Zoo of Quantum Phases and Excitations of Cold Bosonic Atoms in Optical Lattices

Quantum phases and phase transitions of weakly to strongly interacting bosonic atoms in deep to shallow optical lattices are described by a single multiorbital mean-field approach in real space. For weakly interacting bosons in one dimension, the critical value of the superfluid to Mott insulator (MI) transition found is in excellent agreement with many-body treatments of the Bose-Hubbard model. For strongly interacting bosons, (i) additional MI phases appear, for which two (or more) atoms residing in each site undergo a Tonks-Girardeau-like transition and localize, and (ii) on-site excitation becomes the excitation lowest in energy. Experimental implications are discussed.

Figure 1

Quantum phase transition from the SF to MI(1) phase and corresponding orbitals $\phi$ and densities $\rho$ (shifted lower curves) for weakly interacting bosons in optical lattice with $V_0=25E_R$ and $N_w=102$ sites. Orbitals and densities are normalized (on the segment of length $2\pi$) and are plotted against the site index i. The optical lattice is illustrated for guidance by the background sinusoidal curve. (a) The phase transition is described by the intersection of the SF and MI(1) energy per particle curves, $\epsilon$, which occurs at ${\gamma }_c=0.007777(3)$. (b) Orbital and density of the SF phase for $\gamma =0.00776002$. (c) Orbitals and density of the MI(1) phase for $\gamma =0.00777731$, slightly above ${\gamma }_c$ (shown are 3 adjacent orbitals). (d) Orbitals and density of the MI(1) phase for strongly interacting bosons in shallower optical lattice with $V_0=E_R$ for $\gamma =3.491$, slightly above the corresponding ${\gamma }_c=3.4(9)$ (shown are 3 adjacent orbitals).

Figure 2

Quantum phase transition from the MI(2) to MI(1,1) phase and corresponding orbitals $\phi$ and densities $\rho$ (shifted lower curves) for weakly interacting bosons in optical lattice with $V_0=25E_R$ and $N_w=51$ sites. Orbitals and densities are normalized (on the segment of length $2\pi$) and are plotted against the site index i. The optical lattice is illustrated for guidance by the background sinusoidal curve. (a) The phase transition is described by the intersection of the MI(2) and MI(1,1) energy per particle curves, $\epsilon$, which occurs at ${\gamma }_c=12.7(6)$. (b) Orbitals and density of the MI(2) phase for $\gamma =12.7418$ (shown are 3 adjacent orbitals). (c) Orbitals and density of the MI(1,1) phase for $\gamma =12.7609$, slightly above ${\gamma }_c$ (shown are 6 adjacent orbitals). Notice the on-site minima in the density of the MI(1,1) phase in comparison to the standard MI(2) phase.

Figure 3

On-site self-consistent excited state of the MI(1) phase in optical lattice with $V_0=25E_R$ and $N_w=102$ sites for strongly interacting bosons, $\gamma =76.64$. Orbitals and density (shifted lower curve) are normalized (on the segment of length $2\pi$) and are plotted against the site index i. The optical lattice is illustrated for guidance by the background sinusoidal curve. Shown are the excitation site and 4 adjacent orbitals. For the present $\gamma$, this is the excitation lowest in energy, accommodating the energy gap of the MI(1) phase calculated here to be $0.90\sqrt{4V_0{E}_R}$.