Finite-energy Gottesman-Kitaev-Preskill state-enhanced optical interferometry

Technical Summary: Finite-energy Gottesman-Kitaev-Preskill state-enhanced optical interferometry

Problem Statement
Precise estimation of physical parameters, such as phase shifts in optical interferometry, is fundamental to sensing technologies ranging from gravitational wave detection to displacement measurement. While quantum-enhanced interferometry using squeezed vacuum states can surpass the shot-noise limit, the performance of these Gaussian resources is highly susceptible to optical losses, which introduce vacuum fluctuations that degrade squeezing. This paper investigates whether non-Gaussian Gottesman-Kitaev-Preskill (GKP) states, known for their robustness against Gaussian noise in quantum error correction, can offer a metrological advantage over squeezed vacuum states in optical interferometry, particularly under realistic conditions involving optical loss and finite energy constraints.

Methodology
The authors analyze the phase sensitivity of two interferometer architectures: the passive SU(2) interferometer (standard beam splitters) and the active SU(1,1) interferometer (optical parametric amplifiers). In both configurations, one input port is driven by a coherent state, while the "dark" port is injected with either:

1. A momentum-squeezed vacuum state (the conventional optimal Gaussian resource).
2. A finite-energy GKP state, which is a damped version of the ideal GKP state, characterized by a lattice of peaks modulated by a Gaussian envelope.

The study employs the Quantum Fisher Information (QFI) as the metric for phase estimation precision. The authors derive analytical expressions for the QFI of both input states propagating through lossy channels. They consider:

• One-parameter GKP states: Where peak width and envelope width are linked by a single damping parameter $\Delta$.
• Two-parameter GKP states: Where the peak width ($\sigma_{peak}$) and envelope width ($\sigma_{env}$) are varied independently.
• Loss models: Optical loss is modeled as a beam splitter with transmissivity $\eta$, mixing the input state with vacuum.

The analysis compares the QFI of GKP states against squeezed vacuum states under two distinct criteria: fixed squeezing parameters and fixed mean photon number (resource-matched comparison).

Key Results

1. Advantage of Broad Envelopes: For finite-energy GKP states with a sufficiently broad envelope (small damping parameter $\Delta$), the QFI exceeds that of a squeezed vacuum state. This advantage arises from the availability of multiple squeezed peaks in the GKP lattice, which enhances robustness.
2. Crossover Condition: There exists a specific crossover point at $\Delta \approx 0.3282$. Below this value (narrower peaks, broader envelope), the GKP state outperforms the squeezed vacuum. Above this value, the squeezed vacuum performs better. This crossover is determined solely by the condition where the position-quadrature variance of the unsqueezed GKP state equals the vacuum variance ($1/2$). Crucially, this crossover point is independent of both the squeezing strength ($\kappa$) and the optical transmissivity ($\eta$).
3. Effect of Optical Loss: Optical loss diminishes the relative advantage of either state. As transmissivity $\eta$ decreases, the QFI ratio between GKP and squeezed vacuum approaches unity. In the limit of complete loss, both states degrade to the vacuum state, erasing any quantum advantage. However, for non-zero transmissivity, the relative advantage of the GKP state (for $\Delta < 0.3282$) persists, though it is suppressed by the introduction of vacuum fluctuations.
4. Resource-Matched Comparison (Fixed Mean Photon Number): When comparing states with equal mean photon numbers, the GKP state does not outperform the squeezed vacuum state. The authors demonstrate that the QFI of a finite-energy GKP state is bounded by the QFI of a squeezed vacuum state with the same energy. This bound is a consequence of the Heisenberg uncertainty principle; the squeezed vacuum state, being a pure minimum-uncertainty state, saturates the upper bound for position-quadrature variance at a fixed energy.
5. Independence from Interferometer Type: The derived QFI expressions and the comparative results hold for both SU(2) and SU(1,1) interferometers, with the specific architecture only affecting a scaling coefficient in the phase-to-displacement conversion.

Significance and Claims
The paper claims to provide the first direct application and detailed analysis of finite-energy GKP states in optical interferometry. Its primary contribution is establishing a unified framework for comparing non-Gaussian GKP resources with conventional Gaussian squeezed states.

The authors modestly conclude that while GKP states offer a phase-sensing advantage over squeezed vacuum states when the envelope is sufficiently broad (specifically when $\Delta < 0.3282$), this advantage is contingent on the energy budget. When constrained to equal mean photon numbers, the squeezed vacuum remains the superior resource. Therefore, the potential utility of GKP states in interferometry lies not in surpassing the fundamental energy-limited bounds set by Gaussian states, but potentially in scenarios where the specific structural robustness of the GKP lattice against certain types of noise or in specific resource regimes (where the envelope width is the primary constraint) offers practical benefits. The work sets a baseline for future investigations into incorporating GKP error-correction properties directly into sensing protocols and assessing performance under more complex experimental imperfections.