Where should we look for the deviation from quantum mechanics?

Jerome Nicolas; Albert Schwarz

AI summary · arXiv: 2610.24857v1 · quant-ph

Where should we look for the deviation from quantum mechanics?

Jerome Nicolas, Albert Schwarz

Original paper Submitted to arXiv on 17 Oct. 2026 11 pages License: CC BY 4.0

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The essentials

The paper models metastable helium with a quartic deformation and predicts altered mean lifetime and nonexponential decay in a short-memory approximation.

Question

Could measuring the lifetime and decay curve of long-lived metastable helium reveal small deviations from quantum mechanics? The paper examines this for a specific quartic deformation of the excited-state energy.

Method

The authors model helium as a two-level atom coupled to quantized radiation in the rotating-wave approximation, then add a quartic term affecting only the metastable-state population. In a nonlinear Wigner–Weisskopf approximation with a local cubic spectral density, they derive the population-dependent decay rate and calculate the survival curve and mean lifetime to first order in ε = |A₁|/ΔE.

Findings

Result For the local cubic spectral density, the mean lifetime relative to the ordinary lifetime at the same gap changes by −3A₁/ΔE to first order.
See the passage in the paper
For the cubic law, expansion of the integrand using (4.16) yields $$\frac{\tau_{\rm red}}{\tau_{0}}=1-\frac{3A_{1}}{\Delta E}+O(\varepsilon^{2}).$$
Result The reduced survival curve changes shape at first order in the quartic coefficient, even when the energy deformation is small.
See the passage in the paper
On bounded intervals of the scaled time $\Gamma_{0}t$, integration of the reduced law (4.14) gives $$\begin{split}x_{\rm red}(t)=\mathrm{e}^{-\Gamma_{0}t}\left[1-\frac{6A_{1}}{\Delta E}(1-\mathrm{e}^{-\Gamma_{0}t})\right]+O(\varepsilon^{2}).\end{split}$$
Result At the corrected stationary energy gap, the mean agrees to first order with the ordinary lifetime, while the cubic reduced decay curve still changes shape at first order.
See the passage in the paper
Thus the mean agrees with that ordinary lifetime to first order, while the cubic reduced curve (4.17) changes shape at first order.
Authors’ interpretation Within the reduced description, a positive, nonconstant population-dependent decay rate makes the decay nonexponential.
See the passage in the paper
More generally, through the frequency dependence of $J$, the population-dependent resonance frequency (4.11) makes the reduced decay nonexponential whenever $\Gamma$ is positive and nonconstant on $[0,1]$.
Authors’ interpretation The authors argue that helium's long lifetime can make the absolute lifetime shift appreciable, although this does not enlarge the fractional shift.
See the passage in the paper
A long lifetime enlarges this absolute change, but not the fractional mean-lifetime change $3|A_{1}|/\Delta E$ at first order.

Limitations

The reduced prediction relies on a sufficiently short-memory reservoir and a transition frequency that changes slowly over the memory time; the cubic spectral-density approximation must hold across the swept frequency range.
See the passage in the paper
The reduced law (4.14) assumes a regular reservoir with a sufficiently short memory and a slowly varying transition frequency over that memory time. The cubic approximation for $J$ must hold throughout the interval swept by $\omega_{\rm tr}(x)$, $0\leq x\leq 1$, whenever that approximation is used.
The paper gives no quantitative bound on the memory error for helium from the cited scale separation alone.
See the passage in the paper
A bound on the memory error also requires information about the completed spectral density. No quantitative memory-error bound for helium is inferred from the ratio $\Gamma_{0}/\omega_{0}$ alone.
Nonexponential decay or the proposed rate–mean relation is not a unique signature of a quantum-mechanical deformation, because ordinary losses can produce similar behavior.
See the passage in the paper
For ${\Gamma^{\prime}(0)\geq 0}$, this is also an ordinary population-loss law with constant one- and two-body loss coefficients. Such losses satisfy relation (4.24) to first order in $\Gamma^{\prime}(0)/\Gamma_{0}$. The relation is therefore an internal consistency check, not a unique signature of a deformation.

Why it matters

The authors propose precise lifetime and decay-law measurements of metastable helium as a test of deviations from quantum mechanics, emphasizing that its theoretically calculable, long lifetime makes absolute changes potentially observable. They also suggest that deviations from exponential decay could be useful for systems whose lifetimes cannot be calculated precisely, while noting that ordinary loss processes can confound that interpretation.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Metastable helium nonexponential decay modified quantum mechanics quartic deformation lifetime measurement

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Comprehensive Summary
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Original abstract

The paper asks whether small departures from ordinary quantum mechanics might be easier to detect in the decay of a long-lived metastable system than through measurements of energy levels or scattering. The authors propose metastable helium as a candidate because its radiative lifetime is about two hours and can be calculated from first principles. A previous measurement found a lifetime of 7,870 ± 510 seconds, with a relative uncertainty of about 6.5%, consistent with the theoretical prediction. The paper examines how the lifetime and decay curve would change under a specific modification of quantum mechanics; it does not claim that this modification has been observed. The authors model helium as a two-level atom, comprising a ground state and a threefold-degenerate metastable triplet, coupled to the quantized electromagnetic field. They use the rotating-wave approximation and describe ordinary decay with a memory equation for the excited-state amplitude. The proposed modification adds a small, rotation-invariant quartic term to the model’s underlying classical Hamiltonian, proportional to the square of the excited-state population. The authors set the corresponding ground-state term to zero, leaving the photon Hamiltonian and transition magnetic moment unchanged. This term makes the transition frequency depend on the excited-state population. In a short-memory, Wigner–Weisskopf approximation, the usual constant decay rate becomes population-dependent and is determined by the reservoir’s spectral density at the shifted transition frequency. For a local cubic spectral density—appropriate when the effective coupling and atomic form factor vary little across the relevant frequency interval—the rate changes to first order in the quartic coefficient $A_1$ as the population falls. Defining the small parameter as $|A_1|$ divided by the energy gap, the paper finds that the reduced survival curve is nonexponential at first order. The mean survival time, defined as the integral of the excited-state population over time, changes by $-3A_1$ divided by the gap relative to the ordinary lifetime at the same unshifted gap. Its fractional change is therefore $3|A_1|$ divided by the gap, and its absolute change is this fraction multiplied by the ordinary lifetime of about two hours. Thus, even a small fractional effect could correspond to a substantial difference in elapsed time. The comparison depends on the energy gap used. The quartic term shifts the stationary excited-state energy by $A_1$, making the corrected gap equal to the original gap plus $A_1$. Compared with an ordinary atom at this corrected gap and with the same coupling, the modified model’s mean lifetime agrees to first order, although its decay curve still changes shape at first order. In the reduced description, the initial rate for the cubic spectral law is the ordinary rate multiplied by $(1+2A_1/\text{gap})^3$, while the late-time reduced rate returns to the ordinary rate. The changing logarithmic decay rate is a signature of the model’s nonexponential intermediate-time decay, but is not, by itself, unique evidence of modified quantum mechanics. These conclusions are conditional on the chosen deformation and approximations. The exact memory equation predicts nonexponential behavior in ordinary quantum mechanics at very short and very long times, so the paper’s comparison concerns the intermediate regime captured by the reduced approximation. The authors note that the small ratio of decay rate to optical frequency alone is insufficient to provide a quantitative bound on memory effects for helium. Realistic form factors, losses from collisions and boundaries, detector calibration, and photon statistics also matter. Ordinary population-dependent losses could imitate the predicted first-order behavior. Accordingly, the paper presents precision measurements of lifetimes and decay curves as a promising test, not as a definitive or uniquely diagnostic one.

Where should we look for a possible deviation from quantum mechanics?

Most tests of quantum mechanics look for tiny shifts in energy levels or changes in how particles scatter. Jerome Nicolas and Albert Schwarz ask whether another kind of measurement might be more revealing: watching how a long-lived excited system decays over time.

Their proposed candidate is metastable helium. Its lifetime is roughly two hours, and theory can predict that lifetime from first principles. A previous measurement found a lifetime of 7,870 ± 510 seconds, with a relative uncertainty of about 6.5%, consistent with the theoretical prediction. The authors explore how both the lifetime and the shape of the decay curve would respond to one specific modification of quantum mechanics. They do not claim that the modification has been observed.

A simple model of helium’s decay

To study the idea, the authors represent helium as a two-level atom: a ground state and a metastable excited state with threefold degeneracy. The atom interacts with the quantized electromagnetic field, which carries away energy as the excited state decays. In their calculation, they use the rotating-wave approximation and describe ordinary decay with a memory equation for the excited-state amplitude.

They then add a small, rotation-invariant quartic term to the model’s underlying classical Hamiltonian. The term is proportional to the square of the excited-state population, while the corresponding ground-state term is set to zero. The photon Hamiltonian and transition magnetic moment are left unchanged.

This addition makes the transition frequency depend on how much population remains in the excited state. As the atom decays, its transition frequency shifts, and so can its decay rate. In a short-memory approximation known as Wigner–Weisskopf, the rate is determined by the electromagnetic reservoir’s spectral density at the shifted transition frequency.

How the decay curve changes

For a local cubic spectral density—an approximation appropriate when the effective coupling and atomic form factor vary little across the relevant frequency range—the authors find a first-order change in the decay rate as the population falls. The resulting survival curve is not a simple exponential, even at first order in the small deformation.

The paper quantifies the effect using the mean survival time, defined as the integral of the excited-state population over time. Let the small parameter be the magnitude of the quartic coefficient, |A1|, divided by the energy gap. Relative to the ordinary lifetime at the same unshifted gap, the mean survival time changes by −3A1 divided by the gap. Its fractional change therefore has magnitude 3|A1| divided by the gap.

Because metastable helium’s ordinary lifetime is about two hours, even a small fractional change could translate into a noticeable difference in elapsed time. That makes precision lifetime measurements potentially interesting. But the comparison needs care: the deformation also shifts the stationary excited-state energy by A1, so the corrected energy gap is the original gap plus A1.

Why the comparison matters

Compare the modified model with an ordinary atom at that corrected energy gap, using the same coupling, and the mean lifetimes agree to first order. The decay curves, however, still differ in shape at first order. In the reduced description, the initial rate for the cubic spectral law is the ordinary rate multiplied by (1 + 2A1/gap)3, while the late-time reduced rate returns to the ordinary rate.

In other words, a lifetime alone may not capture the whole prediction. Measuring the decay curve in detail could reveal a changing logarithmic decay rate: a sign of the model’s nonexponential intermediate-time behavior. But that pattern, by itself, would not uniquely point to modified quantum mechanics.

Important limits on the claim

The result depends on the chosen deformation and on the approximations used to derive it. The exact memory equation predicts nonexponential behavior in ordinary quantum mechanics at very short and very long times. The paper’s comparison concerns the intermediate regime described by the reduced approximation, not a general claim that any nonexponential decay signals new physics.

The authors also caution that the small ratio of decay rate to optical frequency does not, on its own, provide a quantitative bound on memory effects for helium. Real experiments bring additional complications, including realistic form factors, losses from collisions and boundaries, detector calibration, and photon statistics. Ordinary population-dependent losses could imitate the first-order behavior predicted by the model.

A promising test, not a verdict

The paper’s central message is that long-lived metastable systems may offer a useful place to search for small deviations from standard quantum mechanics. Helium’s long lifetime and calculable properties make precise measurements appealing, especially measurements of the full decay curve rather than just a single lifetime value.

Still, any unusual decay pattern would need to be tested against ordinary physical effects and the limits of the approximations. Nicolas and Schwarz present precision measurements as a promising test—not a definitive or uniquely diagnostic one.

Created on 30 Oct. 2026

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Where should we look for the deviation from quantum mechanics?

The essentials

Question

Method

Findings

Limitations

Why it matters

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Definitions

Where should we look for a possible deviation from quantum mechanics?

A simple model of helium’s decay

How the decay curve changes

Why the comparison matters

Important limits on the claim

A promising test, not a verdict

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