Deterministic generation of cat states with more than $100$ photons under dissipation

This paper proposes a deterministic protocol based on universal quantum control theory and dynamical invariants to generate large cat states with over 120 mean photons in hybrid qubit-bosonic systems, achieving perfect fidelity in the Hermitian case and over 0.962 fidelity under non-Hermitian dissipation.

The Gist

Imagine you are trying to bake the perfect, giant "quantum cake." In the world of quantum physics, this cake is called a Schrödinger's cat state. Just like the famous thought experiment where a cat is both dead and alive at the same time, this quantum cake is a superposition of two very different states existing simultaneously.

The problem is that these cakes are incredibly fragile. The bigger you try to make them (adding more "photons," or particles of light), the more likely they are to crumble before you can serve them. Usually, if you try to make a giant one, the environment (noise, heat, loss) ruins it, and you end up with a tiny, messy crumb instead of a cake.

This paper by Zhu-yao Jin and Jun Jing proposes a new, foolproof recipe to bake these giant quantum cakes deterministically—meaning it works every time, not just by luck.

The Secret Ingredient: The "Dynamical Invariant"

The authors use a mathematical tool called a dynamical invariant. Think of this as a "magic compass" or a "GPS track" that the system must follow.

Usually, when you try to steer a quantum system, it's like trying to drive a car on a bumpy road while blindfolded; you might drift off course. But this "invariant" is like a train on a track. No matter how the engine (the energy source) changes speed or direction, the train must stay on the rails. The authors designed a specific set of rules (a Hamiltonian) that forces the quantum system to stay on this perfect track, guiding it from a simple starting point (an empty vacuum) straight to a giant cat state without ever getting lost or crumbling.

The Recipe: A Two-Part System

To bake this cake, they use a hybrid kitchen with two main ingredients:

1. A Qubit (The Chef): A tiny two-state system (like a switch that is either On or Off).
2. A Bosonic Mode (The Mixing Bowl): A container that holds the light particles (photons).

The Chef (qubit) is connected to the Mixing Bowl (bosonic mode). The magic happens because the Chef is in a special state where it is "both On and Off" at the same time. Because of this, the Mixing Bowl is forced to follow two different paths simultaneously, creating the giant superposition (the cat state).

The Two Scenarios: Perfect Kitchen vs. Leaky Kitchen

1. The Perfect Kitchen (Hermitian Case)
In an ideal world with no noise or energy loss, the authors show that their recipe creates a perfect cat state.

• The Result: They successfully grew a cat state with an average of 120 photons.
• The Quality: The fidelity (how perfect the cake is) is 100%. It is exactly what they intended.

2. The Leaky Kitchen (Non-Hermitian Case)
In the real world, things leak. Energy escapes, or sometimes extra energy accidentally gets in (gain). This usually destroys the quantum state.

• The Trick: The authors realized they could use this "leakiness" to their advantage. They split the baking process into two stages.
  • Stage 1: They let the system "lose" energy (like letting steam escape).
  • Stage 2: They switch to "gain" (adding energy back in).
• The Result: By carefully timing this switch, they cancel out the errors. Even with the leaks, they managed to bake a cat state with 120 photons and a 96.2% fidelity. It's not quite perfect, but it's incredibly close, and it works deterministically.

Bigger Cakes: The "Four-Legged" Cat

The paper also shows how to make even more complex cakes.

• Intrinsic Cat States: These are like a three-way entangled cake involving the cat, the poison bottle, and the radioactive atom all at once.
• Four-Legged Cat States (Compass States): Imagine a cat that is not just "dead and alive," but also "dead-alive" and "alive-dead" in four different directions at once. The authors showed how to bake these large, four-part cakes with high fidelity as well.

Why This Matters

The paper claims this is a major step forward because:

1. Size: They broke the barrier of 100 photons, which is huge for these types of states.
2. Reliability: They did it "deterministically." Previous methods were often like rolling dice—you might get a big cat state once in a while, but mostly you get nothing. This method works every time.
3. Versatility: The same "magic compass" (dynamical invariant) works whether the system is perfect or leaking, and it can make different shapes of quantum cakes (two-legged, four-legged, etc.).

In short, the authors have built a robust, automated assembly line that can reliably manufacture massive, complex quantum states, overcoming the usual fragility that has made this difficult for so long.

Technical Summary

Technical Summary: Deterministic Generation of Cat States with More Than 100 Photons Under Dissipation

Problem Statement
Large-size Schrödinger cat states are fundamental resources for exploring the quantum-to-classical transition, quantum metrology, and fault-tolerant quantum computation. However, generating these states with large amplitudes remains a significant challenge due to their increasing fragility under decoherence. Existing methods face specific limitations:

• Atomic systems: Preparation is hindered by systematic errors and dissipation on individual atoms, resulting in low fidelities (e.g., ~0.54 for 20 Rydberg atoms).
• Bosonic systems: While more versatile, current protocols struggle to achieve large amplitudes ($|\alpha|$) deterministically.
  • Adiabatic protocols are limited by prolonged environmental exposure.
  • Shortcuts to adiabaticity (STA) often require extreme squeezing levels (~15 dB) or yield limited amplitudes ($|\alpha| < 2$).
  • Gate-based deterministic methods have demonstrated amplitudes around $|\alpha| \sim \sqrt{7}$ with fidelities ~0.91.
  • Probabilistic methods (e.g., photon subtraction) can predict high fidelities and large amplitudes but suffer from low success probabilities (~0.02).
  • Steady-state stabilization requires specific two-photon loss dominance, which is difficult to engineer.

There is a need for a unified theoretical framework capable of deterministically generating large-size cat states in both closed and open quantum systems with accessible control.

Methodology
The authors propose a protocol based on Universal Quantum Control (UQC) theory, utilizing dynamical invariants to control the dynamics of hybrid qubit-bosonic systems. The approach involves:

1. System Model: A hybrid system where a qubit is longitudinally coupled to a bosonic mode. The system can be closed (Hermitian) or open (non-Hermitian, derived from the Lindblad master equation under postselection).
2. Ancillary Picture Transformation: The system dynamics are analyzed in an ancillary representation via a time-dependent unitary transformation, $V(t)$, conditional on the qubit state. This transformation maps the original Hamiltonian $H(t)$ to a rotated Hamiltonian $H_{rot}(t)$.
3. Dynamical Invariants: The protocol constructs stationary dynamical invariants, $J(0)$, which satisfy the Heisenberg equation with the rotated Hamiltonian. This imposes specific constraints on the time-dependent parameters of the original Hamiltonian (eigenfrequencies, coupling strengths, and phases).
4. Evolution Control: By ensuring the system evolves according to these invariants, the bosonic mode can be driven deterministically from the vacuum state to a target cat state.
   • Hermitian Case: The constraints ensure perfect evolution to the target state.
   • Non-Hermitian Case (Open Systems): The protocol accounts for gain and loss rates. To preserve probability conservation at the target time, the evolution is divided into two stages: one dominated by gain and the other by loss. The transition between stages is managed by tuning a phase parameter $\beta_i(t)$ to ensure the continuity of the dynamical invariant, effectively normalizing the wavefunction at the final time.

Key Contributions

• Unified Framework: The paper extends UQC theory to hybrid discrete-continuous variable systems, providing a method to generate large cat states in both Hermitian (closed) and non-Hermitian (open) regimes.
• Deterministic Generation: Unlike probabilistic methods, this protocol offers deterministic generation of cat states.
• Scalability: The method is not limited by the ratio of driving strength to nonlinearity in the same way as STA protocols, allowing for the generation of ultra-large amplitudes.
• Versatility: The protocol is generalized to generate not only standard two-component cat states but also "intrinsic" cat states (entangling the bosonic mode with two qubits) and four-legged (compass) cat states.

Results
Numerical simulations based on the proposed protocol demonstrate the following:

• Closed Systems (Hermitian): Starting from a vacuum state and a qubit in a balanced superposition, the system evolves to a cat state with unit fidelity ($F=1$) and a mean photon number $\bar{n} \sim 120$ (amplitude $|\alpha| \approx 3.5\pi$).
• Open Systems (Non-Hermitian): Under dissipation, the protocol achieves a cat state with a mean photon number $\bar{n} \sim 120$ and a fidelity $F > 0.962$. The trace of the density matrix is restored to unity at the target time through the two-stage gain/loss switching.
• Intrinsic and Four-Legged States:
  • Intrinsic Cat States: Generated with close-to-unit fidelity and $\bar{n} > 100$.
  • Four-Legged Cat States: The protocol successfully generates four-component cat states ($|\alpha\rangle + |-\alpha\rangle + |i\alpha\rangle + |-i\alpha\rangle$) with a fidelity $F > 0.965$ and a mean photon number $\bar{n} \sim 95$.

Significance
The paper claims that this work essentially advances Universal Quantum Control to hybrid discrete-continuous variable systems. By utilizing dynamical invariants, the protocol overcomes the fragility of large cat states under decoherence, enabling the deterministic preparation of macroscopic quantum states with record-high fidelities and amplitudes (over 100 photons). This provides a significant pathway for creating the macroscopic quantum states necessary for fault-tolerant quantum computation and high-precision quantum metrology. The model is presented as directly realizable or implementable via displacement operations in various experimental platforms, including circuit QED and trapped-ion systems.