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.     

Instanton approach to Josephson tunneling between trapped condensates

An instanton method is proposed to investigate the quantum tunneling between two weakly-linked Bose-Einstein condensates confined in double-well potential traps. We point out some intrinsic pathologies in the earlier treatments of other authors and make an effort to go beyond these very simple zero order models. The tunneling amplitude may be calculated in the Thomas-Fermi approximation and beyond it; we find it depends on the number of the trapped atoms, through the chemical potential. Some suggestions are given for the observation of the Josephson oscillation and the MQST. Received 29 June 2001 and Received in final form 17 September 2001     

有效质量法调控原子玻色-爱因斯坦凝聚体的双阱动力学

操控原子玻色-爱因斯坦凝聚体在双势阱中的动力学通常是通过改变势阱深度来实现,本文提出了一种基于调节原子有效质量的控制方案,可以在不改变双阱势的前提下操控凝聚体的双阱动力学.利用双模近似,本文解析地导出了超冷原子在双阱势中的隧穿强度和相互作用强度对有效质量的依赖关系,并基于平均场近似数值模拟了在有效质量调节下的凝聚体动力学演化,展示了隧穿振荡和自束缚等典型的双阱动力学行为.此外,本文的研究还发现,借助负有效质量效应,这一方案甚至可以等效地实现对负散射长度原子凝聚体双阱动力学行为的操控.     

Decoherence due to nodal quasiparticles in <Emphasis Type="Italic">d</Emphasis>-wave qubits

We study the Josephson junction between two d-wave superconductors, which is discussed as an implementation of a qubit. We propose an approach to calculate the decoherence time due to an intrinsic dissipative process: quantum tunneling between the two minima of the double-well potential excites nodal quasiparticles, which lead to incoherent damping of quantum oscillations. The decoherence is weakest in the mirror junction, where the contribution of nodal quasiparticles corresponds to the superohmic dissipation and becomes small at small tunnel splitting of the energy level in the double-well potential. For available experimental data, we estimate the quality factor.     

Shape of recombination lines for exciton complexes in quantum dots with in-plane electric field

We study electron–hole recombination lines of exciton (X) and exciton complexes (X−, X+, 2X) in planar quantum dots with the electric field oriented within the plane of confinement. A model of a two-dimensional circular infinite quantum well is applied and the ground state of the complexes is found using an exact diagonalization method. We demonstrate that for each of the exciton complexes the recombination lines become non-monotonic for some material and sample parameters as a result of Coulomb interactions. A phase diagram for the line shape is presented. The relation of the exact results to the mean field approach is also discussed.     

The generalized Fubini instanton

We show that (1+2) nonlinear Klein-Gordon equation with negative coupling admits an exact solution which appears to be the linear superposition of the plane wave and the nonsingular rational soliton. We show that the same approach allows to construct the solution of similar properties for the Euclidean ?⁴ model with broken symmetry. Interestingly, this regular solution will be of instanton type only in the D?5 Euclidean space. Thus one can use the generalized Fubini instantons (in quantum cosmology for example) only for the case of the single infinite extra dimension.     

A “superlattice” model for a smooth GaAs/AlAs (001) heterointerface

The effect of a smooth interface potential on the electronic states in GaAs/AlAs (001) structures is investigated using the pseudopotential method. In this approach, the transition region between GaAs and AlAs is assumed to be a layer corresponding to a half-period of the (AlAs)₂(GaAs)₂ superlattice, with the potential of this layer being close to the real potential near the heterointerface. In this case, the intervalley mixing occurs at two boundaries and in the transition layer rather than at one boundary, as in the model with a sharply cut-off potential. It is shown that a smooth potential has an appreciable effect on electron tunneling in structures with thin layers. This effect is especially important in the case where short-wavelength X states are involved. For one GaAs/AlAs (001) boundary, the transition layer acts as a quantum well localizing the charge density of a mixed Γ-X state near the boundary. In structures with a layer thickness of less than 2 nm, the differences in the resonance energies obtained in the models with a smooth heterointerface and with a sharp heterointerface can be as high as ～0.1 eV. The envelopes of the wave functions associated with Γ ₁ ⁽¹⁾ , Γ ₁ ⁽²⁾ , and Γ ₃ ⁽¹⁾ superlattice valleys and with Γ₁, X ₁, and X ₃ valleys of GaAs and AlAs are analyzed. It is shown that the matching matrices for the envelope functions at the GaAs/(AlAs)₂(GaAs)₂ and (AlAs)₂(GaAs)₂/AlAs boundaries depend only weakly on the electron energy near the bottom of the conduction band and that the probability densities calculated using these functions agree with the results of many-band calculations. Therefore, these functions can be used to construct a model with a smooth interface potential in the framework of the effective-mass method.     

The effect of an external electric field on the optical properties of a quantum molecule with a resonance D(-)-state

The average binding energy and the level width for the resonant D^(-)-state in a quantum molecule have been calculated in the presence of an external electric field. The calculations were performed in the zeroradius potential model with allowance for the tunneling decay of the resonant state. The external electric field is shown to stimulate the decay of resonant D^(-)-states under conditions of dissipative tunneling. It was found that the curve of the probability of photoionization of the D^(-)-center as a function of the external electric field strength has two peaks that are connected with a change in the symmetry of the double-well oscillator potential of the quantum molecule and with the transformation (caused by the electric field) of envelope wave functions, respectively.     

Phase transition in quantum tunneling for a parameterized double-well potential

The transition from the quantum tunneling regime to the classical crossover regime for a parameterized double-well potential is investigated. Using the different methods, the critical value of the potential parameter at which the first-order transition occurs is obtained.     

Relatively-long time decay of Loschmidt echo of a Bose-Einstein condensate in a double-well potential

We study the long-time decay of quantum Loschmidt echo (LE) of a Bose-Einstein condensate (BEC) in a double-well potential. In the tunneling and self-trapping phases of the BEC, the LE has exponential and Gaussian decays, respectively, for relatively-long times. In the crossover region, the LE behaves differently from both the tunneling and the self-trapping phases. These results indicate that relatively-long time decay of the LE is suitable for characterizing the dynamical phase transition of the BEC.     

Controllable dissipative tunneling in an external electric field

We have studied the problem of controllable dissipative tunneling in the system of tunnel-binding quantum dots (quantum molecules) and in the “AFM/STM cantilever tip-quantum dot” system, which was simulated by a double-well oscillator potential interacting with a heat-bath in an external electric field. We show that theoretical results qualitatively describe some experimental I–V curves for “the AFM/STM cantilever tip-zirconium quantum dot” system. These experimental curves were obtained in the Research Institute of Physics and Technologies at the State University of Nizhniy Novgorod.     

Quantum Kaluza-Klein cosmologies (III)

The “ground state” proposal for the quantum state of the universe is generalized to the case of a noncompact spacelike three-hyperboloid as the configuration space. The most probable evolution of the universe must come from a gravitational instanton by quantum tunneling. We show that under some minisuperspace ansatz, there exists only S₄ × S₇ gravitational instanton in d = 11 supergravity. From the point of view of quantum cosmology this fact must be related to the fact that our observed spacetime is four-dimensional.     

Nonlinear Landau-Zener tunneling of Bose-Einstein condensate in a spatially magnetic modulated trap

We study tunneling of a Bose-Einstein condensates confined in a effective double-well potential (a single well with a spatially magnetic modulated scattering length, actually), called pseudo double-well trap, in which the interaction of atoms characterized by the s-wave scattering length a _s can be widely tuned with a magnetic-field Feshbach resonance. As a result, corresponding to different nonlinear parameters, the energy levels of the nonlinear Schrödinger equation can have complex structures in their dependence on the bias between the wells. We discuss the emergence of looped levels, which lead to a breakdown of adiabaticity that the Landau-Zener transition probability does not vanish even in the adiabatic limit. Moreover, we also find that the Landau-Zener tunneling in the pseudo trap show many striking properties distinguished from that of the real trap model (where the barrier is created by the external potential). Possible experimental observation in an opticallyinduced photonic lattices in a photorefractive material is suggested.     

Vibrational coherences in single electron tunneling through nanoscale oscillators

Charging a nanoscale oscillator by single electron tunneling leads to an effective double-well potential due to image charges. We combine exact numerical diagonalizations with generalized master equations and show that the resulting quantum tunneling of the mechanical degree of freedom can be visualized in the electronic current noise spectrum.     

A negatively charged exciton in a quantum disc

The properties of the bound states of the negatively charged exciton X⁻ in a quantum disc with a confined parabolic potential are studied using exact diagonalization techniques. The binding energy spectra of the ground state and the first excited state are calculated as a function of the confinement strength and the effective electron-to-hole mass ratio. The results we have obtained show that the binding energies are closely correlated to the strength of the confinement potential and the effective electron-to-hole mass ratio.     

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations

In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton contributions to the partition function, using the formalism introduced in the first part of the treatise [Ann. Phys. (N. Y.) (previous issue) (2004)]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker-Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is non-zero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton A-function to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.     

Large order behavior for self-avoiding membranes

We derive the large order behavior of the perturbative expansion for the continuous model of tethered self-avoiding membranes. It is controlled by a classical configuration for an effective potential in bulk space, which is the analog of the Lipatov instanton, solution of a highly non-local equation. The nth order is shown to have factorial growth as (−cst)ⁿ (n!)^(1−ϵ/D), where D is the “internal” dimension of the membrane and ϵ the engineering dimension of the coupling constant for self-avoidance. The instanton is calculated within a variational approximation, which is shown to become exact in the limit of large dimension d of bulk space. This is the starting point of a systematic 1/d expansion. As a consequence, the ϵ-expansion of self-avoidinng membranes has a factorial growth, like the ϵ-expansion of polymers and standard critical phenomena, suggesting Borel summability. Consequences for the applicability of the 2-loop calculations are examined.     

Instanton approximation to the graded nonlinear sigma model for the integer quantum hall effect

We construct the multi-instanton solutions for the graded nonlinear σ model with symmetry U(1,1/2)/U(1/1) ⊗ U(1/1), and we calculate the quantum fluctuations around these solutions. The determinant of the fluctuation operator for a fixed multi-instanton solution turns out to be UV finite. However, the integration over instanton parameters contains an integral, ∫d|a| |a|⁻³, over the size, |a|, of each instanton, which is quadratically singular at |a|=0. It is shown that these quadratic divergences cancel exactly in the calculation of all Green functions. The applicability of the present results to the integer quantum Hall effect is discussed.