Assembling a Bose-Hubbard superfluid from tweezer-controlled single atoms

TL;DR

The study addresses bottom-up preparation of low-entropy itinerant many-body states in a two-dimensional Bose-Hubbard system by rearranging laser-cooled ${}^{86}$Sr atoms in a programmable tweezer array and then adiabatically linking them to a lattice-plus-pancake trap. The authors assemble a near-ground-state product state, transfer atoms into a 2D lattice, and infer thermodynamic properties by comparing parity-projected density profiles to finite-temperature Quantum Monte Carlo (QMC) calculations, finding a trap-averaged entropy per particle of about $S/N \approx 2 k_B$ and a temperature regime compatible with a sizable superfluid fraction. Time-of-flight measurements show diffraction peaks indicating phase coherence, while QMC for a homogeneous $10\times10$ system at $n_{\rm hom}=0.324$ yields a superfluid fraction of roughly $f_s \approx 0.5$, though finite-size effects blur the thermodynamic transition. Overall, the work demonstrates a path to bottom-up assembly of itinerant quantum matter with tunable entropy, enabling future explorations of low-entropy Fermi-Hubbard/SU($N$) physics and dynamical superfluid properties in neutral-atom and molecular platforms.

Abstract

Quantum simulation relies on the preparation and control of low-entropy many-body systems to reveal the behavior of classically intractable models. The development of new approaches for realizing such systems therefore represents a frontier in quantum science. Here we experimentally demonstrate a new protocol for generating ultracold, itinerant many-body states in a tunnel-coupled two-dimensional optical lattice. We do this by adiabatically connecting a near-ground-state-cooled array of up to 50 single strontium-86 atoms with a Bose-Hubbard superfluid. Through comparison with finite-temperature quantum-Monte-Carlo calculations, we estimate that the entropy per particle of the prepared many-body states is approximately $2 k_B$, and that the achieved temperatures are consistent with a significant superfluid fraction. This represents the first time that itinerant many-body systems have been prepared from rearranged atoms, opening the door to bottom-up assembly of a wide range of neutral-atom and molecular systems.

Figures (5)

• Figure 1: Tweezer prepared states of interacting atoms.(a) Schematic illustration of the experimental protocol. Roman numerals indicate and label a corresponding series of ramps that take potentials from one panel to the next. ${{}^\mathrm{86}\mathrm{Sr}}$ atoms (blue circles) near their motional ground state are initially trapped in a combination of optical tweezers (green double cone, wavelength $515\,\rm{nm}$) and a two-dimensional (2D) square optical lattice (gray lines, with lattice constant $a_L$). A highly anisotropic 'pancake trap' (light-red shading) is realized by overlapping a single plane of a one-dimensional (1D) optical lattice along the axis $\hat{z} = \hat{x} \times \hat{y}$young2023programmable. $J$ and $U$ correspond to the hopping and interaction parameters, respectively, in the Bose-Hubbard model, which describes the system when the 2D lattice is in the tight-binding regime. (b) Black curves (not drawn to scale) depict 1D cuts through the total potentials shown in (a), including the lattice, tweezers, and pancake trap. (c) From left to right: a single-shot image of the random initial fill for atoms in the array; a single-shot image of atoms after successful rearrangement; arrows representing adiabatic ramps of the optical potentials, and ket-vectors indicating the adiabatic transformation from a classical product state to a superfluid; finally, a single shot image of the itinerant state in the 2D lattice. (d) Phase diagram for the Bose-Hubbard model. Values from a zero-temperature mean-field calculation (see Methods) of the norm-squared order parameter at the center of the trap (where we take $i=0$) $|\psi|^2 = |\langle \hat{b}_0 \rangle|^2$ as a function of the chemical potential $\mu_0$ and lattice depth $V_L$ are indicated by the black-gray-white colorbar. Blue, purple, and red points are the locations in the phase diagram corresponding to experimental data, and the color represents the temperature of the system at each point. The black region labeled $\rho = 0$ corresponds to vacuum states, and the region labeled $\rho = 1$ corresponds to unit-filling Mott-insulator states.
• Figure 1: Sideband thermometry before and after Hubbard state preparation. Measured excitation probability of ${}^3{\rm{P}}_1$ atoms for spectroscopy performed before (black squares) and after (red circles) the Hubbard state preparation protocol. Panel (a) [(b)] shows the spectrum for one of the in-plane directions [the out-of-plane direction]. The "before" thermometry probes the initially stochastically filled ensemble shown in the first panel of Fig. \ref{['fig:1']}(c). The "after" thermometry is taken by preparing an $N=50$ ensemble at $20.8(3)\,E_R$, and then ramping both lattices up quickly to pin the atoms while remaining adiabatic with respect to the band gap. Solid lines are fits to a sum of three Lorentzians, with the amplitudes $A_{\rm rsb}$ ($A_{\rm bsb}$) of the red (blue) sidebands related to the inferred mean motional occupation number $\bar{n}$ by the relation $\bar{n} = 1/(A_{\rm bsb}/A_{\rm rsb}-1)$.
• Figure 2: Many-body thermometry by comparison to quantum Monte Carlo.(a)Row 1: Mean fluorescence images of lattice-prepared states with $N=18$ and lattice depths indicated in units of $E_R$. The density $\rho(x,y)$ is parity-projected by the imaging protocol. Row 2: Quantum-Monte-Carlo (QMC) simulations of the parity-projected density $\rho(x,y)$ at temperatures quoted in the top left of each image, and $N=18$. Lattice depths match those stated in the corresponding panels in the first row. Row 3: Mean fluorescence images of atoms after a time of flight (TOF) performed by quenching the 2D lattice off for $1.5\,\rm{ms}$, and then quenching it back on to pin atoms in deep potential wells for imaging. (b) Gray circles with black edges show the parity projected density $\rho(x,y)$ for the central row of data with $N=18, V_L = 5.1(1)\,E_R$ [top left panel of part (a)]. Solid lines come from QMC calculations at temperatures quoted in the legend. (c)Right axis: The thermal energy $k_B T$ for temperatures of the QMC results shown in row 2 of (a). Left axis: The hopping parameter $J$ plotted continuously versus lattice depth. (d) The maximum value of $\rho$ for the data in the top row of (a) is shown versus 2D lattice depth.
• Figure 3: Temperature and time-of-flight expansions versus system size. All data and simulations in this figure are performed with a lattice depth of $V_L = 5.1\,E_R$. (a) Temperature extracted by comparison to QMC calculations (as in Fig. \ref{['fig:2']}) versus $N$. (b) peak values of the real-space momentum distributions shown in (c). (c) QMC calculations of the momentum distributions for the $N$ and $T$ in (a). (d)Row 1: Using the momentum distributions in (c), we perform a semi-classical calculation (described in the main text) to simulate a $1.5\,\rm{ms}$ time-of-flight (TOF) expansion for the system in the pancake trap. The simulation results are shown in the top row. Row 2 displays mean fluorescence images for a TOF expansion of $1.5\,\rm{ms}$ in the pancake trap. Row 3 compares the parity-projected density $\rho(x,y)$ for the central rows of lattice sites. The open black circles correspond to experimental data, and the green curve is the result of the semi-classical simulation.
• Figure 4: Entropy.(a) Curves correspond to the total entropy $S$ divided by $N$ as a function of lattice depth, and for temperatures in Fig. \ref{['fig:3']}(a). The black dashed line indicates the expected $S/N$ for initially laser-cooled atoms, and the vertical width of the gray shaded region corresponds to 2$\sigma$ uncertainty in this estimate (see Methods). (b) The inferred entropy per particle at the center of the harmonic confinement -- denoted $(S/N)_{\rm cent.}$ in the main text -- plotted as a function of lattice depth. For both (a) and (b), gray, blue, and red points represent $N=4, 18,$ and $50$ respectively.