Shor's algorithm for the prime factorization of numbers provides an exponential speedup over the best known classical algorithms. However, nontrivial practical applications have remained out of reach due to experimental limitations. The bottleneck of the experimental realization of the algorithm is the modular exponentiation operation. In this paper, based on a relation between the modular multiplication operator and generalizations of discrete Fourier transforms, we propose a quantum circuit for modular exponentiation. A distinctive feature of our proposal is that our circuit can be entirely implemented in terms of the standard quantum circuit for the discrete Fourier transform and its variants. The gate complexity of our proposal is $O\left(L^3\right)$ where $L$ is the number of bits required to store the number being factorized. It is possible that such a proposal may provide easier avenues for near-term generic implementations of Shor's algorithm, in contrast to existing realizations which have often explicitly adapted the circuit to the number being factorized.

Quantum-optical descriptions of strong-field processes have attracted significant attention in recent years. Typically, the theoretical modeling has been conducted in the Schrödinger picture, where results are only obtainable under certain approximations, while, in contrast, the Heisenberg picture has remained relatively unexplored. In this work, we develop an accurately controlled perturbative expansion of the time-evolution operator in the Heisenberg picture and derive beyond-semiclassical corrections to the emitter dynamics due to the coupling to the quantized electromagnetic field, capturing effects of the quantum fluctuations present in the latter. We focus on high-order harmonic generation (HHG), where the approach is accurate in parameter regimes of current interest and gives closed-form expressions for key observables. This formulation not only simplifies numerical calculations compared to the Schrödinger-picture approach but also provides a clear correspondence between nonclassical features of the emitted light and the underlying induced dynamics of the generating medium including quantum fluctuations. Moreover, the Heisenberg framework naturally yields scaling relations with the number of independent emitters, enabling us to assess whether nonclassical behavior should persist under typical experimental conditions involving large emitter ensembles. Interestingly, we find that the degree of squeezing increases with the number of emitters, whereas the photon statistics approaches a classical Poissonian distribution in the many-emitter limit. We also find that the beyond-semiclassical emitter dynamics significantly enhances the degree of squeezing of the emitted light. Our work advances the theoretical understanding of quantum-optical HHG and introduces an accessible and well-controlled framework to describe realistic experiments.

Integrating two chemically distinct atomic species into a miniaturized, high-flux cold-atom source is a key challenge for modern experiments with ultracold atomic mixtures and polar molecules. Here, we report a compact setup with in-series two-dimensional magneto-optical traps (2D MOTs) that provides high-flux cold lithium and rubidium atoms. Thanks to the efficient short-distance Zeeman slowing, the maximum 3D MOT loading rate of lithium atoms reaches a record value of $6.6\times {10}^9$ atoms/s at a moderate lithium-oven temperature of 372 degrees Celsius, which is 44 times higher than that without the Zeeman slowing light. The flux of rubidium is also as high as $2.3\times {10}^9$ atoms/s with the rubidium oven held at room temperature. Meanwhile, the entire vacuum-chamber system, including an ultra-high-vacuum science cell, is within a small volume of $65\times 55\times 70{\mathrm{cm}}^3$. Our work represents a substantial improvement over traditional bulky and complex dual-species cold-atom setups. It provides a good starting point for the fast production of a double-degenerate lithium-rubidium atomic mixture and large samples of ultracold lithium-rubidium ground-state molecules.

Nonlinear classical mechanics has established rich phenomena. These include limit tori defined by toroidal attractors supporting quasiperiodic motion with incommensurate frequencies. We study the fate of such structures in open quantum systems using two coupled driven-dissipative Kerr cavities modeled via the Lindblad master equation. Combining Liouvillian spectral theory with the truncated Wigner approximation, we characterize the quantum-to-classical crossover. In the classical limit, two pairs of purely imaginary Liouvillian eigenvalues signal persistent quasiperiodic modes. Quantum fluctuations induce small negative real parts to these eigenvalues, giving rise to finite lifetimes and leading to the quantum melting of the torus. The associated Liouvillian gaps vanish algebraically in the classical limit, indicating a dynamical critical crossover with spontaneous breaking of time-translational symmetry. Quantum trajectory analysis reveals that this melting is driven by fluctuation-induced dephasing. Using a circular-variance-based order parameter, we uncover universal scaling in system size and time. These results establish quantum melting of limit tori as a distinct and robust nonequilibrium critical phenomenon, with clear experimental signatures in trapped ions and superconducting circuits.

We study the decay of a dense, ultracold sample of weakly bound DyK dimers stored in an optical dipole trap. Our bosonic dimers are composed of the fermionic isotopes ${}^{161}\mathrm{Dy}$ and ${}^{40}\mathrm{K}$, which is of particular interest for experiments related to pairing and superfluidity in fermionic systems with mass imbalance. We have realized dipole traps with near-infrared laser light in four different wavelength regions between 1050 and 2002 $\mathrm{nm}$. We have identified trap-light-induced processes as the overall dominant source of losses, except for wavelengths around 2000 $\mathrm{nm}$, where light-induced losses appeared to be much weaker. In a trap near 1550 $\mathrm{nm}$, we found a plateau of minimal light-induced losses, and by carefully tuning the wavelength we reached conditions where losses from inelastic collisions between the trapped dimers became observable. For very weakly bound dimers close to the center of a magnetically tuned Feshbach resonance, we demonstrate the Pauli suppression of collisional losses by about an order of magnitude.

Quasinormal modes characterize the transient response of static optical cavities. Here we introduce the notion of a Floquet quasinormal mode to describe transient responses in photonic time crystals. Contrasting their static counterparts, exceptional points associated with symmetry transitions are an inherent feature, as modes spontaneously and nonperturbatively lock their phase to the oscillations of the material. We further investigate the limiting behavior of the Floquet quasinormal modes in large cavities. Distinct nonperturbative behavior arises in time-modulated systems as increasingly large time-crystal cavities come closer to achieving the maximum gain predicted from a bulk wave number band gap.

We present a scheme for laser cooling and magneto-optical trapping of the group-IV (a.k.a. group-14 or tetrel) atoms silicon (Si), germanium (Ge), tin (Sn), and lead (Pb). These elements each possess a strong type-II transition $(J\to J^{\prime }=J-1)$ between the metastable $s^2{p}^2{}^3{P}_1$ state and the excited $s^{2}p{s}^{\prime }{}^3{P}_0^{\circ }$ state at an accessible laser wavelength, making them amenable to laser cooling and trapping. We focus on the application of this scheme to Sn, which has several features that make it attractive for precision measurement applications. We perform numerical simulations of atomic beam slowing, capture into a magneto-optical trap (MOT), and subsequent sub-Doppler cooling and compression in a blue-detuned MOT of Sn atoms. We also discuss a realistic experimental setup for realizing a high phase-space density sample of Sn atoms.

We present a single-photon transduction scheme using four-wave-mixing and quantum scattering in planar, cooperative Rydberg arrays that is both efficient and highly directional and may allow for terahertz-to-optical transduction. In the four-wave-mixing scheme, two lasers drive the system, coherently trapping the system in a dark ground-state manifold and coupling a signal transition, which may be in the terahertz, to an idler transition that may be in the optical regime. The photon-mediated dipole–dipole interaction between different emitters generates collective super- and subradiant dipolar surface modes, both on the signal and the idler transition. As the array is cooperative with respect to the signal transition, an incident signal photon can efficiently couple into the array and is admixed into dipolar idler modes by way of the drive. Under specific criticality conditions, this admixture is coupled into a superradiant idler mode, which primarily decays into a specific, highly directional optical photon that propagates within the array plane. Outside of the array, this photon may then be coupled into existing quantum devices for further processing. Using a scattering-operator formalism, we derive resonance and criticality conditions that govern this two-step process and obtain analytic transduction efficiencies. For lattices of infinite extension, we predict transduction efficiencies into specific spatial directions of up to 50%, while the overall, undirected transduction efficiency can be higher, beyond 90%. An analysis for finite arrays of $N^2$ emitters shows that the output is collimated into lobes that narrow as $1/\sqrt{N}$. Our scheme combines the broadband acceptance of free-space four-wave mixing with the efficiency, directionality, and tunability of cooperative metasurfaces, offering a route toward quantum coherent THz detection and processing for astronomical spectroscopy, quantum-networked sparse-aperture imaging, and other quantum-sensing applications.

An Auger-electron–ion coincidence experiment on ${\mathrm{N}}_2$ was performed using a recently developed electron–ion coincidence imaging spectrometer equipped with a photoemission electron gun. Clear electron–ion energy correlations were observed and used to determine the state-selected dissociation channels and their corresponding dissociation limits. Molecular-frame Auger-electron angular distributions were also obtained and used to determine the symmetry of the contributing dicationic states. Triple-coincidence measurements further demonstrate that Auger electrons in the 350–360 eV region arise predominantly from $1s$ shake-off ionization, followed by Auger decay into low-lying ${\mathrm{N}}_2{}^{3+}$ states that dissociate into ${\mathrm{N}}^{2+}+{\mathrm{N}}^{+}$. These results establish electron-impact-induced Auger-electron–ion coincidence spectroscopy as a powerful tool for state-selective studies of molecular dication and trication fragmentation dynamics.

When the dynamics of a quantum system of interest is known, an informationally complete set of observables is not needed for state reconstruction via tomographic techniques: letting the system evolve before performing the measurement allows one to effectively extend the available ways to probe the system. This idea leads to dynamical quantum tomography, whose feasibility we characterize for general quantum dynamics using Krylov-based methods. Specializing in Markovian ones, we also provide deterministic tests and randomized ones to effectively assess parametric dynamics. The limits of the methods are explored by comparing unitary and open dynamics when a single observable is available, and the set of observables whose expectation can be reconstructed from the available ones characterized. The framework is illustrated with applications to a spin chain (with or without dissipation) and an electron-nuclear system.

We consider coherently coupled bosonic mixtures scattering at low energies in the presence of an external confinement along either one or two directions. We exactly solve the two-body scattering problem, showing that for large Rabi coupling the confinement-induced resonance can be displaced towards scattering lengths much smaller than the oscillator length. Our results make the observation of confinement-induced resonances more tunable and accessible, offering yet another handle for the efficient control of strong interactions in ultracold quantum gases.

Nonlocality is one of the key features of quantum physics, which is revealed through the violation of a Bell inequality. In large multipartite systems, nonlocality characterization quickly becomes a challenging task. A common practice is to make use of symmetries, low-order correlators, or the exploitation of local geometries, to restrict the class of inequalities. In this paper, we characterize translation-invariant (TI) Bell inequalities with finite-range correlators in one-dimensional geometries. We introduce a methodology based on tropical algebra tensor networks and highlight its connection to graph theory. Surprisingly, we find that the TI Bell polytope has a number of extremal points that can be uniformly upper-bounded with respect to the system size. We give an efficient method to list all vertices of the polytope for a particular system size, and characterize the tightness of a given TI Bell inequality. The connections highlighted in our work allow us to re-interpret concepts developed in the fields of tropical algebra and graph theory in the context of Bell nonlocality, and vice versa. This work extends a parallel article [M. Hu and J. Tura, Phys. Rev. Lett. 136, 100202 (2026)] on the same subject.

The onset of a quantum phase transition in the Jaynes-Cummings model (JCM) and beyond has been a central goal in quantum light-matter interactions, yet the role of dissipation—often considered detrimental—remains less explored. Here, we demonstrate the proactive creation and control of such phase transitions in a tunneling-driven two-photon JCM via dissipation engineering. We show that single-photon loss triggers a first-order quantum phase transition accompanied by the bistability within a mean-field theory, while two-photon loss stabilizes it into a continuous second-order transition with a power-law behavior near the critical point. By designing electronic dissipation, we can further realize dynamic switching of the transition order, and its balance with the photonic component provides a mechanism for tailoring photon statistics in the phase transitions. Our work transforms dissipation from a disruptive force into a resource for sculpting quantum criticality, establishing a new paradigm for designing nonequilibrium phase diagrams in open quantum systems.

Diatomic molecules with an energetically low-lying ${}^3\mathrm{\Delta }_1$ state are attractive platforms to detect new physics beyond the Standard Model, such as parity- and time-reversal-violating phenomena. One of the advantages of using a ${}^3\mathrm{\Delta }_1$ state is its tiny $\mathrm{\Lambda }$ splitting due to the coupling between the electronic and rotational angular momenta, which facilitates polarizing the molecules in small external electric fields. Theoretical estimation of the magnitude of the $\mathrm{\Lambda }$ splitting is helpful for planning new experiments. In this study, we present a theoretical model to calculate the $\mathrm{\Lambda }$ splitting. Our model integrates the relativistic four-component wave function and the traditional rotational Hamiltonian based on Hund's case (a). The multireference character of the wave function is taken into account. Our calculations for PtH and ${\mathrm{ThF}}^{+}$ molecules qualitatively agree with experiment. The $\mathrm{\Lambda }$ splitting of ${\mathrm{TaO}}^{+}$ for the rotational ground state is predicted to be around 9 kHz. This tiny splitting can reduce the systematic uncertainty, but in a practical experiment, it may cause depolarization during rotation ramp-up.

We experimentally demonstrate spatial confinement of optically driven atomic excitation by broadband composite pulses in a rare-earth ion-doped ${\text{Pr}}^{3+}\text{:}{\text{Y}}_2{\text{SiO}}_5$ crystal. The experimental data confirm that such sequences enable spatially localized excitation well below the diameter of the applied laser beam, with a substantial improvement compared to previous developments and applications of other classes of composite pulses. In particular, we derived and applied broadband composite sequences, which also work efficiently in inhomogeneously broadened media, while also maintaining robustness with regard to fluctuations of the driving laser intensity. We perform extensive measurements to assess and compare the performance of several classes of broadband composite sequences. The findings are confirmed by numerical simulations, which agree very well with the experimental data. As the concepts are also applicable to spatially confined excitations below the diffraction limit of the laser beam, the results serve as an important step towards high-resolution imaging, preparation and addressing in quantum technologies and classical applied optics.

We theoretically study the driven-dissipative dynamics of an array of two-level emitters coupled to a single photonic mode in the presence of disorder in the resonant frequencies. We introduce the notion of subradiant correlations in the dynamics, corresponding to the eigenstates of the Liouvillian with a low decay rate, that can also oscillate in time. While the usual collective subradiant states do not survive under emitter resonant frequency fluctuations, these subradiant correlations are immune to this type of disorder. We employ the group representation theory to find the number of nontrivial oscillating subradiant correlations in the system. These long-living correlations exist in finite-size systems, when their lifetime is parametrically longer than in the so-called Dicke time crystal phase.

In the present work, we introduce and explore a technique for the efficient removal of vortices from an atomic Bose-Einstein condensate through the application and subsequent removal of a one-dimensional optical lattice. We showcase a prototypical experimental realization of the technique that motivates a detailed theoretical study of vortex removal mechanisms. Through simulations of the condensate dynamics during application of the optical lattice, we also discover a vortex removal mechanism that arises in narrow, optical-lattice-induced atomic density channels for which the channel width is on the order of the nominal vortex core size and healing length. This mechanism involves the density profile typically associated with a vortex core spatially separating from the phase singularity associated with the vortex. By analyzing numerical experiments covering a wide range of variations of the optical lattice amplitude and fringe periodicity, we identify the existence of an optimal set of parameters that enables the efficient removal of all vortices from the condensate. This analysis paves the way for further studies aimed at understanding vortex dynamics in narrow channels and adds to an experimental toolkit for working with vortices and controlling the dynamical states of condensates.

High-accuracy calculations of electric dipole polarizabilities and quadrupole moments ($\mathrm{\Theta }$) of the clock states of the singly charged calcium (${\mathrm{Ca}}^{+}$), strontium (${\mathrm{Sr}}^{+}$), and barium (${\mathrm{Ba}}^{+}$) alkaline-earth ions are estimated by employing relativistic coupled-cluster (RCC) theory. It demonstrates the importance of the triple excitations in the RCC method for precise determination of the above quantities. We also observe a different trend of correlations in the $\mathrm{\Theta }$ values than an earlier study with respect to orbitals from higher angular momenta. Reliability of the results is verified by comparing the calculated energies, magnetic dipole hyperfine structure constants, and lifetimes of the atomic states with the experimental values of the ${}^{43}\mathrm{Ca}^{+}, {}^{87}\mathrm{Sr}^{+}$, and ${}^{137}\mathrm{Ba}^{+}$ ions. Nuclear quadrupole moments of these isotopes are also estimated by combining calculations with the measured electric quadrupole hyperfine structure constants, showing large deviations from the literature values.

In disordered systems, directly describing electron dynamics under a driving field—as is commonly done for atoms or crystals—remains a significant challenge. However, high-order harmonic generation (HHG), which inherently encodes information about electron motion, provides a powerful tool to qualitatively probe the dependence of electron quiver amplitude on laser intensity and wavelength. In this work, we investigate the quiver amplitude of electrons in one-dimensional disordered systems designed to mimic liquids ($\alpha$ phase) and the thermal effects in crystals ($\beta$ phase). Using the minimum chain length required to converge HHG spectra as a proxy for the quiver amplitude, we find that in the $\alpha$ phase, the quiver amplitude decreases with increasing disorder but is largely insensitive to wavelength. For both phases, the quiver amplitude decreases with increasing laser intensity under weak disorder, but this trend saturates at strong disorder. Furthermore, when the nearest-neighbor distributions match, the two phases exhibit similar HHG spectra under strong fields.

This work theoretically investigates possibilities of using the Stimulated Raman Adiabatic Passage (STIRAP) and its variants to control a coherent superposition of quantum states. We present a generalization of the so-called fractional STIRAP (f-STIRAP), demonstrating precise control over the mixing ratio of quantum states in the wave packet. In contrast to conventional f-STIRAP, designed to drive a system from an eigenstate into a coherent superposition, our scheme enables arbitrary control over the composition of an already existing superposition state. We demonstrate that an approximate version of this technique—where analytically designed laser pulses with composite envelopes are replaced by simple Gaussian pulses—achieves comparable performance in controlling the dynamics of the wave packet. A limiting case of this scheme, utilizing two pulses with identical Gaussian envelopes and tuned delay and relative phase, is also explored, revealing experimentally accessible pathways for manipulating quantum coherence. We apply our developed techniques to control the ultrafast charge migration in the spin-orbit split ground electronic states of xenon cations via intermediate valence- and core-excited states. Finally, we propose concrete experimental realizations of the developed control schemes in combination with attosecond transient absorption spectroscopy as a method to probe the system.