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Angular frequency ω of linear oscillations near the stable fixed point of equation (8), whenever it exists, as a function of α and different values of the interaction parameter η = γ/α. Upper, dotted curve: η = 0; mid, dashed curve: η = 0.1; lower, continuous curve: η = 0.2.
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The nonlinear dynamics of self-bound Bose–Einstein matter waves under the action of an attractive nonlocal and a repulsive local interaction is analyzed by means of a time-dependent variational formalism. The mean-field model described by the Gross–Pitaevskii equation (GPE) is reduced to a single second-order conservative ordinary differential equa...
Citations
... Moreover, for spinor condensates, this approximation does not reproduce the complete structure of the single-particle excitation spectrum [16]. The role of non-local interaction has been recently discussed in physics of ultracold Bose [17,18] and Fermi [19][20][21] gases. Besides that, the applicability of the s-wave scattering-length approximation is limited by the choice of the interaction potential. ...
We theoretically study a weakly interacting gas of spin-1 atoms with Bose-Einstein condensate in external magnetic field within the Bogoliubov approach. To this end, in contrast to previous studies, we employ the general Hamiltonian, which includes both spin and quadrupole exchange interactions as well as the couplings of the spin and quadrupole moment with the external magnetic field (the linear and quadratic Zeeman terms). The latter is responsible for the emergence of the broken-axisymmetry state. We also reexamine ferromagnetic, quadrupolar, and paramagnetic states employing the proposed Hamiltonian. For all magnetic states, we find the relevant thermodynamic characteristics such as magnetization, quadrupole moment, thermodynamic potential. We also obtain three-branch excitation spectrum of the broken-axisymmetry state. We show that this state can be prepared at three different regimes of applied magnetic field. Finally, we present the magnetic state diagrams for each regime of realizing the broken-axisymmetry state.
... This procedure reduces the problem to a set of coupled ordinary differential equations, which can be adapted to axially and radially symmetric MOTs. The analysis of nonlinear systems using variational methods is traditional and simplifies the dynamical equations, as in quantum electron gases [23][24][25] and Bose-Einstein condensates [26][27][28][29][30][31][32]. Due to the Doppler cooling, the lower temperature reached is delimited by the Doppler cooling limit, and the proposed Lagrangian density has a time-dependent exponential factor and is linearly dependent on the velocity potential. ...
... present in the associated Lagrangian function L = exp(νt)[3α 2 r /2 − U]. Figure 2 shows the simulation results for Equation (26) for the same parameters as in the previous section. Solving ∂U ∂α r = 0 (29) or ...
We briefly review some recent advances in the field of nonlinear dynamics of atomic clouds in magneto-optical traps. A hydrodynamical model in a three-dimensional geometry is applied and analyzed using a variational approach. A Lagrangian density is proposed in the case where thermal and multiple scattering effects are both relevant, where the confinement damping and harmonic potential are both included. For generality, a general polytropic equation of state is assumed. After adopting a Gaussian profile for the fluid density and appropriate spatial dependencies of the scalar potential and potential fluid velocity field, a set of ordinary differential equations is derived. These equations are applied to compare cylindrical and spherical geometry approximations. The results are restricted to potential flows.
... Moreover, for spinor condensates, this approximation does not reproduce the complete structure of the single-particle excitation spectrum [16]. The role of non-local interaction has been recently discussed in physics of ultracold Bose [17,18] and Fermi [19][20][21] gases. ...
... Therefore, removing moduli and using Eq. (18) to fix the phases as φ + + φ − − 2φ 0 = 0 for χ 2 − h 2 ≤ 0 and φ + + φ − − 2φ 0 = π for χ 2 − h 2 ≥ 0, we come to the final expressions for S x and S y . The results are summarized in the Table II. ...
... As a result, we get (a − χ)e iφ0 = 2cχ a + c − χ |ζ + ||ζ − |e i(φ++φ−−φ0) . (B5) Therefore, we obtain the constraint on phases (18) and equation for determining a (or chemical potentialμ). The latter has only one solution, ...
We theoretically study a weakly interacting gas of spin-1 atoms with Bose-Einstein condensate in external magnetic field within the Bogoliubov approach. To this end, in contrast to previous studies, we employ the general Hamiltonian, which includes both spin and quadrupole exchange interactions as well as the couplings of the spin and quadrupole moment with the external magnetic field (the linear and quadratic Zeeman terms). The latter is responsible for the emergence of the broken-axisymmetry state. We also re-examine ferromagnetic, quadrupolar, and paramagnetic states employing the proposed Hamiltonian. For all magnetic states, we find the relevant thermodynamic characteristics such as magnetization, quadrupole moment, thermodynamic potential, as well as excitation energies for broken-axisymmetry state. We show that the broken-axisymmetry state can be prepared at three different regimes of applied magnetic field. We also present the magnetic state diagrams for each regime of realizing the broken-axisymmetry state.
... is the many-body Hamiltonian. This model can describe many phenomena, such as the thermal self-interaction of beams inside a plasma [37] and the Bose-Einstein condensation (BEC) with magnetic dipole-dipole forces [12], in which the nonlocal interactions are the dipole-dipole interactions between the dilute atoms; consequently, it has been studied in various articles [5,13,26,31,33,36]. If the interactions between the atoms are of short range (small particle density) and only binary collisions [27] are considered, then the two-body interacting potential K (x − x ) can then be described by a singular response; in this case, we may use a Dirac delta function to model such a response so that the nonlocal effect becomes negligible [6], i.e. ...
... The second step in the Lie's splitting (13) involves only a temporal integral of the wave function, and it preserves the total mass in the system. To see this, we first multiply the differential equation by U * and integrate the resulting equation over the whole domain. ...
We present efficient numerical methods for solving a class of nonlinear Schrödinger equations involving a nonlocal potential. Such a nonlocal potential is governed by Gaussian convolution of the intensity modeling nonlocal mutual interactions among particles. The method extends the Fast Huygens Sweeping Method (FHSM) that we developed in Leung et al. (Methods Appl Anal 21(1):31–66, 2014) for the linear Schrödinger equation in the semi-classical regime to the nonlinear case with nonlocal potentials. To apply the methodology of FHSM effectively, we propose two schemes by using the Lie’s and the Strang’s operator splitting, respectively, so that one can handle the nonlinear nonlocal interaction term using the fast Fourier transform. The resulting algorithm can then enjoy the same computational complexity as in the linear case. Numerical examples demonstrate that the two operator splitting schemes achieve the expected first-order and second-order accuracy, respectively. We will also give one-, two- and three-dimensional examples to demonstrate the efficiency of the proposed algorithm.
... The dynamical study of nonlinear systems can be simplified using variational methods, as in Bose-Einstein condensates [37][38][39][40][41][42][43] and quantum electron gases [15,44,45]. In this context, the timedependent variational method allows us to assess nonlinear and time-dependent dynamics by adopting a trial function, frequently a Gaussian Ansatz. ...
... For spinor condensates, the scattering-length approximation does not allow taking into account the interaction effects associated with the quadrupole degrees of freedom and can lead to an incomplete structure of the single-particle excitation spectra [27] so that the non-local character of interaction cannot be ignored in some problems. The role of non-local interaction has been recently discussed in context of both ultracold Bose [26,28] and Fermi [29][30][31] gases. ...
... Therefore, we only need to diagonalize H (2) 2 (n 0 ). This can be done by employing the general Bogoliubov procedure [46] for diagonalizing a quadratic form given by equation (28). Following it, H (2) 2 (n 0 ) can be reduced to the diagonal form, ...
... The second part of the Hamiltonian H (2) 2 (n 0 ), which can again be diagonalized separately, has the form of equation (28) in which the 2 × 2 matrices A = A † and B = B T have the following matrix elements: ...
We obtain and justify a many-body Hamiltonian of pairwise interacting spin-1 atoms, which includes eight generators of the SU(3) group associated with spin and quadrupole degrees of freedom. It is shown that this Hamiltonian is valid for non-local interaction potential, whereas for local interaction specified by s -wave scattering length, the Hamiltonian should be bilinear in spin operators only (of the Heisenberg type). We apply the obtained Hamiltonian to study the ground-state properties and single-particle excitations of a weakly interacting gas of spin-1 atoms with Bose–Einstein condensate (BEC) taking into account the quadrupole degrees of freedom. It is shown that the system under consideration can be in ferromagnetic, quadrupole, and paramagnetic phases. The corresponding phase diagram is constructed and discussed. The main characteristics such as the density of the grand thermodynamic potential, condensate density, and single-particle excitation spectra modified by quadrupole degrees of freedom are determined in different phases.
... The stationary solution of Eq. (3) in the form ψ = qe ibs can be numerically found by the squared-operator method [20]. Integration of Eq. (3) is implemented by the spilled-step Fourier method [21]. Monitoring the process of evolution, we use the spatial and temporal width of STs ...
A scheme is proposed to generate stable spatiotemporal solitons (STs) in a cold Rydberg atomic system with a Bessel Lattice (BL) potential, by utilizing electromagnetically induced transparency (EIT). We investigate the existence and stability of these three-dimensional (3D) STs supported by BL in nonlocal nonlinear media. The results show that the system can support slow STs with low light intensity. We also find that the BL strength and the degree of nonlocality of the nonlinearity, which can be tuned in the system, can be used to control these STs and their propagation behaviors.
... To summarize, we would like to note that the approximate analytical approach offered here for RD population dynamical systems with long-range interactions can have wider applications in solving various problems of nonlinear physics ranging from quantum matter wave models to cosmology (see, e.g., [30,31] and references therein). ...
... Let us find the general solution of the Einstein-Ehrenfest dynamical system (30) and (31) for the first moments σ (0) (t) and x (0) (t) without imposing the initial conditions (33) and (34). Denote the general solution by ...
... The function w(s)[ϕ] in (43) is given by (30), (33), and (43). For instance, if we take a 0 (t) = a = const, b 0 (t) = b = const, and A = (πα) − 1 2 in (85), then (34) yields σ (0) (0) = 1 and from (36) and (43) we have ...
We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.
We provide the general arguments that quantum atomic gases of interacting high-spin atoms represent a physical system in which the multipole (hidden) degrees of freedom may be manifested. Their manifestation occurs when the interatomic interaction is of non-local type. For a local interaction described by the s-wave scattering length, the multipole degrees of freedom do not reveal themselves. To illustrate our findings, we theoretically examine the phenomenon of Bose–Einstein condensation in an interacting gas of spin-1 atoms in an external magnetic field. This study is based on the SU(2) invariant Hamiltonian, which has a bilinear structure in the spin and quadrupole operators along with the scalar term. It is shown that depending on the conditions imposed on the interaction amplitudes (stability conditions), the ground state of the system may exhibit three different phases: quadrupolar, ferromagnetic, and paramagnetic. The basic thermodynamic characteristics affected by hidden degrees of freedom are found for all phases.
