Tunneling and energy splitting in an asymmetric double-well potential

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of ?ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential.     

Effects of barrier penetration on energy spectrum and wave functions in the path integral formalism

We present a systematic approximation scheme for the non-relativistic Green function in the path integral representation which takes into account barrier penetration effects on the energy spectrum and wave functions for potentials having several minima. The method is applied to the cases of a symmetric potential with two minima and to a periodic potential. The energy spectrum is shown to agree with the usual WKB results. For the first mentioned case we also determine the wave functions in the classically allowed and forbidden regions from the residues of the Green function. They agree with those obtained in the standard WKB approximation.     

Multidimensional WKB Approximation for Tunneling Along Curved Escape Paths

Asymptotics of the perturbation series for the ground state energy of the coupled anharmonic oscillators for the positive coupling constant is related to the lifetime of the quasistationary states for the negative coupling constant. The latter is determined by means of the multidimensional WKB approximation for tunneling along curved escape paths. General method for obtaining such approximation is described. The cartesian coordinates (x,y) are choosen in such a way that the x-axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by the simultaneous expansion of the wave function in the coordinate y and the parameter γ determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. Several simplifications in the integrations of equations are pointed out. It is shown that to calculate outgoing probability flux it is not necessary to deal with inadequacy of the WKB approximation at the classical turning point. The WKB formulas for the large-order behavior of the perturbation series are compared with numerical results and an excellent agreement between the two is found.     

双势阱产生正负电子对过程中的正电子波干涉与克莱因隧穿现象

通过将狄拉克场量子化并且数值求解狄拉克方程的方法研究在一维情况下双势阱激发正负电子对产生的过程.研究发现双势阱激发的正电子波在双势阱之间会出现干涉现象,过程中伴随着克莱因隧穿效应,并且在双势阱之间的距离取正电子波对应的驻波条件时,在双势阱之间会表现出驻波形式的正电子干涉波.并且驻波的出现对正负电子对的产生过程也存在相应的影响,驻波最终会通过克莱因隧穿效应衰减消失.     

Stochastic resonance and energy optimization in spatially extended dynamical systems

We investigate a class of nonlinear wave equations subject to periodic forcing and noise, and address the issue of energy optimization. Numerically, we use a pseudo-spectral method to solve the nonlinear stochastic partial differential equation and compute the energy of the system as a function of the driving amplitude in the presence of noise. In the fairly general setting where the system possesses two coexisting states, one with low and another with high energy, noise can induce intermittent switchings between the two states. A striking finding is that, for fixed noise, the system energy can be optimized by the driving in a form of resonance. The phenomenon can be explained by the Langevin dynamics of particle motion in a double-well potential system with symmetry breaking. The finding can have applications to small-size devices such as microelectromechanical resonators and to waves in fluid and plasma.     

Staggered excitonic resonances of Stark-ladder transitions in an asymmetric double-well GaAs/AlAs superlattice

Excitonic effects on Stark-ladder transitions have been investigated experimentally and theoretically in a novel asymmetric double-well superlattice consisting of wide and narrow GaAs quantum wells separated by a constant AlAs barrier. In this superlattice strong electron resonance can occur under the applied electric field between the wide and narrow wells. It is found that due to existence of the two different heavy-hole localized states two types of excitonic resonances which are staggered in field are observed in the low-temperature photocurrent spectra. This field difference in the staggered exciton resonances is rigorously explained by variational calculations of the changes in the direct and indirect exciton binding energies with the field.     

WKB Wave Function of the General Time-Dependent Quadratic Hamiltonian System

We derived the WKB wave function for the general time-dependent quadratic Hamiltonian system using a unitary transformation method. We applied our research to sinusodially drived Caldirola–Kanai oscillator and confirmed that the time evolution of our approximated WKB wave function is similar to that of the exact one. This wave function can be used to analyze the interference between the probability amplitudes contributed by the area of overlap in phase space of quantum states.     

Condensates in double-well potential with synthetic gauge potentials and vortex seeding

We demonstrate an enhancement in the vortex generation when artificial gauge potential is introduced to condensates confined in a double well potential. This is due to the lower energy required to create a vortex in the low condensate density region within the barrier. Furthermore, we study the transport of vortices between the two wells, and show that the traverse time for vortices is longer for the lower height of the well. We also show that the critical value of synthetic magnetic field to inject vortices into the bulk of the condensate is lower in the double-well potential compared to the harmonic confining potential.     

Computational study of cold ions trapped in a double-well potential

We report a rigorous computational treatment of quantum dynamics of cold ions in a double-well trap using the time-dependent Schrödinger equation. Our method employs a numerically accurate approach that avoids approximations, such as assumption of weak coupling between the wells; normal mode nature of vibrations; or harmonic approximation for energy spectrum of the double-well system. Our goal is to reproduce, from first principles, the process of energy swaps between the wells observed in the experiments at NIST [Nature 471, 196 (2011)] and Innsbruck [Nature 471, 200 (2011)]. The model parameters and the initial conditions are carefully chosen to mimic experimental conditions. We obtain accurate energies and wave functions of the system numerically, and study the evolution of motional wave packets to provide new insight. This model reproduces experimental results obtained by NIST and Innsbruck in detail. We explain the energy transfer in terms of wave packet dynamics in the asymmetric potential energy well. We also show that, for a localised initial wave packet, this phenomenon can be interpreted using the terms of classical dynamics, such as trajectory of motion governed by the well-known simple principle: the angle of reflection equals the angle of incidence.     

WKB approach to calculating the lifetime of quasistationary states

A novel WKB approach to calculating the lifetime of quasistationary states in the potential wells of the form V(x)=P(x)-muQ(x), where P(x) is the radial part of the potential for the spherically symmetric harmonic oscillator or the hydrogen atom and Q(x) is a polynomial, is suggested. In this approach, the usual explicit procedure of the asymptotic matching of the perturbative and WKB wave functions is avoided and a simple formula for the imaginary part of the energy is found. The leading and the first correction terms for the imaginary part of the energy and the related lifetime are analytically calculated.     

Electronic structure of two-electron quantum dot with parabolic potential

In this study, we investigate the parabolic potential effects on the ground and excited energy states of two-electron quantum dot with impurity inside an infinite spherical confining potential well. The wave function and energy eigenvalues were calculated using a modified variational optimization procedure based mainly on quantum genetic algorithm and Hartree–Fock–Roothaan method. The results show that the parabolic potential and impurity charge have a strong effect on the energy states and ionization energies. It is worth pointing out that as impurity charge increases, the ionization energy rises, but the ionization dot radius decreases. On the other hand, as parabolic potential increases, the ionization energy decreases, but the ionization dot radius increases.     

Interference of Atomic Bose-Einstein Condensate Interacting with Laser Field

Interference of an atomic Bose-Einstein condensate interacting with a laser field in a double-well potential with dissipation is investigated. If properly selecting the laser field and the initial states of the atoms in the two wells, we find that the intensity exhibits revivals and collapses. The fidelity of interference is affected by the total number of atoms in the two wells and dissipation.     

对称GaAs/Al0.3Ga0.7As双量子阱中激子的束缚能

在有效质量近似下采用简单的尝试波函数变分地计算了对称GaAs/Al_0.3Ga_0.7As双量子阱中激子体系束缚能，研究了体系束缚能随阱宽和垒宽的变化情况. 发现双量子阱中激子体系束缚能随阱宽变化同单量子阱情况类似，但束缚能的峰值出现在阱宽为10?左右，峰值位置小于单阱的情况；束缚能随垒宽的增加有一极小值，这与波函数向垒中的渗透有关.     

Generating periodic interference in Bose–Einstein condensates

The interference between two condensates with repulsive interaction is investigated numerically by solving the onedimensional time-dependent Gross–Pitaevskii equation.The periodic interference pattern forms in two condensates,which are prepared in a double-well potential consisting of two truncated harmonic wells centered at different positions.Dark solitons are observed when two condensates overlap.Due to the existence of atom–atom interactions,atoms are transferred among the ground state and the excited states,which coincides with the condensate energy change.     

Semi-Analytical Solution of Dirac Equation in Schwarzschild-de Sitter Spacetime

In the present paper, we solve the radial parts of Dirac equation between the inner and the outer horizon in the Schwarzschild-de Sitter (SdS for short) geometry. Complete physical parameter space is divided into two regions depending on the height of the potential barrier and the energy of the incoming particle. In each region, we concentrate on two limiting cases. The first case is when the two horizons are close to each other and the second case is when the horizons are far apart. In each case, we give the semi-analytical solution by using WKB (Wentzel-Krames-Brillouin) approximation and show the instantaneous reflection and transmission coefficients as well as the radial wave functions graphically. PACS: 04.20.-q, 04.70.-s, 04.70.Dy, 95.30.Sf     

Dynamics of a cold quantum gas in a δ-split one-dimensional potential well

Dynamics of a wave function in a non-symmetrically split (spatially asymmetric) doublewell potential is considered. We study the dependence of the probability of well-to-well transitions on the degree of spatial asymmetry of well sizes and show that the quantum tunneling between the wells is significantly suppressed by this asymmetry. Practically complete suppression occurs at five-ten percent asymmetry. This is close to the threshold of sensitivity of contemporary experimental schemes for creating two-well potentials. We predict the phenomenon of resonance in quantum tunneling of considered states. We have also shown that an incoherently prepared superposition state tunnels in a double-well potential almost in the same way as a perfectly coherent state.     

Quasi-bound states, resonance tunnelling, and tunnelling times generated by twin symmetric barriers

In analogy with the definition of resonant or quasi-bound states used in three-dimensional quantal scattering, we define the quasi-bound states that occur in one-dimensional transmission generated by twin symmetric potential barriers and evaluate their energies and widths using two typical examples: (i) twin rectangular barrier and (ii) twin Gaussian-type barrier. The energies at which reflectionless transmission occurs correspond to these states and the widths of the transmission peaks are also the same as those of quasi-bound states. We compare the behaviour of the magnitude of wave functions of quasi-bound states with those for bound states and with the above-barrier state wave function. We deduce a Breit-Wigner-type resonance formula which neatly describes the variation of transmission coefficient as a function of energy at below-barrier energies. Similar formula with additional empirical term explains approximately the peaks of transmission coefficients at above-barrier energies as well. Further, we study the variation of tunnelling time as a function of energy and compare the same with transmission, reflection time and Breit-Wigner delay time around a quasi-bound state energy. We also find that tunnelling time is of the same order of magnitude as lifetime of the quasi-bound state, but somewhat larger.