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Probing the Superfluid–to–Mott Insulator Transition at the Single-Atom Level

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Science  30 Jul 2010:
Vol. 329, Issue 5991, pp. 547-550
DOI: 10.1126/science.1192368

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 (57).

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 (79). 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) 87Rb 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 16Er, where Er is the recoil energy of the effective lattice wavelength given by h2/8ma2 (where m is the mass of 87Rb 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 (1719), corresponding to a lattice depth of 12.2Er. 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 podd 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, podd is given by ½[1 – exp(−2λ)] < ½. In a Mott-insulating region in the zero temperature and zero tunneling limit, podd = 1 for shells with an odd atom number per site, and podd = 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).

Fig. 1

Single-site imaging of atom number fluctuations across the superfluid–Mott insulator transition. (A to D) Images within each column are taken at the same final 2D lattice depth of 6Er (A), 10Er (B), 12Er (C), and 16Er (D). Top row: In situ fluorescence images from a region of 10 × 8 lattice sites within the n = 1 Mott shell that forms in a deep lattice. In the superfluid regime [(A) and (B)], sites can be occupied with odd or even atom numbers, which appear as full or empty sites, respectively, in the images. In the Mott insulator, occupancies other than 1 are highly suppressed (D). Middle row: results of the atom detection algorithm (16) for images in the top row. Solid and open circles indicate the presence and absence, respectively, of an atom on a site. Bottom row: Time-of-flight fluorescence images after 8-ms expansion of the cloud in the 2D plane as a result of nonadiabatically turning off the lattice and the transverse confinement (averaged over five shots and binned over 5 × 5 lattice sites). (E) Measured value of podd versus the interaction-to-tunneling ratio U/J. Data sets, with 1σ error bars, are shown for regions that form part of the n = 1 (squares) and n = 2 (circles) Mott shells in a deep lattice. The lines are based on finite-temperature Monte Carlo simulations in a homogeneous system at constant temperature-to-interaction ratio (T/U) of 0.20 (dotted red line), 0.15 (solid black line), and 0.05 (dashed blue line). The axis on the right is the corresponding odd-even variance given by podd(1 − podd).

We used 24 images at each final lattice depth to determine podd 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 podd 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 16Er. 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 (2325), 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 podd 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.

Fig. 2

Single-site imaging of the shell structure in a Mott insulator. (A to D) The images show podd on each site determined by averaging 20 analyzed fluorescence images. The lattice depth is 22Er and the transverse confinement is 45 Hz. As the atom number is increased, the number of shells in the insulator increases from one to four. The value of podd for odd-numbered shells is close to 1; for even-numbered shells, it is close to 0. The atom numbers, determined by in situ imaging of clouds expanded in the plane, are 120 ± 10 (A), 460 ± 20 (B), 870 ± 40 (C), and 1350 ± 70 (D). (E and F) Long-wavelength disorder can be corrected by projecting an appropriate compensation light pattern onto the atoms, resulting in nearly circular shells. (E) podd (average of 20 analyzed images); (F) a single-shot raw image (arbitrary units).

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 11Er, still in the superfluid regime, using the same ramp described previously. Next, the depth was ramped linearly to 16Er, where, for an adiabatic ramp, a Mott insulator should form. The ramp time was varied from 0.2 to 20 ms, and podd 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 podd 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.

Fig. 3

Dynamics of on-site number statistics for a fast ramp from the superfluid regime to the Mott regime. podd at the end of the ramp versus ramp time is shown in the n = 1 (squares) and n = 2 (circles) shells, averaged over 19 data sets with 1σ error bars. Red lines are exponential fits. Inset shows the two-part ramp used in this experiment. The first part is a fixed adiabatic exponential ramp (t = 81 ms) and the second is a linear ramp starting at 11Er and ending at 16Er. The duration of the second ramp is varied in the experiment.

Relative to the critical value of the tunneling time h/Jc = 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/Uc = 4.1 ms, where Uc 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).

Fig. 4

Low-entropy Mott domains observed in a steep potential gradient. (A) Single-shot in situ image of a Mott insulator in a 16Er deep lattice with 25-Hz transverse confinement. The ring is an n = 1 insulator enclosing an n = 2 region. (B) Average podd over 24 images. Each pixel corresponds to a single lattice site. The red rectangle encloses a region containing a Mott insulator with n = 1, a few lattice sites wide. (C) Column average of podd over the sites within the red rectangle in (B), with 1σ error bars.

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

  1. See supporting material on Science Online.
  2. 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.
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