Coherent Excitation Transfer in a Spin Chain of Three Rydberg Atoms

We study coherent excitation hopping in a spin chain realized using highly excited individually addressable Rydberg atoms. The dynamics are fully described in terms of an XY spin Hamiltonian with a long range resonant dipole-dipole coupling that scales as the inverse third power of the lattice spacing, $C_3/R^3$. The experimental data demonstrate the importance of next neighbor interactions which are manifest as revivals in the excitation dynamics. The results suggest that arrays of Rydberg atoms are ideally suited to large scale, high-fidelity quantum simulation of spin dynamics.

Spin hamiltonians, introduced in the early days of quantum mechanics by Heisenberg to explain ferromagnetism, are widely used in the study of quantum magnetism [1]. Assemblies of interacting, localized spins form a paradigm for quantum many-body systems, where the interplay between interactions, geometry, and frustration is at the origin of a wealth of intriguing quantum phases. Moreover many other phenomena, such as coherent energy transfer in quantum systems, photochemistry or photosynthesis [2], can also naturally be described in terms of spin hamiltonians. However, despite this fundamental significance exact analytical solutions for many interacting spins are known only for the simplest cases, and direct numerical simulations of the strongly correlated spin systems are notoriously difficult.
For those reasons, the quantum simulation of spin Hamiltonians by fully controllable systems has raised a lot of interest. Recently, various approaches have been followed to simulate spin systems using the tools of atomic physics [3], such as cold atoms [4][5][6] or polar molecules [7] loaded into optical lattices, interacting via weak exchange or magnetic dipoledipole interactions, or one-dimensional systems of a few trapped ions with engineered effective interaction between the sites [8][9][10]. As compared to their condensed-matter physics counterparts, the spin-spin couplings can be longrange, which gives rise to new, interesting properties [11][12][13].
Cold Rydberg atoms are a promising alternative platform for quantum simulation and quantum computing due to their controlled interactions over exaggerated length scales [14,15]. In particular, they allow the implementation of various spin-1/2 hamiltonians with very strong couplings, in the MHz frequency range. Rydberg systems interacting through the van der Waals interactions can be described by Ising-type hamiltonians H = i,j V ij σ z i σ z j where σ z is the z-Pauli matrix acting in the (pseudo)-spin Hilbert space, and V ij ∼ |r i − r j | −6 , where r i denotes the position of atom i [16][17][18][19][20]. On the other hand, spin-exchange, or XY, spin hamiltonians of the form , where σ ± = σ x ± iσ y are spin-flip operators and V ij ∼ |r i − r j | −3 , can be realized by using two different Rydberg states, interacting directly via the resonant dipole-dipole interaction. However in this latter case, only the incoherent transfer of excitations has been observed so far, due to either loss [21] or the random positions of the atoms in the ensembles used in experiments [22][23][24][25].
In this Letter, we study the coherent dynamics of a spin excitation in a chain of three Rydberg atoms. The dipole-dipole interaction between the atoms is given by the XY spin hamiltonian where R ij = |r i − r j | is the distance between atoms i and j. We first describe the scheme used to prepare the atoms in a desired spin state using microwave transfer in the Rydberg manifold, and calibrate the spin-spin coupling strength between two Rydberg atoms by investigating the temporal evolution of a system of two Rydberg atoms prepared in the state |↑↓ , as a function of distance R between the atoms. We use a system of three Rydberg atoms prepared in |↑↓↓ and study the propagation of the spin excitation through this minimalistic spin chain, directly observing the effect of long-range hopping of the excitation. The excellent agreement between the experimental data and the XY model with no adjustable parameters validates the usefulness of our setup as a future quantum simulator for systems of a many spins in arbitrary two-dimensional arrays.
The experimental setup, shown schematically in Fig. 1(a), is described in detail in Ref. [26]. Briefly, we focus a reddetuned dipole trap beam with an aspheric lens into a MOT of 87 Rb atoms, to a waist of approximately 1 µm. Multiple traps at arbitrary distances are created by imprinting an appropriate phase on the dipole trap beam with a spatial light modulator prior to focusing [27]. Due to fast light-assisted collisions in the small trapping volume, at most one atom is present in each trap at any time. The temperature of the trapped single atoms is approximately 50 µK. An external magnetic field of 6 G in the vertical direction defines the quantization axis.
We encode the two spin states in the Rydberg states |↑ = 62D 3/2 , m j = 3/2 and |↓ = 63P  . We trigger an experimental sequence when an atom is detected in each of the traps. To prepare the atoms in a desired spin state, we first optically pump them in 5S 1/2 , F = 2, m F = 2 . We then switch off the dipole traps to avoid inhomogeneous light-shifts, and excite the atoms to the Rydberg state |↑ = 62D 3/2 , m J = 3/2 via a twophoton transition (wavelengths 795 and 474 nm, with polarizations π and σ + , respectively), detuned from the intermediate state 5P 1/2 , F = 2, m F = 2 by ∆ 2π × 740 MHz. From the |↑ state the atom can be transferred to the state |↓ = 63P 1/2 , m J = 1/2 using a resonant microwave pulse with a frequency of 9.131 GHz emitted by a dipole antenna placed outside the vacuum chamber [28].
To read out the final state of an atom at the end of an experimental sequence, we again switch on the excitation lasers, coupling only the |↑ state back to the ground state. We then turn on again the dipole traps to recapture the ground-state atoms, while atoms in Rydberg states remain untrapped, and detect atoms in |g by fluorescence. Therefore if we detect an atom in its trap at the end of a sequence, we assume that it was in the |↑ state, while a loss of the atom corresponds to the |↓ state. We reconstruct all the 2 N probabilities P i1...i N of having i k atom in trap k, with i k = 0 or 1, for our Ntrap system (with N = 1, 2, or 3) by repeating the experiment typically 100 times: for instance for N = 3, P 100 is the probability to recapture an atom in trap 1 at the end of the sequence, while recapturing none in traps 2 and 3. The statistical error bar on the determination of the probabilities is typically below 5%. Excitation hopping between the pair states |↑↓ (blue disks) and |↓↑ (red disks) of two Rydberg atoms separated by R = 30 µm. Solid lines are sinusoidal fits to the data, with a frequency 2E/h, to extract the spin coupling strength E. (c) Interaction energy E (circles) as a function of the distance R between the two atoms. The relative uncertainty in the determination of E is below 5%, giving error bars that are smaller than the size of the symbols. The solid line represents the theoretical prediction C3/R 3 with C3 = 7965 MHz · µm 3 , the shaded area corresponding to our systematic 5% uncertainty in the calibration of R. a single atom, by showing Rabi oscillations between |↑ and |↓ : the probability P 1 to recapture the atom at the end of the sequence oscillates with a frequency Ω MW 2π × 4.6 MHz. In 4 µs, we can induce more than 35 spin flips, without observing any noticeable damping in the oscillations.
In a first experiment, we use two atoms, aligned along the quantization axis, to directly probe the coupling between two spins as a function of their distance. The pulse sequence to prepare the atoms in the initial state |↑↓ is shown in Fig. 2(a). We illuminate atom 1 with an addressing beam [29] which induces a light shift of ∼ 20 MHz, shifting it off-resonant to the global Rydberg excitations. Atom 2 is excited to |↑ , and , with no adjustable parameters. For perfect detection and readout, the probabilities P ↑↓↓ (resp. P ↓↑↓ , P ↓↓↑ ) and P100 (resp. P010, P001) would coincide. then transferred to |↓ using a microwave pulse. Subsequently, atom 1 is optically excited to the |↑ state with the addressing beam switched off (atom 2 in |↓ is not affected by the optical Rydberg excitation pulse). We let the system evolve for an adjustable time T and read out the final state by de-exciting the |↑ state back to the ground state. If the state preparation and readout were ideal, P 10 (resp. P 01 ) would give the population of |↑↓ (resp. |↓↑ ).
The evolution of P 10 and P 01 as a function of T for two atoms prepared in the |↑↓ state separated by 30 µm is shown in Fig. 2(b). We see the spin excitation oscillating back and forth between the two atoms several times, with a frequency twice the dipolar coupling strength 2E/h ≈ 0.52 MHz. The finite contrast of the oscillations is essentially due to spontaneous emission via the intermediate 5P 1/2 state during preparation and readout, which limits the amplitude of the oscillations to about 60%. We then repeat the same experiment for several values of the distance R between the atoms, and we observe coherent spin-exchange oscillations for interatomic distances as large as 50 µm. Figure 2(c) shows the measured interaction energies as a function of R, together with the expected C 3 /R 3 behaviour (solid line) for the theoretical value 7965 MHz · µm 3 of the C 3 coefficient, calculated from the dipole matrix elements ↑ |d ±1 | ↓ [26,30]. The agreement between the data and the theoretical expectation is excellent 1 .
We now extend the system to three spins aligned along a 1 A power-law fit to the data (not shown) gives an exponent −2.93 ± 0.20. chain, with a distance R = 20 µm between the atoms. The state preparation sequence is similar to that in figure 2(a) for two atoms, except that we now use the microwave transfer for atoms 2 and 3 to prepare the state |g ↓↓ . Here, the van der Waals interaction between the two atoms in |↑ is very small for R = 20 µm, about 10 kHz, and thus no blockade effect arises during the excitation step. We then excite atom 1 to prepare the state |↑↓↓ . We first analyze theoretically the evolution of the system. Assuming that the initial state is |ψ(0) = |↑↓↓ , the dynamics induced by the XY Hamiltonian (1), which conserves the total magnetization i σ z i , occurs within the subspace spanned by {|↑↓↓ , |↓↑↓ , |↓↓↑ }. Figures 3(a) and (b) show the calculated dynamics of the spin excitation, which moves back and forth between the extreme sites of the chain.  (1), only the terms corresponding to nearest-neighbor interactions are retained. Periodic, fully contrasted oscillations at a frequency √ 2C 3 /R 3 are expected for the population of the extreme sites, while the population of |↓↑↓ oscillates twice as fast between 0 and 1/2. In contrast, in Fig.3(b), the full Hamiltonian (1) is simulated, including the term corresponding to the long-range interaction between the extreme sites. One observes a clear signature of this long-range coupling, as the dynamics now becomes aperiodic for the populations of |↑↓↓ and |↓↓↑ . The interplay of the couplings C 3 /R 3 and C 3 /(8R 3 ), between nearest and next-nearest neighbors respectively, makes the eigenvalues of the Hamiltonian incommensurate. The back-and-forth exchange of excitation is thus modulated by a slowly varying envelope due to the beating of these two frequencies. Figure 3(c) shows the experimental results for P 100 , P 010 and P 001 (symbols). We observe qualitative agreement with Fig. 3(b), in particular the "collapse and revival" in the dynamics show the effects of the long-range coupling. However, one notices two differences with the ideal case: (i) the preparation is imperfect, as one starts with a significant population in |↓↑↓ , and (ii) the oscillations show some damping, which becomes quite significant beyond an interaction time of ∼ 4 µs.
The imperfect preparation stems from the fact that, in addition to the spontaneous emission via the intermediate state during excitation, the Rabi frequency for the optical excitation ( 5.3 MHz) of atom 1 from |g to |↑ is not much higher than the dipolar interaction ( 0.92 MHz for R = 20 µm). Thus, during the excitation of atom 1, the spin excitation already has a significant probability to hop to atom 2. The observed damping essentially arises from the finite temperature of the atoms, which leads to changes in the distances between the atoms, and thus in the couplings.
To go beyond this qualitative understanding of the limitations of our "quantum simulator", we add all the known experimental imperfections to the XY model. The result of this calculation, shown by the solid lines on Fig. 3(c) accurately reproduces the experimental data with no free parameters. To obtain these curves, we simulate the full sequence, i.e. all three optical (de-)excitation pulses with or without the addressing beam, the microwave pulse, and evolution time, by solving the optical Bloch equations (OBEs) describing the dynamics of the internal states of the atoms, restricted to three states: |g , |↑ , and |↓ . Dissipation comes from both the finite lifetimes of the Rydberg states (101 and 135 µs for |↑ and |↓ , respectively [31]), and from off-resonant excitation of the intermediate 5P 1/2 state during the optical excitation pulse. This latter effect is treated as an effective damping of the |g ↔ |↑ transition, present only during the optical pulses, and with a damping rate chosen to match the damping observed on the single-atom Rabi oscillations that we perform to calibrate the excitation Rabi frequency Ω opt [19].
We then take into account the effects due to the thermal motion of trapped atoms. A first consequence of the finite temperature (T 50 µK) is that at the beginning of the sequence, the atoms have random positions (the rms extension of the atomic thermal motion in each microtrap, of radial frequency 90 kHz, is about 120 nm) and random velocities (with an rms value of 70 nm/µs). During the sequence, the traps are switched off and the atoms are thus in free flight with their initial thermal velocity. When solving the OBEs, we thus first draw the initial positions r 0 i and velocities v 0 i of each atom i according to a thermal distribution, and use time-dependent dipolar couplings This yields a dephasing of the oscillations, resulting in a significant reduction of the contrast at long interaction times. A second effect of the finite temperature is that an atom has a small probability ε(t) to leave the trap region during the excitation sequence and is not recaptured. In this case, we mistakenly infer that the atom was in a Rydberg state at the end of the sequence. This effect leads to a small distortion of the measured populations P ijk (i, j, k = 0, 1) [32], that we compute from the actual ones as we described in [19]. We measure ε(t) (which increases with the duration t of the sequence, starting from ∼ 1% at t = 0 and increasing up to ∼ 20% for t = 7 µs) in a calibration experiment and then use it to calculate the expected populations from the simulated ones. Figure 4 shows how those two consequences of the finite temperature of the atoms contribute to the observed damping in the dynamics of P 001 : both have sizable effects, but the dephasing due to fluctuations in the coupling dominates at long times. Reducing the atomic temperature using e.g. Raman cooling [33,34] would render those motional effects negligible for the timescales used here and allow the realization of nearly ideal quantum simulator of spin dynamics.
In summary, we have demonstrated fast coherent manipulation of a single Rydberg atom with a microwave field. Using this tool, we prepared a spin state of two atoms and measured the mutual spin coupling over distances as large as 50 µm. The coupling strength follows the C 3 /R 3 dependence of the resonant dipole-dipole interaction, and the measured values agree well with the calculated value of the C 3 coefficient. Subsequently, we have measured spin excitation dynamics in a minimal linear spin chain consisting of three Rydberg atoms. The evolution of this spin system is accurately described by an XY-Hamiltonian with no adjustable parameters. This spin-exchange hamiltonian with long-range interactions is equivalent to a system of hard-core bosons on a lattice with long-range hopping, a feature hard to realize with ultracold atoms in optical lattices. The presented results set the basis for experiments with spin systems in larger, two dimensional ar-rays [27], where the anisotropic character of the dipole-dipole interaction also plays a crucial role. Another interesting extension of the present work would consist in using resonant dipole-dipole interactions involving more than two Rydberg states at an electrically-tuned Förster resonance [35]. Such systems would be a perfect platform to study of exotic phases and frustration in quantum magnetism, excitation hopping in complex networks [36,37] or quantum random walks with long-range hopping [38].