Probing $n$-spin correlations in optical lattices

We propose a technique to measure multispin correlation functions of arbitrary range as determined by the ground states of spinful cold atoms in optical lattices. We show that an observation of the atomic version of the Stokes parameters, using focused lasers and microwave pulsing, can be related to $n$-spin correlators. We discuss the possibility of detecting not only ground state static spin correlations, but also time-dependent spin-wave dynamics as a demonstrative example using our proposed technique.

Figure 1

(Color online) Noise correlation plotted as a function of wave vector of the one-dimensional $XXZ$ model. The solid (dashed) line corresponds to a ground state with $\eta =0$, long-range ($\eta =2$, short-range) spin correlator. The amplitudes are normalized by the maximum for antiferromagnetic order $(\eta =0)$. The number of atoms in panels (a) and (b) are $N=20$ and $N=200$, respectively.

Figure 2

(Color online) Schematic plot of the experimental procedure used to measure $n$-spin correlation functions. $A$ indicates one of the target atoms. $B$ indicates all other atoms. (a) Target atoms $A$ are transferred to a suitable measurement basis using a combination of focused lasers and microwave pulses [see step (II)]. (b) Atoms at state $\mid \downarrow ⟩$ are transferred to state $\mid 2⟩$ by two microwave $\pi$ pulses, then atoms at state $\mid \uparrow ⟩$ are transferred to state $\mid \downarrow ⟩$ by another microwave $\pi$ pulse [see step (III)]. (c) Target atoms $A$ are transferred back to state $\mid \uparrow ⟩$ from $\mid \downarrow ⟩$ [see step (IV)]. (d) A detection laser is applied to detect the probability of finding target atoms at $\mid \uparrow ⟩$ [see step (IV)].

Figure 3

(a) Time evolution of the probability for the target atoms $A$ to be in the excited state $\mid 3⟩$. (b) The number of scattering photons $n_p^A$ versus time for atoms $A$ at state $\mid \uparrow ⟩$. (c) The number of scattering photons $n_p^B$ versus time for atoms ($A$ or $B$) at states $\mid \downarrow ⟩$ and $\mid 2⟩$. (d) The number of scattering photons $n_p^A$ versus the Rabi frequency of the resonant laser for atoms $A$ at state $\mid \uparrow ⟩$. (e) The number of scattering photons $n_p^B$ versus the Rabi frequency of the resonant laser for atoms ($A$ or $B$) at states $\mid \downarrow ⟩$ and $\mid 2⟩$. The time period for (d) and (e) is $50∕\Gamma$.

Figure 4

Site occupation probability (a) and density-density correlation (b) with respect to scaled spin interaction parameter $\alpha \propto$ time for $N=30$ and $j=0$ [(a) solid line]; $j=1$ [(a) dashed line]; $j=0$, $j^{\prime }=3$ [(b) dashed line]; and $j=2$, $j^{\prime }=3$ [(b) solid line]. The site index $j$ ranges from $-N$ to $N$.