Generating Fock state exceeding 10000 excitations with near unit fidelity by adaptive generalized-parity measurement

This paper proposes an adaptive generalized-parity measurement protocol that deterministically converts large coherent or displaced thermal states into macroscopic Fock states with over 10,000 excitations and near-unit fidelity by converting measurement randomness into adaptive updates, thereby avoiding the limitations of probabilistic postselection.

The Gist

Imagine you are trying to find a single, specific grain of sand on a massive beach. Usually, if you just start digging randomly, you might get lucky, but the odds are terrible. If you try to filter the sand by shaking it through a sieve, you might catch the right grain, but you'd have to throw away almost all the other sand you caught along the way. This is how most current methods for creating specific quantum states work: they are like a sieve that only keeps the "lucky" results and throws away the rest.

This paper proposes a smarter, more efficient way to find that specific grain of sand (a "Fock state" with over 10,000 grains of energy) without throwing anything away.

The Problem: The "Lucky Sieve"

In the quantum world, scientists want to create "macroscopic Fock states." Think of these as containers holding a very precise, huge number of energy packets (photons), like exactly 10,000.

• Old Method: Scientists use a process called "post-selection." Imagine you have a machine that tries to sort sand. It only keeps the sand if it comes out in a very specific order. If the machine makes a mistake, you have to start over. As the number of grains you want increases, the chance of getting the right order by luck drops to almost zero. It's like trying to guess a 10,000-digit password by random guessing; you will never succeed.

The Solution: The "Adaptive GPS"

The authors, Chen-yi Zhang and Jun Jing, propose a new method called Adaptive Generalized-Parity Measurement.

Here is the analogy:
Imagine you are navigating a maze to find a specific room.

• The Old Way: You walk down a path. If you hit a dead end, you go back to the start and try a different path. Most paths are dead ends, so you waste a lot of time.
• The New Way (This Paper): You have a GPS (the "ancillary qubit") that talks to you at every intersection.
  1. You take a step.
  2. The GPS tells you, "You went left."
  3. Instead of saying "Wrong, go back," the GPS says, "Okay, since you went left, the next turn should be right."
  4. You adjust your next step based on that answer.

In this paper, the "GPS" is a tiny quantum bit (a qubit) connected to the big system (the resonator). The scientists measure the qubit. If the qubit says "Up" (outcome $e$), they keep the measurement settings the same for the next step. If it says "Down" (outcome $g$), they slightly shift the timing of the next measurement.

The Magic Trick:
This adaptive rule turns the "randomness" of the measurement into a guide. Instead of discarding the "wrong" answers, the system uses them to update the map. No matter what the qubit says, the process keeps moving forward. You never throw away a measurement; you just use the result to refine the next step.

The Results: Finding the Needle in the Haystack

The authors tested this idea using a standard quantum model (the Jaynes-Cummings model). Here is what they found:

1. Huge Numbers: They successfully created Fock states with over 10,000 excitations (photons). This is a "macroscopic" number, meaning it's huge for the quantum world.
2. Speed: They did this in just 10 rounds of measurement. Because the method is so efficient, the number of steps needed grows very slowly (logarithmically) even as the target number gets massive.
3. Success Rate:
   • On average, the final state was about 80% accurate (fidelity).
   • More impressively, about 35% of the time, they got a state that was 99% perfect.
   • This is a massive improvement over old methods, where the success rate for such large numbers would be practically zero.

Robustness: It Works Even When "Dirty"

Usually, quantum experiments require a perfectly clean, cold starting point. The authors showed their method is tough. Even if they started with a "displaced thermal state" (imagine the sand is a bit warm and jiggly, not perfectly still), the method still worked.

• At moderate temperatures, they could still create a 3,000-photon state with 99% accuracy about 10% of the time.
• This means the method doesn't need a perfectly pristine environment to work, making it more practical for real-world labs.

Summary

The paper presents a new "navigation system" for quantum states. Instead of hoping for a lucky break and throwing away failures, it uses every single measurement result to steer the system toward a massive, precise target. It allows scientists to generate huge, precise quantum states quickly and reliably, even if the starting conditions aren't perfect.

Technical Summary

Technical Summary: Generating Fock States Exceeding 10,000 Excitations via Adaptive Generalized-Parity Measurement

Problem Statement
Macroscopic Fock states (states with a large, definite number of excitations) are critical resources for quantum metrology, simulation, and enhanced sensing. While various protocols exist for generating Fock states in cavity and circuit QED, most are restricted to small excitation numbers. Extending these to the macroscopic regime (e.g., $>10^4$ photons) remains challenging. Existing methods often rely on probabilistic postselection, where success depends on a specific sequence of measurement outcomes, leading to an exponentially decreasing success probability as the target excitation number increases. Furthermore, many protocols require pure-state initialization, precise external driving, or complex gate operations, limiting their scalability and robustness against thermal noise.

Methodology
The authors propose a deterministic protocol based on adaptive generalized-parity measurement (GPM) to generate macroscopic Fock states without postselection. The core framework involves:

1. System Architecture: A main system with a discrete, non-degenerate spectrum (e.g., a bosonic mode) coupled to an ancillary qubit via an exchange-type interaction (modeled by the Jaynes-Cummings Hamiltonian).
2. Adaptive Protocol: Instead of a fixed sequence, the protocol employs repeated projective measurements on the ancillary qubit. Crucially, the time interval ($\tau_k$) between the $k$-th and $(k+1)$-th measurement is adaptively updated based on the outcome of the previous measurement ($m_k$).
3. Generalized Parity Construction:
   • The authors derive a construction rule where the measurement intervals are chosen such that the evolution operator acts as a filter.
   • If the measurement outcome is $|e\rangle$, the system undergoes a "diagonal" GPM, projecting onto a subspace defined by $n_m \equiv n_t(k) \pmod{2^k}$.
   • If the outcome is $|g\rangle$, the system undergoes a "displaced" GPM, projecting onto a shifted subspace.
   • A label $n_t(k)$ is updated deterministically according to the rule $n_t(k+1) = n_t(k) + (1 - \delta_{m_{k+1}, m_k})2^k$.
4. Trajectory Retention: Unlike probabilistic protocols that discard trajectories failing a specific postselection condition, this protocol retains every measurement trajectory. The randomness of the ancilla measurement is converted into an adaptive update of the GPM parameters, ensuring unit success probability in principle (all trajectories lead to a filtered state, though the specific final Fock state index may vary slightly).

Key Contributions

• Deterministic Scaling: The protocol eliminates the exponential suppression of success probability associated with postselection. It achieves a logarithmic scaling of operation rounds ($N \sim \log_2 \sqrt{n_t(0)}$) relative to the target excitation number.
• Macroscopic Generation: The method is theoretically capable of generating Fock states with excitation numbers up to $n_t = O(10^4)$ within approximately 10 measurement rounds.
• Robustness: The protocol is shown to be effective starting from displaced thermal states, demonstrating resilience against moderate thermal mixtures without requiring pure-state initialization.
• Simplicity: The implementation requires no external driving fields, gate operations, or changes in the measurement basis, relying solely on free evolution and projective measurements.

Results
Numerical simulations based on the resonant Jaynes-Cummings model yield the following results:

• Fidelity: For an initial coherent state with $n_t(0) \approx 10^4$, the protocol generates a Fock state with an averaged fidelity of approximately 80% within 10 rounds.
• High-Fidelity Probability: The probability of obtaining a final state with fidelity exceeding 99% is approximately 35% for large $n_t(0)$ (specifically $n_t(0) \ge 5000$). This is significantly higher than postselection-based protocols.
• Thermal Robustness: When initialized with a displaced thermal state, the protocol maintains high performance. For an inverse temperature $\beta = 2/\omega_c$, the probability of achieving $>99\%$ fidelity remains around 29% for $n_t(0) \approx 3000$. While higher temperatures (lower $\beta$) reduce the filtering efficiency and final fidelity, the protocol remains functional and scalable.
• Time Complexity: The total running time scales as $T \sim \sqrt{n_t(0)} \log_2 \sqrt{n_t(0)}$, which is efficient for macroscopic targets.

Significance and Claims
The paper claims to offer a convenient and practical route toward generating macroscopic nonclassical states. By converting measurement randomness into adaptive control, the protocol overcomes the scalability bottleneck of postselection-based methods. The authors emphasize that their approach does not demand the pure-state initialization or complex control sequences (like adaptive driving or gate operations) typical of other high-fidelity generation schemes. Consequently, this method is presented as a robust tool for quantum state engineering in systems with discrete spectra, particularly for applications requiring large photon numbers where current methods are limited by low success probabilities or initialization constraints.