Quantum-Ising Hamiltonian programming in trio, quartet, and sextet qubit systems

Rydberg-atom quantum simulators are of keen interest because of their possibilities towards high-dimensional qubit architectures. Here we report three-dimensional conformation spectra of quantum-Ising Hamiltonian systems with programmed qubit connections. With a Rydberg-atom quantum simulator, various connected graphs, in which vertices and edges represent atoms and blockaded couplings, respectively, are constructed in two or three-dimensional space and their eigenenergies are probed during their topological transformations. Star, complete, cyclic, and diamond graphs, and their geometric intermediates, are tested for four atoms and antiprism structures for six atoms. Spectroscopic resolution (dE/E) less than 10% is achieved and the observed energy level shifts and merges through structural transformations are in good agreement with the model few-body quantum-Ising Hamiltonian.


I. INTRODUCTION
Well-calibrated quantum many-body systems are currently in high demand because of their essential necessities for quantum applications such as quantum computing and quantum simulation [1][2][3]. Among many promising physical platforms [4][5][6][7][8], Rydberg-atom quantum simulators, which use a mesoscopic-scale, deterministic arrangement of neutral atoms with controllable strong local interactions induced by Rydberg-atom excitation, draw latest attentions [9]. They have well-defined energy levels, relatively long coherence and lifetimes, and entanglements in these systems are generated with relative ease through giant dipole-dipole couplings in the Rydberg-atom blockade regime [10][11][12]. In recent demonstrations, these systems are used to generate as-manyas 20 qubit GHZ entangled states [13], to observe the quantum many-body phenomena involved with localization [14] and thermalization [15], and also to investigate the critical phenomena of Ising-type or XY quantum spin models across phase transitions [16,17]. Of particular importance in the context relevant to the present work, Rydberg-atom quantum simulators are expected to realize the possibilities of high-dimensional qubit architectures [9,[18][19][20][21].
Quantum simulation uses a controllable Hamiltonian H of a quantum N -body system to reproduce or predict the behavior of other equivalent systems of model Hamiltonian H G . The system size N increases computational power exponentially, and the controllability in H increases the simulation accuracy as well as the diversity of the systems to be simulated. Currently, Rydberg-atom quantum simulators have scaled up the number of qubits [16], approaching the regime of quantum supremacy [22]. In that regards, dexterous control of the parameters of H, as many as possible and with high fidelities, is of great importance for a programmable quantum simulator. In this paper, we use a Rydbergatom quantum simulator, of Hamiltonian H, to produce and tune a set of two-or three-dimensional arrangements of atoms, that are isomorphic to connected-graphs {G} of N = 3 − 6 qubits, to model quantum-Ising Hamiltonian H G , and probe their topology-dependent eigenspectra, in particular, during structural transformations from one G to another.

II. QUANTUM-ISING HAMILTONIAN
The Hamiltonian of N atoms that are coherently excited to a Rydberg energy state is given bŷ where |0 j and |1 j denote the ground and Rydberg energy states, respectively, of an atom j located at r j , Ω is the Rabi oscillation frequency, U (r jk ) = C 6 /| r j − r k | 6 is the van der Waals interaction between two Rydberg atoms, andn j = |1 j 1| j is the excitation number [10]. In the following, we will consider arrangements of atoms (e.g., see Fig. 1), in which only the nearest neighboring pairs, of the same inter-atom distance d, are within the Rydberg-blockade radius, i.e., d < r b = |C 6 / Ω| 1/6 , and thus all other long-ranged pairs are ignorable. Such an arrangement of atoms can be isomorphically represented by an undirected connected graph G(V = N, E), in which the vertices V denote atoms and the edges E the nearest-neighbor couplings. Then, for a graph G, the Hamiltonian H is given by the quantum-Ising Hamiltonian H G (with inhomogeneous longitudinal field): in whichσ x,z are Pauli spins, J = U (d)/4 is the coupling, h x = Ω/2 is the transverse field, and h of the Rydberg-atom quantum simulator is adequately modeled by the quantum-Ising Hamiltonian H G for a connected graph G(V, E) that isomorphically represents an atom arrangement of equal nearest-neighbor couplings and ignorable long-range interactions. In order to investigate the topological change of a strongly-coupled Rydberg-atom system, we consider, for example, four-atom (N = 4) arrangements and their structural transformations. As shown in Fig. 1(a), there are four nonisomorphic 4-vertex-connected graphs: the star graph, denoted by S 4 , has one atom at the center and three at the ends of three claws, and three nearestneighbor edges (E = 3); the complete graph, K 4 , has four atoms in the tetrahedron configuration with six edges (E = 6); the cyclic graph, C 4 , is the square configuration with four edges (E = 4), and the diamond graph, K 4 -e, has five edges (E = 5). Here, S 4 , C 4 , and K 4 -e are two-dimensional (e.g., in the xy plane) and K 4 is threedimensional, of the tetrahedron shape. Structural transformations among them can be proceeded, for example, in the sequence of S 4 → K 4 , K 4 → C 4 , and C 4 → K 4 -e, as respectively shown in Figs. 1(b-d).
As a pedagogical example, let us consider C 4 , the cyclic graph, which has four nearest-neighbor couplings. Due to the symmetry of the given graph, most of the possible 2 N = 16 eigenstates of H G are dark states, inaccessible from the initial state, |Ψ(t = 0) = |0000 , the bareatom ground state, in our consideration. There are three bright eigenstates, of respective energies (in = 1 unit hereafter) λ 1 = − 3 2 Ω, λ 5 = 0, and λ 7 = 3 2 Ω, where the index j denotes the energy ordering among all energy states, bright and dark. The corresponding eigenstates are given by which are represented with symmetric base states, |W 0 = |0000 , |W 1 = (|1000 + |0100 + |0010 + |0001 )/ √ 4, and |W C 2 = (|1010 + |0101 )/ √ 2, each labeled with an excitation number (the number of atoms in the Rydberg-state). Likewise, there are four bright eigenstates for S 4 , two for K 4 , and four for K 4 -e. Table I summarizes the eigenenergies and eigenstates of H G , for the all four graphs.

III. RYDBERG-ATOM QUANTUM SIMULATOR
In order to probe the eigenspectra of an N -body quantum-Ising Hamiltonian, we used a Rydberg-atom quantum simulator, which can (i) arrange N single atoms isomorphically to a connected graph, (ii) implement the Hamiltonian H in Eq. (1) through creating Rydberg atoms, and (iii) readout the final-state |Ψ(t) after time t. We used the probability P 0 (t) of all atom back to the initial state, defined by where |W 0 is the initial zero-excitation state, e.g., |W 0 = |000 for N = 3, |0000 for N = 4, and |000000 for N = 6. With eigenenergies λ j , P 0 (t) is given by with A j = W 0 |λ j and B jk = 2 |A j | 2 |A k | 2 . So the Fourier transform, F[P 0 (t)], retrieves the energy differences, λ jk = λ j − λ k for all pairs of eigenenergies.
Experiments were performed with an updated version of the machine previously used in our earlier works for deterministic multi-atom arrangements [19,23,24], quantum many-body thermalization [15], and entanglement generations [25]. For the current work, we have increased the Rabi coherence time from 2.5 µs [26] to about 10 µs, and developed three-dimensional atom arrangements, through technical improvements to be discussed in Sec. V. The procedures of (i)-(iii) are summarized below: (i) Atom arrangement: Single atoms are trapped with optical tweezers and arranged in three dimensional space. Rubidium ( 87 Rb) atoms first are cooled below 30 µK by Doppler and polarization gradient cooling in a magnetooptical trap (MOT), and optically pumped to the ground hyperfine state |0 = |5S 1/2 , F = 2, m F = 2 . Then, a I: Quantum-Ising eigenstates of N = 4 atom systems (bright states only), in symmetric base states defined by |W0 = |0000 , |W1 = (|1000 + |0100 Configuration Eigenenergies ( = 1) Eigenstates (bright states only) spatial light modulator (SLM, Meadowlarks 512×512 XY modulator) turns on as-many-as 250 optical tweezers (offresonant optical dipole traps) to capture and rearrange N single atoms deterministically to target positions, with 5-10 µm spacing [24,28]. The wavelength of the optical tweezers is 820 nm and an objective lens (Mitutoyo G Plan Apo 50×) of a high numerical aperture (NA = 0.5) is used. The trap depth and diameter are 1 mK and 2 µm, respectively, and the lifetime of each trapped atom is about 40 s.
(iii) Final state readout: In the detection stage, the optical tweezers are turned back on and the fluores-cence from trapped ground-state atoms in |0 is collected through the same objective lens and imaged onto an EM-CCD camera. The tomographic images of a 3D atomic array structure are obtained with an electrically focustunable lens (ETL, EL-16-40-TC from Optotune), located after the tube lens, by sequentially shifting the focal length. At the same time, the EMCCD is triggered on and off with a period of 40 ms (camera exposure time), for each tomogram. After the interaction H of duration t, up to 5 µs with time step ∆t = 0.1 µs, the events of all atoms back in the bare-atom ground states are collected, and the whole procedure, (i)-(iii), is repeated about 100-200 times of data accumulation to obtain the |W 0 -state probability in Eq. (5).

IV. RESULTS AND ANALYSIS
In the first experiment, we probe three-atom configurations. As in Fig. 2(a), three atoms, ABC, are initially arranged in the triangle configuration, with AB = BC = d = 8 µm, and the bending angle θ = ∠ABC is gradually changed from 60 • (a triangle) to 180 • (a linear chain). The corresponding atom positions are A(−d, 0, 0), B(0, 0, 0), and C(−d cos θ, d sin θ, 0) in Cartesian coordinates. The energy levels, given by the direct diagonalization of H, are shown in Fig. 2(b), in which the bright states (|λ 1 , |λ 2 , |λ 4 , and |λ 5 ) are depicted with solid lines and the dark state (|λ 3 , others are out of the given spectral range) with a dashed line.
In the final experiment, we probe the structural transformation of an N = 6 atom system, from a hexagon to an antiprism (a set of upright and inverted triangles, separated by z). As illustrated in Fig. 4(a), six atoms are initially arranged at the vertices of a hexagon, with positions d(cos θ i , sin θ i , 0) with θ j = jπ/3 (for j = 1, · · · , 6) and the axial z positions of the even numbered atoms (j = 2, 4, 6) are axially translated from z = 0 to z = 3d/2. During the transformation, the length of each triangle is kept constant, AC = d = 8 µm, and the distance AB is changed as AB(z) = d 2 /3 + z 2 . The energy levels calculated with H are plotted in Fig. 4(b), which shows three distinct coupling regimes: (i) U AB Ω, the hexagon regime at z = 0, (ii) U AB ∼ Ω, the AB doubleexcitation regime around z = 3d/4, and (iii) U AB Ω, In the hexagon regime, (i) U AB Ω, there are three eigenstates, constructed with symmetric base states |W 0 = |000000 , |W 1 (the superposition of singleexcitation states), and |W d 2 = (|100100 + |010010 + |001001 )/ √ 3 (the superposition of diagonal double excitations). We denote the eigenstates by |λ 1 , |λ 7 , and |λ 16 . The results for the six atoms in the hexagon configuration at z = 0 are shown in Figs. 4(c,d,e) with the fluorescence image, the measured |W 0 probability, and the Fourier transform, respectively. Likewise, the results for z = 3d/4 and z = 3d/2 are given in Figs. 4(f-h) and 4(i-k), respectively. At z = 3d/4, which we refer to as (ii) the AB double-excitation regime (U AB ∼ Ω), the distance between AB atoms is bigger than the blockade radius, i.e., AB > r b , so AB can be excited together, while they are weakly coupled (U AB ∼ Ω). Therefore, besides the above base states, |W 0 , |W 1 , and |W d 2 , an additional symmetric base state, |W AB 2 (the superposition of adjacent double excitations), is allowed. The spectrum in Fig. 4(h) shows 4 C 2 = 6 peaks, in a reasonable agreement with the numerical calculation. In the regime (iii) U AB Ω, at z = 3d/2, the atom planes are well separated, and, as a result, the two sets of three atoms are decoupled, each constructed with its own symmetric basis, |000 and (|100 + |010 + |001 )/ √ 3. So the eigenenergies, of the decoupled trios, are given by λ 1 = − √ 3Ω, λ 7 = 0, and λ 22 = √ 3Ω.

V. TECHNICAL DETAILS AND IMPROVEMENTS
The spectral resolution of the given spectroscopy is limited by the coherence time of the Rydberg-atom quantum simulator. It is discussed in Refs. [26,27] that the coherent operation time of the machine is dominantly limited by the non-intrinsic dephasing due to laser spectral phase noises. In order to suppress the laser spectral phase noises, we adopted the laser frequency stabilization method described in Ref. [29] without using intracavity electro-optic modulations in this work. The laser frequency was locked to a resonance of a high-finesse cavity using Pound-Drever-Hall (PDH) technique. The reflected light from the cavity was directed into a PDH module (Stable Laser System PDH-1000-20D) which included a fast photo-detector and electronics to demodulate the detected beat signal. The demodulated signal was fed into a fast analog proportional-integral-derivative (PID) controller (Toptica FALC 110) and two separate ('slow' and 'fast') servo loops were implemented; 'slow' for changing the angle of the grating in the extra-cavity diode laser, mostly to compensate frequency drift, and 'fast' for changing the current through the laser diode, mostly to reduce the linewidth. By setting the PID parameters (to get the highest possible gain for low frequency and relatively low gain for high frequency with a proper amount of 90 • phase shifted signals) and optimizing the transfer function of the servo loop, we re-strained the oscillation of the servo-loop at the margin of the bandwidth (which otherwise caused a servo-bump) and achieved the sufficiently low frequency noise level, S ν (f ) < 10 3 Hz 2 /Hz, even at a servo-bump around 1 MHz. With the described method, the coherence time measured from single-atom Rabi oscillation decay is improved from 2.5 µs [26] to 10 µs.
To make the atomic arrangements of the given geometries, we extended the method of dynamic holographic optical tweezers [28], previously restricted to 2D arrangements, to a 3D version. The hologram on demand for an optical tweezer arrangement was calculated with a 3D Gerchberg-Saxton (GS) algorithm, along with the methods of weighted-GS and phase induction [28] for fast convergence. For each cycle of atom rearrangement, positions of about 26 optical tweezers were simultaneously shifted by differential displacements frame-byframe, throughout a serial sequence of 35 successive phase patterns in 700 ms, to achieve the occupation probability per site > 0.94. While the transverse fluctuation of trap positions was below the imaging resolution limit, the axial position fluctuation was about 1 µm, due to limited phase convergence. The number of GS-algorithm iterations was set to five in experiments, compromising between the quality of the optical tweezers and the calculation time.
A typical time budget for an experiment with a singleplane arrangement of atoms is less than one second, given by the sum of the times for atom loading (100 ms), initial occupancy checking (40 ms), atom rearrangements (700 ms), final occupancy checking (40 ms), optical pumping (2 ms), Rydberg-atom excitation (5 µs), and final state detection (40 ms). When atom arrangements are repeated for a multi-plane geometry, the overal time increases but is little significant compared to the 40-s trap life time.

VI. CONCLUSION
In summary, we have utilized three-dimensional arrangements of neutral atoms, of adjustable inter-atom distances, to obtain the conformation energy landscape of strongly-interacting, small-scale Rydberg atom systems, in particular, during their structural transformations. We probed all possible nonisomorphic, connected graph configurations programmed for N = 3, 4 atoms, and partial graphs for N = 6. The experimentally measured topology-dependent eigenspecta are in good agreement with the model calculation of the few-body quantum-Ising Hamiltonian. It is hoped that high-dimensional programming of qubit connectivities demonstrated in this paper shall be useful for further applications of programmable quantum simulators.