Quantitative relation between spectral properties and decay exponent

Establish a quantitative relation between the decay exponent of coherent spin-mixing oscillations in trapped spin-3/2 Fermi gases and specific properties of the underlying single-particle spectrum.

Background

The paper studies coherent spin-mixing dynamics of a trapped spin-3/2 Fermi gas within the time-dependent Hartree–Fock approximation in three one-dimensional trapping potentials: the infinite square well, the harmonic oscillator trap, and the Pöschl–Teller potential. Although the traps have increasing, equally spaced, and decreasing level spacings, respectively, the oscillation envelopes in the initial regular-oscillation regime are fitted by a power law, A(t)=A₀−γt^α.

The decay exponent α varies systematically with the spectral structure: it is smallest for the infinite square well, intermediate for the harmonic trap, and largest for the Pöschl–Teller potential. The authors interpret these differences qualitatively as consequences of coherent dephasing among dynamical modes with different characteristic frequencies. However, they do not derive a quantitative mapping from identifiable features of the single-particle spectrum to the value of α, leaving the precise spectral origin of the exponent unresolved.