Programmable optical parametric amplifier synthesizer for cubic phase states and amplified Schrodinger cat states

Technical Summary: Programmable Optical Parametric Amplifier Synthesizer for Cubic Phase and Amplified Schrödinger Cat States

Problem Statement
Continuous-variable (CV) quantum information processing relies heavily on the generation and manipulation of non-Gaussian states. Two critical resources are cubic phase states (CPS), which provide the necessary nonlinearity for universal Gaussian-only architectures, and large-amplitude squeezed Schrödinger cat states, which are essential for fault-tolerant bosonic error correction. Existing methods face significant limitations: generating high-fidelity CPS often requires complex non-Gaussian seeds (e.g., trisqueezed states) or suffers from extremely low success probabilities. Similarly, amplifying small-amplitude cat states to the large-amplitude regime ($\alpha \ge 2$) typically demands iterative schemes with high cumulative loss, extreme squeezing, or high-order photon detection. While Optical Parametric Amplifiers (OPAs) with heralded photon detection have been used for state generation, previous configurations were restricted to specific idler inputs (typically vacuum or fixed Fock states) and did not fully explore the potential of varying input and output photon numbers to achieve new functionalities like catalytic state preservation or parity-flipping amplification.

Methodology
The authors propose a programmable OPA synthesizer framework operating under a heralded photon-number-resolving detection scheme. The protocol involves:

1. Input Configuration: A signal mode (prepared in either a coherent state or a Schrödinger cat state) and an idler mode (prepared in a Fock state $|m\rangle$) are injected into an OPA.
2. Interaction: The OPA, characterized by a tunable gain parameter $g$, entangles the two modes via the two-mode squeezing operator $S(\tau)$.
3. Heralding: A photon-number-resolving detector measures the idler output, post-selecting events where exactly $n$ photons are detected. This conditional measurement projects the signal mode into a non-Gaussian state.
4. Parameter Space Exploration: The study systematically investigates the full parameter space of input idler photons ($m$) and detected idler photons ($n$). It distinguishes between catalytic configurations ($m = n$), where the idler state is effectively restored and input parity is preserved, and non-catalytic configurations ($m \neq n$), which allow for different parity outcomes and potentially higher success rates.
5. Optimization: Numerical simulations (using QuTiP) optimize the OPA gain $g$, input amplitudes, and Gaussian correction parameters (squeezing and displacement) to maximize fidelity against target states (ideal CPS or squeezed cat states).

Key Contributions
The paper establishes that the OPA heralding architecture functions as a unified, programmable synthesizer capable of two distinct, qualitatively different functionalities depending on the input state and $(m, n)$ configuration:

1. High-Fidelity Cubic Phase State Generation:

   • Using a coherent state as the signal input, the protocol generates cubic phase states with fidelities exceeding 0.99.
   • Crucially, this is achieved not only under the catalytic condition ($m=n$) but also across a broad range of non-catalytic configurations ($m \neq n$).
   • The framework converts Gaussian inputs (coherent states) into highly non-Gaussian outputs using only moderate-gain OPA operation and low-order photon-number-resolving detection, avoiding the need for pre-existing complex non-Gaussian seeds.

2. Amplification of Schrödinger Cat States:

   • Using an unsqueezed Schrödinger cat state ($\alpha_{in} \le 1$) as the signal input, the protocol amplifies the state to a large-amplitude squeezed cat state ($\alpha_{out} \ge 2$) while maintaining fidelity above 0.99.
   • Catalytic Configurations ($m=n$): These preserve the input parity (even remains even, odd remains odd) and restore the idler state.
   • Non-Catalytic Configurations ($m \neq n$): These enable "parity-flipping" amplification (e.g., converting an even cat to an odd cat) and often offer higher success probabilities.
   • The amplified output can serve as a seed for subsequent amplification rounds, offering a self-seeding pathway to progressively larger cat states without requiring increasingly extreme resources.

Results

• Cubic Phase States: The authors demonstrate that configurations such as $(0,4)$, $(1,1)$, $(2,2)$, and $(1,4)$ yield fidelities $F \ge 0.99$. The optimal OPA gain $g$ and input amplitude $\alpha$ are identified for each configuration. The generated states exhibit the characteristic Wigner function negativity and asymmetry of cubic phase states. Success probabilities range from $10^{-2}$ (for vacuum idler inputs) to $10^{-7}$ (for higher-order catalytic cases), with the authors noting that high repetition rates can mitigate low single-trial probabilities.
• Cat State Amplification:
  • For Odd Cat States (OCS), catalytic configurations $(1,1)$ through $(4,4)$ successfully amplify inputs with $\alpha_{in} \lesssim 1.5$ to outputs with $\alpha_{out} \ge 2.0$ with $F > 0.99$.
  • For Even Cat States (ECS), the $(1,1)$ configuration acts as a noiseless attenuator rather than an amplifier, while configurations like $(2,2)$ and $(4,4)$ achieve amplification.
  • Non-catalytic configurations, such as $(1,2)$ for ECS inputs, achieve high fidelity ($F \approx 0.996$) with significantly higher success probabilities ($\sim 1.7\%$) compared to catalytic cases, enabling the conversion of small even "kittens" to large-amplitude odd cat states.
• Robustness: Analysis of photon loss (modeled via a Lindblad master equation) shows that the generated states maintain high fidelity ($F > 0.9$) even under moderate loss rates ($\kappa t \le 0.02$ for cat states, up to $0.50$ for CPS). Notably, the structural complexity of the states persists even after Wigner negativity vanishes.

Significance and Claims
The authors claim that this work uncovers a unified and programmable OPA platform for non-Gaussian state engineering that was previously inaccessible. Key claims include:

• Novelty: Unlike previous schemes restricted to vacuum idler inputs or specific Fock states, this work explores the full $(m, n)$ space, revealing that high-fidelity CPS generation and cat state amplification are possible across diverse configurations, not just the catalytic limit.
• Experimental Accessibility: The protocol requires only moderate-gain OPA operation and low-order photon-number-resolving detection (typically 1–3 photons), making it experimentally feasible with current technology. It avoids the need for extreme squeezing or multi-photon subtraction schemes that suffer from cumulative loss and low success rates.
• Scalability: The "self-seeding" capability, where an amplified cat state serves as the input for the next round, offers a scalable strategy to generate increasingly large cat states without escalating resource demands (e.g., higher-order detection).
• Versatility: The ability to toggle between parity-preserving (catalytic) and parity-flipping (non-catalytic) modes provides a flexible tool for tailored quantum information tasks, such as preparing specific resources for GKP state generation or bosonic error correction.

The paper concludes that this framework provides a direct, experimentally accessible route to the non-Gaussian resources required for universal CV quantum computation and fault-tolerant error correction.