GoKawiilExperimental setup
A detailed description of our experimental setup has been given in previous works35,60,61. In short, we trap individual 88Sr atoms in a programmable one-dimensional array of optical tweezers (813 nm) generated by acousto-optic deflectors (AODs). The atoms are initialized in the 5s2 1S 0 state and cooled on the narrow-line red transition 5s2 1S 0 ↔ 5s5p 3P 1 (689 nm) to near their motional ground state. After rearrangement to a defect-free array with the desired atom number, all atoms are driven to the 5s5p 3P 0 clock state with a combination of direct π pulse (698 nm) and incoherent optical pumping60. Limited by the total number of available tweezers (laser power), we use up to two rounds of dark-state enhanced loading for system sizes of L = 26, 27, 35 to ensure high defect-free state preparation fidelity62. In our quantum simulator, the metastable clock state 5s5p 3P 0 is defined as the ground state |0⟩, which is coupled to the Rydberg state \(5s61{s}^{3}{S}_{1}\equiv |1\rangle \) with a single-photon transition (317 nm). After evolution under the Hamiltonian in equation (1), an auto-ionization beam (408 nm) is applied to push out the atoms in the Rydberg state35. Finally, the remaining atoms are optically pumped out of the clock state and imaged through the 5s2 1S 0 ↔ 5s5p 1P 1 (461 nm) transition63.
Data for both Ising CFT configurations (configurations (1) and (2) in Fig. 4; data shown in Figs. 1–3 and 4b) are taken with a Rydberg Rabi frequency Ω = 2π × 6.0 MHz (with the exception of data presented in Fig. 1d, where Ω = 2π × 7.5 MHz) and a next-nearest-neighbour interaction V 2 = 2π × 3.06(1) MHz at a lattice spacing of a = 3.3 μm for the Ising transition. For the TCI point (configuration (3)), data are taken with Ω = 2π × 5.5 MHz, V 2 = 2π × 8.96(15) MHz at a spacing of a = 2.8 μm for the TCI point. We also experimentally measure the nearest-neighbour interaction V 1 = 2π × 164.6(9) MHz at a spacing of a = 3.3 μm for the Ising case. The critical detunings for configuration (1), (2) and (3) are Δ c = 2π × 10.2 MHz, −0.9 MHz and −8.3 MHz, respectively.
For analysis of the experimental data, we apply a post-selection protocol based on four aspects. First, we post-select on rearrangement success, keeping experimental shots that contain a defect-free array of the correct system size. Second, we post-select based on erasure detection. In the experimental sequence, we perform erasure detection twice: first, immediately after the initial state preparation, and second, after the Rydberg pulse60. If there are atoms detected in the erasure images, they indicate leakage into the 5s2 1S 0 state, which is outside the qubit subspace. These shots are discarded in the post-selection process. Third, we discard measurement runs in which double Rydberg excitations might have occurred. As we work in the Rydberg blockade regime (V 1 ≫ V 2 , Ω, Δ), it is unlikely that two nearest-neighbouring atoms are both excited to the Rydberg state. We discard all the shots in which no atom is detected in consecutive sites after the Rydberg pulse. Note that we do not distinguish the Rydberg occupation from atom loss due to the readout scheme, so this step also filters out shots with loss outside of 5s2 1S 0 . Last, we post-select based on whether the Rydberg pulse is successfully delivered. The pulse is monitored by a photodiode and recorded on an oscilloscope during the experiment; about 0.6% of the shots show no detectable Rydberg pulse because of the arbitrary waveform generator board failing to output the programmed radiofrequency waveform, and are thus discarded61. For the data shown in the main text, 20−70% of experimental runs are post-selected for analysis.
For the adiabatic sweep, we modulate the Rabi frequency and detuning of the global Rydberg laser beam with AODs. The Rydberg interaction in the system is repulsive (V ij > 0) at the operating magnetic field B = 70 G. In this case, the global detuning is shaped to start from a large negative value and evolve to the critical point under a tangent function in time, which ensures that the detuning sweep rate is slower when getting closer to the critical point. Starting from all atoms in state |0⟩, we prepare the ground state at the critical point in this way. To realize an effective attractive interaction (V ij < 0 for |i − j| ≥ 2) sector of equation (1) in the same system, we can flip the sign of all the terms in the Hamiltonian, which means reversing the sign of detuning relative to the critical point in the ramp (Δ − Δ c ). As the nearest-neighbour interaction V 1 is much greater than any other energies in the Hamiltonian, the eigenenergies of the Hamiltonian are clustered into sectors separated by about V 1 . Then we prepare the highest-energy state within the blockade-violation-free sector of the full native Hamiltonian, which is the ground state of the blockade-enforced Hamiltonian equation (1) with the sign of interaction and detuning reversed. We call this procedure a backward sweep. Depending on the experiment, the global detuning is then set to either hold at the critical point (for σ measurement), or ramp to the \({{\mathbb{Z}}}_{2}\) or disordered phase in a symmetric way after modulation (for spectroscopy). These choices, together with the sweep parameters, are optimized such that the system has a minimal number of excitations after the sweep.
For experiments that require specific local detuning terms (δΔ i in the Hamiltonian) during the Rydberg pulse, we apply the same set of tweezers (813 nm) as for trapping. As a reference, for experiments that do not require local detuning terms, the tweezer light is turned off with a fast acousto-optic modulator (AOM)64 before the Rydberg pulse for two reasons: first, intensity noise in the tweezer translates into unwanted detuning noise on the Rydberg-qubit manifold; second, the Rydberg state is anti-trapped in 813-nm light. The tweezer light induces a light shift on the 5s5p 3P 0 ↔ 5s61s 3S 1 transition that is tunable with control of the local light intensity. We experimentally measure the light shift to be −3.545(54) MHz for tweezers with 1.4(2) mW power per spot and a waist of 0.75(5) μm. The time constant of the atom loss due to the anti-trapping of the Rydberg state at this tweezer trap intensity is measured to be 63(12) μs. We calibrate the local detuning for each tweezer pattern with a Ramsey sequence before taking the data. Extended Data Fig. 5 shows an example of the calibration in the case of L = 26 at the tricritical point with the fixed boundary condition.
Determining the critical point
To determine the parameters required to prepare our experimental setup along the Ising critical line, we use exact diagonalization of the effective Hamiltonian in equation (1). Specifically, we consider the second-order PXP Hamiltonian, first derived in ref. 65, using a Schrieffer–Wolf transformation. The resulting Hamiltonian corresponds to equation (1) with δΔ i = 0:
$${V}_{ij}=\left\{\begin{array}{ll}{V}_{1} & | i-j| =1\\ \frac{64{V}_{2}}{| i-j{|}^{6}} & | i-j| \ge 2\end{array}\right.$$
and
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