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arXiv:1405.0966 · Physical Review A

Trimer superfluid induced by photoassocation on the state-dependent optical lattice

Wanzhou Zhang, Ran Li, W. X. Zhang, Cheng-Bo Duan, T. C. Scott

Published Sep 22, 2014• 9 citations
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Abstract

We use the mean-field method, the quantum Monte Carlo method and the density-matrix-renormalization-group method to study the trimer superfluid phase and the quantum phase diagram of the Bose-Hubbard model in a one-dimensional optical lattice, with an explicit trimer-tunneling term. Theoretically, we derive the explicit trimer hopping terms, such as ${a}_{i}^{3\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{3}$, by the Schrieffer-Wolf transformation. In practice, the trimer superfluid described by these terms is driven by photoassociation. The phase transition between the trimer superfluid phase and other phases are also studied. Without the on-site interaction, the phase transition between the trimer superfluid phase and the Mott insulator phase is continuous. Turning on the on-site interaction, the phase transitions are first order with Mott insulators of atoms filling 1 and 2. With nonzero atom tunneling, the phase transition is first order from the atom superfluid to the trimer superfluid. In the trimer superfluid phase, the winding numbers can be divided by three without any remainder. The power-law decay exponent is $1/2$ for the nondiagonal correlation ${a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}3}{a}_{j}^{3}$, i.e., the same as the exponent of the correlation ${a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}$ in hardcore bosons. The density-dependent atom-tunneling term ${n}_{i}^{2}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}$ and pair-tunneling term ${n}_{i}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}2}{a}_{j}^{2}$ are also studied. With these terms, the phase transition from the empty phase to atom superfluid is first order and different from the cases without the density-dependent terms. The effects of finite three-body interactions and finite temperatures are also studied. Our results will be helpful in realizing the trimer superfluid by a cold-atom experiment.

DOI
10.1103/physreva.90.033622

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We use the mean-field method, the quantum Monte Carlo method and the density-matrix-renormalization-group method to study the trimer superfluid phase and the quantum phase diagram of the Bose-Hubbard model in a one-dimensional optical lattice, with an explicit trimer-tunneling term. Theoretically, we derive the explicit trimer hopping terms, such as ai3\ifmmode†\else\textdagger\fiaj3{a}_{i}^{3\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{3}ai3\ifmmode†\else\textdagger\fi​aj3​, by the Schrieffer-Wolf transformation. In practice, the trimer superfluid described by these terms is driven by photoassociation. The phase transition between the trimer superfluid phase and other phases are also studied. Without the on-site interaction, the phase transition between the trimer superfluid phase and the Mott insulator phase is continuous. Turning on the on-site interaction, the phase transitions are first order with Mott insulators of atoms filling 1 and 2. With nonzero atom tunneling, the phase transition is first order from the atom superfluid to the trimer superfluid. In the trimer superfluid phase, the winding numbers can be divided by three without any remainder. The power-law decay exponent is 1/21/21/2 for the nondiagonal correlation ai\ifmmode†\else\textdagger\fi3aj3{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}3}{a}_{j}^{3}ai\ifmmode†\else\textdagger\fi3​aj3​, i.e., the same as the exponent of the correlation ai\ifmmode†\else\textdagger\fiaj{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}ai\ifmmode†\else\textdagger\fi​aj​ in hardcore bosons. The density-dependent atom-tunneling term ni2ai\ifmmode†\else\textdagger\fiaj{n}_{i}^{2}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}ni2​ai\ifmmode†\else\textdagger\fi​aj​ and pair-tunneling term niai\ifmmode†\else\textdagger\fi2aj2{n}_{i}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}2}{a}_{j}^{2}ni​ai\ifmmode†\else\textdagger\fi2​aj2​ are also studied. With these terms, the phase transition from the empty phase to atom superfluid is first order and different from the cases without the density-dependent terms. The effects of finite three-body interactions and finite temperatures are also studied. Our results will be helpful in realizing the trimer superfluid by a cold-atom experiment.
The Bose–Hubbard model with an effective off-site three-body tunneling, characterized by jumps towards one another, between one atom on a site and a pair atoms on the neighborhood site, is studied systematically on a one-dimensional (1D) lattice, by using the density matrix renormalization group method. The off-site trimer superfluid, condensing at momentum k = 0 , emerges in the softcore Bose–Hubbard model but it disappears in the hardcore Bose–Hubbard model. Our results numerically verify that the off-site trimer superfluid phase derived in the momentum space from [ Phys. Rev. A 81 , 011601(R) (2010)] is stable in the thermodynamic limit. The off-site trimer superfluid phase, the partially off-site trimer superfluid phase and the Mott insulator phase are found, as well as interesting phase transitions, such as the continuous or first-order phase transition from the trimer superfluid phase to the Mott insulator phase. Our results are helpful in realizing this novel off-site trimer superfluid phase by cold atom experiments.
By the photoassociation method, the trimer superfluid phase disappears in the one-dimensional state-dependent optical lattice if the ratio of the three-body interaction WWW to the trimer tunneling JJJ is kept at W/J=12W/J=12W/J=12 [W. Z. Zhang et al., Phys. Rev. A 90, 033622 (2014)]. To search for a trimer superfluid and trimer supersolid, we load the cold atom into two-dimensional lattices, whose coordinate number zzz and kinetic energy $\ensuremath{-}zJ$ are respectively larger and lower than those of a one-dimensional lattice. Herein, we study the Bose-Hubbard model, which has an additional trimer tunneling term, a three-body interaction, and a next-nearest repulsion. The on-site trimer and trimer superfluid exist if the on-site two-body repulsion and three-body repulsion are smaller than some thresholds. With atom-tunneling terms, the phase transitions from trimer superfluid phase to both atom superfluid and atom supersolid phases are first order. With W/J=12W/J=12W/J=12, in a one-dimensional lattice, the trimer superfluid phase does not exist at all. In contrast, the trimer superfluid phase exists in the lower density regions $0<\ensuremath{\rho}<2$ on the square lattice if JJJ is not very large. The trimer superfluid phase emerges in a wider range $0<\ensuremath{\rho}<3$ in the triangular lattice, or in the cubic lattice (z=6)(z=6)(z=6). When the three-body interaction is turned on, a trimer supersolid phase emerges due to the classical degeneracy between the quasitrimer solid and the trimer solid being broken by the quantum fluctuation. The phase transitions from the trimer supersolid phase to the quasitrimer solid are first order and the phase transition from the trimer supersolid phase to trimer solid is continuous. Our results, obtained by mean-field and quantum Monte Carlo methods, will be helpful in realizing the trimer superfluid and supersolid by cold-atom experiments.
We study the two-dimensional Bose-Hubbard model in the presence of a three-body interaction term, both at a mean-field level and via quantum Monte Carlo simulations. The three-body term is tuned by coupling the triply occupied states to a trapped universal trimer. We find that, for a sufficiently attractive three-body interaction, the n=2 Mott lobe disappears and the system displays first-order phase transitions separating the n=1 from the n=3 lobes and the n=1 and n=3 Mott insulator from the superfluid. We also analyze the effect of finite temperature and find that transitions are still of first order at temperatures T~J, where J is the hopping matrix element.
For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the lattice attenuation factor. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on-site occupation numbers. In the mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero-temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is, in principle, possible, but completely suppressed at the lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.
By generalizing the traditional single-site strong coupling expansion approach to a cluster one, we study the zero-temperature phase diagram of bosonic atoms in a trimerized Kagomé optical lattice. Some new features are present in this system. Due to the strong intra-trimer hopping interaction, there will be a new Mott insulator (MI), which is by definition incompressible but with a fractional filling per trimer. This is different from the traditional MI, which has an integral filling and originates only from the repulsive interaction between particles. We investigate the MI-to-superfluid transition and the nature of the fractional MI by calculating the critical exponents of phase transitions and the low-lying energy excitation spectra of quasiparticles (quasihole). We will show how the low-energy properties of this system can be understood qualitatively as a Bose-Hubbard model in triangular lattice from the point of view of the cluster strong coupling expansion. We also discuss how our results are related to experiment by studying the Bragg spectroscopy.
The Bose-Hubbard Hamiltonian of spin-2 cold bosons with repulsive interaction in an optical lattice is proposed. After neglecting the hopping term, the site-independent Hamiltonian and its energy eigenvalues and eigenstates are obtained. We consider the hopping term as a perturbation to do the calculations in second order and draw the phase diagrams for different cases. The phase diagrams show that there is a phase transition from Mott insulator with integer number bosons to superfluid when the ratio c0/t{c}_{0}/tc0​/t (c0{(c}_{0}(c0​ is the spin-independent on-site interaction and t is the hopping matrix element between adjacent lattice sites) is decreased to a critical value and that there is a different phase boundary between superfluid and Mott-insulator phase for different Zeeman level component in some ground states. We find that the position of phase boundary for different Zeeman level component is related to its average population in the Mott ground state.