Engineering the dynamics of effective spin-chain models for strongly interacting atomic gases

We consider a one-dimensional gas of cold atoms with strong contact interactions and construct an effective spin-chain Hamiltonian for a two-component system. The resulting Heisenberg spin model can be engineered by manipulating the shape of the external confining potential of the atomic gas. We find that bosonic atoms offer more flexibility for independently tuning the parameters of the spin Hamiltonian through interatomic (intraspecies) interaction, which is absent for fermions due to the Pauli exclusion principle. Our formalism can have important implications for control and manipulation of the dynamics of few- and many-body quantum systems; as an illustrative example relevant to quantum computation and communication, we consider state transfer in the simplest nontrivial system of four particles representing exchange-coupled qubits.

Figure 1

(a) A system of four atoms in a 1D trap is initialized by changing the internal state (flipping spin) of one of the atoms. (b) Trapping potential of Eq. (10) for $V_0=50\varepsilon$ and $u=(0,1,2,4)\times u_{\mathrm{p}}$ (top to bottom) with $u_{\mathrm{p}}\simeq 12.5\varepsilon$.

Figure 2

Energy eigenvalues ${\lambda }_n$ of $H_s$ (less the $E_0\mathbf{I}$ term) vs $u$ of Eq. (10), for $N_{\downarrow }=1,N_{\uparrow }=3$ and (a) $\kappa \to \infty$ (fermions), (b) $\kappa =\frac{1}{2}$, and (c) $\kappa =2$. The ratio $J_2/J_1$ of the coupling constants is shown in (d), with dashed lines marking $J_2/J_1=\sqrt{4/3}$ and $u=u_{\mathrm{p}}$, corresponding to the equidistant spectrum in (c). $\lambda$'s and $u$ are in units of $\varepsilon$ and $g=100$.

Figure 3

Fidelity $F\left(t\right)$ of state transfer in a four-spin system with (a) $\kappa \to \infty$ (fermions), (b) $\kappa =\frac{1}{2}$, and (c) $\kappa =2$ [cf. Figs. 2, 2, and 2], for $u=0$ (dashed, blue), $u_{\mathrm{p}}$ (solid, orange), and $2u_{\mathrm{p}}$ (dotted, black). For visual aid, the values of $F=2/3,0.9,0.99$ are marked with thin dashed horizontal lines. Time is in units of $\hslash /\varepsilon$.