Exact solvability and two-frequency Rabi oscillation in cavity-QED setup with moving emitter

In this paper, we investigate the energy spectrum and coherent dynamical process in a cavity-QED setup with a moving emitter, which is subject to a harmonic potential. We find that the vibration of the emitter will induce the effective Kerr and optomechanical interactions. With the assistance of Bogliubov operators approach, we obtain the energy spectrum of the system exactly. Furthermore, we show that the dynamics of the system exhibit a two-frequency Rabi oscillation behavior. We explain such behavior by optomechanical interaction induced quantum transition between emitter-cavity dressed states. We hope that the interaction between cavity mode and moving emitter will provide a versatile platform to explore more exotic effects and potential applications in cavity-QED scenario.

PDM Damped-Driven Oscillators: Exact Solvability, Classical States Crossings, and Self-Crossings

Within the standard Lagrangian and Hamiltonian setting, we consider a position-dependent mass (PDM) classical particle performing a damped driven oscillatory (DDO) motion under the influence of a conservative harmonic oscillator force field $V\left( x\right) =\frac{1}{2}\omega ^{2}Q\left( x\right) x^{2}$ and subjected to a Rayleigh dissipative force field $\mathcal{R}\left( x,\dot{x}\right) =\frac{1}{2}b\,m\left( x\right) \dot{x}^{2}$ in the presence of an external periodic (non-autonomous) force $F\left( t\right) =F_{\circ }\,\cos \left( \Omega t\right) $. Where, the correlation between the coordinate deformation $\sqrt{Q(x)}$ and the velocity deformation $\sqrt{m(x)}$ is governed by a point canonical transformation $q\left( x\right) =\int \sqrt{m\left( x\right) }dx=\sqrt{%Q\left( x\right) }x$. Two illustrative examples are used: a non-singular PDM-DDO, and a power-law PDM-DDO models. Classical-states $\{x(t),p(t)\}$ crossings are analysed and reported. Yet, we observed/reported that as a classical state $\{x_{i}(t),p_{i}(t)\}$ evolves in time it may cross itself at an earlier and/or a later time/s.

Extended study on the application of the sextic potential in the frame of X(3)-Sextic

The main aim of the present paper is to study extensively the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a finite number of eigenvalues are found explicitly, by algebraic means, so-called Quasi-Exact Solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and first two β bands in the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, also the shape evolution within an isotopic chain. Numerical results are given for 39 nuclei, namely, 98-108Ru, 100-102Mo, 116-130Xe, 180-196Pt, 172Os, 146-150Nd, 132-134Ce, 152-154Gd, 154-156Dy, 150-152Sm, 190Hg and 222Ra. Across this study, it seems that the higher quasi-exact solvability order improves our results by decreasing the rms, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly in the critical point.

Benign ghosts in higher-derivative systems

A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however: ghosts are benign. This happens, in particular, in exactly solvable higher-derivative theories, but exact solvability seems to be a sufficient but not a necessary condition for the benign nature of the ghosts. We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe represents then a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

SOME BOUNDARY VALUE PROBLEMS WITH INVOLUTION FOR THE NONLOCAL POISSON EQUATION

In this paper, we study new classes of boundary value problems for a nonlocal analogue of the Poisson equation. The boundary conditions, as well as the nonlocal Poisson operator, are specified using transformation operators with orthogonal matrices. The paper investigates the questions of solvability of analogues of boundary value problems of the Dirichlet and Neumann type. It is proved that, as in the classical case, the analogue of the Dirichlet problem is unconditionally solvable. For it, theorems on the existence and uniqueness of the solution to the problem are proved. An explicit form of the Green's function, a generalized Poisson kernel, and an integral representation of the solution are found. For an analogue of the Neumann problem, an exact solvability condition is found in the form of a connection between integrals of given functions. The Green's function and an integral representation of the solution of the problem under study are also constructed.