Abstract
In quantum simulation, many-body phenomena are probed in controllable quantum systems. Recently, simulation of Bose–Hubbard Hamiltonians using cold atoms revealed previously hidden local correlations. However, fermionic many-body Hubbard phenomena such as unconventional superconductivity and spin liquids are more difficult to simulate using cold atoms. To date the required single-site measurements and cooling remain problematic, while only ensemble measurements have been achieved. Here we simulate a two-site Hubbard Hamiltonian at low effective temperatures with single-site resolution using subsurface dopants in silicon. We measure quasi-particle tunnelling maps of spin-resolved states with atomic resolution, finding interference processes from which the entanglement entropy and Hubbard interactions are quantified. Entanglement, determined by spin and orbital degrees of freedom, increases with increasing valence bond length. We find separation-tunable Hubbard interaction strengths that are suitable for simulating strongly correlated phenomena in larger arrays of dopants, establishing dopants as a platform for quantum simulation of the Hubbard model.
Highlights
In quantum simulation, many-body phenomena are probed in controllable quantum systems
We find that interference of atomic orbitals directly contained in the quasi-particle wavefunction (QPWF) allows us to quantify the electron–electron correlations and the entanglement entropy
The entanglement entropy is directly related to the Hubbard interactions U=t, and we find that U=t is tunable with dopant separation, increasing from 4-14 for d/ aB 1⁄4 2.2-3.7, where aB 1⁄4 1.3 nm is the effective Bohr radius
Summary
Many-body phenomena are probed in controllable quantum systems. Fermionic many-body Hubbard phenomena such as unconventional superconductivity and spin liquids are more difficult to simulate using cold atoms. In the analogue approach to quantum simulation exemplified by cold atoms in optical lattices[4,5], the simulator’s Hamiltonian maps to the desired. Experimentally resolving individual lattice sites, crucial elsewhere in Bose–Hubbard simulation[4], remains very challenging in quantum simulation of the Hubbard model[5]. We perform atomic resolution measurements resolving spin–spin interactions of individual dopants, realizing an analogue quantum simulation of a two-site Hubbard system. We demonstrate the much desired combination of low effective temperatures, single-site spatial resolution, and non-perturbative interaction strengths of great importance in condensed matter[9,10,11]
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