## Atoms behaving in an orderly manner

In physics, interactions between components of a system can cause it to become more orderly in an attempt to minimize energy. Such ordered phases appear, for example, in magnetic systems. Schauss *et al*. simulated these phenomena using a collection of neutral atoms at low temperatures. By shining laser light on the atoms, the authors brought some of them into a high energy state called the Rydberg state. By carefully varying the experimental parameters, they coaxed these Rydberg atoms into patterns reminiscent of crystal lattices in rod- and disk-shaped atomic samples.

*Science*, this issue p. 1455

## Abstract

Dominating finite-range interactions in many-body systems can lead to intriguing self-ordered phases of matter. For quantum magnets, Ising models with power-law interactions are among the most elementary systems that support such phases. These models can be implemented by laser coupling ensembles of ultracold atoms to Rydberg states. Here, we report on the experimental preparation of crystalline ground states of such spin systems. We observe a magnetization staircase as a function of the system size and show directly the emergence of crystalline states with vanishing susceptibility. Our results demonstrate the precise control of Rydberg many-body systems and may enable future studies of phase transitions and quantum correlations in interacting quantum magnets.

Quantum spin systems governed by interactions with a power-law dependence on distance are predicted to show intriguing physics very different from systems with at most next-neighbor interactions (*1*, *2*). Such interactions can lead, for example, to the realization of quantum spin glasses (*3*), quantum crystals (*4*), or strong modifications of the light-cone–like propagation of correlations (*5*, *6*). Rydberg atoms offer the possibility to realize such systems with neutral atoms because of the strong van der Waals interaction between them (*7*). The magnitude of the interactions between the Rydberg atoms is determined by the choice of the excited state, and it can exceed all other relevant energy scales on distances of several micrometers, thereby leading to an ensemble dominated by power-law interactions. The resulting Ising-type Hamiltonian is expected to support crystalline magnetic phases. To prepare the system in the crystalline phase, a dynamical approach based on controlled laser coupling has been suggested that adiabatically connects the ground state containing no Rydberg atoms (corresponding to all spins pointing downward) with the targeted crystalline state. At the heart of this dynamical crystallization technique is the coherent control of the many-body system (*8*–*14*).Here, we report on the deterministic ground-state preparation in quantum Ising spin systems composed of several hundred strongly interacting spins via coherently controlled coupling as proposed in (*8*–*10*). The physical system studied is a well-defined line- or disc-shaped atomic sample in an optical lattice with one atom per site. The ^{87}Rb atoms are coupled to the Rydberg state 43*S*_{1/2} with a controlled time-dependent Rabi frequency and detuning of the laser frequency from the atomic resonance . The corresponding theoretical model describing this system is the so-called frozen Rydberg gas, in which only the internal electronic degrees of freedom are considered. This is justified by the short time scale of our experiments, during which the motion of the atoms in the lattice is negligible (*15*, *16*). Adopting a pseudo spin-1/2 description, the system maps to an Ising-type Hamiltonian

Here, the vectors label the position of the atoms on the lattice. The spin-1/2 operators on each site are defined as and , where we omitted the site label to simplify the notation. The operators and describe a spin flip from the ground state to the Rydberg state and vice versa, whereas the operators and represent the local Rydberg and ground-state population, respectively. The first two terms of the Hamiltonian describe a transverse and longitudinal magnetic field, respectively. The former is controlled by the coherent coupling between ground and Rydberg state with the time-dependent Rabi frequency . The detuning determines the longitudinal field and can be used to counteract the energy offset (positive for our parameters). The third term represents the van der Waals interaction potential between two atoms in the Rydberg state. For the 43*S*_{1/2} state, is repulsive because the van der Waals coefficient . Here, is the distance between two atoms on the lattice with period .

This system exhibits a rich variety of strongly correlated magnetic phases (*4*, *5*, *6*, *17*–*20*). In the classical limit () and for , the many-body ground state corresponds to crystalline states with vanishing fluctuations in the total magnetization , which, for fixed total atom number , is determined by the spin- component . In a one-dimensional(1D) chain of lattice sites, the number of spin- atoms increases by one at the critical detunings separating successive crystal states (*8*) (Fig. 1A). The laser coupling introduces quantum fluctuations that can destroy the crystalline order (*4*, *17*, *18*, *21*). Although finite-size effects naturally broaden the transitions in the parameter space (the plot in Fig. 1A was calculated for ), extended lobes corresponding to crystalline states can be identified.

The preparation of the crystalline states requires fast dynamical control because of the short lifetime of the Rydberg states of typically several tens of microseconds. The underlying idea is based on the well-known quantum optical technique of rapid adiabatic passage, here realized on a many-body level. Simultaneous temporal control of the Rabi frequency and laser detuning permits us to dynamically connect the many-body ground states in two distinct parameter regimes, while assuring a finite energy gap to the first excited state along the passage. Our initial state with all atoms in their electronic ground state () coincides with the many-body ground state of the system for negative detuning and vanishing Rabi frequency. For a small coupling strength , the energy gap to the first excited state closes at the transition points between successive manifolds; thus, both and have to be varied to maximize the adiabaticity of the preparation scheme. An intuitive and simple choice of the path starts by slowly switching on the coupling at a large negative detuning (*8*–*10*). Next, the detuning is increased to the desired final blue-detuned value , followed by a gradual reduction of the coupling strength to zero. Choosing between the critical detunings of adjacent manifolds thus yields a crystalline state with a well-defined and controllable magnetization. In the final stage of the last step, the energies of several many-body states become nearly degenerate, as illustrated in Fig. 1B for a representative system of five atoms. These lowest many-body excited states all belong to the same manifold but feature a finite density of dislocations with respect to the perfectly ordered classical ground state. In practice, this leads to unavoidable nonadiabatic transitions at the end of the laser pulse, which in 1D lead to a slight broadening of the characteristic spatial correlations.

Our experiment started from a 2D degenerate gas of 250 to 700 ^{87}Rb atoms confined to a single antinode of a vertical (*z* axis) optical lattice. The gas was driven deep into the Mott-insulating phase by adiabatically turning on a square optical lattice with period in the *xy* plane (*22*). We used a deconfining beam to reduce the harmonic potential induced by the lattice beams and thereby enlarged the spatial extension of a single occupancy Mott insulating state (*23*). Next, we prepared the initial atomic density distribution precisely by cutting out the desired cloud shape from the initial Mott insulator using a spatial light modulator (Fig. 1, C and D) (*24*). For our measurements, we chose line- or disc-shaped atomic samples of well-controlled lengths or radii. The line had a width of three lattice sites and a variable length . Because this width was much smaller than the blockade radius of approximately nine sites, this geometry can theoretically be described by an effective 1D chain with a collectively enhanced Rabi frequency . The average filling was 0.8_{}atoms per site, and at the edge it dropped to below 0.1_{}atoms per site, within one lattice site. The coupling to the Rydberg state was realized by a two-photon process via the intermediate state , using laser wavelengths of 780 and 480 nm with and polarizations, respectively (*25*). Detailed coupling beam parameters are summarized in table S1. Fast control of the Rabi frequency and the detuning was implemented by tuning intensity and frequency of the 780-nm coupling laser using a calibrated acousto-optical modulator (*24*). Finally, the Rydberg atoms were detected locally by fluorescence imaging after removing the ground-state atoms from the trap and depumping the Rydberg state back to the ground state (*24*, *25*). The spatial distribution of Rydberg atoms and, therefore, the magnetization profile were measured by averaging over at least 40 realizations (Fig. 1, E and F).

In a first series of experiments, we prepared crystalline states in the elongated geometry. For a fixed system size, the experimentally realizable number of spin- atoms is limited by the interaction energy as the longitudinal magnetic field scales weakly with . Hence, instead of varying the detuning, we changed the length of the initial system to explore the characteristics of the Rydberg crystals (*8*). We measured the mean number of Rydberg atoms for varying length using a numerically optimized sweep (*24*). In the optimization, the sweep duration was set to 4 μs, which is a reasonable compromise between the decreasing detection efficiency for longer sweeps and adiabaticity (fig. S1A). The results for the sweep to kHz (Fig. 2A) exhibit clear plateaus in and agree well with numerical predictions that take into account the measured initial atomic density, the laser sweep, and the detection efficiency; the latter is the only free parameter, with a fitted value of (*24*). On the plateaus, the theory predicts strong overlap with states of fixed total magnetization (fig. S3). Using the fact that varying the system size is approximately equivalent to varying the detuning , we extract the susceptibility from our data (*24*). is found to vanish in the plateau regions (Fig. 2B), as expected for crystalline magnetic states. The finite values in between result from the small energy gaps between crystalline states of different magnetization around , leading to the preparation of compressible superposition states.

The adiabatic preparation requires the crossing of a phase transition (*4*, *17*, *18*) and, thus, the system undergoes complex correlated quantum dynamics. To study the crystallization process along the sweep trajectory , we abruptly switched off the coupling at different times, thereby projecting the many-body state onto the eigenstates of the uncoupled system . For the measurement, we chose the optimized sweep for the crystalline manifold in a system of 3 by 23 sites. The path through the phase diagram is shown in Fig. 3A. For each evolution time, we measured the Rydberg number histogram, from which we extracted its mean and its normalized variance (Fig. 3B). During the sweep, increases until we observe a saturation behavior that we interpret as the onset of crystallization (fig. S2). Simultaneously, the factor decreases from the Poissonian value to , which reflects the approach to the crystalline state. The expected value is increased to due to our detection efficiency. The measurement of the full counting statistics along the sweep trajectory allows for a more quantitative comparison with theory (Fig. 3C). However, the finite detection efficiency strongly affects the observed histograms and leads to a tail of the distributions toward lower spin- atom numbers (*24*). Nevertheless, we find very good agreement between theory and experiment for the previously fitted .

The high-resolution detection scheme allows for an even more detailed study of the dynamics via the spatial magnetization density, which is largely unaffected by the detection efficiency. For these elongated finite-size systems, crystallization is directly apparent in the magnetization density and provides similar information as the correlation function . This is because of a breakdown of the translational invariance: For a spin- atom localized at the edge is energetically favorable. At the beginning of the pulse, we observe delocalized Rydberg atoms throughout the cloud (Fig. 3D), characteristic for the magnetically disordered phase in this parameter regime. For longer times, the spin- atoms start to accumulate at both ends of the line-shaped cloud and finally crystallize to the expected triple-peak configuration. The dynamics of this crystallization process match well with the theoretical prediction. The observed width of the peaks is compatible with the spatial resolution of one lattice site (*25*) (which was included in the theory).

In a different set of experiments, we investigated the adiabatic preparation in a disc-shaped spin system of up to 400 spins. We used the spatial light modulator to prepare the initial distribution with a controlled radius, whose value fluctuated by only one lattice site (*24*). Here, the dynamical preparation turned out to be more challenging, because the effects of the fluctuating boundary are much more pronounced in 2D than in the effective 1D geometry discussed above. Nevertheless, a proper frequency chirp of the coupling laser offers substantial control of the many-body dynamics and the preparation of energetically low-lying many-body states. This is demonstrated in Fig. 4, where we compare the magnetization density at a constant detuning to the result of a chirped coupling from to (fig. S1B). In the former case, the magnetization is almost uniformly distributed across the atomic sample, whereas in the latter, low-energy states with a localized magnetization density are prepared. The initial system size permits us to control the number of spin- atoms. With increasing , the configuration with all Rydberg atoms located along the circumference becomes energetically unfavorable compared to configurations with an extra Rydberg atom in the center. This structural change is directly visible in the observed patterns shown in Fig. 4.

In conclusion, we have prepared and studied the many-body ground state in a strongly interacting Ising model across the transition to the crystalline phase. This should be contrasted with (*25*), where the observation of ordered structures in highly excited many-body states relied on postselection of the pictures with above average Rydberg number. Such states can be realized by straightforward pulsed laser excitation, whereas preparation of the low-energy states requires precise coherent control of the many-body system via laser coupling and initial density engineering. In the future, our technique might be used to explore quantum phase transitions and the predicted intriguing dynamics when crossing them, such as two-stage melting via a floating crystal phase (*17*, *18*). More generally, our results enable studies of long-range quantum correlations and dissipative quantum magnets in Ising-type spin systems (*26*–*28*). The demonstrated level of control over Rydberg many-body systems is an important step toward the control of multi-atom correlations required for the quantum simulation of dynamical gauge theories (*29*).