A time-to-digital converter with steady calibration through single-photon detection

This paper presents a scalable, FPGA-based Time-to-Digital Converter (TDC) for Quantum Key Distribution that achieves a 27 ps residual jitter and maintains steady performance through a continuous, single-photon detection-based calibration method that eliminates data loss and operates reliably across a wide temperature range.

The Gist

Imagine you are trying to catch a fly with a net, but the fly is moving so fast that it's invisible to the naked eye. To catch it, you need a stopwatch that is incredibly precise—so precise it can measure time in "picoseconds" (one trillionth of a second). This is what a Time-to-Digital Converter (TDC) does. It's the high-speed stopwatch used in advanced technologies like Quantum Key Distribution (QKD), which is essentially an unhackable way of sending secret messages using light particles (photons).

However, building these super-precise stopwatches out of standard computer chips (FPGAs) is tricky. They are like cheap, mass-produced watches: they work well, but they drift. If the temperature changes or the chip gets a little warm, the "ticks" of the stopwatch get slightly uneven. One tick might be 18 picoseconds, and the next might be 20. This "jitter" ruins the accuracy, making it hard to distinguish a real message from background noise.

The Old Problem: The "Pause and Reset" Dilemma

Traditionally, to fix this drift, engineers had to stop the stopwatch, run a calibration test, and then start again.

• The Analogy: Imagine a race car driver trying to set their lap times. To check if their watch is accurate, they have to pull over, stop the car, run a diagnostic, and then get back on the track.
• The Problem: In the world of quantum communication (especially with satellites), you can't just "pull over." You only have a few minutes of contact with a satellite. If you stop to calibrate, you lose precious data. If you don't calibrate, your data is garbage because the watch is drifting.

The New Solution: "Steady Calibration"

The authors of this paper, working at the University of Padua, built a new device called MARTY. Their breakthrough is a method they call "Steady Calibration."

Instead of stopping the race to check the watch, MARTY uses the race itself to fix the watch in real-time.

• The Analogy: Imagine a runner who is also their own coach. Every time the runner takes a step, they instantly check their foot placement against a mental map. If they notice they are leaning slightly left, they adjust their stride immediately for the very next step. They never stop running; they just get better with every single step.
• How it works: MARTY uses the actual light particles (photons) it is supposed to be timing. It assumes that these particles arrive at random, uniform intervals. By constantly analyzing the pattern of these arrivals, the device can detect if its internal "ticks" are getting too short or too long due to temperature changes, and it instantly corrects the math behind the scenes.

The "Thermometer" and the "Bubble"

To understand how precise this is, imagine the chip has a long line of dominoes (a "delay line"). When a signal hits the first domino, it starts falling down the line.

• The Problem: Sometimes, due to electrical quirks, a domino might fall slightly out of order (a "bubble error"), or the distance between dominoes might change if the room gets hot.
• The Fix: MARTY counts exactly which dominoes have fallen. It uses a "thermometer code" (a way of counting where the signal is in the line) to figure out exactly how much time has passed. The "Steady Calibration" constantly re-maps the distance between these dominoes as the temperature changes, ensuring the measurement stays accurate even if the chip heats up from 5°C to 80°C (from a chilly fridge to a hot summer day).

Why This Matters: The Satellite Connection

The paper highlights a specific use case: Satellite Quantum Communication.

• The Scenario: A satellite flies over a ground station. It has a very short window (maybe 5 minutes) to send a massive amount of secret keys before it disappears over the horizon.
• The Challenge: Space is a harsh environment. The temperature swings wildly. A normal TDC would drift during those 5 minutes, causing errors and losing the secret keys.
• The MARTY Advantage: Because MARTY calibrates itself continuously using the data it's already collecting, it doesn't need to stop. It can stream data for a whole week without overflowing its memory, and it keeps its accuracy steady even as the temperature changes.

The Results

The team tested MARTY against a top-tier commercial device (the QuTAG).

• Precision: MARTY achieved a "jitter" (uncertainty) of about 27 picoseconds. That is incredibly fast.
• Performance: When used in a real Quantum Key Distribution test, it performed just as well as the expensive commercial device, with a low error rate (2.2%).
• Efficiency: It can handle up to 12 million events per second without breaking a sweat.

In a Nutshell

The authors created a self-correcting stopwatch that never needs to pause. It uses the very data it's measuring to fix its own internal errors in real-time. This makes it perfect for high-stakes, high-speed applications like satellite internet, where you can't afford to stop the data stream to fix a broken watch. It's like a runner who gets faster and more accurate with every single step, ensuring you never miss a beat, no matter how hot the track gets.

Technical Summary

1. Problem Statement

Time-to-Digital Converters (TDCs) are essential for applications requiring picosecond resolution, such as Quantum Key Distribution (QKD), LiDAR, and particle physics. While Field-Programmable Gate Arrays (FPGAs) offer a cost-effective and flexible alternative to Application-Specific Integrated Circuits (ASICs), they suffer from intrinsic non-linearities and environmental drift (particularly temperature).

Existing calibration methods face significant trade-offs:

• Static Code-Density Test: Requires stopping data acquisition to inject a uniform signal (usually from a Ring Oscillator), causing data loss and making it unsuitable for continuous or time-sensitive applications like satellite QKD.
• Drift Estimation: Tracks oscillator frequency changes but assumes uniform environmental effects, leading to cumulative inaccuracies under thermal gradients.
• Dual-Chain Architectures: Enable continuous calibration but double the logic resources per channel, which is impractical for multi-channel systems.
• Event-Driven Calibration: While promising, previous implementations often relied on specific sources (e.g., radioactive decay in PET) or entangled photons, limiting broader applicability.

The core challenge is achieving continuous, high-accuracy calibration without interrupting data acquisition or doubling hardware resources, specifically for environments with varying temperatures (e.g., space).

2. Methodology

The authors designed and implemented a TDC named MARTY on a Zynq-7020 SoC (FPGA + ARM CPU).

Hardware Design

• Architecture: The TDC uses a coarse counter (using DSP48 blocks clocked at 412.5 MHz) for macro-time estimation and a Tapped Delay Line (TDL) for fine-time estimation.
• TDL Implementation: Utilizes CARRY4 fast-carry chains (36 elements, 144 total stages) to create a propagation line. The output is captured as a "thermometer code" and decoded using a pipelined adder tree algorithm (Adamič et al.) to handle "bubble errors" (metastability artifacts).
• Data Streaming: The system employs a double-buffering scheme in Block RAM (BRAM) with Central Direct Memory Access (CDMA). The CPU extracts data from the BRAM while the FPGA continues filling the buffer, enabling continuous streaming to a PC via Ethernet at up to 12 Mevents/s for approximately one week without overflow.

Calibration Strategy: "Steady Calibration"

The paper introduces a novel steady calibration method, an extension of event-driven calibration:

1. Principle: Instead of stopping acquisition to run a code-density test with a Ring Oscillator (RO), the system uses the natural stream of single-photon detection events from the experimental setup (e.g., a pulsed laser in a QKD system).
2. Mechanism: The system maintains a sliding window (histogram) of detection events. As new events arrive, the oldest bin is removed, and the new one is appended. The calibration curve is recalculated in real-time based on this moving window.
3. Assumption: The input events (single photons from a pulsed laser) are assumed to be uniformly distributed over the TDL delay line. This uniformity is ensured by using non-commensurable clock frequencies for the laser repetition rate and the TDC sampling clock, combined with natural clock drift.
4. Initialization: A standard static calibration is performed at startup to establish the initial baseline, after which the system self-corrects continuously.

3. Key Contributions

• Steady Calibration Algorithm: A method that utilizes the actual measurement data (single-photon events) to update calibration parameters dynamically. This eliminates the need for data acquisition pauses and removes the distinction between "data collection" and "calibration."
• Hardware Efficiency: The design achieves continuous calibration without requiring a second delay line (dual-chain), preserving FPGA logic resources for multi-channel scaling.
• Validation of Single-Photon Calibration: The authors experimentally demonstrated that using a laser and Single-Photon Detector (SPD) for calibration yields results statistically equivalent to using an internal FPGA Ring Oscillator ($\chi^2 = 0.026$).
• High-Throughput Streaming: A robust data transfer architecture capable of sustaining ~12 Mevents/s for a week, addressing the bottleneck of continuous high-speed data acquisition.

4. Results

The device was characterized under various conditions, including a temperature sweep from 5°C to 80°C.

• Performance Metrics (Room Temp):
  • Resolution ($\tau_{res}$): 18.37 ps.
  • Differential Non-Linearity (DNL): $[-0.97, 2.86]$ (only 21 out of 132 bins exceeded DNL > 1).
  • FWHM Jitter: 27.63 ps.
• Temperature Stability:
  • Without Calibration: Jitter increased monotonically with temperature.
  • Static Calibration (RO at 5°C): Jitter increased significantly as temperature rose.
  • Steady Calibration: The FWHM jitter remained stable and low across the entire temperature range, with an average standard deviation of 0.61 ps. This performance was comparable to recalibrating manually at every 1°C step but without any data loss.
• QKD Application Test:
  • MARTY was tested in a QKD setup (BB84 protocol) against a commercial high-end TDC (QuTAG).
  • Quantum Bit Error Rate (QBER): Both devices achieved a mean QBER of 2.2%.
  • Pulse Width: The detected pulse width using MARTY (107 ps) was comparable to the QuTAG (108 ps), despite MARTY's slightly higher intrinsic jitter. This is because the detector's jitter (~100 ps) dominates the total system jitter.
• Longevity: The system successfully streamed data for ~1 week without overflow.

5. Significance and Future Outlook

• Satellite Quantum Communication: The ability to maintain calibration in harsh, varying thermal environments without stopping data acquisition makes MARTY ideal for satellite-based QKD, where communication windows are short and environmental conditions are extreme.
• Scalability: The resource-efficient design allows for easy scaling to multi-channel systems, which is critical for high-rate quantum networks.
• Future Improvements: The authors suggest migrating the design to Ultrascale+ (16 nm FinFET) FPGAs. This would reduce the intrinsic delay of the carry chains, potentially lowering the resolution to the single-digit picosecond range, which would better leverage the performance of superconducting nanowire single-photon detectors (SNSPDs).

In conclusion, the paper presents a robust, FPGA-based TDC solution that solves the critical trade-off between calibration accuracy and data continuity, enabling high-performance quantum communication in real-world, dynamic environments.