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The nonlinear dynamics of a quasi-1D BEC loaded in a double-well potential is studied. The beyond mean-field corrections to the energy in the form of the Lee-Huang-Yang term are taken into account. One-dimensional geometry is considered. The problem is described in the scalar approximation by the extended Gross-Pitaevskii (EGP) equation with the attractive quadratic nonlinearity, due to the Lee-Huang-Yang correction, and the effective cubic mean-field nonlinearity describing the residual intra- and inter-species interactions. To describe tunneling and localization phenomena, a two-mode model was obtained. The frequencies of the Josephson oscillations are found and confirmed by the full numerical simulations of the EGP equation. The parametric resonance in the Josephson oscillations, when the height of the barrier is periodically modulated, is studied. The predictions of the dimer model, including the case of the one-dimensional Lee-Huang-Yang superfluid, have been proven., Comment: 13 pages, 7 figures

The recent claim of superconductivity above room temperature in Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O with 0.9 < $x$ < 1 (referred to as LK-99) has sparked considerable interest. To minimize the influence of structural defects and impurity phases on the physical properties, we have synthesized phase-pure single crystals with $x \sim 1$. We find that the crystals are highly insulating and optically transparent. X-ray analysis reveals an uneven distribution of the substituted Cu throughout the sample. Temperature ($T$) dependent magnetization measurements for $ 2 \leq T \leq 800$ K reveal the diamagnetic response characteristic of a non-magnetic insulator, as well as a small ferromagnetic component, possibly originating from frustrated exchange interactions in Cu-rich clusters in the Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O structure. No anomalies indicative of phase transitions are observed. We therefore rule out the presence of superconductivity in Pb$_{9}$Cu(PO$_4$)$_6$O crystals, and provide some considerations on the origin of anomalies previously reported in experiments on polycrystalline specimen.

We study the influence of quantum fluctuations on the macroscopic quantum tunneling and self-trapping of a two-component Bose-Einstein condensate in a double-well trap. Quantum fluctuations are described by the Lee-Huang-Yang term in the modified Gross-Pitaevskii equation. Employing the modified Gross-Pitaevskii equation in scalar approximation, we derive the dimer model using a two-mode approximation. The frequencies of Josephson oscillations and self-trapping conditions under quantum fluctuations are found analytically and proven by numerical simulations of the modified Gross-Pitaevskii equation. The tunneling and localization phenomena are investigated also for the case of the Lee-Huang-Yang fluid loaded in the double-well potential., Comment: 14 pages, 9 figures

We investigate the relationship between Orthogonal Time Frequency Space (OTFS) modulation and Orthogonal Short Time Fourier (OSTF) signaling. OTFS was recently proposed as a new scheme for high Doppler scenarios and builds on OSTF. We first show that the two schemes are unitarily equivalent in the digital domain. However, OSTF defines the analog-digital interface with the waveform domain. We then develop a critically sampled matrix-vector model for the two systems and consider linear minimum mean-squared error (MMSE) filtering at the receiver to suppress inter-symbol interference. Initial comparison of capacity and (uncoded) probability of error reveals a surprising observation: OTFS under-performs OSTF in capacity but over-performs in probability of error. This result can be attributed to characteristics of the channel matrices induced by the two systems. In particular, the diagonal entries of OTFS matrix exhibit nearly identical magnitude, whereas those of the OSTF matrix exhibit wild fluctuations induced by multipath randomness. It is observed that by simply replacing the unitary matrix, relating OTFS to OSTF, by an arbitrary unitary matrix results in performance nearly identical to OTFS. We then extend our analysis to orthogonal frequency division multiplexing (OFDM) and also consider a more extreme scenario of relatively large delay and Doppler spreads. Our results demonstrate the significance of using OSTF basis waveforms rather than sinusoidal ones in OFDM in highly dynamic environments, and also highlight the impact of the level of channel state information used at the receiver.

For static fluid spheres, the condition of hydrostatic equilibrium is given by the generalized Tolman--Oppenheimer--Volkoff (TOV) equation, a Riccati equation in the radial pressure. For a perfect fluid source, it is known that finding a new solution from an existing solution requires solving a Bernoulli equation, if the density profile is kept the same. In this paper, we consider maps between static (an)isotropic fluid spheres with the same (arbitrary) density profile and present solution-generating techniques to find new solutions from existing ones. The maps, in general, require solving an associated Riccati equation, which, unlike the Bernoulli equation, cannot be solved by quadrature. In any case, it can be shown that the output solution is not, in general, regular for a given regular input solution. However, if pressure anisotropy is kept the same, the new solution is both regular and can be found by solving a Bernoulli equation. We give a few examples where the generalized TOV equation, under algebraic constraints, can be converted into a Bernoulli equation and thus, solved exactly. We discuss the physical significance of these Bernoulli equations. Since the density profile remains the same in our approach, the spatial line element is identical for all solutions, which facilitates direct comparison between various equilibrium configurations using fluid variables as functions of the radial coordinate. Finally, combining with the previous study on generation algorithms, we show how this study leads us to a new three-parameter family of exact solutions that satisfy all desirable physical conditions., Comment: 18 pages. 2 Figures

An open question in the theory of word-representable graphs for the past decade has been whether the line graph of a non-word-representable graph is always non-word-representable. By formulating an appropriate optimization problem for the decision problem of 3-semi-transitive graphs, we show that the line graph of a non-word-representable graph can be word-representable. Using IBM's CPLEX solver, we demonstrate for several known word-representable and non-word-representable graphs that the line graph of a graph is 3-semi-transitive when there is a solution to the optimization problem. This results in an example where the line graph of a non-word-representable graph is both 3-semi-transitive and semi-transitive and thus is word-representable., Comment: 11 pages, 7 figures