Quantum control of spin correlations in ultracold lattice gases

We describe a technique for the preparation of quantum spin correlations in a lattice gas of ultracold atoms using an atom-light interaction of the kind routinely employed in quantum spin polarization spectroscopy. Our method is based on entropic cooling via quantum nondemolition measurement and feedback, and allows the creation and detection of quantum spin correlations, as well as a certain degree of multipartite entanglement which we verify using a generalization of the entanglement witness described previously M. Cramer et al., Phys. Rev. Lett. 106, 020401 (2011). We illustrate the procedure with examples drawn from the bilinear-biquadratic Hamiltonian, which can be modeled by a one-dimensional chain of spin-1 atoms.

Figure 1

Proposed experimental setup. Atoms trapped in an optical lattice (black) are probed with a far-detuned, linearly polarized standing-wave light pulse with wave vector $k_p$ (1), which is outcoupled to record $S_2$ (2). The measurement induces spin squeezing, introducing quantum correlations among the atoms in spatial mode $k=2k_p$. Feedback is applied via optical pumping to set $\langle J_{\alpha }\left(k\right)\rangle =0$ (3). Successive spin components $J_{\alpha }$ can then be separately squeezed by coherently rotating the atomic spin between measurements. Steps (1)–(3) are repeated for a set of wave vectors $k_p$, with interaction strengths $C_p$ weighted according to the cosine Fourier transform of the desired spatial correlation signature (4).

Figure 2

Spin correlation functions for (a) exponential decay and (b) algebraic decay. (i) Real space (in units of lattice sites): The lines from darker to lighter shades are for optical depth $d=\infty$, 300, 99, and 33. The insets are log-linear and log-log plots, respectively, where for clarity we subtract $G(\delta r\to \infty )$, extracted from a fit. In all panels, the $d=\infty$ data is plotted with open circles to allow comparison with fitted curves (green dashed lines). In (a.i), without decoherence, the decay follows an exponential fit. Deviations at large distance become stronger with increasing decoherence (decreasing $d$), and $G(\delta r\to \infty )$ cannot be reliably determined, yielding deviations from straight lines. In (b.i), the curves are straight lines for all values of $d$, and algebraic fits are very accurate. (ii) $k$ space (in units of reciprocal lattice sites): The covariances $\Gamma (k,-k)$ closely follow the cosine Fourier transform of the desired correlation signature $G_{\mathrm{des}}\left(\delta r\right)$, even for small $d$. Deviations occur only primarily at small $k$ where the pulse strength $C_p$ is high. Insets: Entanglement witness $W(\delta r,\varphi )$ for $d=\infty$, minimized at $\varphi =0$. If $W<0$ the system displays nonclassical correlations.

Figure 3

Spin correlations similar to the critical phase of the bilinear-biquadratic Hamiltonian, with an algebraic decay and characteristic period-3 oscillations. The resulting covariances $\tilde{\Gamma }(k,-k)$, shown in the inset, follow the input coupling strengths. These are chosen to heuristically model the structure factor of this phase. Real-space correlations are plotted in units of lattice sites, and $k$-space correlations in units of reciprocal lattice sites.