Preparation of conditionally-squeezed states in qubit-oscillator systems

This paper proposes a protocol for generating superpositions of orthogonally squeezed states in a qubit-oscillator system via quadratic coupling, evaluates its robustness against decoherence, and demonstrates its potential application in a new quantum error-correcting code.

The Gist

Here is an explanation of the paper, translated into everyday language using analogies to make the complex quantum physics concepts accessible.

The Big Picture: Tuning a Quantum Guitar String

Imagine you have a tiny, invisible guitar string (a quantum oscillator) that vibrates in a room. In the quantum world, this string doesn't just vibrate normally; it can exist in strange, "squeezed" states where its uncertainty is shifted. Think of it like a balloon: usually, it's round, but if you squeeze it, it becomes long and thin in one direction and fat in another.

The scientists in this paper figured out a new way to "squeeze" this quantum string into very specific shapes and, even cooler, to create a superposition of two different squeezed shapes at the same time. It's like having a balloon that is simultaneously long-and-thin and short-and-fat, existing in two realities at once.

Here is how they did it, broken down step-by-step:

1. The Setup: The String and the Conductor

• The Oscillator (The String): This is the thing they want to manipulate. In real experiments, this could be a tiny mechanical drum or a vibrating beam.
• The Qubit (The Conductor): This is a tiny quantum switch (like a light switch that can be both ON and OFF at the same time).
• The Connection: The paper focuses on a system where the Conductor and the String are connected by a quadratic coupling.
  • Analogy: Imagine the Conductor is a drummer, and the String is a drum. In normal setups, the drummer hits the drum directly. In this setup, the drummer doesn't hit the drum; instead, they change the tension of the drum skin based on whether they are standing up or sitting down. This change in tension "squeezes" the drum's vibration.

2. The Protocol: The "Squeeze" Dance

The researchers proposed a specific routine to get the string into the desired state:

1. The Setup: They start with the string perfectly still (the "ground state") and the Conductor in a special "superposition" state (halfway between ON and OFF).
2. The Drive: They tap the Conductor with a specific rhythm (a microwave signal). This is like the Conductor doing a specific dance move that changes the drum's tension.
3. The Magic Moment: Because the Conductor is in a superposition, the drum gets squeezed in two different directions simultaneously.
   • If the Conductor is "ON," the drum gets squeezed horizontally.
   • If the Conductor is "OFF," the drum gets squeezed vertically.
   • Since the Conductor is both, the drum is squeezed in a superposition of horizontal and vertical.
4. The Measurement: Finally, they check the Conductor's state. Based on what they find, the drum collapses into a specific, beautiful pattern of "squeezed" energy.

3. The Result: "Squeezed Cats"

You might have heard of Schrödinger's Cat, a famous thought experiment where a cat is both dead and alive. In quantum physics, scientists often try to make "Cat States" (superpositions of two very different things).

This paper creates "Squeezed Cat States."

• Analogy: Instead of a cat that is dead and alive, imagine a cat that is simultaneously stretched out like a rubber band and squished like a pancake.
• These states are special because they are very robust. They can hold onto their quantum information even when the environment tries to mess them up.

4. Why Does This Matter? (The Error-Correcting Code)

The most exciting part of the paper is the application: Quantum Error Correction.

• The Problem: Quantum computers are fragile. A tiny bit of heat or noise (like a sneeze in the room) can ruin the calculation.
• The Solution: The scientists propose using these "Squeezed Cat States" as a new way to store data.
  • Analogy: Imagine you are trying to send a message across a stormy ocean. If you send a single small boat (a normal bit), a wave will flip it over. But if you send a massive, double-hulled ship (the Squeezed Cat State), the waves might rock it, but it won't sink.
• How it works: Because the "Squeezed Cat" states have a very specific mathematical symmetry, the computer can detect if a wave (error) hit the ship and fix it automatically without needing to look at the message directly (which would destroy the quantum state).

5. Will It Work in Real Life?

The authors ran computer simulations to see if this is possible with real hardware.

• The Good News: They found that if the "Conductor" (the qubit) is high-quality and doesn't get too noisy, the "Squeezed Cat" states can be created with high accuracy.
• The Challenge: The main enemy is decoherence (noise). If the Conductor gets tired or confused (decoheres) too quickly, the magic connection breaks, and the drum stops vibrating in the special pattern.
• The Verdict: They believe current technology (using superconducting circuits and mechanical oscillators) is just good enough to try this experiment.

Summary

This paper is a recipe for a new type of quantum magic trick. By using a quantum switch to "squeeze" a vibrating string in two directions at once, the scientists can create a super-stable state of matter. This state acts like a fortress against errors, potentially helping us build the first truly reliable quantum computers that can survive the noisy real world.

Technical Summary

Here is a detailed technical summary of the paper "Preparation of conditionally-squeezed states in qubit-oscillator systems" by Hope, Lidal, and Massel.

1. Problem Statement

The paper addresses the challenge of generating specific non-classical quantum states in bosonic systems, which are crucial for Bosonic Quantum Error Correction (bQEC). While significant progress has been made in preparing "cat states" (superpositions of coherent states) and GKP states in superconducting circuits, there is a need for alternative encoding schemes that leverage different quantum resources. Specifically, the authors aim to generate superpositions of orthogonally squeezed states (analogous to cat states but in the squeezing domain) in a system where a qubit is coupled to a harmonic oscillator via a quadratic interaction.

2. Methodology

System Model

The authors propose a system consisting of a two-level system (qubit) coupled to a harmonic oscillator (mechanical or microwave mode) with a quadratic coupling. The Hamiltonian is given by:
$$H(t) = \frac{\omega_q}{2}\sigma_z + \omega_m b^\dagger b + g(b + b^\dagger)^2\sigma_z + H_d(t)$$
Where:

• $\omega_q, \omega_m$: Qubit and oscillator frequencies.
• $g$: Quadratic coupling strength.
• $H_d(t)$: An external drive applied to the qubit.

Protocol Design

The protocol relies on conditional squeezing driven by an external qubit drive.

1. Drive Configuration: The qubit is driven with a harmonic field at frequency $\omega_d \approx \omega_m$.
2. Rotating Wave Approximation (RWA): By transforming to a rotating frame and applying RWA (assuming $\omega_m \gg g$), the effective Hamiltonian contains terms dependent on Bessel functions of the drive amplitude $\bar{A} = 2A/\omega_d$.
3. Condition for Pure Squeezing: The authors identify a specific drive amplitude where the first-order Bessel function vanishes ($J_0(\bar{A}) = 0$, specifically $\bar{A} \approx 2.405$). This eliminates the photon-number-dependent qubit splitting term, leaving a pure conditional squeezing Hamiltonian:
   $$H_{cs} = g_{cs}(b^2 + b^{\dagger 2})\sigma_z$$
   where $g_{cs} = g J_2(\bar{A})$.

State Preparation Steps

1. Initialization: The system starts in the product state $|0\rangle_{osc} \otimes (|e\rangle + |g\rangle)_q$ (oscillator ground state, qubit in a superposition of $\sigma_z$ eigenstates).
2. Entanglement: Under $H_{cs}$, the qubit state dictates the squeezing direction of the oscillator. The state evolves into an entangled superposition:
   $$|\psi(t)\rangle \propto S(\xi)|0\rangle \otimes |e\rangle + S(-\xi)|0\rangle \otimes |g\rangle$$
   Here, $S(\xi)$ and $S(-\xi)$ represent squeezing along orthogonal axes in phase space.
3. Measurement: A projective measurement of the qubit in the $\sigma_x$ basis collapses the oscillator into a symmetric or antisymmetric superposition of orthogonally squeezed vacuum states.

3. Key Contributions

• Novel Protocol: A deterministic method to generate superpositions of orthogonally squeezed states using quadratic coupling, distinct from linear coupling methods used for cat states.
• New Quantum Code: The proposal of a Bosonic Rotation Code based on these squeezed superpositions. The logical states are defined as:
  $$|0_L\rangle \propto [S(\xi) + S(-\xi)]|0\rangle, \quad |1_L\rangle \propto [S(\xi) - S(-\xi)]|0\rangle$$
  This code exhibits a 4-fold symmetry in phase space (similar to a 4-component cat code) and belongs to the class of approximate number-phase codes.
• Error Correction Analysis: The authors demonstrate that in the limit of large squeezing, the code satisfies the Knill-Laflamme (KL) conditions for correcting single boson loss and dephasing errors. The logical states become indistinguishable under error operators as squeezing increases, analogous to how large-displacement cat codes work.
• Robustness Analysis: Numerical evaluation of the protocol's fidelity under realistic experimental conditions, including qubit decoherence ($\gamma_1, \gamma_\phi$) and mechanical thermal noise.

4. Results

• State Fidelity: Numerical simulations show that high-fidelity preparation of the target states is achievable. The fidelity is sensitive to the drive amplitude (must satisfy the $J_0=0$ condition) and the ratio of coupling strength to oscillator frequency.
• Open System Dynamics:
  • The protocol was tested with parameters mimicking recent experiments (e.g., $\omega_m = 1$ GHz, $g = 100$ kHz).
  • Decoherence Limits: The fidelity is primarily limited by qubit decoherence. Successful state preparation requires qubit decay and dephasing rates ($\gamma_1, \gamma_\phi$) to be less than or comparable to the coupling strength ($g$).
  • Thermal Noise: With a mechanical thermal occupation of $n_{m,th} \approx 1$ (at 10 mK), the protocol remains robust, provided the qubit coherence is maintained.
• Code Performance: The ratio of expectation values $\langle (b^\dagger b)^p \rangle$ between logical states approaches 1 as the squeezing parameter $r$ increases, confirming the code's ability to correct errors in the high-squeezing limit.

5. Significance

• Expansion of bQEC: This work expands the toolbox for bosonic quantum error correction beyond cat and GKP codes. It introduces a "squeezed" analogue to cat states, offering a new encoding paradigm that may be advantageous in specific noise environments or hardware architectures.
• Experimental Feasibility: The protocol is directly applicable to existing and emerging platforms, specifically superconducting circuits coupled to mechanical oscillators (quantum acoustics) or nanopillars, where strong quadratic coupling has recently been demonstrated.
• Theoretical Insight: It bridges the gap between conditional displacement (used for cat states) and conditional squeezing, providing a unified theoretical framework for generating non-Gaussian states in qubit-oscillator systems.
• Path to Fault Tolerance: By demonstrating that these states can form a valid error-correcting code, the paper contributes to the broader goal of achieving fault-tolerant quantum computing using continuous-variable systems.

In summary, the paper presents a theoretically sound and experimentally viable protocol for generating complex non-Gaussian states in hybrid quantum systems, proposing a new avenue for robust quantum information storage and processing.