Transverse-Field Ising Dynamics in a Rydberg-Dressed Atomic Gas

We report on the realization of long-range Ising interactions in a cold gas of cesium atoms by Rydberg dressing. The interactions are enhanced by coupling to Rydberg states in the vicinity of a Förster resonance. We characterize the interactions by measuring the mean-field shift of the clock transition via Ramsey spectroscopy, observing one-axis twisting dynamics. We furthermore emulate a transverse-field Ising model by periodic application of a microwave field and detect dynamical signatures of the paramagnetic-ferromagnetic phase transition. Our results highlight the power of optical addressing for achieving local and dynamical control of interactions, enabling prospects ranging from investigating Floquet quantum criticality to producing tunable-range spin squeezing.

Figure 1

Experimental setup and Rydberg dressing scheme. (a) A cloud of cesium atoms is held in an optical dipole trap and locally illuminated with 319 nm light to generate Ising interactions of characteristic range $r_c$ and strength $J_0$. The quantization axis is set by a 1 G magnetic field $\mathbf{B}$. (b) Energy level diagrams for a single atom (left) and for a pair of atoms (right). (c) Alternating between interactions ($H_{ZZ}$) and microwave rotations ($H_X$) produces an effective transverse-field Ising model.

Figure 2

Measuring Ising interactions. (a) Ramsey sequence with spin echo. Bloch spheres show average spin $⟨\mathbf{S}⟩$ at select times for two different initial states $|\theta ⟩$ (blue and red). (b) Interference fringe for $|\theta ⟩=|3\pi /4⟩$ showing spatial dependence of interaction-induced phase shift. Black dashed lines show analysis region for subplots (c),(d). (c) Phase shift $\varphi$ vs initial tilt $\theta$ for different interaction times ${\tau }_R$ with fit curves $\varphi =-Q\cos \theta$. (d) Twisting strength $Q$ (blue circles) vs time, extracted from fits in (c). The slope of the linear fit (solid blue) gives the mean-field interaction energy $\chi =2\pi \times 15(1)\mathrm{kHz}$. Also shown are interference contrast $\mathcal{C}$ (red diamonds with fit curve) and atom number $N$ (magenta squares) remaining after Rydberg dressing, normalized to initial atom number $N_0$.

Figure 3

Transverse-field Ising dynamics. (a) Trajectories $\mathbf{S}(k)$ for initial states $|\theta ,\varphi ⟩$ (square data points) and up to $k=4$ Floquet cycles, obtained with dressing parameters $(\mathrm{\Omega },\mathrm{\Delta })=2\pi \times (2.8,25)\mathrm{MHz}$. Plots (i)-(iv) are for ${\mathrm{\Lambda }}_{\mathrm{eff}}=0,1.2(2),1.8(3),2.7(4)$. Blue flow lines show mean-field theory for best fit $\mathrm{\Lambda }=0$, 1.1, 1.5, 2.2. (b) Sequence of microwave (purple) and Rydberg dressing (blue) pulses for $k$ Floquet cycles. The first application of $H_{ZZ}$ is split into two, with the second Rydberg pulse after the last microwave rotation, to keep the fixed points along the $\varphi =0$ meridian. (c) Twisting strength $Q$ vs $k$ measured with $({\tau }_R,{\tau }_X)=(10,0)\mu \mathrm{s}$ in the four regions of the atomic cloud (i)-(iv) used in (a).

Figure 4

Bifurcation of fixed points, signifying paramagnetic and ferromagnetic ground states. We measure the phase $\varphi$ after $k=4$ Floquet cycles with (a) $h{\tau }_X=0.14(1)$ or (b) $h{\tau }_X=0$, as a function of initial tilt $\theta$ and position. The $\varphi =0$ contour reveals fixed points of the mean-field dynamics, matching the theoretical prediction (purple dot-dashed, purple dashed, and gray dotted curves for the ferromagnetic ground states, paramagnetic ground state, and unstable fixed points, respectively). Fitting the phase evolution in (b) yields the average mean-field interaction $\chi {\tau }_R$ per cycle. (c) Green points and fit curve show $\mathcal{C}\chi {\tau }_R$ vs position, compared with rotation angle $h{\tau }_x$ (brown line). (d) Final phase $\varphi$ vs initial tilt $\theta$ for cuts labeled $A$ (yellow diamonds), $B$ (red circles), $C$ (blue squares), and $D$ (green triangles), in order of increasing $|\mathrm{\Lambda }|$. Solid lines show Floquet mean-field model for the measured values $\chi {\tau }_R$ and $h{\tau }_X$ with no contrast loss, while edge of shaded region accounts for contrast $\mathcal{C}$.