External-field-modified entropic exchange in confined quantum systems: static versus time-dependent electric fields

Abstract

We review how external electric fields reshape the information content of spatially confined atoms, and we contrast the response to a uniform static field with that to a monochromatic, time-dependent laser field. The central quantities are the Shannon entropies in position and momentum space, Sr and Sp, and their sum ST = Sr +Sp, which obeys the BBM bound. Using the confined hydrogen atom (CHA) and the particle in a spherical box (PISB), its tight confinement limit, as reference systems, we first show that the field-free total entropy of the CHA depends non-monotonically on the confining radius whereas that of the PISB is strictly scale invariant. We then present Stark maps of the entropies for the ten lowest m = 0 levels at the confining radius of 6 a.u. The ground-state polarizability is reduced from the free-atom value 4.5 a.u. to 4.06 a.u. in the CHA but is 46.8 a.u. in the PISB, and the entropic response of the two systems differs correspondingly.

The paper examines how static electric fields and monochromatic driving change position- and momentum-space Shannon entropies in hard-wall confined hydrogen (CHA) and a particle in a spherical box (PISB). It combines a review with numerical calculations, principally at confinement radius $R_c=6$ a.u. for $m=0$, using finite-difference radial states, static-field matrix diagonalization, and Floquet theory. Laser-driven entropies are computed from cycle-averaged densities. Without fields, the PISB entropy sum $S_T=S_r+S_p$ is radius-independent by scaling, whereas CHA values depend non-monotonically on radius; its ground-state minimum is $6.492$ near $R_c=2.5$ a.u., above the BBM bound $3(1+\ln\pi)=6.434$. At $R_c=6$, ground-state polarizabilities are $4.06$ a.u. for CHA and $46.8$ a.u. for PISB. Static fields produce opposite signs of position- and momentum-entropy changes in the two systems, with substantial cancellation in $S_T$. Entropy redistribution accompanies narrow avoided crossings and field-induced mixing of accidentally degenerate levels. Near the CHA $1s$–$2p$ laser resonance, the Floquet gap is approximately $E_0d_{12}$, with $d_{12}=0.9225$ a.u. Increasing $E_0$ from $0.01$ to $0.04$ a.u. shifts the resonance from $0.4437$ to $0.4490$ a.u. and broadens entropy exchange. Both branches’ entropy sums exceed their off-resonant values near resonance. The stated limitations include angular-basis truncation affecting higher static-field levels, arbitrary zero-field mixtures within a degenerate subspace, restriction to hard walls and $m=0$, and cycle averaging rather than pulse-resolved entropy dynamics.