Directional Goldstone waves in polariton condensates close to equilibrium

TL;DR

This paper investigates the excitation spectrum of thermalized polariton condensates, demonstrating a transition to superfluid-like behavior with directional Goldstone waves influenced by asymmetric pumping.

Contribution

It introduces a microscopic measurement technique to observe the transition from free particles to superfluid excitations in polariton condensates, highlighting directional effects.

Findings

Transition from parabolic to linear dispersion with increased density

Creation of directional excitations analogous to sea breeze surface waves

Measurement of critical sound speed near equilibrium

Abstract

Quantum fluids of light are realized in semiconductor microcavities by exciton-polaritons, solid-state quasi-particles with a light mass and sizeable interactions. Here, we use the microscopic analogue of oceanographic techniques to measure the excitation spectrum of a thermalised polariton condensate. Increasing the fluid density, we demonstrate the transition from a free-particle parabolic dispersion to a linear, sound-like Goldstone mode characteristic of superfluids at equilibrium. Notably, we show that excitations are created with a definite direction with respect to the condensate, analogous to how a sea breeze develops surface waves aligned with the wind. These results reveal the effect of asymmetric pumping on the collective excitations of a condensate. Furthermore, we measure the critical sound speed for polariton superfluids close to equilibrium.

Figures

Equations

$$\hslash {\omega }_{bog}(k)$$

$${\omega }_{ee}(k)$$

$$i\hslash \frac{\partial \psi (\mathbf{r},t)}{\partial t}$$

$$\displaystyle+g|\psi(\mathbf{r},t)|^{2}+V(\mathbf{r})\Bigg{]}\psi(\mathbf{r},t)$$

$$V(\mathbf{r})=\frac{V_0}{L_0}(L-x)$$

Full text

Exciton-polaritons (hereafter polaritons) are bosons that condense in a driven-dissipative environment, where the steady state is achieved through a balance between gain and losses Imamoglu et al. (1996); Carusotto and Ciuti (2013). In this two-dimensional system, the pump is usually a non-resonant laser that creates a large population of excitons (in a reservoir), which quickly relax into lower-energy polariton states. Dissipation mainly occurs through leakage from the cavity mirrors, requiring constant feeding from the pump and allowing the optical detection of the polariton field by photoluminescence (PL) measurements. Above a density threshold, macroscopic quantum degeneracy of polaritons has been demonstrated in a variety of materials and structures up to room temperature Kasprzak et al. (2006); Balili et al. (2007); Daskalakis et al. (2015); Lerario et al. (2017). However, the fundamental dynamics of the collective excitations of polariton quantum fluids are often hidden by the microscopic details of the disorder environment and by the effect of pumping and dissipation.

The excitation spectrum of polariton condensates is modified by drive and decay with respect to the equilibrium case of cold atoms, and is diffusive rather than linear at small momenta Keeling et al. (2004); Szymańska et al. (2006). This is a general feature of driven-dissipative systems and prevents the Landau criterion from being fulfilled in non-resonantly pumped polariton condensates Wouters and Carusotto (2007a). Recently, it has been suggested that the diffusive character of low-energy excitations can be strongly suppressed for long-lived polaritons (lifetimes longer than $100\text{\,}\mathrm{ps}$) and further reduced if the polariton condensate is spatially separated from the exciton reservoir Sun et al. (2017); Caputo et al. (2018).

Although in principle the excitation spectrum of polariton condensates is accessible through PL experiments, in practice its detection is seriously hindered by the low signal-to-noise ratio, the strong emission intensity from the condensate itself and the relatively broad polariton resonances Utsunomiya et al. (2008); Pieczarka et al. (2015); Byrnes et al. (2012); Nakayama and Ueda (2017). First indications of the existence of a soft Goldstone mode in time resolved experiments were obtained by the observation of a critical slowing down of the dynamics of an optical parametric amplifier Wouters and Carusotto (2007b); Ballarini et al. (2009). More recently, four-wave mixing experiments have highlighted the presence of the ghost branch, which appears with negative energies with respect to the condensate Wouters and Carusotto (2009); Kohnle et al. (2011, 2012); Byrnes et al. (2012). However, so far any quantitative comparison has been challenging due to the short polariton lifetime and the small condensate size, limiting in energy and wavevector resolution a direct observation of the Goldstone spectrum.

In trapped atomic gases, Bragg scattering of two photons is used to detect collective excitations, allowing very accurate measurements of their energy and wavevector Anderson et al. (1995); Jin et al. (1996); Steinhauer et al. (2002). In the opposite limit of wave-particle duality, the Bogoliubov spectrum of photons in a hot atomic vapour has been recently measured by estimating the group velocity of transverse perturbations Fontaine et al. (2018). For quantum fluids of light, the excitation spectrum $S(k,\omega)$ can be directly measured by performing the Fourier transform (FT) of the intensity pattern in space and time, $S(k,\omega)=\text{FT}\left[\left|\psi(\mathbf{r},t)\right|^{2}\right]$ Vocke et al. (2015). A related technique is used to measure the dispersion of ocean waves by the FT of a time series of pictures taken from an aircraft: instead of measuring directly the water displacement, the light diffracted by the surface waves at different times is used to reconstruct the frequency-wavevector relation with high resolution Dugan et al. (2001).

In this Letter, we employ a high-quality semiconductor microcavity with a reduced density of defects and long polariton lifetime to form a condensate close to equilibrium and to measure the spectrum of its collective excitations by interferometric measurements of temporal and spatial coherence. This represents the optical analogue of the oceanographic technique used to obtain the dispersion of surface waves (Supplemental Information). In our case, the spatio-temporal oscillations of the polariton condensate are obtained through the fluctuations of the first-order correlation function. Using this technique we observe, for the first time, the transition to a linearised dispersion with increasing particle density, showing the dominant phonon character of the Goldstone modes in a thermalised polariton superfluid. Interestingly, collective excitations form in the condensate with a preferential direction, much like surface waves in the ocean are oriented along the wind blowing in a specific direction. The origin of this peculiar effect in polariton condensates is due to the asymmetric pumping configuration which results in an asymmetrically populated Goldstone dispersion.

The sample used in these experiments is a high quality factor ($Q>10^{5}$) GaAs microcavity with a polariton lifetime of $\approx$100\text{,}\mathrm{ps} and twelve 7-nm quantum wells placed at the positions of maximum field enhancement within the cavity Nelsen et al. ([2013](#bib.bib27)); Ballarini et al. ([2017](#bib.bib28)); Estrecho et al. ([2018](#bib.bib29)). The pump laser (tuned to the first minimum of the stop band, at $E=$1690\text{\,}\mathrm{meV}, much higher than the polariton resonance) is focussed into a Gaussian spot with diameter $\approx$10\text{,}\mathrm{\SIUnitSymbolMicro m}. After fast energy relaxation, a high density of excitons and polaritons is formed under the pump spot. As a consequence, an expanding cloud of polaritons ballistically propagate radially outwards from the injection point, while excitons, much heavier, are mainly localised within the pumping spot. In Fig. [1](#S0.F1)a-b, (showing momentum- and real-space cross sections, respectively) the $k$-delocalised emission at $E\approx$4\text{\,}\mathrm{meV} comes from the high polariton density within the pump spot and the peaks at $|k|=$2\text{,}\mathrm{\SIUnitSymbolMicro m}$^{-1}$ correspond to polaritons radially expanding from the pump spot outward. Above a critical density, energy relaxation occurs efficiently in the expanding polariton cloud through stimulated scattering mediated by phonons, reducing the kinetic energy towards the bottom of the polariton dispersion and triggering the formation of a large polariton condensate with $k\approx 0$ all around the pumping spot Doan et al. (2005); Wouters et al. (2010); Ballarini et al. (2017). The highest intensity peak (red rectangle) is measured at the bottom of the dispersion ($k=0$) and is due to the emission coming from the extended condensate formed outside of the pump spot.

The signal coming from a region of the bottom condensate on the right side of the pump spot is sent to a Michelson interferometer with a corner reflector in one arm (R) and with a long delay line (M) in the other arm (Fig. 1c). An energy filter is applied to keep only the signal close ($0.5\text{\,}\mathrm{meV}$) to the bottom of the dispersion and avoid the contribution of higher-energy polaritons. The corner reflector is used to obtain the symmetric image around the central point (autocorrelation point), while the long delay line is used to measure the spatial correlations at different times. Selecting only the direction passing through the center of the image, the one dimensional interferogram is recorded for increasing delays to obtain, from the fringe visibility, the two-dimensional spatio-temporal correlation map $g^{(1)}(\Delta{x},\Delta t)$. The condensate density can be controlled by externally tuning the intensity of the exciting laser and, in the left column of Fig. 2, $g^{(1)}(\Delta{x},\Delta t)$ is shown for increasing polariton density from top to bottom. In the right column of Fig. 2, the two-dimensional Fourier transforms of $g^{(1)}(\Delta{x},\Delta t)$ directly show the momentum-energy relations with increasing densities from top to bottom.

For a weakly-interacting many-boson system without drive and dissipation, the role of interactions can be addressed in the Bogoliubov approximation by introducing new quasi-particles, defined as a superposition of condensed particles, corresponding to forward and backward propagating waves Bogolyubov (1947). The associated Bogoliubov dispersion has a positive and a negative branch of the form:

[TABLE]

where $m$ is the particle mass, $\mu$ the self-interaction energy ($\mu=g\left|\psi\right|^{2}$ with $g$ the particle-particle interaction strength and $\psi$ the condensate wavefunction) and $k=\frac{p}{\hbar}$ the wavevector of the matter wave. The effect of the out-of-equilibrium configuration, with pumping and decay, is instead reflected in the following spectrum of elementary excitations

[TABLE]

by the term $\Gamma$, which is a combination of pumping and dissipation rates Szymańska et al. (2006); Wouters and Carusotto (2007a).

Experimental data are compared in Fig. 3 with the dispersion relation of Eq. 2 and with the single-particle dispersion $\frac{\hbar^{2}k^{2}}{2m}$. The single-particle parabolic dispersion is measured from the momentum space PL well below the threshold power for condensation (Fig. 3a). To take into account the small, but still measurable, velocity of the condensate that tilts the excitation spectrum with respect to the reference frame, the Doppler term $\omega_{dop}=(k-k_{c})v_{c}$ is added to Eq. 2. Moreover, to center the dispersion at the energy and wavevector of the condensate, the angular frequency $\omega_{offset}=v_{c}k_{c}$ is used, giving a total frequency term of $\omega_{tot}=\omega_{dop}+\omega_{ee}+\omega_{offset}$. Approaching the threshold (Fig. 3b), the measured excitation spectrum (points are obtained from the positions of the maximum intensity in the data shown in the right column of Fig. 2) cannot be distinguished from the bare dispersion (dashed line). Linearisation can instead be seen in Fig. 3c and more clearly in Fig 3d. Bogoliubov quasi-particles behave as phonons ($\omega=c_{s}k$, with $c_{s}=\sqrt{\mu/m}$ the speed of sound) at small momentum $p\ll mc$ and as $\frac{\hbar^{2}k^{2}}{2m}+2\mu$, i.e. the single-particle dispersion with an additional frequency shift due to interactions, in the opposite limit of large momentum. From Fig. 3c to Fig. 3d, the sound speed extracted from the fitting curves increases from $0.35\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{s}$ to $0.55\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{s}$ and the healing length decreases from $\xi=$4\text{,}\mathrm{\SIUnitSymbolMicro m} to $\xi=$2.5\text{\,}\mathrm{\SIUnitSymbolMicro m}, giving cutoff wavevectors of $\frac{1}{\xi}\approx$0.24\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1} and $\frac{1}{\xi}\approx$0.38\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}, respectively. The fluid velocity also increases, from $v_{f}=$0.04\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{s} in Fig [3](#S0.F3)c to $v_{f}=$0.13\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{s} Fig 3d.

In Fig. 4, the points of Fig. 3d close to the condensate energy (energy resolution is given by $\delta E=\hbar\frac{2\pi}{T}=$0.02\text{,}\mathrm{meV}, with $T\approx$200\text{\,}\mathrm{ps} the total time delay in the interferometric measurements) and wavevector are shown zoomed in. The sound dispersion $\omega=ck$ follows the experimental data for $k<$0.3\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}, while for larger wavevectors the excitation spectrum tends to recover the parabolic dispersion (dashed-gray line). At all densities, $\Gamma<$0.005\text{\,}\mathrm{p}\mathrm{s}^{-1}, that is, the diffusive part of the excitation spectrum, where $\text{Re}\left[\omega_{ee}(k)\right]=0$ (flat dispersion), is limited to only very small wavevectors $k<$0.007\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}, i.e. much smaller than the momentum resolution of the measurements ($\delta k=\frac{2\pi}{L}$, with $L\approx$100\text{\,}\mathrm{\SIUnitSymbolMicro m} the lateral size of the condensate). This means that the diffusive character of the excitation spectrum emerges only for distances larger than the condensate itself.

As can be seen in Fig. 2 and Fig. 3, only one branch of the collective excitations, corresponding to forward propagating positive frequencies (and negative backward frequencies), is detected in the experiments. It is interesting to note that a finite condensate velocity only tilts the excitation spectrum in momentum space, and leaves the dispersion population symmetric. Our observations are instead very similar to measurements obtained for surface gravity waves in the presence of wind, for which only one branch can be observed (see SI). In our case, the directional driving force is ascribed to the asymmetric pumping configuration, where we consider only a portion of the condensate placed on the right-hand side of the pumping laser. The quantum and thermal fluctuations which populate the collective excitations of the condensate develop within the pump spot region where the exciton reservoir is localised and are mostly directed radially outwards, resulting in a negligible “up-wind” contribution to the excitation spectrum in our measurements.

To understand the physical origin of these observations, a numerical analysis based on the driven-dissipative Gross-Pitaevskii equation (GPE) for the polariton field $\psi$ was performed with the same microscopic parameters. Specifically, the GPE is given by

[TABLE]

where $m\approx 3.8\cdot 10^{-5}m_{e}$ is the polariton effective mass, $\gamma(\mathbf{r})$ is the Gaussian pump, $\kappa\approx$1\text{/}200\text{,}\mathrm{ps}$^{-1}$ is the effective decay rate, and g=$0.004\text{\,}\mathrm{meV}\text{\,}\mathrm{\SIUnitSymbolMicro m}$2 and $n_{s}$=$1000\text{\,}\mathrm{\SIUnitSymbolMicro m}$-2 are the effective polariton-polariton interaction strength and the saturation density, respectively. In the spirit of truncated Wigner methods, we include thermal fluctuations by adding white noise as an initial condition and averaging over different realisations.

In Fig. 5, the asymmetric experimental configuration is replicated by appropriate numerical simulations. The wedge potential, given by

[TABLE]

where $V_{0}=0.1meV$ and $L_{0}=400.0\mu m$, is used in the simulations shown in Fig. 5 to fully reproduce the experimental conditions but it is inessential to the physics observed here, as explicitly demonstrated in the SI. The pump is located at the bright spot where the dashed lines cross in Fig. 5a, while we measure the condensate on the right-hand side of the pump (see also Fig. 5b, showing the phase of the condensate in 2D space with the region under consideration marked by a semitransparent rectangle). In the inset of Fig. 5a, we plot the polariton density along the horizontal and vertical cross sections passing through the pump, with a decreasing polariton density going from left to right in the region under consideration (indicated by the solid, red line). In Fig. 5c, we present the excitation spectrum obtained following the same procedure as for the experimental data, and showing that only one Bogoliubov branch is occupied, as in experiment. Note that the ghost branch appears populated in this case as a result of applying the Fourier transform to $g^{(1)}(\Delta\mathbf{r},\Delta t)$, which is symmetric with respect to frequency. The true population of the ghost branch is shown in Fig. 5d, where the FT is applied to $\left|\psi(\mathbf{r},t)\right|^{2}$ and the ghost branch appears much less populated than in Fig. 5c (but still visible). However, Fig. 5d also shows the experimentally observed directionality of excitations, which is therefore independent on the technique used in the experiments. In Fig. 5e, we show the PL obtained analytically in the Bogoliubov approximation for homogeneous (spatially uniform) pumping. The other parameters are the same as those in the inhomogeneously pumped simulations, including the same finite velocity of the condensate, but nonetheless the directionality of the excitations is lost.

In conclusion, we have shown that the collective excitations typical of equilibrium BEC survive in a driven-dissipative environment, demonstrating the phonon-like character of a Goldstone mode. To obtain the frequency-wavevector relation of the collective excitations of the condensate, we have measured the fluctuations of the correlation function in space and time, in analogy to oceanographic techniques measuring the dispersion of surface waves. Moreover, we have investigated the origin of the asymmetry in the population of the excitation spectrum, finding that the off-axis pumping configuration is responsible for the directionality of the Goldstone waves as it leads to a breeze of quantum fluctuations blowing on the condensate. These results show that long-lived polaritons manifest universal features typical of interacting bosons at equilibrium and open the door to the optical investigation and control of the collective excitations in open-dissipative condensates by engineering the reservoir topology.