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XYZ quantum Heisenberg models with p-orbital bosons
Fernanda Pinheiro1,2, Georg M. Bruun3, Jani-Petri Martikainen4, and Jonas Larson1
1
Stockholm University, Stockholm - Sweden, 2NORDITA, Stockholm - Sweden
3
Aarhus University, Aarhus - Denmark, 4Aalto University, Helsinki - Finland
Introduction
We demonstrate how the spin-1/2 XYZ quantum Heisenberg model
can be realized with bosonic atoms loaded in the p-band of an optical lattice in the Mott regime. The combination of Bose statistics and the symmetry of the p-orbital wave functions leads to a
non-integrable Heisenberg model with anti-ferromagnetic couplings.
Moreover, the sign and the relative strength of the couplings characterizing the model are shown to be experimentally tunable. We
display the phase diagram in the one dimensional case, and discuss
finite size effects relevant for trapped systems. We also consider
issues related to preparation, manipulation, detection and imperfections.
Hamiltonian for p-orbital bosons..
From the expression of the Hamiltonian of weakly interacting Bose gases, we expand the fields in terms of the site-localized Wannier functions of the p-band.
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• We assume a 2D lattice potential given by Vlatt(~
r ) = Vx sin kxx + Vy sin ky y which has amplitudes and wave vector given, respectively, by Vα,
α ∈ {x, y}, and kα = 2π/λα, with λα being the wave length of the lasers.
p-orbitals and the anisotropic tunneling..
From superposition of beams of linearly polarized lasers it is possible to
create spatially periodic potentials that can be used to trap cold atoms.
Spin mapping & effective Hamiltonian
• An atom in the px orbital has much larger tunneling rate in the
x-direction than in the y-direction. The opposite is valid for an
atom in the py -orbital.
Manipulation & detection schemes..
Spin is encoded in an external spatial degree of freedom.
Ground-state properties..
Making use of the symmetries of the px and py orbitals, stimulated
Raman transitions can drive both side-band and carrier transitions for
the chosen orbitals.
Spin rotations can be implemented driving side-band transitions:
• A rotation around the x direction can be achieved by driving the
red-sidebands for both orbitals.
• A rotation around z can be achieved by Stark shifting one of the
orbitals.
The chain with imperfections..
• Current experimental realizations with p-orbital atoms report 80%
fidelity for transferring atoms from the s-band to the p-band.
Conclusions and Prospects..
• The present proposal shows that the physics of the non-integrable XYZ Heisenberg model can be studied with the use of cold atoms loaded in the p
band of an optical lattice. It provides a controllable environment where different phases can in principle be accessed. In particular, this system could
be used to investigate properties of the XYZ model in 2D, where little is known.
• An interesting prospect is the three orbital case in the 3D lattice, where the effective Hamiltonian describing the Mott phase with a unit filling can be
mapped into a SU (3) chain.
References
[1] Fernanda Pinheiro, Jani-Petri Martikainen, Jonas Larson, “Confined p-band Bose-Einstein condensates”, Phys. Rev. A 85, 033638 (2012).
[2] Fernanda Pinheiro, Georg M. Bruun, Jani-Petri Martikainen and Jonas Larson, "XYZ quantum Heisenberg models with p-orbital bosons",
arXiv:1304.3178 (2013).
Work is supported by the Swedish Research Council (Vetenskapsrådet).