Probing excited-state dynamics of transmon ionization

This paper experimentally probes and characterizes the multiphoton ionization dynamics of transmons during strong dispersive readout, quantifying critical photon thresholds and state populations while verifying the Landau-Zener nature of the transition through pulse-shaping techniques and theoretical modeling.

The Gist

Imagine you are trying to listen to a very shy, quiet friend (a qubit) in a crowded, noisy room. To hear them clearly, you need to shout a bit louder (send a readout signal). But here's the catch: if you shout too loud, you don't just hear your friend better; you accidentally scare them into running away to the roof or jumping out the window.

In the world of quantum computing, this "running away" is called ionization. It's a major problem because it ruins the measurement and destroys the delicate quantum information.

This paper is like a detective story where the researchers figure out exactly why their friend runs away, where they go, and how to stop it. Here is the breakdown in simple terms:

1. The Problem: The "Too Loud" Shout

In quantum computers, scientists use a special type of atom-like circuit called a transmon to store information. To read what the transmon is thinking, they send a microwave signal (a "tone") into a nearby resonator (a kind of echo chamber).

• The Goal: Make the signal loud enough to get a clear answer quickly.
• The Mistake: If the signal gets too strong, the transmon absorbs so much energy that it gets "ionized." It jumps out of its safe, comfortable home (the two-level system) and flies into a chaotic, high-energy state where it can't be controlled anymore.

2. The Special Tool: The "Deep Well" Transmon

Usually, these transmons are like shallow bowls. If you push the ball (the quantum state) too hard, it rolls right over the edge.

• The Innovation: The researchers used a special, super-strong transmon (called a High-EJ/EC transmon). Think of this as a deep, reinforced well. Because the well is so deep, the ball can bounce around inside it many times without falling out.
• The Benefit: This allowed the team to see exactly what happens when the ball gets pushed hard. They could watch it climb the walls of the well, see which specific "floor" it lands on, and even watch it bounce back down.

3. The Discovery: It's a "Trapdoor" Effect

The team discovered that the transmon doesn't just randomly fly away. It happens because of a specific resonance.

• The Analogy: Imagine pushing a child on a swing. If you push at just the right rhythm, the swing goes higher and higher. In this experiment, the microwave signal acts like the pusher. At a specific number of "pushes" (photons), the energy lines up perfectly with a "trapdoor" in the energy levels of the transmon.
• The Result: Once the signal hits this specific threshold, the transmon snaps open the trapdoor and falls into a high-energy state. The researchers found they could predict exactly how many "pushes" (photons) it takes to open this door.

4. The "Slow Walk" vs. The "Fast Run" (Landau-Zener)

One of the coolest parts of the paper is how they tested how the transmon falls through the trapdoor. They used a technique called pulse shaping to control the speed of the signal.

• The Analogy: Imagine a hiker approaching a narrow bridge over a canyon.
  • Fast Walk (Diabatic): If the hiker runs across the bridge quickly, they might not notice the gap and just keep walking straight. The transmon stays safe.
  • Slow Walk (Adiabatic): If the hiker walks very slowly, they have time to feel the bridge sway and eventually step off into the canyon.
• The Finding: The researchers proved that if they increased the signal slowly, the transmon was more likely to fall into the high-energy state. If they increased it quickly, it stayed put. This confirmed that the ionization is a specific type of quantum jump known as a Landau-Zener transition.

5. The "Ghost" in the Machine (Offset Charge)

Finally, they tested a standard, weaker transmon (the kind used in most current quantum computers). They found something tricky: the "trapdoor" didn't stay in the same place.

• The Analogy: Imagine the floor of the room is shifting slightly because of invisible vibrations (called offset charge or charge noise). Sometimes the trapdoor is right in front of you; sometimes it's a few feet to the left.
• The Result: Because of these tiny vibrations, the point at which the transmon gets ionized changes over time. The researchers mapped out exactly how these invisible vibrations move the trapdoor, which helps explain why some measurements fail unpredictably.

Why Does This Matter?

This research is a roadmap for building better quantum computers.

1. Safety First: By knowing exactly how many "pushes" (photons) are safe, engineers can set a "speed limit" for reading qubits so they don't accidentally scare them away.
2. Better Design: They proved that including specific details about the transmon's internal structure (Josephson harmonics) is crucial for predicting these jumps.
3. Future Control: Understanding that "slow" signals cause more trouble than "fast" ones gives engineers a new tool to design readout pulses that are both fast and safe.

In a nutshell: The researchers built a super-deep well to watch a quantum particle get scared by a loud noise. They figured out exactly how loud the noise has to be to make it jump, proved that moving the noise slowly makes the jump more likely, and mapped out how invisible vibrations shift the danger zone. This helps us build quantum computers that are less likely to crash.

Technical Summary

Here is a detailed technical summary of the paper "Probing excited-state dynamics of transmon ionization":

1. Problem Statement

In circuit Quantum Electrodynamics (cQED), dispersive readout is the standard method for measuring superconducting qubits (e.g., transmons). While high-fidelity readout is essential for quantum error correction, it is limited by measurement-induced state transitions (MIST), often referred to as transmon ionization.

• The Issue: Strong readout drives can induce multiphoton resonances that transition the qubit from its computational subspace (ground and first excited states) to highly excited states.
• Consequences: This leads to leakage, loss of quantum information, and reduced readout fidelity.
• Knowledge Gap: While the critical photon number for ionization is theoretically predicted, key features remain unverified experimentally due to the difficulty of controlling and measuring highly excited states in typical transmons. Specifically, the final state reached, the population transfer dynamics, and the nature of the transition (e.g., Landau-Zener behavior) were not fully characterized.

2. Methodology

The authors employed a combination of advanced experimental techniques and theoretical modeling to probe these dynamics:

• Device Selection:
  • High-$E_J/E_C$ Transmons: They utilized transmons with a large ratio of Josephson energy to charging energy ($E_J/E_C \approx 235-275$). These devices have deep potentials that confine many energy levels (up to 10 resolvable states), making highly excited states accessible and insensitive to charge noise.
  • Typical Transmon: For comparison, they also used a standard transmon ($E_J/E_C \approx 55$) to study offset-charge dependence.
• Experimental Sequences:
  • Ionization Mapping: Prepared the transmon in specific eigenstates ($|0\rangle$ to $|7\rangle$) and applied a square stimulation pulse to the readout resonator. They measured the final state populations as a function of the maximum mean photon number ($\bar{n}_{r,max}$).
  • Pulse Shaping (Landau-Zener): Developed a pulse-shaping technique to control the photon number trajectory in the resonator. This allowed them to sweep the system through multiphoton resonances at variable speeds (adiabatic vs. diabatic) to test Landau-Zener physics.
  • Time-Resolved Measurements: For the typical transmon, they interleaved Ramsey experiments (to measure offset charge $n_g$) with ionization experiments over 8 hours to correlate charge noise with critical photon numbers.
• Theoretical Modeling:
  • Semiclassical Driven Model: Used a semiclassical Hamiltonian where the resonator field acts as a classical drive on the transmon.
  • Floquet Analysis: Analyzed the quasienergies of the driven system to identify avoided crossings corresponding to multiphoton resonances.
  • Josephson Harmonics: Incorporated higher-order Josephson harmonics ($E_{J8}$ model) into simulations, which proved crucial for accurately predicting transition frequencies.

3. Key Contributions

1. Direct Observation of Excited-State Dynamics: For the first time, the authors directly measured the population of highly excited states (up to $|9+\rangle$) during ionization, rather than inferring it from leakage out of the qubit subspace.
2. Identification of Final States: They identified specific final states (e.g., $|7\rangle$) resulting from ionization from $|1\rangle$ and verified this via the reverse "deionization" process ($|7\rangle \to |1\rangle$), confirming the resonant nature of the transition.
3. Verification of Landau-Zener Physics: By controlling the sweep speed of the photon number, they demonstrated that transmon ionization is a Landau-Zener-type transition. They showed that adiabatic passage through the resonance leads to significantly higher ionization probabilities than diabatic passage.
4. Offset Charge Dependence: They quantified how the critical photon number ($n_{r,crit}$) fluctuates with the offset charge ($n_g$) in typical transmons, a phenomenon often overlooked in standard readout models.
5. Model Validation: Validated the semiclassical driven transmon model and highlighted the necessity of including Josephson harmonics to accurately predict the spectrum of highly excited states.

4. Key Results

• Critical Photon Numbers: Ionization occurs at specific critical photon numbers (e.g., $\sim 880$ photons for the $|1\rangle \leftrightarrow |7\rangle$ transition in high-$E_J/E_C$ devices).
• State Transfer: Upon ionization, population is transferred to specific highly excited states (e.g., $|7\rangle$) before relaxing to lower states or ionizing further to unresolvable states ($|9+\rangle$).
• Landau-Zener Dynamics:
  • In adiabatic regimes (slow sweep), the system follows the eigenstate branch, leading to a high probability of ionization (population transfer out of the ground state).
  • In diabatic regimes (fast sweep), the system jumps across the avoided crossing, remaining in the initial state.
  • Experimental data matched theoretical predictions using the Landau-Zener formula $P_{LZ} = \exp(-\pi \Delta_{ac}^2 / 2v)$, where $v$ is the sweep speed.
• Charge Noise Sensitivity: In typical transmons ($E_J/E_C \approx 55$), $n_{r,crit}$ fluctuates significantly over time due to charge noise. The experimental time traces of $n_{r,crit}$ correlated perfectly with the measured offset charge $n_g$ and theoretical Floquet predictions.
• Model Accuracy: Simulations using the $E_{J8}$ model (including 8 Josephson harmonics) showed excellent agreement with experimental data, whereas the conventional single-harmonic model ($E_{J1}$) failed to predict the correct critical photon numbers and final states.

5. Significance

• Quantum Error Correction: Understanding and mitigating ionization is critical for achieving fault-tolerant quantum computing. This work provides the data needed to optimize readout parameters (power, duration) to stay below the ionization threshold.
• Qudit Applications: As superconducting circuits are increasingly used as high-dimensional qudits, understanding the complex resonance structure of highly excited states is vital for designing robust multi-tone readout schemes.
• Fundamental Physics: The study confirms that strongly driven nonlinear oscillators exhibit universal behaviors like Landau-Zener transitions and multiphoton ionization, bridging the gap between atomic physics and circuit QED.
• Future Directions: The findings suggest that optimizing transmon parameters (specifically $E_J/E_C$ and harmonics) and using pulse shaping to control adiabaticity can mitigate ionization errors. It also opens avenues for exploring longitudinal readout and other strong-drive scenarios in superconducting circuits.