Research Article

Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms

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Science  23 Aug 2019:
Vol. 365, Issue 6455, pp. 775-780
DOI: 10.1126/science.aav9105

An interacting topological phase

Most topologically nontrivial systems discovered to date consist of noninteracting particles. Their properties can therefore be explained within a single-particle picture. De Léséleuc et al. engineered a topological phase of bosonic atoms in which interactions play a crucial role. The atoms, which were in highly excited Rydberg states, were held in an array of optical tweezers. Depending on the spatial arrangement of the tweezers, the dipole-dipole interactions between the atoms gave rise to two configurations with different topological properties.

Science, this issue p. 775


The concept of topological phases is a powerful framework for characterizing ground states of quantum many-body systems that goes beyond the paradigm of symmetry breaking. Topological phases can appear in condensed-matter systems naturally, whereas the implementation and study of such quantum many-body ground states in artificial matter require careful engineering. Here, we report the experimental realization of a symmetry-protected topological phase of interacting bosons in a one-dimensional lattice and demonstrate a robust ground state degeneracy attributed to protected zero-energy edge states. The experimental setup is based on atoms trapped in an array of optical tweezers and excited into Rydberg levels, which gives rise to hard-core bosons with an effective hopping generated by dipolar exchange interaction.

The concept of symmetry breaking can be used to characterize a variety of quantum phases. Topological quantum phases, however, do not fit into this paradigm. The most prominent example of a topological phase is the integer quantum Hall state with its robust edge states, giving rise to a quantized Hall conductance (1). The prediction of topological insulators (2) revealed another class of topological states of matter, denoted as symmetry-protected topological phases (SPTs). They occur in systems that display an excitation gap in the bulk (i.e., bulk insulators) and are invariant under a global symmetry. Their defining property is that the ground state at zero temperature cannot be transformed into a conventional insulating state upon deformations of the system that do not close the excitation gap or violate the symmetry. In particular, the zero-energy edge states are robust to any perturbation commuting with the symmetry operators.

SPT phases were first predicted and observed in materials in which the interaction between electrons can be effectively neglected (3, 4). In this specific case of noninteracting fermions, SPT phases can be classified based on the action of the Hamiltonian on a single particle (5, 6). Thus, the appearance of robust edge states is fully understood from the single-particle eigenstates. This notable simplification motivated experimental studies of topological phenomena at the single-particle level in classical systems such as coupled mechanical oscillators (7, 8), radiofrequency circuits (9), optical devices (1013), and plasmonic systems (14, 15). Similar studies were conducted also in quantum systems such as ultracold atoms (1622), where atomic interactions were either negligible (1621) or used only to fill up Bloch bands (22), but the observed physics could be explained by a noninteracting model.

By contrast, the ground state of noninteracting bosons is a Bose-Einstein condensate. Therefore, bosonic SPT phases only appear in systems with interactions between the particles. Their classification requires the analysis of the quantum many-body ground state and is based on group cohomology (23, 24). A notable example is the Haldane phase of the antiferromagnetic spin-1 chain (25), which has been experimentally observed in some solid-state materials (26, 27). In the context of artificial matter, interactions have been introduced between bosonic particles in a system with a topological band structure (28, 29), but studies were restricted to the two-body limit, still far from the many-body regime.

Here, we report the realization of a many-body SPT phase of interacting bosonic particles in an artificial system. Our setup is based on a staggered one-dimensional chain of Rydberg atoms, each restricted to a two-level system, resonantly coupled together by the dipolar interaction (30). We use this system to encode hard-core bosons—i.e., bosonic particles with infinite on-site interaction energy–coherently hopping along the chain. The system then realizes a bosonic version of the Su-Schrieffer-Heeger (SSH) model (31), which was originally formulated to describe fermionic particles hopping on a dimerized lattice. Our bosonic setup gives rise to two distinct phases of a bulk–half-filled chain: a trivial one with a single ground state and an excitation gap; and an SPT phase, with a fourfold ground state degeneracy due to edge states, and a bulk excitation gap. After an adiabatic preparation of a bulk–half-filled chain, we detect the ground state degeneracy in the topological phase and probe the zero-energy edge states. Furthermore, we experimentally demonstrate the robustness of the SPT phase under a perturbation respecting the protecting symmetry and show that this robustness cannot be explained at the single-particle level, a feature that distinguishes our system from noninteracting SPT phases.

SSH model for hard-core bosons

The SSH model is formulated on a one-dimensional lattice with an even number of sites N and staggered hopping of particles (Fig. 1A). It is convenient to divide the lattice into two sublattices: A={1,3,,N1}, involving odd lattice sites, and B={2,4,,N}, with even sites. Then, a particle on site i of one sublattice can hop to a site j of the other sublattice with a hopping amplitude Jij (we do not restrict the system to nearest-neighbor hopping). The many-body Hamiltonian is H=iA,jBJij[bibj+bjbi](1)with bi (bi) being the creation (annihilation) operator of a particle on site i. In the original formulation of the SSH model, the particles are noninteracting fermions. Here we consider hard-core bosons for which the operators bi (bi) satisfy bosonic commutation relations on different sites ij, and additionally the hard-core constraint (bi)2=0, as two bosons cannot occupy the same site i. In our realization, the nearest-neighbor hoppings are dominating—with J denoting the strong link and J the weak link, i.e., |J|<|J| (Fig. 1, C and D)—and are sufficient to describe the qualitative behavior of the model. For quantitative comparison between theory and experiment, we do not restrict our system to the nearest-neigbor approximation.

Fig. 1 Bosonic SSH model.

(A) Dimerized one-dimensional lattice and the two sublattices A and B. The staggered nearest-neighbor hopping energies are denoted as J and J with |J|>|J|. (B) Each lattice site hosts a Rydberg atom with two relevant levels: 60S1/2 being the vacuum state |0 and 60P1/2 describing a bosonic particle bi|0. The dipolar exchange interaction provides the hopping mechanism. Inset in (A): Angular dependence of the hopping amplitude measured between two sites; filled (empty) circle: positive (negative) amplitude. The amplitude vanishes and changes sign at the angle θm54.7°. The solid line is the theoretical prediction of the angular dependence. Standard error of the mean (SEM) of the data is smaller than the symbol size. (C and D) Single-shot fluorescence images of the atoms assembled in the artificial structure for the topological (C) and the trivial (D) configuration. The chain is tilted by the angle θm to cancel couplings between sites in the same sublattice.

At the single-particle level, the spectrum of the Hamiltonian in Eq. 1, shown in Fig. 2A, is obtained by diagonalizing the coupling matrix Jij. It displays two bands separated by a spectral gap 2(|J||J|); depending on the end links of the chain, there may or may not be in-gap localized zero-energy edge modes. There are two such modes for a chain ending with weak links J (topological configuration, Fig. 1C) and none if the chain ends with strong links J (trivial configuration, Fig. 1D). The topology of the bands emerges from the sublattice (or chiral) symmetry of the SSH Hamiltonian (5, 6), which notably constrains the hopping matrix Jij to connect only sites of different sublattices—e.g., next–nearest-neighbor hoppings Ji,i+2=J are forbidden (see below for the experimental realization). The existence and degeneracy of edge modes are topologically protected from any perturbation that does not break the sublattice symmetry. These single-particle properties of the coupling matrix Jij defining the SSH model have been observed in many platforms, such as ultracold atoms (32, 33), polaritons in an array of micropillars (14), or mechanical granular chains (34).

Fig. 2 Single-particle properties.

(A) Single-particle spectrum for the trivial and the topological configuration probed by microwave spectroscopy. Right: A selection of single-particle wave functions. (B) Experimental site-resolved spectra showing the averaged occupancy of each site as a function of Δμw. The lower-band bulk states are always observed, whereas states in the upper band are not visible as their microwave coupling to |0 is either weak or completely inhibited. Edge states at zero energy appear only for the topological configuration. The white dashed lines indicate the calculated gap. (C) Spatial distribution of the edge states, observed (red bars) and calculated (black crosses), showing an exponential localization on the edges. The dashed line indicates the 2% noise level caused by preparation and detection errors. (D) Particle transfer between the two edges. A particle on the left edge is essentially a superposition of the symmetric and antisymmetric zero-energy modes, split in energy by Ehyb owing to the hybridization for finite chains. (E) Observation of the transfer for chains of N = 4, 6, and 8 sites after injecting a particle on the leftmost site. (F) From the transfer frequency, we obtain the hybridization energy Ehyb (red circles) and compare it to calculations, keeping only nearest-neighbor hoppings (dashed line) and including the full dipolar interaction (solid line). All theoretical results are obtained by diagonalizing the coupling matrix.

We now turn to the properties of the quantum many-body ground state corresponding to a bulk–half-filled chain. For noninteracting fermions, the properties of the SSH chain follow from the Fermi sea picture based on the single-particle eigenstates: In the trivial configuration one obtains a single insulating ground state, whereas in the topological configuration, (i) an excitation gap appears in the bulk, and (ii) the ground state is fourfold degenerate as the two zero-energy edge modes can be either empty or occupied. For interacting bosons, the description of the many-body ground state is much more challenging. In the special case with only nearest-neighbor hoppings J and J, the bosonic many-body ground states for hard-core bosons can be derived via a Jordan-Wigner transformation from the fermionic ones and inherits the properties of a bulk excitation gap and a fourfold ground state degeneracy in the topological configuration (35). These properties are robust against the addition of extra terms to the Hamiltonian as long as the bulk gap remains finite and the protecting symmetry is respected, as required to describe a state in the SPT phase. Here, two symmetries have to be respected: the particle number conservation and the symmetry represented by the anti-unitary operatorSB=i=1N[bi+bi]K(2)where bi+bi is a particle-hole transformation and K denotes the complex conjugation. Indeed, the general classification of the bosonic SPT phases (23, 24) allows for a topological and a trivial phase for the symmetry described by Eq. 2. Moreover, in contrast to the chiral symmetry (protecting the single-particle properties and the fermionic SPT phase), next–nearest-neighbor hoppings J=Ji,i+2 are symmetry allowed, i.e., [H,SB]=0 even when including J. We will demonstrate this fundamental difference by engineering a perturbation that shifts the edge modes away from zero energy at the single-particle level but preserves the ground state degeneracy in the bosonic many-body system. In general, our protecting symmetries allow for arbitrary and even complex hopping terms and interactions such as ZiZj with Zi=12bibi. However, a relaxation of the hard-core constraint and introducing interaction terms bibibibi violates the symmetry and leads to a splitting of the ground state degeneracy (36).

Experimental realization with Rydberg atoms

Our realization of the bosonic SSH model is performed on an artificial structure with N = 14 sites of individually trapped 87Rb atoms (3739) (Fig. 1, C and D). The motion of the atoms is frozen during our experiment, occurring in a few microseconds. Coupling between the atoms is achieved, despite the large interatomic distance (~10 µm), by preparing the atoms in Rydberg states for which the dipole-dipole coupling is enhanced to a few megahertz (30, 40).

We first prepare each atom in a Rydberg s-level, |60S1/2,mJ=1/2, using a two-photon stimulated Raman adiabatic passage (STIRAP) with an efficiency of 95% (35). From there, the atom can be coherently transferred to a Rydberg p-level, |60P1/2,mJ=1/2, by using a microwave field tuned to the transition between the two Rydberg levels (E0/h16.7 GHz). The detuning from the transition is denoted Δμw, and the Rabi frequency is Ωμw/(2π)0.1 to 20 MHz. We denote the state with all Rydberg atoms in the s-level as the “vacuum” |0 of the many-body system, whereas a Rydberg atom at site i excited in a p-level is described as a bosonic particle, bi|0. Because each Rydberg atom can only be excited once to the p-level, we obtain the hard-core constraint naturally. The resonant dipolar interaction occurring between the s- and p-levels of two Rydberg atoms at site i and j gives rise to hopping of these particles (30) (Fig. 1B); we use this to engineer the hopping matrix Jij. At the end of the experiment, we deexcite atoms in the Rydberg s-level to the electronic ground state and detect them by fluorescence imaging; an atom in the Rydberg p-level is lost from the structure. The detection errors are at the few percent level (35). For each experimental run, we thus obtain the occupancy of each site, which is then averaged by repeating the experiment every ~0.3 s.

To implement the sub-lattice symmetry, we use the angular dependence of the dipolar coupling Jij=d2(3cos2θij1)/Rij3 with d the transition dipole moment between the two Rydberg levels. The hopping depends on the separation Rij, as well as the angle θij with respect to the quantization axis defined by the magnetic field Bz50 G. In Fig. 1A, we show the measured angular dependence, vanishing at the “magic angle” θm=arccos(1/3)54.7°, which allows us to suppress the hopping along this direction. By arranging the atoms in two subchains aligned along the magic angle, we satisfy the sublattice symmetry. The measured nearest-neighbor couplings are J/h=2.42(2) MHz and J'/h=0.92(2) MHz, in full agreement with numerical determination of the pair potential (41). The dipolar interaction also gives rise to longer-range hopping on the order of ~0.2 MHz to third neighbors, which do not qualitatively change the properties of our system but are fully taken into account for quantitative comparison between theory and experiment.

Single-particle spectrum

As a benchmark of our system, we first study the properties of a single particle in the chain. The single-particle spectrum is probed by microwave spectroscopy (Fig. 2A). Initializing the vacuum state |0 with all Rydberg atoms in the s-level, a weak microwave probe with a Rabi frequency Ωμw/(2π)=0.2 MHz applied for a time t=0.75 μs can lead to the coherent creation of a particle only if an eigenstate energy matches the microwave detuning Δμw and if this state is coupled to |0 by the microwave field. We show in Fig. 2B the site-resolved probability of finding a particle on a given site for the two different chain configurations. In both cases, we observe a clear signal for Δμw<|J||J| from lower band modes delocalized along the chain. States in the upper band are not observed as the microwave coupling from |0 to these states is very small. Only in the topological configuration do we observe an additional signal localized at the boundaries around zero energy, corresponding to the two edge modes. The finite width of the signal is caused by microwave power broadening (35). In Fig. 2C, we quantitatively show the localization of edge modes at the boundary by postselecting experimental runs where at most one particle was created. We observe that the particle populates significantly only the leftmost and rightmost sites, and their second neighbors, as expected from the sublattice symmetry (edge states have support on one of the two subchains only) and in good agreement with a parameter-free calculation (black crosses) (35).

For any finite chain, the left and right edge modes hybridize to form symmetric and antisymmetric states with an energy difference EhybJ|J/J|N, which breaks the degeneracy of the edge modes but decreases exponentially with the chain length N. This remains negligible compared to our experimental time scale for a long chain of 14 sites (Ehybh×20 kHz), but the hybridization is observable for smaller chains. Notably, it gives rise to a coherent transfer of a particle between the two boundaries without involving the bulk modes, as sketched in Fig. 2D. To observe this, we prepare a particle on the leftmost site using a combination of an addressing beam and microwaves sweeps (42), and then let the system evolve freely. We show in Fig. 2E the experimental results for three chains of four, six, and eight sites. The energy Ehyb is determined from the frequency of transfer and exhibits the expected exponential scaling (Fig. 2F), in excellent agreement with theoretical calculations, including the full hopping matrix Jij (35).

Many-body ground state

We now turn to the study of the many-body system. In Fig. 3, B and C, we show the full energy spectrum in the trivial and topological configurations, calculated using exact diagonalization and ordered by increasing number of particles (35). In the trivial case, there is a single ground state at half-filling. By contrast, the topological configuration exhibits four degenerate ground states corresponding to the bulk half-filled, and which are characterized by zero, one, or two additional particles mainly residing at the edges. To prepare the ground state, we perform a microwave adiabatic sweep (Fig. 3A), where the final detuning Δf plays the role of a chemical potential, tuning the number of particles loaded in the chain. From a theoretical analysis simulating the full time-evolution, we expect that our ramping procedure ending at a final detuning |Δf|<|J||J| prepares the ground state with high fidelity (35).

Fig. 3 Preparing the many-body phase at half-filling.

(A) Microwave sweep with time-varying Rabi frequency Ωμw and detuning Δμw; the latter ends at Δf. (B and C) Energy spectrum of the many-body system in the trivial (top) and the topological (bottom) configuration for different particle numbers. The trivial chain exhibits a single gapped ground state with seven particles, whereas the topological configuration exhibits a fourfold degeneracy involving six, seven (twofold degenerate), and eight particles. Starting from the empty chain, the microwave adiabatic sweep loads hard-core bosons in the lattice and prepares the lowest-energy states. (D and E) We measure the occupancy of bulk (blue) and edge sites (green and brown) as a function of the final detuning Δf. Solid lines are the results of simulations, without free parameters, including preparation and detection errors. For a sweep ending in the single-particle gap (between the dashed vertical lines), the bulk of the chain is half-filled. Bosons are loaded in the edge sites of the topological configuration when Δf>0. The error bars represent SEMs.

We present in Fig. 3, D and E, the dependence of the local density of particles on Δf: The bulk sites’ occupancy (blue symbols) exhibits a characteristic plateau at half-filling within the single-particle gap. The fluctuations of the number of particles in the bulk are strongly reduced with a measured probability of 48% to find exactly six particles on the 12 bulk sites [mainly decreased from 100% by detection errors (35)]. The local bulk properties are independent of the topology of the setup, but the situation is drastically different for the edge occupancy: In the trivial configuration, the edge sites behave as the bulk sites, whereas for the topological chain, the boundaries remain depleted for Δf<0 and exhibit a sharp transition to full occupancy for Δf>0. This behavior is consistent with the expected ground state degeneracy. Finally, the small increase in the bulk density around Δf=0, even in the trivial configuration, is caused by preparation errors creating lattice defects. A defect gives rise to two chains, one of which starts with a weak link, and thus supports a zero-energy edge state, which gets populated when Δf becomes positive. Including preparation and detection errors in the simulation (solid lines) gives an excellent agreement with the experimental data (35).

We gain more insight into the many-body state by analyzing the correlations between particles, which we can measure because our detection scheme provides the full site-resolved particle distribution. In the strongly dimerized regime |J||J|, we expect the N/2 particles in the bulk to be highly correlated as they can minimize their energy by each delocalizing on a dimer (two sites connected by a strong link J). The picture remains valid even in our regime where |J|2.6|J|. We measure a large and negative density-density correlation Cz(2i,2i+1)=Z2iZ2i+10.67(1) with Zi=12bibi, corresponding to a suppressed probability of finding two particles on the same dimer. We also access the off-diagonal correlations, Cx(i,j)=XiXj with Xi=bi+bi measuring the coherence between two sites i and j, by applying a strong microwave pulse before the detection, which rotates the local measurement basis around the Bloch sphere. We obtain Cx(2i,2i+1)+0.48(2), indicating that a particle is coherently and symmetrically delocalized on two sites, forming a dimer. Furthermore, our detection scheme allows us to determine string order parameters. For our experimental implementation, the theory predicts that we can distinguish the topological state from the trivial state by a finite value of the string order parameters Cx and Cz (23, 43) withCstringz=Z2eiπ2k=3N2ZkZN1(3)and in analogy for Cstringx. Indeed, we measure a finite string order in the topological phase with Cstringz=0.11(2) and Cstringx=0.05(2), whereas in the trivial phase they are consistent with zero, e.g., Cstringz=0.02(3). All measured correlators are in good agreement with simulations.

We now demonstrate the degeneracy of the many-body ground state in the topological phase and the bulk excitation gap. We first prepare a six-particle many-body ground state with the bulk at half-filling but empty edge states by an adiabatic sweep ending at Δf/(2π)=1 MHz. We then apply a weak microwave probe at various detunings Δμw (Fig. 4A) and observe when particles are created or annihilated to probe the excitation spectrum of the many-body ground state. Figure 4, B and C, shows the three expected and measured transitions: (i) A particle is added at zero energy at the edge, and we reach a seven-particle ground state; (ii) particles are added to the bulk, which requires at least the bulk gap in energy; (iii) particles are removed from the bulk, which appears as a dip at negative detuning.

Fig. 4 Probing the SPT phase degeneracy and bulk excitation gap.

(A) A microwave sweep ending at Δf/(2π)=1 MHz first prepares the many-body ground state with six particles, and we then apply for 2 µs a microwave probe with a Rabi frequency Ωμw/(2π)=0.3 MHz and a variable detuning Δμw. (B) Magnified view of the bottom of the energy spectrum of a chain in the topological configuration. Starting from the ground state with six particles (solid circle), we can (i) reach one of the other degenerate ground states by adding a particle at the edge for zero energy cost. In addition, we can probe the bulk excitation gap by (ii) adding a particle to, or (iii) removing a particle from, the bulk. (C) Measured occupancy of bulk (blue circles) and edge sites (green and brown circles) showing the three expected transitions. Error bars are SEMs.

Probing the protecting symmetry

We now probe the robustness of the fourfold ground state degeneracy to small perturbations that respect the protecting symmetry represented by SB. To do so, we distort the chain on one side by moving the rightmost site out of the sublattice B (Fig. 5A). Because the edge site and its second neighbor are no longer at the “magic angle,” this creates a coupling J/h0.26 MHz between them. This perturbation breaks the chiral symmetry protecting the fermionic SSH model and correspondingly leads to a splitting of the single-particle edge modes. However, such a perturbation commutes with SB and therefore should not break the many-body ground state degeneracy. To check these expectations, we first repeat the spectroscopic measurement in the single-particle regime (applying the microwave probe on an empty chain, as shown in Fig. 2A), and observe a splitting of the edge modes (Fig. 5B). By contrast, the spectroscopic measurement for the bosonic many-body ground state (applying the probe after the adiabatic preparation reaching half-filling of the bulk, as done in Fig. 4) indeed reveals a quasi-degenerate ground state (Fig. 5C). The small remaining shift of 0.03(2) MHz is explained in (35) by using simulations that include van der Waals interactions. We also checked that when we prepare the ground state with a half-filled bulk, i.e., whenever Δf lies in the region |Δf|<|J||J|, the spectroscopic measurement reveals a symmetry-protected ground state degeneracy.

Fig. 5 Perturbation and robustness of the bosonic topological phase.

(A) The rightmost site is shifted upward to give a finite hopping amplitude J to the second neighbor. (B and C) Probability of finding a particle in the left (green circles) and right (brown circles) edge sites when scanning the detuning Δμw of the microwave probe. The experiment is performed either on (B) an initially empty chain to observe the energy difference between the two single-particle edge modes caused by the perturbation J or (C) on the many-body ground state with a half-filled bulk (six particles in a 14-site chain) to observe the protection of the ground state degeneracy. Solid lines are Gaussian fits from which we extract an energy difference of 0.21(1) MHz in (B) and 0.03(2) MHz in (C).

The above experiment illustrates that, in contrast to a noninteracting SPT phase, the robustness of the bosonic many-body ground state at half-filling cannot be understood at the single-particle level. To gain an intuition for the differences between the SPT phase of noninteracting fermions and of hard-core bosons, we use the following simple picture. Considering only the three rightmost sites (the edge and a dimer), and taking the perturbative limit (JJ,J), we first obtain the energy of having no particle on the edge site and one delocalized on the dimer: J(J+J)2/(2J) (the second term is an energy correction stemming from virtual hopping of the particle from the bulk to the edge). By contrast, when there is one particle on the dimer and one on the edge, we obtain J(J±J)2/(2J) with an energy correction now depending on the particle quantum statistics (+ sign for bosons, − for fermions, because of commutation rules). More details can be found in section 3.3 of (35). This simplified model captures why the fermionic degeneracy is broken by the J term, which is not the case for hard-core bosons.

Discussion and outlook

We prepared quantum many-body states in two topologically different phases and observed four signatures of an SPT phase for interacting bosons: (i) a ground state degeneracy characterized by zero-energy edge states, (ii) an excitation gap in the bulk, (iii) a nonvanishing string order, and (iv) a robustness of these properties against perturbations respecting the protecting symmetry SB.

The symmetry protecting our SPT phase defines the symmetry group U(1)×2T, which can also protect the Haldane phase of an antiferromagnetic spin-1 chain (25). We show numerically in (35) that our quantum many-body ground state is in the same SPT phase as the Haldane state, as they can be smoothly connected by adding hopping terms with complex amplitudes as well as additional Ising-type interactions VijZiZj, both terms respecting SB.

Our work demonstrates that Rydberg platforms, which combine flexible geometries and a natural access to the strongly correlated regime via the hard-core constraint, can explore unconventional quantum many-body states of matter. In combination with schemes for obtaining topological band structures in higher dimensions (44), it opens the prospect for the realization of long-range entangled states such as fractional Chern insulators with anyonic excitations.

Supplementary Materials

Supplementary Text

Figs. S1 to S8

Table S1

References (4671)

References and Notes

  1. See supplementary materials.
Acknowledgments: Funding: This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no. 817842 (PASQuanS), from “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM, projects QUANTICA and XYLOS) and from the IXCORE-Fondation pour la Recherche. H.P.B. is supported by the European Union under the ERC consolidator grant SIRPOL (grant no. 681208). Author contributions: S.d.L., V.L., P.S., and D.B. carried out the experimental measurements and data analysis; S.W. and N.L. performed the theoretical analysis. All work was supervised by H.P.B., T.L., and A.B. All authors contributed to the writing of the manuscript. Competing interests: The authors declare no competing interests. Data and materials availability: The experimental data and theory code of this study are available at (45).

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