Probing spin-motion coupling of two Rydberg atoms by a Stern-Gerlach-like experiment

Abstract

We propose and implement a protocol to measure the state-dependent motion of Rydberg atoms induced by dipole-dipole interactions. Our setup enables simultaneous readout of both the atomic internal state and position on a one-dimensional array of optical tweezers. We benchmark the protocol using two atoms in the same Rydberg state, which experience van der Waals repulsion, and measure velocities in agreement with theoretical predictions. When preparing the atoms in a different pair state, we observe an oscillatory dynamics that we attribute to the proximity of a macrodimer bound state. Finally, we perform a Stern-Gerlach-like experiment in which a superposition of the two previous pair states results in the separation of the atomic wavepacket into two macroscopically distinct trajectories, thereby demonstrating spin-motion coupling mediated by the interactions.

Figures (8)

• Figure 1: Experimental sequence. The sequence (a) consists of four steps which are illustrated on the right. (b) Loading: a 1D array of optical tweezers separated by $a \simeq 2$ µm is filled with two ground-state $^{87}\text{Rb}$ atoms (gray balls). The tweezers depth is then adiabatically ramped down by a factor $\sim 100$. (c) State preparation: after switching off the tweezers, a set of optical beams (at $420$ and $1014$ nm) and microwave pulses (MW at $\sim 17$ GHz) prepare the atoms in the pair states $\ket{S,S}$ (blue balls), $\ket{+} = \left( \ket{S,P} + \ket{P,S} \right)/\sqrt{2}$ (red balls) or in a superposition of both (see text). (d) Spin-motion coupling: the atoms interact under the Rydberg-Rydberg interactions, without any external force. The forces $\mathbf{F}_{SS}$ and $\mathbf{F}_{+}$ depend on the internal state, leading to a coupling between internal and external degrees of freedom. (e) Readout: each atom's internal state is measured by imaging the state $\ket{S}$, as well as its position in the optical tweezer. $A$ (resp. $B$) are the sites on the left (resp. right) part of the array.
• Figure 2: Trajectories of Rydberg atoms under a repulsive van der Waals interaction, when prepared in the same Rydberg state $\ket{S}$. (a)--(d) Each panel shows the recapture probability (color scale) per optical tweezer (labeled by their position $x$ on the vertical axis) as a function of the evolution time $t$ (horizontal axis). Dotted white lines indicate the predicted 1D classical trajectories under $V_{SS}$. The top panels are experimental data and the bottom panels are numerical simulations taking into account various experimental imperfections (see text). The white data points are the extracted experimental trajectories (see text). (a) Single-atom reference. (b,c,d) Two atoms separated by various initial distances $r_0 \in \{6a, 5a, 4a\}$; the van der Waals repulsion results in symmetric trajectories away from the origin. (e) Calculated potential of the state $\ket{S,S}$ along $x$ using Weber_2017 (solid black line) and fit by the functional form $V_{SS}(r) = C_6/r^6$. The red data points indicate the values of $V_{SS}$ extracted from panels (c) and (d) (see text). (f) Chain of three atoms with nearest-neighbor distance $r_0 = 5a$; the central atom stays nearly fixed due to balanced forces from its neighbors.
• Figure 3: Oscillations around a Rydberg macrodimer. (a) Calculated Born-Oppenheimer potential curves as a function of interatomic distance, along $x$. The color bar displays the overlap on the pair state $\ket{+}$. The dashed blue line is the potential energy $V_+$ given by effective Hamiltonian theory (see text). The dotted red line $V_\uparrow$ is the eigenenergy of the state $\ket{\uparrow(r)}$, that governs the adiabatic dynamics of a system initially prepared in $\ket{\uparrow(r)} \simeq \ket{+}$ at $r=5a$ (black dot). It shows an attractive branch that becomes repulsive at short distance, with a minimum corresponding to a Rydberg macrodimer. (b) Trajectory of two atoms prepared in $\ket{+}$: the atoms move inward until $t \sim 12$ µs, then bounce back and partially return to their initial separation by $t \sim 42$ µs. The dotted line is a numerical simulation of the classical motion under $V_\uparrow$. The upper (lower) panel in (b) corresponds to the experiment (numerical simulation).
• Figure 4: Stern-Gerlach-like experiment. (a,b,c) Trajectory of two atoms prepared in the superposition $\tfrac{1}{\sqrt{2}}(\ket{S,S}+\ket{+})$, analyzed with: (a) the site-resolved recapture probability $\left\langle \hat{n}_S(x) \right\rangle$; (b) the site-resolved recapture probability correlated with experimental shots where two atoms are recaptured, $\left\langle \hat{n}_S(x) \hat{\mathbbm{1}}_{SS}\right\rangle$; (c) the site-resolved recapture probability correlated with experimental shots where one out of the two atoms is recaptured, i.e. $\left\langle \hat{n}_S(x) \hat{\mathbbm{1}}_{\uparrow}\right\rangle$. The dotted lines are classical simulations of the trajectories. (d,e,f) Rabi oscillation between the states $\ket{S,S}$ and $\ket{\uparrow}$ followed by an evolution time $t=14$ µs, analyzed with: (d) the total probability of the events $\hat{\mathbbm{1}}_{SS}$ and $\hat{\mathbbm{1}}_{\uparrow}$, summed over all tweezers; (e) the site-resolved version $\left\langle \hat{n}_S(x) \hat{\mathbbm{1}}_{SS}\right\rangle$; (f) $\left\langle \hat{n}_S(x) \hat{\mathbbm{1}}_{\uparrow}\right\rangle$.
• Figure S1: Estimation of the optical tweezers' positions. (a) Intensity distribution of the tweezers, imaged on a diagnostics camera after the vacuum chamber. (b) Ideal positions (black crosses) and fitted positions (colored disks). The green disks indicate the initial positions in the benchmark experiment described in Fig. \ref{['fig:vdw']}(c), in which two atoms initially separated by $r_0 = 5a$ evolve under a repulsive van der Waals interaction. (c) Benchmark experiment [same data as Fig. \ref{['fig:vdw']}(c)] compared to simulations assuming either ideal or disordered tweezers' positions.