## Abstract

Quantum gases in optical lattices offer an opportunity to experimentally realize and explore condensed matter models in a clean, tunable system. We used single atom–single lattice site imaging to investigate the Bose-Hubbard model on a microscopic level. Our technique enables space- and time-resolved characterization of the number statistics across the superfluid–Mott insulator quantum phase transition. Site-resolved probing of fluctuations provides us with a sensitive local thermometer, allows us to identify microscopic heterostructures of low-entropy Mott domains, and enables us to measure local quantum dynamics, revealing surprisingly fast transition time scales. Our results may serve as a benchmark for theoretical studies of quantum dynamics, and may guide the engineering of low-entropy phases in a lattice.

Microscopic measurements can reveal properties of complex systems that are not accessible through statistical ensemble measurements. For example, scanning tunneling microscopy has allowed physicists to identify the importance of nanoscale spatial inhomogeneities in high-temperature superconductivity (*1*), and single-molecule microscopy (*2*) has enabled studies of local dynamics in chemical reactions, revealing the importance of multiple reaction pathways (*3*). Whereas previous ultracold quantum gas experiments focused primarily on statistical ensemble measurements, the recently introduced single atom–single lattice site imaging technique in a quantum gas microscope (*4*) opens the door for probing and controlling quantum gases on a microscopic level. Here, we present a microscopic study of an atom-lattice system that realizes the bosonic Hubbard model and exhibits a quantum phase transition from a superfluid to a Mott insulator (*5*–*7*).

In the weakly interacting superfluid regime, the many-body wave function factorizes into a product of states with well-defined phase on each lattice site, known as coherent states, with Poissonian number fluctuations. As the strength of the interaction increases, the number distribution is narrowed, resulting in a fixed atom number state on each site deep in the Mott insulator regime. We study this change in the number statistics across the transition; these microscopic studies are complementary to previous experiments that have focused on measuring ensemble properties such as long-range phase coherence, excitation spectra, or compressibility (*7*–*9*). Local properties such as on-site number statistics (*10*) had been accessible only indirectly (*8*, *11*, *12*) and were averaged over several shells of superfluid and Mott insulating domains in the inhomogeneous system, complicating quantitative interpretation. More recently, the shell structure was imaged through tomographic (*13*), spectroscopic (*14*), and in situ imaging techniques, coarse-grained over several lattice sites (*15*).

We started with a two-dimensional (2D) ^{87}Rb Bose-Einstein condensate of a few thousand atoms confined in a single well of a standing wave, with a harmonic oscillator length of 130 nm (*16*). The condensate resided 9 μm from an in-vacuum lens that was part of an imaging system with a resolution of ~600 nm. This high-resolution system was used to project a square lattice potential onto the pancake cloud with a periodicity of *a* = 680 nm, as described in (*4*). The lattice depth was ramped exponentially with a time constant of 81 ms up to a maximum depth of 16*E*_{r}, where *E*_{r} is the recoil energy of the effective lattice wavelength given by *h*^{2}*/*8*ma*^{2} (where *m* is the mass of ^{87}Rb and* h* is Planck’s constant). In a homogeneous system in two dimensions, the transition to a Mott insulator with one atom per site occurs at a ratio of interaction energy to tunneling rate of *U/J* = 16.7 (*17*–*19*), corresponding to a lattice depth of 12.2*E*_{r}. During this ramp, the initial transverse confinement of 9.5 Hz was increased such that the cloud size remained approximately constant. After preparing the many-body state, we imaged the atoms by increasing the lattice depth by a factor of several hundred, and then illuminated the atoms with laser cooling light that served to localize the atoms while fluorescence photons were collected by high-resolution optics. As a result of the imaging process, the many-body wave function was projected onto number states on each lattice site. In addition, light-assisted collisions immediately ejected atoms in pairs from each lattice site, leaving behind an atom on a site only if its initial occupation was odd (*20*). The remaining atoms scattered several thousand photons during the exposure time and could be detected with high fidelity. By preparing the sample repeatedly under the same conditions, we deduced the probability *p*_{odd} of having an odd number of atoms on a site before the measurement.

For a coherent state on a lattice site with mean atom number λ, *p*_{odd} is given by ½[1 – exp(−2λ)] < ½. In a Mott-insulating region in the zero temperature and zero tunneling limit, *p*_{odd} = 1 for shells with an odd atom number per site, and *p*_{odd} = 0 for shells with an even atom number per site. Figure 1, A to D, shows fluorescence images in a region of the cloud as the final depth of the lattice is increased. The initial superfluid density was chosen to obtain an insulator with two shells on the Mott side of the transition; the region shown is in the outer shell containing one atom per site. For high filling fractions, the lattice sites in the images were barely resolved, but the known geometry of the lattice and imaging system point-spread function obtained from images at sparser fillings allowed reliable extraction of site occupations (*16*).

We used 24 images at each final lattice depth to determine *p*_{odd} for each site. The transverse confining potential varied slowly relative to the lattice spacing, and the system was, to a good approximation, locally homogeneous. We made use of this to improve the error in our determination of *p*_{odd} by averaging over a group of lattice sites—in this case, 51 sites for regions in the first shell and 30 sites for regions in the second shell (Fig. 1E). In the *n* = 1 shell, we detected an atom on a site with probability 94.9 ± 0.7% at a lattice depth of 16*E*_{r}. We measured the lifetime of the gas in the imaging lattice and determined that 1.75 ± 0.02% of the occupied sites were detected as unoccupied, as a result of atoms lost during the imaging exposure time (1 s) because of background gas collisions. The average occupation numbers and error bars shown in Fig. 1E include corrections for this effect.

Measuring the defect density in the Mott insulator provides sensitive local thermometry deep in the Mott regime. Thermometry in the Mott state has been a long-standing experimental challenge (*21*, *22*) and has acquired greater importance as experiments approach the regime of quantum magnetism (*23*–*25*), where the temperature scale should be on the order of the superexchange interaction energy. Our method directly images the excitations of the *n* = 1 Mott insulator, holes and doublons, because they both appear as missing atoms in the images. Similarly, for Mott insulators with higher fillings *n*, sites with excitations (*n* + 1, *n* − 1) can be detected through their opposite-parity signal. For a finite tunneling rate *J* much smaller than the interaction energy *U*, the admixture fraction of coherent hole-doublon pair excitations is ~(*J/U*)^{2}, whereas any other excitations are due to incoherent thermal fluctuations and are suppressed by a Boltzmann factor, exp(–*U/T*).

The theory curves presented in Fig. 1E are the predicted *p*_{odd} in the two shells for different values of *T/U*. The curves were obtained using a quantum Monte Carlo “worm” algorithm (*26*, *27*), and the average temperature extracted using the data points at the three highest *U/J* ratios was *T/U* ≈ 0.16 ± 0.03. At the transition point for *n* = 1, this corresponds to a temperature of 1.8 nK. Assuming this value of *T/U* to be the overall temperature, the thin layer between the Mott shells should be superfluid, and the transition to a normal gas is expected around a critical temperature of *zJ* = 2.8 nK, where *z* is the number of nearest neighbors in the lattice (*28*).

Next, we studied the global structure of the Mott insulator. The high-resolution images provide an atom-by-atom picture of the concentric shell structure, including the transition layers between the insulating shells. In Fig. 2, A to D, the formation of the various shells, up to the fourth, is shown as the atom number in the trap is increased. Slowly varying optical potential disorder causes deviation from circular symmetry in the shells. The contour lines of the potential are directly seen in the images in Fig. 2. In Fig. 2, E and F, we compensated for this disorder by projecting a light pattern, generated using a digital micromirror device, through the objective (*16*), resulting in a nearly circular shell structure.

In a second series of experiments, we used on-site number statistics to probe the adiabaticity time scale for the transition, focusing on the local dynamics responsible for narrowing the number distribution. We started by increasing the lattice depth adiabatically to 11*E*_{r}, still in the superfluid regime, using the same ramp described previously. Next, the depth was ramped linearly to 16*E*_{r}, where, for an adiabatic ramp, a Mott insulator should form. The ramp time was varied from 0.2 to 20 ms, and *p*_{odd} was measured in the first and second shells as before (Fig. 3). We found that the data fit well to exponential curves that asymptotically approach the value of *p*_{odd} obtained in the adiabatic case. The fitted time constant in the first shell is 3.5 ± 0.5 ms; in the second shell, the constant is 3.9 ± 1.3 ms.

Relative to the critical value of the tunneling time *h/J*_{c} = 68 ms for the first shell, the observed dynamics were counterintuitively fast. This can be understood by using a simple picture of two atoms in a double well. In this system, as the tunneling is varied, the minimal gap between the ground state and the first excited state is *U*, which sets the adiabaticity time scale. It is an open question whether this argument can be generalized to a lattice. In an infinite system, the appearance of Goldstone modes in the superfluid regime leads to a vanishing gap at the transition point, but the density of states is low for energies much less than *U* (*29*). In fact, the 1*/e* time scale observed experimentally is comparable to *h/U*_{c} = 4.1 ms, where *U*_{c} is the critical interaction energy for an *n* = 1 insulator.

Although the local number statistics change on a fast time scale of *h/U*, entropy redistribution in the inhomogeneous potential should occur on a much slower time scale of *h/J*. Because superfluid and normal domains have a larger specific heat capacity than Mott domains, in an inhomogeneous system, entropy is expelled from the Mott domains and accumulates in the transition regions after crossing the phase transition if the system is in thermal equilibrium (*30*). However, in bulk Mott regions, the insulating behavior makes entropy transport difficult, and global thermalization is slow on experimental time scales (*31*). In our system, optical potential corrugations produce sizable potential gradients in some regions, leading to a heterostructure of almost 1D Mott domains, about one to two lattice sites thick, surrounded by transition layers (Fig. 4). We found remarkably low defect densities and sharp transitions between superfluid and Mott states in these regions. The measured defect probability per site in the domain shown is 0.8 ± 0.8%. In these microscopic domains, each site of a Mott domain is in contact with a superfluid region. Such a configuration is likely to lead to fast thermalization, which would explain the low defect density we observed. This suggests that the lowest entropies in a Mott insulator might be obtained under conditions where the chemical potential is engineered so as to obtain alternating stripes (2D) or layers (3D) of insulating and superfluid regions (*32*, *19*).

In addition to the number statistics studied in this work, single-site imaging could be applied to study spatial correlations in strongly correlated quantum gases (*33*), and to directly measure entanglement in a quantum information context. The low-defect Mott states we detect would provide an ideal starting point for quantum magnetism experiments; if the low entropy in the Mott domains can be carried over to spin models, it should be possible to realize magnetically ordered states such as antiferromagnets, which could be directly detected with single-site imaging.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1192368/DC1

Materials and Methods

References

## References and Notes

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- We thank G. Jotzu, E. Demler, D. Pekker, B. Wunsch, T. Kitagawa, E. Manousakis, W. Ketterle, and M. D. Lukin for stimulating discussions. Supported by a grant from the Army Research Office with funding from the Defense Advanced Research Projects Agency Optical Lattice Emulator program, and grants from Air Force Office of Scientific Research Multidisciplinary University Research Initiative, NSF, an Alfred P. Sloan Fellowship (M.G.), and the Swiss National Science Foundation. The simulations were run on the Brutus cluster at ETH Zürich.