Near-deterministic photon entanglement from a spin qudit in silicon using third quantisation

This paper proposes a near-term experiment using an antimony donor in a silicon chip to realize Rudolph's third quantization framework, enabling the generation of nearly deterministic multipartite entanglement among 56 random pairs with up to 87.5% efficiency without relying on non-deterministic entangling gates.

The Gist

The Big Picture: The "Photon Traffic Jam" Problem

Imagine you are trying to build a super-fast computer using light particles (photons) instead of silicon chips. This is a great idea because light is fast and doesn't get hot. However, there is a huge problem: photons are shy.

In the quantum world, particles usually need to "bump into" each other to interact and do math. Electrons in a normal computer do this easily. But photons? They pass right through each other like ghosts. Because they don't interact, building a logic gate (a switch) with them is incredibly hard. Usually, you have to try a million times to get them to interact just once. This makes the process slow and unreliable (non-deterministic).

The Solution: The authors propose a clever trick called "Third Quantization." Instead of forcing photons to bump into each other, they use the photons' ability to be in many places at once to create a connection without ever touching.

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The Star of the Show: The Antimony "Orchestra Conductor"

To pull off this trick, the team uses a specific atom: Antimony (Sb), implanted inside a silicon chip.

Think of a normal atom as a simple drum with two beats: Hit or Miss.
The Antimony atom, however, is like a giant, complex orchestra. Because of its unique nuclear spin, it doesn't just have two states; it has 16 different energy levels (like 16 different instruments in the orchestra).

This high-dimensional nature is the key. It allows the atom to act as a "conductor" that can split a single photon into a superposition of many different possibilities simultaneously.

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The Magic Trick: The "Time-Traveling Photon"

Here is how they generate the entanglement without the photons ever meeting:

1. The Setup: The Antimony atom is placed in a tiny microwave cavity (a box that traps light).
2. The Split: The team prepares the Antimony atom in a state where it is "superposed" (simultaneously in all 8 of its main energy levels).
3. The Emission: They trigger the atom to release a single photon. But because the atom was in a superposition, the photon doesn't just go into one time slot. It is emitted into a superposition of 8 different time slots.
   • Analogy: Imagine throwing a ball. Usually, it lands at 1:00 PM. In this experiment, the ball lands at 1:00, 1:01, 1:02... all the way to 1:07 PM at the same time.
4. The Result: This creates a special state called a W-state. It's a single photon that is "spread out" across 8 different moments in time.

The Party Game: Distributing the "Time-Slices"

Now, imagine you have this single photon spread across 8 time slots. You send these 8 "time-slices" to 8 different people (Parties A through H).

• The Rule: Each person gets a chance to look at their specific time slot.
• The Twist: You do this with two independent Antimony atoms, each creating their own "spread-out" photon.
• The Outcome: When you look at the results, you find that the two photons have created a Bell State (the strongest possible quantum link) between two of the people, even though the photons never touched.

The Analogy:
Imagine two people, Alice and Bob, each have a deck of 8 cards. They shuffle their decks and deal one card to 8 different people in a room.

• Because the cards were dealt from a "super-deck" (the superposition), if you look at the room, you will find that Alice and Bob are perfectly linked in 56 out of 64 possible scenarios.
• They didn't need to shake hands or talk. The link was created simply by how the cards were distributed.

Why This is a Big Deal

1. No "Shy" Photons Needed: They didn't force the photons to interact. They used the "spread" of the photon to do the work.
2. High Efficiency: In standard photon experiments, you might succeed only 50% of the time (or much less). This method has a theoretical success rate of 87.5%. That is "near-deterministic"—meaning it works almost every time.
3. Scalability: Because the Antimony atom is so versatile (it has 16 levels), this system can be scaled up. If you add more atoms, you can create even more complex entanglement, potentially leading to a universal quantum computer.

The "Real-World" Device

The paper proposes a physical device that looks like a tiny silicon chip with:

• Antimony Donors: The "orchestra conductors."
• Microwave Cavity: The "box" that catches the photon.
• Transmon Qubits: These act as the "receivers" or detectors that catch the photon after it travels through the time slots.

Summary in One Sentence

By using a high-dimensional Antimony atom to split a single photon into many time slots, the researchers created a way to link quantum particles together with 87.5% efficiency, bypassing the need for photons to physically bump into each other.

The Takeaway: They found a way to make photons "dance together" without ever touching, using the unique "superpowers" of an Antimony atom in silicon. This opens a new, faster path toward building quantum computers.

Technical Summary

1. Problem Statement

The paper addresses a fundamental bottleneck in photonic quantum computing: the difficulty of controlling photons due to their weak interactions.

• The Challenge: In matter-based architectures (e.g., superconducting qubits, spins), qubits interact strongly, allowing for deterministic gates. In contrast, photonic qubits typically require non-deterministic, measurement-induced gates (like linear optics) to create entanglement. These gates have low success probabilities and do not scale efficiently.
• The Limitation of Current Approaches: While a single photon can deterministically spread over multiple modes to create entanglement, purely linear optical approaches relying solely on this mechanism face severe scaling challenges (exponential circuit complexity) or fail to provide universality.
• The Gap: There is a need for a scalable, near-deterministic method to generate multipartite entanglement without relying on probabilistic photon-photon interactions or complex multi-photon input states.

2. Methodology

The authors propose a near-term experimental realization of Third Quantisation, a framework introduced by Rudolph, using a high-spin antimony (Sb) donor embedded in a silicon chip.

A. The Physical Platform: Antimony Donor in Silicon

• System: A neutral $^{123}\text{Sb}$ donor atom in a silicon lattice.
• Degrees of Freedom:
  • Nuclear Spin ($I=7/2$): Provides an 8-dimensional Hilbert space.
  • Electron Spin ($S=1/2$): Coupled to the nucleus via hyperfine interaction, creating a 16-dimensional total Hilbert space.
• Advantages over other donors: Compared to Bismuth ($I=9/2$) or Arsenic ($I=3/2$), Antimony offers a balance of high spin dimensionality, manageable hyperfine coupling (avoiding strong measurement back-action), and low lattice damage.
• Control Mechanism:
  • EDSR (Electric Dipole Spin Resonance): Used to drive transitions between electron-nuclear states. This leverages the Stark effect to modulate hyperfine interaction, enabling faster coupling to microwave cavities (MHz range) compared to magnetic ESR.
  • Time-Bin Multiplexing: Instead of using multiple physical cavities (which is currently infeasible), the protocol uses a single microwave cavity. The photon is emitted into a superposition of different time bins ($t_1, t_2, \dots, t_8$), corresponding to the 8 nuclear spin states.

B. The Protocol: Third Quantisation

1. State Preparation:
   • Initialize the Sb donor in a specific state.
   • Apply a qudit Hadamard operation to create a uniform superposition of the 8 nuclear spin states.
   • Apply an EDSR pulse conditioned on the nuclear state to flip the electron spin and emit a photon into the cavity.
   • Apply permutation operations (NMR pulses) to swap populations between nuclear states, ensuring the photon is emitted at distinct time steps.
   • Result: A single photon is deterministically spread across 8 time bins, creating an 8-mode $|W_8\rangle$ state (a high-dimensional entangled state).
2. Decoupling: A final Hadamard operation and measurement of the nuclear spin decouple the matter qubit, leaving a pure photonic $|W_8\rangle$ state.
3. Entanglement Generation:
   • Two independent $|W_8\rangle$ states are generated (using two sequential emissions or two donors).
   • These states are distributed to $K=8$ parties (modes).
   • Post-selection: The protocol selects events where the two photons land in different parties.
   • Outcome: This distribution creates a maximally entangled Bell state ($|\Psi^+\rangle$) between a random pair of the 8 parties.

3. Key Contributions

• Experimental Realization of Third Quantisation: This is the first proposal to implement Rudolph's third quantisation framework using a solid-state spin qudit, moving the concept from theory to a feasible near-term experiment.
• Deterministic Entanglement without Interaction: The method generates entanglement between photons that never interact. The entanglement arises purely from the deterministic spreading of single photons into multiple modes (time bins) and classical communication.
• High Efficiency: The protocol achieves a theoretical upper-bound efficiency of 87.5% for generating a Bell state between random pairs, significantly outperforming standard linear optics methods (which typically cap at 50% for Bell states).
• Scalability: The approach is inherently scalable. By increasing the number of donors and the dimensionality of the spin system, the framework can generate complex high-dimensional cluster states suitable for universal quantum computing.

4. Results and Performance Analysis

• Success Probability:
  • Theoretical Limit: 87.5% (56 valid outcomes out of 64 possible distributions for 2 photons and 8 parties).
  • With Losses: Simulations show that even with photon loss rates of 1% to 10% per mode, the success rate remains robust (dropping to ~70-80% in worst-case scenarios).
• Error Sources:
  • Decoherence: The electron spin coherence time ($T_2^* \approx 510 \mu s$) is orders of magnitude longer than the photon emission time (~333 ns), ensuring high fidelity.
  • Gate Fidelity: Operations like NMR and ESR pulses have demonstrated fidelities >99% in previous experiments.
  • Photon Loss: Dominated by cavity quality factors ($Q_i, Q_c$). Even with realistic cavity losses, the protocol remains viable.
• Hardware Requirements:
  • A single silicon chip with an Sb donor, a microwave antenna, and a superconducting cavity.
  • 8 superconducting transmon qubits (acting as detectors/parties) to read out the time-bin states.
  • Fast microwave switches to route photons to specific transmons.

5. Significance and Outlook

• Paradigm Shift: The work challenges the conventional wisdom that photonic quantum computing requires probabilistic gates. It demonstrates that deterministic multi-mode entanglement combined with classical communication is sufficient for universal quantum computing.
• Silicon Integration: By utilizing antimony donors in silicon, the proposal leverages existing semiconductor manufacturing capabilities, offering a pathway to integrate quantum emitters with control electronics on a single chip.
• Future Directions:
  • Scaling: Moving from $N=2$ photons to $N>2$ to generate complex graph states and cluster states.
  • Loophole-Free Tests: The compact nature of the device allows for potential loophole-free Bell tests within a single dilution refrigerator.
  • Frequency Multiplexing: While the current proposal uses time-bin multiplexing, future advancements could enable frequency multiplexing (using multiple cavities) to reduce the temporal footprint.

In conclusion, this paper presents a compelling, experimentally feasible route to near-deterministic quantum entanglement, bridging the gap between matter-based control and photonic scalability through the innovative application of third quantisation in a silicon spin qudit.