Quantum simulation of material-specific Hubbard models with alkaline-earth atoms

The gist: This architecture introduces an integrated quantum-simulation approach using fermionic alkaline-earth atoms to programmatically control multi-orbital material Hamiltonians by encoding orbital degrees of freedom in long-lived clock states, utilizing state-dependent optical potentials for hopping control, and employing spatially structured coherent driving for symmetry-selective hybridization<ref:2610.10713#pg6>

Material Target and Model

The paper focuses on reproducing the low-energy Hamiltonians of strongly correlated quantum materials by using the bilayer nickelate La3Ni2O7 as a demanding benchmark<ref:2610.10713#pg5> The material is chosen because it is the only other family of correlated materials, after cuprates, to superconduct above the boiling point of liquid nitrogen and requires treatment of multiple active orbitals near the Fermi level<ref:2610.10713#pg6> The single-particle Hamiltonian for La3Ni2O7 involves two active orbitals per Ni atom, specifically the partially filled dx2−y2 and dz2 orbitals, which are labeled x and z respectively<ref:2610.10713#pg5> The non-interacting tight-binding Hamiltonian is defined with hopping terms like txx∥, tzz∥, and the inter-orbital hybridization term txzij that breaks orbital number conservation<ref:2610.10713#pg5>

Integrated Simulation Architecture

The proposed architecture combines several ingredients to realize material-specific requirements in a single microscopic framework<ref:2610.10713#pg6> The key components are summarized in Fig. 1, which includes long-lived clock states encode synthetic orbitals, their differential polarizabilities generate orbital-selective tunneling, and contact interactions provide multi-orbital Hubbard and Hund couplings<ref:2610.10713#pg6>

The control mechanisms are detailed as follows:

1. Orbital Encoding: Two long-lived clock states in alkaline-earth atoms (AEA), such as 87Sr, encode the two synthetic orbitals, with the excited state e=3P0 mapping to dx2−y2 and the ground state g=1S0 mapping to dz2<ref:2610.10713#pg5>

2. Orbital-Selective Tunneling: This is achieved through state-dependent optical lattices where the differential polarizability of the clock states assigns each synthetic orbital its own tunneling amplitudes, allowing independent control over in-plane and vertical confinement to realize anisotropic hopping hierarchies<ref:2610.10713#pg6>

3. Symmetry-Selective Hybridization: This is realized via a spatially structured coherent driving on the ultra-narrow clock transition, which acts as an optical “mask” with amplitude and phase imprinting the desired inter-orbital tunneling structure, such as the B1g form factor required by nickelate hybridization<ref:2610.10713#pg6>

4. Crystal Field Splitting: The clock detuning independently controls orbital energy offsets, which directly corresponds to the crystal-field splitting of the multi-orbital Hubbard model<ref:2610.10713#pg6>

Implementation Details with Alkaline-Earth Atoms

The implementation utilizes fermionic 87Sr, where the electronic ground state g = 1S0 and excited clock state e = 3P0 provide the synthetic orbitals<ref:2610.10713#pg5>

**-Orbital-Selective Tunneling: The hopping hierarchy is controlled by tuning the in-plane lattice wavelength λL, which selects the state-dependent polarizability ratio αe/αg to match target ratios like txx∥/tzz∥<ref:2610.10713#pg6> The vertical direction requires a state-dependent superlattice formed by short (λs) and long (λl) wavelength components, which allows for independent tuning of the vertical hopping ratios tgg⊥/te∥ and tgg⊥/tee∥<ref:2610.10713#pg6> The optimization ensures that all hopping, hybridization, and interaction matrix elements are obtained from a single common microscopic lattice configuration through self-consistent recalculations<ref:2610.10713#pg6> 4. Crystal Field Control: The detuning term in the clock drive provides independent control over the relative orbital energy, directly controlling the crystal-field splitting ∆, which is equivalent to ħδ<ref:2610.10713#pg6> 5. Interaction Structure: Contact interactions generate orbital-resolved Hubbard repulsions, direct inter-orbital terms, and intra-atomic Hund exchange, with the ratio of exchange to direct interactions naturally set by the atomic scattering lengths in 87Sr, which yields a ferromagnetic Hund coupling Jex > 0<ref:2610.10713#pg6> The nuclear-spin encoding allows for engineering the Hund sector, where the large I = 9/2 nuclear spin of 87Sr allows for a wide selection of spin encodings to realize exotic interorbital hybridization<ref:2610.10713#pg6> 6. Spin Encoding: The paper explores three encodings for electron spins: time-reversal pair mF = ±9/2, adjacent pair mF = 9/2, 7/2, and Floquet-engineered pair mF = 9/2, 7/2<ref:2610.10713#pg6> The Floquet engineering of the Raman-driven Hamiltonian is proposed to achieve an effective txz↑ ≈ txz↓ by using a pulse sequence with rapid π-pulses<ref:2610.10713#pg6> 7. Self-consistent Wannierization: The hopping and hybridization targets must be imposed self-consistently because the clock field modifies the optical potential through its off-resonant Stark shift, requiring iterative calculations to find the required Rabi frequency for a given lattice depth<ref:2610.10713#pg6> 8. Conclusion: The toolbox provides a general blueprint for engineering multi-orbital Hamiltonians by combining these ingredients into a single microscopic architecture<ref:2610.10713#pg6> This platform enables the study of how pairing physics responds to modified Hund sectors and probes the robustness of magnetic and pairing phenomena in nickelate superconductors<ref:2610.10713#pg6> The resulting simulator reproduces the three key features of the target single-particle Hamiltonian: a strongly dispersive dx2−y2 orbital in the plane, a dz2 orbital with dominant tunneling across the bilayer, and the nearest-neighbor B1g inter-orbital hybridization<ref:2610.10713#pg6> The target hopping ratios, including the clock-laserinduced inter-orbital hybridization, can be maintained over a continuous range of in-plane lattice depths<ref:2610.10713#pg6> The same variation of lattice depth changes the interaction strengths substantially while preserving the desired interaction hierarchy<ref:2610.10713#pg6> The large I = 9/2 nuclear spin of 87Sr allows for a wide selection of spin encodings to realize exotic interorbital hybridization<ref:2610.10713#pg6> The structured clock drive provides a direct route to momentum-dependent hybridization form factors by combining different spatial Fourier components of the optical field<ref:2610.10713#pg6> Additionally, interactions can be tuned from the native interatomic scattering lengths by means of optical Feshbach resonances to realize a wide range of interaction hierarchies<ref:2610.10713#pg6> The toolbox provides a particularly flexible framework for moving beyond paradigmatic single-band Hubbard models, toward material-realistic quantum simulation<ref:2610.10713#pg6> The paper demonstrates that the full target hopping structure from Tab. II can be realized over a wide range of in-plane lattice depths<ref:2610.10713#pg6> Over the displayed range, the dominant inplane hopping is tee∥/h ≃ 50-100 Hz<ref:2610.10713#pg6> The vertical lattice depths required to maintain the target intra-bilayer hoppings are shown in Fig. 7(b), in units of the vertical recoil energy ER,z for λs = 750 nm<ref:2610.10713#pg6> The required depth remains in a standard experimentally accessible range throughout the scan<ref:2610.10713#pg6> The lower end of the in-plane lattice depth scan in Figs. 3 and 7 is set by the χγ Although the required clock drive decreases toward shallower lattices, the bare lattice confinement decreases more rapidly, so that the relative deformation χγ increases (primarily of the excited clock state, χe)<ref:2610.10713#pg6> The bond center becomes locally flat at χγ = 1/2, and turns into an additional local minimum for χγ > 1/2<ref:2610.10713#pg6> The clock-induced modification of the lattice acts in the same direction as the desired in-plane hopping hierarchy<ref:2610.

1. Bold orbital encoding capability allows AI systems to directly simulate material-specific low-energy Hamiltonians, such as those of bilayer nickelate La3Ni2O7. This enables the AI to predict complex correlated phenomena like dominant pairing channel and magnetic and pairing correlations by solving models where the relevant low-energy degrees of freedom are often distributed across several active orbitals.

2. State-dependent optical potentials enable AI to implement orbital-selective tunneling, where hopping amplitudes are assigned based on the state, allowing simulation of anisotropic hopping hierarchies like those dictated by t xx∥ ≫ t zz∥ and t zz⊥ ≫ t xx⊥.

3. Spatially structured coherent driving enables AI to realize symmetry-selective inter-orbital hybridization, allowing the simulation to imprint material symmetry directly onto the effective Hamiltonian through a spatially structured Rabi mask that generates coupling terms proportional to cos(kxa) − cos(kya).

4. Clock detuning independently controls orbital energy offsets, which allows AI to simulate effects on the crystal-field splitting and control the occupation of specific orbitals, such as those in the γ pocket.

5. The ability to vary lattice depth while keeping the hopping Hamiltonian fixed allows AI to explore different correlation regimes by changing interaction strengths while preserving the desired hierarchy between intra-orbital repulsion, inter-orbital repulsion, and ferromagnetic Hund exchange.