⚛️ quantum physics

Robust phase sensitivity in Mach-Zehnder interferometer using photon added and subtracted squeezed coherent state

This paper demonstrates that using photon-added squeezed coherent states as inputs in a Mach-Zehnder interferometer significantly enhances phase estimation precision and offers robustness against low photon loss, particularly when analyzed via quantum Fisher information and compared against intensity-based detection schemes.

Original authors: Shivani Singh, Priya Malpani, Anirban Pathak

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shivani Singh, Priya Malpani, Anirban Pathak

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: Measuring the Invisible

Imagine you are trying to measure the exact length of a very thin thread, but you can't touch it. You have to use a special tool called a Mach-Zehnder Interferometer (MZI). Think of this tool as a fork in the road for light. You send a beam of light down two paths, twist one path slightly (this is the "phase" you want to measure), and then bring them back together.

When the two light beams meet, they create a pattern of bright and dark spots (like ripples in a pond). By looking at how bright or dark the spots are, you can figure out how much you twisted the path. The goal of this paper is to make that measurement as precise as possible.

The Problem: Classical Light vs. Quantum Light

Usually, we use standard light (like a laser pointer) for these measurements. However, standard light has a "fuzziness" limit, like trying to measure a room with a ruler that has blurry markings. This is called the Standard Quantum Limit.

To get better precision, scientists use "non-classical" light—light that behaves in weird, quantum ways. The authors of this paper are testing different types of these "super-light" states to see which one gives the sharpest measurement.

The Ingredients: Adding and Subtracting Photons

The paper focuses on a specific type of quantum light called Squeezed Coherent States.

The Analogy: Imagine a balloon filled with air (the light). "Squeezing" it means you push the air into a specific shape to make it more organized.
The Twist: The authors take this squeezed balloon and perform two tricks:
1. Photon Added (PASCS): They magically inject extra tiny particles of light (photons) into the balloon.
2. Photon Subtracted (PSSCS): They magically remove some tiny particles from the balloon.

They wanted to know: Is it better to add particles, remove them, or mix both to get the best measurement?

The Experiment: The Race for Precision

The researchers set up a virtual race. They sent different combinations of these "added" and "subtracted" light states into their interferometer and measured how well they could detect the twist (the phase).

They used a mathematical tool called Quantum Fisher Information (QFI) as their "scorecard."

High Score: The state is very sensitive and can detect tiny changes.
Low Score: The state is fuzzy and misses small changes.

The Results:

The Winner: The state where they added photons to the squeezed light (PASCS) in both paths of the interferometer was the clear winner. It provided the most precise measurement.
The Loser: The state where they subtracted photons (PSSCS) performed the worst.
The Mixed Team: When they mixed one "added" path with one "subtracted" path, the performance was somewhere in the middle, fluctuating between the best and worst results.

Key Takeaway: If you want the most precise measurement, you should pump extra photons into your squeezed light, not take them away.

The Measurement Trap: Looking at the Wrong Thing

The paper also looked at how you measure the light at the end. They compared two methods:

Counting the Difference (IDD): Measuring how many more photons are in path A than in path B.
Counting One Path (SID): Just measuring the brightness of path A.

The Surprise:
The authors found that simply counting the brightness (intensity) isn't always the best way to measure the twist.

The Analogy: Imagine trying to measure the speed of a car by looking at how much gas is in the tank. It's a related number, but not the speed itself. In quantum physics, "Intensity" (brightness) and "Phase" (the twist) are like a see-saw. If you measure the brightness very precisely, the uncertainty about the twist gets worse.
The Finding: While counting photons is easy to do, it isn't the optimal way to find the phase, especially when the light is highly "squeezed." The theoretical limit (the best possible precision) is often better than what you get just by counting the light intensity.

The Challenge: Losing Particles (Photon Loss)

In the real world, light doesn't travel perfectly; some of it gets lost along the way (like a leaky hose). The researchers tested what happens if some photons disappear in one of the paths.

The Good News:
They found that the "Photon Added" state (PASCS) is robust.

The Analogy: Imagine you are trying to hear a whisper in a noisy room. If you shout a little louder (add photons), you can still hear the whisper even if the room gets a bit noisier (some light is lost).
The Finding: As long as the light loss isn't total, the PASCS state holds up well. It maintains its high precision even when a few photons go missing.

Summary

This paper is a guide for building the most sensitive quantum sensors. The authors conclude that:

Add, don't subtract: To get the best precision, use light where you have added extra photons to a squeezed state.
Don't just count: Simply measuring how bright the light is might not give you the best answer; there are better ways to measure the "twist" in the light.
It's tough but resilient: This special type of light is strong enough to handle small amounts of signal loss without ruining the measurement.

The paper suggests that this specific setup (PASCS) is a strong candidate for high-precision sensing, provided you can find the right way to measure the output.

Technical Summary: Robust Phase Sensitivity in Mach-Zehnder Interferometer using Photon Added and Subtracted Squeezed Coherent States

Problem Statement
Precision measurement of unknown phases is a fundamental challenge in quantum metrology. While Mach-Zehnder interferometry (MZI) is a standard technique for phase estimation, its precision is fundamentally limited by the Standard Quantum Limit (SQL) when using classical resources. Although non-classical states (e.g., NOON, squeezed, cat states) can breach the SQL to approach the Heisenberg Limit (HL), many of these states are difficult to generate and are highly susceptible to environmental noise and photon losses. Furthermore, while Gaussian states are easy to engineer, non-Gaussian states are often required for optimal parameter estimation. This paper investigates the role of specific non-Gaussian states—Photon Added Squeezed Coherent States (PASCS) and Photon Subtracted Squeezed Coherent States (PSSCS)—in enhancing MZI phase sensitivity and evaluates their robustness against photon loss.

Methodology
The authors analyze the phase sensitivity of a standard MZI where the input modes are populated by various combinations of PASCS and PSSCS. The study employs two primary analytical frameworks:

Quantum Fisher Information (QFI): Used to determine the theoretical lower bound on phase sensitivity (Quantum Cramér-Rao Bound, QCRB), independent of the specific measurement scheme. This provides the maximum achievable precision for a given input state.
Specific Detection Schemes: The practical phase sensitivity is calculated for two intensity-based detection schemes:
- Intensity Difference Detection (IDD): Measuring the difference in photon numbers at the two output ports.
- Single Intensity Detection (SID): Measuring the photon number at a single output port.

The analysis involves deriving analytical expressions for higher-order moments of the input states to calculate the expectation values of observables. The study also models photon loss in one arm of the interferometer using a fictitious beam splitter with transmittance $t$ and Kraus operator formalism to evaluate the degradation of QFI under lossy conditions.

Key Contributions and Results

Optimization of Input States: The systematic analysis reveals that the choice of input state significantly impacts phase sensitivity.
- PASCS Superiority: When PASCS is used in both input modes, the system achieves the maximum Quantum Fisher Information ( $I_{dd}$ ), yielding the lowest bound on phase sensitivity.
- PSSCS Performance: Conversely, using PSSCS in both modes results in the minimum QFI and the lowest phase sensitivity.
- Hybrid States: For combined inputs (one mode PASCS, one mode PSSCS), the sensitivity oscillates between the maximum and minimum values observed in the pure states. The QFI increases as the number of added photons exceeds the number of subtracted photons.
Parameter Dependence:
- Displacement ( $\alpha, \beta$ ): Phase sensitivity improves (decreases) as the displacement parameters increase.
- Squeezing Phase ( $\phi$ ): Sensitivity is optimized when the squeezing phase is an integral multiple of $\pi$ .
- Squeezing Amplitude ( $r$ ): While increasing the squeezing amplitude generally improves QFI-based sensitivity, the behavior differs under practical intensity measurements.
Detection Scheme Limitations:
- The study highlights a critical discrepancy between the theoretical QCRB and practical intensity measurements. Due to the non-commuting nature of phase and intensity (Heisenberg uncertainty), intensity measurements (IDD and SID) are not always optimal.
- Specifically, in IDD and SID schemes, phase sensitivity ( $\Delta\theta$ ) increases (worsens) as the squeezing amplitude $r$ increases. This is because the intensity in the MZI grows exponentially with $r$ ( $I \approx \sinh^2 r$ ), and precise intensity measurement leads to higher phase uncertainty.
- IDD reaches the QCRB only when the relative phase $\theta$ is an integral multiple of $\pi$ , whereas SID performance varies based on which port is detected and the specific phase range.
Robustness Against Photon Loss:
- The paper demonstrates that PASCS is robust against photon loss, particularly when the loss is low (high transmittance $t \to 1$ ).
- The presence of added photons increases the average photon number in the system, which helps maintain the lower bound on phase sensitivity even in the presence of loss.

Significance and Claims
The paper establishes that PASCS is a superior input state for MZI-based phase estimation compared to PSSCS or hybrid combinations, offering the highest theoretical precision. The authors emphasize that while non-Gaussian states like PASCS can significantly enhance sensitivity, the choice of detection scheme is equally critical; intensity-based measurements may not be optimal for states with large squeezing parameters due to the phase-intensity uncertainty relation.

The work concludes that PASCS offers a robust solution for phase estimation in low-loss environments, making it a candidate for practical applications where photon loss is a concern. The authors suggest that while their theoretical model shows promise, finding an optimal detection technique for PASCS inputs remains an important open question, and further exploration of these states in adaptive or feedback-based schemes (e.g., for quantum radar applications involving attenuation constants) is warranted.