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Physical Review A

Topological quenching of the tunnel splitting for a particle in a double-well potential on a planar loop

Stefan Weigert

Published Dec 1, 1994• 16 citations
Publisher

Abstract

The motion of a particle along a one-dimensional closed curve in a plane is considered. The only restriction on the shape of the loop is that it must be invariant under a twofold rotation about an axis perpendicular to the plane of motion. Along the curve a symmetric double-well potential is present leading to a twofold degeneracy of the classical ground state. In quantum mechanics, this degeneracy is lifted: the energies of the ground state and the first excited state are separated from each other by a slight difference \ensuremath{\Delta}E, the tunnel splitting. Although a magnetic field perpendicular to the plane of the loop does not influence the classical motion of the charged particle, the quantum-mechanical separation of levels turns out to be a function of its strength B. The dependence of \ensuremath{\Delta}E on the field B is oscillatory: for specific discrete values ${\mathit{B}}_{\mathit{n}}$ the splitting drops to zero, indicating a twofold degeneracy of the ground state. This result is obtained within the path-integral formulation of quantum mechanics; in particular, the semiclassical instanton method is used. The origin of the quenched splitting is intuitively obvious: it is due to the fact that the configuration space of the system is not simply connected, thus allowing for destructive interference of quantum-mechanical amplitudes. From an abstract point of view this phenomenon can be traced back to the existence of a topological term in the Lagrangian and a nonsimply connected configuration space. In principle, it should be possible to observe the splitting in appropriately fabricated mesoscopic rings consisting of normally conducting metal.

DOI
10.1103/physreva.50.4572

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The ground-state energy splitting due to tunneling in two-dimensional double wells of the form $V(x,y)=\frac{{({x}^{2}\ensuremath{-}{R}^{2})}^{2}}{8{R}^{2}}+\frac{{x}^{2}\ensuremath{-}{R}^{2}}{{R}^{2}}\ensuremath{\gamma}y+\frac{{\ensuremath{\omega}}^{2}}{2}{y}^{2}$ is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of ${R}^{\ensuremath{-}2}$, each term of the series being a finite polynomial in ${\ensuremath{\gamma}}^{2}$. We find an ascending sequence of the values of the parameter $\ensuremath{\gamma}$ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of ${R}^{\ensuremath{-}2}$ being an infinite series in powers of ${\overline{\ensuremath{\gamma}}}^{2}$, $\overline{\ensuremath{\gamma}}=\ensuremath{\gamma}/R$. Third, we find that the radius of convergence of the series is determined by the singularity at ${\overline{\ensuremath{\gamma}}}_{s}=\ensuremath{\omega}/2$. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of $\overline{\ensuremath{\gamma}}$, most likely of the form ${[1\ensuremath{-}{(\overline{\ensuremath{\gamma}}/{\overline{\ensuremath{\gamma}}}_{s})}^{2}]}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=1/2$ and $\ensuremath{\alpha}=\ensuremath{-}1/2$ for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large $R$ the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter $\overline{\ensuremath{\gamma}}$ entering the Hamiltonian linearly.

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Tunneling and energy splitting in an asymmetric double-well potential

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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. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1

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The motion of a particle along a one-dimensional closed curve in a plane is considered. The only restriction on the shape of the loop is that it must be invariant under a twofold rotation about an axis perpendicular to the plane of motion. Along the curve a symmetric double-well potential is present leading to a twofold degeneracy of the classical ground state. In quantum mechanics, this degeneracy is lifted: the energies of the ground state and the first excited state are separated from each other by a slight difference \ensuremath{\Delta}E, the tunnel splitting. Although a magnetic field perpendicular to the plane of the loop does not influence the classical motion of the charged particle, the quantum-mechanical separation of levels turns out to be a function of its strength B. The dependence of \ensuremath{\Delta}E on the field B is oscillatory: for specific discrete values Bn{\mathit{B}}_{\mathit{n}}Bn​ the splitting drops to zero, indicating a twofold degeneracy of the ground state. This result is obtained within the path-integral formulation of quantum mechanics; in particular, the semiclassical instanton method is used. The origin of the quenched splitting is intuitively obvious: it is due to the fact that the configuration space of the system is not simply connected, thus allowing for destructive interference of quantum-mechanical amplitudes. From an abstract point of view this phenomenon can be traced back to the existence of a topological term in the Lagrangian and a nonsimply connected configuration space. In principle, it should be possible to observe the splitting in appropriately fabricated mesoscopic rings consisting of normally conducting metal.
A particle is assumed to move along a one-dimensional loop such as an ellipse that rotates in a plane. Because of the centrifugal force the particle is subjected to a symmetric double-well potential. Classically, the Coriolis force does not affect the motion of the particle, whereas the corresponding term in the Lagrangian influences the properties of the quantum system: its ground state turns out to be degenerate for a discrete set of angular velocities. The analogy between a constant magnetic field and a uniform rotation is used to propose, in addition, a variant of the Aharonov-Bohm experiment, which can be performed also with neutral particles.
Quantum interference is studied for a charged particle on a multiply connected configuration space. The holes of the space are assumed to accommodate independent magnetic fluxes. As follows from an instanton calculation, tunneling of the particle is suppressed completely in such a geometry for appropriately chosen field strengths. This set of values defines an algebraic variety in parameter space, not a manifold. The quenching persists even if the geometric symmetry of the system is broken. Aharonov-Bohm–type experiments and mesoscopic tunneling devices are natural candidates to observe many-path interference.
Quantum tunneling of the ground and first excited states in a quantum superposition driven by a novel analytical configuration of a double-well (DW) potential is investigated. Symmetric and asymmetric potentials are considered as to support quantum mechanical zero mode and first excited state analytical solutions. Reporting about a symmetry breaking that supports the quantum conversion of a zero-mode stable vacuum into an unstable tachyonic quantum state, two inequivalent topological scenarios are supposed to drive stable tunneling and coherent tunneling destruction respectively. A complete prospect of the Wigner function dynamics, vector field fluxes and the time dependence of stagnation points is obtained for the analytical potentials that support stable and tachyonic modes.
The ground-state energy splitting due to tunneling in two-dimensional double wells of the form $V(x,y)=\frac{{({x}^{2}\ensuremath{-}{R}^{2})}^{2}}{8{R}^{2}}+\frac{{x}^{2}\ensuremath{-}{R}^{2}}{{R}^{2}}\ensuremath{\gamma}y+\frac{{\ensuremath{\omega}}^{2}}{2}{y}^{2}$ is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of ${R}^{\ensuremath{-}2}$, each term of the series being a finite polynomial in ${\ensuremath{\gamma}}^{2}$. We find an ascending sequence of the values of the parameter $\ensuremath{\gamma}$ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of ${R}^{\ensuremath{-}2}$ being an infinite series in powers of ${\overline{\ensuremath{\gamma}}}^{2}$, $\overline{\ensuremath{\gamma}}=\ensuremath{\gamma}/R$. Third, we find that the radius of convergence of the series is determined by the singularity at ${\overline{\ensuremath{\gamma}}}_{s}=\ensuremath{\omega}/2$. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of $\overline{\ensuremath{\gamma}}$, most likely of the form ${[1\ensuremath{-}{(\overline{\ensuremath{\gamma}}/{\overline{\ensuremath{\gamma}}}_{s})}^{2}]}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=1/2$ and $\ensuremath{\alpha}=\ensuremath{-}1/2$ for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large RRR the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter $\overline{\ensuremath{\gamma}}$ entering the Hamiltonian linearly.
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. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1