Temporal coherence of spatially indirect excitons across Bose–Einstein condensation: the role of free carriers

In semiconductor physics, Bose–Einstein condensation of excitons, i.e. Coulomb bound electron–hole pairs, has received a large attention since original theoretical predictions were formulated in the 1960s [1–3]. Indeed, excitons are very attractive because their effective mass, much lighter than atoms, yields a cryogenically accessible critical temperature for the phase transition. However, the excitons electronic structure has hidden for a long time their condensation [4]. This property is well illustrated in widely studied GaAs heterostructures where excitons have four accessible 'spin' states, two optically dark states with spins (±2) and two optically bright ones with spins (±1). Bright excitons lying at a slightly higher energy than dark ones [5, 6], Bose–Einstein condensation leads to a macroscopic occupation of lowest energy dark states so that the condensate is not detectable easily. In fact, it is only in double quantum wells (DQWs) that a condensate can be imaged directly [7], when a small fraction (∼20%) of bright excitons is possibly introduced coherently in a dark condensate by exciton–exciton interactions at sufficiently high densities (∼10¹⁰cm⁻²) [7–9]. The condensate then becomes gray and is signaled by the weak photoluminescence it radiates.

Recently, we have reported a series of experiments combining the signatures expected for a gray condensate of indirect excitons [10] in GaAs DQWs. These signatures include first a (quantum) darkening of the photoluminescence below a critical temperature of about 1 K and for a narrow range of densities only (∼(2–3) × 10¹⁰ cm⁻²) [11]. This darkening marks a quantum statistical occupation of dark excitons because the energy splitting between bright and dark states amounts to only a few μeV in DQWs [5, 6], i.e. at least 10 times less than the thermal energy at 1 K. We have then confirmed the gray nature of the condensation, i.e. the coherent coupling between dark and bright excitons. For that purpose we used spatial interferometry and showed that the weak photoluminescence radiated in the regime of quantum darkening exhibits macroscopic (quantum) spatial coherence [12], below the same critical temperature of about 1 K and for the same narrow density range.

Here, we scrutinize another fingerprint expected for exciton condensation, namely a steep increase of temporal coherence across the condensation threshold [9]. This behavior is understood by noting that in the condensed phase optically bright excitons merely occupy k ∼ 0 states, directly coupled to photons. At sub-Kelvin temperatures, the excitons coherence time is then limited by interactions between condensed and uncondensed (${\bf{k}}\ne 0$) excitons, as well as interactions between excitons and excess carriers which are strong in GaAs heterostructures [13]. Indeed, we note that exciton–phonon interactions are negligible in the sub-Kelvin regime [14]. Bose–Einstein condensation then has to increase the excitons coherence time because interactions between excitons of the condensate cannot contribute to collisional broadening due to energy and momentum conservation. In the following experiments, for a measured condensate fraction of about 80% [12] we show that the photoluminescence temporal coherence is increased by two-fold at threshold, the scattering rate of k ∼ 0 bright excitons being then reduced by the same fraction. Our studies indicate that this amplitude is limited by the amount of free carriers, due to both the level of unintentional doping during the epitaxial growth of our heterostructure, and a possible exciton ionization due to residual in-plane electric fields reminiscent of electrostatic trapping for indirect excitons [15, 16].

We report time and spatially resolved interferometry to probe the temporal coherence of indirect excitons confined in a 10 μm trap. We study the same device as in [11, 12] in the regime where we have shown signatures of excitonic condensation. In this heterostructure, indirect excitons (IXs) are made by the Coulomb attraction between electrons and holes confined in two 8 nm GaAs quantum wells separated by a 4 nm AlGaAs barrier. The trap is engineered by using two metallic electrodes, a central 10 μm wide disk surrounded by an outer guard, both deposited at the surface of a field-effect device embedding the DQW. We apply the same potential on both electrodes (−4.7 V), so that the 200 nm gap between them leads to a rectification of potential leading to an electrostatic barrier, i.e. a hollow-trap [12]. Moreover, note that IXs exhibit a long lifetime (∼100 ns) in our studies and reach thermal equilibrium [17] in a mostly electrically neutral environment about 100 ns after optical injection, when a transient photocurrent (≲100 pA D.C.) is evacuated. Indeed, we inject electronic carriers optically, using a 100 ns long laser excitation, repeated at a rate of 1.5 MHz and tuned on resonance with the direct exciton absorption of the two quantum wells (figure 1). IXs are thus obtained once electrons and holes have tunneled towards their minimum energy states, located in each quantum well. Let us then note that the laser beam covers most of the trap area.

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To study the temporal coherence of bright IXs confined in the trap, we magnified the real image of the photoluminescence (PL) and sent it to a Mach–Zehnder interferometer. The PL is divided equally between a fixed and a mobile arm, and the two beams are superposed at the output of the interferometer, i.e. without any spatial displacement. Horizontal interference fringes are induced by a vertical tilt angle between the two outputs, of about 25°. To evaluate the temporal coherence of the PL, we vary the longitudinal path length difference, i.e. we translate one arm by δL. The two outputs are then shifted temporally by δt = 2δL/c, where c is the speed of light in air, the path length difference being actively stabilized with ∼20 nm precision. At the output, the contrast of the interference pattern measures directly the coherence time τ_c of optically bright excitons. Indeed, the interference visibility is given by the modulus of the first-order correlation function $| {g}^{(1)}({\bf{r}},\delta t)| \propto | \langle {\psi }_{0}^{* }({\bf{r}},t){\psi }_{0}({\bf{r}},t+\delta t){\rangle }_{t}|$. Here, ψ₀(r, t) is the photoluminescence field which reflects the wave function of bright excitons with an in-plane momentum k ∼ 0 since the exciton–photon coupling is linear close to resonance [18]. Moreover $\langle ..{\rangle }_{t}$ denotes the time averaging and r = (x, y) reads the coordinate in the plane of the electrostatic trap. For a homogeneously broadened gas, the g⁽¹⁾-function is exponentially decaying with δt, at a rate inversely proportional to the k = 0 excitons coherence time τ_c. In a thermal regime above a few Kelvin, τ_c is controlled by exciton–phonon interactions [14]; however, below a few Kelvin, we expect a dramatic change because these interactions become inefficient.

Unlike studies performed with atomic gases [19], our experiments require an accumulation of several millions of single-shot realizations, during around 10–20 s, each realization starting by a loading laser pulse after which the PL is interferometrically analyzed (figure 1). This approach is inevitable due to the weak photoluminescence signal radiated from the trap, but it constitutes a severe limitation. Indeed, we observe large spectral fluctuations, with a characteristic timescale of about 10 s, as shown in figure 1(a) at a bath temperature T_b = 330 mK and for a trapped exciton density n_X about 2.5 × 10¹⁰ cm⁻². Particularly, figure 1(a) displays two spectra measured successively; a first spectrum consists mostly of a central narrow line, with a width limited by the spectrometer resolution. By contrast, another spectrum obtained for the same experimental settings can be about 1 meV broad with no recognizable shape (dashed line in figure 1(a)).

We attribute the spectral diffusion evidenced in figure 1(a) as the result of the large electric dipole moment (∼500 Debye) characterizing indirect excitons and which makes the energy of their photoluminescence very sensitive to the electrostatic environment. On the one hand this property is useful since it allows us to extract the density of excitons in the trap n_X, indirect excitons experiencing repulsive dipolar interactions that yield a blue shift of the photoluminescence energy scaling as u₀n_X [20, 21], where u₀ ∼ 0.5 meV/10¹⁰ cm⁻². On the other hand, indirect excitons interact strongly with free carriers, due to either the laser induced photocurrent or to the level of residual doping during the growth of our heterostructure. In both cases, the exciton-free carrier interactions provide a direct source for spectral diffusion because the local concentration of excess carriers cannot be controlled and a priori varies randomly during our measurements. To minimize this effect, we typically analyze the PL long after extinction of the laser excitation, i.e. when the photocurrent is damped. Nevertheless, even a density of excess carriers small compared to n_X can yield a significant electrostatic noise, because in GaAs quantum wells the interaction strength between excitons and free carriers is about 10 times larger than the one between excitons [13].

Spectral fluctuations are also evidenced interferometrically, with actually better dynamics since the acquisition time is reduced from about 20 to 10 s once the PL is not filtered spectrally. For fixed experimental conditions at T_b = 330 mK, figure 1(c) shows that either a clear interference pattern can be observed across the center of the trap, or a rather blurred pattern (figure 1(d)). For these experiments where the path length difference is set such that δ_t = 1.8 ps, in the former case the contrast along the center of the trap (x = 5 μm) amounts to 35% (figure 1(e)) whereas in the latter case it reduces to about 23% (figure 1(f)) manifesting an increased spectral width. Such variation is quantified in figure 1(b) where we display the interference visibility measured at T_b = 330 mK for a statistical ensemble of 100 experiments performed all under the same conditions. We note that the contrast exhibits an asymmetric dispersion, from 17% to 37%, marked by a tail at high contrasts (30%–37%) and a sharp growth at low contrasts (from 17% to 22%). While we cannot quantify the origin of such asymmetry, it indicates that the spectral diffusion is complex and not simply due to a Gaussian noise process. Based on the calibration of our interferometer, we nevertheless deduce that the photoluminescence exhibits a spectral width in average lower than 700 μeV and that for the most regular electrostatic environment, e.g. for a lowest concentration of excess free carriers, the spectral width can become as narrow as 300 μeV.

We now study the variation of the photoluminescence coherence time as a function of the exciton density n_X and bath temperature T_b. To limit the influence of spectral fluctuations over our conclusions, we restrict our analysis to the realizations with highest visibility for each experimental settings. Precisely, we average the contrasts measured for the best 10% of experimental realizations, post-selected within a statistical ensemble of a minimum of 50 measurements for each experimental conditions (see for instance the orange colored region in figure 1(b)). Thus, we aim at minimizing the amount of measurements where inhomogeneous broadening plays a significant role. Instead, we focus onto experiments where indirect excitons are confined in the most regular trapping potential, combined to a minimum amount of excess charges interacting with them. Post-selecting these optimum conditions is actually necessary to study the excitons gray condensation, since we have shown in [12] that a gray condensate is only obtained for such optimized experimental settings and cannot form otherwise.

Let us first discuss the dilute regime, i.e. an exciton concentration n_X of about 2.5 × 10¹⁰ cm⁻² obtained for a 170 ns delay to the extinction of the loading laser pulse. Figure 2 quantifies the variation of the interference contrast measured along the vertical axis at the center of the trap (x = 5 μm), as a function of the time delay δt introduced between the two arms of the interferometer. We note that the interference visibility follows a mono-exponential decay ${{\rm{e}}}^{-| \delta t| /{\tau }_{c}}$, at every bath temperature T_b below 2.36 K. According to the Wiener–Kintchine theorem [22], the photoluminescence spectrum then consists of a single Lorentzian. This shows that we study a homogeneously broadened gas which constitutes a major improvement compared to previous works [23–28]. To the best of our knowledge, such a single Lorentzian PL profile had never been observed for a gas of indirect excitons. For the measurements shown in figure 2, the photoluminescence spectral width Γ = 2ℏ/τ_c reaches then 550 μeV at T_b = 2.4 K. This value is nevertheless ∼300 μeV greater than the theoretically expected linewidth due to exciton–exciton interactions [20], indicating that these latter are not the only mechanism governing the homogeneous broadening.

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We now discuss the photoluminescence temporal coherence as a function of the exciton density. Experimentally, this is achieved simply by varying the delay to the loading laser pulse. Indeed, the exciton density decreases monotonously in the trap due to radiative recombination, and let us stress that in our studies the total density of excitons n_X, including both bright and dark states, is the same at a given delay for every bath temperature below 3 K [11]. For a fixed delay δt = 1.8 ps, figure 3 shows that three regimes emerge: (i) in the most dilute regime, i.e. for longest delays to the end of the loading laser pulse, the interference contrast does not vary with T_b. This behavior indicates that the gas is localized, as expected since in this range of densities the blue shift of the photoluminescence, i.e. the excitons mean interaction energy, does not exceed the electrostatic disorder in the trap (∼500 μeV). At higher densities (regime (ii)) the excitons mean-field energy compensates the strength of potential disorder so that dipolar repulsions delocalize the exciton gas. Then, the interference contrast varies strongly with T_b: while the visibility keeps a value similar to regime (i) for T_b ≥ 1.2 K, it increases steeply at T_b = 330 mK, from about 24% to 30% when n_X ≳ 2 × 10¹⁰ cm⁻². Increasing further the density, n_X ≳ 3.5 × 10¹⁰ cm⁻² in regime (iii), leads to a breakdown of this behavior and the interference contrast depends again weakly on the bath temperature.

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From the visibility measured for δt = 1.8 ps, we can deduce the spectral width Γ = 2ℏ/τ_c in regime (ii) where we have shown in figure 2 that the photoluminescence is homogeneously broadened. At T_b = 1.2 K, figure 3 shows that the homogeneous spectral width is about 500 μeV at 1.2 K and varies weakly with the exciton density n_X. At T_b = 2.4 K , Γ is slightly larger (∼600 μeV), possibly due to the increased role of acoustic phonons, while it also varies weakly with n_X. Such weak variations with n_X are theoretically expected [20, 29], since interactions between indirect excitons vary Γ by less than 100 μeV in the density range explored here, i.e. by less than our instrumental precision. However, the magnitude of Γ lies about 300 μeV above its theoretical expectation. We attribute this discrepancy as a manifestation of the interaction between indirect excitons and residual excess charges, although we minimize the concentration of these latter in regime (ii) since the delay to the laser excitation exceeds 120 ns. Let us then note that interactions between excitons and free carriers yields a lorentzian broadening of the photoluminescence [30].

To study the role of excess carriers in the DQW, let us first estimate the fraction of free carriers induced by the laser excitation. For that purpose, we consider the dynamics of the photocurrent, which average $\bar{I}$ amounts to 100 pA, as measured in a 20 ms time interval with the source-measure device used to polarize the exciton trap. Moreover, we have previously shown that the photocurrent has a transient time of the order of 70 ns [11]. In this case we deduce that $\bar{I}={{fI}}_{m}{T}_{p}$ where I_m is the maximum current generated at the termination of the laser excitation, while f and T_p denote the 1.5 MHz repetition rate of our sequence and the length of the loading pulse, respectively. This expression for $\bar{I}$ directly converts into n_m, the maximum density of excess carriers induced at the termination of the laser excitation, since n_m = I_m/($e\pi {{fr}}^{2}$), where e is the electron charge. For the radius r = 5 μm of our trap electrode, we conclude that n_m = 3.5 × 10⁹ cm⁻² so that the density of photo-induced excess carriers reduces exponentially to a few 10⁸ cm⁻² in regime (ii) of figure 3. On the other hand, excess carriers also result from the level of unintentional doping during the growth of our heterostructure. Indeed, the concentration of impurities for the molecular beam epitaxy of our sample is at most 10¹⁵ cm⁻³, yielding a density of free carriers of about 10⁹ cm⁻² for two 8 nm wide quantum wells. Unintentional doping contributes then much more significantly to the concentration of free carriers interacting with excitons long after the loading laser pulse. Finally, let us note that free carriers can also result from exciton ionization at the boundary of our electrostatic trap. Indeed, we use the rectification of the electrostatic potential at the interface between the two surface electrodes to define the trap area. This rectification leads to an electric field component normal to the quantum wells, thus inducing a potential barrier for indirect excitons. However, it also leads to field components in the plane of the quantum wells [15]. Although these latter are normally small for our heterostructure design, we cannot exclude that some fraction of excitons is ionized in the outer rim of the trap, in which case excess holes would remain confined under the trap area [31].

To estimate the impact of excess carriers onto the photoluminescence spectral width, we compare the strength of collisional broadening due to exciton–exciton interactions, in single and DQWs. In the former case, experiments by Honold et al [13] have shown that the photoluminescence spectral width is increased by about 100 μeV when the exciton density grows from 10¹⁰ to 2 × 10¹⁰ cm⁻². This collisional strength is of the same order as the calculations made by Zimmermann for indirect excitons in DQWs [20], neither of which contradict our observations where collisional broadening between indirect excitons is not observed with our ∼100 μeV resolution (regime (ii) in figure 3). This reasonable agreement suggests that in both geometries exciton–exciton interactions have a comparable strength. Let us then extend to DQWs the strength of interactions between excitons and excess carriers measured in single quantum wells, which is 8 times larger than interactions between excitons [13]. To account for the 300 μeV mismatch between the homogeneous linewidth predicted in [20] and our measurements at T_b ≳ 1.2 K in regime (ii), we deduce that a free carrier density of about a few 10⁹ cm⁻² is sufficient. Remarkably, this concentration is of the order of the level of unintentional doping thus suggesting that this limitation is potentially responsible for about 300 μeV of the measured photoluminescence spectral width. Finally, let us note that by considering n_m and the level of unintentional doping, together with the strength of interactions between excitons and excess carriers measured by Honold et al [13], we deduce that the photoluminescence spectral width at the termination of the laser excitation is mostly governed by interactions between excitons and photo-injected free carriers, and amounts to about 1 meV. This value is in good agreement with our observations [11], thus supporting further the previous assumptions. In fact, we attribute the rapid and temperature independent increase of Γ in regime (iii) to excitons-free carriers interactions. Indeed, the latter regime corresponds to delays ranging from 120 to 20 ns after extinction of the loading laser pulse. While the exciton density is increased by analyzing shorter delays, the concentration of excess carriers is also increased since the transient photocurrent is not fully suppressed and therefore Γ increases in a temperature independent fashion governed by excitons-free carriers scatterings.

The most striking feature of figure 3 is certainly the threshold increase of the interference contrast at T_b = 330 mK and n_X ∼ 2 × 10¹⁰ cm², so that Γ ∼ 300 μeV and remains rather constant until n_X ∼ 3.5 × 10¹⁰ cm⁻². Remarkably, this density range coincides with the one where we have demonstrated a gray Bose–Einstein condensate for identical experimental conditions [12]. It is then natural to attribute the observed threshold as the result of the exciton condensation. Furthermore, Γ ∼ 300 μeV matches the contribution of interactions between excitons and excess carriers to the homogeneous broadening, thus suggesting that τ_c is governed by scatterings with free carriers in this condensed regime. Finding this limitation is somehow not very surprising since longer coherence times are expected from exciton–exciton interactions alone [9]. Finally, at T_b = 330 mK figure 3 shows that τ_c abruptly decreases to its value at 1.2 K at the onset of regime (iii). This indicates that the gray condensation is accessible in a density range possibly limited by the increased concentration of excess carriers at lower delays to the loading laser pulse. Nevertheless, we do not exclude that other processes become significant at n_X ∼ 3.5 × 10¹⁰ cm⁻².

To relate further the threshold increase of τ_c at T_b = 330 mK to the gray condensation, we measured $| {g}^{(1)}|$(r,δt) along the center of the trap and as a function of T_b for n_X ∼ 2.5 × 10¹⁰ cm⁻², i.e. at the center of regime (ii). For each bath temperature τ_c was first extracted from the exponential decay of the interference visibility with δt. As shown in figure 4(a), a rapid increase is revealed below a bath temperature of about 800 mK, τ_c passing from about 2 ps at 1 K to 4 ps at 330 mK. To measure more precisely the increase of τ_c, we performed an additional experiment where the interference contrast was evaluated for δt = 1.8 ps only. Thus, the stability of our measurements was largely increased since the experiments could be performed during a single day, compared to 3 d for the data displayed in figure 4(a). As shown in figure 4(b) we then recovered that two regimes emerge: for T_b ≥ 1.2 K, $| {g}^{(1)}|$ varies weakly, it is about 22% corresponding to τ_c ∼ 2 ps. By contrast, below T_c ∼ 1.2 K, the interference contrast increases abruptly to 32% at T_b = 330 mK, manifesting that τ_c is doubled at sub-Kelvin bath temperatures. Let us then stress that this value of T_c reproduces quantitatively the critical temperature measured for the appearance of quantum spatial coherence for the same exciton density in the trap [12]. This strongly supports our interpretation that the increase of excitonic temporal coherence is due to their quantum condensation in the trap.

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We have shown that the gray condensation of indirect excitons is associated to a threshold increase of the photoluminescence temporal coherence. Our experiments reveal that the condensate has a coherence time limited to around 4 ps, due to interactions between indirect excitons and excess carriers. These latter have a low concentration long after a laser excitation, about 10⁹ cm⁻², i.e. of the order of the level of unintentional doping during the epitaxial growth of our heterostructure. In this situation we do not find any indication for the formation of trions, i.e. bound states between excitons and excess carriers, neither in the normal nor in the condensed regime. Furthermore, from our observations we anticipate that reducing the density of free carriers to a minimum value of a few 10⁸ cm⁻² would lead to a maximum time coherence of several tens of ps for gray condensates. This may be accessed following the technology developed for electrically connected bilayer heterostructures [32], i.e. by using additional gate electrodes contacting directly each quantum well. Moreover, our experiments indicate that quantum condensation is excluded for densities of free carriers exceeding ∼5 × 10⁹ cm⁻², as expected theoretically [33]. In every case, we note that the coherence time of a gray condensate remains one to two orders of magnitude smaller than for quantum fluids of polaritons [34]. We attribute this difference as the manifestation of the matter-like nature of exciton condensates which are then very sensitive to their electrostatic environment.

The authors are grateful to Kamel Merghem for technical support. Our experiments have been financially supported by the projects XBEC (EU-FP7-CIG), by OBELIX from the French Agency for Research (ANR-15-CE30-0020).