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arXiv:2208.13030 · Pure and Applied Analysis

Quantum tunneling in deep potential wells and strong magnetic field revisited

Bernard Helffer, Ayman Kachmar

Published May 16, 2024• 10 citations
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Abstract

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).]

DOI
10.2140/paa.2024.6.319

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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.
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.
We have studied simple tunneling problems in two dimensions in the presence of a high transverse magnetic field both by numerical integration of the Schr\"odinger equation and by semiclassical evaluation of the path integral. We have chosen three model potentials: (i) asymmetric single well, (ii) symmetric double well, and (iii) quadruple well. We find that the semiclassical approach is analytically tractable and gives a very accurate description of the exponential and oscillatory behaviors of the tunneling matrix elements. A precise definition of the Aharonov-Bohm phase for the tunneling paths is given. In addition to the Aharonov-Bohm phase, there is also a geometrical phase coming from the fluctuation determinant, and we find that for every closed loop it is exactly \ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}.
In a previous work [H. Pu, W. Zhang, and P. Meystre, Phys. Rev. Lett. 87, 140405 (2001)]], we have shown that a spinor condensate confined in a periodic or double-well potential exhibits ferromagnetic behavior due to the magnetic dipole-dipole interactions between different wells, and in the absence of external magnetic field, the ground state has a twofold degeneracy. In this work, we demonstrate the possibility of observing macroscopic quantum spin tunneling between these two degenerate states and show how the tunneling rate critically depends on the strength of the transverse field.