Probing magnetic order in ultracold lattice gases

A forthcoming challenge in ultracold lattice gases is the simulation of quantum magnetism. That involves both the preparation of the lattice atomic gas in the desired spin state and the probing of the state. Here we demonstrate how a probing scheme based on atom-light interfaces gives access to the order parameters of nontrivial quantum magnetic phases, allowing us to characterize univocally strongly correlated magnetic systems produced in ultracold gases. This method, which is also nondemolishing, yields spatially resolved spin correlations and can be applied to bosons or fermions. As a proof of principle, we apply this method to detect the complete phase diagram displayed by a chain of (rotationally invariant) spin-1 bosons.

Figure 1

Schematic detection setup. Atoms placed in an optical lattice of periodicity $d/2$ (not shown) are illuminated by a laser beam in a standing-wave configuration (yellow region) shifted by $\alpha$ from the optical lattice configuration. The output light is redirected through a polarimeter that measures its polarization through a homodyne detection (HD).

Figure 2

(a) Phase diagram of the model in Eq. (6). (b) The quantity $\varepsilon (\pi /3d,\alpha )$ (in units of ${\kappa }^2$) for $\theta =0.3\pi$ for $L=132$ as a function of $\alpha$.

Figure 3

(left) The function $\varepsilon (k_P,\alpha )$ (in units of ${\kappa }^2$) for different values of $\theta$ in the three phases for $L=132$: top, $\theta =-0.5\pi$ (dimer); middle, $\theta =0$ (Haldane); bottom, $\theta =0.3\pi$ (critical). (right) The same plots but restricted to $\alpha =0$.

Figure 4

(a) The quantity ${\mathcal{C}}_{\varepsilon }=\Delta \varepsilon (\pi /3d,5/4,1/2)$ (circles), in units of ${\kappa }^2$, as a function of $\theta$ for $L=132$ and the rescaled magnetic structure factor $[S(2\pi /3d)-1]/5$ (squares). Inset shows finite-size scaling of ${\mathcal{C}}_{\varepsilon }$ for $\theta =0.2\pi$. The solid line scales as $1/L$. (b) The quantity ${\mathcal{D}}_{\varepsilon }=\Delta \varepsilon (\pi /4d,1/2,3/2)$ (circles), in units of ${\kappa }^2$, for different values of $\theta$ for $L=132$ and the rescaled dimer order parameter $D_R=D/5.38$. In both plots, we distinguish the model phases with different shading: horizontal lines, dimer; no shading, Haldane; oblique lines, critical.