First-hand information for everyone
Quantum Fisher information (QFI), a key metric of how well a parameter can be estimated from a quantum state, depends on the electromagnetic gauge chosen for ultrastrongly coupled light‑matter systems. The study shows that the QFI for a ground state of a cavity–two‑level system traced an exact parabola when the gauge parameter was varied, and that the minimum of the QFI occurs at a universal angle θ_* = Ω/(Ω+ω) for harmonic matter, set only by the matter and cavity frequencies.
Using a non‑perturbative derivation, the authors present an exact relation for the QFI as a function of the gauge parameter:
F_g(θ)=F_g(0)+4θ²Var(̂x̂X)−8θW,
where ̂x̂X is the gauge‑generator operator and W its covariance with the derivative of the ground state with respect to the coupling strength. The relation holds for all values of the coupling and all gauge angles, and it relies only on ground‑state properties in the minimal‑coupling frame.
For harmonic matter, the QFI in the dipole gauge can exceed the minimal‑coupling value by a factor up to 9.6, with the direction of the difference depending on the specific gauge choice. This means that, depending on how the system is described, precision can appear to improve or degrade when the light‑matter coupling is increased.
The analysis also addresses proposals that virtual excitations at the Dicke point yield diverging metrological gains. By comparing a truncated gauge‑inconsistent model with a gauge‑consistent completion that includes the full diamagnetic term, the authors show that the apparent divergence in the static QFI disappears. The table below demonstrates that the static QFI saturates in the consistent family, whereas it diverges in the truncated family. This clarifies that the claimed critical enhancement is a gauge artifact.
| κ | ∂ωω−|tr | ∂ωω−|cons | Iωtr | Iωcons |
|---|---|---|---|---|
| 0.2 | 0.503 | 0.402 | 0.0028 | 0.0025 |
| 0.5 | 0.530 | 0.265 | 0.036 | 0.017 |
| 0.9 | 0.870 | 0.087 | 2.60 | 0.063 |
| 0.99 | 2.53 | 0.025 | 305 | 0.072 |
<table> <caption>Comparison of static QFI for truncated (tr) and gauge‑consistent (cons) families at ω=1. The factor R(κ) (Eq. 14) shows how the dynamic‑protocol gain is reduced and the static QFI saturates. </caption> <thead> <tr><th>κ</th><th>∂_ωω₋|tr</th><th>∂_ωω₋|cons</th><th>I_ω^tr</th><th>I_ω^cons</th></tr> </thead> <tbody> <tr><td>0.2</td><td>0.503</td><td>0.402</td><td>0.0028</td><td>0.0025</td></tr> <tr><td>0.5</td><td>0.530</td><td>0.265</td><td>0.036</td><td>0.017</td></tr> <tr><td>0.9</td><td>0.870</td><td>0.087</td><td>2.60</td><td>0.063</td></tr> <tr><td>0.99</td><td>2.53</td><td>0.025</td><td>305</td><td>0.072</td></tr> </tbody> </table>
The study highlights that gauge consistency is essential when evaluating quantum metrology in ultrastrong coupling regimes, as inconsistent gauge choices can lead to misleading predictions of precision gains.