Two-photon coupling via Josephson element II: Interaction dressing, cross-Kerr coupling, and limits of low-energy bosonic model

This paper investigates the renormalization of interactions mediated by a symmetric SQUID in a coupled phase qubit system, demonstrating that cross-Kerr coupling persists due to potential asymmetry and coupler nonlinearity, while establishing the limits of the low-energy bosonic model and providing verifiable predictions for two-photon detection and quantum-nondemolition readout applications.

The Gist

Imagine you have two musical instruments: a simple, steady drum (the resonator) and a quirky, slightly broken piano key (the phase qubit). You want them to talk to each other, but not in the usual way. Usually, if you hit the drum once, the piano key jumps once. But in this paper, the authors are trying to make a special connection where two hits on the drum are required to make the piano key jump once.

This is called two-photon coupling. It's like a bouncer at a club who only lets you in if you bring a friend; you can't get in alone.

The Magic Bridge: The SQUID

To make these two instruments talk, the authors use a special bridge made of a superconducting loop called a SQUID. Think of this SQUID as a very sensitive, adjustable door between the drum and the piano. By tweaking the magnetic field on this door, they can change how the two instruments interact.

The Problem: The "Ghost" Interactions

In the world of quantum physics, things don't just happen directly. Sometimes, invisible "ghost" steps happen in between.

• The Goal: They wanted to create a clean connection where two drum hits equal one piano jump.
• The Surprise: They found that even when they tried to set the door perfectly to block unwanted interactions, a "ghost" interaction kept sneaking in. This ghost interaction is called cross-Kerr coupling.

The Analogy: Imagine you are trying to have a private conversation with a friend (the two-photon interaction). You think you've found a soundproof room. But because your friend's voice is slightly hoarse (asymmetry in the potential) and the room has weird echoes (nonlinearity), your voice accidentally changes the pitch of their voice, even when you aren't speaking directly to them. You can't turn this off just by closing the door; the room's shape itself causes it.

The Main Discoveries

1. The Ghost Can't Be Erased
The authors discovered that this unwanted "pitch change" (cross-Kerr coupling) never completely disappears. Even if you tune the bridge perfectly to maximize the "two hits for one jump" effect, the ghost interaction remains. It gets "dressed up" or reinforced by the quirks of the system. It's like trying to stop a leak in a boat by plugging one hole, only to find the water pressure forces it out of a different, smaller crack that you can't seal.

2. How High Can the Piano Jump?
To make these calculations work, the authors treated the piano key as if it had an infinite number of keys it could jump to (a "bosonic" model). But in reality, a real piano key can only jump so high before it breaks or falls off the piano.

• They calculated exactly how many "virtual jumps" (ghost steps) the system needs to take to create these effects.
• The Result: They found that the system only needs to be able to reach about three or four high notes above its resting state for their math to be accurate. Since their specific "piano" (the rf SQUID) has about seven safe notes before it falls, their theory holds up perfectly.

3. The "Dressing" Effect
The authors explain that the strength of the connection isn't just what you see on the surface. It's "dressed" by these invisible ghost steps.

• Two-Photon Coupling: The main connection (two hits = one jump) stays very close to what you expect. The ghost steps barely change it.
• Cross-Kerr Coupling: The unwanted connection gets significantly stronger because of these ghost steps. It's like the ghost steps act as a megaphone for the unwanted noise.

Why Does This Matter? (According to the Paper)

The paper suggests two main ways to use this specific setup:

1. Detecting Pairs of Particles: Because the system is tuned to react only when two things happen at once, it could act as a detector for pairs of microwave photons (particles of light). It's like a security camera that only triggers if two people walk in together, ignoring anyone walking alone.
2. Reading a Qubit Without Breaking It: They propose using this setup to "read" the state of a quantum bit (qubit) without destroying its delicate state. By using the "ghost" pitch change (cross-Kerr coupling) while turning off the direct "knock" (linear coupling), they can listen to the qubit's state indirectly. It's like checking if a bird is in a nest by listening to the branches sway, rather than reaching in and scaring it away.

Summary

The paper is a detailed map of a very specific quantum machine. It tells us that while we can build a bridge that forces two inputs to create one output, we can't fully eliminate a side-effect where the inputs accidentally change the tone of the system. However, this side-effect is predictable, calculable, and actually useful for reading quantum information without destroying it. The authors also confirmed that their math works because the system doesn't need to jump higher than its physical limits allow.

Technical Summary

Technical Summary: Two-photon coupling via Josephson element. II. Interaction dressing, cross-Kerr coupling, and limits of low-energy bosonic model

Problem Statement
This work investigates the interactions mediated by a symmetric Superconducting Quantum Interference Device (SQUID) coupling a resonator and a phase qubit (modeled as an artificial atom in a metastable well). While previous research established the existence of a bare two-photon coupling arising directly from Josephson nonlinearity, this study addresses three critical gaps:

1. Renormalization: Do virtual transitions induced by the system's plethora of nonlinearities and interactions significantly correct the bare two-photon coupling rate?
2. Cross-Kerr Coupling: Does a cross-Kerr coupling arise in this symmetric setup, and can it be mitigated? Previous assumptions suggested it could be switched off by setting the coupler bias phase $\delta = \pi/2$, as the bare cross-Kerr term vanishes at this point.
3. Model Validity: What is the minimum number of coherent energy states required in the qubit's metastable well for the bosonic (anharmonic oscillator) approximation to accurately describe the dressing processes?

Methodology
The authors employ a systematic perturbative approach based on the Schrieffer-Wolff transformation to derive an effective Hamiltonian near the two-photon resonance ($\omega_a \approx 2\omega_r$).

• Expansion: The Josephson coupling energy is expanded to the fourth order in the phase variables of the resonator and the atom.
• Bosonic Approximation: The artificial atom is modeled as an anharmonic oscillator with a specific anharmonicity $\Xi_a$, rather than a strict two-level system, allowing for the analysis of virtual transitions climbing the energy ladder.
• Diagrammatic Analysis: The authors utilize diagrams to visualize virtual processes. These diagrams track the system's path through energy states, specifically identifying processes that climb the qubit ladder as high as possible to determine the minimum number of required metastable states.
• Wick's Theorem and Equations of Motion: To verify the origin of corrections and their quantum nature, the authors cross-check results using Wick's theorem for normal ordering and Heisenberg equations of motion (demonstrating that single-loop corrections also appear in the classical limit).

Key Contributions and Results

• Cross-Kerr Renormalization: Contrary to the expectation that cross-Kerr coupling vanishes when the bare term is zero (at $\delta = \pi/2$), the authors find that the cross-Kerr coupling never vanishes in the presence of potential asymmetry and nonlinearity.

  • The effective cross-Kerr rate is renormalized as $K_{0,X} = K_0 + 24g_2 X_a / \omega_{p,a}$.
  • This correction arises from virtual processes where the optomechanical interaction ($g_2$) chains with the atom's cubic nonlinearity ($X_a$).
  • Numerical estimates for a realistic rf SQUID phase qubit show this correction can reach approximately 15 MHz, which is significant compared to typical coupling rates. This dressing persists even when the bare inductive coupling is tuned to zero.

• Two-Photon Coupling Stability: The study finds that the two-photon coupling rate ($\tilde{g}_2$) acquires only small corrections relative to its bare value ($g_2$).

  • While higher-order virtual processes exist, their amplitudes are suppressed under reasonable conditions (specifically when the linear coupling is below the ultrastrong regime and zero-point fluctuations are comparable).
  • The authors conclude that the bare two-photon coupling is a robust approximation for this system, validating the findings of their preceding work.

• Limits of the Bosonic Model: By analyzing the virtual processes required for the dressing corrections, the authors determine the minimum number of energy states needed for the theory to hold.

  • The dominant cross-Kerr correction requires at least three coherent energy states.
  • Other corrections require at most four states.
  • For the specific rf SQUID parameters analyzed (which possess seven metastable states), the bosonic approximation is deemed valid. The authors note that if the system were driven beyond these states (e.g., into the true ground state of the double-well potential), the model would break down.

• Linear Interaction Dressing: The linear inductive coupling is also renormalized by the atom's cubic nonlinearity. This allows for the precise tuning of the coupler bias to switch off single-photon interactions while retaining the cross-Kerr interaction, a feature useful for specific readout protocols.

Significance and Applications
The paper claims that these findings are crucial for the design and application of superconducting circuits involving two-photon interactions:

• Two-Photon Detection: The system can function as a Josephson photomultiplier for detecting photon pairs. The estimated two-photon coupling rate ($\sim 30$ MHz) suggests a detection time of $\sim 10$ ns.
• Quantum-Nondemolition (QND) Readout: The ability to suppress single-photon coupling while maintaining a strong, dressed cross-Kerr interaction enables high-fidelity QND readout of qubits with asymmetric potentials. The authors estimate a readout fidelity of over 99.9% within 30 ns for a system with 30% quantum efficiency.
• Theoretical Validity: The work provides a quantitative assessment of the bosonic approximation's limits, offering a recipe for determining the necessary number of energy levels for accurate modeling in similar circuit QED systems.

The authors emphasize that while the cross-Kerr coupling is unavoidable due to asymmetry and nonlinearity, this "dressing" can be leveraged for applications where such coupling is required, rather than viewed solely as a detrimental effect. The theory is presented as applicable to systems where the qubit remains within its metastable well, with specific numerical estimates provided for an rf SQUID phase qubit.