## Abstract

We report experimental studies on the dynamics of excited-state condensate for exciton-polaritons confined in an optically generated trap. The three-dimensionally confined trap was realized by imposing two optical barriers onto a one-dimensional ZnO whispering gallery microcavity. Experimentally, we characterized the confined polariton condensate by varying the trap width and the barrier height. Theoretically, we calculated the spatial overlap between the polariton wavefunction and the excitonic reservoir. Direct comparison of these results verified that such polariton-reservoir overlap was responsible for the observed excited-state polariton condensate.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The strong coupling between semiconductor excitons and microcavity photons leads to the formation of composite particles, well-known as exciton-polaritons [1–4]. They are continuously attracting lots of attention in the past decades due to the rich physics they carry and their potential application in optoelectronics. Among the fascinating physics, perhaps the most appealing characteristic for polaritons is their ability to condense, like a Bose Einstein gas, at temperature that is many orders of magnitude higher than that for real atoms [5–7]. This is mainly due to their unique half-light half-matter nature. Indeed, the effective mass of a typical polariton is only 10^{−5} m_{e} (m_{e}: free electron mass) [8,9], elevating the condensation temperature up to room temperature or even higher. On the other hand, the excitonic fraction of polariton leads to strong interparticle interactions, which generates a wide variety of nonlinear effects.

Prior to their practical applications, effective manipulation of the polariton condensate and its flow is essential. To this end, one efficient way is to confine them in potential traps. Spatial confinement, together with the short lifetime of polaritons (typically a few picoseconds), leads to many interesting physics, such as evaporative cooling [10], binding and antibinding polariton states [11]. Compared with traditional static traps which are usually realized by mechanically patterning the samples, more and more attention is being paid to the optically generated traps in recent years. Unlike those static traps, the optical traps are created by means of nonresonant optical pumping which injects a dense cloud of uncondensed excitons in addition to polaritons. As a result of the inherited excitonic fraction, polaritons experience strong repulsive Coulomb interaction with the dense excitonic cloud (*i.e.*, excitonic reservoir), making the reservoir a barrier for polaritons [9,12,13]. Based on the flexibility of optical traps, recent studies have found a variety of new polariton physics, including vortex lattices [14], superfluid phase transitions [15], tunable magnetic alignment [16], etc. Besides these, another amazing characteristic for optically confined polaritons was found recently by Askitopoulos *et al* [17]. By means of a ring-shaped excitation beam, Askitopoulos and his associates demonstrated that polaritons in optical traps tend to condense in excited states. These observations indicate the complexity of polariton condensate in optical traps. An in-depth investigation of their dynamics is thus highly desirable.

In this work, we report experimental studies on the dynamics of excited-state condensate for exciton-polaritons confined in an optically generated trap. Unlike the ring-shaped excitation scheme which was frequently employed in pioneering literatures, the trap in this work was realized by imposing two potential barriers onto a one-dimensional ZnO microcavity using two Gaussian laser beams. Experimentally, we characterized the confined condensate by varying the trap width and barrier height. Theoretically, we calculated the spatial overlap between the polariton wavefunction and the excitonic reservoir. Direct comparison of these data verifies that the polariton-reservoir overlap was responsible for the observed excited-state condensation phenomena.

## 2. Sample and experimental details

The samples we used are one-dimensional (1D) ZnO microrods grown by chemical vapor deposition. Typical diameter of these microrods is around 1 ~2 μm and length exceeding 100 μm. The as-grown microrods possess regular hexagonal cross-sections which form naturally whispering gallery (WG) microcavities for photons [9,10,18–20]. The strong coupling between these WG modes and the excitonic transition of ZnO leads to the formation of exciton-polaritons.

Typical angle-resolved photoluminescence (PL) image of such microrods is shown in Fig. 1(c). An angle dependent parabolic emission pattern is clearly visible. The excellent agreement of this emission pattern with the theoretical dispersion curve (black dashed curve, Fig. 1(c)) fitted using the coupled oscillator model confirms, that this emission pattern comes from the lower branch of polaritons in 1D ZnO cavity. For comparison, the red dash-dotted curve in Fig. 1(c) shows the bare WG cavity mode which corresponds to the observed polariton branch. However, when we pump the microrod using two laser beams with power exceeding lasing threshold (as shown in Fig. 1(a)), the parabolic dispersion turns into a set of regular emission spots, as shown in Fig. 1(d). Further examination shows that these spots are equally spaced in energy, telling us that polaritons are now occupying simple harmonic oscillator states. The formation of simple harmonic oscillator states stems from the half-light half-matter nature of exciton polaritons. Under nonresonant optical pumping, the laser beam injects a localized excitonic reservoir in addition to polaritons. Due to the relatively large effective masses of excitons and the polariton-exciton repulsive interaction, the excitonic reservoir behaves essentially as a barrier for polaritons [9,12,13]. In our two-beam configuration, the two excitonic reservoirs form a parabolic trap for polaritons and lead to the formation of simple harmonic oscillator states [13].

At a first glance, the image in Fig. 1(d) shows that the confined polariton states are all populated. However, this is not the case. Recent time-resolved study by Askitopoulos *et al* reveals that exciton polaritons confined in optical traps tend to condense at the highest available state [17]. Based on this excited-state condensate picture [17], the scenario for our observation in Fig. 1(d) is shown schematically in Fig. 2. After the initial injection of an optical trap by a laser pulse, polariton condensate occupies only the highest *N* = 4 confined state (*N*: number of antinodes), as depicted by the blue-filled curve in Fig. 2(a). However, as the barriers are decreasing due to the dissipation of excitons, the *N* = 3 quantum state becomes the highest available state at the next moment and becomes populated. Such transition continues until the lowest *N* = 1 state is populated. As we are taking time-integrated and ensemble-averaged measurements over many laser pulses, we could observe the occupation of all these confined states, as depicted in Fig. 1(d) and Fig. 2(d).

Besides the equally spaced emission spots, there is another significant feature for the image shown in Fig. 1(d). If examined more carefully, one would find that the emission intensity of the confined states seems to decrease monotonically for lower energy states. This suggests that there should be some parameter for the system whose value decreases for lower energy states. Considering the excitation scheme we employed, we conjecture that this parameter should be the spatial overlap between polariton wavefunction and the excitonic reservoir. Indeed, as polaritons are fed by the barriers in an optical trap, the highest confined state, whose wavefunction has the largest overlap with the reservoir, will have the highest gain. The smaller overlap for lower energy states thus leads to weaker emissions.

## 3. Results and discussion

To verify our conjecture, one efficient way is to vary the polariton-reservoir overlap and monitor the evolution of the emission intensities. For this purpose, we first changed the trap width by varying the distance *D* between the two laser beams. The angle-resolved PL images for four representative trap widths are shown in Fig. 3. As one can see clearly, as the trap width decreased from *D* ≈6.2 μm in Fig. 3(a) to *D* ≈4.0 μm in Fig. 3(d), the highest available confined state transits from *N* = 6 to *N* = 3, consecutively. What’s more, the emission intensity decreases from high- to low-energy states for all these representative images. This phenomenon is in consistence with the data shown in Fig. 1(d), which again supports our conjecture that there should be some parameter whose value decreases for lower-lying confined states.

As pointed out in previous text, only the highest available confined state is populated at each transient moment. The appearance of multi-mode condensate originates from the time integration and ensemble averaging over many laser pulses. Therefore, we only paid our attention to the highest confined state which corresponds to the initial condition when the optical barrier is highest. The emission intensity for these highest confined states is shown by the red half-filled circles in Fig. 4(a), as a function of their corresponding quantum number *N*. As demonstrated, the emission intensity decreases for wider traps.

For a quantitative analysis, we calculated the wavefunction of polaritons confined in optical traps using the standard simple harmonic oscillator model. The validity of using this harmonic oscillator model can be supported by the density distribution of excitons. As there are no dark excitonic states in ZnO microrod, the spatially resolved excitonic emission gives a direct estimation of the spatial distribution of excitons. Here, it should also be noted that the energy upper bound of polariton emission is very closed to the exciton emission. Based on the fact that diffusion length of polaritons is much larger than that of excitons, the energy range of polariton (and thus exciton) emission can be measured directly by moving the detection points to areas far away from the laser pumping spot. The spatial distribution of excitons measured in this way is shown in Fig. 1(b). Obviously, the excitonic distribution indeed has a parabolic profile in the trap region. For polaritons confined by such harmonic potential *V*(*x*) = *m**ω^{2}*x*^{2} / 2, their wavefunction can be written as

*N*is a normalization constant,

_{n}*H*(ζ) is the standard Hermite polynomial and

_{n}*m** is the effective mass of polaritons. Using this wavefunction and the experimental parameters, the spatial probability distribution of optically confined polaritons can be calculated directly. Typical calculated results are shown by the white dashed curves in Fig. 1(d) (for all confined states) and Fig. 3 (for highest confined states), which show very good agreement with the experimental results.

To further study the role of polariton-reservoir overlap, we define an overlap factor Π to quantify the spatial overlap, as follow:

*(*

_{r}*x*)>|

^{2}represents the spatial density distribution of excitons. With the experimentally measured |<ψ

*(*

_{r}*x*)>|

^{2}and theoretically calculated |<ψ

*(*

_{n}*x*)>|

^{2}, the overlap factor Π for the highest confined states in Figs. 3(a-d) can be calculated. The results are shown by the black half-filled circles in Fig. 4(a), as a function of their corresponding quantum number

*N*. As one can see clearly, the polariton-reservoir overlap decreases monotonically as the trap width was increased. More importantly, the curve is almost in excellent agreement with that of the emission intensity, thus supporting our conjecture that the polariton-reservoir overlap is strongly correlated to the excited-state condensation behavior.

In addition to the trap width, the polariton-reservoir overlap can also be varied by changing the trap depth via the excitation power. By increasing the barrier height, the highest available confined state can be tuned from *N* = 1 to *N* = 6. Again, we calculated the overlap factor using Eq. (2) with the known parameters. The results are plotted by the black half-filled circles in Fig. 4(b), together with the emission intensity of the highest available confined states (red half-filled circles). As one can see unambiguously, the two curves for the overlap factor and the emission intensity are again in very good agreement with each other.

The quantitative agreement found here supports strongly that the polariton-reservoir overlap is critical for the excited-state condensation behavior. The physical picture for this correlation is quite straightforward. In optically generated traps, hot excitons (*i.e*., reservoir) are localized around the laser spot area. Final state stimulated scattering of these hot excitons into polariton condensate is the main way for polariton injection. In this case, the highest confined state, which has the largest spatial overlap with the excitonic reservoir, will have significantly higher gain than other states and become populated. The consistent behavior for the overlap factor and the emission intensities of uppermost confined states reported here is thus a strong support for this scenario.

## 4. Summary

In conclusion, we report experimental studies on the dynamics of excited state condensate for polaritons confined in an optically generated trap. By calculating the spatial overlap between polariton condensate and excitonic reservoir, we were able to verify the role of the polariton-reservoir overlap. Polariton injection from trap barriers and optical confinement are responsible for the excited-state condensation behavior. Excited-state condensate is a unique characteristic for polaritons confined in optical traps. Experimentally clarifying their mechanism and dynamics would thus facilitate the development of polariton-based physics and devices.

## Funding

National Natural Science Foundation of China (11404120, 61875062); Fundamental Research Funds for the Central Universities, HUST (2018KFYXKJC011, 2019kfyXKJC005).

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