Fractional decay in the spontaneous emission of a two-level system

We find that when the environment of a two-level system has an energy spectrum with a lower bound but without an upper one, the survival probability of the spontaneous emission of the two-level system scales with the spatial dimension $D$ and the exponent $n$ of the energy dispersion $|\vec{k}|^n$ of the environment in the form $1-αt^{2-D/n}$ in the short-time and in the form $αt^{D/n-2}$ in the long-time regime. The former fractional scaling of the survival probability leads to a quantum Zeno effect with a different scaling of the Zeno time.

💡 Research Summary

In this paper the authors investigate the spontaneous emission dynamics of a two‑level system (TLS) coupled to a D‑dimensional environment whose dispersion relation is ω(k)=ω₀|k|ⁿ. Crucially, the environmental spectrum possesses a lower bound but no upper bound, which makes the usual variance ⟨H²⟩‑⟨H⟩² diverge and invalidates the standard short‑time quadratic expansion. By employing a Dyson series expansion of the time‑evolution operator, they obtain a dimensionless combination g² ω₀^{‑ν} t^{2‑ν} (with ν≡D/n) that governs the survival amplitude. For 0 < ν < 1 the survival probability behaves as

 p(t) ≃ 1 − C t^{2‑ν}

in the short‑time regime, where C depends on the coupling strength g, the cutoff‑free dispersion parameters, and geometric factors. This “fractional exponent” replaces the usual quadratic term and originates from an autocorrelation function η(τ)∝τ^{‑ν}, which reflects a non‑Markovian memory kernel: the environment retains a power‑law memory of the emitted excitation rather than an instantaneous delta‑correlation.

The fractional short‑time decay leads to a modified quantum Zeno effect. The conventional Zeno time τ_Z scales as c^{‑1/2} for quadratic decay (c being the coefficient of t²). Here, because p(t)≈1‑c t^{2‑ν}, the Zeno time scales as

 τ_Z ∝ c^{‑1/(2‑ν)}.

Thus, the characteristic time can be tuned by the ratio D/n: larger ν (higher dimension or lower dispersion exponent) shortens τ_Z, requiring more frequent measurements to observe the Zeno slowdown, while smaller ν lengthens τ_Z, making the effect easier to detect.

The authors also consider a finite momentum cutoff Λ. By keeping terms up to second order in g, they show that for times satisfying (ω₀Λⁿ)^{‑1} ≪ t ≪ τ_Z the same fractional law persists, provided the cutoff is sufficiently large (ω₀Λⁿ ≫ ω_S). Hence the fractional regime is robust against realistic high‑energy cutoffs.

For the long‑time behavior they employ the Feshbach‑PQR projection formalism, deriving an effective non‑Hermitian Hamiltonian H_eff(z)=ω_S−Σ(z) with self‑energy Σ(z)∝g² z^{ν‑1}. Analyzing the complex z‑plane, they identify a branch cut (originating from the fractional power) and, depending on parameters, a real pole z₀ on the negative axis. The contour integral for the survival amplitude separates into a pole contribution (producing an undamped oscillatory term) and a branch‑cut contribution that decays as t^{‑(2‑ν)}. Consequently, the survival probability at long times follows

 p(t) ≃ |α₀|² + K t^{‑(2‑ν)} cos(z₀ t + φ),

where K and φ are determined by ν, the coupling, and geometric factors. This power‑law decay with superimposed oscillations starkly contrasts with the simple exponential law predicted by Markovian master equations.

The paper discusses experimental relevance: recent realizations of higher‑order dispersions (n≥3) in multi‑Weyl semimetals, as well as controllable photonic or cold‑atom platforms, allow tuning of D and n to explore the predicted regimes. The fractional Zeno scaling and the long‑time power‑law with oscillations provide clear signatures that can be probed by repeated projective measurements or time‑resolved spectroscopy.

In summary, the work establishes a new class of non‑Markovian decay dynamics arising from environments with a lower‑bounded but unbounded spectrum. It derives universal scaling laws for both short‑ and long‑time regimes, introduces the concept of a “fractional quantum Zeno effect,” and highlights how dimensionality and dispersion exponent dictate the decay exponent. These insights broaden our understanding of open quantum systems beyond the conventional exponential paradigm and open avenues for experimental verification in modern quantum materials and engineered photonic structures.