Abstract
Although recent studies have established a powerful framework to search for and classify topological phases based on symmetry indicators, there exists a large class of fragile topology beyond the description. The Euler class characterizing the topology of twodimensional real wave functions is an archetypal fragile topology underlying some important properties. However, as a minimum model of fragile topology, the twodimensional topological Euler insulator consisting of three bands remains a significant challenge to be implemented in experiments. Here, we experimentally realize a threeband Hamiltonian to simulate a topological Euler insulator with a trappedion quantum simulator. Through quantum state tomography, we successfully evaluate the Euler class, Wilson loop flow, entanglement spectra and Berry phases to show the topological properties of the Hamiltonian. The flexibility of the trappedion quantum simulator further allows us to probe dynamical topological features including skyrmionantiskyrmion pairs and Hopf links in momentumtime space from quench dynamics.
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Introduction
Topological phases have seen a rapid progress over the past two decades^{1,2,3,4,5}. In particular, the tenfold classification based on the Ktheory represents a cornerstone in the description of topological phases^{6,7,8}. Besides the internal symmetries, crystalline symmetries greatly enrich the classification of topological phases^{9,10,11,12}. Remarkably, recent efforts have led to the development of a powerful framework based on symmetry indicators to categorize topological crystalline insulators^{13,14,15}. However, a large class of topological phases falls outside the description^{16,17,18,19,20,21,22}. Such a phase belongs to the category of socalled fragile topology that can be trivialized by adding trivial bands^{16}, in stark contrast to a stable topology which remains nontrivial upon adding trivial bands. In this context, a class of topological phases protected by spacetime inversion symmetry is highlighted^{20,22,23,24,25,26,27,28,29}. Among them, the Euler class characterizing the topological property of twodimensional real wave functions underlies the failure of the NielsenNinomiya theorem^{20} and the existence of Wilson loop winding^{22}. Yet, adding a trivial band can annihilate crossing nodes through braiding and remove the Wilson loop winding, showing a fragile topology of the Euler class. Such a fragile topology is theoretically shown to protect the nonzero superfluid weight in twisted bilayer graphene^{30}. Despite recent important progress on experimental characterizations of the fragile topology in an acoustic metamaterial^{31}, the implementation of the topological Euler insulator^{32,33} as a minimum model of fragile topology poses a significant experimental challenge.
Quantum simulators have been proven to be powerful platforms for experimentally studying topological phases. During the past decade, there have been great advances in simulating various topological phases via different quantum simulators including cold atom systems^{34,35,36,37,38,39,40}, solidstate spin systems^{41,42,43,44,45}, and superconducting circuits^{46,47,48,49}. Trapped ions provide an alternative flexible platform to perform quantum simulations due to its stateoftheart technologies to control and measure^{50,51}, enabling us to use it to simulate exotic topological phases and directly probe their intriguing topological properties through measurements with high precision.
In the article, we experimentally implement a threeband topological Euler Hamiltonian in momentum space using a single ^{171}Yb^{+} ion trapped in an electrodesurface trap as shown in Fig. 1. By measuring the momentumresolved eigenstates through quantum state tomography, we evaluate the Euler class ξ, Wilson loop flow, and entanglement spectra to identify the band topology. With the Euler class, there are 2ξ protected crossing nodes between the two lowest energy bands. Such nodes can be annihilated either by closing the gap between the second and third bands or by adding a trivial band, followed by intricate braiding of crossing nodes^{20,26,29}. Our further measurements of the Berry phases along four distinct closed trajectories illustrate the existence of four crossing points. Apart from the equilibrium topological properties, it has been theoretically demonstrated that skyrmionantiskyrmion pairs and Hopf links appear in momentumtime space from quench dynamics for the postquench threeband topological Euler Hamiltonian^{32}. We experimentally observe the skyrmionantiskyrmion pairs and Hopf links by measuring the timeevolving states under the Euler Hamiltonian.
Results
Model Hamiltonian and experimental realization
We start by considering the following threeband Hamiltonian for Euler insulators in momentum space, which will be experimentally engineered,
where
is a real unit vector at k = (k_{x}, k_{y}) in the twodimensional (2D) Brillouin zone. Here, \({{{{{{{\mathcal{N}}}}}}}}\) is the normalization factor, and m is a parameter with ∣m∣ ≠ 0, 2 for H(k) to be welldefined. The Hamiltonian is a real symmetric matrix and thus respects \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) (composition of twofold rotational and timereversal operators) symmetry, which can be represented by the complex conjugation \({{{{{{{\mathcal{K}}}}}}}}\) in a suitable basis^{26}. For simplicity, we have flattened the spectrum of the Hamiltonian without affecting the band topology. The Hamiltonian has two degenerate bands \({u}_{1,2}({{{{{{{\boldsymbol{k}}}}}}}})\rangle ={{{{{{{{\boldsymbol{u}}}}}}}}}_{1,2}({{{{{{{\boldsymbol{k}}}}}}}})\) with eigenenergy E_{1,2} = − 1 and one band \({u}_{3}({{{{{{{\boldsymbol{k}}}}}}}})\rangle ={{{{{{{\boldsymbol{n}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})={{{{{{{{\boldsymbol{u}}}}}}}}}_{1}({{{{{{{\boldsymbol{k}}}}}}}})\times {{{{{{{{\boldsymbol{u}}}}}}}}}_{2}({{{{{{{\boldsymbol{k}}}}}}}})\) with eigenenergy E_{3} = 1. The eigenstates are real unit vectors because of the reality of the Hamiltonian.
We implement the Euler Hamiltonian H(k) in momentum space through microwave operations on three hyperfine states \(\left1\right\rangle =\leftF=0,{m}_{F}=0\right\rangle\), \(\left2\right\rangle =\leftF=1,{m}_{F}=1\right\rangle\), and \(\left3\right\rangle =\leftF=1,{m}_{F}=0\right\rangle\) in the ^{2}S_{1/2} manifold using a single ^{171}Yb^{+} ion trapped in an electrodesurface chip trap as shown in Fig. 1 (see Subsection "Experimental setup" in Methods for details). Specifically, we drive the transition between the \(\left1\right\rangle\) and \(\left2\right\rangle\) levels or the transition between the \(\left1\right\rangle\) and \(\left3\right\rangle\) levels by near resonant microwaves and drive the transition between the \(\left2\right\rangle\) and \(\left3\right\rangle\) levels by two fardetuned microwaves through microwave Raman transitions. In the experiment, we implement the Hamiltonian \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\) which has the same eigenstates as H(k) and thus is topologically equivalent to H(k) (see Subsection "Microwave operations in the trappedion system" in Methods for details). To measure the band topology of \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\), we first prepare the ion in the \(\left1\right\rangle\) state and then slowly vary the Hamiltonian to \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\). Since \(\left1\right\rangle\) is the highest energy eigenstate \(\left{u}_{3}({{{{{{{{\boldsymbol{k}}}}}}}}}^{* })\right\rangle\) of our initial Hamiltonian H(k^{*}) at highsymmetry points k^{*} in momentum space, the state can evolve to its highest energy eigenstate \(\left{u}_{3}({{{{{{{\boldsymbol{k}}}}}}}})\right\rangle\) of \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\) at the momentum k over the adiabatic passage (see Subsection "Adiabatic passage" in Methods for details). At the end, we employ quantum state tomography to measure the density matrix ρ(k) (see Subsection "Quantum state tomography for a qutrit system" in Methods for details) and take the real state closest to ρ(k) as the measured state for \(\left{u}_{3}({{{{{{{\boldsymbol{k}}}}}}}})\right\rangle\) (see Subsection "The real state from the measured density matrix" in Methods for details). For the adiabatic preparation, the average fidelities are \(\overline{F}=97.1 \%\) and \(\overline{F}=96.9 \%\) for the topologically nontrivial (m = 1) and trivial phases (m = 3), respectively (see Subsection "The fidelity of the measured state" in Methods for detailed discussion). Since \(\left{u}_{3}({{{{{{{\boldsymbol{k}}}}}}}})\right\rangle\) contains full information of the flattened Hamiltonian H(k), we can thus determine the topological properties of the Euler Hamiltonian using these measured states.
Euler class
The band topology of a Euler insulator can be characterized by the Euler class^{20,23,24,26}
for the two occupied bands \(\left{u}_{1}\right\rangle\) and \(\left{u}_{2}\right\rangle\). It is enforced to be quantized by the reality of eigenstates due to \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) symmetry. To well define the Euler class, we require that the two occupied bands form an orientable real vector bundle, which is ensured by the vanishing of Berry phases along any noncontractible loops for the occupied space^{20,22,26}. For a threeband Euler Hamiltonian, the Euler class can be reduced to another form^{26}
showing that ξ is equal to twice of the winding number (Pontryagin number) of n(k) over the 2D Brillouin zone (see Supplementary Note 1). In other words, the Euler class is determined by twice of times that n(k) wraps the sphere S^{2}, which also implies that the Euler class for a threeband Hamiltonian should be an even integer. Note that if the orientations of all vectors n(k) are reversed, we obtain the same Hamiltonian but the opposite winding for n(k), showing that the sign of ξ is ambiguous and only its absolute value characterizes the topology^{22}. With the relation between the Euler class ξ and the winding number of n(k), we can obtain the phase diagram that the Euler Hamiltonian H(k) in Eq. (1) is topologically nontrivial with ξ = 2 for 0 < ∣m∣ < 2 and trivial with ξ = 0 for ∣m∣ > 2.
Our experimentally measured vectors n(k) in a topological phase indeed exhibit a skyrmion structure over the Brillouin zone [see Fig. 2(a)] wrapping the entire sphere once [see Fig. 2(b)], which suggests that the Euler class ξ = 2 for the experimentally realized Hamiltonian. To be more quantitative, we map the measured vectors to a twoband Chern insulator by H_{C}(k) = n(k)⋅σ with Pauli matrices σ = (σ_{x}, σ_{y}, σ_{z}). The fact that the Euler class is equal to twice of the Chern number of H_{C}(k) allows us to determine the Euler class by computing the Chern number, which is much more efficient than directly performing the integral for Eq. (4) (see more details in Supplementary Note 1). We find that the Chern number calculated using the measured n(k) is equal to 1 so that ξ = 2. In comparison, we also display the measured vectors n(k) in a trivial phase, which do not form a skyrmion structure [see Fig. 2(c)] and cover only parts of the sphere [see Fig. 2(d)], indicating that ξ = 0.
Wilson loops and entanglement spectra
The Wilson loop provides a powerful framework to characterize the fragile topology^{17,19,20,22}. However, it is very challenging to make an experimental measurement of it. In the trappedion quantum simulator, the quantum state tomography technique allows us to evaluate the Wilson loop based on the measured eigenstates of a Euler Hamiltonian.
Specifically, the xdirected Wilson loop \({{{{{{{{\mathcal{W}}}}}}}}}_{x}\) with the base point k_{0} = (k_{x}, k_{y}) can be computed in a N × N discretized Brillouin zone as^{52,53}
where \({{{{{{{{\boldsymbol{k}}}}}}}}}_{j}=({k}_{x}+\frac{2\pi j}{N},{k}_{y})\), N is the number of discrete momenta on the loop, and P_{occ}(k_{j}) is the projector on the occupied bands at the momentum k_{j}. The Wilson loop operator \({{{{{{{{\mathcal{W}}}}}}}}}_{x}\) is unitary so that its eigenvalues take the form of \({e}^{i{\theta }_{x}({k}_{y})}\) which only depends on k_{y}; θ_{x}(k_{y}) as a function of k_{y} is known as the Wilson loop spectrum for \({{{{{{{{\mathcal{W}}}}}}}}}_{x}\). The ydirected Wilson loop \({{{{{{{{\mathcal{W}}}}}}}}}_{y}\) and the corresponding spectrum θ_{y}(k_{x}) can be defined similarly. For our threeband model, we only need the experimentally measured highest energy eigenstates to construct the occupied projectors in the Wilson loop,
The Wilson loop spectrum can be determined by diagonalizing the matrix
which gives us three eigenvalues as \(\{{e}^{i{\theta }_{x}^{(1)}},{e}^{i{\theta }_{x}^{(2)}},0\}\). Discarding the zero eigenvalue contributed by the unoccupied subspace, we obtain the Wilson loop eigenvalues \(\{{\theta }_{x}^{(1)},{\theta }_{x}^{(2)}\}\) for the two occupied bands.
For the Euler Hamiltonian with two occupied bands, due to the reality of eigenstates, the Wilson loop operator takes the form of \({e}^{i\theta {\sigma }_{y}}\), which has a pair of eigenvalues e^{±iθ}. For a topological Euler insulator, both θ_{x}(k_{y}) and θ_{y}(k_{x}) exhibit a nontrivial winding, indicating an obstruction to the Wannier representation. The winding number is equivalent to the Euler class ξ^{22}.
Figure 3(a, b) shows the experimentally measured Wilson loop spectra θ_{x}(k_{y}) [θ_{y}(k_{x}) has similar behaviors], which are evaluated based on the measured highest energy eigenstates. In the topological phase, each branch of the Wilson loop spectra exhibits a winding number (±2), whereas in the trivial phase, the winding patterns are not observed. All the experimental results are in excellent agreement with the theoretical ones. We remark that such a winding can be removed by adding a trivial band, which reveals the fragile topology feature of the system (see Supplementary Note 2 for more details).
Although we experimentally realize the topological Euler Hamiltonian in momentum space, we can extract the edge state information through the singleparticle entanglement spectra evaluated based on the measured states; such spectra can exhibit more robust nontrivial features than those for physical boundaries in a topological band insulator^{10,54,55}.
Figure 3c displays the entanglement spectra ES_{x}(k_{y}) obtained by partially tracing out the right part of the system using the experimentally measured unoccupied eigenstates \(\left{u}_{3}({{{{{{{\boldsymbol{k}}}}}}}})\right\rangle\) for the Euler Hamiltonian H(k) (see Supplementary Note 3 for more details). In the topological phase, an ingap spectrum with midgap modes near ξ_{n} = 0.5 arises in the entanglement spectra, which agrees very well with the theoretical prediction. The experimental results also support the theoretical prediction of the parabolic dispersion for the entanglement spectra near the midgap modes. In the trivial phase, our experimental results do not reveal the existence of gapless entanglement spectra, indicating that the phase is adiabatically connected to a trivial phase with zero entanglement entropy.
Dirac points
The Euler class ξ is also manifested in the existence of 2ξ stable Dirac points between the two occupied bands^{20,26}. They are protected by the \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) symmetry and cannot be annihilated without the gap closing with the third band. To see this feature, we consider the following model by adding an extra term to H(k) in Eq. (1) with m = 1,
with \({h}_{0}({{{{{{{\boldsymbol{k}}}}}}}})=0.1[\cos ({k}_{y})\cos ({k}_{x})]\) and h_{±}(k) = h_{0}(k) ± 0.5. The additional term lifts the degeneracy of the two occupied bands for the flattened Hamiltonian H(k) except at the Dirac points as shown in Fig. 4(a). Due to the \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) symmetry, a Dirac point between the two lowest bands yields a quantized Berry phase γ = π for the lowest eigenstates \(\left{u}_{1}({{{{{{{\boldsymbol{k}}}}}}}})\right\rangle\) along a closed path l enclosing it.
Since the energy gap between the two lowest eigenstates on a path enclosing the Dirac point is opened, we can still use the adiabatic passage to realize the eigenstate \(\left{u}_{1}({{{{{{{\boldsymbol{k}}}}}}}})\right\rangle\) at the momenta on the closed path. After that, we measure the states by quantum state tomography and then evaluate the Berry phase based on the measured states. We find that the experimentally evaluated Berry phase γ = π on the four closed paths [see Fig. 4(b)], indicating the presence of a Dirac point inside each closed path.
Quench dynamics
Besides the equilibrium features, it has been theoretically shown that nonequilibrium dynamics provides another tool to uncover the static band topology of an Euler insulator^{32}. Let us start from an initial state ψ(k, t = 0) = ψ_{0} = (0, 0, 1)^{T} for each momentum k, which can be seen as the eigenstate of a topologically trivial Euler Hamiltonian H_{0}(k) = diag( −1, −1, 1). We consider the quench dynamics for the trivial initial state evolving under a postquench Euler Hamiltonian H(k) in Eq. (1). Due to the flatness of H(k) in Eq. (1), the state evolves as
which is periodic about time t with a period T = π. The periodicity of the evolving state both in time t and momentum k in 2D Brillouin zone makes the space of (k_{x}, k_{y}, t) form a 3torus T^{3}. In analogy to the quench dynamics of a twoband Chern insulator^{56}, one can construct a map f from the (k_{x}, k_{y}, t) space as a T^{3} to a 2sphere S^{2} as follows. For a point (k_{x}, k_{y}, t) on T^{3}, the image of the map f(k_{x}, k_{y}, t) is a unit vector \(\hat{{{{{{{{\boldsymbol{p}}}}}}}}}=({p}_{x},{p}_{y},{p}_{z})\) on S^{2} as \(\hat{{{{{{{{\boldsymbol{p}}}}}}}}}={\psi }^{{{{\dagger}}} }({k}_{x},{k}_{y},t){{{{{{{\boldsymbol{\mu }}}}}}}}\psi ({k}_{x},{k}_{y},t)\), where μ = (μ_{x}, μ_{y}, μ_{z}) with μ_{ν} (ν = x, y, z) being a 3 × 3 matrix^{32} (see Supplementary Note 4). Because the map f from any T^{2} crosssection of (k_{x}, k_{y}, t) space to S^{2} is trivial with zero Chern number, the map f is equivalent to the form of a Hopf map from S^{3} to S^{2} classified by an integer called Hopf invariant^{57,58}. In the quench dynamics of a topological Euler Hamiltonian, the Hopf invariant determines the linking number of a linking structure for the inverse images of the Hopf map^{32}.
The nontrivial linking structure for the quench dynamics of an Euler insulator is directly related to the static band topology of the postquench Hamiltonian^{32}. To see this relation, we write the evolving state as
Here a(k) = H(k)ψ_{0} = iψ(k, t = π/2), which defines a map from the 2D Brillouin zone to S^{2}. Based on the static Hamiltonian H(k) in Eq. (1), we obtain \({{{{{{{\boldsymbol{a}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})={(2{n}_{x}({{{{{{{\boldsymbol{k}}}}}}}}){n}_{z}({{{{{{{\boldsymbol{k}}}}}}}}),2{n}_{y}({{{{{{{\boldsymbol{k}}}}}}}}){n}_{z}({{{{{{{\boldsymbol{k}}}}}}}}),2{n}_{z}^{2}({{{{{{{\boldsymbol{k}}}}}}}})1)}^{T}\). By parameterizing n(k) with spherical coordinates as \({{{{{{{\boldsymbol{n}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})=(\sin \alpha \cos \beta ,\sin \alpha \cos \beta ,\cos \alpha )\), we have \({{{{{{{\boldsymbol{a}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})=(\sin 2\alpha \cos \beta , \sin 2\alpha \cos \beta ,\cos 2\alpha )\). For a nontrivial Euler Hamiltonian with ξ = 2, n(k) fully cover the S^{2} so that there exist 1D curves with \({n}_{z}({{{{{{{\boldsymbol{k}}}}}}}})=\cos \alpha ({{{{{{{\boldsymbol{k}}}}}}}})=0\) in the Brillouin zone on which a(k) = (0, 0, −1); the curves divide the entire Brillouin zone into two patches. The curves also serve as fixed points for the dynamics where the initial state only picks up a global phase during the time evolution. By shrinking the curve into a single point, each of the two patches can be seen as a sphere. In this case, a(k) defines a map from S^{2} to S^{2} characterized by the winding number for each of these patches. Though the winding number of a(k) over the entire Brillouin zone is zero since the state ψ(k, t = π/2) is trivial, a(k) can wrap the sphere S^{2} once in each patch^{32}. The nontrivial winding of a(k) over each patch is associated with the Hopf link in the quench dynamics of the patch, similar to the correspondence between the static Chern number and the existence of the dynamical Hopf link for quench dynamics of a Chern insulator (see Supplementary Note 4).
Figure 5 shows our experimentally measured vectors a(k) and linking structures. Specifically, we first prepare the ion in the \(\left3\right\rangle\) level and then measure the density matrix ρ(k, t) of the timeevolving state for a momentum in the Brillouin zone via quantum state tomography after the unitary time evolution under the experimentally engineered Euler Hamiltonian. We then evaluate a(k) and the images of the Hopf map, that is, \(\langle {\mu }_{i}\rangle ={{{{{{{\rm{Tr}}}}}}}}(\rho ({{{{{{{\boldsymbol{k}}}}}}}},t){\mu }_{i})\) with i = x, y, z based on the measured density matrices. For the nontrivial postquench Euler Hamiltonian, the experimentally measured a(k) exhibit a nontrivial skyrmion and antiskyrmion structure in the upper and lower halves of the Brillouin zone divided by the curves k_{y} = 0, π with n_{z}(k) = 0, as shown in Fig. 5(a). To quantitatively identify the skyrmion and antiskyrmion structure of the measured a(k), we construct a model H_{C}(k) = a(k)⋅σ for each of the two patches and find that the Chern numbers are equal to ±1, which are in excellent agreement with the theoretical results. The pair of skyrmions in the Brillouin zone leads to a pair of links with opposite signs for the inverse images in the corresponding regions of the (k_{x}, k_{y}, t) space (see Supplementary Note 4), which are experimentally demonstrated in Fig. 5(b). This also indicates the nontrivial band topology of the postquench Hamiltonian. For a trivial postquench Hamiltonian, the measured a(k) do not fully cover S^{2} for the two patches of the Brillouin zone, and the inverse images have no linking structures, as shown in Fig. 5(c) and (d).
Conclusions
We have experimentally realized a minimum Bloch Hamiltonian for topological Euler insulators protected by \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) symmetry in a trappedion qutrit system and identified its band topology by evaluating the Euler class, Wilson loop flow, entanglement spectra and the Berry phases based on the measured states via quantum state tomography. We further observed the nontrivial dynamical topological structures including the skyrmionantiskyrmion structures and Hopf links during the unitary evolution under the topological Euler Hamiltonian. Our work opens the door for further studying fragile topological phases using quantum simulation technologies.
Methods
Experimental setup
We use 399 nm and 370 nm laser beams to ionize the ytterbium (Yb) atom and use a linear radiofrequency Paul trap driven at 20.27 MHz (realized by a surfaceelectrode chip) to trap a single ^{171}Yb^{+} ion. The surface trap is designed and fabricated in our group at CQI, IIIS, Tsinghua University by following previous works^{59,60,61}. In our case, we make some modifications to dimensions of electrodes and the number of segmented electrodes in order to achieve a higher ability in controlling ion’s motion (see detailed presentation in Section A of Supplementary Note 5). The Doppler cooling, the optical pumping and the detection are implemented by 370 nm laser beams, which are modulated by acoustooptic modulators and electrooptic modulators. We also shine a 935 nm laser beam to repump the ion from the ^{2}D_{3/2} manifold back to the ^{2}S_{1/2} manifold in case that the ion decays to the ^{2}D_{3/2} manifold through spontaneous emission. After the ion is prepared in the dark state via optical pumping, we slowly vary the Hamiltonian through microwave operations. In the experiment, a magnetic field is applied to the system to split the \(\left2\right\rangle\) and \(\left3\right\rangle\) levels so that we can individually control the couplings between the hyperfine states through microwave operations. To control the amplitude and phase of microwaves, we use an arbitrary waveform generator mixed with a highfrequency signal to modulate them. In order to implement an arbitrary experimental sequence, a field programmable gate array is employed to control acoustooptic modulators, electrooptic modulators, arbitrary waveform generators and a photomultiplier tube. At the end, we perform measurements by collecting the statedependent fluorescence emission by an objective with a 0.33 numerical aperture and a photomultiplier tube; the detection fidelities for the dark state and the bright state are 99.4% and 98.1%, respectively (see Section C of Supplementary Note 5 for detailed discussions on how the detection fidelities and their statistical uncertainties are estimated). For more detailed discussion on laser and microwave setups and detection, see Section B and C of Supplementary Note 5.
Microwave operations in the trappedion system
In our trappedion system, we simultaneously shine microwave radiations with four different frequencies on the trapped ion to implement the threeband Hamiltonian. The microwave pulses are generated by a highfrequency generator and modulated by an arbitrary waveform generator mixed by an inphase and quadrature mixer. In the presence of a magnetic field and the microwaves, the trappedion qutrit system is described by the Hamiltonian
where the atomic part is
with ω_{hf} being the central transition frequency between \(\left1\right\rangle\) and \(\left3\right\rangle\), and ω_{z} (ω_{q}) being the frequency of the firstorder (secondorder) Zeeman energy determined by magnetic fields. The interaction part due to the microwaves is
where \({\hat{\sigma }}_{x}^{(j)}=\left1\right\rangle \left\langle j\right+H.c.\), ω_{n} and ϕ_{n} are the frequency and initial phase of each microwave, and \({{{\Omega }}}_{ij}^{(n)}\) is the Rabi frequency for the \(\lefti\right\rangle \leftrightarrow \leftj\right\rangle\) transition driven by the nth microwave. We then derive the effective Hamiltonian \({H}_{\exp }\) (see Eq. S41 in Supplementary Note 6) by following the method introduced in ref. ^{62} (see the detailed derivation in Supplementary Note 6). To ensure that the Rabi rate is always a linear function of the power of a radiofrequency signal, we set the maximum Rabi rates in our experiments as Ω_{12} = (2π) × 55.6 kHz and Ω_{13} = (2π) × 50.2 kHz. The errors of Ω_{12} and Ω_{13} are 0.036 kHz and 0.038 kHz, respectively, obtained by fitting the Rabi oscillations with time (the time step is 0.5 μs and the number of repetitions at each data point is 5000).
To simulate the Hamiltonian H(k) at each k, in the experiment, we in fact implement the Hamiltonian
where I_{3} is the 3 × 3 identity matrix. Clearly, this Hamiltonian has the same eigenstates as H(k) and thus is topologically equivalent to H(k). Here, c(k) is a real positive parameter that needs to be numerically determined, and it may be different at different k. Given the fact that the required evolution time for the adiabatic passage in the experiment is proportional to 1/c(k) and our coherent time is finite, for a fixed H(k), we find the minimum of 1/c(k) [or the maximum of c(k)] by solving Eq. (14) via the fmincon function in MATLAB. Since \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\) is a function of the Rabi frequencies, the detuning and the phases of microwaves, by solving Eq. (14), we obtain their values. In solving the equation, we also add some constraints on the microwave parameters. The far detuning for the Raman transition is set in the range from (2π) × 175.7 kHz to (2π) × 326.3 kHz, and the near detuning is set below (2π) × 10.0 kHz in most cases. In light of the fact that the Rabi frequency of the Raman transition is only about 1/10 of the direct resonant Rabi flopping, we usually need much higher power for the far detuning microwaves than that of the near detuning microwaves. Based on this fact, we set the power of the two far detuning microwaves below W_{1(2)}, where W_{1(2)} corresponds to the maximal power of a resonant microwave used to generate the maximal Rabi rate Ω_{12(13)}. Meanwhile, we set the two near detuning microwaves below 0.2W_{1(2)} in most cases. With these constraints, we obtain a set of microwave parameters, and based on these parameters we realize the Hamiltonian \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\) in experiments.
In addition, the maximum value of the coupling between \(\left2\right\rangle\) and \(\left3\right\rangle\) that we can reach in the experiment is much smaller than the values of the couplings between \(\left1\right\rangle\) and \(\left2\right\rangle\) or \(\left1\right\rangle\) and \(\left3\right\rangle\), since the former coupling is realized through microwave Raman transitions. When H_{23} is much larger than H_{12} or H_{13} in H(k), the energy scale c is very small so that a much longer period of time is required for the adiabatic evolution. To overcome the difficulty, we apply a πrotation between \(\left1\right\rangle\) and \(\left2\right\rangle\) (or \(\left1\right\rangle\) and \(\left3\right\rangle\)) at an appropriate moment during the adiabatic passage to change the basis for the qutrit system. The resultant effective Hamiltonian relative to the new basis has a smaller entry H_{23} so that a larger value of the coefficient c is obtained, which effectively reduces the time for the adiabatic evolution. Meanwhile, we continue the adiabatic passage for the Hamiltonian relative to the new basis and apply another πrotation in detections for quantum state tomography.
Adiabatic passage
To measure the topological properties of the engineered Euler Hamiltonian, we first prepare the ion in the dark state \(\left1\right\rangle\), which is the highest energy eigenstate of the Hamiltonian H(k) at highsymmetry points in momentum space,
To prepare the ion in the highest energy eigenstate of H(k), we choose a starting point k^{*} in the set of highsymmetry points so that the path in the Brillouin zone from k^{*} to the final point k is the shortest. We then slowly vary the Hamiltonian to the final one \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\) through the shortest path. Specifically, we divide the path into N = ∣k − k^{*}∣/Δk parts (\({{\Delta }}k=\pi \sqrt{2}/400\)), and at each part k_{n} = [(k − k^{*})/∣k − k^{*}∣]nΔk + k^{*} with n = 1, 2, ⋯ , N, the associated Hamiltonian is \({H}_{\exp }({{{{{{{{\boldsymbol{k}}}}}}}}}_{n})\), where c(k_{n}) is numerically calculated by solving Eq. (14). We then vary the microwave parameters so that at time \({t}_{n}=\mathop{\sum }\nolimits_{j = 1}^{n1}{T}_{j}\) [T_{j} indicates the duration over which the Hamiltonian stays at \({H}_{\exp }({{{{{{{{\boldsymbol{k}}}}}}}}}_{j+1})\)], the implemented Hamiltonian is \({H}_{\exp }({{{{{{{{\boldsymbol{k}}}}}}}}}_{n})\). To optimize the adiabatic evolution, we set T_{n} = α_{n}/c(k_{n}) where α_{n} may vary from 10 to 50. Each segment typically takes 1–2 μs. The total process typically takes 200–300 μs, and at some momentum points it may take up to 500 μs. After the adiabatic passage, we obtain a state which is very close to the highest eigenstate of \({H}_{\exp }({{{{{{{\boldsymbol{k}}}}}}}})\). We then measure the final states by quantum state tomography, based on which the topological properties are identified.
Quantum state tomography for a qutrit system
At the end of microwave operations, we perform a quantum state tomography to obtain the full density matrix of the qutrit system^{42,63}. The density matrix can be written in terms of eight unknown real variables as
Since the probability P_{1} in the dark state relative to eight different bases realized by applying π or π/2 or both rotations on Bloch spheres is a linear function of these variables (see Table I in Supplementary Note 7), we can determine them by solving the linear equations after we obtain the probabilities. In experiments, each probability is measured by counting the number of occurrences of the dark state over 4400 repeated experiments; the dark state is experimentally identified through the threshold method (see Section C of Supplementary Note 5 and the references therein for the introduction of the threshold method). We also employ the maximum likelihood estimation to find the best estimate for the physical density matrix. For the quantum state tomography and the maximum likelihood estimation, see Supplementary Note 7 for more details.
The fidelity of the measured state
We calculate the fidelity for the measured density matrix by
where ψ(k) is the theoretically obtained state, and ρ(k) is the measured density matrix optimized by the maximum likelihood estimation. With the optimization, the average fidelities over 20 × 20 momentum points are \(\overline{F}=97.1 \%\) and \(\overline{F}=96.9 \%\) for the topologically nontrivial (m = 1) and trivial phases (m = 3), respectively (see the details on how to determine the average fidelity in Supplementary Note 7).
The infidelity may arise from the detection infidelity of the dark state or the bright state, microwave pulse errors caused by nonlinear effects of experimental equipments and environment fluctuations. In addition, decoherence is also a factor for infidelity when the microwave operations take a long period of time. Note that decoherence usually does not occur significantly within 600 μs in our trappedion system, which is mainly restricted by the Zeeman state. In the experiment, we perform calibrations and optimize the experiment setups per hour in order to obtain a high fidelity.
The real state from the measured density matrix
To identify the band topology of the Euler Hamiltonian, we need to transform the measured density matrix ρ into a real state \(\left\psi \right\rangle ={(\alpha ,\beta ,\gamma )}^{T}\) by maximizing the function
so that \(\left\psi \right\rangle\) is the closest real state to the measured density matrix ρ. We find that the average fidelities between the real state \(\left\psi \right\rangle\) and the density matrix ρ are 97.4% and 97.1% when m = 1 (nontrivial) and m = 3 (trivial), respectively. We also find that the fidelities between the closest complex pure states and the density matrix are 97.5% and 97.3%, respectively. We see that the infidelity resulted from the restriction to a real state rather than a general complex pure state in the above maximization is below 1%, suggesting that the measured density matrix may not correspond to a pure state due to decoherence and detection errors.
Data availability
The data that support the findings of this study are available from the authors upon request.
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Acknowledgements
We thank W.Q. Lian for helpful discussions. This work was supported by the Beijing Academy of Quantum Information Sciences, the Frontier Science Center for Quantum Information of the Ministry of Education of China, and Tsinghua University Initiative Scientific Research Program. Y. Xu also acknowledges the support from the National Natural Science Foundation of China (Grant no. 11974201).
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L.M.D., Y. X., and Z.C.Z. supervised the project. Y.B.Y. and Y. X. performed the theoretical calculation. Y.J., Z.C.M., W.D.Z., L.H. did the fabrication of the surface trap, W.D.Z., W.X.G., L.Y.Q., G.X.W., L.Y., Z.C.Z. carried out the experiment. Y.X., W.D.Z, Y.B.Y., L.M.D. wrote the manuscript.
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Zhao, W., Yang, YB., Jiang, Y. et al. Quantum simulation for topological Euler insulators. Commun Phys 5, 223 (2022). https://doi.org/10.1038/s42005022010012
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DOI: https://doi.org/10.1038/s42005022010012
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