Indistinguishability of photonic qubits emitted from trapped $^{40}$Ca$^+$ ions via pulsed excitation

This paper investigates the indistinguishability of Raman photons from two trapped $^{40}$Ca$^+$ ions under pulsed excitation, demonstrating that the mean number of spontaneous back-decays to the initial state is a key single-emitter metric that directly correlates with the achievable Hong-Ou-Mandel interference visibility.

The Gist

Imagine you are trying to build a super-secure internet using light. To do this, you need to send individual "packets" of light (photons) from two different sources and make them meet at a crossroads. If these two packets are truly identical—like two perfect twins—they will interfere with each other in a very specific, magical way called the Hong-Ou-Mandel (HOM) effect. This interference is the key to linking quantum computers together.

However, if the twins aren't perfect—if one has a slightly different heartbeat or a tiny scar—they won't interfere correctly, and the connection fails.

This paper is about how the researchers at the University of Saarland tried to make these "twin" photons from trapped calcium ions as identical as possible, and how they figured out what was ruining their perfection.

The Setup: The Ion Factory

Think of the researchers' lab as a high-tech factory. Inside a vacuum chamber, they trap a single atom of Calcium-40 (an ion) using invisible electric fields, holding it like a fly in a jar.

To make a photon, they hit the ion with a very short, sharp "tap" of laser light (a pulse lasting just a few billionths of a second).

1. The Tap: This kick pushes the ion into an excited state.
2. The Drop: The ion immediately falls back down to a lower energy state, releasing a photon (a packet of light) in the process.
3. The Goal: They want to do this twice, once for one ion and once for another, and then bring the two resulting photons together to see if they are identical twins.

The Problem: The "Back-Step"

Here is where things get tricky. When the ion is excited, it doesn't always fall directly to the final destination. Sometimes, it takes a "back-step."

Imagine the ion is a hiker trying to reach a summit (the final state). The laser pushes them up a cliff.

• The Ideal Path: The hiker jumps up, slides down the other side, and drops a flag (the photon) at the bottom. Done.
• The Back-Step: The hiker jumps up, slips, falls back down to the starting point, scrambles up the cliff again, and then finally drops the flag.

Every time the ion slips back down and has to climb up again, it adds a tiny delay and a bit of "jitter" to the photon it eventually releases. If the ion slips back multiple times, the photon becomes a bit "fuzzy" or "stretched out" in time.

If you have two ions, and one of them took a few extra back-steps while the other didn't, their photons won't be identical twins anymore. They will be like a well-rested twin and a tired, stumbling twin. When they meet at the crossroads, they won't interfere perfectly, and the quantum connection fails.

The Discovery: Counting the Stumbles

The researchers wanted to know: How many times does the ion stumble back before it finally succeeds?

They developed a clever way to count these "back-steps" (which they call back-decays).

• Every time the ion slips back down, it emits a different color of light (393 nm) before it finally emits the main photon (854 nm).
• By watching for these "warning flashes" of 393 nm light right before the main photon arrives, they could count how many times the ion stumbled.

They found a direct link: The more back-stumbles an ion makes, the less identical the photons become.

The Experiment: Two Ions, One Beam Splitter

To prove this, they trapped two ions side-by-side.

1. They hit both ions with laser pulses of different lengths (some short, some long).
2. They counted the back-stumbles for each ion.
3. They sent the main photons from both ions into a 50:50 beam splitter (a mirror that splits light in half).
4. They measured the HOM Visibility: This is a score from 0 to 100% that tells you how well the photons interfered. A score of 100% means they are perfect twins; 0% means they are strangers.

The Result:
They found a perfect correlation. When the excitation pulses were short and weak, the ions stumbled very little (low back-decay count), and the photons interfered beautifully (high visibility). When the pulses were long and strong, the ions stumbled more often, and the interference score dropped.

The Takeaway

The paper concludes that you don't need to measure the complex quantum wave of the photon to know if it's good. You just need to count the "back-stumbles" (the 393 nm flashes) of a single ion.

• Low back-stumbles = High-quality, identical photons.
• High back-stumbles = Messy, non-identical photons.

This is a huge practical tool. It means scientists can easily check the quality of their quantum light source by simply counting the "warning flashes" on a single ion, rather than doing complex interference tests every time. This helps them tune their lasers to find the "sweet spot" where they get the most photons without making them too messy to use for quantum networking.

Why This Matters (According to the Paper)

The paper explicitly mentions that this ability to generate high-quality, identical photons is the "keystone" for:

• Quantum Repeaters: These are devices needed to send quantum information over long distances (like a quantum internet).
• Entanglement Swapping: A process where two distant quantum memories (like the ions) become entangled just by meeting their photons in the middle.

The researchers also note that their setup, using flexible laser pulses, could eventually help connect different types of quantum computers (like ions and diamond defects) into a single, heterogeneous network.

Technical Summary

Technical Summary: Indistinguishability of Photonic Qubits from Trapped $^{40}\text{Ca}^+$ Ions via Pulsed Excitation

Problem Statement
The generation of indistinguishable single photons is a fundamental requirement for implementing entanglement swapping protocols based on Bell state measurements, which are essential for quantum repeaters and distributed quantum computing. While Raman transitions in trapped ions offer a pathway to generate single photons, the indistinguishability of photons from two separate emitters is often compromised by spontaneous scattering events. Specifically, an ion may spontaneously decay back to the initial ground state and undergo re-excitation before the final Raman photon is emitted. While these "back-decays" do not alter the spectral properties of the photon, they incoherently add time-shifted contributions to the photon's temporal wave-packet, broadening it beyond the Fourier limit. This broadening reduces the visibility of Hong-Ou-Mandel (HOM) interference, thereby limiting the fidelity of entanglement swapping operations.

Previous approaches to mitigate this involved using excitation pulses shorter than the excited state lifetime (sub-nanosecond pulses). However, generating such pulses is technically challenging, often requiring complex modulation of near-infrared light before frequency doubling, and demands high laser powers that scale inversely with the square of the pulse length.

Methodology
The authors investigate the trade-off between photon generation probability and indistinguishability using acousto-optic modulators (AOMs) to generate excitation pulses with lengths in the few-nanosecond range (6.5 ns to 56.6 ns). This approach offers greater scalability and control over pulse shape and repetition rates compared to pulsed laser systems.

The experimental setup utilizes a linear trap for $^{40}\text{Ca}^+$ ions. The ions are excited via the $S_{1/2} \to P_{3/2}$ transition at 393 nm using $\pi$-polarized pulses. The system is designed to detect Raman photons emitted at 854 nm (from the $P_{3/2} \to D_{5/2}$ transition) and parasitic back-decay photons at 393 nm.

The study is divided into two main experimental phases:

1. Single-Ion Characterization: A single trapped ion is excited by a train of 80 pulses. The researchers measure the arrival times of both 393-nm (back-decay) and 854-nm (Raman) photons. By analyzing the cross-correlation between these detection events, they determine the mean number of back-decays ($\langle N \rangle$) occurring before the emission of a successful Raman photon.
2. Two-Ion Interference: Two co-trapped ions are excited simultaneously. The 854-nm photons collected from each ion are combined on a 50:50 fiber beam splitter. HOM interference is measured by comparing detection time differences for photons with parallel versus orthogonal polarizations (inferred from ion state readouts).

Key Contributions and Results

• Quantification of Back-Decays: The authors introduce the mean number of back-decays, $\langle N \rangle$, as a measurable single-emitter quantity. They demonstrate that $\langle N \rangle$ can be extracted from the cross-correlation of 393-nm and 854-nm detection events. For the pulse lengths tested, $\langle N \rangle$ remains below 0.5, indicating that most Raman photons are generated without prior back-decays, with non-zero values primarily arising from single back-decay events ($N=1$).
• Correlation with HOM Visibility: The study establishes a direct anti-correlation between the mean number of back-decays ($\langle N \rangle$) and the achievable HOM visibility ($V$). As pulse length and strength increase, $\langle N \rangle$ increases, leading to a reduction in HOM visibility.
• Experimental Validation: The measured HOM visibility for two ions is compared against numerical simulations that account for experimental imperfections, including background counts, detector efficiency imbalances, decoherence, imperfect beam splitter ratios, and polarization angle mismatches. The simulations, which use the measured pulse parameters as inputs, show good agreement with experimental data.
• Optimization Trade-off: The results empirically illustrate the trade-off between the success probability of generating a Raman photon ($P_{854}$) and the indistinguishability of the photons (HOM visibility). Longer, stronger pulses increase $P_{854}$ but degrade indistinguishability due to higher $\langle N \rangle$.

Significance and Claims
The paper claims that the mean number of back-decays ($\langle N \rangle$) serves as a robust, easily accessible indicator for predicting the HOM visibility in experiments involving two identical emitters. This allows researchers to optimize excitation pulse parameters based on single-ion measurements to achieve desired interference properties in multi-ion setups.

The authors assert that their setup, utilizing AOM-controlled few-nanosecond pulses, is suitable for realizing quantum repeater (QR) segments. They note that their system has already demonstrated entanglement swapping with two co-trapped ions and that the flexibility of AOM-based timing facilitates the interfacing of heterogeneous quantum platforms (e.g., combining trapped ions with diamond color centers) in future quantum networks. Furthermore, the use of short pulses is highlighted as beneficial for quantum key distribution (QKD) protocols and the generation of time-bin qubits, as the pulse timing is compatible with interferometer-type converters.

The work does not claim to have solved the back-decay problem entirely but rather provides a method to characterize and manage the resulting trade-offs, supporting the theoretical description of these phenomena across a broader parameter space.