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1. Ground states in spatially discrete non-linear Schrödinger models

   https://doi.org/10.1088/1361-6544/acdbfc  

   Stefanov, Atanas G.; Ross, Ryan M.; Kevrekidis, Panayotis G. (June 2023, Nonlinearity)

   Abstract In the seminal work (Weinstein 1999Nonlinearity12673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on ${\mathbb{Z}}^d$. More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed powerP(i.e.l²mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedkeet al1994Phys. Rev. Lett.731055 and Laedkeet al1996Phys. Rev.E544299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g.l²supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity ${\mathrm{\partial }}_{\omega }\parallel {\phi }_{\omega }{\parallel }_{l^2}^2$. In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameterωvaries, in contrast with the corresponding continuous NLS model. 

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2. Haake–Lewenstein–Wilkens approach to spin-glasses revisited

   https://doi.org/10.1088/1751-8121/ac9d10  

   Lewenstein, Maciej; Cirauqui, David; Ángel García-March, Miguel; Guigó i Corominas, Guillem; Grzybowski, Przemysław; Saavedra, José R; Wilkens, Martin; Wehr, Jan (November 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We revisit the Haake–Lewenstein–Wilkens approach to Edwards–Anderson (EA) model of Ising spin glass (SG) (Haakeet al1985Phys. Rev. Lett.552606). This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter, $q_{\text{EA}}$. We use saddle point/steepest descent (SPSD) method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that $q_{\text{EA}}>0$at $0<T<T_c$in 3D and 4D. The case of 2D seems to be a little more subtle, since in the present approach energy increase for a domain wall competes with boundary/edge effects more strongly in 2D; still our approach predicts SG order at sufficiently low temperature. We speculate, how these predictions confirm/contradict widely spread opinions that: (i) There exist only one (up to the spin flip) ground state in EA model in 2D, 3D and 4D; (ii) there is (no) SG transition in 3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and Marek Cieplak. 

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3. Existence and stability for the travelling waves of the Benjamin equation

   https://doi.org/10.1088/1361-6544/adabaa  

   Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas G (January 2025, Nonlinearity)

   Abstract In the seminal work of Benjamin (1974Nonlinear Wave Motion(American Mathematical Society)), in the late 70s, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin–Ono equation, During the 90s, there was renewed interest in it. Benjamin (1992J. Fluid Mech.245401–11; 1996Phil. Trans. R. Soc.A3541775–806) studied the problem for existence of solitary waves, followed by works of Bona–Chen (1998Adv. Differ. Equ.351–84), Albert–Bona–Restrepo (1999SIAM J. Appl. Math.592139–61), Pava (1999J. Differ. Equ.152136–59), who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in 2024 (arXiv:2404.04711 [math.AP]), Abdallahet al, proved existence, orbital stability and uniqueness results for these waves, but only for large values of $\frac{c}{{\gamma }^2}\gg 1$. In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to allL²subcritical cases (i.e. power nonlinearities $(|u{|}^{p-2}u{)}_x$, $2<p\text{⩽}6$). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers $2<p<\mathrm{\infty }$, but produces some unstable waves, for largep. Some open questions and a conjecture regarding this last result are proposed for further investigation. 

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4. Magneto-optical trap of titanium atoms

   https://doi.org/10.1103/PhysRevResearch.7.023025  

   Eustice, Scott; Schrott, Jackson; Stöltzel, Anke; Wolf, Julian; Novoa, Diego; Cassella, Kayleigh; Stamper-Kurn, Dan M (April 2025, Physical Review Research)

   We realize a magneto-optical trap (MOT) of titanium (Ti) atoms, performing laser cooling on the 498 nm transition between the long-lived $3{\mathrm{d}}^3\left({}^4\mathrm{F}\right)4\mathrm{s}{\mathrm{a}}^5{\mathrm{F}}_5$metastable state and the $3{\mathrm{d}}^3\left({}^4\mathrm{F}\right)4\mathrm{p}{\mathrm{y}}^5{\mathrm{G}}_6^{\mathrm{o}}$excited state. Without the addition of any repumping light, we observe MOTs of the three stable, $I=0$bosonic isotopes, ${}^{46}\mathrm{Ti}, {}^{48}\mathrm{Ti}$, and ${}^{50}\mathrm{Ti}$. Up to $8.30\left(26\right)\times {10}^5{}^{48}\mathrm{Ti}$atoms are trapped at a maximum density of $1.3\left(4\right)\times {10}^{11}{\mathrm{cm}}^{-3}$and at a temperature of $90\left(15\right)\mu \mathrm{K}$. By measuring the decay of the MOT, we constrain the leakage branching ratio of the cooling transition ( $\le 2.5\times {10}^{-6}$) and the two-body loss coefficient ( $\le 2\times {10}^{-10}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$). Our approach to laser cooling Ti can be applied to other transition metals, enabling a significant expansion of the elements that can be laser cooled. Published by the American Physical Society2025 

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5. An optical tweezer array of ground-state polar molecules

   https://doi.org/10.1088/2058-9565/ac676c  

   Zhang, Jessie T; Picard, Lewis R; Cairncross, William B; Wang, Kenneth; Yu, Yichao; Fang, Fang; Ni, Kang-Kuen (May 2022, Quantum Science and Technology)

   Abstract Fully internal and motional state controlled and individually manipulable polar molecules are desirable for many quantum science applications leveraging the rich state space and intrinsic interactions of molecules. While prior efforts at assembling molecules from their constituent atoms individually trapped in optical tweezers achieved such a goal for exactly one molecule (Zhang J T et al 2020 Phys. Rev. Lett. 124 253401; Cairncross W B et al 2021 Phys. Rev. Lett. 126 123402; He X et al 2020 Science 370 331–5), here we extend the technique to an array of five molecules, unlocking the ability to study molecular interactions. We detail the technical challenges and solutions inherent in scaling this system up. With parallel preparation and control of multiple molecules in hand, this platform now serves as a starting point to harness the vast resources and long-range dipolar interactions of molecules. 

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