Response to Comment on "Pushing the frontiers of density functionals by solving the fractional electron problem".

Gerasimov et al. claim that the ability of DM21 to respect fractional charge (FC) and fractional spin (FS) conditions outside of the training set has not been demonstrated in our paper. This is based on (i) asserting that the training set has a ~50% overlap with our bond-breaking benchmark (BBB) and (ii) questioning the validity and accuracy of our other generalization examples. We disagree with their analysis and believe that the points raised are either incorrect or not relevant to the main conclusions of the paper and to the assessment of general quality of DM21.

Pushing the frontiers of density functionals by solving the fractional electron problem.

Density functional theory describes matter at the quantum level, but all popular approximations suffer from systematic errors that arise from the violation of mathematical properties of the exact functional. We overcame this fundamental limitation by training a neural network on molecular data and on fictitious systems with fractional charge and spin. The resulting functional, DM21 (DeepMind 21), correctly describes typical examples of artificial charge delocalization and strong correlation and performs better than traditional functionals on thorough benchmarks for main-group atoms and molecules. DM21 accurately models complex systems such as hydrogen chains, charged DNA base pairs, and diradical transition states. More crucially for the field, because our methodology relies on data and constraints, which are continually improving, it represents a viable pathway toward the exact universal functional.

A deep learning model for predicting next-generation sequencing depth from DNA sequence.

Targeted high-throughput DNA sequencing is a primary approach for genomics and molecular diagnostics, and more recently as a readout for DNA information storage. Oligonucleotide probes used to enrich gene loci of interest have different hybridization kinetics, resulting in non-uniform coverage that increases sequencing costs and decreases sequencing sensitivities. Here, we present a deep learning model (DLM) for predicting Next-Generation Sequencing (NGS) depth from DNA probe sequences. Our DLM includes a bidirectional recurrent neural network that takes as input both DNA nucleotide identities as well as the calculated probability of the nucleotide being unpaired. We apply our DLM to three different NGS panels: a 39,145-plex panel for human single nucleotide polymorphisms (SNP), a 2000-plex panel for human long non-coding RNA (lncRNA), and a 7373-plex panel targeting non-human sequences for DNA information storage. In cross-validation, our DLM predicts sequencing depth to within a factor of 3 with 93% accuracy for the SNP panel, and 99% accuracy for the non-human panel. In independent testing, the DLM predicts the lncRNA panel with 89% accuracy when trained on the SNP panel. The same model is also effective at predicting the measured single-plex kinetic rate constants of DNA hybridization and strand displacement.

Observing Localization in a 2D Quasicrystalline Optical Lattice.

Quasicrystals are long-range ordered but not periodic, representing an interesting middle ground between order and disorder. We experimentally and numerically study the localization transition in the ground state of noninteracting and weakly interacting bosons in an eightfold symmetric quasicrystalline optical lattice. In contrast to typically used real space in situ techniques, we probe the system in momentum space by recording matter wave diffraction patterns. Shallow lattices lead to extended states whereas we observe a localization transition at a critical lattice depth of V_{0}≈1.78(2)E_{rec} for the noninteracting system. Our measurements and Gross-Pitaevskii simulations demonstrate that in interacting systems the transition is shifted to deeper lattices, as expected from superfluid order counteracting localization. Quasiperiodic potentials, lacking conventional rare regions, provide the ideal testing ground to realize many-body localization in 2D.

Synthetic dissipation and cascade fluxes in a turbulent quantum gas.

Scale-invariant fluxes are the defining property of turbulent cascades, but their direct measurement is a challenging experimental problem. Here we perform such a measurement for a direct energy cascade in a turbulent quantum gas. Using a time-periodic force, we inject energy at a large length scale and generate a cascade in a uniformly trapped three-dimensional Bose gas. The adjustable trap depth provides a high-momentum cutoff k D, which realizes a synthetic dissipation scale. This gives us direct access to the particle flux across a momentum shell of radius k D, and the tunability of k D allows for a clear demonstration of the zeroth law of turbulence. Moreover, our time-resolved measurements give unique access to the pre-steady-state dynamics, when the cascade front propagates in momentum space.

Emergence of a turbulent cascade in a quantum gas.

A central concept in the modern understanding of turbulence is the existence of cascades of excitations from large to small length scales, or vice versa. This concept was introduced in 1941 by Kolmogorov and Obukhov, and such cascades have since been observed in various systems, including interplanetary plasmas, supernovae, ocean waves and financial markets. Despite much progress, a quantitative understanding of turbulence remains a challenge, owing to the interplay between many length scales that makes theoretical simulations of realistic experimental conditions difficult. Here we observe the emergence of a turbulent cascade in a weakly interacting homogeneous Bose gas-a quantum fluid that can be theoretically described on all relevant length scales. We prepare a Bose-Einstein condensate in an optical box, drive it out of equilibrium with an oscillating force that pumps energy into the system at the largest length scale, study its nonlinear response to the periodic drive, and observe a gradual development of a cascade characterized by an isotropic power-law distribution in momentum space. We numerically model our experiments using the Gross-Pitaevskii equation and find excellent agreement with the measurements. Our experiments establish the uniform Bose gas as a promising new medium for investigating many aspects of turbulence, including the interplay between vortex and wave turbulence, and the relative importance of quantum and classical effects.

Possible rules for the ancestral origin of Hox gene collinearity.

The Hox gene cluster is believed to have formed from a single ProtoHox gene by repeated cycles of the following events: tandem gene duplication, mutation to generate a new expression boundary along the embryonic axis, and acquisition of a new Hox patterning function. The Hox cluster in Bilateria evolved in compliance with the so-called collinearity rule. That is, the order of the genes along the chromosome corresponds with the order of their embryonic expression domains along the head-tail axis. Gaunt (2015) suggested that collinearity may have arisen as a mechanism to minimise the incidence of boundaries between active and inactive genes within the Hox cluster. We now attempt to clarify the model by presenting it in the form of three rules: 1) no two Hox genes may persist in the same cluster with the same anterior boundary of activity in the same tissue; 2) an inactive Hox gene must not be flanked by two active Hox genes; 3) an active Hox gene must not be flanked by two inactive genes. We provide evidence and illustrative computer simulations to show that these rules, which can apply only to partially overlapping patterns of Hox activity, may account for the ancestral origin of Hox gene collinearity.

Quantum gases. Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas.

Kibble-Zurek theory models the dynamics of spontaneous symmetry breaking, which plays an important role in a wide variety of physical contexts, ranging from cosmology to superconductors. We explored these dynamics in a homogeneous system by thermally quenching an atomic gas with short-range interactions through the Bose-Einstein phase transition. Using homodyne matter-wave interferometry to measure first-order correlation functions, we verified the central quantitative prediction of the Kibble-Zurek theory, namely the homogeneous-system power-law scaling of the coherence length with the quench rate. Moreover, we directly confirmed its underlying hypothesis, the freezing of the correlation length near the transition. Our measurements agree with a beyond-mean-field theory and support the expectation that the dynamical critical exponent for this universality class is z = 3/2.

Quantum Joule-Thomson effect in a saturated homogeneous Bose gas.

We study the thermodynamics of Bose-Einstein condensation in a weakly interacting quasihomogeneous atomic gas, prepared in an optical-box trap. We characterize the critical point for condensation and observe saturation of the thermal component in a partially condensed cloud, in agreement with Einstein's textbook picture of a purely statistical phase transition. Finally, we observe the quantum Joule-Thomson effect, namely isoenthalpic cooling of an (essentially) ideal gas. In our experiments this cooling occurs spontaneously, due to energy-independent collisions with the background gas in the vacuum chamber. We extract a Joule-Thomson coefficient µJT>10(9) K/bar, about 10 orders of magnitude larger than observed in classical gases.

Stability of a unitary Bose gas.

We study the stability of a thermal (39)K Bose gas across a broad Feshbach resonance, focusing on the unitary regime, where the scattering length a exceeds the thermal wavelength λ. We measure the general scaling laws relating the particle-loss and heating rates to the temperature, scattering length, and atom number. Both at unitarity and for positive a<<λ we find agreement with three-body theory. However, for a<0 and away from unitarity, we observe significant four-body decay. At unitarity, the three-body loss coefficient, L(3) proportional λ(4), is 3 times lower than the universal theoretical upper bound. This reduction is a consequence of species-specific Efimov physics and makes (39)K particularly promising for studies of many-body physics in a unitary Bose gas.

Bose-Einstein condensation of atoms in a uniform potential.

We have observed the Bose-Einstein condensation of an atomic gas in the (quasi)uniform three-dimensional potential of an optical box trap. Condensation is seen in the bimodal momentum distribution and the anisotropic time-of-flight expansion of the condensate. The critical temperature agrees with the theoretical prediction for a uniform Bose gas. The momentum distribution of a noncondensed quantum-degenerate gas is also clearly distinct from the conventional case of a harmonically trapped sample and close to the expected distribution in a uniform system. We confirm the coherence of our condensate in a matter-wave interference experiment. Our experiments open many new possibilities for fundamental studies of many-body physics.

Robust digital holography for ultracold atom trapping.

We have formulated and experimentally demonstrated an improved algorithm for design of arbitrary two-dimensional holographic traps for ultracold atoms. Our method builds on the best previously available algorithm, MRAF, and improves on it in two ways. First, it allows for creation of holographic atom traps with a well defined background potential. Second, we experimentally show that for creating trapping potentials free of fringing artifacts it is important to go beyond the Fourier approximation in modelling light propagation. To this end, we incorporate full Helmholtz propagation into our calculations.