This paper introduces a novel numerical method based on an energy-minimizing normalized residual network(EMNorm Res Net)to compute the ground-state solution of Bose-Einstein condensates at zero or low temperatures.Sta...This paper introduces a novel numerical method based on an energy-minimizing normalized residual network(EMNorm Res Net)to compute the ground-state solution of Bose-Einstein condensates at zero or low temperatures.Starting from the three-dimensional Gross-Pitaevskii equation(GPE),we reduce it to the 1D and 2D GPEs because of the radial symmetry and cylindrical symmetry.The ground-state solution is formulated by minimizing the energy functional under constraints,which is directly solved using the EM-Norm Res Net approach.The paper provides detailed solutions for the ground states in 1D,2D(with radial symmetry),and 3D(with cylindrical symmetry).We use the Thomas-Fermi approximation as the target function to pre-train the neural network.Then,the formal network is trained using the energy minimization method.In contrast to traditional numerical methods,our neural network approach introduces two key innovations:(i)a novel normalization technique designed for high-dimensional systems within an energy-based loss function;(ii)improved training efficiency and model robustness by incorporating gradient stabilization techniques into residual networks.Extensive numerical experiments validate the method's accuracy across different spatial dimensions.展开更多
Utilizing the dissipative Gross-Pitaevskii equation,we investigated the splitting dynamics of triply quantized vortices at finite temperature.Through linear perturbation analysis,we determined the excitation modes of ...Utilizing the dissipative Gross-Pitaevskii equation,we investigated the splitting dynamics of triply quantized vortices at finite temperature.Through linear perturbation analysis,we determined the excitation modes of these vortices across various dissipation parameters.We identified three unstable modes with p=2-,3-and 4-fold rotational symmetries,revealing a significant dynamic transition of the most unstable mode.That is,as the dissipation parameter increases the most unstable mode transitions from the p=2 mode to the p=3 mode.Throughout the entire range of dissipation parameters,the p=4 unstable mode is never the dominant mode.Subsequently,we performed nonlinear numerical simulations of the vortex splitting process.Under random perturbations we confirmed the dynamical transition,and under specific perturbations we confirmed the instability of the p=4 mode.Our findings on the finite temperature dependence of the splitting dynamics of triply quantized vortices are expected to be verifiable in experiments.展开更多
The Josephson effect,an important quantum supercurrent phenomenon,has been extensively studied in superconductors and superfluids.In this paper,we investigate the rich physics of one-dimensional Josephson junctions in...The Josephson effect,an important quantum supercurrent phenomenon,has been extensively studied in superconductors and superfluids.In this paper,we investigate the rich physics of one-dimensional Josephson junctions in a red-detuned optical lattice with sodium(Na)quantum gas.A one-dimensional Josephson array is formed by setting up an optical lattice using a red-detuned laser.By characterizing the dependence of Josephson oscillations of the lattice depth,we experimentally demonstrate the Josephson current.The lattice depth is controlled by altering the lattice power,and our observations are consistent with theoretical predictions.These findings offer valuable insights into quantum coherent transport and the intricate dynamics inherent to superfluidity.展开更多
Introducing PT-symmetric generalized Scarf-Ⅱpotentials into the three-coupled nonlinear Gross-Pitaevskii equations offers a new way to seek stable soliton states in quasi-onedimensional spin-1 Bose-Einstein condensat...Introducing PT-symmetric generalized Scarf-Ⅱpotentials into the three-coupled nonlinear Gross-Pitaevskii equations offers a new way to seek stable soliton states in quasi-onedimensional spin-1 Bose-Einstein condensates.In scenarios where the spin-independent parameter c_(0)and the spin-dependent parameter c_(2)vary,we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross-Pitaevskii equations with PT-symmetric generalized Scarf-Ⅱpotentials.We obtain analytical soliton states and find that simply modulating c_(2)may change the analytical soliton states from unstable to stable.Additionally,we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations,exhibiting distinct behavior in energy exchange.For further investigation,we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components.These findings may contribute to a deeper understanding of solitons in Bose-Einstein condensates with PT-symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.展开更多
Kármán vortex street not only exists in nature,but also widely appears in engineering practice,which is of great significance for understanding superfluid.Parity-time(PT)symmetric potential provides a good p...Kármán vortex street not only exists in nature,but also widely appears in engineering practice,which is of great significance for understanding superfluid.Parity-time(PT)symmetric potential provides a good platform for the study of Kármán vortex streets.In this paper,different patterns of vortex shedding formed behind PT symmetric potential in Bose-Einstein condensate(BEC)are simulated numerically.Kármán vortex streets and others are discovered to emerge in the wake of a moving obstacle with appropriate parameters.Compared with BEC without PT symmetric potential,the frequency and amplitude of the drag force are more complex.The parametric regions of the combined modes are scattered around the Kármán vortex street.Numerical simulations indicate that the imaginary part of the PT symmetric potential affects the vortex structure patterns.Finally,we proposed an experimental protocol that may observe a Kármán vortex street.展开更多
The problem of an adequate description of the wave processes in Bose-Einstein condensates (CBE), including space-temporal evolution of CBE in the electron CBE condensate in the self-consistent electrical field and CBE...The problem of an adequate description of the wave processes in Bose-Einstein condensates (CBE), including space-temporal evolution of CBE in the electron CBE condensate in the self-consistent electrical field and CBE atomic condensate in the self-consistent gravitational field is considered. The complete nonlocal system for the CBE evolution is delivered including particular case and analytical solutions.展开更多
We investigate the Landau damping of the collective mode in a quasi-two-dimension repulsive Bose-Einstein condensate by using the self-consistent time-dependent Hatree-Fock-Bogoliubov approximation and a complete and ...We investigate the Landau damping of the collective mode in a quasi-two-dimension repulsive Bose-Einstein condensate by using the self-consistent time-dependent Hatree-Fock-Bogoliubov approximation and a complete and orthogonal eigenfunction set for the elementary excitation of the system. We calculate the three-mode coupling matrix element between the collective mode and the thermal excited quasi-particles and the Landau damping rate of the collective mode. We discuss the dependence of the Landau damping on temperature, on atom number in the condensate, on transverse trapping frequency and on the length of the condensate. The energy width of the collective mode is taken into account in our calculation. With little approximation, our theoretic calculation results agree well with the experimental ones and are helpful for deducing the damping mechanics and the inter-particle interaction.展开更多
The modulational instability of two-component Bose-Einstein condensates(BECs)under an external parabolic potential is discussed.Based on the trapped two-component Gross-Pitaevskill equations,a time-dependent dispersio...The modulational instability of two-component Bose-Einstein condensates(BECs)under an external parabolic potential is discussed.Based on the trapped two-component Gross-Pitaevskill equations,a time-dependent dispersion relation is obtained analytically by means of the modified lens-type transformation and linear stability analysis.It is shown that a modulational unstable time scale exists for trapped two-component BECs.The modulational properties-which are determined by the wave number,external trapping parameter,intraand inter-species atomic interactions-are modified significantly.The analytical results are confirmed by direct numerical simulation.Our results provide a criterion for judging the occurrence of instability of the trapped two-component BECs in experiment.展开更多
This paper proposes a method for calculating the Landau damping of a low-energy collective mode in a harmonically trapped Bose-Einstein condensate. Based on the divergence-free analytical solutions for ground-state wa...This paper proposes a method for calculating the Landau damping of a low-energy collective mode in a harmonically trapped Bose-Einstein condensate. Based on the divergence-free analytical solutions for ground-state wavefunction of the condensate and eigenvalues and eigenfunctions for thermally excited quasiparticles, obtained beyond Thomas-Fermi approximation, this paper calculates the coupling matrix elements describing the interaction between the collective mode and the quasiparticles. With these analytical results this paper evaluates the Landau damping rate of a monopole mode in a spherical trap and discusses its dependence on temperature, particle number and trapping frequency of the system.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.11971411)。
文摘This paper introduces a novel numerical method based on an energy-minimizing normalized residual network(EMNorm Res Net)to compute the ground-state solution of Bose-Einstein condensates at zero or low temperatures.Starting from the three-dimensional Gross-Pitaevskii equation(GPE),we reduce it to the 1D and 2D GPEs because of the radial symmetry and cylindrical symmetry.The ground-state solution is formulated by minimizing the energy functional under constraints,which is directly solved using the EM-Norm Res Net approach.The paper provides detailed solutions for the ground states in 1D,2D(with radial symmetry),and 3D(with cylindrical symmetry).We use the Thomas-Fermi approximation as the target function to pre-train the neural network.Then,the formal network is trained using the energy minimization method.In contrast to traditional numerical methods,our neural network approach introduces two key innovations:(i)a novel normalization technique designed for high-dimensional systems within an energy-based loss function;(ii)improved training efficiency and model robustness by incorporating gradient stabilization techniques into residual networks.Extensive numerical experiments validate the method's accuracy across different spatial dimensions.
基金provided by the Guangdong Basic and Applied Basic Research Foundation of China(Grant Nos.2024A1515012552,2022A1515011938,2022A1515012425)the National Natural Science Foundation of China(Grant No.12005088)the support received from Lingnan Normal University(Grant Nos.YL20200203,ZL1930)。
文摘Utilizing the dissipative Gross-Pitaevskii equation,we investigated the splitting dynamics of triply quantized vortices at finite temperature.Through linear perturbation analysis,we determined the excitation modes of these vortices across various dissipation parameters.We identified three unstable modes with p=2-,3-and 4-fold rotational symmetries,revealing a significant dynamic transition of the most unstable mode.That is,as the dissipation parameter increases the most unstable mode transitions from the p=2 mode to the p=3 mode.Throughout the entire range of dissipation parameters,the p=4 unstable mode is never the dominant mode.Subsequently,we performed nonlinear numerical simulations of the vortex splitting process.Under random perturbations we confirmed the dynamical transition,and under specific perturbations we confirmed the instability of the p=4 mode.Our findings on the finite temperature dependence of the splitting dynamics of triply quantized vortices are expected to be verifiable in experiments.
基金Project supported by the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0302103)the National Natural Science Foundation of China(Grant Nos.62325505,62020106014,62175140,62475138,92165106,12104276)the Shanxi Province Graduate Student Research Innovation Project(Grant No.2024KY105)。
文摘The Josephson effect,an important quantum supercurrent phenomenon,has been extensively studied in superconductors and superfluids.In this paper,we investigate the rich physics of one-dimensional Josephson junctions in a red-detuned optical lattice with sodium(Na)quantum gas.A one-dimensional Josephson array is formed by setting up an optical lattice using a red-detuned laser.By characterizing the dependence of Josephson oscillations of the lattice depth,we experimentally demonstrate the Josephson current.The lattice depth is controlled by altering the lattice power,and our observations are consistent with theoretical predictions.These findings offer valuable insights into quantum coherent transport and the intricate dynamics inherent to superfluidity.
基金supported by NSFC under Grant No.12272403Beijing Training Program of Innovation under Grant No.S202410019024。
文摘Introducing PT-symmetric generalized Scarf-Ⅱpotentials into the three-coupled nonlinear Gross-Pitaevskii equations offers a new way to seek stable soliton states in quasi-onedimensional spin-1 Bose-Einstein condensates.In scenarios where the spin-independent parameter c_(0)and the spin-dependent parameter c_(2)vary,we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross-Pitaevskii equations with PT-symmetric generalized Scarf-Ⅱpotentials.We obtain analytical soliton states and find that simply modulating c_(2)may change the analytical soliton states from unstable to stable.Additionally,we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations,exhibiting distinct behavior in energy exchange.For further investigation,we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components.These findings may contribute to a deeper understanding of solitons in Bose-Einstein condensates with PT-symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.
基金supported by the National Natural Science Foundation of China under Grant Nos.12065022,12147213。
文摘Kármán vortex street not only exists in nature,but also widely appears in engineering practice,which is of great significance for understanding superfluid.Parity-time(PT)symmetric potential provides a good platform for the study of Kármán vortex streets.In this paper,different patterns of vortex shedding formed behind PT symmetric potential in Bose-Einstein condensate(BEC)are simulated numerically.Kármán vortex streets and others are discovered to emerge in the wake of a moving obstacle with appropriate parameters.Compared with BEC without PT symmetric potential,the frequency and amplitude of the drag force are more complex.The parametric regions of the combined modes are scattered around the Kármán vortex street.Numerical simulations indicate that the imaginary part of the PT symmetric potential affects the vortex structure patterns.Finally,we proposed an experimental protocol that may observe a Kármán vortex street.
文摘The problem of an adequate description of the wave processes in Bose-Einstein condensates (CBE), including space-temporal evolution of CBE in the electron CBE condensate in the self-consistent electrical field and CBE atomic condensate in the self-consistent gravitational field is considered. The complete nonlocal system for the CBE evolution is delivered including particular case and analytical solutions.
基金Project supported by National Natural Science Foundation of China (Grant No.10864006)the Key Research Project of Xinjiang Higher Education,China (Grant No.XJED2010141),the Key Discipline of Theoretical Physics of Xinjiang,China,and the Prior Development Subject of Theoretical Physics of Xinjiang Normal University,China
文摘We investigate the Landau damping of the collective mode in a quasi-two-dimension repulsive Bose-Einstein condensate by using the self-consistent time-dependent Hatree-Fock-Bogoliubov approximation and a complete and orthogonal eigenfunction set for the elementary excitation of the system. We calculate the three-mode coupling matrix element between the collective mode and the thermal excited quasi-particles and the Landau damping rate of the collective mode. We discuss the dependence of the Landau damping on temperature, on atom number in the condensate, on transverse trapping frequency and on the length of the condensate. The energy width of the collective mode is taken into account in our calculation. With little approximation, our theoretic calculation results agree well with the experimental ones and are helpful for deducing the damping mechanics and the inter-particle interaction.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11764039,11847304,11865014,11475027,11274255 and 11305132the Natural Science Foundation of Gansu Province under Grant No 17JR5RA076the Scientific Research Project of Gansu Higher Education under Grant No 2016A-005
文摘The modulational instability of two-component Bose-Einstein condensates(BECs)under an external parabolic potential is discussed.Based on the trapped two-component Gross-Pitaevskill equations,a time-dependent dispersion relation is obtained analytically by means of the modified lens-type transformation and linear stability analysis.It is shown that a modulational unstable time scale exists for trapped two-component BECs.The modulational properties-which are determined by the wave number,external trapping parameter,intraand inter-species atomic interactions-are modified significantly.The analytical results are confirmed by direct numerical simulation.Our results provide a criterion for judging the occurrence of instability of the trapped two-component BECs in experiment.
基金Project supported by the National Nature Science Foundation of China （Grant Nos 90403008 and 10434060）, and State Key Development Program for Basic Research of China （Grant No 2005CB724508）.
文摘This paper proposes a method for calculating the Landau damping of a low-energy collective mode in a harmonically trapped Bose-Einstein condensate. Based on the divergence-free analytical solutions for ground-state wavefunction of the condensate and eigenvalues and eigenfunctions for thermally excited quasiparticles, obtained beyond Thomas-Fermi approximation, this paper calculates the coupling matrix elements describing the interaction between the collective mode and the quasiparticles. With these analytical results this paper evaluates the Landau damping rate of a monopole mode in a spherical trap and discusses its dependence on temperature, particle number and trapping frequency of the system.
