⚛️ quantum physics

Efficient optical cat state generation using squeezed few-photon superposition states

This paper proposes a highly efficient and loss-tolerant protocol using linear optics and squeezed few-photon superposition states to generate high-fidelity optical Schrödinger cat states with a success probability exceeding 50%, offering a practical solution for current quantum optics experiments.

Original authors: Haoyuan Luo, Sahand Mahmoodian

Published 2026-02-17

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Haoyuan Luo, Sahand Mahmoodian

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a very delicate, high-tech house of cards. In the world of quantum computing, these "cards" are special states of light called Schrödinger cat states.

Why "cat"? Because, just like the famous thought experiment where a cat is both alive and dead at the same time, these light states are a superposition of two opposite things (like a wave peaking and a wave troughing) existing simultaneously. These states are the "bricks" needed to build future quantum computers that can fix their own errors.

The Problem:
Building these cat states out of light is incredibly hard. It's like trying to catch a specific, rare butterfly in a hurricane using a tiny net. Usually, you have to squeeze light (compress it) and then try to subtract a photon (a particle of light) from it. But this process is:

Inefficient: You often fail to catch the butterfly (low success rate).
Fragile: If you lose even a tiny bit of light (due to imperfect detectors), the whole thing falls apart (low fidelity).
Complex: It often requires complicated, multi-step setups.

The Solution:
The authors of this paper, Haoyuan Luo and Sahand Mahmoodian, have come up with a clever new recipe. Instead of starting with a blank slate (vacuum) or a single photon, they suggest starting with a "few-photon superposition."

Think of this as starting with a "pre-mixed" ingredient rather than raw flour. Specifically, they use a state that is a mix of zero photons (nothing) and two photons (a pair). They call this the $\theta$ state.

Here is how their new method works, broken down into simple analogies:

1. The "Pre-Mixed" Ingredient ( $\theta$ State)

Imagine you want to bake a perfect cake (the Cat State).

Old Way: You start with an empty bowl and try to add exactly the right amount of flour and sugar by guessing. It's messy and often fails.
New Way: You start with a bowl that already has a perfect, pre-measured mix of "nothing" and "two scoops of sugar." This is the $\theta$ state. Because it's already a mix, it's much easier to transform into the final cake.

2. The Two Cooking Methods

The paper proposes two ways to turn this pre-mixed ingredient into the final Cat State:

Method A: The Linear Optics "High-Five" (The Easy Way)

The Setup: You take two of these pre-mixed ingredients and smash them together using a beam splitter (a mirror that splits light).
The Trick: You look at one of the output paths. If you see exactly one photon appear there (like a signal flare), you know the other path now contains your perfect Cat State.
The Win: This method is surprisingly robust. Even if your detectors are a bit "leaky" (losing 2% of the light), you still get a high-quality cake 99% of the time. It works about 50% of the time, which is huge in this field.

Method B: The "Atomic Bouncer" (The Deterministic Way)

The Setup: Instead of smashing two ingredients together, you send just one pre-mixed ingredient past a single atom (a two-level system) acting like a bouncer at a club.
The Trick: The atom interacts with the light. If the atom successfully "kicks out" a single photon into a side channel, the main beam is transformed into a Cat State.
The Win: This is even more efficient. It works with a single ingredient and has a success rate of over 80%. It's like having a magic door that almost always lets the right people through.

3. Why This is a Big Deal

Resilience: Previous methods were like glass houses; a little bit of dust (loss) broke them. This new method is like a tent; it can handle a little wind (2% loss) and still stand strong.
Speed: Because the starting ingredients ( $\theta$ states) can be generated quickly by quantum emitters (like tiny light bulbs made of atoms), the whole process can be much faster.
Scalability: This is the missing link. If we can make these Cat States reliably, we can use them to build the massive, error-correcting quantum computers of the future.

The Bottom Line

The authors found a way to stop trying to build a complex quantum state from scratch every time. Instead, they found a "starter pack" (the squeezed $\theta$ state) that, when combined with simple tools (mirrors or a single atom), reliably produces the high-quality "bricks" needed for the next generation of quantum technology.

It's the difference between trying to sculpt a statue out of a raw rock versus having a pre-carved block that just needs a little polishing. They've found the perfect pre-carved block.

1. Problem Statement

The generation of high-fidelity, large-amplitude optical Schrödinger cat states (superpositions of coherent states $|\alpha\rangle \pm |-\alpha\rangle$ ) remains a significant challenge in quantum optics. These states are critical resources for photonic quantum computing, particularly for generating error-correcting codes like Gottesman-Kitaev-Preskill (GKP) states.

Current Limitations: Existing methods typically rely on photon subtraction from squeezed vacuum states using linear optics. These protocols suffer from low success probabilities (highly non-deterministic) and are extremely sensitive to photon loss and detector inefficiencies.
The Trade-off: Achieving high fidelity usually requires sacrificing success probability, and vice versa. Furthermore, generating large cat states often requires complex multi-stage protocols that compound errors.

2. Methodology

The authors propose a novel resource state: squeezed superpositions of zero and two photons, denoted as $|r, \theta\rangle = \hat{S}(r) [\cos(\theta/2)|0\rangle + \sin(\theta/2)|2\rangle]$ . These states can be deterministically generated by quantum emitters (e.g., quantum dots or atoms in cavities).

The paper presents two distinct protocols to convert these resource states into high-fidelity cat states via photon subtraction:

A. Linear Photon Subtraction Protocol

Setup: Two squeezed $\theta$ states are interfered on a beam splitter.
Mechanism: A single photon is detected in one output port, heralding the generation of a cat state in the other output port.
Optimization: The parameters (squeezing strengths $r_1, r_2$ , input angles $\theta_1, \theta_2$ , and beam splitter transmissivity $\gamma$ ) are numerically optimized to maximize both success probability and fidelity against target cat states of size $|\alpha|^2 \in [2, 6]$ .
Advantage over Fock States: Unlike using fixed two-photon Fock states ( $|2\rangle$ ), the $\theta$ states offer tunable parameters that allow for a simultaneous optimization of success probability and fidelity, overcoming the limitations of fixed-input protocols.

B. Nonlinear Photon Subtraction Protocol

Setup: A single squeezed $\theta$ state is scattered off a Two-Level System (TLS) (e.g., an atom) chirally coupled to a waveguide.
Mechanism: The TLS scatters the input wave packet into a product state where the dominant mode contains the cat state, and an orthogonal temporal mode contains a single photon. Detecting this single photon heralds the cat state.
Modeling: The dynamics are modeled using Matrix Product States (MPS) to handle the multi-mode nature of the photon field and the non-perturbative interaction between the TLS and the field.
Optimization: The input angle $\theta$ is specifically chosen as $\theta_{\text{max}}(r)$ to maximize the mean photon number of the resource state under squeezing, thereby maximizing the size of the resulting cat state.

C. Generation of $\theta$ States

The paper also details methods to generate the initial $|0\rangle + |2\rangle$ superposition states:

Probabilistic: Using linear optics with single-photon inputs and post-selection.
Deterministic: Using two three-level atoms coupled to a cavity. By preparing atoms in a superposition of ground and shelving states and performing controlled operations, a superradiant two-photon state is emitted. The authors propose using a Quantum Pulse Gate (QPG) to purify this state into a single-mode Fock state if necessary.

3. Key Results

Performance Metrics

Success Probability: Both protocols achieve success probabilities $P_{\text{suc}} \gtrsim 50\%$ (Linear) and $P_{\text{suc}} > 0.8$ (Nonlinear for specific parameters), significantly higher than previous linear subtraction schemes for large cat sizes.
Fidelity:
- Linear Protocol: Achieves fidelity $F > 0.99$ for cat states with size $|\alpha|^2 = 5$ .
- Nonlinear Protocol: Achieves $F \geq 0.99$ for $|\alpha|^2 \leq 4.3$ and $F \geq 0.664$ success probability. For larger sizes, fidelity is limited by the stellar rank of the input state but remains high.
Loss Tolerance: The linear protocol is remarkably robust. Even with 2% detection loss (detector efficiency $\epsilon = 0.98$ ), the protocol maintains $F > 0.98$ for $|\alpha|^2 = 5$ . This is a significant improvement over existing protocols which degrade rapidly with loss.

Stellar Rank Analysis

The authors analyze the "stellar rank" (a measure of non-Gaussianity) of the output states.

The squeezed $\theta$ states have a stellar rank of 5.
The linear subtraction protocol preserves a high stellar rank (5) in the output, allowing for better approximation of large cat states compared to protocols using fixed two-photon inputs, which offer less flexibility in tuning the output wavefunction.

Comparison with Alternatives

Vs. Squeezed Vacuum: The $\theta$ states provide a better starting point than squeezed vacuum because they already possess the necessary non-Gaussianity (finite stellar rank) without requiring strong nonlinearities to create the initial superposition.
Vs. Fixed Fock States: Using $\theta$ states instead of fixed $|2\rangle$ states allows for higher success probabilities at the same fidelity levels.

4. Significance and Impact

Scalability: The proposed protocols bridge the gap between deterministic quantum emitter sources and the generation of large, high-fidelity cat states required for fault-tolerant quantum computing.
Experimental Feasibility: The schemes rely on current state-of-the-art technologies: high-quality single-photon sources, linear optics, and chiral waveguide QED systems. The high tolerance to loss makes these protocols viable for near-term experimental realization.
Resource for GKP States: By enabling the efficient generation of large cat states, this work provides a critical building block for "breeding" GKP states, which are essential for error-corrected quantum computation with propagating light.
Deterministic Potential: The nonlinear protocol approaches near-deterministic success, a major step forward from the probabilistic nature of traditional photon subtraction.

In conclusion, the paper demonstrates that by utilizing squeezed superpositions of vacuum and two-photon states, one can overcome the traditional trade-offs between success probability, fidelity, and loss tolerance in optical cat state generation, paving the way for practical photonic quantum computing.

1. The "Pre-Mixed" Ingredient (θ\thetaθ State)