Optimal initial states for quantum Fisher information in linearized cavity optomechanics

Technical Summary: Optimal Initial States for Quantum Fisher Information in Linearized Cavity Optomechanics

Problem Statement
The paper addresses the metrological question of how to optimally estimate the single-photon radiation-pressure coupling strength ($g_0$) in a linearized cavity-optomechanical system. While previous metrological studies in optomechanics have focused on estimating optical phase, displacement, or force, the estimation of $g_0$ itself—crucial for state exchange, transduction, and entanglement generation—has received limited attention, particularly regarding which initial quantum states of the optical and mechanical oscillators maximize the information extraction. The authors seek to identify the optimal initial states for pure inputs that maximize the Quantum Fisher Information (QFI) for $g_0$ under the linearized beam-splitter (red-detuned) interaction.

Methodology
The authors employ the framework of quantum metrology, specifically utilizing the Quantum Cramér–Rao bound. For a pure input state $|\psi_0\rangle$ undergoing unitary encoding $U(\theta) = \exp(-i\theta \hat{n})$, where $\theta = g_0 \alpha t$ is the rotation angle and $\hat{n} = \hat{d}\hat{b}^\dagger + \hat{d}^\dagger\hat{b}$ is the excitation-exchange operator, the QFI is given by $F_Q(\theta) = 4 \text{Var}_{\psi_0}(\hat{n})$.

The core of the methodology involves:

1. Deriving a Master Formula: The authors derive an exact expression for the variance of $\hat{n}$ for product input states $|\psi\rangle_d \otimes |\phi\rangle_b$. This formula relates the QFI to the first and second moments (mean occupations, coherent amplitudes, and pair coherences) of the initial optical and mechanical states.
2. Optimization Strategies: The optimization is tackled in three regimes:
   • Fixed Mechanical Reference: Optimizing the optical probe for a fixed mechanical state.
   • Fixed Optical Reference: Optimizing the mechanical probe for a fixed optical state.
   • Joint Optimization: Optimizing both modes simultaneously under a fixed total excitation number constraint.
3. Analytical Tools: The analysis utilizes the Schwinger representation of angular momentum to map the problem onto SU(2) manifolds. For general states, the authors employ a Lagrange-dual argument to prove an exact moment bound on the two-photon coherence $|\langle \hat{a}^2 \rangle| \leq \sqrt{N(N+1)}$ at fixed mean photon number $N$.
4. Extensions: The analysis is extended to the blue-sideband (two-mode squeezing) regime, open-system dynamics under cavity loss and mechanical decoherence, and a realistic parameter benchmark against state-of-the-art experiments.

Key Contributions and Results

• Optimal Probes for Fixed References:

  • Number-State Reference: If the mechanical mode is in a Fock state $|m\rangle$, the QFI depends only on the mean photon number of the optical probe. The optimum is degenerate: any state with the same energy is optimal. The per-photon sensitivity is enhanced by a factor of $(2m+1)$, meaning a Fock reference with many phonons linearly enhances sensitivity.
  • General Reference: For an arbitrary fixed mechanical reference, the optimal optical probe is a phase-matched squeezed vacuum. The squeezing axis must be aligned with the complex conjugate of the reference's pair coherence. The authors prove that squeezed vacua are the unique global optima, saturating a new moment bound on two-photon coherence.

• Joint Optimization at Fixed Total Energy:

  • When both modes are optimized with a fixed total excitation number $E$, the QFI is maximized by an entangled superposition of the two extremal eigenstates of the generator $\hat{n}$. This state is equivalent to a two-mode NOON state transformed by a balanced beam splitter.
  • This strategy saturates a Heisenberg bound scaling quadratically with energy ($F_Q \propto 4E^2$).
  • In contrast, balanced product Fock states (e.g., $|E/2, E/2\rangle$) only achieve a linear enhancement ($F_Q \propto 2E^2$), reaching exactly half the Heisenberg limit for large $E$.

• Blue-Sideband Regime:

  • For the counter-rotating (blue-sideband) interaction, the generator has an unbounded spectrum, making the NOON-type entangled optimum impossible.
  • The optimal state at fixed definite energy is the balanced product Fock state $|E/2, E/2\rangle$, yielding $F_Q = 2E^2 + 4E + 4$.
  • At fixed mean occupations, the phase-matched double-squeezed vacuum remains the optimal product input.

• Open-System Dynamics and Decoherence:

  • Under cavity loss, the ideal quadratic-in-time ($t^2$) accumulation of QFI transitions to a linear-in-time ($t$) accumulation.
  • The optimal sensing strategy involves repeating pulsed trials of a specific duration $\tau^* \approx 2.51 \kappa^{-1}$ (where $\kappa$ is the cavity decay rate).
  • Robustness: Strategies relying on number-state or thermal references are robust against decoherence. However, strategies relying on pair coherences (squeezed states) or entangled NOON states are highly fragile; mechanical decoherence suppresses the gains from squeezing, and photon loss destroys the coherence of entangled states (which-path information), reverting them to linear scaling.

• Experimental Benchmark:

  • Applying these results to the parameters of a state-of-the-art quantum-coherent coupling experiment (Verhagen et al.), the authors find that per-hour relative precisions for $g_0$ in the range of $10^{-6}$ to $10^{-5}$ are achievable.
  • At the few-quanta level accessible in current experiments, number-reference and thermal-reference strategies are the most robust. While entangled states offer theoretical advantages, they provide no practical advantage over simpler strategies under realistic decoherence conditions.

Significance
The paper provides complete, closed-form solutions to the optimization of initial states for estimating the single-photon coupling in linearized optomechanics. By reducing the problem to variance maximization of the excitation-exchange operator, the authors establish a hierarchy of optimal strategies:

1. Number-state references offer a simple, degenerate optimum with linear sensitivity enhancement.
2. Phase-matched squeezed vacua provide the unique global optimum for general references.
3. Entangled NOON-like states saturate the Heisenberg bound for joint optimization but are fragile to loss.

The work bridges the gap between theoretical quantum metrology and experimental optomechanics by explicitly quantifying the trade-offs between ideal quantum enhancements and the robustness required in the presence of cavity loss and mechanical decoherence. It identifies that while entangled states offer the highest theoretical precision, the most practical and robust strategies for current experiments involve number-state or thermal references, potentially augmented by squeezed probes if the reference is also phase-coherent.