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

AI summary

Overview

Research area: Quantum metrology and continuous-variable quantum optics — specifically, whether non-Gaussian Gottesman-Kitaev-Preskill (GKP) states can outperform squeezed vacuum as the non-classical input to optical interferometers. Technical level: Advanced (assumes familiarity with quantum Fisher information, Gaussian/non-Gaussian continuous-variable states, and interferometer input-output formalism). Scope: A theoretical comparison of finite-energy GKP states and squeezed vacuum injected at the dark port of lossy SU(2) and SU(1,1) interferometers driven by coherent light, quantified through the quantum Fisher information (QFI).

What This Paper Is About

Quantum-enhanced interferometry usually improves phase sensitivity by injecting squeezed vacuum into the unused input port, but squeezing degrades under optical loss. The authors ask whether a finite-energy GKP state — a non-Gaussian, lattice-structured state known for robustness to Gaussian noise — can beat squeezed vacuum in the same interferometer setting, and under what conditions. The goal is to derive QFI expressions for both input states in lossy SU(2) and SU(1,1) interferometers and map out exactly when the GKP advantage appears or disappears.

Key Contributions

1. Derives analytical QFI expressions for momentum-squeezed finite-energy GKP states and squeezed vacuum at the dark port of both SU(2) and SU(1,1) interferometers, in lossless and lossy cases, and compiles them in two summary tables (one in terms of state parameters, one in terms of mean photon number).
2. Shows the GKP-versus-squeezed-vacuum crossover reduces to a single condition on the unsqueezed GKP position variance, Var(q) = 1/2, which is independent of both squeezing strength κ and transmissivity η (for η > 0), and fixes the crossover numerically at Δ ≃ 0.3282.
3. Provides a methodology for estimating the QFI of non-Gaussian states in interferometers using a common effective-displacement framework, F_Q = 8A² Var(q), where A = α/2 for SU(2) and A = Gα with G = sinh r cosh r for SU(1,1).
4. Generalizes the analysis to a two-parameter GKP state with independently variable peak width σ_peak and envelope width σ_env, showing the equal-QFI condition becomes a boundary in that two-dimensional parameter space rather than a single number.

Main Findings

• Broad-envelope GKP can beat squeezed vacuum even with loss: For Δ below the crossover (Δ < Δ_equalQFI ≃ 0.3282), the finite-energy GKP state has larger QFI than squeezed vacuum, and this ordering survives any nonzero transmissivity η.
• Crossover is universal: The two states have equal QFI exactly when the unsqueezed GKP position variance equals the vacuum variance, Var_GKP(q) = 1/2. This holds for all κ and all η > 0, so loss does not move the crossover in Δ.
• Loss shrinks, but does not erase, the advantage: The ratio F_GKP/F_SQ is written as 1 plus a term proportional to η e^{2κ}[2Var_GKP(q) − 1] divided by η e^{2κ} + 1 − η. Because loss pulls both output variances toward the vacuum value 1/2, the limit η → 0 gives F_GKP/F_SQ → 1.
• Squeezing does not change relative sensitivity at fixed Δ: In the lossless case (η = 1) the factor e^{2κ} cancels in the ratio, so curves for different κ overlap; squeezing changes the absolute QFI of both states but not their ratio.
• Higher squeezing helps under loss: For η < 1, increasing κ increases the separation between the GKP and squeezed-vacuum QFIs, as shown in the panels for κ = 0, 3, and 10 dB.
• Physical origin is multiple squeezed peaks: The advantage is attributed to the GKP state's multiple squeezed lattice peaks. At the crossover, the nearest-neighbour overlap factor is exp[−π/(2(0.3282)²)] ≃ 4.6 × 10⁻⁷, so peaks are effectively non-overlapping, while one standard deviation of the envelope spans only σ_env/d_lattice = 1/Δ_equalQFI ≃ 3.04 lattice spacings — a few sites, not the asymptotic many-peak regime.
• Envelope width dominates the two-parameter behavior: In the generalized state, increasing σ_env raises the QFI ratio, reflecting both a larger local contribution and the increased spatial extent of the populated lattice; σ_peak has two competing effects — direct broadening of peaks and increased overlap between neighbouring lattice components.
• Energy constraint reverses the conclusion: Because lowering the mean photon number reduces the GKP envelope width, the squeezed vacuum input performs better when compared with a GKP state of equal or lower mean photon number.
• The one-parameter GKP family is a curve, not a region: It corresponds to σ_peak σ_env = 1/√(2π) inside the two-dimensional (σ_peak, σ_env) space studied in the generalized analysis.

Methodology in Plain English

The authors model a realistic, finite-energy GKP state by damping the ideal, infinitely energetic GKP lattice with a Gaussian operator, which gives a comb of finite-width peaks under a broad Gaussian envelope (widths σ_peak = Δ and σ_env = 1/(√(2π)Δ) in the one-parameter case). They place this state, or a squeezed vacuum state, at the dark input port of an interferometer fed by a coherent carrier, and treat both interferometer types in one common small-phase, high-amplitude description: the unknown phase appears as a displacement of the momentum quadrature of the dark-port field, with a coefficient A that absorbs all the differences between SU(2) and SU(1,1) architectures. The quantum Fisher information then reduces to a quadrature variance, F_Q = 8A² Var(q), so comparing the two resource states becomes a matter of comparing their position variances. They compute the GKP variance analytically, including an infinite double sum over lattice indices that is evaluated numerically, then apply squeezing and a loss channel with transmissivity η. Finally they plot the QFI ratio in decibels, 10 log₁₀(F_GKP/F_SQ), against Δ, η, and the two independent widths, and translate the results into mean-photon-number form to compare at fixed energy.

Why This Matters

The work identifies a concrete, testable regime where a non-Gaussian state beats the best Gaussian resource for the same interferometer — and, just as importantly, the regime where it does not, which is when energy rather than variance is held fixed. It gives experimentalists a design rule (the GKP envelope must be broad enough, i.e., Δ < 0.3282) and theoretical metrologists a reusable QFI template for non-Gaussian probes.

Real-world applications:

• Gravitational-wave strain sensing, where squeezed light is already used and loss is a limiting factor.
• Displacement sensing, the setting where GKP states were previously shown to offer advantages.
• Acceleration and rotation sensing, both of which can be encoded in optical phase shifts.
• Fundamental tests and precision measurement technologies that rely on sub-shot-noise interferometric phase readout.

Industry relevance spans gravitational-wave observatories that deploy squeezed-light injection, photonic sensing and metrology companies, and quantum-hardware efforts where GKP states are pursued as bosonic error-correcting codes and could serve a dual metrological role.

Future Directions

• Resolving the energy-constrained comparison: establishing rigorously when a GKP state at equal or lower mean photon number than squeezed vacuum can still win, given that the envelope width shrinks with photon number.
• Going beyond the small-phase, effective-displacement description used to define the QFI, and checking whether the predicted advantage survives at larger phase shifts.
• Specifying and benchmarking an actual measurement strategy that approaches the QFI for these non-Gaussian states, rather than only the ultimate bound.
• Assessing whether GKP states with a broad enough envelope (Δ < 0.3282) can be generated and maintained with realistic squeezing, loss, and decoherence in an interferometer.

Target Audience

Quantum metrology and continuous-variable quantum optics researchers; graduate students and postdocs working on non-Gaussian states, QFI, or interferometer design; and experimentalists in precision sensing or gravitational-wave detection evaluating alternative non-classical input states. Readers should be comfortable with quadrature variance, squeezing parameters, and the quantum Cramér-Rao bound, since the paper is built around them.

Authors’ abstract

We present the case of a Gottesman-Kitaev-Preskill (GKP) state-enhanced optical interferometry with detailed analysis of the phase sensitivity for both the SU(2) and SU(1,1) interferometers. The conventional quantum-enhanced SU(2) interferometer, employing coherent light at one input port and squeezed light at the other, is compared with a modified configuration using coherent light and a GKP state. While it is known that the squeezed vacuum state is the optimal Gaussian resource input mode when paired with the coherent state, we show that the finite-energy GKP state with sufficiently broad envelope outperforms the squeezed vacuum injection, irrespective of the presence of optical losses. This can be attributed to the enhanced robustness coming from the availability of multiple squeezed peaks in the GKP case. However, because lowering the mean photon number reduces the GKP envelope width, the squeezed vacuum input performs better when compared with a GKP state of equal or lower mean photon number. We also observe that optical losses tend to diminish the relative advantage of either input state, since both states approach the (unsqueezed) vacuum state asymptotically. Our work demonstrates the direct application of finite-energy GKP states in optical interferometry along with a methodology for estimating the quantum Fisher information (QFI) and presents a phase estimation procedure using non-Gaussian resources in comparison with conventional Gaussian states.