Typical density-functional theory (DFT) and approximations thereto solve the many-electron ground-state problem by working from a numerically efficient noninteracting Kohn-Sham reference system and benefit from useful minimization conditions that allow iteration (i.e., self-consistency) to the optimal energy and density. This work reveals that ensembles of ground and excited states can also benefit from similar minimization conditions within ensemble DFT (EDFT) and potential functional theory (EPFT) frameworks. It then reveals that individual excited states also have state-specific stationary conditions that can be deduced from the ensemble solution and apply to EDFT and its EPFT counterpart. However, the state-specific stationary condition for the noninteracting Kohn-Sham EPFT is revealed to be more complicated than the ground-state problem, due in part to a contribution from density-driven correlations that are neglected in typical “$\mathrm{\Delta }\mathrm{SCF}$” approaches. The importance of this term is scrutinized through exact and approximate examples. Some implications for self-consistency in exact theory and approximations are discussed.

The Zeeman splittings and electron paramagnetic resonance frequencies of alkali-metal atoms are shifted in the presence of a polarized noble gas. For a spherical geometry, the shift is enhanced over what is expected classically by a dimensionless atomic parameter ${\kappa }_0$ that is unique to each alkali-metal-atom–noble-gas pair. We present a precise measurement of ${\kappa }_0$ for the ${}^{39}\mathrm{K}\text{−}{}^3\mathrm{He}$ system with a relative accuracy of better than 1%. A critical component of achieving subpercent accuracy involved characterizing the shape of our samples using both MRI and CT medical-imaging techniques. The parameter ${\kappa }_0$ plays an important role in establishing the absolute polarization of ${}^3\mathrm{He}$ in a variety of contexts, including polarized targets for electron-scattering experiments and MRI of the gas space of the lungs. Our measurement more than doubles the accuracy possible when using ${\kappa }_0$ for polarimetry purposes. Just as important, the work presented here represents a direct measurement of ${\kappa }_0$ for the ${}^{39}\mathrm{K}\text{−}{}^3\mathrm{He}$ system; previous values for ${\kappa }_0$ in the ${}^{39}\mathrm{K}\text{−}{}^3\mathrm{He}$ system relied on a chain of measurements that were benchmarked by previous measurements of ${\kappa }_0$ in the Rb-${}^3\mathrm{He}$ system.

Spin masers are a prototype nonlinear dynamic system. They undergo a bifurcation at a critical amplification factor, transiting into a limit cycle phase characterized by a Larmor precession around the external bias magnetic field, thereby serving as a key frequency reference for precision measurements. Recently, a system of coupled twin spin masers placed in dual bias magnetic fields, involving simultaneously two intrinsic Larmor frequencies, has been studied in Commun. Phys. 8, 41 (2025). Compared with previous spin masers, this setup exhibits new attractors such as quasiperiodic orbits and chaos in addition to the usual limit cycles and the trivial no signal fixed point. The richer dynamic phases imply the existence of bifurcations, whose nature has not been fully analyzed. Here, to shed light on the nature of the bifurcations, we turn to a closely related system and systematically study the various bifurcations therein along different routes in parameter space. We identify the bifurcations as of the types including pitchfork, Hopf, homoclinic bifurcations, and saddle-node bifurcations of cycles. By both analytical and numerical methods, we reveal how various attractors interplay with each other and change their stability. We also quantitatively evaluate the locations where these bifurcations occur by tracking the both stable and even not easily detected unstable limit cycles. These findings deepen our understanding of the underlying mechanisms resulting in the rich dynamic phases in the coupled twin spin masers.

Theoretical studies on the spin dynamics of hot alkali-metal vapors typically require numerical simulations of the density-matrix equation, which is nonlinear because of spin-exchange collisions. In this study we derive a set of effective hyperfine Bloch equations (EHBEs) for systems with arbitrary polarization in the spin-exchange-relaxation-free (SERF) regime. These EHBEs provide an intuitive framework for understanding the spin dynamics in the SERF regime and yield simple solutions that are in excellent agreement with the results of density-matrix simulations. A stochastic EHBE model is created by adding spin noise caused by fundamental quantum fluctuations. This yields an optical-spin-noise spectrum that includes the intrinsic spin-projection noise and the backaction light-shift noise from Faraday rotation measurements. The spectrum is a summation of several absorptive and dispersive components, each of which has a clear physical origin and a simple algebraic expression. This provides a benchmark for calibrating future spin-squeezing experiments in the SERF regime.

Quantum addition based on the quantum Fourier transform can be an integral part of a quantum circuit and proved to be more efficient than the existing classical ripple carry adder. Our study includes identifying the quantum resource required in a quantum adder in any arbitrary dimension and its relationship with the performance indicator in the presence of local noise acting on the circuit and when a limited number of controlled rotation operations is permitted, a procedure known as banding. We analytically prove an upper bound on the number of the controlled rotation gates required to accomplish the quantum addition up to an arbitrary defect in the fidelity between the desired and imperfect output. When the environment interacts with individual qudits, we establish a connection between quantum coherence and fidelity of the output. Interestingly, we demonstrate that when banding is employed in the presence of noise, approximate circuits of constant depth outperform circuits with a higher number of controlled rotations, establishing a complementary relationship between the approximate quantum adder and the strength of the noise. We exhibit that utilizing magnetic fields to prepare an initial state that evolves according to a one-dimensional spin chain for a specific amount of time can be a potential technique to implement quantum addition circuits in many-body systems.

We establish the profound equivalence between the coherence measures defined via the convex roof method and two classes of genuine multipartite entanglement (GME) under certain conditions. Initially we construct two distinct classes of measures for genuine multipartite entanglement utilizing real symmetric concave functions and the convex roof technique. We then demonstrate that all coherence measures for any qudit states, defined through the convex roof approach, are identical to our two classes of GME measures of the states combined with an incoherent ancilla under an incoherent unitary operation. This relationship implies that genuine multipartite entanglement can be generated from the coherence incoherent in an initial state through the incoherent unitary operations. Furthermore, we conduct a preliminary exploration into the interplay between coherence and genuine quantum correlations such as genuine multipartite steering and genuine multipartite nonlocality. We show that the coherence exists in a qubit state if and only if genuine multipartite nonlocality is present for the special three-qubit $X$ states, obtained by performing unitary incoherent operations on the tensor product of the qubit state and the incoherent ancilla. Moreover, under the above process, we prove that the coherence of the qubit state exists if and only if the corresponding special $X$ state is genuinely multipartite steerable.

The generation of a high-dimensional quantum entangled state provides a crucial resource for achieving more complex and diverse quantum information processing tasks. In this paper, we present two deterministic protocols designed for the generation of a four-dimensional two-qudit Bell state and three-qudit Greenberger-Horne-Zeilinger (GHZ) state assisted by cross-Kerr nonlinearity, where the four-dimensional qudit is encoded in the hybrid polarization-spatial state of the single photon to conserve quantum resources and alleviate environmental noise. The proposed protocols for generating a four-dimensional two-qudit Bell state and three-qudit GHZ state are circuit concise, high success, robust fidelity, noise resistant, and resource saving. Moreover, the four-dimensional $N$-qudit ($N>3$) GHZ states can be generated with the help of the $N-1$ generating devices of the four-dimensional Bell state. Consequently, its efficiency and fidelity decay exponentially by a factor of $N-1$. Further, the arbitrary high-dimensional qudit-encoded entangled states can be achieved with linear-optics four-dimensional single-qudit $X$ gates and $Z$ gates. These protocols offer a reference for further advancements in high-dimensional state generation technology and providing insights for the practical implementations of metrological enhancement, quantum simulation, high-dimensional quantum computation, and quantum error correction.

Energy exchanges among internal energy modes in ${\mathrm{O}}_2+{\mathrm{N}}_2$ collisions directly affect the microscopic dynamical evolution of nonequilibrium phenomena in these gases. Quantum calculations of energy exchanges, especially those involving high molecular levels, are challenging due to their daunting computational costs. In this study, by capturing quantum effects associated with vibrational motions, we employ the improved mixed quantum-classical (MQC) method to obtain rate coefficients of both vibration-to-vibration (V-V) and vibration-to-translation or rotation (V-T–R) energy exchanges for ${\mathrm{O}}_2+{\mathrm{N}}_2$ collisions over a wide temperature range (100–9000 K). This approach permits the achievement of results up to vibrational levels close to the molecular dissociation limit. The adopted potential-energy surface (PES) is formulated using a proper representation of both intermolecular and intramolecular components to provide the dependence of the interaction energy on molecular separation distances and molecular deformations. Notably, the MQC calculations with the new PES yield vibrational relaxation rate coefficients consistent with experimental data. In addition to the comprehensive MQC data, we utilize Gaussian process regression to predict rates for processes not directly computed, thus completing the V-V and V-T–R datasets for selected processes across full vibrational levels of ${\mathrm{O}}_2$ and ${\mathrm{N}}_2$. At low temperatures, notable anti-Arrhenius behavior is observed, which would enhance relaxation processes in cold environments typical of upper and planetary atmospheres. The extensive datasets are valuable for describing the kinetic behaviors of ${\mathrm{O}}_2$ and ${\mathrm{N}}_2$ across various vibrational levels and temperatures, aiding in addressing non-local thermodynamic equilibrium conditions where the influence of vibrationally excited air molecules cannot be ignored.

Doubly differential and total cross sections for single ionization in ${\mathrm{H}}^{+}-{\mathrm{H}}_2$ collisions are evaluated in the continuum distorted wave with eikonal initial state model. One- and two-center models are used to represent the target molecule. In the two-center model, the initial orbitals are represented by Hartree-Fock (HF) or density functional methods, while the continuum wave function of the ejected electron is determined on the spherically averaged model potential derived from the ground-state configuration. In the one-center description of the ${\mathrm{H}}_2$ molecule, both the initial and final orbitals are represented on the same potential. Cross sections evaluated with the HF description of the target are generally in good agreement with both experimental data and available theoretical calculations. Models describing the motion of the ejected electron in a potential field derived from a two-center description of the molecule underestimate the electron yield at backward emission angles. The inadequacy of these applications is related to the scattering of the electron by the target nucleus, which is more realistically considered in a calculation based on a one-center description of the molecule.

Manipulating quantum states is at the heart of quantum information processing and quantum metrology. Landau-Zener Rabi oscillation (LZRO), which arises from a quantum two-level system swept repeatedly across the avoided crossing point in the time domain, has been suggested for widespread use in manipulating quantum states. Cold atoms are one of the most prominent platforms for quantum computing and precision measurement. However, LZRO has never been observed in cold atoms due to its stringent requirements. By compensating for the linear drift of the clock laser and optimizing experimental parameters, we successfully measured LZRO on the strontium atomic optical clock platform under both fast- and slow-passage limits within four to six driving periods. Compared to previous results on other platforms, the duration of the plateau is ${10}^4$ times longer in the optical lattice clock. The experimental data also suggests that destructive Landau-Zener interference can effectively suppress dephasing effects in the optical lattice clock, paving the way for manipulating quantum states against various environmental effects in cold atomic systems.

Light can be described either by its field or potential, which are often interchangeable. We theoretically investigate the distinct roles of the electric field and the vector potential of an ultrashort laser pulse in strong-field tunneling ionization, based on their distinct central frequencies. Employing the conservation law between the angular momentum and energy, we demonstrate that quantum tunneling dynamics is driven by the electric field, while the subsequent classical motion in the continuum is governed by the vector potential. Our work casts new light on the fundamental laser-matter interactions and offers a fresh perspective on the application of ultrashort laser pulses in strong-field physics.

Phase matching lies at the heart of the classical nonlinear process for frequency conversion and all-optical processing, as well as the spontaneous parametric process for the quantum light sources generation. In this work, we experimentally demonstrate the coupled-mode phase-matching (CMPM) engineered by the two-paralleled-coupled waveguides on a silicon-on-insulator (SOI) chip, extending the converted beam and entangled photons to the S band and L band. We measure the spectrum and efficiency of wavelength conversion, which successfully validates the designed CMPM scheme. We further measure the pairs generation rate of the biphoton based on the spontaneous four-wave mixing, which features a narrow bandwidth consistent well with the theoretical simulations. Multiple CMPM conditions can be achieved by exploiting the combinations of even and odd supermodes of the coupled system and tuned by designing a proper coupling gap between the two waveguides. The CMPM can extend the FWM nonlinear process ranging from the O band to the U band and even the midinfrared, enriching the working wavelength to a large extent. In addition, the CMPM features a discrete-band narrow bandwidth, which will greatly enhance the encoding density of frequency and the spectral purity of generated photon. In addition, our methods can be easily extended to a wide range of materials such as silicon nitride, dispensing with the dispersion engineering for phase matching.

Coherent control of many-body interactions allows the possibility of generating complex quantum states for future quantum technology. Specifically, the quantum states of the center-of-mass motion of a trapped macroscopic particle, being highly isolated from the environment, in hybrid systems is a potential platform for testing macroscopic quantum mechanics. We present a many-body interaction platform to enhance multimode Gaussian entanglement where the center-of-mass motion is coupled to a magnon through a microwave field. The strong and ultrastrong cavity-magnon couplings allow the generation of simultaneous entanglement in all bipartitions, without any pairwise entanglement, through numerical optimization of parameters, akin to quantum phase transitions in spin systems. Moreover, the ultrastrong coupling facilitates the generation of such multimode entanglement for a low-frequency oscillator in the unresolved sideband. This scheme provides a potential platform to explore the quantum effects in a large massive oscillator and paves the way for creating complex states for quantum simulation, sensing, exploring exotic quantum phases, and quantum computing.

We report the formation of on-axis energy backflow in the focus of tightly focused hybrid vector beams featuring noncircular mapping tracks on the Poincaré sphere for azimuthal polarization states. Using the Richards-Wolf vectorial diffraction integral, we give the general expressions for all components of the electric and magnetic fields, as well as the expression for the longitudinal component of the Poynting vector in the focal plane under $xy$ basis vectors. We reveal that the on-axis energy backflow will appear when the amplitude modulation factor $l$ and the topological charge $m$ of the vortex phase carried by basis vectors satisfy the condition $\left|l\right|-\left|m\right|=\pm 2$. And, we show that the longitudinal Poynting vector is not circularly symmetric but related to the azimuthal angle in the focal plane for the hybrid vector beams with nonzero $l$ and $m$. Furthermore, it is found that the on-axis energy backflow can also be observed in the focus of the hybrid vector beams under other basis vectors beyond the $xy$ case. More intriguingly, we additionally uncover that there is on-axis energy backflow in the focus of a linearly polarized optical vortex with the topological charge $\left|m\right|=2$, where the near-axis energy flows spirally along the optical axis. The numerical simulation results are in good agreement with the theoretical analyses.

In the present work we explore nonclassical effects in the phase space of two superposed kitten states induced by photon addition and subtraction operations applied in different sequences. We investigate two scenarios: In the first, photon addition is applied to the state, followed by photon subtraction, while in the second, the order of operations is reversed. We demonstrate that applying multiphoton operations to the state results in notable nearly isotropic sub-Planck structures, with the characteristics of these structures being influenced by the photon addition and subtraction. Increasing the number of added photons compresses the sub-Planck structures in both cases. Photon subtraction, however, has the opposite effect on the sub-Planck structures in the first case and no effect in the second. Furthermore, we observe that the optimal choices of multiphoton operations lead to improved isotropy of sub-Planck structures in our cases. The presence of the sub-Planck structures in our states leads to improved sensitivity to displacements, exceeding the standard quantum limit, as verified across all the depicted scenarios.