Multiphoton heralding generates large-amplitude squeezed Schrödinger cat states and parity-selective Fock superpositions from squeezed vacuum via an OPA

This paper proposes a multiphoton heralding scheme using an optical parametric amplifier to convert squeezed vacuum into large-amplitude squeezed Schrödinger cat states and parity-selective Fock superpositions, which exhibit strong Wigner negativity, loss-resilient quantum complexity, and Heisenberg-limited phase estimation capabilities.

The Gist

The Big Idea: A "Quantum Magic Trick" with Light

Imagine you have a machine that generates a very special kind of light called "squeezed vacuum." Think of this light as a smooth, calm ocean wave. In the world of quantum physics, this smooth wave is useful, but it's a bit too "classical" (boring) to do the most advanced quantum computing tasks. To do those advanced tasks, you need "non-Gaussian" states—think of these as choppy, wild, or complex waves with strange shapes, like Schrödinger's famous "cat" (which is both alive and dead at the same time).

The problem is that making these wild, complex waves is usually like trying to catch a specific fish in a dark ocean using a tiny net. It's incredibly difficult, slow, and you often catch nothing.

The Solution:
The authors of this paper propose a new machine setup using an Optical Parametric Amplifier (OPA). Think of the OPA not just as a light amplifier, but as a quantum blender that can mix and reshape light in very precise ways.

Their new method is called "Multiphoton Heralding." Here is how it works:

1. The Setup: They shoot a "squeezed vacuum" (the smooth ocean) into one side of the blender.
2. The Trigger: On the other side, they inject a specific number of photons (particles of light) and then count exactly how many come out the other side.
3. The "Herald": If they count a specific number (like 2 or 4), a signal goes off (a "herald") saying, "Success! The light on the other side has been transformed into the wild, complex wave we wanted."

The Magic Rules: Parity and Selection

The paper discovered a surprising rule about how this blender works, which they call a parity selection rule.

Imagine you have a deck of cards with only red and black cards.

• If you add an odd number of cards to the deck and remove an odd number, the remaining deck has a specific "odd" flavor.
• If you add an even number and remove an even number, the flavor is "even."

In this experiment, the "flavor" is whether the resulting light wave is an Odd Cat State (like a wave with a dip in the middle) or an Even Cat State (like a wave with a bump in the middle).

The authors found that by carefully choosing how many photons they put in ($m$) and how many they count coming out ($n$), they can force the machine to produce specific types of these "Cat states."

• Example: If they put in 1 photon and count 2 coming out, they get a "large" Odd Cat state.
• Example: If they put in 4 photons and count 1 coming out, they get an even "larger" Odd Cat state.

This is a big deal because previous methods could only make small Cat states, or they required catching 4 or 5 photons to get a big one, which happened so rarely it was practically impossible. This new method gets the same big results with much higher success rates.

Why is this "Cat" important?

In quantum computing, these "Cat states" are like the building blocks for error correction.

• The Problem: Quantum computers are fragile. If a single photon is lost (like a drop of water evaporating from a wave), the information can get corrupted.
• The Fix: Large Cat states are robust. They are like a wave with two distinct peaks far apart. Even if the wave gets a little shaky or loses a little water, it's still clearly a "two-peak" wave, not a mess. This makes them perfect for fault-tolerant quantum computing (computers that don't break easily).

The paper also mentions these states can be used to create GKP qubits, which are a specific type of quantum code designed to fix errors automatically.

Measuring Success: Negativity vs. Complexity

The authors used two ways to measure how "quantum" and "complex" their light waves are:

1. Wigner Negativity: This is like checking for "magic." If the math shows negative values, it proves the light is truly quantum and not just a classical wave.
2. Phase-Space Complexity: This measures how intricate and detailed the shape of the wave is.

The Surprise:
Usually, if you lose photons (light leaks out), the "magic" (negativity) disappears first. However, the authors found that even when the "magic" is gone due to loss, the complexity of the wave remains high.

• Analogy: Imagine a complex origami crane. If you tear a small piece off, it might lose its "perfect" status (negativity), but it still looks like a complex, folded shape (complexity) rather than a flat piece of paper. This means the light retains useful structure even when it's not perfect, making it a resilient resource for quantum tasks.

Real-World Feasibility: Is it doable?

The paper does a reality check on whether this can actually be built in a lab.

• The Odds: The chance of getting a "win" in a single try is low for the most complex states (like putting in 4 photons and getting 1 out). It's roughly 1 in a million.
• The Fix: However, lasers can fire millions of times per second. If you run the machine at a high speed (like a machine gun firing light), you can still generate thousands of these special states every second.
• Conclusion: The authors conclude that with current technology (fast lasers and good detectors), this method is experimentally feasible. It offers a faster and more flexible way to make these difficult quantum states compared to older methods.

Summary

This paper proposes a new, efficient way to turn smooth, boring light into wild, complex "Schrödinger's cat" states using a special light blender (OPA). By counting photons in a specific way, they can create large, robust quantum states that are essential for building future quantum computers that don't break easily. Even when these states lose some energy, they keep their complex structure, making them a promising tool for the future of quantum technology.

Technical Summary

Technical Summary: Multiphoton Heralding of Large-Amplitude Squeezed Schrödinger Cat States and Parity-Selective Fock Superpositions via OPA

Problem Statement
Continuous-variable (CV) quantum information relies heavily on Gaussian states (e.g., squeezed vacua, coherent states), which are insufficient for universal quantum computation or fault-tolerant error correction without non-Gaussian resources. While Schrödinger cat (SC) states and Fock superpositions are essential for these applications, generating them with large amplitudes ($\alpha \ge 2$) is challenging. Conventional methods using beam splitters (BS) for photon subtraction require detecting a high number of photons ($k$) to achieve amplitudes $\alpha \sim \sqrt{k}$. However, the success probability of such schemes scales as $R^k$ (where $R$ is the beam splitter reflectivity), leading to extremely low generation rates for high-order subtractions. Existing alternatives like generalized photon subtraction or quantum optical catalysis often lack flexibility or are tailored to specific target states.

Methodology
The authors propose a multiphoton heralding scheme utilizing an Optical Parametric Amplifier (OPA) instead of a beam splitter. The protocol involves:

1. Input Configuration: Injecting a squeezed vacuum (SV) state into the signal port of the OPA and a specific Fock state $|m\rangle$ (containing $m$ photons) into the idler port.
2. Heralding: Performing photon-number-resolving detection (PNRD) on the idler output to detect exactly $n$ photons.
3. State Engineering: The conditional detection heralds a non-Gaussian state in the signal output. The OPA unitary $S(\tau)$ simultaneously enables photon addition and subtraction, with the gain $g$ acting as a tunable parameter.
4. Theoretical Framework: The unnormalized output state is derived analytically as a superposition of photon-added/attenuated squeezed states. The authors systematically vary $m$ and $n$ to identify configurations that approximate specific target states (squeezed SC states or Fock superpositions) with high fidelity.
5. Metrics: The generated states are characterized using Wigner negativity ($N$) to measure non-classicality and a newly introduced "phase-space complexity" metric ($C$) to quantify structural richness independent of negativity.
6. Applications: The protocol's utility is evaluated through phase estimation (calculating Quantum Fisher Information, QFI) and robustness analysis against photon loss and dephasing.

Key Contributions and Results

• Parity-Selective Generation Rule: The study reveals a fundamental parity selection rule unique to the OPA scheme. The parity of the heralded signal state is determined by $(-1)^{m+n}$.

  • If $m+n$ is even, the output contains only even Fock components.
  • If $m+n$ is odd, the output contains only odd Fock components.
  • Crucially, not all pairs $(m, n)$ satisfying $m+n=k$ yield a Schrödinger cat state. Specific configurations are required to approximate SC states, while others yield multi-Fock superpositions.

• High-Fidelity Large-Amplitude SC States:

  • Configuration (1, 2): By injecting 1 photon and detecting 2, the scheme effectively realizes 3-photon subtraction, generating a squeezed odd SC (SOSC) state with amplitude $\alpha \approx \sqrt{3} \approx 1.73$. The fidelity exceeds 0.99.
  • Configuration (4, 1): By injecting 4 photons and detecting 1, the scheme effectively realizes 5-photon subtraction, generating an SOSC state with amplitude $\alpha \approx \sqrt{5} \approx 2.24$. This achieves the critical $\alpha \ge 2$ threshold required for fault-tolerant bosonic error correction with a fidelity of 0.991.
  • Comparison: These amplitudes are comparable to those achieved by 4- or 5-photon subtraction in conventional BS schemes but with significantly higher success probabilities (e.g., $\sim 10^{-2}$ for the (1,2) case vs. $\sim 10^{-4}$ for BS subtraction).

• Diverse Non-Gaussian States:

  • Fock Superpositions: Configurations like (1, 3) and (1, 4) generate even- and odd-parity Fock superpositions (e.g., $|0\rangle + |2\rangle + |4\rangle$) rather than SC states, offering a new class of resources.
  • Quantum Catalysis ($m=n$): When $m=n$, the idler mode acts as a quantum catalyst. The (2, 2) configuration generates a state approximating a finite-energy Gottesman-Kitaev-Preskill (GKP) codeword with a fidelity of 0.842.

• Quantum Metrology Performance:

  • The heralded states are applied to Mach-Zehnder interferometry for phase estimation.
  • Configurations with high complexity and negativity, such as (1, 4), (4, 1), and (1, 3), exhibit Quantum Fisher Information (QFI) that surpasses the Standard Quantum Limit (SQL) and, in certain regimes, exceeds the Heisenberg Limit (HL) defined as $N^2$.
  • In the low-photon-number regime, catalyzed states ($m=n$) demonstrate super-Heisenberg scaling.

• Robustness to Loss:

  • Under photon loss, Wigner negativity ($N$) vanishes faster than phase-space complexity ($C$).
  • Even when $N$ drops to zero, $C$ remains significantly above its minimum value (1), indicating that the structural complexity of the state persists. This suggests these states retain utility as quantum resources even under realistic loss conditions where traditional non-classicality markers fail.

• Experimental Feasibility:

  • The single-trial success probability for the high-fidelity (1, 2) configuration is calculated to be $\approx 1.15\%$.
  • While the (4, 1) configuration has a lower single-trial probability ($\sim 10^{-7}$) due to the difficulty of generating 4-photon Fock states, the authors note that high-repetition-rate lasers (GHz to THz) can compensate, yielding heralding rates in the kilohertz range, making the scheme experimentally viable with current technology.

Significance
The paper establishes the OPA as a versatile, reconfigurable platform for non-Gaussian state engineering, surpassing the limitations of conventional beam-splitter-based subtraction. By uncovering a parity-dependent selection rule, the authors provide a systematic method to generate both large-amplitude squeezed SC states (essential for fault-tolerant error correction) and diverse Fock superpositions. The introduction of phase-space complexity as a resilience metric highlights the potential of these states for loss-tolerant quantum metrology and information processing. The protocol offers a practical pathway to generating high-fidelity, large-amplitude non-Gaussian states with significantly improved generation rates compared to existing methods.