While the phenomenon of the exact crossing of energy levels is a rarely occurring event, in the case of quantum resonances associated with metastable states this phenomenon is much more frequent and various scenarios can occur. When there is an exact crossing of the imaginary parts of the resonances in a two-level quantum system subject to an external DC electric field, then a damped beating phenomenon occurs, which is absent if there is no such crossing. This fact, tested numerically on an explicit one-dimensional model, suggests the possibility of designing quantum sensors to determine in a very simple way whether the external field strength has an assigned value or not.
Abstract We propose a theoretical model for a quantum sensor that can determine in a very simple way whether the intensity of an electric field has an assigned value or not. It is based on the fact that when an exact crossing of the imaginary parts of the resonances occurs in a double-well quantum system subject to an external DC electric field, a damped beating phenomenon occurs, which is absent if there is no such a crossing. This result is then tested numerically on an explicit one-dimensional model.
Since the 1990s, experiments on double-well heterostructures have highlighted a key phenomenon: applying an external DC electric field induces pronounced wave packet oscillations at specific field strengths. \ To gain a deeper insight into this effect, a comprehensive quantitative analysis of the properties of quantum resonances near a crossing point is required. \ This approach lays the theoretical foundation for devising a quantum sensor capable of rapidly and efficiently determining whether the strength of an external DC electric field lies within a narrow target range.
In the case of crossings of isolated pairs of quantum resonances for double-well potentials it is possible to observe that, under appropriate conditions, the associated metastable states are equally localised on both wells. This property, together with the fact that the imaginary parts of the two resonances coincide, results in the establishment of an interference phenomenon between the two metastable states that characterises the time dynamics of certain physical observables. In this way, it is possible to lay the theoretical foundations for the design of a quantum device to assess whether or not the model’s parameters assume pre-assigned values.
A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methods to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair $^{23}\mathrm{Na}\text{\ensuremath{-}}^{85}\mathrm{Rb}$ which are of current interest.
We have observed the crossing of the atomic hydrogen levels ${\ensuremath{\beta}}_{B}(2{S}_{\frac{1}{2}}, {m}_{J}=\ensuremath{-}\frac{1}{2}, {m}_{I}=\ensuremath{-}\frac{1}{2})$ and ${e}_{B}(2{P}_{\frac{1}{2}}, {m}_{J}=+\frac{1}{2}, {m}_{I}=\ensuremath{-}\frac{1}{2})$ near 605 G. A new atomic-beam method is used. H atoms are excited to the metastable
2S state by electron bombardment. The $2S({m}_{J}=\ensuremath{-}\frac{1}{2})$ state is quenched in the bombardment region. The
2S(mJ=+21) beam passes through a zero-field region where
31 of it is converted to the single hyperfine level ${\ensuremath{\beta}}_{B}$. The ${\ensuremath{\beta}}_{B}$ atoms enter a uniform magnetic field parallel to the beam. With no externally applied electric field, 92% of them reach the detector. When we apply an electrostatic field of about 0.7 V/cm perpendicular to the beam, the ${\ensuremath{\beta}}_{B}$ atoms are strongly quenched near the ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ crossing point. Observed quenching agrees with the Bethe-Lamb theory of the
2S lifetime in external fields. We measure magnetic field by the nuclear magnetic resonance frequency of protons in water. Natural asymmetry and minor experimental corrections reduce the observed center of the ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ quenching resonance by 1 part in
104. The corrected ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ crossing occurs at 2577.57\ifmmode\pm\else\textpm\fi{}0.25 kc/sec, proton nmr in water. From this, we calculate the Lamb shift
S in H,
n=2. With a diamagnetic shielding correction, and the accurately known
2S hyperfine interval, we get $\mathcal{S}=1058.07\ifmmode\pm\else\textpm\fi{}0.10$ Mc/sec. This disagrees with $\mathcal{S}=1057.77\ifmmode\pm\else\textpm\fi{}0.10$ Mc/sec from the Lamb experiments.