We demonstrate laser cooling of barium monofluoride (${}^{138}\mathrm{BaF}$) molecules. We use serrodynes to synthesize time-sequenced optical spectra that can be precisely tailored to the hyperfine structure of this heaviest nonradioactive alkaline earth monofluoride. By optimizing these optical spectra, we realize strong Sisyphus cooling forces that efficiently collimate a molecular beam. Our technique is an important step toward using intense beams of barium monofluoride for precision measurement applications, and will be useful for cooling other molecular species with complex level structure.

Autocorrelation is ubiquitous in real-life time series. For trend analysis, positive autocorrelation mainly increases the statistical uncertainty of the detected trend. The theoretical framework has been established to quantify how the autocorrelation can impact trend analysis if its characteristic is known precisely. However, if the autocorrelation is also uncertain like that estimated in real-life data, then the consequences are yet unclear. The situation is further complicated by the concurrence of both short- and long-range correlation (SLRC), e.g., in air temperature. In this study, we use an appropriate minimal model with one short-range autoregressive parameter and one long-range fractional parameter to account for such SLRC, and then accordingly propose a framework to integrate the effect of the uncertainty of autocorrelation estimation into trend analysis, on the basis of the detrended fluctuation analysis and Akaike weights. This framework can also obtain a joint probability density function of the estimated autoregressive and fractional parameters for measuring their uncertainty, identifying their combined optimal estimates, uncovering their strong interdependence, and determining their two-dimensional confidence region. The effectiveness and applicability of the proposed framework are verified by a numerical experiment and a case study of daily air temperature anomalies at the Potsdam station. This study can provide a solid theoretical basis to underpin our understanding of the autocorrelation effect on trend analysis in theory and applications.

We introduce a diagrammatic multi-scale approach to the Hubbard model based on the interaction-irreducible (multi-boson) vertex of a small cluster embedded in a self-consistent medium. The vertex captures short-range correlations up to the length scale of the cluster, while long-range correlations are recovered from a set of diagrammatic equations for the Hedin three-leg vertex. By virtue of the crossing symmetry, the Fierz decoupling ambiguity of the Hubbard interaction is resolved exactly. Our benchmarks for the half-filled Hubbard model on the square lattice are in very good agreement with numerically exact diagrammatic Monte Carlo simulations.

We determine the Dzyaloshinskii-Moriya interaction within and between two magnetic cobalt layers separated by a nonmagnetic spacer through ab initio calculations. We investigate different materials for the nonmagnetic layer, focusing on the experimentally realized Co/Ag/Co system. We laterally shift the atoms in the nonmagnetic layer to achieve the symmetry breaking required for the interlayer Dzyaloshinskii-Moriya interaction. We compare the resulting interactions with the Levy-Fert model and observe a good overall agreement between the model and the ab initio calculations for the dependence on the atomic positions. Additionally, we derive a formula for the strength of the interlayer isotropic exchange interaction depending on the position of the atoms in the nonmagnetic layer and compare it to the first-principles results. We investigate the limitations of the Levy-Fert model by turning off the spin-orbit coupling separately on the nonmagnetic and magnetic atoms and by studying the effect of band filling. Our work advances the understanding of the microscopic mechanisms of the interlayer Dzyaloshinskii-Moriya interaction and gives insight into possible new material combinations with strong interlayer Dzyaloshinskii-Moriya interaction.

We show that hematite, $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$, below its Morin transition, has a ferroic ordering of rank-5 magnetic triakontadipoles on the Fe ions. In the absence of spin-orbit coupling, these are the lowest-order ferroically aligned magnetic multipoles, and they give rise to the $g$-wave nonrelativistic spin splitting in hematite. We find that the ferroically ordered magnetic triakontadipoles result from the simultaneous antiferroic ordering of the charge hexadecapoles and the magnetic dipoles, providing a route to manipulating the magnitude and the sign of the magnetic triakontadipoles as well as the spin splitting. Furthermore, we find that both the ferroic ordering of the magnetic triakontadipoles and many of the spin-split features persist in the weak ferromagnetic phase above the Morin transition temperature.

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal the following: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization and, thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely.

We calculated the structural, electronic, and magnetic properties of FeSe within density-functional theory at the generalized gradient approximation level. First, we studied how the bandwidth of the $d$-bands at the Fermi energy is renormalized by adding simple corrections: Hubbard $U$, Hund's $J$, and by introducing long-range magnetic orders. We found that introducing either a striped or a staggered dimer antiferromagnetic order brings the bandwidths—which are starkly overestimated at the generalized gradient approximation level—closer to those experimentally observed. Second, for the ferromagnetic, the striped, checkerboard, and staggered dimer antiferromagnetic order, we investigate the change in magnetic formation energy with local magnetic moment of Fe at a pressure up to 6 GPa. The bilinear and biquadratic exchange energies are derived from the Heisenberg model and noncollinear first-principles calculations, respectively. We found a nontrivial behavior of the spin-exchange parameters on the magnetization, and we put forward a field-theory model that rationalizes these results in terms of two-dimensional spin and orbital fluctuations. The character of these fluctuations can be either that of a standard density wave or a topological vortex. Topological vortices can result in mesoscopic magnetization structures.

We show how a laser driven two-level system including quantized external degrees of freedom for each state can be decoupled into a set of oscillator equations acting only on the external degrees of freedom with operator valued damping representing the detuning. We give a way of characterizing the solvability of this family of problems by appealing to a classical oscillator with time-dependent damping. As a consequence of this classification we (i) obtain analytic and representation-free expressions for Rabi oscillations including external degrees of freedom with and without an external linear potential, (ii) show that whenever the detuning operator can be diagonalized (analytically or numerically) the problem decomposes into a set of classical equations, and (iii) we can use the oscillator equations as a perturbative basis to describe Rabi oscillations in weak but otherwise arbitrary external potentials. Moreover, chirping of the driving fields phase emerges naturally as a means of compensating the Ehrenfest/mean-value part of the detuning operator's dynamics while in the presence of driving phase noise leads to a stochastic evolution equation of Langevin type. Lastly, our approach is representation free with respect to the external degrees of freedom and as consequence a suitable representation or basis expansion can be chosen a posteriori depending on the desired application at hand.

We examine the dynamical properties of a single-layer convolutional recurrent network with a smooth sigmoidal activation function, for small values of the inputs and when the convolution kernel is unitary, so all eigenvalues lie exactly at the unit circle. Such networks have a variety of hallmark properties: the outputs depend on the inputs via compressive nonlinearities such as cubic roots and both the timescales of relaxation and the length scales of signal propagation depend sensitively on the inputs as power laws, both diverging as the input $\to 0$. The basic dynamical mechanism is that inputs to the network generate ongoing activity, which in turn controls how additional inputs or signals propagate spatially or attenuate in time. We present analytical solutions for the steady states when the network is forced with a single oscillation and when a background value creates a steady state of “ongoing activity” and derive the relationships shaping the value of the temporal decay and spatial propagation length as a function of this background value.

A Gaussian quantum theory of bosonic modes has been widely used to describe quantum optical systems, including coherent Ising machines (CIMs) that consist of ${\chi }^{\left(2\right)}$ degenerate optical parametric oscillators (DOPOs) as nonlinear elements. However, Gaussian models have been thought to be invalid in the extremely strong-gain-saturation limit. Here, we develop an extended Gaussian model including two third-order fluctuation products, $\langle \delta {\widehat{X}}^3\rangle$ and $\langle \delta \widehat{X}\delta {\widehat{P}}^2\rangle$, which we call self-skewness and cross-skewness, respectively. This new model which we call the skew-Gaussian model more precisely replicates the success probability predicted by the quantum master equation (QME), relative to Gaussian models. We also discuss the impact of skew variables on the performance of CIMs.

We theoretically study the extreme ultraviolet (XUV) photoemission of the asymmetric molecular ion ${\mathrm{HeH}}^{2+}$ beyond the dipole approximation. Based on the double-slit interference of the photoelectron emitted from two atomic cores, we propose a scheme to obtain the XUV photoemission time delays originated from different physical mechanisms. In our study, two types of sub-attosecond photoemission time delays are identified and extracted from the observable photoelectron momentum distributions. One is the so-called birth time delay, originated from the travel time of the photon across the molecule, and the other is the escaping time delay of the photoelectron from two different cores. The present study shows the observable evidence demonstrating that in single-photon ionization, it takes more time for the photoelectron escaping from the helium nucleus than that from the proton.

Experimental indications have been reported suggesting that laser pulses shining on materials with relativistic dispersion can produce currents that survive long after the illumination has died out. Such residual currents, i.e., remnant currents, have applications in petahertz logical gates. The remnant currents' strength strongly depends on the pulse shape. We develop an analytical formula that allows one to optimize the pulse shape for remnant current production; we predict remnant currents exceeding the values observed so far by up to five orders of magnitude. This can be achieved by using single-cycle pulses instead of the previously employed multicycle pulses. In fact, remnant currents can be almost as strong as the peak current under irradiation.

Precision optical measurements of the electron-spin precession of nitrogen-vacancy (NV) centers in diamond form the basis of numerous applications. The most sensitivity-demanding applications, such as femtotesla magnetometry, require the ability to measure changes in GHz spin transition frequencies at the sub-millihertz level, corresponding to a fractional resolution of better than ${10}^{-12}$. Here we study the impact of microwave (MW) phase noise on the response of an NV sensor. Fluctuations of the phase of the MW waveform cause undesired rotations of the NV spin state. These fluctuations are imprinted in the optical readout signal and, left unmitigated, are indistinguishable from magnetic-field noise. We show that the phase noise of several common commercial MW generators results in an effective $\mathrm{pT}{\mathrm{s}}^{1/2}$-range noise floor that varies with the MW carrier frequency and the detection frequency of the pulse sequence. The data are described by a frequency-domain model incorporating the MW phase-noise spectrum and the filter-function response of the sensing protocol. For controlled injection of white and random-walk phase noise, the observed NV magnetic noise floor is described by simple analytic expressions that accurately capture the scaling with pulse sequence length and the number of $\pi$ pulses. We outline several strategies to suppress the impact of MW phase noise and implement a version, based on gradiometry, that realizes a $>10$-fold suppression. Our study highlights an important challenge in the pursuit of sensitive diamond quantum sensors and is applicable to other qubit systems with a large transition frequency.

In the search of quantum spin liquid (QSL), spin-1/2 triangular-lattice Heisenberg antiferromagnets have always been viewed as fertile soils. Despite the true magnetically ordered ground state, anisotropy has been considered to play a significant role in stabilizing a QSL state. However, the nature and ground state of the most anisotropic case, the triangular-lattice Ising antiferromagnet (TLIAF), remain elusive and controversial. Here, we report specific-heat and thermal-conductivity measurements on a newly discovered Ising-type QSL candidate ${\mathrm{PrMgAl}}_{11}{\mathrm{O}}_{19}$. At zero field, the magnetic specific heat shows a quadratic temperature dependence. On the contrary, no direct positive magnetic contribution to thermal conductivity is detected, ruling out the presence of mobile gapless fermionic excitations. Further analysis of phonon thermal conductivity reveals that the phonons are strongly scattered by thermally activated magnetic excitations out of a gap, which exhibits a linear dependence with magnetic field. These results demonstrate that the spin-1/2 TLIAF ${\mathrm{PrMgAl}}_{11}{\mathrm{O}}_{19}$ has a gapped $Z_2$ QSL ground state.

We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in the two-dimensional $\alpha \text{−}{T}_3$ lattice. By tuning the hopping parameter $\alpha \in [0,1]$, the $\alpha \text{−}{T}_3$ lattice interpolates between pseudospin $S=1/2$ (graphene) and $S=1$ ($T_3$ or dice lattice), for $\alpha =0$ and 1, respectively, which is followed by a continuous change of the Berry phase from $\pi$ to 0. The range of existence for edge states in the momentum space is determined by solving tight-binding equations at the boundaries of the ZRs. We find that the transitions of in-gap bands from bulk to edge states in the momentum space do not only occur at the positions of the Dirac cones but also at additional points depending on $\alpha \in (0,1)$. The $\alpha \text{−}{T}_3$ ZRs are mapped onto stub Su-Schrieffer-Heeger chains by performing unitary transformations of the bulk Hamiltonian. The nontrivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the topological invariant is manifested by winding numbers on the complex plane and the Bloch sphere.

We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a self-similar and universal pattern termed an “entanglement fractal” in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems.

The ubiquitous domain wall kinetics under magnetic field or current application describes the dynamic properties in nanostructured magnets. However, when the geometrical size of a nanomagnetic system is constricted to the limiting domain wall length scale, the competing energetics between anisotropy, exchange, and dipolar interactions can cause emergent kinetics due to quasiparticle relaxation, similar to bulk magnets of atomic origin. This paper presents a joint experimental and theoretical study to support this argument: constricted nanomagnets, made of antiferromagnetic and paramagnetic neodymium thin film with honeycomb motif, reveal fast kinetic events at picosecond timescales due to the relaxation of topological quasiparticles that persist to low temperature in the absence of any external stimuli. This discovery is especially important considering the fact that paramagnets or antiferromagnets have no net magnetization. Yet, the kinetics in neodymium nanostructures is quantitatively similar to that found in ferromagnetic counterparts and only varies with the thickness of the specimen. This suggests that a universal, topological quasiparticle-mediated dynamical behavior can be prevalent in nanoscopic magnets, irrespective of the nature of the underlying magnetic material.

Photonic parity projection plays an important role in photonic quantum information processing. Nondestructive parity projections normally require high-fidelity controlled-$Z$ gates between photonic and matter qubits, which can be experimentally demanding. In this paper, we propose a nearly deterministic parity projection protocol on two photonic qubits which only requires stable matter-photon controlled-phase gates. We also demonstrate that our protocol can tolerate moderate Gaussian phase errors in the controlled-phase gates as well as Pauli errors on the matter qubits. The fact that our protocol does not require perfect controlled-$Z$ gates makes it more amenable to experimental implementation. Although we focus on photonic qubits, our protocol can be applied to any physical system or circuit with imperfect controlled-$Z$ gates. Our protocol also provides a new optimization space for parity projection operations on various physical platforms, which is potentially beneficial for achieving high-fidelity parity projection operations.