Effective spin-spin interactions in bilayers of Rydberg atoms and polar molecules

We show that indirect spin-spin interactions between effective spin-1/2 systems can be realized in two parallel one-dimensional optical lattices loaded with polar molecules and/or Rydberg atoms. The effective spin can be encoded into low-energy rotational states of polar molecules or long-lived states of Rydberg atoms, tightly trapped in a deep optical lattice. The spin-spin interactions can be mediated by Rydberg atoms, placed in a parallel shallow optical lattice, interacting with the effective spins by charge-dipole (for polar molecules) or dipole-dipole (for Rydberg atoms) interaction. Indirect $XX$, Ising, and $XXZ$ interactions with interaction coefficients $J^{\perp }$ and $J^{zz}$ sign varying with interspin distance can be realized, in particular, the $J_1-J_2 XXZ$ model with frustrated ferro-(antiferro-)magnetic nearest (next-nearest) neighbor interactions.

Figure 1

Setup schematic. (a) Illustration of the bilayer setup, in which spin-encoding polar molecules or Rydberg atoms are trapped in a deep optical lattice or trap array, and the mediating Rydberg atom(s) is placed in a shallow optical lattice such that its spatial wave function is delocalized over several sites, and it can simultaneously interact with several spins. (b) Geometry of the setup: An effective $m\mathrm{th}$ spin with a dipole moment ${\vec{d}}_m$ interacts with a mediator atom via charge-dipole interaction Eq. (1a) in the case of the polar molecule spin encoding or dipole-dipole interaction Eq. (1b) in the case of the Rydberg atom spin encoding. The distance between the two parallel lattices is $|\vec{\rho }|, {\vec{R}}_{qm}$ is the vector connecting the ionic core of the $q\mathrm{th}$ mediator atom to the spin, $X_{qm}=X_q-X_m$ is the distance between the mediator atom and the spin along the $X$ axis, and $\vec{r}$ is the vector connecting the mediator Rydberg electron and the ionic core.

Figure 2

Schemes of different types of spin encoding into low-energy rotational states of polar molecules (left side) or long-lived states of Rydberg atoms (right side), allowing us to model indirect interactions in a 1D effective spin chain: (a) Spin encoding into low-energy rotational states of a polar molecule and near resonant interaction between molecular rotational and mediator Rydberg atomic states, allowing us to model $XX$ type of interaction. (b) Example of spin encoding into long-lived states of a Rydberg atom. (c) Polar molecule rotational states not connected by a dipole-allowed transition can be used as spin states to realize effective Ising-type spin-spin interaction. Spin state dipole moments can be induced by near-resonant coupling to opposite parity rotational states by MW fields. (d) The same as in (c) for spins encoded in Rydberg states. (e) Spin encoding in nearly degenerate rotational sublevels of a polar molecule such that $|E_{\mathrm{spin}}|\ll |{\mathcal{E}}_{n{p}_{j^{\prime }}}-{\mathcal{E}}_{ns}|$ is satisfied allows us to realize an effective $XXZ$ interaction. (f) The same as (e) for Rydberg atom spin encoding. The Rydberg encoded spin states can be initially additionally split by an external magnetic field.

Figure 3

Spin encoding in low-energy rotational states of polar molecules near resonantly interacting with a mediator atom Rydberg transition allows us to model $XX$ interaction. Numerically calculated from Eqs. (21): (a) Interaction coefficient $J_{im}^{\perp }$ and (b) effective magnetic field $b_i^z$ for spins encoded in rotational states of NaCs $\left|\downarrow \right.〉=\left|J=0,m_J=0\right.〉, \left|\uparrow \right.〉=\left|J=1,m_J=0\right.〉$, near resonant with the Rb transition $\left|99p_{3/2},m_j=\pm 1/2,3/2\right.〉-\left|99s_{1/2},m_j=1/2\right.〉$. The mediator atoms are assumed to be initially prepared in the superatom state Eq. (11) in the $\left|99s_{1/2},m_j=1/2\right.〉$ internal and in a BEC motional state with $k_0=0, {\nu }_0=1$. In Eqs. (21), summation over quasimomenta in the first Brillouin zone for $\nu =1,\cdots ,5$ lowest Bloch bands of the mediator lattice with the depth $V_0=-E_{\mathrm{rec}}$ was performed. The red (circles) and blue (squares) curves in (a) correspond to $B_M=0$ G and $B_M=0.92$ G, respectively; in (b) the effective magnetic field was calculated for $B_M=0$ G. Setup dimensions: $\rho =1.5\mu \mathrm{m}$, spin and mediator lattice periods $L_{\mathrm{spin}}=L_{\mathrm{at}}=1.5\mu \mathrm{m}, N=N_{\mathrm{at}}=100$. The calculations were done for a spin in the center of the array $m=0$.

Figure 4

In the $XX$ model next-nearest and more distant interaction strengths relative to the nearest-neighbor interaction strength can be controlled by (a) the width of the quasimomenta distribution of the initial superposition of Bloch states of mediator atoms in Eq. (14) and (b) external dc magnetic field. (a) Assuming a Gaussian initial distribution of the quasimomenta in the second Bloch band $|c_{k_{0{\nu }_0=2}}{|}^2\sim e^{-k_0^2/{\kappa }_0^2}$, the ratios of interaction coefficients $J_{im}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ can be controlled by the distribution width ${\kappa }_0$: $J_{m+2,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (red solid curve), $J_{m+3,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (green dashed curve), $J_{m+4,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (blue dotted curve), and $J_{m+5,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (pink dashed-dotted curve). (b) The external magnetic field can tune the energies of the initial $\left|n{s}_{1/2},m_j=\pm 1/2\right.〉$ and virtually excited $\left|n{p}_{3/2},m_j=\pm 1/2,3/2\right.〉$ mediator states, which can be used to tune the ratios of interaction coefficients $J_{im}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$.

Figure 5

Spin encoding in MW dressed states of a Rydberg atom interacting with a MW dressed mediator Rydberg atom allows us to realize the $XXZ$ interaction. (a) Spin-1/2 system can be encoded in long-lived $\left|\tilde{n}{s}_{1/2},m_j=\pm 1/2\right.〉$ Rydberg states dressed by MW fields near resonantly coupling them to $\left|\tilde{n}{p}_{1/2},m_j=\pm 1/2\right.〉$ states. The sublevels of the $\tilde{n}{p}_{1/2}$ and $\tilde{n}{s}_{1/2}$ states can be additionally split by a dc magnetic field to make the spin transition frequency different from frequencies between other dressed states. (b) The $J_{im}^{\perp }$ and $J_{im}^{zz}$ interaction coefficients can be made $\sim 100$ kHz by using mediator virtual transitions between the initial $\left|n{s}_{1/2},m_j=-1/2\right.〉$ and dressed by a MW field states ${\left|\pm \right.〉}_{\mathrm{med}}$, which can be separated by $\lesssim 100$ MHz.

Figure 6

Tilting the effective spins with respect to the axis of the spin array allows us to cancel direct dipole-dipole interaction in the case of Rydberg atom spin encoding. In the case of spin encoding into long-lived Rydberg states the direct dipole-dipole interaction between the effective spins can be canceled by tilting the spin quantization axis $Z$ with respect to the line, connecting the spins. For the angle $\theta =arccos(1/\sqrt{3})$ between the $-Z$ and $X^{\prime }$ axes the resonant part of the direct dipole-dipole interaction is zero.

Figure 7

Interaction coefficients in the case of spin encoding into long-lived Rydberg states, allowing us to realize $XXZ$ interaction: (a) $J_{im}^{\perp }$ (circles, red curve), $J_{im}^{zz}$ (squares, blue curve) and (b) effective magnetic field $b_i^z$ for spins encoded in $\left|\uparrow \right.〉={\left|+\right.〉}_{\uparrow }=(\left|\tilde{n}{p}_{1/2},m_j=1/2\right.〉+\left|\tilde{n}{s}_{1/2},m_j=1/2\right.〉)/\sqrt{2}, \left|\downarrow \right.〉={\left|+\right.〉}_{\downarrow }=(\left|\tilde{n}{p}_{1/2},m_j=-1/2\right.〉+\left|\tilde{n}{s}_{1/2},m_j=-1/2\right.〉)/\sqrt{2}$ states of Rb with $\tilde{n}=50$. The mediator atoms are initially prepared in a superatom state (11) in the $\left|100s_{1/2},m_j=-1/2\right.〉$ internal state and a BEC motional state with $k_0=0, {\nu }_0=1$. In calculations of $J_{im}^{\perp ,zz}, b_i^z$ of Eqs. (22) quasimomenta in the first Brillouin zone were summed for $\nu =1,\cdots ,5$ lowest Bloch bands of the mediator atom lattice. Setup dimensions: interlayer distance $\rho =7\mu \mathrm{m}$, spin and mediator lattice periods $L_{\mathrm{spin}}=L_{\mathrm{at}}=7\mu \mathrm{m}$. The calculations were done for a spin in the center of the array $m=0$. Other parameters were as follows: $E_{{ns}_{1/2},m_j=1/2}-E_{{ns}_{1/2},m_j=-1/2}=E_{\tilde{n}{s}_{1/2},m_j=1/2}-E_{\tilde{n}{s}_{1/2},m_j=-1/2}=148.5$ MHz, ${\mathrm{\Omega }}_{\uparrow }=2.5$ MHz, ${\mathrm{\Omega }}_{\downarrow }=1$ MHz, ${\mathrm{\Omega }}_{\mathrm{med}}=2.655$ MHz, ${\mathrm{\Delta }}_{\uparrow }={\mathrm{\Delta }}_{\downarrow }={\mathrm{\Delta }}_{\mathrm{med}}=0$, resulting in $E_{+\mathrm{med}}-E_{{ns}_{1/2},m_j=-1/2}=151.155$ MHz, $E_{-\mathrm{med}}-E_{{ns}_{1/2},m_j=-1/2}=145.845$ MHz.

Figure 8

In the $XXZ$ model with interaction coefficients Eqs. (14) and (22) the strengths of next-nearest and more distant neighbors relative to the nearest-neighbor interaction can be controlled by the width of the quasimomenta distribution of the initial superposition of Bloch states of mediator atoms. Assuming the atoms initially prepared in the lowest Bloch band with the Gaussian initial distribution $|c_{k_{0{\nu }_0=1}}{|}^2\sim e^{-k_0^2/{\kappa }_0^2}$ of quasimomenta, the ratios of interaction coefficients $J_{im}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ can be controlled by the distribution width ${\kappa }_0$: $J_{m+2,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (red solid curve), $J_{m+3,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (green dashed curve), $J_{m+4,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (blue dotted curve), and $J_{m+5,m}^{\perp }/\left|J_{m+1,m}^{\perp }\right|$ (pink dashed-dotted curve). The interaction coefficients shown in Fig. 7 correspond to ${\kappa }_0=0$.

Figure 9

Angles of the vectors ${\vec{R}}_{qm}$ and $\vec{r}$ in the case of a general orientation of the Rydberg atom with respect to the molecule.

Figure 10

Bloch energies of the lowest five bands of the 1D optical lattice described by the trapping potential $V\left(X_q\right)=V_0{cos}^2\left(K_{\mathrm{at}}{X}_q\right)$ with $V_0=-E_{\mathrm{rec}}$.