Abstract
It is known that the overlap of two energy eigenstates in a decaying quantum system is bounded from above by a function of the energy detuning and the individual decay rates. This is usually traced back to the positive definiteness of an appropriately defined decay operator. Here, we show that the weaker and more realistic condition of positive semi-definiteness is sufficient. We prove also that the bound becomes an equality for the case of single-channel decay. However, we show that the condition of positive semi-definiteness can be spoiled by quantum backflow. Hence, the overlap of quasibound quantum states subjected to outgoing-wave conditions can be larger than expected from the bound. A modified and less stringent bound, however, can be introduced. For electromagnetic systems, it turns out that a modification of the bound is not necessary due to the linear free-space dispersion relation. Finally, a geometric interpretation of the nonorthogonality bound is given which reveals that in this context the complex energy space can seen as a surface of constant negative curvature.
Highlights
Any realistic quantum or wave system is an open system because it can never be perfectly isolated from its environment
The current strong interest in open systems arises from the new fields of non-Hermitian physics and paritytime (PT) symmetry [1,2,3]
The nonHermiticity or non-self-adjointness H = H † implies that its eigenvalues E j are complexvalued with the imaginary part determining a decay rate
Summary
Any realistic quantum or wave system is an open system because it can never be perfectly isolated from its environment. It has been shown that nonorthogonality can lead to nonexponential transient decay [26,27,28], chirality in perturbed whispering-gallery microcavities [29,30], power oscillations in optical waveguides [31,32], sensitivity of resonance widths under perturbation [33], limitation of mode selectivity [34], interesting transport properties [35,36], and quantum excess noise in lasers [37,38,39,40] It has been known for a long time that in decaying quantum systems there is an upper bound for the squared modulus of the overlap of two normalized energy eigenstates |ψl and |ψ j.
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