Quantum Simulations With Trapped Ions

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In this project we propose how to realize a quantum simulation of the Haldane phase in trapped ions. More specically, we would like to simulate the Haldane phase which exists in the Heisenberg region of the spin-one XXZ antiferromagnetic chain, where the spin is modelled via the three-level hyperfine structure in the microwave regime, and the interaction is achieved by the use of large magnetic field gradients that compensate the presence of a very small Lamb-Dicke parameter when using microwave sources. By reverse engineering we show how to generate all the terms in the simulated Hamiltonian using five microwave driving fields.
     We explain how to reach the Haldane phase adiabatically, starting from the large D phase where the ground states are robust to magnetic and Rabi frequency uctuations, namely, they belong to the decoherence-free subspace.
     The verication of the Haldane phase can be achieved by measuring its characteristics: an excitation gap and exponentially decaying correlations, a nonvanishing nonlocal string order and a double-degenerated entanglement spectrum.
In higher dimensions this platform gives rise to new research exploring quantum spin liquid phases that exhibit long-range entanglement patterns and hidden global topological orders. Their highly nonlocal ground states would be robust to local noise sources, giving rise to the realization of topologically protected quantum computation.