Experimental realization of a symmetry protected topological phase of interacting bosons with Rydberg atoms

TL;DR

This paper reports the experimental creation of a symmetry protected topological phase of interacting bosons using Rydberg atoms in optical tweezers, demonstrating robust edge states and ground state degeneracy.

Contribution

First experimental realization of a symmetry protected topological phase of interacting bosons with Rydberg atom arrays.

Findings

Observation of robust ground state degeneracy

Detection of protected edge states

Implementation of topological phase in artificial matter

Abstract

The concept of topological phases is a powerful framework to characterize ground states of quantum many-body systems that goes beyond the paradigm of symmetry breaking. While a few topological phases appear in condensed matter systems, a current challenge is the implementation and study of such quantum many-body ground states in artificial matter. 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 edge states. The 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 by dipolar exchange interaction.

Equations

$$H=-\sum _{i\in A,j\in B}{J}_{ij}\left[b_i^{†}{b}_j+b_j^{†}{b}_i\right],$$

$${\mathcal{S}}_{\mathrm{B}}=\prod _{i=1}^N\left[b_i+b_i^{†}\right]K,$$

$$C_{\mathrm{string}}^z=-⟨Z_2{e}^{i\frac{\pi }{2}{\sum }_{k=3}^{N-2}{Z}_k}{Z}_{N-1}⟩$$

Full text

Introduction

The paradigm of symmetry breaking has proven very successful for characterizing quantum phases. However, not all of them follow this paradigm, and some of these phases are nowadays characterized in the framework of topological phases. The most prominent example of a topological phase is the integer quantum Hall state with its remarkably robust edge states giving rise to a quantized Hall conductance Klitzing et al. (1980). For a long time, it was believed that such phases occurred only in the presence of a magnetic field, until the prediction of topological insulators Kane and Mele (2005) revealed a novel class of topological states of matter, nowadays denoted as symmetry protected topological phases (SPT). They occur in systems displaying an excitation gap in the bulk, i.e., bulk insulators, and an invariance 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 edge states are robust to any perturbation commuting with the symmetry operators.

SPT phases were first predicted and observed in materials where the interaction between electrons can be effectively neglected Hasan and Kane (2010); Qi and Zhang (2011). In this specific case of non-interacting fermions, SPT phases can be classified based on the action of the Hamiltonian on a single particle Chiu et al. (2016); Ludwig (2016). Thus, the appearance of robust edge states is fully understood from the single-particle eigenstates. This remarkable simplification motivated experimental studies of topological phenomena at the single-particle level with artificial quantum matter realized on ultracold atoms platforms Jotzu et al. (2014); Flaschner et al. (2016); Aidelsburger et al. (2014); Stuhl et al. (2015); Mancini et al. (2015); Lohse et al. (2015); Tai et al. (2017), and in classical systems of coupled mechanical oscillators Süsstrunk and Huber (2015); Nash et al. (2015), as well as optical Hafezi et al. (2013); Rechtsman et al. (2013); Ling et al. (2015); Lu et al. (2016) or radio-frequency circuits Ningyuan et al. (2015), and plasmonic systems St-Jean et al. (2017); Bandres et al. (2018).

In contrast, the situation is different for bosonic SPT phases as the ground state of non-interacting bosons is a Bose-Einstein condensate. Therefore, it is well established that strong interactions between the particles are required for the appearance of topological phases. Their classification is not derived from single-particle properties, but requires the analysis of the quantum many-body ground state; a classification of bosonic interacting SPT phases based in terms of group cohomology has been achieved Chen et al. (2012). A notable example is the Haldane phase of the anti-ferromagnetic spin-1 chain Haldane (1983), which has been experimentally observed in some solid-state materials Hagiwara et al. (1990); Glarum et al. (1991). However, the realization of topological phases in artificial matter, where one has full microscopic control on the particles, would allow to gain a deeper understanding on the nature of such topological states of matter. A first step has recently been achieved by introducing interactions between bosonic particles in a system with a topological band structure Tai et al. (2017); Di Liberto et al. (2016), but studies were restricted to the two-body limit, still far from the many-body regime.

Here, we report the first 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 Browaeys et al. (2016). We use this 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 Su et al. (1979); the latter originally described fermionic particles hopping on a dimerized lattice, giving rise to a SPT phase of non-interacting fermions. Similarly, our bosonic setup gives rise to two distinct phases of an half-filled chain: a trivial one with a single ground state and an excitation gap, and a SPT phase, with a four-fold ground state degeneracy due to edge states, and a bulk excitation gap. Following an adiabatic preparation of a 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 non-interacting 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, see Fig. 1A. It is convenient to divide the lattice into two sub-lattices: $A=\{1,3,\ldots,N-1\}$, involving odd lattice sites, and $B=\{2,4,\dots,N\}$, with even sites. Then, a particle on site $i$ of one sub-lattice can hop to a site $j$ of the other sub-lattice with a hopping amplitude $J_{ij}$ (we do not restrict the system to nearest neighbor hopping). The many-body Hamiltonian is

[TABLE]

with $b^{\dagger}_{i}$ ($b_{i}$) the creation (annihilation) operator of a particle on site $i$. In the original formulation of the SSH model, the particles are non-interacting fermions. Here we consider hard-core bosons and the operators $b^{\dagger}_{i}$ ($b_{i}$) satisfy bosonic commutation relations on different sites $i\neq j$, and additionally the hard-core constraint $(b_{i}^{\dagger})^{2}=0$, as two bosons cannot occupy the same site $i$. In our realization, the nearest neighbor hoppings are dominant with their energies denoted as $J_{2i,2i+1}=J$ and $J_{2i-1,2i}=J^{\prime}$ with $|J^{\prime}|<|J|$, and are sufficient to describe the qualitative behavior of the model.

At the single particle level, the spectrum of the Hamiltonian in Eq. (1), shown in Fig. 2A, is obtained by diagonalizing the coupling matrix $J_{ij}$. It displays two bands separated by a spectral gap $2(|J|-|J^{\prime}|)$ and, depending on the boundaries of the chain, localized zero-energy edge modes. There are two such modes for a chain ending with weak links $J^{\prime}$ (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 sub-lattice (or chiral) symmetry of the SSH Hamiltonian Chiu et al. (2016); Ludwig (2016), which notably constrains the hopping matrix $J_{ij}$ to connect only sites of different sub-lattices, e.g., next nearest neighbor hoppings $J_{i,i+2}=J^{\prime\prime}$ are forbidden. The existence and degeneracy of edge modes are topologically protected from any perturbation that does not break the sub-lattice symmetry. These single-particle properties of the coupling matrix $J_{ij}$ defining the SSH model have been observed in many platforms such as, e.g., ultracold atoms Atala et al. (2013); Meier et al. (2016), polaritons in array of micropillars St-Jean et al. (2017) or mechanical granular chains Chaunsali et al. (2017).

We now turn to the properties of the quantum many-body ground state corresponding to an half-filled chain. For non-interacting fermions, the properties of the SSH chain follows from the Fermi sea picture based on the single-particle eigenstates: in the trivial configuration one obtains a single insulating ground state, while, in the topological configuration, (i) an excitation gap appears in the bulk, and (ii) the ground state is four-fold 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 $J^{\prime}$, 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 excitations gap and a four-fold ground state degeneracy in the topological configuration sup . Based on the general classification of bosonic SPT phases Chen et al. (2012), these properties are robust by adding longer-range hoppings as well as additional interactions between the bosons as long as the bulk gap remains finite and the protecting symmetry is respected. Here, the protecting symmetries are the particle number conservation and the anti-unitary operator

[TABLE]

where $b_{i}+b_{i}^{{\dagger}}$ is a particle-hole transformation and $K$ denotes the complex conjugation. In contrast to the chiral symmetry (protecting the single-particle properties and the fermionic SPT phase), next nearest neighbor hoppings $J^{\prime\prime}=J_{i,i+2}$ are symmetry allowed, i.e., $\left[H,\mathcal{S}_{\rm{\scriptscriptstyle B}}\right]=0$ even when including $J^{\prime\prime}$. We will demonstrate this fundamental difference by engineering a perturbation which 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.

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 Kim et al. (2016); Endres et al. (2016); Barredo et al. (2016) (Fig. 1C-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 inter-atomic distance ($\sim 10\,\mu$m), by considering Rydberg states for which the dipole-dipole coupling is enhanced to a few MHz Saffman et al. (2010); Browaeys et al. (2016).

We first prepare each atom in a Rydberg s-level, $|60S_{1/2},m_{J}=1/2\rangle$, using a two-photon stimulated Raman adiabatic passage (STIRAP) with an efficiency of 95% sup . From there, the atom can be coherently transferred to a Rydberg p-level, $|60P_{1/2},m_{J}=-1/2\rangle$, using a microwave field tuned to the transition between the two Rydberg levels ($E_{0}/h\sim 16.7$ GHz). The detuning from the transition is $\Delta_{\mu\mathrm{w}}$, and the Rabi frequency is $\Omega_{\mu\mathrm{w}}/(2\pi)\sim 0.1-20$ MHz. We denote the state with all Rydberg atoms in the s-level as the ‘vacuum’ $|0\rangle$ of the many-body system, while a Rydberg atom at site $i$ excited in a p-level is described as a bosonic particle $b^{{\dagger}}_{i}|0\rangle$. Since each Rydberg atom can only be excited once to the p-level, we obtain naturally the hard-core constraint. The resonant dipolar interaction occuring between the s- and p-levels of two Rydberg atoms at site $i$ and $j$ gives rise to hopping of these particles Browaeys et al. (2016), as illustrated in Fig. 1B. We use this to engineer the hopping matrix $J_{ij}$. At the end of the experiment, we de-excite atoms in the Rydberg s-level to the electronic ground state and detect them by fluorescence imaging, while an atom in the Rydberg p-level is lost from the structure. The detection errors are at the few percent level sup . For each experimental run, we thus obtain the occupancy of each site, which is then averaged by repeating the experiment every $\sim 0.3$ s.

In order to implement the sub-lattice symmetry, we use the angular dependence of the dipolar coupling $J_{ij}=d^{2}(3\cos^{2}\theta_{ij}\!-\!1)/R_{ij}^{3}$ with $d$ the transition dipole moment between the two Rydberg levels. The hopping depends on the separation $R_{ij}$, as well as the angle $\theta_{ij}$ with respect to the quantization axis defined by the magnetic field $B_{z}\simeq 50$ G. In Fig. 1A, we show the measured angular dependence, vanishing at the ‘magic angle’ $\theta_{m}=\arccos(1/\sqrt{3})\approx 54.7^{\circ}$, which allows us to suppress the hopping along this direction. By arranging the atoms in two sub-chains aligned along the magic angle, we thus satisfy the sub-lattice symmetry. The measured nearest neighbor couplings are $J/h=2.42(2)$ MHz and $J^{\prime}/h=-0.92(2)$ MHz, in full agreement with numerical determination of the pair potential Weber et al. (2017). The dipolar interaction also gives rise to longer range hopping on the order of $\sim 0.2$ 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 $|0\rangle$ with all Rydberg atoms in the s-level, a weak microwave probe with a Rabi frequency $\Omega_{\mu\mathrm{w}}/(2\pi)=0.2$ MHz applied for a time $t=0.75\,\mu$s can lead to the coherent creation of a particle only if an eigenstate energy matches the microwave detuning $\Delta_{\mu\mathrm{w}}$ and if this state is coupled to $|0\rangle$ by the microwave field. We show in Fig. 2B the site-resolved probability to find a particle on a given site for the two different chain configurations. In both cases, we observe a clear signal for $\hbar\Delta_{\mu\mathrm{w}}<|J^{\prime}|-|J|$ from lower band modes delocalized along the chain. States in the upper band are not observed as the microwave coupling from $|0\rangle$ 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 due to microwave power broadening sup . In Fig. 2C, we quantitatively show the localization of edge modes at the boundary by post-selecting 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 sub-lattice symmetry (edge states have support on one of the two sub-chains only) and in a good agreement with a parameter-free calculation (black crosses).

For any finite chain, the left and right edge modes hybridize to form symmetric and anti-symmetric states with an energy difference $E_{\rm hyb}\propto J^{\prime}|J^{\prime}/J|^{N}$, which breaks the degeneracy of the edge modes but decreases exponentially with the chain length $N$. While this remains negligible compared to our experimental time scale for a long chain of 14 sites ($E_{\rm hyb}\simeq h\times 20$ kHz), 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 de Léséleuc et al. (2017), and then let the system evolve freely. We show in Fig. 2E the experimental results for three chains of 4, 6 and 8 sites. The energy $E_{\rm hyb}$ is determined from the frequency of transfer and exhibits the expected exponential scaling, see Fig. 2F, in excellent agreement with theoretical calculations including the full hopping matrix $J_{ij}$.

Many-body ground state

We now turn to the study of the many-body system. In Fig. 3B-C, we show the full energy spectrum in the trivial and topological configurations, calculated using exact diagonalization, and ordered by increasing number of particles. In the trivial case, there is a single ground state at half-filling. In contrast, the topological configuration exhibits four degenerate ground states corresponding to the bulk half-filled, and which are characterized by additional or missing particles mainly residing at the edges. In order to prepare the ground state, we perform a microwave adiabatic sweep, shown in Fig. 3A, where the final detuning $\Delta_{\mathrm{f}}$ 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 $|\hbar\Delta_{\mathrm{f}}|<|J|-|J^{\prime}|$ prepares the ground state with high fidelity sup .

We present in Fig. 3D-E the dependence of the local density of particles on $\Delta_{\mathrm{f}}$: the bulk sites occupancy (blue curves) exhibits a characteristic plateau at half-filling within the single particle gap. Especially, the fluctuations of the number of particles in the bulk are strongly reduced with a probability of 48 % to find exactly 6 particles on the 12 bulk sites (mainly decreased from 100 % by detection errors sup ). While the local bulk properties are independent of the topology of the setup, 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 $\Delta_{\mathrm{f}}<0$ and exhibit a sharp transition to full occupancy for $\Delta_{\mathrm{f}}>0$. This behavior is consistent with the expected ground state degeneracy.

We gain more insight about the many-body state by analyzing the correlations between particles, that we can measure as our detection scheme provides the full site-resolved particle distribution. In the strongly dimerized regime $|J|\gg|J^{\prime}|$, we expect the $\sim N/2$ 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 $|J|\simeq 2.6|J^{\prime}|$. We measure a large and negative density-density correlation $C^{z}(2i,2i+1)=\langle Z_{2i}Z_{2i+1}\rangle\simeq-0.67(1)$ with $Z_{i}=1-2b_{i}^{\dagger}b_{i}$, corresponding to a suppressed probability to find two particles on the same dimer. We also access the off-diagonal correlations, $C^{x}(i,j)=\langle X_{i}X_{j}\rangle$ with $X_{i}=b_{i}+b_{i}^{{\dagger}}$ 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 $C^{x}(2i,2i+1)\simeq+0.48(2)$ 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, which have emerged as an indicator of topological states Kennedy and Tasaki (1992); Hida (1992):

[TABLE]

and in analogy for $C^{x}_{\rm{\scriptscriptstyle string}}$. Indeed, we measure a finite string order in the topological phase with $C^{z}_{\rm{\scriptscriptstyle string}}=0.11(2)$ and $C^{x}_{\rm{\scriptscriptstyle string}}=0.05(2)$, while in the trivial phase they are consistent with zero, e.g., $C^{z}_{\rm{\scriptscriptstyle string}}=-0.02(3)$. All measured correlators are in good agreement with simulations sup .

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

Probing the protecting symmetry

We finally probe the robustness of the four-fold ground state degeneracy to small perturbations, which respect the protecting symmetry $\mathcal{S}_{\rm{\scriptscriptstyle B}}$. To do so, we distort the chain on one side by moving the rightmost site out of the sub-lattice $B$, see Fig. 5A. As the edge site and its second neighbor are not at the ‘magic angle’ anymore, this creates a coupling $J^{\prime\prime}/h\simeq 0.26$ 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 the symmetry $\mathcal{S}_{\rm{\scriptscriptstyle B}}$ 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, see Fig. 5B. In 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 degenerate ground state, see Fig. 5C. In sup , we checked that when we prepare the ground state with a half-filled bulk, i.e., when $\Delta_{\mathrm{f}}$ lies in the region $|\hbar\Delta_{\mathrm{f}}|<|J|-|J^{\prime}|$, the spectroscopic measurement reveals a symmetry protected ground state degeneracy.

The above experiment illustrates that, in contrast to a non-interacting 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 non-interacting 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 ($J\gg J^{\prime},J^{\prime\prime}$), we first obtain the energy of having no particle on the edge site and one delocalized on the dimer: $-J-(J^{\prime}+J^{\prime\prime})^{2}/\left(2J\right)$ (the second term is an energy correction due to virtual hopping of the particle from the bulk to the edge). On the contrary, when there is one particle on the dimer and one on the edge, we obtain $-J-(J^{\prime}\pm J^{\prime\prime})^{2}/\left(2J\right)$ with an energy correction now depending on the particle quantum statistics ($+$ sign for bosons, $-$ for fermions, due to commutation rules). More details can be found in S3.3 of sup . This simplified model captures why the fermionic degeneracy is broken by the $J^{\prime\prime}$ term, while this is not the case for hard-core bosons.

Discussion and Outlook

In this work, we prepared quantum many-body states in two topologically different phases, and observed four characteristic signatures of a SPT phase for interacting bosons in one-dimension: (i) a ground state degeneracy characterized by zero-energy edge states, (ii) an excitation gap in the bulk, (iii) a non-vanishing string order, and (iv) a robustness of these properties against perturbations respecting the protecting symmetry $\mathcal{S}_{\rm{\scriptscriptstyle B}}$.

Furthermore, we point out that the symmetry protecting our SPT phase defines the symmetry group $U(1)\times\mathbb{Z}_{2}^{T}$, which can also protect the Haldane phase of an anti-ferromagnetic spin-1 chain Haldane (1983). We show numerically in sup that our quantum many-body ground state is in the same SPT phase as the celebrated Haldane state, as they can be smoothly connected by adding hopping terms with complex amplitudes as well as additional Ising type interactions $V_{ij}\propto Z_{i}Z_{j}$, both terms respecting $\mathcal{S}_{\rm{\scriptscriptstyle B}}$.

Our work demonstrates that Rydberg platforms, which combine flexible geometries and a natural access to the strongly correlated regime via the hard-core constraint, are capable to explore unconventional quantum many-body states of matter. Combining ideas for obtaining topological band structures in higher dimensions Weber et al. (2018), it opens the prospect for the realization of long-range entangled states such as fractional Chern insulators with anyonic excitations.

Acknowledgment

This work benefited from financial support by the EU (FET-Flag 817482, PASQUANS), and “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM, projects QUANTICA and XYLOS), H.P. B is supported by the European Union under the ERC consolidator grant SIRPOL (grant N. 681208).