitation power, the dynamics is characterized by dipole oscillations of constant amplitude. A damping of these oscillations is observed at a high excitation power. It is attributed to collisional relaxation within the coherent polariton gas. We investigate the dependence of this effect on the excitation power, polarization, and polariton excitonic content to highlight the role of polariton-polariton scattering. The experiments are described in the frame of a Gross-Pitaevskii mean-field theory. We find a good agreement between the theoretical simulations and the experimental observations. Analysis of the theoretical model reveals that multiple parametric scattering and final-state stimulation are responsible for the damping of the oscillations. DOI: 10.1103/PhysRevB.83.155304
PACS number(s): 78.47.J−, 71.36.+c, 67.85.−d
I. INTRODUCTION
II. EXPERIMENTAL SETUP
Semiconductor microcavities in the strong coupling regime are unique systems for investigating the many-body physics of an interacting Bose gas in a solid-state environment. During the past decade, many efforts have been made to understand and manipulate the nonlinear properties of polaritons; the most prominent examples are the observation of parametric amplification,1 polariton condensation,2 and superfluidity.3 In polariton condensation experiments with GaAs-based microcavities, it was demonstrated that the trapping potential plays an important role.4,5 Polariton trapping has been achieved with various techniques such as conventional micropillars,4 mesa structures,6 and mechanical stress.5 An elegant route to experiments with polaritons is their creation by a laser pulse in resonance with the polariton frequency. Under continuous wave excitation, the phase of the polariton gas is mainly fixed by the excitation laser, allowing for all-optical coherent control of the wave-function patterns of a single confined polaritons state.7 In contrast, under pulsed resonant excitation, the polariton dynamics is free as soon as the excitation pulse has passed. We recently reported on the spatial dynamics of microcavity polaritons in traps of different sizes and showed that the description of the dynamics strongly depends on the number of populated states.8 In the present paper, we demonstrate the observation of nonlinear effects on the dynamics of a trapped polariton gas. We create the polariton gas in a superposition of the three lowest energy states, that is, in a dipole oscillation state. For a Bose gas confined in a harmonic potential, the dipole state should remain undamped even if the particles are interacting.9–11 Our cylindrical trapping potential being strongly anharmonic, polariton collisional relaxation results in the damping of the dipole oscillations. We investigate this relaxation as a function of the polariton density and polarization. The latter study highlights the crucial role of interactions in the damping mechanism.12
The investigated sample is a λ GaAs/AlAs microcavity with an embedded InGaAs quantum well. The traps consist of cylindrical extensions (mesas) patterned on top of the cavity spacer. The efficiency of our trapping technique is demonstrated in Ref. 6. Quantum confinement leads to the discretization of the polariton spectrum, with energy spacing between confined states ranging from 500 μeV in small traps to a few micro–electron volts in large traps. Due to the conservation of the in-plane momentum, the in-plane wave vector k of microcavity polaritons is directly related to the excitation (or emission) angle θ of the extracavity photons by E the relation k = h¯ cp sin(θ ), where Ep is the polariton energy. This relation allows highly selective excitation of individual states by adjusting both the energy and the angle of the excitation beam. The sample is held in a cold finger cryostat at 5 K. The experiments are done with 9-μm-diameter mesas at different detunings to highlight the role of the excitonic content on polariton-polariton interactions. The initial 150-fs excitation pulses are shaped to obtain 10-ps pulses, with a spectral width at half-maximum of less than 500 μeV. The large spot size (50 μm) allows us to have a well-defined excitation spot in momentum space. The polarization of the pulses can be changed between circular and linear using a quarter-wave plate.
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III. POLARITON DIPOLE OSCILLATIONS
We excite the mesas resonantly in a coherent superposition of the ground state (GS) and first and second excited states (E1 and E2). The intensity distributions of these three states are shown in Fig. 1(a). In cylindrical mesas, the confined polariton modes are labeled with a radial quantum number, n = 1, 2, 3, . . . , and an angular quantum number, m = 0,±1,±2, . . . .13 In the angular direction φ, the phase of the states evolves as eimφ . Under resonant excitation, the phase of the +m and −m states is locked to the phase of
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FIG. 1. (Color online) Polariton dipole oscillation dynamics at different excitation powers. a: Momentum space (kx ,ky ) probability distributions of the ground state (GS; n = 1, m = 0) and first and second excited states, E1 (n = 1, m = 1) and E2 (n = 1, m = 2). b–j: Time evolution (kx ,t) of the polariton density for 1, 10, and 20 μW excitation power. Dashed lines track the position of the maximum. (b–e) Slightly negative detuning. (b–d) Under circular excitation (σ +), the oscillation amplitude decreases with increasing excitation power but remains constant during the whole evolution. (e) The effect is significantly weaker under linear excitation (lin. exc.), showing the role of collisions. (f–j) Positive detuning. (f–h) Circular excitation. At 10 and 20 μW, the oscillations damp continuously and the polariton density stabilizes around kx = 0, suggesting a continuous population transfer to the GS. (i, j) Linear excitation. Damping occurs at a much higher excitation power (40 μW). The red arrow on the y axis in (b) and (f) indicates the pulse arrival time for the experiments at slightly negative and positive detuning, respectively.
the laser. This gives rise to lobe patterns for the m = 0 states [Fig. 1(a)],7 with a π phase shift between two consecutive lobes in the azimuthal direction.14 Under pulsed excitation, the system is not driven anymore after the end of the excitation pulse. The phase relation between the excited states evolves periodically according to their energy differences. The resulting interferences lead to a propagation of the polariton density that can be observed in both real and momentum space.8 To get a simple idea of the resulting dynamics, one may first consider the time evolution of GS and E1 only. During the time evolution, the two lobes of E1 will alternatively be in and out of phase with the GS lobe.15 Therefore, one expects to observe dipole oscillations with a period related to their energy difference. By scanning the wave front of the emitted light perpendicularly to the slits of the streak camera, we are able to reconstruct videos of the dipole oscillations in momentum space (kx ,ky ,t). We study the modifications of the dipole oscillations upon increasing the polariton density in a mesa at a slightly negative detuning and in a mesa at a large positive detuning. Data are shown in Figs. 1(b)–1(j). For the sake of clarity, the data were integrated along ky to obtain two-dimensional
images (kx ,t). In addition, we normalized the density profiles along kx at each time t. IV. COLLISIONAL RELAXATION
We first discuss the case of a mesa at slightly negative detuning, δ = −0.1 meV (48% exciton), under circularly polarized excitation [Fig. 1(b)–1(d)]. The laser is incident at kx = +0.6 μm−1 . At a low excitation power [Fig. 1(b)], we observe clear dipole oscillations with a period of 28 ps and an amplitude of 1 μm−1 . The amplitude of the oscillations depends on the instantaneous population ratio between the first excited state and the GS. In the linear regime, this ratio is constant and determined by the central frequency of the excitation pulse. When the excitation power is increased to 10 and 20 μW [Figs. 1(c) and 1(d)] we observe a narrowing of the oscillation amplitude to 0.8 and 0.6 μm−1 , respectively. The reason is that the high polariton density induces a blueshift responsible for an increase in the GS population in the superposition state. Note that this blueshift has to be very fast and to occur during the excitation pulse to change the population ratio from the beginning of the time evolution. Since the blueshift is different for each confined state, one
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may expect changes in the oscillation period as a function of the excitation power. We observed that this effect is actually very small (changes of less than 5 ps between 1 and 20 μW) and has no impact on the dynamics. To highlight the role of collisions in the narrowing of the oscillation amplitude, we repeat the experiment under linearly polarized excitation. Because the interactions between cocircularly polarized polaritons are stronger than interactions between counter-circularly polarized ones, the interactions between polaritons created using a linearly polarized excitation are expected to be weaker than the ones between polaritons created with a circularly polarized excitation. This is illustrated in Fig. 1(e), where we show that the reduction of the oscillation amplitude is negligible under linearly polarized excitation. The measured decay time of the emission intensity is the same (22 ps) for all excitation powers. At a large positive detuning, the behavior is drastically different. In Figs. 1(f), 1(g), and 1(h), we show the evolution of the oscillations amplitude for three excitation powers, under circular excitation, for a mesa at δ = +1.55 meV (70% exciton). For this experiment the laser was incident at kx = −0.7 μm−1 . In the linear regime [Fig. 1(f)], the dynamics is the same as described previously. We observe dipole oscillations of constant amplitude between kx = −0.7 and k = +0.7μm−1 . Strikingly, the data taken at 10 and 20 μW [Figs. 1(g) and 1(h), respectively] reveal a continuous damping of the oscillation amplitude. In the nonlinear regime, the population ratio is influenced both by the blueshift and by relaxation processes. The blueshift causes an increase in the initial GS population. The amplitude of the first oscillation (at t 30 ps) decreases for increasing excitation power: the first lobe is located at 0.45 and 0.38 μm−1 , for excitation powers of 10 and 20 μW, respectively. In the absence of any relaxation process, the oscillation amplitude would remain constant during the subsequent evolution, similarly to the negative detuning experiments. Here, the observation of a damping of the dipole oscillations denotes a continuous population transfer to the GS of the system. Again, we demonstrate the crucial influence of collisions by repeating the measurements using a linearly polarized pump. The results are given in Figs. 1(i) and 1(j). The damping of the oscillations is negligible even for an excitation power of 40 μW. Another important observation is the change in the decay time τr of the emission intensity. Between 1 and 20 μW excitation power, we measured an increase in τr from 14 to 27 ps, while it was constant in the experiments at negative detuning. This observation indicates that at positive detuning, the dynamics involves the contribution of high-energy states with a large excitonic content,16 probably populated by multiple scattering processes. More detailed information on the occupation of the different modes can be accessed through time-resolved photoluminescence experiments. Figure 2 shows the population evolution per state for four excitation powers, under circular excitation. At 500 nW and 5 μW [Figs. 2(a) and 2(b)], the emission is dominated by E1 and the population ratio among GS, E1, and E2 is constant during the whole time evolution. At high excitation powers [Figs. 2(c) and 2(d)], we observe a transient, after which the emission from the GS overcomes the emission from the excited states. This confirms the presence
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FIG. 2. (Color online) Time-resolved occupation of the ground state (GS), E1, and E2 (positive detuning). a–d: Experiments. (a) 500 nW; the emission is dominated by E1 and E2. (b) 5 μW; the intensity of the GS remains below the intensity of E1. At higher excitation power, (c) 10 μW and (d) 25 μW, a population transfer from the excited states to the GS causes a damping of the dipole oscillations. e, f: Calculations are in good qualitative agreement with experimental data. At a low excitation power (e), the states evolve with the same dynamics. At a high excitation power (f), the GS becomes dominant because of multiple parametric processes.
of important relaxation processes in the high-density regime. Beats in the intensity profiles are secondary dynamics due to the coupling between the three states. The measured period (35 ps) corresponds to an energy spacing of 120 μeV, which is close to the energy spacing between two consecutive levels. Also, one can see that the oscillations of E1 and E2 are in phase at 10 μW [Fig. 2(c)] and in antiphase at 25 μW [Fig. 2(d)]. The oscillation contrast is better at a high exitation power. All this reflects the coherent nature of the relaxation processes. Our simulations [Figs. 2(e) and 2(f)], performed with the model described here, are in good qualitative agreement with the experimental data. Our model shows that parametric scattering processes are responsible for the continuous population transfer from the excited states to the GS. V. THEORETICAL MODEL
Theoretically, polariton oscillation dynamics can be qualitatively understood within a classical field (Gross-Pitaevskii) framework, because the population of the relevant modes is much larger than unity due to the coherent excitation. Thanks to the weak coupling between states of opposite circular polarization, spin degrees of freedom can be neglected. Polaritons are described as two coupled fields constituted by the excitons (ψX ) and photons (ψC ) i
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where the cavity photon dispersion is given by C (k) = ωC 1 + k/kz2 , the linewidth of the excitons and photons is denoted γX,C , respectively, and exciton-exciton interactions are modeled by a contact interaction g. The values of γX,C are estimated from the fits of experimental decay times. To simplify the model, we use the same damping rate for excitons and photons, γX,C = 0.04 meV. The external laser field FL creates the photons that are reversibly converted into excitons through the Rabi coupling term ωR . The external potential Vext describes the 9-μm-diameter cylindrical photonic trap that confines the polaritons. The potential depth Vext is 6 meV. Note that the simulations are not very sensitive to this value as long as it is much higher that the other energies in the system (blueshift, Rabi splitting, damping rates). Numerical simulations with this Gross-Pitaevskii model are reported in Fig. 3. The theoretical simulations show a trend that is in qualitative agreement with the experimental observations. At a low density, the beatings between the different states give rise to separated lobes in k space, whereas at a higher density, the polariton fluid follows a smooth trajectory. Moreover, at a higher density, the oscillation amplitude decays with time, albeit less pronounced than in the experiments. We suspect that additional relaxation due to interactions with phonons and/or localized excitons is responsible for this deviation.16 The comparison between the experimental observations and the theoretical simulations for the occupation of the different single-particle states is shown in Fig. 2. The overall qualitative agreement is satisfactory: at a low intensity, there is almost no population transfer between the different states. The discrepancy in the rise times between theory and experiments is due to the temporal smearing in the experiments, imposed by the time-frequency uncertainty in the spectrally resolved measurements. In contrast, good agreement with the experiments is obtained for what concerns the transfer of population between the different states when the laser intensity is increased. At the highest power, the GS population becomes the dominant
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one at late times. The fast population oscillations seen in the simulations originating from the beatings between all the modes are, again, not resolved in the experimental measurements. For the slower features of the dynamics, in contrast, good agreement between theory and experiments is obtained. Notice, for example, the correspondence between the theoretical and the experimental slow oscillation dynamics in the second excited state at the highest densities. The relaxation process from the first excited state to the GS is complex, but the dominant mechanism is a parametric process, 2E1 → G + E2 . Because this process is highly nonlinear due to stimulated scattering, a first estimation of the relaxation time can be obtained by neglecting the change in the occupation of the modes. Within a three-mode approximation, where only the modes that participate in the parametric scattering are retained, the typical time for the relaxation can be estimated. Keeping only the resonant term, we find that the typical time for the relaxation scales as
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where is the energetic mismatch for the parametric process and ψX,G(E1 ) is the excitonic density in the ground (first excited) state. This explains the dependence on the detuning in the experiments. Indeed, going from negative to positive detuning causes a decrease in and an increase of g due to the higher excitonic content. Therefore, τ is much smaller at positive detuning (nearly 1 order of magnitude). Consequently, the relaxation due to parametric processes (quadratic in g) becomes significant enough to overcome the effect of the blueshift (linear in g) and to cause a damping of the oscillations.
VI. CONCLUSION
Our experiments demonstrate the effect of nonlinear interactions on the spatial dynamics of a trapped polariton gas. When excited in a superposition of its three lowest energy states, the system manifests dipole oscillations. In the linear regime, the amplitude of these oscillations is constant during the whole evolution. In the nonlinear regime, we observe a continuous damping of the dipole oscillation amplitude. The damping increases in time, even after the excitation pulse. This is the consequence of a continuous population transfer to the GS due to collisional relaxation. To underline the role of collisions, we demonstrate that the oscillation damping is less efficient under linear excitation than circular excitation. We reproduce the most important features of the experiments with a theoretical model based on the GrossPitaevskii equation.
ACKNOWLEDGMENTS
The authors would like to thank V. Savona and G. Nardin for fruitful discussions. This work was supported by the Quantum Photonics NCCR and the Swiss National Science Foundation.
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F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999), 10 J. F. Dobson, Phys. Rev. Lett. 73, 2244 (1994). 11 W. Kohn, Phys. Rev. 123, 1242 (1961). 12 I. A. Shelykh, A. Kavokin, and G. Malpuech, Phys. Status Solidi B 242, 2271 (2005). 13 G. Nardin, T. K. Para¨ıso, R. Cerna, B. Pietka, Y. L´eger, O. E. Daif, F. Morier-Genoud, and B. Deveaud-Pl´edran, Appl. Phys. Lett. 94, 181103 (2009). 14 G. Nardin, Y. L´eger, B. Pietka, F. Morier-Genoud, and B. Deveaud-Pl´edran, Phys. Rev. B 82, 045304 (2010). 15 See supplemental material at [http://link.aps.org/supplemental/ 10.1103/PhysRevB.83.155304] for videos of the experiments. 16 T. K. Para¨ıso et al., Phys. Rev. B 79, 045319 (2009).
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