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Three-spin interactions in optical lattices and cluster Hamiltonians

Jiannis K. Pachos, Martin Bodo Plenio

Published Jan 19, 2004• 0 citations
OpenAlex

Abstract

We demonstrate that a triangular optical lattice of two atomic species can be employed to generate a variety of novel spin-1/2 Hamiltonians. They include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. This motivates the study of the ground state properties of various three-spin Hamiltonians in terms of their two-point and n-point correlations and the more detailed localizable entanglement. In particular, we present a cluster Hamiltonian with a finite energy gap above its unique ground state for which the localizable entanglement exhibits a diverging entanglement length while the classical correlation length remains finite.

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We demonstrate that a triangular optical lattice of two atomic species can be employed to generate a variety of novel spin-1/2 Hamiltonians. They include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. This motivates the study of the ground state properties of various three-spin Hamiltonians in terms of their two-point and n-point correlations and the more detailed localizable entanglement. In particular, we present a cluster Hamiltonian with a finite energy gap above its unique ground state for which the localizable entanglement exhibits a diverging entanglement length while the classical correlation length remains finite.
We demonstrate that in a triangular configuration of an optical lattice of two atomic species a variety of novel spin-1/2 Hamiltonians can be generated. They include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. This motivates the study of ground state properties of various three-spin Hamiltonians in terms of their two-point and n-point correlations as well as the localizable entanglement. We present a Hamiltonian with a finite energy gap above its unique ground state for which the localizable entanglement length diverges for a wide interval of applied external fields, while at the same time the classical correlation length remains finite.
We demonstrate that a triangular optical lattice of two atomic species, bosonic or fermionic, can be employed to generate a variety of spin-1∕21∕21∕2 Hamiltonians. These include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. Such interactions can be employed to simulate particular one- or two-dimensional physical systems with ground states that possess a rich structure and undergo a variety of quantum phase transitions. In addition, tunneling can be activated by employing Raman transitions, thus creating an effective Hamiltonian that does not preserve the number of atoms of each species. In the presence of external electromagnetic fields, resulting in complex tunneling couplings, we obtain effective Hamiltonians that break chiral symmetry. The ground states of these Hamiltonians can be used for the physical implementation of geometrical or topological objects.
The entanglement properties of some novel quantum systems are studied that are inspired by recent developments in cold-atom technology. A triangular optical lattice of two atomic species can be employed to generate a variety of spin-1/2 Hamiltonians including effective three-spin interactions. A variety of one- or two-dimensional systems can thus be realized that possess multi-degenerate ground states or non-vanishing chirality. The properties of these ground states and their phase transitions are probed with appropriate measures such as the entropic entanglement and the spin chirality.
The regular structures obtained by optical lattice technology and their behavior are analyzed from the quantum information perspective. Initially, we demonstrate that a triangular optical lattice of two atomic species, bosonic or fermionic, can be employed to generate a variety of novel spin-1/2 models that include effective three-spin interactions. Such interactions can be employed to simulate specific one-or two-dimensional physical systems that are of particular interest for their condensed matter and entanglement properties. In particular, connections between the scaling behavior of entanglement and the entanglement properties of closely spaced spins are drawn. Moreover, three-spin interactions are well suited to support quantum computing without the need to manipulate individual qubits. By employing Raman transitions or the interaction of the atomic electric dipole moment with magnetic field gradients, one can generate Hamiltonians that can be used for the physical implementation of geometric or topological objects. This work serves as a review article that also includes many new results.
A spin-1∕21∕21∕2 chain model that includes three-spin interactions can effectively describe the dynamics of two species of bosons trapped in an optical lattice with a triangular-ladder configuration. A perturbative theoretical approach and numerical study of its ground state is performed that reveals a rich variety of phases and criticalities. We identify phases with periodicity one, two, or three, as well as critical points that belong in the same universality class as the Ising or the three-state Potts model. We establish a range of parameters, corresponding to a large degeneracy present between phases with period 2 and 3, that nests a gapless incommensurate chiral phase.
We present a general procedure for constructing lattices of qubits with a Hamiltonian composed of nearest-neighbor two-body interactions such that the ground state encodes a cluster state. We give specific details for lattices in one, two, and three dimensions, investigating both periodic and fixed boundary conditions, as well as present a proof for the applicability of this procedure to any graph. We determine the energy gap of these systems, which is shown to be independent of the size of the lattice but dependent on the type of lattice (in particular, the coordination number), and investigate the scaling of this gap in terms of the coupling constants of the Hamiltonian. We provide a comparative analysis of the different lattice types with respect to their usefulness for measurement-based quantum computation.