Spin-1 atoms in optical superlattices: Single-atom tunneling and entanglement

We examine spinor Bose-Einstein condensates in optical superlattices theoretically using a Bose-Hubbard Hamiltonian that takes spin effects into account. Assuming that a small number of spin-1 bosons is loaded in an optical potential, we study single-particle tunneling that occurs when one lattice site is ramped up relative to a neighboring site. Spin-dependent effects modify the tunneling events in a qualitative and quantitative way. Depending on the asymmetry of the double well, different types of magnetic order occur, making the system of spin-1 bosons in an optical superlattice a model for mesoscopic magnetism. We use a double-well potential as a unit cell for a one-dimensional superlattice. Homogeneous and inhomogeneous magnetic fields are applied, and the effects of the linear and the quadratic Zeeman shifts are examined. We also investigate the bipartite entanglement between the sites and construct states of maximal entanglement. The entanglement in our system is due to both orbital and spin degrees of freedom. We calculate the contribution of orbital and spin entanglements and show that the sum of these two terms gives a lower bound for the total entanglement.

Figure 1

Two spin-1 bosons with antiferromagnetic ordering in a double-well potential. Here, $n_R$ is the occupation number of the right well, and $\epsilon$ characterizes the energy offset between the two wells ($t/U_0=0.05$ and $U_2/U_0=0.1$). Depending on the total spin of the system, bosonic staircase transitions occur at different bias voltages. Note that both states with $S_{\mathrm{tot}}=0$ (red, solid line) and $S_{\mathrm{tot}}=2$ (blue, dashed line) have symmetric orbital wave functions with respect to particle exchange. The difference in the occupation numbers arises due to spin-dependent interactions and not due to a different orbital symmetry of the states. Thus, a measurement of the spin-dependent bosonic staircases provides a demonstration of mesoscopic magnetism.

Figure 2

Bosonic staircase for three spin-1 bosons with antiferromagnetic ordering in a double-well potential ($t/U_0=0.05$ and $U_2/U_0=0.1$): $S_{\mathrm{tot}}=1$ (red, solid line) and $S_{\mathrm{tot}}=3$ (blue, dashed line). (Inset) Variance in the particle number in the left well for the step around $\epsilon =0$.

Figure 3

A possible way to separate the $S_{\mathrm{tot}}=2$ spin component from the $S_{\mathrm{tot}}=0$ spin component in the case of antiferromagnetic interactions between the atoms: The potentials in the $x$ and $y$ directions are manipulated separately. In the first step (b), the energy offset between the double wells is lifted until the bosons combining with the total spin $S_{\mathrm{tot}}=0$ separate while the bosons belonging to $S_{\mathrm{tot}}=2$ still remain in the same site. (c) Next, the wells are separated by a large potential barrier, and tunneling is suppressed. (d) An additional laser is switched on, and the bosons coupling to $S_{\mathrm{tot}}=2$ distribute in the resulting double well. The switching is assumed to happen adiabatically such that the system can be regarded to be in the ground state at every instant.

Figure 4

Energy spectrum of two spin-1 bosons in a double well with strong tunneling ($t/U_0=0.5$ and $U_2/U_0=0.1$): $S_{\mathrm{tot}}=0$ subspace (red, solid lines), $S_{\mathrm{tot}}=2$ (blue, dashed lines), and $S_{\mathrm{tot}}=1$ (dotted line).

Figure 5

Energy spectrum of two spin-1 bosons in a double well with weak tunneling ($t/U_0=0.05$ and $U_2/U_0=0.1$). Color code as in Fig. 4.

Figure 6

Linear Zeeman shift of the energy levels in the energy spectrum of two spin-1 bosons ($B/U_0=0.05,t/U_0=0.05$, and $U_2/U_0=0.1$). The energy levels of Fig. 5 split into spin multiplets. The red arrows denote ground-state level crossings. Color code as in Fig. 4.

Figure 7

Critical magnetic field $p=g{\mu }_{B}B$ above which the staircase for two bosons shows a discontinuous behavior signifying spin-flip transitions ($U_2/U_0=0.1$).

Figure 8

Two spin-1 bosons with antiferromagnetic ordering in a double-well potential ($t/U_0=0.05$ and $U_2/U_0=0.1$). (Dashed lines) $S_z^{\mathrm{tot}}=0$ for different magnetic fields $q=q_0{B}^2$ (short dashes $q/U_0=0.2$, long dashes $q/U_0=0.1$, and medium-sized dashes $q/U_0=0$). (Solid line) $S_z^{\mathrm{tot}}=2$. In this case, the staircase does not depend on the magnetic field. The difference in this staircase to the ones with $S_z^{\mathrm{tot}}=0$, which is the main manifestation of mesoscopic magnetism, persists in the presence of the quadratic Zeeman effect.

Figure 9

Expectation value $\langle S_{\mathrm{tot}}^2\rangle$ of the system for four bosons for different magnetic fields $q=q_0{B}^2$ ($S_z^{\mathrm{tot}}=0,t/U_0=0.05$, and $U_2/U_0=0.1$).

Figure 10

EOF between two wells for two bosons with antiferromagnetic interactions ($t/U_0=0.1$ and $U_2/U_0=0.1$) for the total spin $S_{\mathrm{tot}}=0$.

Figure 11

EOF between two wells for four particles in the presence of a magnetic field ($S_z^{\mathrm{tot}}=0$, $t/U_0=0.05$ and $U_2/U_0=0.1$). For $q>0$, even small fields will eliminate the contribution of the spin degrees of freedom to the entanglement.

Figure 12

EOF between two wells for two bosons with very weak tunneling and antiferromagnetic interactions ($t/U_0=0.001$ and $U_2/U_0=0.1$) for the total spin $S_{\mathrm{tot}}=0$.

Figure 13

EOF between two symmetric wells ($\varepsilon =0$) for different values of $U_2$, i.e., different spin interactions (solid line $U_2/U_0=0.2$, dotted line $U_2/U_0=0.01$, and dashed line $U_2/U_0=-0.2$).

Figure 14

EOF, $E_{\mathrm{spin}}$, and $E_{\mathrm{orbital}}$ between two symmetric wells for $U_2/U_0=0.1$.

Figure 15

EOF between two wells for three bosons with antiferromagnetic interactions ($t/U_0=0.1$ and $U_2/U_0=0.1$) for the total spin $S_{\mathrm{tot}}=1$.

Figure 16

EOF between two wells for three bosons with antiferromagnetic interactions ($t/U_0=0.005$ and $U_2/U_0=0.1$) for the total spin $S_{\mathrm{tot}}=1$.

Figure 17

EOF, $E_{\mathrm{spin}}$, $E_{\mathrm{orbital}}$, and $E_{\mathrm{spin}}+E_{\mathrm{orbital}}$ between two symmetric wells for three bosons with antiferromagnetic interactions ($U_2/U_0=0.1$) in a symmetric double-well potential ($\varepsilon /U_0=0$).