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 $\hbar ω$. 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.
The level splitting formula of an asymmetric double well potential is calculated taking into account the multi-instanton contributions (dilute gas approximation). Results can be related with known semiclassical ones obtained with a truncated hamiltonian, and the symmetric case is easily recovered.
We propose an asymmetric double quantum well structure with a common continuum and investigate the effect of resonant tunneling on the control of coherent electron population transfer between the two quantum wells. By numerically solving the motion equations of element moments, the almost complete electron population transfer from initial subband to the target subband could be realized due to the constructive interference via flexibly adjusting the structure parameters.
We present a simple non-hermitian model to describe the phenomenon of asymmetric tunneling between two energy-degenerate sites coupled by a non-reciprocal interaction without dissipation. The system was described using a biorthogonal family of energy eigenvectors, the dynamics of the system was determined by the Schrödinger equation, and unitarity was effectively restored by proper normalization of the state vectors. The results show that the tunneling rates are indeed asymmetrical in this model, leading to an equilibrium that displays unequal occupation of the degenerate systems even in the a…
We have studied dynamical properties and quantum tunneling in asymmetric double-well (DW) systems, by solving Schrödinger equation with the use of two kinds of spectral methods for initially squeezed Gaussian wavepackets. Time dependences of wavefunction, averages of position and momentum, the auto-correlation function, an uncertainty product and the tunneling probability have been calculated. Our calculations have shown that (i) the tunneling probability is considerably reduced by a potential asymmetry $ΔU$, (ii) a resonant tunneling with $|ΔU| \simeq κ\:\hbar ω$ is realized for motion starti…
We present an improved Wentzel-Kramers-Brillouin (WKB) calculation of tunnel splitting in one dimensional asymmetric double well potentials. We show that the tunnel splitting in general can have linear dependence on the bias energy, beside the well-known quadratic dependence. We demonstrate that the linear correction is greater than previously thought.
We discuss the energy level splitting $Δε$ due to quantum tunneling between congruent tori in phase space. In analytic cases, it is well known that $Δε$ decays faster than power of $\hbar$ in the semi-classical limit. This is not true in non-smooth cases, specifically, when the tori are connected by line on which the Hamiltonian is not smooth. Under the assumption that the non-smoothness depends only upon the x- or p-coordinate, the leading term in the semi-classical expansion of $Δε$ is derived, which shows that $Δε$ decays as $\hbar^{k+1}$ when $\hbar\to 0$ with k being the order of non-smoo…
Despite quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, we derive an succinct analytical formula based on the WKB semi-classical approach. Two different examples of asymmetric potentials are discussed: when the two localized levels are degenerate or not. For the first case, we also discuss a time-dependent problem showing quantum Zeno effect.
An asymmetric double-well potential is considered, assuming that the wells are parabolic around the minima. The WKB wave function of a given energy is constructed inside the barrier between the wells. By matching the WKB function to the exact wave functions of the parabolic wells on both sides of the barrier, for two almost degenerate states, we find a quantization condition for the energy levels which reproduces the known energy splitting formula between the two states. For the other low-lying non-degenerate states, we show that the eigenfunction should be primarily localized in one of the we…
Spin-1 tunneling and splitting of energy levels as a result of tunneling are observed explicitly on IBM's quantum computer, ibmq-bogota. The spin-1 is realized with two spins-1/2. We detect oscillations of spin-1 between the states $|1\rangle$, $|-1\rangle$ in the result of tunneling on the basis of studies of the time dependence of the mean value of z-component of spin-1 on the quantum device. The energy level splitting is observed quantifying on the IBM's quantum computer the eigenvalues of Hamiltonian which describes the spin tunneling.
Using a simple model for nonlinear Landau-Zener tunneling between two energy bands of a Bose-Einstein condensate in a periodic potential, we find that the tunneling rates for the two directions of tunneling are not the same. Tunneling from the ground state to the excited state is enhanced by the nonlinearity, whereas in the opposite direction it is suppressed. These findings are confirmed by numerical simulations of the condensate dynamics. Measuring the tunneling rates for a condensate of rubidium atoms in an optical lattice, we have found experimental evidence for this asymmetry.
The study of tunneling splitting is fundamental to get insight into the dynamics of a multitude of molecular systems. In this paper, a novel approach to the analysis of the ground-state tunneling splitting is presented and explicitly applied to one-dimensional systems. The isomorphism between the Fokker-Planck-Smoluchowski operator and the Born-Oppenheimer quantum Hamiltonian is the key element of this method. The localization function approach, used in the field of stochastic processes to study the Kramers problem, leads to a simple, yet asymptotically justified, integral approximation for th…
An asymmetric double quantum wells (QWs) structure with resonant tunneling is suggested to achieve high efficient four wave mixing (FWM). We analytically demonstrate that the resonant tunneling can induce high efficient mixing wave in such a semiconductor structure with a low light pump wave. In particular, the FWM conversion efficiency can be enhanced dramatically in the vicinity of the center frequency. This interesting scheme may be used to generate coherent long-wavelength radiation in solid-state system.
We report the first experimental observation of the resonance-assisted dynamical tunneling (RADT) in the inter-mode interaction in an asymmetric-deformed microcavity. A selection rule for strong inter-mode coupling induced by RADT was observed on angular mode numbers as predicted by the RADT theory. In addition, the coupling strength was measured to be proportional to the square of the phase-space area associated with the nonlinear resonance involved in RADT. The proportionality constant was found to depend only on the nonlinear resonance, supporting the semiclassical nature of RADT.
We present a scheme for asymmetric quantum information splitting, where a sender distributes asymmetrically a qubit to distant agents in a network. The asymmetric distribution leads to that the agents have different powers to reconstruct the sender's qubit. In other words, the authorities of the agents for getting the quantum secret are hierarchized. The scheme does not need the agents to get together and make nonlocal operations. Our scheme can also be modified to implement controlled teleportation against uncooperation of part of supervisors.
We theoretically investigate the tunneling-induced transparency (TIT) and the Autler-Townes (AT) doublet and triplet in a triple-quantum-dot system. For the resonant tunneling case, we show that the TIT induces a transparency dip in a weak-tunneling regime and no anticrossing occurs in the eigenenergies of the system Hamiltonian. However, in a strong-tunneling regime, we show that the TIT evolves to the AT splitting, which results in a well-resolved doublet and double anticrossings. For the off-resonance case, we demonstrate that, in the weak-tunneling regime, the double TIT is realized with a…
The tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approxi…
We study the full-fledged microscopic dynamics of two interacting, ultracold bosons in a one- dimensional double-well potential, through the numerically exact diagonalization of the many-body Hamiltonian. With the particles initially prepared in the left well, we increase the width of the right well in subsequent trap realizations and witness how the tunneling oscillations evolve into particle loss. In this closed system, we analyze the spectral signatures of single- and two-particle tunneling for the entire range of repulsive interactions. We conclude that for comparable widths of the two wel…
We investigate quantum tunneling in smooth symmetric and asymmetric double-well potentials. Exact solutions for the ground and first excited states are used to study the dynamics. We introduce Wigner's quasi-probability distribution function to highlight and visualize the non-classical nature of spatial correlations arising in tunneling.
The complex of a methane molecule and a fluoride anion represents a 12-dimensional (12D), four-well vibrational problem with multiple large-amplitude motions, which has challenged the quantum dynamics community for years. The present work reports vibrational band origins and tunneling splittings obtained in a full-dimensional variational vibrational computation using the GENIUSH program and the Smolyak quadrature scheme. The converged 12D vibrational band origins and tunneling splittings confirm complementary aspects of the earlier full- and reduced-dimensionality studies: (1) the tunneling sp…
We examine tunneling of topological charge between non-Abelian anyons as a perturbation of the long-range effective theory of a topologically ordered system. We obtain energy corrections in terms of the anyons' universal algebraic structure and non-universal tunneling amplitudes. We find that generic tunneling completely lifts the topological degeneracy of non-Abelian anyons. This degeneracy splitting is exponentially suppressed for long distances between anyons, but leaves no topological protection at shorter distances. We also show that general interactions of anyons can be expressed in term…
Large quantum dot energy level shifts in anomalous photon-assisted tunneling
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Delft University of Technology（代尔夫特理工大学）；Netherlands Organisation for Applied Scientific Research（荷兰应用科学研究组织）
A local impurity usually only strongly affects few single-particle energy levels, thus cannot induce a quantum phase transition (QPT), or any macroscopic quantum phenomena in a many-body system within the Hermitian regime. However, it may happen for a non-Hermitian impurity. We investigate the many-body ground state property of a one-dimensional tight-binding ring with an embedded single asymmetrical dimer based on exact solutions. We introduce the concept of semi-localization state to describe a new quantum phase, which is a crossover from extended to localized state. The peculiar feature is …
The so-called $δ'$-interaction as a particular example in Kurasov's distribution theory developed on the space of discontinuous (at the point of singularity) test functions, is identified with the diagonal transmission matrix, {\em continuously} depending on the strength of this interaction. On the other hand, in several recent publications, the $δ'$-potential has been shown to be transparent at some {\em discrete} values of the strength constant and opaque beyond these values. This discrepancy is resolved here on the simple physical example, namely the heterostructure consisting of two extrem…
The double-well potential is a good example, where we can compute the splitting in the bound state energy of the system due to the tunneling effect with various methods, namely WKB or instanton calculations. All these methods are non-perturbative and there is a common belief that it is difficult to find the splitting in the energy due to the barrier penetration from a perturbative analysis. However, we will illustrate by explicit examples containing singular potentials (e.g., Dirac delta potentials supported by points and curves and their relativistic extensions)that it is possible to find the…
We study tunneling in various shaped, closed, two-dimensional, flat potential, double wells by calculating the energy splitting between symmetric and anti-symmetric state pairs. For shapes that have regular or nearly regular classical behavior (e.g. rectangular or circular) the tunneling rates vary greatly over wide ranges often by several orders of magnitude. However, for well shapes that admit more classically chaotic behavior (e.g. the stadium, the Sinai billiard) the range of tunneling rates narrows, often by orders of magnitude. This dramatic narrowing appears to come from destabilization…
The perfect quenching of spin tunneling first predicted for a model with biaxial symmetry, and recently observed in the magnetic molecule Fe_8, is further studied using the discrete phase integral (or Wentzel-Kramers-Brillouin) method. The analysis of the previous paper is extended to the case where the magnetic field has both hard and easy components, so that the Hamiltonian has no obvious symmetry. Herring's formula is now inapplicable, so the problem is solved by finding the wavefunction and using connection formulas at every turning point. A general formula for the energy surface in the vi…
We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest two eigenvalues--the eigenvalue splitting--is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping probability between the wells, a key parameter in tight binding models of solid state physics.
We report on measurements of splitting Bose-Einstein condensates (BEC) by using a time-dependent optical lattice potential. First, we demonstrate the division of a BEC into a set of equally populated components by means of time dependent control of Landau-Zener tunneling in a vertical lattice potential. Next, we apply time dependent optical Bragg mirrors to a BEC oscillating in a harmonic trap. We demonstrate high-order Bragg reflection of the condensate due to multi-photon Raman transitions, where the depth of the optical lattice potential allows for a choice of the order of the transition. F…
An effect of paramagnetic impurity located in a vicinity of a quantum well (QW) on spin polarization of the carriers in the QW is analyzed theoretically. Within approach of Bardeen's tunneling Hamiltonian the problem is formulated in terms of Anderson-Fano model of configuration interaction between a localized hole state at Mn and continuum of heavy hole states in the InGaAs-based QW. The hybridization between the localized state and the QW leads to resonant enhancement of interband radiative recombination. The splitting of the configuration resonances induced by splitting of the localized sta…
The perfect quenching of spin tunneling that has previously been discussed in terms of interfering instantons, and has recently been observed in the magnetic molecule \Fe8, is treated using a discrete phase integral (or Wentzel-Kramers-Brillouin) method. The simplest model Hamiltonian for the phenomenon leads to a Schrödinger equation that is a five-term recursion relation. This recursion relation is reflection-symmetric when the magnetic field applied to the molecule is along the hard magnetic axis. A completely general Herring formula for the tunnel splittings for all reflection-symmetric fi…
