Effective Two-State Model and NOON States for Double-Well Bose-Einstein Condensates in Strong-Interaction Regime

The model of double-well Bose-Einstein condensates in the strong-interaction regime is shown to reduce adiabatically to an effective two-state model describing the Rabi oscillations between the two atomic Fock states ｜N, 0〉 and [0, N〉, and the NOON states of arbitrary ultracold atoms can therefore be generated periodically from the initial state of either one of the Foek states.

Analytical Expression of Ground State of Double-Well Bose-Einstein Condensates

We derive an approximate analytical expression for the ground state of double-well BEC＇s, which reproduces highly accurately the numerical solution for the whole parameter regimes of the two-body repulsive interaction strength, the total number of atoms, and the hopping parameter.

Revised Iterative Solution of Ground State of Double-Well Potential

The revised new iterative method for solving the ground state of Schroedingerequation is deduced. Based on Green functions defined by quadratures along a single trajectory thisiterative method is applied to solve the ground state of the double-well potential. The result iscompared to the one based on the original iterative method. The limitation of the asymptoticexpansion is also discussed.

Squeezing via coupling of Bose–Einstein condensates in a double-well potential with a cavity light field

Squeezing via the interaction between the cavity light field and the Bose Einstein Condensate （BEC） in a doublewell potential is considered within the context of the two-mode approximation. For the cavity light field initially in a coherent state, it is shown that by choosing appropriate parameters, quadrature squeezing of the cavity light field can be achieved and it exhibits periodic oscillation. We also study the case in which BEC is tuned to resonance by periodically modulating the trapping potentiaL and the quadrature squeezing of the cavity field exhibits periodic collapse and revival effect. Both analytic and numerical calculations are performed, and they are found to be in good agreement with each other. The result shows that the quantum statistical properties of the cavity light field can be manipulated by its coupling with the condensates in the double-well potential. On the other hand, dynamical properties of the condensates in the double-well potential will be reflected by the quadrature squeezing of the light field.

Adiabatic tunneling of Bose–Einstein condensates with modulated atom interaction in a double-well potential

We study the adiabatic tunneling of Bose-Einstein condensates in a symmetric double-well potential when the inter- action strength between the atoms is modulated linearly or in a cosine periodic form. It is shown that the system evolves along a nonlinear eigenstate path. In the case of linear modulation under the adiabatic approximation conditions, the tun- neling probability of the condensate atoms to the other potential well is half. However, when the system is periodically scanned in the adiabatic process, we find an interesting phenomenon. A small change in the cycle period can lead to the condensate atoms returning to the right well or tunneling to the left well. The system comes from a linear eigenstate back to a nonlinear one, which is completely different from the linear eigenstate evolution. We explain the results by using the energy level and the phase diagram.

Spontaneous symmetry breaking of a Bose-Fermi mixture in a two-dimensional double-well potential

We study the spontaneous symmetry breaking of a superfluid Bose-Fermi mixture in a two-dimensional double- well potential. The mixture is described by a set of coupled Gross-Pitaevskii equations. The symmetry breaking phenomenon is demonstrated in the two-dimensional double-well potential in the mixture. The results are summarized in the phase diagrams of the mixture particle numbers, which are divided into symmetric and asymmetric regions by the asymmetry ratios. The dynamical pictures of the spontaneous symmetry breaking induced by a gradual transformation of the single-well potential into a double-well one are also illustrated. The properties of the quantum degenerate mixture are explored using the realistic parameters for a ^40K-^87Rb system.

Dynamics of Bose-Einstein condensates in a one-dimensional optical lattice with double-well potential

We study dynamical behaviors of the weakly interacting Bose-Einstein condensate in the one- dimensional optical lattice with an overall double-well potential by solving the time-dependent Gross-Pitaevskii equation. It is observed that the double-well potential dominates the dynamics of such a system even if the lattice depth is several times larger than the height of the double-well potential. This result suggests that the condensate flows without resistance in the periodic lattice just like the case of a single particle moving in periodic potentials. Nevertheless, the effective mass of atoms is increased, which can be experimentally verified since it is connected to the Josephson oscillation frequency. Moreover, the periodic lattice enhances the nonlinearity of the double-well condensate, making the condensate more ＂self-trapped＂ in the ^-mode self-trapping regime.

Fundamental modes in waveguide pipe twisted by saturated double-well potential

We study fundamental modes trapped in a rotating ring with a saturated nonlinear double-well potential. This model, which is based on the nonlinear Schrodinger equation, can be constructed in a twisted waveguide pipe in terms of light propagation, or in a Bose-Einstein condensate （BEC） loaded into a toroidal trap under a combination of a rotating π-out-of-phase linear potential and nonlinear pseudopotential induced by means of a rotating optical field and the Feshbach resonance. Three types of fundamental modes are identified in this model, one symmetric and the other two asymmetric. The shape and stability of the modes and the transitions between different modes are investigated in the first rotational Brillouin zone. A similar model used a Kerr medium to build its nonlinear potential, but we replace it with a saturated nonlinear medium. The model exhibits not only symmetry breaking, but also symmetry recovery. A specific type of unstable asymmetric mode is also found, and the evolution of the unstable asymmetric mode features Josephson oscillation between two linear wells. By considering the model as a configuration of a BEC system, the ground state mode is identified among these three types, which characterize a specific distribution of the BEC atoms around the trap.

Chaotic Driven Tunneling in Rectangular Double-Well

We consider a charged particle confined in a one-dimensional rectangular doublewell potential,driven by an external periodic excitation at frequencyΩand with amplitude A.We find that there is the regime of the parametric resonance due to the modulation of the amplitude A at the frequencyωprm,which results in the change in the population dynamics of the energy levels.The analysis relies on the Dirac system of Hamiltonian equations that are equivalent to the Schr¨odinger equation.Considering a finite dimensional approximation to the Dirac system,we construct the foliation of its phase space by subsets F_(ab)given by constraints a≤N_(0)≤b on the occupation probabilities N0 of the ground state,and describe the tunneling by frequenciesνab of the system’s visiting subsets F_(ab).The frequenciesνab determine the probability density and thus the Shannon entropy,which has the maximum value at the resonant frequencyω=ωprm.The reconstruction of the state-space of the system’s dynamics with the help of the Shaw-Takens method indicates that the quasi-periodic motion breaks down at the resonant valueωprm.

The Pionic Deuterium and the Pion Tetrahedron Vacuum Polarization

A double-well potential model is proposed for the pionic deuterium that enables to calculate the energy split, the potential barrier height and estimate the pion tetrahedron edge length. We propose that pion tetrahedrons, πTd = ud~dũ, play a central role in the Yukawa interaction by enabling quark exchange reactions between protons and neutrons by tunneling through a potential barrier. A vacuum polarization Feynman diagram is proposed for the πTd having chains of fermion loops for the two valence quarks and anti-quarks connected by gluons. With a higher order vacuum polarization diagram, the d and u quark loops are interleaved and the chiral symmetry is broken dynamically. The proposed πTd vacuum polarization integral does not diverge in both the IR and UV limits and vanishes in the limit of an infinite pion tetrahedron condensate. We propose a new Delbruck scattering Feynman diagram that includes d and u quark and anti-quark interleaved loops. We further propose that conversion of gravitons to photons may occur via quark and anti-quark loops that describe the pion tetrahedrons dynamics in the vacuum and may also transfer gravitational waves.

Inversion Vibrational Energy Levels of PH3^＋（^～X^2A″2） Calculated by a New Two-dimension Variational Method

A new 2-D variational method is proposed to calculate the vibrational energy levels of the symmetric P-H stretching vibration （vl） and the symmetric umbrella vibration （inversion vibration） （v2） of PH3^＋（^～X^2A″2） that has the tunneling effect. Because the symmetric internal Cartesian coordinates were employed in the calculations, the kinetic energy operator is very simple and the inversion vibrational mode is well characterized. In comparison with the often used I-D model to calculate the inversion vibrational energy levels, this 2-D method does not require an assmnption of reduced mass, and the interactions between the vl and v2 vibrational modes are taken into consideration. The calculated vibrational energy levels of PH3^＋ are the first reported 2-D calculation, and the average deviation to the experimental data is less than 3 cm^-1 for the first seven inversion vibrational energy levels. This method has also been applied to calculate the vibrational energy levels of NH3. The application to NH3 is less successful, which shows some limitations of the method compared with a full dimension computation.

Mean First-Passage Time in Optical Bistable Systems

The transient properties of an optical bistable system driven by multiplicative and additive noises are investigated. The effects of multiplicative and additive noises intensities on the mean first-passage times （MFPTs） in two opposite directions are discussed. The results show that the intensities of multiplicative and additive noises affect the MFPTs on two directions in the same way.

Quantum Standing Waves and Tunneling Through a Finite Range Potential

We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing Mathematica we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.

Self-Trapping of Two Weakly Coupled Bose-Einstein Condensates with Oscillating Atomic Scattering Length

We investigate the self-tapping phenomena for two weakly coupled Bose-Einstein condensates with a rapid periodic modulation of the atomic scattering length. By using an averaging method, the equations of motion of the slow dynamics are derived to analyze the self-trapping behavior. It is shown numerically that under certain conditions, an alternative self-trapping in either well appears.

Directed transport of coupled Brownian ratchets with time-delayed feedback

A time-delayed feedback ratchet consisting of two Brownian particles interacting through the elastic spring is consid ered. The model describes the directed transport of coupled Brownian particles in an asymmetric two-well ratchet potential which can be calculated theoretically and implemented experimentally. We explore how the centre-of-mass velocity is af fected by the time delay, natural length of the spring, amplitude strength, angular frequency, external force, and the structure of the potential. It is found that the enhancement of the current can be obtained by varying the coupling strength of the delayed feedback system. When the thermal fluctuation and the harmonic potential match appropriately, directed current evolves periodically with the natural length of the spring and can achieve a higher transport coherence. Moreover, the external force and the amplitude strength can enhance the directed transport of coupled Brownian particles under certain conditions. It is expected that the polymer of large biological molecules may demonstrate a variety of novel cooperative effects in real propelling devices.

Feedback control in a coupled Brownian ratchet

On the basis of the double-well ratchet potential which can be calculated theoretically and implemented experimentally, the influences of the time delay, the coupling constant, and the asymmetric parameter of the potential on the performance of a delayed feedback ratchet consisting of two Brownian particles coupled mutually with a linear elastic force are investigated. The centre-of-mass velocity of two coupled Brownian particles, the average effective diffusion coefficient, and the Pe number are calculated. It is found that the parameters are affected by not only the time delay and coupling constant but also the asymmetric parameter of the double-well ratchet potential. It is also found that the enhancement of the current may be obtained by varying the coupling constant of the system for the weak coupling case. It is expected that the results obtained here may be observed in some physical and biological systems.

Extended Wronskian Determinant Approach and Iterative Solutions ofOne-Dimensional Dirac Equation

An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation, which can be solved by iterative procedure to find the wave functions, is established. We employ this approach to study the one-dimensional Dirac equation with one-well potential,and give the energy levels and wave functions up to the first order iterative approximation. For double-well potential,the energy levels up to the first order approximation are given.

Revisiting the dynamics of Bose-Einstein condensates in a double well by deep learning with a hybrid network

Deep learning,accounting for the use of an elaborate neural network,has recently been developed as an efficient and powerful tool to solve diverse problems in physics and other sciences.In the present work,we propose a novel learning method based on a hybrid network integrating two different kinds of neural networks:Long Short-Term Memory(LSTM)and Deep Residual Network(ResNet),in order to overcome the difficulty met in numerically simulating strongly-oscillating dynamical evolutions of physical systems.By taking the dynamics of Bose-Einstein condensates in a double-well potential as an example,we show that our new method makes a highly efficient pre-learning and a high-fidelity prediction about the whole dynamics.This benefits from the advantage of the combination of the LSTM and the ResNet and is impossibly achieved with a single network in the case of direct learning.Our method can be applied for simulating complex cooperative dynamics in a system with fast multiplefrequency oscillations with the aid of auxiliary spectrum analysis.