Generation of large Fock states from coherent states using Kerr interaction and displacement

This paper proposes a feasible scheme for generating large Fock states with near-unity fidelity from coherent states in circuit QED and cavity field systems by repeatedly applying a unitary transformation that combines Kerr nonlinearity and pulsed coherent displacements.

The Gist

The Big Idea: Turning a "Fuzzy Cloud" into a "Perfect Marble"

Imagine you are trying to create a specific number of marbles.

• The Goal: You want exactly 5 marbles (or 10, or 20). In the quantum world, these are called Fock states. They are perfect, precise packets of light energy.
• The Problem: Nature usually gives you a coherent state. Think of this as a "fuzzy cloud" of marbles. It's like a bag where you expect to have 5 marbles, but sometimes you pull out 4, sometimes 6, sometimes 3. It's a probability cloud, not a precise count.
• The Challenge: Getting that fuzzy cloud to collapse into a single, perfect number (like exactly 5) without losing any marbles or breaking the system is incredibly hard. Usually, you need a "super-powerful" magnet (giant nonlinearity) to force the marbles into place, but we don't have magnets that strong yet.

The Solution: The "Kerr-Displacement" Dance

The authors propose a clever trick to turn that fuzzy cloud into a perfect marble count. Instead of using one giant magnet, they use a repeated dance of two simple moves.

Think of the quantum light field as a ball of dough on a table.

1. Move 1: The Kerr Twist (The Non-Gaussian Operation)

   • Imagine you have a special rolling pin that doesn't just flatten the dough; it twists it in a weird, non-linear way. It doesn't change how much dough you have (the total number of marbles), but it scrambles the shape of the dough. It makes the "fuzzy cloud" wobble and stretch in a specific way.
   • In physics: This is the Kerr interaction. It changes the phase of the light without changing the average number of photons.

2. Move 2: The Displacement (The Push)

   • Now, imagine you give the dough a gentle push with your hand. This adds a little bit of extra dough or takes a little bit away, shifting the whole ball to a new spot on the table.
   • In physics: This is the Displacement operation. It adds or subtracts energy (photons) to the system.

The Magic: Doing It Over and Over

If you do just one "Twist" and one "Push," you get a slightly better shape, but it's still a bit fuzzy. You might get 70% of the way to having exactly 5 marbles.

The breakthrough in this paper is repetition.
The authors realized that if you do the Twist-Push-Twist-Push sequence multiple times (like 2 or 3 times), the dough starts to tighten up.

• Iteration 1: The dough is still a blob, but it's starting to look like a ball.
• Iteration 2: The blob is getting smaller and more defined.
• Iteration 3: The dough has collapsed into a perfect, tight marble.

By carefully tuning how hard you twist (Kerr strength) and how far you push (Displacement amplitude) at each step, you can squeeze the "fuzzy cloud" until it becomes a state with a definite number of photons (a Fock state) with near-perfect accuracy (95% to 99% fidelity).

Why This Matters

• No Giant Magnets Needed: Previous methods required "giant" nonlinearities (super-strong magnets) that are hard to build. This method works with the "magnets" we already have in labs today, provided we repeat the dance enough times.
• Scalability: They showed you can create states with up to 20 photons (which is a lot in the quantum world) with high accuracy.
• Real-World Use: This isn't just math. They showed it works in Circuit QED (superconducting circuits used in quantum computers) and Optical Cavities (mirrors trapping light).

The "Noise" Problem (The Leaky Bucket)

In the real world, nothing is perfect. Imagine your dough is in a leaky bucket. Every time you twist and push, a few crumbs fall out (photon loss).

• The authors ran simulations to see how much leaking the system could handle.
• The Result: Even if the bucket is leaking (which happens in real experiments), as long as the "twisting" happens fast enough compared to the leaking, you can still get a perfect marble. They found that even with significant loss, they could still get 90%+ accuracy for up to 20 photons.

Summary Analogy

Imagine you are trying to tune a radio to a specific station (the Fock state).

• Old Way: You try to find the station by turning the dial to a "super-powerful" frequency that doesn't exist yet.
• This Paper's Way: You start with a fuzzy signal. You gently nudge the dial, then twist the antenna, then nudge again, then twist again. With each small, repeated adjustment, the static clears up, and the music becomes crystal clear. You don't need a super-powerful radio; you just need to be patient and precise with your adjustments.

In a nutshell: The paper presents a recipe to turn a "maybe" (a coherent state) into a "definitely" (a Fock state) by repeatedly applying two simple quantum operations. This makes it much easier to build the precise building blocks needed for future quantum computers and communication networks.

Technical Summary

1. Problem Statement

The generation of high-photon-number Fock states (number states $|N\rangle$) is a critical challenge in quantum optics and quantum information processing. Fock states are fundamental building blocks for quantum computing, simulation, and metrology due to their definite photon number and robustness.

• Current Limitations: Existing deterministic methods, such as atom-cavity interactions or photon blockade in Kerr nonlinear cavities, struggle to generate high-order Fock states ($N > 3$) with high fidelity.
• Specific Bottlenecks:
  • Photon Blockade: Requires "giant" Kerr nonlinearities to create sufficient anharmonicity, which is experimentally difficult to achieve for large $N$.
  • Dissipation: High-fidelity generation is often limited by cavity decay rates.
  • State Purity: Some theoretical proposals generate mixed states or states that are not exact Fock states even without dissipation.

The authors aim to propose a scheme that can deterministically generate large Fock states (up to $N=20$ and beyond) with near-unity fidelity using experimentally feasible parameters, without requiring giant nonlinearities.

2. Methodology

The proposed scheme transforms an initial coherent state $|\alpha\rangle$ into a target Fock state $|N\rangle$ through a multi-step iterative process.

Core Mechanism

The transformation relies on the repeated application of a combined unitary operator $\hat{U}(\beta, \chi)$, which consists of two distinct operations:

1. Kerr Interaction ($\hat{U}_K$): A non-Gaussian operation described by the unitary $\hat{U}_K(\chi) = e^{-i\chi \hat{a}^{\dagger 2}\hat{a}^2}$. This introduces nonlinearity and modifies the phase-space distribution without changing the photon number statistics initially.
2. Displacement ($\hat{D}$): A linear operation described by $\hat{D}(\beta) = e^{\beta \hat{a}^\dagger - \beta^* \hat{a}}$, which shifts the state in phase space.

The Iterative Process

The process involves $M$ iterations. Starting with an initial coherent state $|\alpha\rangle$ where the amplitude is chosen as $|\alpha| \approx \sqrt{N}$:
$$|\Psi^{(M)}\rangle = \prod_{k=1}^{M} \left[ \hat{D}(\beta_k) e^{-i\chi_k \hat{a}^{\dagger 2}\hat{a}^2} \right] |\alpha\rangle$$

• Optimization: For each step $k$, the displacement amplitude $\beta_k$ and the Kerr strength parameter $\chi_k$ (related to interaction time and nonlinearity strength) are numerically optimized via a grid search.
• Goal: To progressively narrow the photon number distribution of the coherent state, suppressing all components except the target $|N\rangle$.

Physical Implementation

The scheme is modeled as a cavity filled with a Kerr nonlinear medium driven by a series of ultra-short coherent pulses.

• The Hamiltonian is $\hat{H} = K\hat{a}^{\dagger 2}\hat{a}^2 + i\sum_j \delta(t-t_j)\beta_j(\hat{a}^\dagger - \hat{a})$.
• In the limit of ultra-short pulses (delta-function approximation), the evolution between pulses corresponds exactly to the unitary sequence defined above.
• The parameters $\chi_k$ are tuned by adjusting the time intervals between pulses, while $\beta_k$ is controlled by the pulse amplitude.

3. Key Contributions

• Scalability: The method successfully generates Fock states up to $N=20$ (and theoretically higher) with fidelities exceeding 0.95, using only 3 iterations ($M=3$).
• Feasibility without Giant Nonlinearity: Unlike photon blockade schemes, this method does not require extreme Kerr nonlinearity. It works with experimentally achievable nonlinear strengths found in current circuit QED systems.
• Robustness to Dissipation: The authors demonstrate that the scheme remains effective even in the presence of cavity decay, provided the ratio of decay rate to nonlinearity ($\gamma/K$) is within realistic bounds (e.g., $10^{-5}$ to $10^{-3}$).
• Platform Agnosticism: The scheme is applicable to both cavity QED (optical cavities with nonlinear media) and circuit QED (superconducting resonators).

4. Results

The authors performed extensive numerical simulations to validate the scheme:

• Fidelity Optimization:

  • For a single iteration ($M=1$), the maximum fidelity for $|N\rangle$ is limited (e.g., $\sim 0.8$ for $N=1$).
  • With $M=2$ iterations, fidelities $>0.9$ are achieved for $N$ up to 6.
  • With $M=3$ iterations, fidelities $>0.95$ are achieved for $N$ up to 20.
  • Table I in the paper provides the specific optimal values for $\beta_k$ and $\chi_k$ for various $N$.

• Phase Space Evolution:

  • The Wigner functions of the intermediate states show a transition from a Gaussian distribution (coherent state) to a ring-like structure characteristic of Fock states.
  • After 3 iterations, the probability distribution becomes highly concentrated around the target photon number $N$, with negligible probability for other numbers.

• Dissipation Analysis:

  • Using the master equation with a decay rate $\gamma$, the authors plotted fidelity against the normalized decay rate $\gamma/K$.
  • Results indicate that for $N=20$, a fidelity $>0.9$ is maintained as long as $\gamma/K \lesssim 10^{-3}$. This aligns with current experimental capabilities in circuit QED (where $\gamma/K \sim 10^{-5} - 10^{-3}$).

5. Significance and Impact

• Practical Quantum Engineering: This work provides a concrete, experimentally feasible roadmap for generating large Fock states, which are essential for encoding qubits in continuous-variable quantum computing and for quantum metrology.
• Overcoming Nonlinearity Limits: By utilizing a multi-step displacement-Kerr protocol, the authors circumvent the need for "giant" nonlinearities, making the generation of high-photon-number states accessible with current technology.
• Circuit QED Applicability: The direct mapping to superconducting circuits suggests immediate experimental realization, potentially accelerating the development of bosonic quantum error correction codes (like GKP qubits) that rely on precise Fock states.
• Future Directions: The authors note that while the current scheme uses Kerr and displacement, future work could incorporate squeezing operations to generate more complex nonclassical states, such as superpositions of number states or GKP qubits.

In conclusion, the paper presents a robust, deterministic protocol for generating large Fock states by iteratively refining a coherent state through Kerr nonlinearity and displacement, achieving high fidelity with realistic experimental parameters.