High-order resonances in a parametrically excited magneto-optical trap

We present analytical and numerical study of high-order parametric resonance in a driven magneto-optical trap of cold atoms. We have obtained the general solutions for parametric resonance of arbitrary order. In particular, the amplitude and phase of atomic limit-cycle motion is expressed as a function of the modulation amplitude and frequency. Moreover, the atomic dynamics for high-order parametric resonance is investigated in terms of the Hamiltonian approach, which is useful in studying transitions between attractors. We find that the analytical results are in good agreement with the numerical calculations.     

Spin dynamics of magnetic resonance with parametric modulation in a potassium vapor cell

A typical magnetic-resonance scheme employs a static bias magnetic field and an orthogonal driving magnetic field oscillating at the Larmor frequency, at which the atomic polarization precesses around the static magnetic field. Here we demonstrate both theoretically and experimentally the variations of the resonance condition and the spin precession dynamics resulting from the parametric modulation of the bias field. We show that the driving magnetic field with the frequency detuned by different harmonics of the parametric modulation frequency can lead to resonance as well. Also, a series of frequency sidebands centered at the driving frequency and spaced by the parametric modulation frequency can be observed in the precession of the atomic polarization. We further show that the resonant amplitudes of the sidebands can be controlled by varying the ratio between the amplitude and the frequency of the parametric modulation. These effects could be used in different atomic magnetometry applications.     

Monte Carlo Simulation of Cooling Induced by Parametric Resonance

We demonstrate that the parametric resonance in a magnetic quadrupole trap can be exploited to cool atoms by using Bird＇s method. In our programme the parametric resonance was realized by anisotropically modulating the trap potential. The modulation frequency dependences of temperature and fraction of the trapped atoms are explored. Furthermore, the temperature after the modulation as functions of the modulation amplitude and the mean elastic collision time are also studied. These results are valuable for the experiment of parametric resonance in a quadrupole trap.     

Parametric resonance and cooling on an atom chip

This paper observes the parametric excitation on atom chip by measuring the trap loss when applying a parametric modulation. By modulating the current in chip wires, it modulates not only the trap frequency but also the trap position. It shows that the strongest resonance occurs when the modulation frequency equals to the trap frequency. The resonance amplitude increases exponentially with modulation depth. Because the Z-trap is an anharmonic trap, there exists energy selective excitation which would cause parametric cooling. We confirm this effect by observing the temperature of atom cloud dropping.     

Parametric instability and Hamiltonian chaos in cavity semiclassical electrodynamics

We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high-Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ/2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high-Q microwave cavity are possible objects for observing parametric instability and dynamical chaos. Zh. éksp. Teor. Fiz. 115, 740–753 (February 1999)     

Nonlinear dynamics of a pipe conveying pulsating fluid with combination,principal parametric and internal resonances

Nonlinear dynamics of a hinged–hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance is investigated. The system has geometric cubic nonlinearity due to stretching effect out of immovable support conditions at both ends. The pipe conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. For appropriate choice of system parameters, the natural frequency of the second mode is approximately three times that of the first mode for a range of mean flow velocity, activating a three-to-one internal resonance. The analysis is carried out using the method of multiple scales by directly attacking the governing nonlinear integro-partial-differential equations and the associated boundary conditions. The set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for combination parametric resonance and principal parametric resonance. Stability, bifurcation and response behavior of the pipe are investigated. The amplitude and frequency detuning of the harmonic velocity perturbation are taken as the control parameters. The system exhibits response in the directly excited and indirectly excited modes due to modal interaction. Dynamic response of the system is presented in the form of phase plane trajectories, Poincare maps and time histories. A wide array of dynamical behavior is observed illustrating the influence of internal resonance.     

Atomic population oscillations between two coupled Bose——Einstein condensates with time-dependent nonlinear interaction

The atomic population oscillations between two Bose--Einstein condensates with time-dependent nonlinear interaction in a double-well potential are studied. We first analyse the stabilities of the system's steady-state solutions. And then in the perturbative regime, the Melnikov chaotic oscillation of atomic population imbalance is investigated and the Melnikov chaotic criterion is obtained. When the system is out of the perturbative regime, numerical calculations reveal that regulating the nonlinear parameter can lead the system to step into chaos via period doubling bifurcations. It is also numerically found that adjusting the nonlinear parameter and asymmetric trap potential can result in the running-phase macroscopic quantum self-trapping (MQST). In the presence of a weak asymmetric trap potential, there exists the parametric resonance in the system.     

Amplitude reduction of parametric resonance by dynamic vibration absorber based on quadratic nonlinear coupling

The paper proposes an amplitude reduction method for parametric resonance with a new type of dynamic vibration absorber utilizing quadratic nonlinear coupling. A main system with asymmetric nonlinear restoring force and harmonic excitation causes parametric resonance in the system. In contrast with autoparametric vibration absorber, the natural frequency of the vibration absorber is tuned to be in the neighborhood of twice that of the main system. For such a vibration absorber, we investigate the effect on the amplitude reduction for a parametrically excited main system. Analytical results using the method of multiple scales show that the amplitude of parametric resonance is reduced by the effect of the vibration absorber. The experimental results by a simple apparatus indicate that the parametric resonance is stabilized by the effects of both vibration absorber and Coulomb friction of the main system. Moreover, numerical results considering the Coulomb friction of the main system show that the amplitude of parametric resonance becomes close to zero by the proposed vibration absorber.     

Dynamics of cold atoms in a quadrupole magnetic trap with an orbiting potential

We study the dynamics of atoms confined to a quadrupole magnetic trap with an orbiting potential. For typical values of the experimental parameters of the trap, the rotating magnetic field is shown to produce high-frequency modulation of atomic momenta with an amplitude comparable to the widths of the momentum distributions for the lowest oscillation states of atoms in the time-averaged potential. We find the quantum-statistical momentum and position distributions of atoms and show that at temperatures much higher than the effective vibrational temperature of the atoms in the trap the quantum-statistical momentum and position distributions are Gaussian. We also establish that at temperatures comparable to the effective vibrational temperature of the atoms the quantum-statistical momentum distribution has an annular structure in the trap’s symmetry plane, which is due to the deep modulation of the atomic momenta caused by the rotating magnetic field. Zh. éksp. Teor. Fiz. 114, 23–36 (July 1998)     

蒙特卡罗模拟四极阱中原子的参变激发

在中性原子的磁囚禁实验中,磁阱线圈的电流噪声会激发磁阱中的原子运动,势必对原子团的温度和寿命产生不可忽视的影响。对于非简谐阱,这种激发具有能量选择特性,它又取决于电流噪声的频谱分布。选择了实验中常用的四极阱为研究对象,用直接模拟蒙特卡罗方法来模拟四极阱中原子运动的参变激发现象,得到了原子温度与原子数损失随激发频率的变化关系,并进一步计算了两个共振峰处原子温度随调制时间和调制深度的变化曲线。此外,还研究了弹性碰撞速率对参变激发过程中原子温度上升的影响。这些结果对四极阱参变激发的实验有较好的参考价值。     

Melnikov's criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation

We consider the problems of chaos and parametric control in nonlinear systems under an asymmetric potential subjected to a multiscale type excitation. The lower bound line for horseshoes chaos is analyzed using the Melnikov's criterion for a transition to permanent or transient nonperiodic motions, complement by the fractal or regular shape of the basin of attraction. Numerical simulations based on the basins of attraction, bifurcation diagrams, Poincare? sections, Lyapunov exponents, and phase portraits are used to show how stationary dissipative chaos occurs in the system. Our attention is focussed on the effects of the asymmetric potential term and the driven frequency. It is shown that the threshold amplitude ∣γ(c)∣ of the excitation decreases for small values of the driven frequency ω and increases for large values of ω. This threshold value decreases with the asymmetric parameter α and becomes constant for sufficiently large values of α. γ(c) has its maximum value for asymmetric load in comparison with the symmetric load. Finally, we apply the Melnikov theorem to the controlled system to explore the gain control parameter dependencies.     

Two cold atoms in a time‐dependent harmonic trap in one dimension

We analyze the dynamics of two atoms with a short‐ranged pair interaction in a one‐dimensional harmonic trap with time‐dependent frequency. Our analysis is focused on two representative cases: (i) a sudden change of the trapping frequency from one value to another, and (ii) a periodic trapping frequency. In case (i), the dynamics of the interacting and the corresponding non‐interacting systems turn out to be similar. In the second case, however, the interacting system can behave quite differently, especially close to parametric resonance. For instance, in the regions where such resonance occurs we find that the interaction can significantly reduce the rate of energy increase. The implications for applications of our findings to cool or heat the system are also discussed.     

Direct measurement of the oscillation frequency in an optical-tweezers trap by parametric excitation

We demonstrate a novel technique for direct measurement of the oscillation frequency in an optical-tweezers trap. The technique uses the phenomenon of parametric resonance in an oscillator when the stiffness of the trapping potential is modulated. The trapped particle is a strongly damped oscillator; hence, the signature of parametric resonance is not an increase in the amplitude but an increase in the size of Brownian fluctuations. The trap frequency is measured with an accuracy of 0.1%, which is better than previous techniques and thus opens up new possibilities in experiments with optical tweezers.     

Chaotic tunneling in a laser field

We study the driven tunneling of a one-dimensional charged particle confined to a rectangular double well. The numerical simulation of the Schrödinger equation based on the Cranck-Nicholson finite-difference scheme shows that the modulation of the amplitude of the external field may result in parametric resonance. The latter is accompanied by the breakdown of the quasi-periodic motion characteristic of the usual driven tunneling and the emergence of irregular dynamics. We describe the above breakdown with the occupation probability for the ground state of the unperturbed system and create the visualization of the irregular dynamics with the help of Shaw-Takens’ reconstruction of the state space. Both approaches agree concerning the values of the resonant frequency for the parametric excitation. Our results indicate that the shape of the laser pulse could be essential for generating chaotic tunneling.     

Noise-enabled precision measurements of a duffing nanomechanical resonator

We report quantitative measurements of the nonlinear response of a radio frequency mechanical resonator with a very high quality factor. We measure the noise-free transitions between the two basins of attraction that appear in the nonlinear regime, and find good agreement with theory. We measure the transition rate response to controlled levels of white noise, and extract the basin activation energy. This allows us to obtain precise values for the relevant frequencies and the cubic nonlinearity in the Duffing oscillator, with applications to parametric sensing.     

Inhibition of light tunneling in chirped and longitudinally modulated semi-infinite waveguide arrays

The inhibition of light tunneling in chirped and longitudinally modulated semi-infinite waveguide arrays where the refractive index is linearly modulated in the transverse direction and harmonically modulated along the light propagation direction is considered. We report on the effect of the refractive index transverse amplitude modulation rate, longitudinal modulation frequency and depth on tunneling inhibition in both linear and nonlinear regimes. We show that in the linear regime an optimal value for the transverse amplitude modulation rate of refractive index exists and can determine the optimal longitudinal modulation frequency or depth leading to a maximum of distance-averaged power fraction. In the nonlinear regime the tunneling inhibition dynamics is affected dramatically by the transverse amplitude modulation rate and the associated electric field amplitude of the input beam.     

Chaotic Oscillations in a Nonlinear Lumped-Parameter Transmission Line

We present the results of numerical simulation of the complex dynamics of a nonlinear radio-technical line having reflections at the boundaries and excited by an external harmonic signal. It is shown that, with increase in the amplitude of the input signal, periodic oscillations at the external-forcing frequency become unstable and are changed to more complex regimes, either quasiperiodic or chaotic. The main scenarios of transition to chaos are studied. The influence of the modulation instability and soliton formation on the complex dynamics is discussed.     

Infection dynamics in structured populations with disease awareness based on neighborhood contact history

It is well-known that the climate system, due to its nonlinearity, can be sensitive to stochastic forcing. New types of dynamical regimes caused by the noise-induced transitions are revealed on the basis of the classical climate model previously developed by Saltzman with co-authors and Nicolis. A complete parametric classification of dynamical regimes of this deterministic model is carried out. On the basis of this analysis, the influence of additive and parametric noises is studied. For weak noise, the climate system is localized nearby deterministic attractors. A mixture of the small and large amplitude oscillations caused by noise-induced transitions between equilibria and cycle attraction basins arise with increasing the noise intensity. The portion of large amplitude oscillations is estimated too. The parametric noise introduced in two system parameters demonstrates quite different system dynamics. Namely, the noise introduced in one system parameter increases its dispersion whereas in the other one leads to the stabilization of the climatic system near its unstable equilibrium with transitions from order to chaos.     

Role of parametric resonance in the inhibition of chaotic escape from a potential well

This paper shows how a periodic parametric modulation can inhibit chaotic escape of a driven oscillator from the cubic potential well that typically models a metastable system close to a fold. Melnikov analysis shows that, depending on its amplitude, period, and initial phase, a periodic parametric modulation of the linear potential term suppresses chaotic escape when certain resonance conditions are met. In particular, it is shown that chaotic escape suppression is impossible under a period-1 parametric perturbation. The effect of nonlinear damping on the inhibition scenario is also studied.