Abstract
The control of strongly interacting manybody systems enables the creation of tailored quantum matter with complex properties. Atomic ensembles that are optically driven to a Rydberg state provide many examples for this: atom–atom entanglement^{1,2}, manybody Rabi oscillations^{3}, strong photon–photon interaction^{4} and spatial pair correlations^{5}. In its most basic form Rydberg quantum matter consists of an isolated ensemble of strongly interacting atoms spatially confined to the blockade volume—a superatom. Here we demonstrate the controlled creation and characterization of an isolated mesoscopic superatom by means of accurate density engineering and excitation to Rydberg pstates. Its variable size allows the investigation of the transition from effective twolevel physics to manybody phenomena. By monitoring continuous laserinduced ionization we observe a strongly antibunched ion emission under blockade conditions and extremely bunched ion emission under offresonant excitation. Our measurements provide insights into both excitation statistics and dynamics. We anticipate applications in quantum optics and quantum information as well as manybody physics experiments.
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Main
Rydberg superatoms combine single and manybody quantum effects in a unique way and have been proposed as fundamental building blocks for quantum simulation and quantum information^{6}. Owing to the phenomenon of Rydberg blockade^{7}, the ensemble collectively forms a system with only two levels of excitation. Provided a range of interaction larger than the sample size, the presence of one excitation shifts all other atoms out of resonance; therefore only one excitation can be created at a time. Changing the size or the driving conditions restores the underlying manybody nature, and the presence of several excited atoms with pronounced correlations becomes possible. This tunability and the possibility of multiple usage within a single experimental sequence make superatoms a promising complement to singleatombased quantum technology. It is therefore important to understand the significance of the superatom concept, the implications of its finite spatial extent and its manybody level structure. Here we investigate the latter by measuring the mean Rydberg excitation as well as its timeresolved twoparticle correlations in an optically excited, mesoscopic superatom for varying excitation strength and under resonant and nonresonant conditions, revealing very different excitation dynamics.
The realization of superatombased quantum systems requires the implementation of arbitrary arrangements of isolated mesoscopic atomic ensembles in a scalable way. Here we prepare an individual superatom by carefully shaping the density distribution of a Bose–Einstein condensate of ^{87}Rb atoms. We first load the condensate into a onedimensional optical lattice with a spacing of 532 nm, to suppress the axial movement of the atoms. We subsequently compress the atomic sample in the radial direction to reduce its size below the blockade radius and empty all but three (or more) lattice sites using a focused electron beam^{8,9,10} (Fig. 1a and Methods). The atom number within the ensemble (N) can be adjusted between 100 and 500 at a temperature of T = (3.5 ± 0.5) μK and the typical size of the sample is ≤3 μm in diameter (Fig. 1b). Our preparation scheme is readily scalable to arrays of superatoms (Fig. 1d).
After preparation we excite the atomic ensemble with a singlephoton transition from the 5s_{1/2}〉 ground state to the 51p_{3/2}, m_{j} = 3/2〉 Rydberg state at a wavelength of 297 nm with a coupling strength Ω. The singlephoton transition circumvents scattering from any intermediate state, therefore allowing a long exposure. The key observable is the string of ions produced by excited atoms that are photoionized by the trap laser^{11} (with ionization rate Γ_{ion} = (45 ± 5) kHz). Specifically, we detect the initial peak ion rate as well as the temporal pair correlation function g^{(2)}(τ), extracted from the timeresolved ion signal (see Methods). The ions continuously emitted from the ensemble lead to a slow decay of the superatom on a timescale between a few milliseconds (see inset Fig. 2) and seconds. Eventually almost 100% of the constituent atoms are converted into ions, of which we detect (40 ± 8)%. For weak resonant driving the superatom mimics an effective twolevel system where the excitation of more than one atom is suppressed as a result of blockade. This can be clearly observed in the experiment: Fig. 2 shows pronounced antibunching, in good agreement with a theoretical rate model (Methods and Supplementary Information). We extrapolate a value of g^{(2)}(0) = 0.08 ± 0.06, taking into account an uncorrelated background signal (see Methods). The background constitutes 10–15% of the signal and originates from atoms which are not removed during the preparation of the superatom. The antibunching amplitude stays constant during the gradual decay of the superatom, which thus acts as a continuously operating singleion source^{12}. The strong suppression of collective oscillations, indicated by the purely exponential shape of g^{(2)}(τ), shows that the system is in the overdamped regime, where the coherent coupling is overcome by decoherence from the laser linewidth, thermal motion of atoms and residual field fluctuations as well as intrinsic dephasing mechanisms^{13}.
The ability to adjust the size of the atomic sample allows a continuous transition from the superatom limit to a manybody system, where blockade conditions break down. First, this can be used to determine the blockade radius. In Fig. 3a the initial ion rate per atom is shown for increasing axial size l of the sample, keeping the groundstate atom density constant. Under blockade conditions the ion rate first decreases as more and more atoms contribute to the same signal. However, above a critical spatial extent, which we identify with the blockade radius, the ion rate remains constant. The corresponding, independently measured, antibunching signal (inset in Fig. 3a) leads to a compatible value of the blockade radius of (2.7 ± 0.8) μm. At first glance it is surprising that we observe blockade at all when we resonantly excite to a pstate. In a pstate the van der Waals interaction is strongly angular dependent (Fig. 3b). For a small interval of θ, the coefficient C_{6} is vanishingly small, potentially leading to a breakdown of the overall blockade. Thus, to understand the observed blockade effect we need to go beyond the standard C_{6} asymptotics. Figure 3b shows the interaction potential curves for the asymptotic 51p_{3/2,3/2}, 51p_{3/2,3/2}〉 pair state, obtained by diagonalization of the interaction Hamiltonian (see Methods). As a result of an avoided crossing with the asymptotic 51p_{3/2,1/2}, 51p_{3/2,1/2}〉 pair state, which is energetically separated by an external magnetic field of 35 G, the potential curves which have a negative C_{6} coefficient for large distances bend into a repulsive interaction for smaller distances. Thus, the interaction potential for all angles becomes repulsive, enabling an overall blockade.
To describe the complex interaction potential structure in an effective but simple way, we assume an isotropic repulsive interaction and solve the manybody problem within an approximate rate equation model with a C_{6}^{eff} coefficient as the only free parameter (see Methods). We chose a van der Waals coefficient of to compare the model to the experimental results. The resulting potential curves are indicated by the green shaded area in Fig. 3b. Throughout this paper, we apply the effective model with these parameters to our data and find good agreement over a wide range of laser intensities and detunings.
Our mesoscopic superatom permits interatomic distances R of up to ≍3 μm. As a consequence, when the excitation is offresonant the blockade conditions can be tuned into an antiblockade^{14} and pronounced bunching of the ion emission can be observed (Fig. 4a). We find bunching values of up to g^{(2)}(0) = 61 ± 8 for large detunings. This behaviour can be understood from the full level structure of the mesoscopic superatom, including excited states with more than one excitation, and is captured well by our rate model: whereas the transition to the first collective Rydberg state is out of resonance, subsequent transitions into doubly excited states are shifted into resonance (Fig. 1c), leading to a cascaded excitation process.
The transition of the mesoscopic superatom from an effective twolevel system to a complex manylevel system is also reflected in its saturation behaviour. In Fig. 4b, we plot the dependence of the initial ion rate on the Rabi frequency for resonant and offresonant (Δ/2π = 4 MHz) excitation through three orders of magnitude of the experimental parameter Ω. The excitation probability on resonance initially grows quadratically and starts to saturate around an ion rate corresponding to the presence of one excitation. Driving the superatom more strongly, the blockade radius is reduced and above the saturation threshold more excitations can be created, resulting in an increasing initial ion rate—however, with a smaller slope. Thus, the blockade is overcome after saturation has been reached. For offresonant excitation the signal again shows a quadratic initial slope at a reduced absolute value, but enters a region where the slope is steeper than quadratic, showing a strong enhancement of excitations. For strong enough driving, the resonant and offresonant excitations eventually reach comparable levels. This happens when the collective coupling strength becomes comparable to the detuning Δ and the difference in the first excitation step for the resonant and offresonant case disappears. The rate equation model reproduces all experimental findings, despite the major simplifications made. Only for the largest Rabi frequencies in Fig. 4b does the model underestimate the excitation rate. Here, the ensemble Rabi frequency is larger than the decoherence rate and coherent manybody dynamics might become visible. However, a comparison of the rate equation model with a fully quantummechanical treatment leads to almost identical predictions for the initial ion rate and g^{(2)}(0) (Supplementary Information). To observe coherent dynamics requires the condition Ω/Γ_{d} > 1 and the blockade condition to be satisfied simultaneously. The experiment is thus limited at present to incoherent dynamics. Reducing decoherence is one avenue towards coherent superatom dynamics. An alternative is excitation to higher Rydberg n levels, which increases the interaction.
The temporal correlation function g^{(2)}(τ) also provides insight into the manybody dynamics of the superatom. Figure 5a shows the dependence of the correlation times τ_{c} of g^{(2)}(τ) on the detuning. Three different physical regimes can be identified. In Regime I, for large detunings, the atoms spend most of the time in the ground state with small probabilities for single and double excitations. The detection of an ion projects the density matrix onto states with one excitation less. Only doubly excited states emit a second ion and contribute to g^{(2)}(τ). The correlation time of g^{(2)}(τ) is thus simply given by the lifetime of the Rydberg excitation. In Regime II, for smaller detunings, we observe a marked slowdown of the relaxation dynamics. In this regime strongly correlated Rydberg aggregates form^{15,16}. An ionization event projects the system onto a state with increased weight on aggregates of a few excited atoms (Fig. 5b). The relaxation to the steady state is set by the lifetimes of these aggregates, which exceed that of Regime I, as several atoms have to decay. In Region III, on resonance, where antibunching occurs, g^{(2)}(τ) reflects the excitation dynamics after the emission of an ion. The timescale is below 5 μs, shorter than all relevant singleparticle timescales, and is therefore a signature of the collectively enhanced excitation rate of the superatom. The corresponding probability distributions are shown in Fig. 5c. Our analysis shows that the temporal pair correlation function is a powerful tool to characterize manybody dynamics and can be used in the future to study quantum phases of driven dissipative systems^{17}.
Single superatoms based on collective Rydberg excitations have great potential for applications in quantum optics. They can be used to build highfidelity photon absorbers^{13} and deterministic ion sources^{12}. An interaction between multiple superatoms can be realized choosing a Förster resonance, which features a longrange R^{−3} dipole–dipole interaction. The increased interaction then enables us to switch to the coherent collective excitation regime, allowing deterministic state manipulation. Strings of superatoms (see Fig. 1d) are an ideal system for the investigation of energy transfer mechanisms^{18,19} and onedimensional spin systems^{20}. Our approach can be straightforwardly extended to arbitrary patterns of superatoms in twodimensional lattice systems^{8,9}. Such quantum systems can then be a resource for further investigations of isotropic and anisotropic longrange interactions^{21}, for the quantum simulation of open spin systems^{22,23}, Bell state measurements^{24,25} or interferometric applications^{26,27}.
Methods
Superatom preparation and laser excitation.
We start with a Bose–Einstein condensate of approximately 1,700 rubidium atoms in a crossed optical dipole trap at a wavelength λ = 1,064 nm. The final trap frequencies are 2π × (180/85/185) Hz. A onedimensional optical lattice is then superimposed by linearly ramping up a retroreflected laser beam along the weak axis of the trap and the axial motion is frozen out. Subsequently, the atomic cloud is radially compressed by increasing the intensity of one of the two dipole trap beams. A focused electron beam ((250 ± 100) nm diameter, 20 nA beam current), which is also used to image the sample, removes the atoms from selected areas^{8,9,10}. In this way, we prepare samples of several hundred atoms with ≤3 μm diameter at a temperature of (3.5 ± 1) μK. After the preparation sequence, a fraction of 10–16% of the atoms reside in the outer regions of the trap.
The superatom is directly excited to a Rydberg state by illuminating it with an ultraviolet laser beam at 297 nm with a waist of 100 μm and a laser power up to 160 mW. The light is produced by frequency doubling a stabilized dye laser (MatisseDR) in a heated caesium lithium borate (CLBO) crystal installed in a Pound–Drever–Hall stabilized bowtie cavity. The frequency of the dye laser can be tuned via an offsetlocked reference laser, resulting in a relative uncertainty of the ultraviolet frequency of ±0.5 MHz. The linewidth of the excitation light is estimated from dye laser control parameters to less than 200 kHz and the power noise is below 10%.
Electric fields, ion detection and signal processing.
The atomic sample is surrounded by quadruply segmented copper rings of 40 mm diameter at a distance of 25 mm, embodying an octupole electrode configuration^{28}. Applying corresponding voltages, residual electric fields in the chamber are compensated in all directions apart from a remaining permanent vertical component of E_{0} ≤ 0.25 V cm^{−1} that is used to extract the produced ions towards the ion optics below. The ion optics guide the ions into a dynode multiplier (ETP 14553). The signal pulses are further processed with a temporal resolution of 100 ns.
Temporal correlation function.
We numerically calculate the secondorder temporal correlation function of the ion signal I(t)
where 〈I(t)I(t + τ)〉 is calculated as the averaged product of counts (0 or 1) in two bins separated by τ and the normalization 〈I(t)〉〈I(t + τ)〉 is given by the averaged ion rate. Several factors affect the data evaluation and have to be taken into account. Artefacts from detector ringing occur for time separations of less then 400 ns. Coulomb repulsion between the ions during time of flight produces further correlations on a timescale which depends on the extraction field. For our parameters, they never occur on timescales longer than 2 μs. We therefore discard all data points with τ ≤ 2 μs. The background atoms (fraction r) contribute an uncorrelated signal to the ion emission of the superatom. This leads to a reduction of the measured amplitude g_{meas}^{(2)} compared to the bare signal from the superatom g_{real}^{(2)}, which we correct for: Note that g^{(2)}(τ) is independent of the detector efficiency.
Calculation of the interaction potential.
The potential curves are calculated by diagonalization of the dipole–dipole interaction Hamiltonian in the presence of an electric and magnetic field. The basis set consists of all pair states which are closer than 15 GHz in energy to the initial 51p_{3/2,3/2}, 51p_{3/2,3/2}〉 pair state. We consider all possible combinations of s, p and dstates, including all Zeeman levels.
Effective rate model.
The superatom dynamics is described by a Lindblad equation , where and L_{ν} are the jump operators for ionization (Γ_{ion}), spontaneous decay into lowlying states which are not ionized (Γ_{sp}) and decoherence (Γ_{d}). The decoherence rate represents the cumulative effect of laser linewidth, thermal atomic motion, fluctuating electric fields and intrinsic dephasing mechanisms^{13}. The rates of ionization (Γ_{ion} = 45 kHz) and internal spontaneous decay (Γ_{sp} = 5 kHz) are known from independent measurements and we extract the excitation rate from the saturation measurements shown in Fig. 4b. For weak driving the ion signal is independent of the interaction term and given by I_{Ω≪γ} = (Γ_{ion}N_{atoms})/(Γ_{ion} + Γ_{sp})Ω^{2}γ/(γ^{2} + 4Δ^{2}), with γ = Γ_{d} + Γ_{ion} + Γ_{sp}, and N_{atoms} being the number of atoms within the superatom. Fits to the saturation measurements at Δ = 0 and Δ/(2π) = 4 MHz yield the relation between laser intensity and Ω and the decoherence rate Γ_{d}/(2π) ≍ 140 kHz and Γ_{d}/(2π) ≍ 340 kHz, for two different sets of parameters used in the experiment.
Beyond the regime of weak driving we describe the superatom by classical rate equations, which is justified by the large decoherence rate present in our setup and validated in previous studies of strongly interacting Rydberg systems using such methods^{15,29,30}. The system of rate equations describes dynamics in classical configuration space, where individual states are connected by singleatom transitions at excitation rate (P_{i}) and deexcitation rate (D_{i}). These rates depend on the effective detuning of the atom through P_{i} = (Ω^{2}γ)/(4δ_{i}^{2} + γ^{2}) and D_{i} = (Ω^{2}γ)/(4δ_{i}^{2} + γ^{2}) + Γ_{ion} + Γ_{sp}. The set of manybody rate equations is solved by stochastic sampling of trajectories. Simulations take into account the spatial distribution of atoms as measured in the experiment by averaging over many realizations. Atomic motion is not included within our description and the intricate pstate interaction is approximated with an effective, isotropic van der Waals potential C_{6}^{eff}/R^{6}.
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Acknowledgements
We acknowledge financial support by the DFG within the SFB/TRR 49. V.G. and G.B. were supported by Marie Curie IntraEuropean Fellowships.
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T.M.W., T.M., T.N., G.B. and H.O. designed and set up the apparatus. G.B. and H.O. conceived the experiment. T.M.W., T.N., O.T., T.M. and G.B. performed the experiment. H.O. supervised the experiment. T.M.W. analysed the data and prepared the manuscript. M.H. and M.F. developed the theoretical model. M.H. performed the numerical simulations. M.F. supervised the numerical simulations. All authors contributed to the data interpretation and manuscript preparation.
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Weber, T., Höning, M., Niederprüm, T. et al. Mesoscopic Rydbergblockaded ensembles in the superatom regime and beyond. Nature Phys 11, 157–161 (2015). https://doi.org/10.1038/nphys3214
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DOI: https://doi.org/10.1038/nphys3214
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