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arXiv:0812.1548 · New Journal of Physics

Emergent timescales in entangled quantum dynamics of ultracold molecules in optical lattices

Michael L. Wall, Lincoln D. Carr

Published May 14, 2009• 50 citations
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Abstract

We derive a novel lattice Hamiltonian, the molecular Hubbard Hamiltonian , which describes the essential many-body physics of closed-shell ultracold heteronuclear molecules in their absolute ground state in a quasi-one-dimensional optical lattice. The molecular Hubbard Hamiltonian is explicitly time dependent, making a dynamic generalization of the concept of quantum phase transitions necessary. Using the time-evolving block decimation algorithm to study entangled dynamics, we demonstrate that, in the case of hard-core bosonic molecules at half-filling, the molecular Hubbard Hamiltonian exhibits emergent timescales over which spatial entanglement grows, crystalline order appears and oscillations between rotational states self-damp into an asymptotic superposition. We show that these timescales are non-monotonic functions of the physical parameters describing the lattice.

DOI
10.1088/1367-2630/11/5/055027

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We derive a novel lattice Hamiltonian, the molecular Hubbard Hamiltonian , which describes the essential many-body physics of closed-shell ultracold heteronuclear molecules in their absolute ground state in a quasi-one-dimensional optical lattice. The molecular Hubbard Hamiltonian is explicitly time dependent, making a dynamic generalization of the concept of quantum phase transitions necessary. Using the time-evolving block decimation algorithm to study entangled dynamics, we demonstrate that, in the case of hard-core bosonic molecules at half-filling, the molecular Hubbard Hamiltonian exhibits emergent timescales over which spatial entanglement grows, crystalline order appears and oscillations between rotational states self-damp into an asymptotic superposition. We show that these timescales are non-monotonic functions of the physical parameters describing the lattice.
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The molecular Hubbard Hamiltonian (MHH) naturally arises for ultracold ground-state polar alkali-metal dimer molecules in optical lattices. We show that, unlike ultracold atoms, different molecules display different many-body phases due to intrinsic variances in molecular structure even when the molecular symmetry is the same. We also demonstrate a wide variety of experimental controls on ${}^{1}\ensuremath{\Sigma}$ molecules via external fields, including applied static electric and magnetic fields, an ac microwave field, and the polarization and strength of optical lattice beams. We provide explicit numerical calculations of the parameters of the MHH, including tunneling and direct and exchange dipole-dipole interaction energies, for the molecules 6{}^{6}6Li133{}^{133}133Cs, 23{}^{23}23Na40{}^{40}40K, 87{}^{87}87Rb133{}^{133}133Cs, 40{}^{40}40K87{}^{87}87Rb, and 6{}^{6}6Li23{}^{23}23Na in weak and strong applied electric fields. As case studies of many-body physics, we use infinite-size matrix product state methods to explore the quantum phase transitions from the superfluid phase to half-filled and third-filled crystalline phases in one dimension.
We derive effective lattice models for ultracold bosonic or fermionic nonreactive molecules (NRMs) in an optical lattice, analogous to the Hubbard model that describes ultracold atoms in a lattice. In stark contrast to the Hubbard model, which is commonly assumed to accurately describe NRMs, we find that the single on-site interaction parameter U is replaced by a multichannel interaction, whose properties we elucidate. Because this arises from complex short-range collisional physics, it requires no dipolar interactions and thus occurs even in the absence of an electric field or for homonuclear molecules. We find a crossover between coherent few-channel models and fully incoherent single-channel models as the lattice depth is increased. We show that the effective model parameters can be determined in lattice modulation experiments, which, consequently, measure molecular collision dynamics with a vastly sharper energy resolution than experiments in a free-space ultracold gas.
In this dissertation I present results for lattice boson systems based on quantum Monte Carlo simulations by the Worm algorithm. Lattice bosons can be realized by ultracold gases in optical lattices and, under certain conditions, they are described by the Bose-Hubbard model. I will discuss the ground state, i.e. the phase diagram and properties of elementary excitations, as well as finite temperature thermodynamic properties of the two- and three-dimensional systems. These results provide bench-marks and guidance for experimentalists. I will also discuss the phase diagram of the one-dimensional hard-core Bose-Hubbard model with three-body interactions. The model is realized by cold polar molecules in optical lattices. The ground state presents a novel solid phase at unconventional filling, characterized by coexisting charge density wave and bond orders. Such phase has not been predicted by the preexisting mean field theory.
An ultracold gas of heteronuclear alkali-metal dimer molecules with hyperfine structure loaded into a one-dimensional optical lattice is investigated. The hyperfine molecular Hubbard Hamiltonian (HMHH), an effective low-energy lattice Hamiltonian, is derived from first principles. The large permanent electric dipole moment of these molecules gives rise to long-range dipole-dipole forces in a dc electric field and allows for transitions between rotational states in an ac microwave field. Additionally, a strong magnetic field can be used to control the hyperfine degrees of freedom independently of the rotational degrees of freedom. By tuning the angle between the dc electric and magnetic fields and the strength of the ac field, it is possible to control the number of internal states involved in the dynamics as well as the degree of correlation between the spatial and internal degrees of freedom. The HMHH's unique features have direct experimental consequences such as quantum dephasing, tunable complexity, and the dependence of the phase diagram on the molecular state.
We present a wide array of quantum measures on numerical solutions of one-dimensional Bose- and Fermi-Hubbard Hamiltonians for finite-size systems with open boundary conditions. Finite-size effects are highly relevant to ultracold quantum gases in optical lattices, where an external trap creates smaller effective regions in the form of the celebrated ``wedding cake'' structure and the local density approximation is often not applicable. Specifically, for the Bose-Hubbard Hamiltonian we calculate number, quantum depletion, local von Neumann entropy, generalized entanglement or Q measure, fidelity, and fidelity susceptibility; for the Fermi-Hubbard Hamiltonian we also calculate the pairing correlations, magnetization, charge-density correlations, and antiferromagnetic structure factor. Our numerical method is imaginary time propagation via time-evolving block decimation. As part of our study we provide a careful comparison of canonical versus grand canonical ensembles and Gutzwiller versus entangled simulations. The most striking effect of finite size occurs for bosons: we observe a strong blurring of the tips of the Mott lobes accompanied by higher depletion, and show how the location of the first Mott lobe tip approaches the thermodynamic value as a function of system size.