🔬 optics

Programming Quantum Measurements of Time inside a Complex Medium

This paper demonstrates a scalable method for programming generalized measurements of high-dimensional photonic time-bin superpositions by exploiting the coupling of spatial and temporal degrees of freedom within a single multi-mode fiber, thereby overcoming the complexity and stability limitations of conventional unbalanced interferometers.

Original authors: Dylan Danese, Vatshal Srivastav, Will McCutcheon, Saroch Leedumrongwatthanakun, Mehul Malik

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dylan Danese, Vatshal Srivastav, Will McCutcheon, Saroch Leedumrongwatthanakun, Mehul Malik

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 Idea: Measuring Time with a "Magic" Fiber

Imagine you are trying to listen to a symphony where the musicians don't play notes, but instead, they send you messages at specific times. Some messages arrive at 1:00 PM, some at 1:01 PM, some at 1:02 PM. In the quantum world, these "messages" are single particles of light (photons), and the "times" are called time-bins.

Scientists want to use these time-bins to build super-fast quantum computers and unbreakable encryption. But there's a huge problem: How do you measure a complex mix of these time-messages?

If you try to listen to a simple mix of two times, it's easy. But if you have a mix of 11 different times, the traditional tools required to measure them are like trying to build a skyscraper out of thousands of tiny, unstable Lego bricks. They are huge, fragile, and need constant adjustment to stay in place.

The Solution: The researchers from Heriot-Watt University found a way to turn a single, ordinary multi-mode fiber optic cable (the kind used for internet) into a super-smart, programmable time-measuring machine.

The Analogy: The "Time-Traveling Hallway"

To understand how they did it, let's use an analogy.

1. The Old Way: The Unstable Bridge

Imagine you want to measure the arrival time of runners. The old method (called a Franson interferometer) is like building a massive, complex bridge with many separate lanes.

To measure 11 different times, you need 11 separate lanes.
Each lane has to be exactly the right length.
If the wind blows or the temperature changes, the bridge sways, and your measurements are ruined.
It's expensive, bulky, and hard to build.

2. The New Way: The "Magic" Fiber

The researchers realized that a multi-mode fiber is like a giant, chaotic hallway with thousands of different paths.

Normally, if you throw a ball into this hallway, it bounces around wildly and comes out at a random time. It's messy.
The Breakthrough: The team figured out how to "program" the hallway. They found specific "secret paths" (which they call $\tau$ -modes) inside the fiber.
If you send a photon down one of these secret paths, it travels through the chaos and arrives at a precise, predictable time, as if the hallway knew exactly when to let it out.

3. Programming the Measurement

Here is the magic trick:

The researchers use a digital mirror device (like a high-tech projector) to shape the light before it enters the fiber.
They can tell the light, "Go down Path A, Path B, and Path C all at the same time, but with different weights."
Because each path has a different "travel time" inside the fiber, the light exits the fiber as a perfectly timed interference pattern.
It's as if they programmed the fiber to act like a giant, stable interferometer that can measure any combination of time-bins they want, simply by changing the shape of the light entering it.

Why This is a Big Deal

1. It's a "Common-Path" Interferometer
In the old method, the light travels down two different bridges. If one bridge moves, the measurement fails. In this new method, the light travels down one single fiber. It's like running a race on a single track; no matter how the wind blows, the track stays the same. This makes the system incredibly stable and easy to use.

2. It Scales Easily
Want to measure 100 time-bins instead of 11? With the old method, you'd need a bridge the size of a city. With this method, you just need a slightly thicker or longer fiber. The fiber itself does the heavy lifting.

3. It's "Programmable"
Think of the fiber as a musical instrument. In the past, to play a different note (measure a different time), you had to build a new instrument. Now, the fiber is like a digital synthesizer. You just press a button (change the hologram on the mirror), and it instantly becomes a tool to measure any time-pattern you need.

The Results

The team tested this by measuring time-bins in dimensions of 2, 4, and even 11.

They proved they could measure complex quantum states with very high accuracy (over 96% fidelity for 4 dimensions).
They showed that even though the fiber is "messy" inside, they can tame it to perform precise quantum measurements.

The Bottom Line

This paper is about taking a chaotic, messy piece of glass (a fiber optic cable) and teaching it to act like a precise, programmable quantum clock. Instead of building fragile, giant machines to measure time, they turned a single fiber into a versatile tool that could revolutionize how we build quantum computers and secure communication networks.

In short: They turned a "messy hallway" into a "precision time-machine" just by learning how to walk through it correctly.

1. Problem Statement

The temporal degree-of-freedom of light is a powerful resource for quantum technologies, enabling high-dimensional quantum states (qudits) for scalable quantum computing and high-rate quantum key distribution (QKD). However, performing generalized measurements on large, complex superpositions of photon time-of-arrival (time-bins) remains a significant experimental bottleneck.

Limitations of Current Methods: The standard approach utilizes unbalanced Franson-type interferometers. Extending these to high dimensions ( $d > 2$ ) requires cascading multiple interferometers with active phase stabilization.
Scalability Issues: These cascaded systems suffer from poor scaling, high experimental overhead, and stability issues. Current demonstrations are limited to dimensions $d \le 8$ (integrated) or $d \le 4$ (bulk optics).
Restricted Capabilities: Conventional methods are often limited to phase-only superposition measurements and cannot easily access arbitrary amplitude and phase transformations required for full quantum state tomography in high dimensions.

2. Methodology

The authors propose a novel approach that harnesses the coupling of spatial and temporal information within a single commercial Multi-Mode Fiber (MMF) to program generalized measurements.

Core Concept: $\tau$ -modes

Transmission Matrix Characterization: The team measures the Multi-Spectral Transmission Matrix (MSTM), $T(\omega)$ , of a 40m-long step-index MMF. This matrix maps input spatial modes to output spatial modes across different wavelengths.
Time-Resolved Transmission Matrix (TRTM): By Fourier-transforming the MSTM, they obtain the TRTM, $\tilde{T}(t)$ , which describes spatial scattering at specific time delays.
Singular Value Decomposition (SVD): They perform an SVD on the TRTM to identify a special set of orthogonal spatial modes called $\tau$ -modes.
- These modes arrive at the fiber output at distinct, well-separated time delays.
- Crucially, they experience minimal temporal/spectral scattering (they are "unscattered" in time relative to their specific delay).
Programmable Interferometry:
- Instead of using cascaded bulk optics, the system uses a Digital Micromirror Device (DMD) to shape the input spatial profile into an arbitrary coherent superposition of these $\tau$ -modes.
- As the light propagates through the MMF, the different $\tau$ -modes acquire distinct time delays, effectively creating a large, unbalanced, multi-mode interferometer inside the fiber.
- A Spatial Light Modulator (SLM) at the output coherently recombines the spatial modes back into a fundamental Gaussian mode, which is coupled into a Single-Mode Fiber (SMF) and detected by a Superconducting Nanowire Single-Photon Detector (SNSPD).

Experimental Setup

Fiber: 40m commercial step-index MMF (50 $\mu$ m core).
Input: DMD generates complex spatial modes (holography) to excite specific $\tau$ -mode superpositions.
Detection: SNSPD with time-tagging logic.
Calibration: Quantum tomography is performed using a tunable Franson interferometer to prepare known time-bin states for verification.

3. Key Contributions

Generalized Measurement in a Single Fiber: The paper demonstrates that a single MMF can function as a scalable, common-path interferometer capable of performing arbitrary projective measurements on high-dimensional time-bin qudits.
Arbitrary Superposition Control: Unlike previous methods limited to phase-only control, this technique allows for the independent control of both amplitude and phase of time-bin components by engineering the spatial superposition of $\tau$ -modes.
Scalability: The method is inherently scalable. Increasing the fiber diameter supports more spatial modes (higher $d$ ), and increasing fiber length provides larger time delays, without the need for complex cascaded optical setups.
High-Dimensional Demonstration: The authors successfully demonstrated programmable measurements in dimensions $d=2, 4,$ and $11$.

4. Experimental Results

Qubit Verification ( $d=2$ ): The system successfully measured time-bin qubits, exhibiting a visibility of 97% (cosine-squared fit), confirming the device acts as a high-quality time-bin "polarizer" following Malus' law.
Mutually Unbiased Bases (MUBs): The team programmed the device to measure MUBs, which are critical for quantum state tomography and entanglement witnessing.
- Dimension $d=4$ : Reconstructed 16 measurement operators targeting MUB projectors with an average fidelity of 96.24%.
- Dimension $d=11$ : Demonstrated measurements in 11 dimensions. While fidelity decreased with dimension, the system successfully reconstructed operators with an average fidelity of 84.73%.
Fidelity Analysis: The drop in fidelity at higher dimensions ( $d=11$ ) is attributed to temperature-induced fluctuations in the Multi-Spectral Transmission Matrix (MSTM) over the long measurement times required for tomography, rather than a fundamental limitation of the fiber itself.

5. Significance and Outlook

Overcoming Stability Barriers: By using a single fiber as a common-path interferometer, the system eliminates the need for active phase stabilization of multiple cascaded interferometers, significantly reducing experimental complexity and environmental sensitivity.
Quantum Network Applications: This technology is vital for long-distance quantum networks where high-dimensional time-bin encoding offers noise resilience and higher information capacity.
Photonic Quantum Computing: The ability to program high-dimensional measurements is a crucial step toward scalable linear optical quantum computing architectures that utilize the temporal degree of freedom.
Future Improvements: The authors suggest that using faster, more efficient spatial modulators (e.g., fast SLMs instead of binary DMDs) and incorporating thermal compensation models could further improve fidelity and measurement speed.

In summary, this work transforms a passive, complex scattering medium (an MMF) into a programmable, high-dimensional quantum measurement device, offering a scalable and robust alternative to traditional bulk-optics interferometry for time-bin quantum technologies.