Ruddlesden-Popper bilayer nickelates provide an emerging platform for studying high-temperature superconductivity, yet the superconducting pairing symmetry remains under debate. Here, we use atomic-resolution scanning tunnelling microscopy and spectroscopy to investigate superconducting 1.5-unit-cell (La,Pr)3Ni2O7 films grown on SrLaAlO4. A cryogenic ultrahigh-vacuum (UHV) sample transfer preserves an ordered sqrt(2) * sqrt(2) surface and yields reproducible U-shaped spectra with two gap scales of ~14 and ~20 meV and extended flat zero-conductance bottoms. By contrast, samples exposed for a longer time in UHV without cooling during transfer show V-shaped spectra despite retaining the surface reconstruction and a transport superconducting transition onset above 40 K. Wide-energy-range spectra indicate that oxygen loss can mix density-wave-related spectral weight. Our measurements provide an atomic-scale observation of the intrinsic nodeless superconducting gap in bilayer nickelate ultrathin films.
How the detailed entanglement structure emerges from quantum dynamics remains a fundamental challenge motivated by recent advances in quantum simulators and information processing. As a central milestone, the small-incremental-entangling (SIE) theorem bounds the entanglement-entropy growth rate but does not control the entanglement spectrum itself. In this work, we define the spectral-entangling strength, which quantifies how strongly an operator can reshape the distribution of Schmidt coefficients across a bipartition. We then prove a spectral-SIE theorem: for Rényi index α≥1/2, the growth rate of Rényi-entanglement entropies admits a universal bound. Remarkably, our bound at α=1/2 is both qualitatively and quantitatively optimal; below this threshold (α<1/2), no universal speed limit on entanglement growth can exist. This result yields a sharp 1/s^{2} threshold in the tail of the ordered squared Schmidt coefficients, or equivalently in the entanglement-spectrum weights (with s the Schmidt index). It enables rigorous truncation-based-error control and establishes a quantitative link between entanglement-spectrum structure and computational complexity. As a practical highlight, for one-dimensional power-law interactions 1/r^{η} with η>2, this implies matrix-product-state approximations with bond dimension polynomial in (n/ε) for ground states, real-time evolved states, and Gibbs states, thereby closing the quasipolynomial gap, where n is the system size, and ε denotes the target approximation error in the relevant state or operator norm. By controlling Rényi entanglement, we further obtain a rigorous a priori truncation-error bound for the time-dependent density-matrix renormalization-group scheme analyzed here. Overall, we extend the SIE paradigm from bounding entanglement entropies to constraining the entanglement spectrum itself.
Molecular hydrogen (H_2) is the simplest neutral molecule and its ionization and dissociation energies are key benchmarks for testing first-principles calculations of nonadiabatic, relativistic, quantum-electrodynamic and finite-nuclear-size effects in molecules. To obtain improved H_2 ionization and dissociation energies, we performed precision measurements of 15 transition frequencies in para-H_2. They connect the X ^1Σ_g^+(v=0,N=0) ground state, the long-lived EF ^1Σ_g^+(v=0,N=0) excited state, and several high members of the np and nd Rydberg series with principal quantum number n in the range between 48 and 70. The new data by itself leads to a five times more accurate value of the ionization energy of the ground state of para-H_2, equivalent to a relative frequency accuracy of 5× 10^-11. Combined with a global network analysis of 45 energy intervals in para- and ortho-H_2 that have been measured in the past three decades, we obtain new statistically robust values for the ionization energy [E_ I(0,0)/(hc)= 124 417.491 1573(58) cm^-1] and the dissociation energy [D_0^N=0=36 118.069 6646(58) cm^-1] of H_2. These values have an unprecedented accuracy equivalent to 175 kHz. The network analysis also leads to improved term values and dissociation energies of the excited states involved in the analysis. By comparing the results with first-principles calculations of the level structures of H_2 and H_2^+ , a significantly reduced upper bound of 540 kHz is derived for putative beyond-the-standard-model contributions to the energy difference between para- and ortho-H_2.
