Strongly interacting two-component coupled Bose gas in optical lattices

Two-component coupled Bose gas in a 1D optical lattice is examined. In addition to the postulated Mott insulator and superfluid phases, multiple bosonic components manifest spin degrees of freedom. Coupling of the components in the Bose gas leads to substantial changes in the previously observed spin phases, giving rise to a new effective spin Hamiltonian and unraveling remarkable spin correlations. The system in the absence of coupling exhibits ferromagnetic and nonferromagnetic spin phases for on-site intracomponent interaction stronger than intercomponent interaction. Upon introduction of coupling, the phase transition switches from first to second order. For comparable on-site inter- and intracomponent interactions, with coupling, instead of one, two spin phases emerge with a second-order phase transition. Exact diagonalization and variational Monte Carlo with stochastic minimization on the entangled-plaquette state bestow a unique and enhanced perspective on the system beyond the scope of a mean-field treatment.

Figure 1

Two-component Bose-Hubbard model with on-site inter- ($U_{ab}$) and intra- ($U$) component interaction ($U\gg U_{ab}$ and $U\sim U_{ab}$) with nearest-neighbor tunneling coefficients $t_a$ and $t_b$. The optical lattice is tilted via external inhomogeneous static electric or magnetic field. Inset: To the system is introduced laser-assisted intercomponent coupling: $J_1$ (same site) and $J_2$ (nearest neighbor). The two components are visualized as two levels of an atom.

Figure 2

Schematic of the effective spin Hamiltonian. (a) The low- and high-energy subspaces for a two-site system, roughly separated by the on-site interaction strength and coupled by the intercomponent coupling and intracomponent tunneling coefficients. (b) The Hamiltonian with low- and high-energy subspaces forming the diagonal with the off-diagonal coupling and tunneling coefficients coupling the two subspaces and the effective decoupling of the two subspaces to obtain the effective low-energy spin Hamiltonian.

Figure 3

Schematic of spin phases hypothesized to emerge in the presence of intercomponent coupling ($J_2$). The phases are shown in the deep Mott regime, (a) $U\gg U_{ab}$ and (b) $U\sim U_{ab}$, and presented as a function of scaled intracomponent tunneling $\frac{t_b}{t_a}$ and intercomponent coupling $\frac{J_2}{t_a}$. The FM phase demonstrates typical behavior, with spin along $x$ or $z$. The non-FM phase demonstrates a typical $z\mathrm{AFM}$, and an unconventional non-FM phase. A key feature of the unconventional phase: a spin along $z$ on one site correlated with a spin along $x$ on another site.

Figure 4

Spin order parameters, describing the spin behavior along [(i) and (ii)] $z$ and [(iii) and (iv)] $x$ at site $i$ in subspace $A$ and $i+1$ in $B$. The orders evaluated via (a) mean-field approximation, (b) exact diagonalization for $N=8$, and (c) ED with the symmetry-breaking term ($\delta h_z=4\times {10}^{-4}{t}_a$) for $N=8$ are shown as a function of scaled tunneling ($\frac{t_b}{t_a}$) and coupling ($\frac{J_2}{t_a}$), in the deep Mott regime $(\frac{U}{U_{ab}}=10,\frac{U_{ab}}{t_a}=20)$.

Figure 5

Spin correlation describing the spin relationship between the nearest-neighbor sites at (a) $\frac{U}{U_{ab}}=10$ and (b) $\frac{U}{U_{ab}}=1.2$ evaluated via (i) mean-field approximation and (ii) exact diagonalization for $N=8$. The correlation is shown as a function of scaled tunneling ($\frac{t_b}{t_a}$) and coupling ($\frac{J_2}{t_a}$), in the deep Mott regime ($\frac{U_{ab}}{t_a}=20$). The dashed curve denotes the mean-field critical values ${\left(\frac{t_b}{t_a}\right)}^C$ from Eq. (5).

Figure 6

Spin correlations describing the spin relationship between site 1 and some site $j$ along (i) $z\mathsf{\text{−}}x$ and (ii) $z$, for (a) $N=12$ via exact diagonalization and (b) $N=20$ via VMC-SM on EPS. The correlations are shown as a function of site $j$ at $\frac{J_2}{t_a}=0.34,\frac{t_b}{t_a}=0.0$ and $\frac{J_2}{t_a}\approx 0.0, \frac{t_b}{t_a}=0.0$ in the deep Mott regime ($\frac{U_{ab}}{t_a}=20, \frac{U}{U_{ab}}=10$).

Figure 7

Probability of the basis states of the (a) effective Hamiltonian and (b) low-energy basis states of the original Hamiltonian (i) ${|a\rangle }_1{|a\rangle }_2$, (ii) ${|b\rangle }_1{|b\rangle }_2$, (iii) ${|a\rangle }_1{|b\rangle }_2$, and (iv) ${|b\rangle }_1{|a\rangle }_2$. The probabilities are determined in the deep Mott regime ($\frac{U_{ab}}{t_a}=20$) $\frac{U}{U_{ab}}=10$ and presented as a function of scaled tunneling ($\frac{t_b}{t_a}$) and coupling ($\frac{J_2}{t_a}$).