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arXiv:2412.21100 · Proceedings of the National Academy of Sciences

Quantum tunneling and its absence in deep wells and strong magnetic fields

Charles Fefferman, Jacob Shapiro, Michael I. Weinstein

Published Feb 21, 2025• 1 citation
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Abstract

We present results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double-well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to antisymmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.

DOI
10.1073/pnas.2420062122

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We present results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double-well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to antisymmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.
Inspired by a recent paper∗^*∗ by C. Fefferman, J. Shapiro and M. Weinstein, we investigate quantum tunneling for a Hamiltonian with a symmetric double well and a uniform magnetic field. In the simultaneous limit of strong magnetic field and deep potential wells with disjoint supports, tunneling occurs and we derive accurate estimates of its magnitude. ∗ ^*\,∗[Lower bound on quantum tunneling for strong magnetic fields. SIAM J. Math. Anal. 54(1), 1105-1130 (2022).]
We study tunneling between two identical, smooth, compactly supported potential wells in a constant perpendicular magnetic field, with the coupling and magnetic field fixed and the well separation tending to infinity. For a sufficiently small positive magnetic field, we prove lower bounds on the hopping coefficient and on the splitting of the two lowest eigenvalues outside a set of separations of density zero. If the well centers are separated by 2d12d_12d1​ and the magnetic field strength is bλbλbλ, the lower bounds have the form e−bλd12−Cd11+δe^{-bλd_1^2-Cd_1^{1+δ}}e−bλd12​−Cd11+δ​ for every fixed δ∈(0,1)δ\in(0,1)δ∈(0,1). No radial or reflection symmetry of the single-well potential is assumed. The proof combines the explicit Landau resolvent kernel with a complex analytic estimate for logarithmic averages of the hopping coefficient.
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
We revisit the problem of laser-induced suppression of quantum dynamical tunneling in a model system studied by Kilin et al. [Phys. Rev. Lett. 76 (1996) 3297]. This quantum system consists of a ground state symmetric double-well potential which is coupled by a strong laser field to an excited state asymmetric double-well potential. By analyzing the assumptions used in their analysis we show that the suppression of quantum dynamical tunneling can be explained with the use of dark and bright states of the system. We also generalize the system and the conditions for suppression of quantum tunneling and show that, in certain cases, suppression can occur regardless the characteristics of the excited potential surface.