Abstract
Spectra of lowlying elementary excitations are critical to characterize properties of bosonic quantum fluids. Usually these spectra are difficult to observe, due to low occupation of noncondensate states compared to the ground state. Recently, lowthreshold BoseEinstein condensation was realised in a symmetryprotected bound state in the continuum, at a saddle point, thanks to coupling of this electromagnetic resonance to semiconductor excitons. While it has opened the door to longliving polariton condensates, their intrinsic collective properties are still unexplored. Here we unveil the peculiar features of the Bogoliubov spectrum of excitations in this system. Thanks to the dark nature of the boundinthecontinuum state, collective excitations lying directly above the condensate become observable in enhanced detail. We reveal interesting aspects, such as energyflat parts of the dispersion characterized by two parallel stripes in photoluminescence pattern, pronounced linearization at nonzero momenta in one of the directions, and a strongly anisotropic velocity of sound.
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Introduction
Bound states in the continuum (BICs) have been originally proposed as a mathematical feature of the Schrödinger equation in presence of specially prepared potentials^{1,2,3}. Similar states have been later found in a wide range of systems like graphene, topological insulators, atomic superlattices, dielectric photonic crystals, metasurfaces and patterned optical waveguides^{4}. In the latter cases, coupling in both the real and the imaginary parts of the two counterpropagating electromagnetic modes leads to the opening of an energy gap, with one of the new eigenmodes being bright and the other exhibiting vanishing radiative losses despite the nonHermiticity of the underlying Hamiltonian^{5,6}. Symmetryprotected optical BICs occuring from two interfering resonances enable light confinement and are routinely used in grating surfaceemitting and distributed feedback lasers^{7,8,9}, where emission from the BIC mode manifests itself in a specific twolobe far field pattern. Interestingly, these quasiinfinitelifetime photonic states can couple to matter excitations, such as surface plasmons^{6} or excitons in monolayer transitionmetal dichalcogenides^{10,11} and quantum wells^{12} and form polaritons with interesting features, such as greatly enhanced tunable lifetimes and strong nonlinearities.
Polaritons are hybrid halflight halfmatter quasiparticles in semiconductor heterostructures, typically embedded in a FabryPérot microcavity to enhance the coupling of electronic excitations (e.g. excitons) with photons and to reduce the losses^{13}. Such lowdimensional bosonic systems have been shown to provide conditions for observation of collective behaviors like BoseEinstein condensation^{14,15,16} and superfluidity^{17,18,19}, at the same time allowing for a direct measurement of their momentum and spatial distributions via the emitted light. The onset of BoseEinstein condensation is accompanied with a narrowed enhanced emission from the ground state at k = 0 of the polariton energy dispersion, which consequently blueshifts due to particleparticle interactions. At the same time, the momentumenergy dispersion on top of the condensate becomes linearized in accordance with the textbook Bogoliubov prediction^{20}, therefore giving origin to superfluidity as follows from the Landau criterion. Multiple experiments with cavity excitonpolaritons have provided evidence of such linearization^{21,22,23,24,25} despite the drivendissipative nature of the system suggesting the diffusive character of the Bogoliubov dispersion^{26,27} with a zero real part in the region of small momenta. Precise observation of the shape of the excitations spectrum due to the thermal and quantum depletion of the macroscopically occupied ground state is also possible, but requires substantial momentumspace filtering covering a much brighter signal from the condensate^{28,29} or using refined interferometric techniques^{25}.
In this work, we study the elementary excitation spectrum of a polariton Bose condensate arising from a BIC state in a planar nanostructured waveguide. It has been recently reported^{12} that due to the band folding effect and the consequent coupling with the quantum well exciton mode, the condensate which is formed on the lowest polariton branch appears, counterintuitively, at a saddle point of the dispersion rather than in a global energy minimum. Since the particles accumulate in the quasiBIC state with a very long lifetime, much longer than the carriers relaxation time, the polariton condensate in such a state appears to be a paradigmatic system for studying the excitation spectrum of a condensate in thermal equilibrium^{30}. In such state the condensate cannot directly radiate and, as we demonstrate in the following, measurements of the shape of the spectrum of elementary excitations are possible in great detail and even without any momentum filtering. We derive theoretically the finitetemperature Bogoliubov dispersion for polaritons accumulating in the BIC and show that a local energy minimum starts to appear around the saddle point k = 0 in the spectrum of excitations which, possibly, could help in the formation of a longliving BoseEinstein condensate, with the real part of the spectrum being nonzero despite the negative effective mass in one of the directions and losses present in the system. We experimentally observe various and very distinct characteristics of collective excitations dispersion strongly dependent on the direction in kspace. The detailed knowledge of the Bogoliubov spectrum allows to investigate the angular profile of the sound velocity in such highly anisotropic system. We believe that this study could open a way of controlling the condensate properties by engineering its excitation spectrum.
Results
Saddlepoint polariton condensate
The sample studied here, sketched in Fig. 1a, is a GaAs/AlGaAs waveguide hosting 12 20nmthick GaAs quantum wells embedded in a Al_{0.4}Ga_{0.6}As core. A set of linear diffraction gratings is etched on the surface of the waveguide to allow for the observation, within the lightcone, of the propagating guided modes. By changing the grating period a and the filling factor it is possible to tune the energy of the BIC (i.e. its excitonic fraction) and the width of the energy gap separating the lowerpolariton BIC, with ultralong lifetimes, and the leaky polaritonic modes. The coupling of the counterpropagating TEmodes of electromagnetic waves to the exciton results in the four branches of polariton dispersion, the analytical derivation of which are provided in the Supplementary Information (SI). Of interest, in this work, are the two anisotropic lowest branches which are shown in Fig. 1b. It is clearly seen that the shape of the upper of the two lowerpolariton modes (ULP) resembles the regular lower cavity excitonpolariton, albeit with differing effective masses along and perpendicular to the grating, whereas the lower (LLP) mode is distinct from the standard dispersion showing different behaviors along k_{x}– and k_{y}–directions. In particular, we note the appearance of a negative effective mass in the x–direction, with an absolute value much lower than the positive mass in the y–direction. It is also important to note that, when expanded at k → 0 in Taylor series up to the second order, the LLP dispersion appears to have an imaginary contribution in the x–direction (for more details, see the SI). This imaginary term provides the k_{x}–dependent loss rate of polaritons on the lowest branch, with the zero radiative loss at k = 0, as was experimentally verified in ref. ^{12}.
The polariton dispersions along k_{x} and k_{y} can be directly measured by energy and momentum resolved photoluminescence (PL). Figure 1c and d show two different cuts (along k_{x} and k_{y} respectively) of the PL of the singleparticle lower polariton dispersions. While along k_{x} (Fig. 1c) the ULP and LLP have the masses of opposite sign, this is not the case along k_{y} where the mass is positive for both (Fig. 1d). The dispersion in Fig. 1d has been taken at a slightly finite wavevector in the x–direction (k_{x} ~ 0.02 μm^{−1}) to avoid the completely dark stripe of the BIC state at k_{x} = 0. These experimental dispersions are in very good agreement with the analytical model which is shown by the overlaid yellow dashed lines in both panels. We note that the range of horizontal axis along k_{y} is four times larger than that of k_{x} and that the effective masses along the two directions differ by about two orders of magnitude, being in quantitative agreement with the theory developed in the SI. Both images in Fig. 1c and d show the PL emission below threshold, when the upper of the two lowerpolariton branches is emitting more light than the lowest one, as it is much more lossy around k = 0 (see SI), while their populations are comparable. When increasing the excitation power, polaritons condense into the BIC at the saddle point of the LLP branch^{12}. Figure 1e (righthand side) shows the crosscut of the dispersion along k_{x} with the timeintegrated PL measured slightly above the condensation threshold. We only show one of the two bright spots forming to either side of the saddlepoint condensate, blueshifted compared to the belowthreshold image shown on the lefthand side of the panel. Figure 1f (righthand side) shows the corresponding dispersion along k_{y} for k_{x} ~ 0.02 μm^{−1}, i.e. close to the inner edge of the bright spot of Fig. 1e above threshold. The latter has an interesting form, with an energyflattened emission roughly extending from −0.5 μm^{−1} (not visible) to +0.5 μm^{−1} continuing into linear tails going up in energy. It is important to note that, as we will show in the following, the flat region in this case is not related to the diffusive character of the condensate. On the contrary, it is a consequence of both the very long polariton lifetime and the saddle shape of the dispersion.
Hartree–Fock–Bogoliubov theory for elementary excitations
To explain these features observed in Fig. 1e, f, we derive the Bogoliubov spectrum of excitations on top of the condensate forming in the BIC state, i.e. in the vicinity of the k = 0 saddle point of the LLP branch for low finite temperatures in the presence of the (dark) exciton reservoir^{30}. The Hamiltonian of lower polaritons is expressed in the second quantization as
where \({\hat{P}}_{\pm }({{{{{{{\bf{r}}}}}}}})\) are the ULP and LLP field operators, respectively, \(\hat{{{{{{{{\bf{p}}}}}}}}}=i\hslash \nabla,{\varepsilon }_{\pm }({{{{{{{\bf{p}}}}}}}})={E}_{\pm }^{{{{{{{{\rm{LP}}}}}}}}}({{{{{{{\bf{p}}}}}}}}){{{{{{{\rm{Re}}}}}}}}\,{E}_{}^{{{{{{{{\rm{LP}}}}}}}}}(0)\) are the corresponding bare particle dispersions (the shape of \({E}_{\pm }^{{{{{{{{\rm{LP}}}}}}}}}\,({{{{{{{\bf{p}}}}}}}})\) is given in the SI) counted from the saddle point of the LLP branch, μ_{±} is the chemical potential, \(\hat{Q}({{{{{{{\bf{r}}}}}}}})=\int\,d{{{{{{{{\bf{r}}}}}}}}}^{{\prime} }[X({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}}){\hat{P}}_{}({{{{{{{{\bf{r}}}}}}}}}^{{\prime} })+C({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}}){\hat{P}}_{+}({{{{{{{{\bf{r}}}}}}}}}^{{\prime} })]\) is the exciton field operator, and \(\hat{\tilde{Q}}({{{{{{{\bf{r}}}}}}}})\) describes the field of background excitons that do not directly convert into polaritons (e.g. dark excitons). In the above, \(X({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}})=(1/S){\sum }_{{{{{{{{\bf{p}}}}}}}}}{X}_{{{{{{{{\bf{p}}}}}}}}}\exp \{i{{{{{{{\bf{p}}}}}}}}\cdot ({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}})/\hslash \},C({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}})=(1/S){\sum }_{{{{{{{{\bf{p}}}}}}}}}{C}_{{{{{{{{\bf{p}}}}}}}}}\exp \{i{{{{{{{\bf{p}}}}}}}}\cdot ({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}})/\hslash \},{X}_{{{{{{{{\bf{p}}}}}}}}}\) and C_{p} are the exciton and photon Hopfield coefficients, S is the quantization area. For simplicity, the excitonexciton pair interaction potential is assumed to be contact: U(r) = gδ(r). The Hamiltonian (1) allows to write the Heisenberg equations for the evolution of both lower polariton fields \({\hat{P}}_{\pm }({{{{{{{\bf{r}}}}}}}},t)\).
To obtain the excitation spectrum on top of the Bose condensate of BIC polaritons, we will assume that above threshold the LLP is macroscopically populated, which is supported by the experimental observation (see Fig. 1e). Since the polaritons occupying the ULP branch do not convert into LLP polaritons because of the different symmetry of the underlying photon and exciton modes, we exclude the ULP operators from consideration when describing the macroscopically occupied saddle point. In this case, one can separate the condensate in the regular Bogoliubov fashion and, by averaging the Heisenberg equation for \({\hat{P}}_{}\) in the HartreeFock meanfield approximation (see “Methods” section), obtain the expression for the chemical potential of the LLP polaritons:
where n_{0} is the condensate density, \({n}_{Q}^{{\prime} }\equiv \langle {\hat{Q}}^{{\prime} {{{\dagger}}} }({{{{{{{\bf{r}}}}}}}}){\hat{Q}}^{{\prime} }({{{{{{{\bf{r}}}}}}}})\rangle\) is the noncondensate exciton density due to finite temperature, and \(\tilde{n}\equiv \langle {\hat{\tilde{Q}}}^{{{{\dagger}}} }({{{{{{{\bf{r}}}}}}}})\hat{\tilde{Q}}({{{{{{{\bf{r}}}}}}}})\rangle\) is the background reservoir density. We note that μ_{−} contains the thermal contribution \({n}_{Q}^{{\prime} }\) and the dark reservoir contribution \(\tilde{n}\) compared to the regularly used expression for the lowerpolariton chemical potential μ_{LP} = gn_{0}∣X_{0}∣^{4}. Importantly, Eq. (2) corresponds to the experimentallyobserved condensate blueshift, which will be used together with the excitation spectrum below to define n_{0} and \(\tilde{n}\) at a given pump power (note that at low temperatures considered here, \({n}_{Q}^{{\prime} }\, \ll \,{n}_{0}\)). Diagonalizing the Hamiltonian (1) within the HartreeFockBogoliubov framework, one obtains the excitation spectrum:
It needs to be underlined that while Eq. (3) has the form looking similar to the textbook Bogoliubov prediction, it is, in fact, substantially different. Besides noticing that \({{{{{{{\mathcal{E}}}}}}}}\) contains the reservoir contributions \({n}_{Q}^{{\prime} },\tilde{n}\) and the renormalization due to the dependence of the Hopfield coefficient X_{p} on momentum, the bare polariton dispersion ε_{−}(p) itself is very unusual: in the k_{x}–direction it features, at the same time, the negative effective mass and momentumdependent photon losses (see Supplementary Fig. 1 and Eq. (8) in the SI). Due to the latter, despite the negativelydefined \({{{{{{{\rm{Re}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}^{2}\) in the vicinity of k_{x} = 0, the real part of Eq. (3) is nowhere zero except the special points of the dispersion where \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}^{2}\) changes sign. The excitation spectrum does not acquire a diffusive character predicted for systems with dissipation^{26,27}. In fact, here the nonzero imaginary part of ε_{−}(p) ensures the existence of lowlying states just above the condensate with \({{{{{{{\rm{Re}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}} > 0\). The analysis of the imaginary part of the spectrum is given below.
Figure 2a shows the real part of the Bogoliubov dispersion of excitations (Eq. 3) on top of the condensate formed in the saddle point of the LLP branch. The linearization at small momenta in the y–direction is clearly seen. Figure 2b, c show the cuts of \({{{{{{{\rm{Re}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) in Fig. 2a along different nearzero k_{x} (b) and k_{y} = 0 (c), displaying the flattening of the dispersion in the k_{y}–direction. As can be anticipated from (3), the extent of the energyflat part along k_{y} depends on the chosen nearzero k_{x}, while the slope of the linear tails of the excitation spectrum at larger k_{y} (for each k_{x}) is unambiguously defined by the polariton condensate density n_{0}. Moreover, the zoomin on the small momenta region displays a local minimum in the k_{x}–direction (see the inset of Fig. 2c). At the values of k_{x} outside this local minimum, the Bogoliubov spectrum recovers the negative slopes characteristic to the singleparticle LLP dispersion.
Excitations luminescence dynamics in the pulsed regime
The experimental PL images of collective excitations obtained in the regime of pulsed excitation are shown in Fig. 2d–f. While the features very close to k_{x} ~ 0 are below the experimental resolution and correspond to the dark region due to the long BIC lifetime, the dispersion along k_{y} shows a striking correspondence to the theory derived above. To perform a better comparison, we have measured the temporal dynamics of the dispersion of excitations in PL along k_{y}. This allows us to rule out any temporal smearing of the dispersion and, at the same time, track the change of the dispersion for different condensate and reservoir densities. Figure 2de display two different snapshots of the emission along k_{y} for k_{x} ~ 0.02 μm^{−1}. These snapshots corresponding to different times clearly exhibit an energyflat region with lesspronounced linear tails at higher values of k_{y}. Thanks to the intrinsic dark nature of the condensate from a BIC state, it is possible to directly measure these features without any filtering in the momentum space. The red solid lines in Fig. 2d, e show the analytical crosscuts of the dispersion (Eq. 3) along k_{x} ~ 0.02 μm^{−1}, for two values of densities calculated from the fitting of the measured linear tails together with the blueshift (as compared to the singleparticle dispersion shown by the lightblue dashed lines). Full temporal dynamics of the PL vs. k_{y} is provided in the Supplementary Movie 1, while additional snapshots corresponding to different densities are shown in the SI. Experimentally following this dynamics allows to observe in time the delayed formation of the dark condensate after the pulse arrival, accompanied by the theoretically predicted energyflat parts of the excitation spectrum, its shifting down with time due to the decreasing blueshift and, finally, disappearance of the flat parts and recovering of the singleparticle nearparabolic LLP dispersion as the system goes below threshold. Finally, Fig. 2f shows the energyresolved PL image along the k_{x}–direction above threshold. While the BIC polariton condensate in the local minimum of the modified dispersion stays dark, the excitations on top of the condensate occupying the very narrow region around k_{x} = 0 are emitting light. Here, since under pulsed nonresonant excitation the blueshift is strongly time dependent, the resulting timeintegrated measurement in Fig. 2f looks apparently broadened in energy (see the corresponding enlarged curves for different densities in the inset of Fig. 2c, getting smeared over time) which appears as a twolobe image.
Dispersion of collective excitations at different k _{x} under continuouswave excitation
In order to see a more pronounced linearisation of the dispersion tails, we turn to continuouswave (c.w.) excitation that allows to reach a density state while avoiding the energy blur induced by the dynamical change of energy of the condensate under timeintegrated measurements. For fixed excitation conditions, we compare the PL from energy dispersion versus k_{y} resolved at different values of k_{x} as shown in Fig. 3: one cut closer to k_{x} = 0 (a), another directly within the bright spot with maximal emission that is seen above threshold in Fig. 2f (b), and a final one when k_{x} is further away from the center (c). As one can observe, while the PL from the noncondensate (outoftheBIC) lower polaritons in panels (a) and (c) is weak enough to allow registering the emission from the upper branch, in panel (b) the PL from the flat part of the dispersion is very bright, making the ULP invisible on the same (normalized) scale. Matching the observed PL with the theoretical expressions for the dispersion of excitations Eq. (3) and blueshift Eq. (2) for all three cases and using the condensate and reservoir densities as fitting parameters allows us to define exactly the values of k_{x} at which those cuts were measured.
The first straightforward outcome of this study is the clearly increased slope of the linear part of the Bogoliubov dispersion for the states lying above the flat stripe, as shown in Fig. 3a, b. While the difference in energy at which the energyflat part is seen in (a) and (b) lies within the linewidth (see the red arrows in Fig. 3d or the inset of Fig. 2c), the length of the stripe is changing with k_{x} and is different for (a) and (b), in agreement with the theoretical Fig. 2b. The tails of the dispersion outside the flat part in both (a) and (b) are clearly more visible than in the case of pulsed excitation studied above, and are linearized in accordance with Eq. (3). The shifted LLP dispersion is plotted in panel (b) together with the Bogoliubov spectrum. When moving away from the condensate, the dispersion of excitations recovers the shape of the lower polariton branch, while still staying blueshifted, as shown in Fig. 3c. For comparison, in the righthand sides of panels Fig. 3a–c we plot the theoretically calculated PL emission, using the excitation spectrum derived above (note that since we neglected the upper polaritons in the theory, we do not reproduce the ULP luminescence). In this calculation, we assume that at finite temperature the normal branch of the Bogoliubov spectrum is predominantly populated (due to thermal depletion of the condensate).
As a more subtle and more meaningful insight from fine resolving the PL in k_{x} close to the BIC, we reveal that the maximal emission (panel b), i.e. the flat line in the Energy vs. k_{y} dispersion, corresponds to the point in k_{x} where the real part of the excitation spectrum E (counted from the blueshifted saddle point) goes to zero. This analysis is presented in Fig. 3d, where we indicate by arrows the two values of k_{x} at which panels (a) and (b) were measured (k_{x} corresponding to panel (c) lies out of the range of values plotted in (d) and is not marked). As discussed above, the only values of k_{x} where \({{{{{{{\rm{Re}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}=0\) correspond to the points in which the imaginary part of \({E}_{{{{{{{{\bf{p}}}}}}}}}^{2}\) changes sign, i.e. the points that lie at the same energy as the dark condensate in the BIC, thus making the scattering from the condensate towards these states energetically effortless. Looking at the imaginary part of the dispersion, plotted as well in Fig. 3d, allows us to conclude that since \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) is negative for all k_{x} > 0, the condensate with \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}=0\) stays nevertheless stable and there is no macroscopic gain in the states indicated by the arrow ‘b’. However, since they correspond to the largest \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\), these states correspond to the maximum PL intensity observed in the experiments. We conclude that the particles residing in the dark condensate at k_{x} = 0 at any scattering event may jump to the states ‘b’ where they leak out of the system due to maximal loss. For completeness of the analysis, we plot \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) vs. k_{y} in Fig. 3e. One can see that the large imaginary part and hence short lifetimes are characteristic only for the states along the energyflat part of the dispersion E_{p}(k_{y}), whereas the linear tails, corresponding to lowlying excitations above the saddlepoint condensate, feature narrow linewidth and can be described adequately within the equilibrium theory.
Anisotropy in the momentum space
To study the anisotropic dispersion of excitations discussed above, we measure the photoluminescence emission under pulsed excitation in the plane (k_{x}, k_{y}) which is shown in Fig. 4 for different energies close to the BIC state. For a clearer visualization, we suggest to watch the Supplementary Movie 2 showing the experimental crosscut of the PL emission in the far field at different energies. The exotic shape with two parallel stripes in the middle at a given energy can be directly related to the populated states of the excitation spectrum where \({{{{{{{\rm{Re}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}=0\) and the absolute value of the negative \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) is large, as discussed above and confirmed by the calculated \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) dependence on k_{x} and k_{y} in Fig. 3d, e. It is important to note that the distribution of the PL in two parallel stripes appears only above threshold: without the macroscopic population of the saddlepoint, there are no states lying at the same energy with the condensate and hence no population of the energyflat states along k_{y} that exhibit strong emission. For comparison, we provide in the SI and the Supplementary Movie 3 the corresponding data taken below threshold, where the energy corresponding to the saddle point stays fully dark. In the abovethreshold case of Fig. 4, due to time averaging in the pulsed excitation setting, the twostripe pattern which lies at the energy of the condensate is visible in more than one panel, as the condensate shifts down over time (due to decreasing blueshift). We note that the observed peculiar far field PL distribution in this case is very different from the twolobe pattern of gratingbased lasers^{8,9}, where, despite the similar dark line along k_{x} = 0, the lasing happens exclusively at the photonic BIC energy and the intensity distribution along k_{y} is a Gaussian centered at k_{y} = 0. In our case, the PL emission along both directions and in energy follows the calculated dispersion Eq. (3), with the length of the parallel lines shorter when the condensate density is lower, and the curved tails at nonzero momenta appearing outside the twostripe region (whose emission is weaker since \({{{{{{{\rm{Im}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) is much smaller). In all panels of Fig. 4, the theoretically calculated crosscuts of the real part of E_{p} at the corresponding energies above the BIC state are overlayed as dashed lines on top of the experimental images. The maximum of the experimental emission appears at those states where the real part of the dispersion E_{p} crosses the given energy (the brighter spots move away from the center in the panels (c) and (d) that correspond to higher energies).
From the point of view of the anisotropy of collective excitations which is illustrated by the above study of the spectrum, it is also interesting to address the limit of very small momenta. Experimentally, since the region of k_{x} corresponding to the positive slope of \({{{{{{{\rm{Re}}}}}}}}{E}_{{{{{{{{\bf{p}}}}}}}}}\) is very small, the shift in energy which is acquired within this narrow Δk_{x} lies within the linewidth. At the same time, it is not possible to follow the change in energy with k_{y} along the positive slope of the saddle (from Fig. 2a one sees that at k_{x} → 0 the energyflat part along k_{y} disappears and the dispersion is linear) due to the darkness of the states corresponding to k_{x} = 0 (at any k_{y}). Theoretically, however, taking the limit p → 0 in Eq. (3) allows to obtain the sound velocity which appears to be strongly dependent on direction (on the angle φ in the polar coordinate system):
where \({m}_{{{{{{{{\rm{LP}}}}}}}}}^{x(y)}\) are the effective masses of LLP in the x(y)–direction, and s_{x} is the masslike parameter defined by the radiative losses rate γ and the radiative coupling of the modes U (for more details, see the SI). In a standard case of real positive masses in both directions, for experimentallyrelevant parameters (resulting in \(1/{m}_{{{{{{{{\rm{LP}}}}}}}}}^{x}\gg 1/{m}_{{{{{{{{\rm{LP}}}}}}}}}^{y}\)) the velocity given by Eq. (4) would significantly increase when approaching the x–axis (the direction corresponding to \(\varphi=\pi m,m\in {\mathbb{Z}}\)). However, here the expression for the sound velocity is modified by the negative \({m}_{{{{{{{{\rm{LP}}}}}}}}}^{x}\) and by the imaginary contribution s_{x}, which makes the definition of the sound velocity more complicated. In Fig. 5b we plot the sound velocity dependence on the angle φ revealing a very anisotropic profile. The increase of c_{s} in the x–direction is clearly seen from the difference of slopes in the linearized parts of the Bogoliubov dispersion (see the zoomin at the lowmomenta region in Fig. 5a): while the slope along k_{y} at k_{x} = 0 corresponds to very low \({c}_{{{{{{{{\rm{s}}}}}}}}}^{y}\), the shallow parts of the spectrum at finite but small k_{x} result in the sound velocity in the directions slightly deviating from the y–axis being very close to zero (see Fig. 5c). Much steeper slopes along k_{x} at k_{y} = 0 provide maxima of c_{s} at φ = πm, as shown in Fig. 5b for various condensate densities. It is important to note that taking into account the nonradiative exciton losses would additionally affect the behavior of the excitation spectrum along k_{x} in a very narrow vicinity of the saddle point. A more detailed discussion of this matter is provided in the SI.
Discussion
In this work, we studied theoretically and experimentally the spectrum of elementary excitations of the BIC polariton condensate arising from the coupling of excitons to the photonic modes in a patterned semiconductor waveguide. The studied system is exotic and very different from the wellstudied case of microcavity polaritons. We show that despite the saddlelike shape of the singleparticle dispersion of lower polaritons, which exhibits a maximum in one of the directions in k–space, with the accumulation of particles in the saddle point, a local minimum is created with the positive slopes of the Bogoliubov dispersion, which may help in further accumulation of polaritons. At the same time, two parallel stripes in momentum space, which correspond to the states at the same energy as the condensate—but being bright—appear. It is, in fact, the interplay of the negative mass and the mometumdependent losses that results in the existence of such unique, extended in momentum, states outside the condensate. This unusual anisotropic shape of the excitation spectrum yields in the k_{y}–direction to two energyflat zones (near k_{x} = 0) separated by the dark line at k_{x} = 0. Thanks to the quasiinfinite lifetime of the BIC state, the condensate itself does not emit light hence allowing to neatly observe the populated states around and above the saddle point. With both the slopes of the linear parts of the dispersion that follow the energyflat regions and the interactioninduced blueshift being directly experimentally accessible, our theory allows to precisely define the macroscopic density of the dark polariton condensate. Thus, while the observation of the excitation spectrum in such a system is striking on its own, it also allows to obtain the information about the saddlepoint polariton condensate which is otherwise invisible.
Remarkably, despite the negative mass in the x–direction and the presence of losses, the real part of the spectrum is positive everywhere at k → 0, allowing to extract the velocity of sound in every direction. The latter appears to be highly anisotropic, dropping to almost zero in the directions next to the yaxis, i.e. slightly misaligned with the grating principal axis, and growing in the direction of propagation along the waveguide. It needs to be noted that the anisotropic sound velocity was reported previously for atomic BECs with anisotropic dipolar interactions^{31,32}. In case of dipolar BECs, however, transport measurements reveal the anisotropy of the critical Landau velocity rather than the velocity of sound^{33}, since the dispersion of excitations is affected by directiondependent roton softening. The situation realized in our work can be better compared with anisotropic superfluidity reported for excitons^{34} and cavity photon BEC^{35} in periodically modulated planar structures. At the same time, we stress that here we deal with a more exotic case of anisotropy: even though there is a welldefined finite sound velocity, superfluidity in the sense of Landau criterion cannot be reached in all directions of propagation. Indeed, no matter how small the velocity would be below c_{s} in the x–direction, due to the negative slopes of the dispersion, it could always elastically create an excitation out of the condensate. However if an obstacle is propagating strictly along the y–axis, the velocity \({c}_{{{{{{{{\rm{s}}}}}}}}}^{y}\) would be a proper Landau sound velocity below which superfluidity can be observed. This illustrates the richness of saddlepoint condensates and their anisotropic behaviors.
Along with the previously proposed topologicaldispersion engineering^{36}, we believe that this work demonstrates the high interest in engineering the excitation spectrum of the condensate e.g. via implementing different grating symmetries—as a tool to impart new properties to the condensate itself—and underlines once again the astonishing richness of polariton systems, as well as multiple possible avenues for further investigations.
Methods
Theoretical methods
Starting from the Hamiltonian (1) in the neglection of the ULP fields, one can separate the macroscopically occupied condensate state as
where n_{0} is the condensate density, \({\hat{Q}}^{{\prime} }({{{{{{{\bf{r}}}}}}}},t)=\int\,X({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }{{{{{{{\bf{r}}}}}}}}){\hat{P}}_{}^{{\prime} }({{{{{{{{\bf{r}}}}}}}}}^{{\prime} },t)d{{{{{{{{\bf{r}}}}}}}}}^{{\prime} }\), and the average \(\langle {\hat{P}}_{}^{{\prime} }({{{{{{{\bf{r}}}}}}}},t)\rangle=\langle {\hat{Q}}^{{\prime} }({{{{{{{\bf{r}}}}}}}},t)\rangle=0\). The substitution Eq. (5) allows to rewrite the triple products of the field operators in Eq. (1) in the HartreeFock meanfield approximation as
with notations \({n}_{Q}^{{\prime} }=\langle {\hat{Q}}^{{\prime} {{{\dagger}}} }({{{{{{{\bf{r}}}}}}}},t){\hat{Q}}^{{\prime} }({{{{{{{\bf{r}}}}}}}},t)\rangle,\tilde{n}=\langle {\hat{\tilde{Q}}}^{{{{\dagger}}} }({{{{{{{\bf{r}}}}}}}},t)\hat{\tilde{Q}}({{{{{{{\bf{r}}}}}}}},t)\rangle\). Using the Hamiltonian Eq. (1) with the substitutions Eqs. (5) and (6), (7), the Heisenberg equation for the noncondensed part of the polariton field \({\hat{P}}_{}^{{\prime} }({{{{{{{\bf{r}}}}}}}},t)\) can be written as
In Fourier space, (8) takes the form \(i\hslash {\partial }_{t}{\hat{P}}_{{{{{{{{\bf{p}}}}}}}}}(t)=[{\varepsilon }_{}({{{{{{{\bf{p}}}}}}}}){\mu }_{}+g(2{n}_{0}{X}_{0}{}^{2}+2{n}_{Q}^{{\prime} }+\tilde{n})]{X}_{{{{{{{{\bf{p}}}}}}}}}{}^{2}{\hat{P}}_{{{{{{{{\bf{p}}}}}}}}}(t)+g{n}_{0}{X}_{0}^{2}{X}_{{{{{{{{\bf{p}}}}}}}}}^{*2}{\hat{P}}_{{{{{{{{\bf{p}}}}}}}}}^{{{{\dagger}}} }(t)\) that allows to perform the Bogoliubov transformation
where \({\hat{\alpha }}_{{{{{{{{\bf{p}}}}}}}}}\) is the annihilation operator of the Bogoliubov excitation with momentum p. Eq. (9) yields the excitation spectrum Eq. (3). We note that due to the underlying photon dispersion whose imaginary part is strongly momentumdependent, the Hopfield coefficients X_{p}, C_{p} as well as the Bogoliubov amplitudes u_{p}, v_{−p} are complex, and careful treatment of complex conjugations throughout the theory is required. Moreover, choosing the Bogoliubov transformation in the specific shape given by Eq. (9) defines the socalled ghost branch (GB) of excitations as \({E}_{{{{{{{{\bf{p}}}}}}}}}^{{{{{{{{\rm{GB}}}}}}}}}={E}_{{{{{{{{\bf{p}}}}}}}}}^{*}\) (this definition of the GB is consistent with that used in ref. ^{27}).
Theoretical calculations of the intensity of photoluminescence from the excitations around the BIC state that are presented in Fig. 4 are performed using the standard expression^{37,38}:
Experiments
All the PL data reported in this work has been acquired using a confocal setup. The objective back focal plane is imaged on the entrance slit of the spectrometer allowing to obtain the energy versus k dispersion. Two different magnification of the objective focal plane are used in order to properly image the dispersions along k_{x} and k_{y} which have very different effective masses and hence very different extension in the kspace. The laser used in the experiments is a fspulsed laser having a repetition rate of 80 MHz and a pulse duration of ~100 fs. Excitation wavelength is set at 780 nm. The time resolved measurements are obtained by scanning the farfield emission and acquiring for each k_{y} an energy versus time temporal trace with a streak camera. The set of temporal traces is then used to reconstruct the dynamics in the Energy vs k_{y} space. Supplementary Movie 1 shows an example of temporal dynamics reconstructed in this way. The emission in the (k_{x}, k_{y})–plane at a given energy like the images shown in Fig. 4 are obtained by scanning the farfield emission and acquiring a set of Energy vs k_{x} dispersions, each one corresponding to a different k_{y}. The data are then merged and cut at a given energy in the plane k_{x}, k_{y}. Supplementary Movies 2 and 3 show a sequence of such cuts for different energies above and below threshold, respectively.
Data availability
Relevant datasets generated and/or analyzed during the current study are available in the Open Science Framework (OSF) repository under the link https://osf.io/8zu5f/?view_only=5d54939f8c584c598308244c34c34348. Data and any other information are also available upon reasonable request.
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Acknowledgements
We are grateful to Paolo Cazzato for technical support. The authors acknowledge the financial support of the Russian Foundation for Basic Research (RFBR) Grant No. 20527816 (joint with CNR) and the MEPhI Priority 2030 Program (A.G. and N.V.), the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (A.G.), the Italian Ministry of University (MIUR) for funding through the PRIN project “Interacting Photons in Polariton Circuits” – INPhoPOL (grant 2017P9FJBS), project “Hardware implementation of a polariton neural network for neuromorphic computing” – Joint Bilateral Agreement CNRRFBR – Triennal Programm 20212023, and the PNRR MUR projects “Integrated Infrastructure Initiative in Photonic and Quantum Sciences” IPHOQS (IR0000016) and “National Quantum Science and Technology Institute” NQSTI (PE0000023). We thank Scott Dhuey at the Molecular Foundry for assistance with the electron beam lithography. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. This research is partly funded by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF9615 to L. N. Pfeiffer, and by the National Science Foundation MRSEC grant DMR 1420541.
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N.V., D.S. and V.A. initiated the research project; A.G. and N.V. performed the calculations and theoretical analysis; F.R. designed the grating structures, growth was performed by K.B. and L.P.; M.E.T., V.A. and D.S. realized the experiments and analyzed the data together with N.V.; N.V., V.A. and D.S. drafted the manuscript, and all the authors, including M.D.G., D.T. and D.B., were involved in the discussion of results and the final manuscript editing.
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Grudinina, A., EfthymiouTsironi, M., Ardizzone, V. et al. Collective excitations of a boundinthecontinuum condensate. Nat Commun 14, 3464 (2023). https://doi.org/10.1038/s4146702338939y
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DOI: https://doi.org/10.1038/s4146702338939y
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