Crystallization in Ising quantum magnets

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Science  27 Mar 2015:
Vol. 347, Issue 6229, pp. 1455-1458
DOI: 10.1126/science.1258351

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


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 (814).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 (810). The physical system studied is a well-defined line- or disc-shaped atomic sample in an optical lattice with one atom per site. The 87Rb atoms are coupled to the Rydberg state 43S1/2 with a controlled time-dependent Rabi frequency Embedded Image and detuning Embedded Image of the laser frequency Embedded Image from the atomic resonance Embedded Image. 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

Embedded Image (1)

Here, the vectors Embedded Image label the position of the atoms on the lattice. The spin-1/2 operators on each site are defined as Embedded Image and Embedded Image, where we omitted the site label to simplify the notation. The operators Embedded Image and Embedded Image describe a spin flip from the ground state Embedded Image to the Rydberg state Embedded Image and vice versa, whereas the operators Embedded Image and Embedded Image 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 Embedded Image. The detuning Embedded Image determines the longitudinal field and can be used to counteract the energy offset Embedded Image (positive for our parameters). The third term represents the van der Waals interaction potential between two atoms in the Rydberg state. For the 43S1/2 state, Embedded Image is repulsive because the van der Waals coefficient Embedded Image. Here, Embedded Image is the distance between two atoms on the lattice with period Embedded Image.

This system exhibits a rich variety of strongly correlated magnetic phases (4, 5, 6, 1720). In the classical limit (Embedded Image) and for Embedded Image, the many-body ground state corresponds to crystalline states with vanishing fluctuations in the total magnetization Embedded Image, which, for fixed total atom number Embedded Image, is determined by the spin-Embedded Image component Embedded Image. In a one-dimensional(1D) chain of Embedded Image lattice sites, the number of spin-Embedded Image atoms increases by one at the critical detunings Embedded Image 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 Embedded Image parameter space (the plot in Fig. 1A was calculated for Embedded Image), extended lobes corresponding to crystalline states can be identified.

Fig. 1 Phase diagram, energy spectrum, and experimental sequence.

(A) Phase diagram calculated for a 1D system of N = 7 atoms. The color scale indicates the number of spin-Embedded Image atoms Embedded Image in the many-body ground state (also visualized in the crystalline phase by the small spheres). The detuning Embedded Image and Rabi frequency Embedded Image axes are rescaled by their sixth root. (B) Evolution of the many-body spectrum during a sweep, where coupling strength Embedded Image and detuning Embedded Image are controlled. The spectrum was calculated for a representative 1D system of five atoms. First, the detuning is changed from Embedded Image to Embedded Image at constant Rabi frequency Embedded Image, with Embedded Image chosen to prepare Embedded Image. Subsequently, the Rabi frequency is reduced from Embedded Image to 0. The inset is a zoom into the end of the sweep highlighting the shrinking gap between the energy levels. The color of each line indicates the mean number of spin-Embedded Image atoms in the many-body state. For strongly negative detuning, the four different manifolds correspond to the crystalline states with fixed magnetization given by Embedded Image, as indicated in the figure. In three limiting cases in which the states become classical, the spatial distribution is shown schematically (blue circles, Rydberg atoms; gray circles, ground-state atoms). (C to E) Representative fluorescence pictures from different times in the experimental cycle. (C) Mott insulator with lattice sites (white dots) and spatial light modulator pattern (semitransparent overlay). (D) Initial atom configuration. (E) Single-shot Rydberg pattern. (F) Spatial Rydberg density (magnetization density) after averaging 40 experimental runs (darker color means more detected atoms).

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 (Embedded Image) coincides with the many-body ground state of the system for negative detuning and vanishing Rabi frequency. For a small coupling strength Embedded Image, the energy gap to the first excited state closes at the transition points Embedded Image between successive Embedded Image manifolds; thus, both Embedded Image and Embedded Image have to be varied to maximize the adiabaticity of the preparation scheme. An intuitive and simple choice of the path Embedded Image starts by slowly switching on the coupling Embedded Image at a large negative detuning Embedded Image (810). Next, the detuning is increased to the desired final blue-detuned value Embedded Image, followed by a gradual reduction of the coupling strength Embedded Image to zero. Choosing Embedded Image between the critical detunings Embedded Image of adjacent Embedded Image 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 Embedded Image 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 87Rb 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 Embedded Image 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 Embedded Image. 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 Embedded Image. The average filling was 0.8atoms per site, and at the edge it dropped to below 0.1atoms per site, within one lattice site. The coupling to the Rydberg state was realized by a two-photon process via the intermediate state Embedded Image, using laser wavelengths of 780 and 480 nm with Embedded Image and Embedded Image polarizations, respectively (25). Detailed coupling beam parameters are summarized in table S1. Fast control of the Rabi frequency Embedded Image and the detuning Embedded Image 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 Embedded Image of spin-Embedded Image atoms is limited by the interaction energy as the longitudinal magnetic field scales weakly with Embedded Image. Hence, instead of varying the detuning, we changed the length Embedded Image of the initial system to explore the characteristics of the Rydberg crystals (8). We measured the mean number of Rydberg atoms Embedded Image for varying length Embedded Image 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 Embedded ImagekHz (Fig. 2A) exhibit clear plateaus in Embedded Image 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 Embedded Image (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 Embedded Image is approximately equivalent to varying the detuning Embedded Image, we extract the susceptibility Embedded Image from our data (24). Embedded Image 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 Embedded Image, leading to the preparation of compressible superposition states.

Fig. 2 Identification of the crystalline phase.

(A) Mean spin-Embedded Image number Embedded Image versus system length Embedded Image for an elongated system. Blue circles correspond to the experimental mean number of Rydberg atoms Embedded Image after the optimized sweep. The green line is the result of the numerical simulation for the experimental initial states, taking into account an initial state filling of 0.8 and length fluctuations on the order of one site. The gray line shows the classical (Embedded Image) prediction. The only free parameter used to fit the theory to the experimental data is the detection efficiency Embedded Image. (Insets) Measured spatial distribution of the magnetization (left) and corresponding theory (right) for system lengths Embedded Image of 12, 23, and 35 sites. The brightness (light to dark) translates to the normalized number of spin-Embedded Image atoms. (B) Susceptibility Embedded Image of the prepared states. Blue circles are derived from the experimental data shown in (A), using a numerical central difference estimator for the derivative (24). The green line is a numerical calculation of Embedded Image for a given system size Embedded Image. All error bars, mean ± SEM.

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 Embedded Image, we abruptly switched off the coupling at different times, thereby projecting the many-body state onto the eigenstates of the uncoupled system Embedded Image. For the measurement, we chose the optimized sweep for the Embedded Image 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 Embedded Image and its normalized variance Embedded Image (Fig. 3B). During the sweep, Embedded Image increases until we observe a saturation behavior that we interpret as the onset of crystallization (fig. S2). Simultaneously, the Embedded Image factor decreases from the Poissonian value Embedded Image to Embedded Image, which reflects the approach to the crystalline state. The expected value Embedded Image is increased to Embedded Image 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-Embedded Image atom numbers (24). Nevertheless, we find very good agreement between theory and experiment for the previously fitted Embedded Image.

Fig. 3 Dynamical crystallization.

(A) Illustration of the laser sweep. The black line shows the path of the sweep through the phase diagram; the numbered positions mark the measurements [see (C)]. The color scale indicates the number of spin-Embedded Image atoms Embedded Image of the ground state for every pair Embedded Image. The gray line indicates the boundary of the crystalline lobes, where the Embedded Image factor drops below –0.9. The phase diagram was calculated for the experimental parameters. (B) Mean number of Rydberg atoms (blue circles) and Embedded Image factor (red circles) for the seven times marked in (A), together with the corresponding theoretical predictions (lines). Each of the seven data points is based on at least 65 independent experimental repetitions. (C) Experimental and theoretical probability distributions of the number of Rydberg atoms along the sweep [see (A)]. Blue boxes show experimental data, and the dashed and solid lines represent the theoretical result for detection efficiencies of Embedded Image and Embedded Image, respectively. (D) Transversally averaged distributions (probability per site) of the spin-Embedded Image atoms for the same times as in (C) with a binning of two sites (blue circles). The slight asymmetry toward the right might be due to a gradient in the Rabi frequency (24). The green line is the numerical result. All error bars, mean ± SEM.

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 Embedded Image. This is because of a breakdown of the translational invariance: For Embedded Image a spin-Embedded Image 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-Embedded Image 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 Embedded Image to Embedded Image (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-Embedded Image atoms. With increasing Embedded Image, 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.

Fig. 4 Dynamical crystallization in disc-shaped samples.

(A) Unprocessed experimental single shot pictures with 6, 7, and 8 Rydberg atoms from the rightmost data set. Each blue point corresponds to a single atom. (B and C) Rydberg densities for pulsed (left gray box) and sweeped laser coupling with increasing cloud size from left to right in the right gray box. The pulsed coupling was done with the same amplitude modulation as for the sweep (fig. S1B), but the detuning Embedded Image was held constant (averaged data for Embedded ImagekHz and Embedded ImagekHz is shown). The cloud radius was Embedded Image lattice sites (left to right). (B) Measured 2D distribution of the magnetization. The color scale represents the normalized counts per site. (C) Azimuthally averaged density distribution (probability per site) of the data shown in (B) (blue dots) and comparison with theory (green line). The theoretical calculation was only feasible for small clouds and takes the atomic ground-state density distribution into account. The experimentally measured initial density is shown in gray on the right axis. Error bars, mean ± SEM.

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 (2628). 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).

Supplementary Materials

Supplementary Text

Figs. S1 to S3

Table S1

References (30, 31)

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. Acknowledgments: We acknowledge funding by Max-Planck-Gesellschaft, Deutsche Forschungsgemeinschaft, European Union (UQUAM, SIQS, ITN-COHERENCE, and HAIRS, and a Marie Curie Fellowship to M.C.).
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