⚛️ quantum physics

Efficient and flexible preparation of photonic NOON states in a superconducting system

This paper proposes an efficient and flexible protocol for generating photonic NOON states in a superconducting system using an auxiliary five-level qudit and adjustable classical fields, achieving high fidelity without requiring nonlinear interactions or complex parameter tuning.

Original authors: Dong-Sheng Li, Yi-Hao Kang, Zhi-Cheng Shi, Yang Xiao, Ye-Hong Chen, Yan Xia

Published 2026-03-19

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CC BY 4.0

Original authors: Dong-Sheng Li, Yi-Hao Kang, Zhi-Cheng Shi, Yang Xiao, Ye-Hong Chen, Yan Xia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

The Big Picture: What is a "NOON State"?

Imagine you have two empty rooms (let's call them Room A and Room B). In the quantum world, these rooms are like microwave cavities where you can store light particles called photons.

A NOON state is a very special, "spooky" arrangement of these photons. It's like saying: "There are exactly 5 photons in Room A and 0 in Room B, OR there are 0 in Room A and 5 in Room B, but we don't know which one it is until we look."

This "superposition" (being in both states at once) is incredibly powerful. It's like having a super-precise ruler that can measure tiny changes in the world much better than any normal ruler. Scientists want to create these states to build better quantum computers and super-sensitive sensors.

The Problem: Why is it Hard?

Making these states is usually like trying to climb a ladder one rung at a time.

The Old Way: To get 5 photons, you have to add one, then another, then another. If you want 100 photons, you have to climb 100 steps. This takes a long time, and every step is a chance to make a mistake (like dropping a photon).
The Requirement: Many old methods also required "magic" ingredients called nonlinear interactions (like forcing photons to bump into each other and change their nature). In the real world, photons usually ignore each other, so creating these "magic" interactions is very difficult and expensive.

The Solution: The "Elevator" Protocol

The authors of this paper propose a new, clever way to do this in a superconducting system (a type of super-fast computer chip that works at extremely cold temperatures).

Instead of climbing a ladder, they built an elevator.

Here is how their 3-step "Elevator" works, using an analogy of Airplanes and Cargo:

The Setup

The Cargo: The photons in the two rooms (Cavities).
The Airplane: A special 5-level "qudit" (a quantum object that acts like a pilot).
The Destination: The target NOON state.

Step 1: Loading the Plane

What happens: The pilot (qudit) starts on the ground floor (level 0). The researchers use a laser pulse to instantly lift the pilot up to a "superposition" of two high floors (levels 3 and 4).
The Analogy: Imagine the pilot is now hovering in the sky, simultaneously in two different seats. The cargo (photons) is still sitting on the ground in the two rooms, but it's now "connected" to the pilot.
The Trick: They use a special, optimized pulse (like a perfectly smooth takeoff) to ensure the pilot doesn't wobble or crash due to small errors.

Step 2: The Flight (Displacement)

What happens: Now, the researchers turn on different controls. Because the pilot is in those high seats, the "wind" (classical fields) pushes the cargo.
The Analogy: The pilot's position acts like a switch. If the pilot is in Seat 3, the wind blows all the cargo into Room A. If the pilot is in Seat 4, the wind blows all the cargo into Room B.
The Magic: Instead of adding one photon at a time, they use a "turbo boost" to instantly fill the rooms with exactly N photons (where N can be any number they want). It's like teleporting a whole truckload of cargo into the air instantly.

Step 3: Landing and Unloading

What happens: The pilot needs to come back down to the ground floor to finish the job. The researchers use a "frequency-matched" pulse (a very specific radio signal) to guide the pilot back down.
The Analogy: As the pilot descends, the "parachute" opens. The connection between the pilot and the cargo is cut. The pilot lands safely on the ground (level 0), and the cargo is left floating in the two rooms in that perfect, spooky "NOON" state.
The Result: The pilot is clean and ready for the next trip, and the two rooms now hold the perfect NOON state.

Why is this Paper a Big Deal?

It's Fast and Fixed: Whether you want 2 photons or 100 photons, the process always takes 3 steps. You don't need to climb more rungs of the ladder just because you want more photons. You just adjust the "turbo boost" in Step 2.
No Magic Required: They don't need those difficult "nonlinear interactions." They only use simple, standard connections (linear couplings) that are easy to build in current superconducting chips.
It's Tough: The researchers tested their method against "noise" (like temperature fluctuations or imperfect lasers). Even with these real-world problems, the system still worked with very high accuracy (over 90-99% success rate).
Flexible: Because it doesn't rely on weird, hard-to-find physics, this method could be used in many different types of quantum systems, not just this specific one.

Summary

Think of this paper as inventing a magic elevator for quantum particles. Instead of slowly carrying heavy boxes up a staircase one by one (the old way), this new method loads the pilot, hits a button to instantly fill the rooms with the right amount of cargo, and lands perfectly. It's faster, safer, and works with the equipment we already have in our labs today.

1. Problem Statement

NOON states ( $|N,0\rangle + |0,N\rangle$ ) are critical resources for quantum metrology (achieving Heisenberg-limited phase sensitivity), quantum lithography, and quantum communication. However, their practical generation faces significant challenges:

Probabilistic Methods: Linear optical approaches often rely on post-selection, resulting in low success probabilities and degraded fidelity due to detector inefficiencies.
Scalability Issues in Deterministic Methods: Unitary-operation-based protocols typically require the number of steps to scale linearly with the photon number $N$ (e.g., $2N$ steps). This leads to complex operational procedures and cumulative error accumulation for large $N$ .
Nonlinearity Constraints: Protocols using fixed-step operations often rely on strong nonlinear interactions (e.g., Kerr nonlinearity), which are difficult to implement in many physical systems and limit flexibility.
Parameter Sensitivity: Existing schemes often require re-designing control fields for every specific photon number $N$ , making them hard to extend.

2. Methodology

The authors propose a three-step deterministic protocol to generate NOON states with an arbitrary photon number $N$ in a superconducting system.

System Architecture:

Components: Two microwave cavities ( $C_1, C_2$ ) coupled to a single auxiliary five-level qudit ( $Q_0$ ).
Levels: The qudit has levels $|0\rangle_0$ (ground), $|1\rangle_0, |2\rangle_0, |3\rangle_0, |4\rangle_0$ .
Coupling: The qudit couples to the cavities via capacitive coupling ( $\lambda_1, \lambda_2$ ).
Driving Fields: Classical fields drive specific transitions:
- Resonant fields drive $|0\rangle_0 \leftrightarrow |3\rangle_0$ and $|0\rangle_0 \leftrightarrow |4\rangle_0$ .
- Off-resonant (dispersive) fields drive $|1\rangle_0 \leftrightarrow |4\rangle_0$ and $|2\rangle_0 \leftrightarrow |3\rangle_0$ .

The Three-Step Protocol:
The protocol relies on large detuning conditions ( $\Delta_k \gg \Omega_k, \lambda_k$ ) to derive effective Hamiltonians, avoiding the need for strong nonlinearities.

Step 1: Superposition Preparation (Loading)
- Action: Turn off off-resonant fields; apply resonant fields to drive the qudit from $|0\rangle_0$ to a superposition of $|3\rangle_0$ and $|4\rangle_0$ .
- Result: The system evolves from $|0,0,0\rangle$ to $\frac{1}{\sqrt{2}}(|3,0,0\rangle + |4,0,0\rangle)$ .
- Optimization: The control field is designed using reverse engineering and invariant-based optimal control to nullify sensitivity to systematic errors (robust $\pi$ -pulse).
Step 2: Conditional Displacement (Transport)
- Action: Turn off resonant fields; turn on off-resonant fields.
- Mechanism: Due to the large detuning, the effective Hamiltonian creates state-dependent displacements.
  - If the qudit is in $|3\rangle_0$ , cavity $C_2$ is displaced.
  - If the qudit is in $|4\rangle_0$ , cavity $C_1$ is displaced.
- Result: The cavities evolve into coherent states $|\alpha_0\rangle$ where $\alpha_0 = \sqrt{N}$ . The state becomes $\frac{1}{\sqrt{2}}(|3, 0, |\alpha_0\rangle\rangle + |4, |\alpha_0\rangle, 0\rangle)$ .
- Flexibility: Only the displacement amplitude $\alpha_0$ needs adjustment for different $N$ ; the pulse shape remains unchanged.
Step 3: Disentanglement and Finalization (Unloading)
- Action: Apply frequency-matched pulses (both resonant and off-resonant fields).
- Mechanism: The pulses drive the qudit back to the ground state $|0\rangle_0$ while transferring the photon number from the coherent states to Fock states $|N\rangle$ .
- Result: The qudit disentangles, leaving the cavities in the target NOON state: $\frac{1}{\sqrt{2}}(|0, N, 0\rangle + |0, 0, N\rangle)$ .

3. Key Contributions

Fixed-Step Efficiency: Unlike previous unitary protocols where steps scale with $N$ , this protocol requires exactly three steps regardless of the photon number $N$ .
Linear Interactions Only: The protocol utilizes only linear qudit-cavity couplings and classical fields. It does not require difficult-to-implement nonlinear interactions (like Kerr nonlinearity).
Scalability and Flexibility: To generate NOON states for different $N$ , only the displacement parameter $\alpha_0 = \sqrt{N}$ and the modulation frequency in Step 3 need to be adjusted. The waveforms of the control fields remain identical.
Robust Control Design: The use of reverse engineering and optimal control techniques in Steps 1 and 3 significantly enhances robustness against systematic errors in Rabi frequencies and coupling strengths.

4. Results

Numerical simulations were performed using parameters accessible in current superconducting systems (flux qudits and 3D microwave cavities) to validate the protocol.

Fidelity:
- For photon numbers $N=2, 3, 4$ , the protocol achieves fidelities $F > 0.99$ (specifically $>0.999$ for Step 1 and $>0.99$ for the final state) under ideal conditions.
- Under systematic errors (Rabi frequency deviations up to 20%, coupling strength drifts), fidelity remains above 96–99%.
- Under inter-cavity crosstalk (up to 1% of coupling strength), fidelity remains $>99.35\%$ .
- Under decoherence (relaxation and dephasing rates up to 25 kHz, photon loss up to 2 kHz), fidelity remains $>90\%$ (specifically $>97\%$ for dephasing/relaxation and $>96\%$ for photon loss).
Combined Disturbances: Even when all disturbing factors (errors, crosstalk, decoherence) are present simultaneously, the protocol maintains a fidelity $>90\%$ , demonstrating high experimental feasibility.

5. Significance

Practical Implementation: The protocol is highly compatible with current superconducting circuit technology, as it avoids the need for strong nonlinearities and complex multi-step sequences.
Versatility: By relying on linear interactions, the method is not limited to superconducting systems and could potentially be adapted to other physical platforms (e.g., trapped ions or atomic ensembles) where similar level structures exist.
Quantum Metrology Advancement: The ability to efficiently generate high-fidelity NOON states with arbitrary $N$ directly facilitates the realization of Heisenberg-limited phase measurements, a key goal in quantum sensing.
Error Resilience: The integration of optimal control theory makes the protocol robust against the inevitable imperfections of experimental hardware, addressing a major bottleneck in scaling quantum information processing tasks.

In conclusion, this work presents a flexible, efficient, and robust solution for NOON state generation, overcoming the scalability and complexity limitations of previous methods while maintaining high fidelity in realistic noisy environments.