Measuring coherent dynamics of a superconducting qubit in an open waveguide

This paper presents a resonator-free experimental method using a two-pulse technique to measure the relaxation, decoherence, and Rabi oscillations of a superconducting transmon qubit coupled to an open waveguide, demonstrating consistency with frequency-domain measurements.

The Gist

Imagine you are trying to listen to a very shy, tiny musician (a superconducting qubit) playing a violin in a giant, empty concert hall (an open waveguide).

Usually, scientists study these musicians by putting them in a small, echoey room (a resonator). The room amplifies the sound, making it easy to hear if the musician is playing the right note or if they are getting tired. But in this new experiment, the musicians are playing in the open hall with no walls to bounce the sound off. This makes them much harder to hear and study.

Here is how the researchers solved this problem, explained simply:

1. The Problem: The "Echo" is Missing

In the old way (using a resonator), the room acts like a microphone that catches every little vibration. Without the room, the sound just flies away. The researchers needed a way to "listen" to the musician without building a room around them. They needed to know two things:

• Relaxation: How fast does the musician stop playing and go to sleep?
• Decoherence: How fast does the musician start playing out of tune or get confused?

2. The Solution: The "Two-Pulse" Detective Trick

Since they couldn't just listen to the musician directly, they invented a clever two-step trick, like a detective interrogating a suspect.

Step 1: The "Wake-Up" Call (The Excitation Pulse)
First, they send a short, specific microwave "tap" to the musician. This tap is tuned to the musician's favorite note. It wakes the musician up and gets them to start playing (moving from the "ground state" to the "first excited state").

Step 2: The "Probe" (The Readout Pulse)
This is the genius part. Instead of trying to listen to the first note (which is hard to hear in the open hall), they send a second tap. This second tap is tuned to a different note—the one the musician would play if they were really excited (the "second excited state").

• If the musician is asleep (ground state): The second tap does nothing. The signal passes through the hall unchanged.
• If the musician is awake (first excited state): The second tap makes the musician react strongly, changing the sound of the signal passing through the hall.

By checking if the second tap caused a reaction, they can tell if the musician was awake or asleep. It's like asking a question that only a person who just woke up would answer.

3. The "Chevron" Dance

The researchers didn't just do this once; they did it over and over, changing the length and strength of the "wake-up" tap.

When they plotted the results, they saw a beautiful pattern that looked like a chevron (a V-shape).

• Think of this like a dance floor. If you push the dancer just right, they spin perfectly. If you push too hard or too long, they get dizzy and stop spinning.
• The "V" shape shows exactly how long the musician can keep dancing before they get tired (decoherence) or fall asleep (relaxation).

4. Why This Matters

This method is like upgrading from a grainy black-and-white photo to a high-definition video.

• Old way: You needed a big, expensive room (resonator) to study the qubit.
• New way: You can study the qubit in the "open field." This is crucial for the future of Quantum Internet.

Imagine the future internet. We want to send quantum information (like secret messages) flying through cables (waveguides) between cities. To do that, we need to control these flying "photons" (particles of light) using qubits. But to control them, we need to know exactly how long the qubit stays "awake" and "in tune" while it's out in the open, not stuck in a box.

The Bottom Line

The researchers successfully built a way to measure how long a superconducting qubit stays "alive" and "coherent" while it's floating in an open wire, without needing a fancy box to hold it. They proved that even without a resonator, we can still perfectly track the qubit's heartbeat, which is a huge step toward building real-world quantum networks.

Technical Summary

1. Problem Statement

The field of quantum information processing increasingly utilizes superconducting circuits coupled to open waveguides (resonator-free settings) to generate, detect, and manipulate itinerant microwave photons. This architecture is essential for quantum networks and routers. However, standard characterization techniques used in Circuit Quantum Electrodynamics (cQED)—which rely on dispersive readout via a resonator—are inapplicable in these open systems due to the absence of a cavity.

The core challenge addressed in this work is the lack of a direct, reliable method to measure relaxation ($T_1$) and dephasing ($T_\phi$) rates, as well as to verify the occupation of the first excited state, in a resonator-free transmon qubit coupled to an open coplanar waveguide.

2. Methodology

The authors propose and demonstrate a two-pulse time-domain technique that exploits the specific anharmonicity properties of transmon qubits.

• System Setup: A flux-tunable transmon qubit is capacitively coupled to an open coplanar waveguide. The sample is fabricated using Manhattan-type technology with aluminum air bridges to suppress parasitic modes and is operated at 10 mK.
• Physical Principle: Unlike flux qubits with large anharmonicity (where the $|1\rangle \to |2\rangle$ transition is often inaccessible), transmon qubits have low anharmonicity. This results in closely spaced transition frequencies ($\omega_{01}$ and $\omega_{12}$), allowing both to be probed using standard microwave equipment.
• The Two-Pulse Protocol:
  1. Excitation Pulse: A pulse at frequency $\omega_{01}$ (close to the qubit eigenfrequency) is applied to excite the qubit from the ground state $|0\rangle$ to the first excited state $|1\rangle$.
  2. Readout Pulse: A second pulse at frequency $\omega_{12}$ probes the transition between the first and second excited states ($|1\rangle \to |2\rangle$).
  • Mechanism: The scattering (transmission) of the readout pulse at $\omega_{12}$ is strongly dependent on the population of the $|1\rangle$ state. If the qubit is in $|1\rangle$, the readout pulse induces a transition to $|2\rangle$, altering the transmission amplitude and phase. This encodes the state of the qubit into the readout signal.
• Measurement Modes:
  • Steady-State Spectroscopy: Used to characterize the qubit parameters via transmission dips at $\omega_{01}$ and $\omega_{02}/2$ (two-photon transition).
  • Time-Domain Dynamics: Used to measure Rabi oscillations and relaxation by varying pulse durations and delays.

3. Key Contributions

• Novel Readout Scheme: The paper introduces a method to separate the drive and readout pulses, utilizing the $\omega_{12}$ transition for readout. This allows the system to operate in the weak-drive limit for state manipulation while optimizing the readout pulse parameters independently.
• Direct Parameter Extraction: The authors successfully extract radiative loss rates ($\Gamma_{10}$), non-radiative relaxation rates ($\Gamma_l$), and pure dephasing rates ($\Gamma_\phi$) without a resonator.
• Model Validation: The study provides a rigorous comparison between two fitting models for the steady-state transmission:
  1. Resonator Fit: Treating the qubit-waveguide coupling as a notch-type resonator.
  2. Qubit Fit: Using a direct equation relating transmission to qubit parameters.
     The consistency between these models and the time-domain results validates the theoretical framework for open waveguide systems.

4. Results

The authors successfully reconstructed the coherent dynamics of the transmon qubit and extracted the following parameters (normalized to $2\pi$):

• Transition Frequencies:
  • $\omega_{01} \approx 4.49$ GHz
  • $\omega_{12} \approx 4.337$ GHz
  • Anharmonicity ($\alpha$): $-153$ MHz.
• Decoherence and Relaxation Rates:
  • Radiative Loss ($\Gamma_{10}$): $\approx 2.2$ MHz (derived from spectral fits).
  • Non-radiative Loss ($\Gamma_l$): $\approx 0.31$ MHz.
  • Pure Dephasing ($\Gamma_\phi$): $\approx 1.28 - 1.38$ MHz.
  • Total Relaxation Rate ($\Gamma_1 = \Gamma_{10} + \Gamma_l$): $\approx 2.5 - 2.7$ MHz, corresponding to a $T_1$ time of 58.86 ns.
  • Total Decoherence Rate ($\gamma_{10}$): $\approx 2.54 - 2.73$ MHz.
• Rabi Dynamics:
  • Rabi oscillations were observed with a decay rate $\Gamma_{Rabi} \approx 2.04$ MHz, corresponding to a coherence time $T_{Rabi} \approx 77.92$ ns.
  • The Rabi drive rate ($\Omega_p$) was calibrated to $2\pi \cdot 22.47$ MHz.
• Consistency: The parameters derived from time-domain pulse experiments showed excellent agreement with those estimated from frequency-domain spectroscopy, confirming the validity of the weak-drive approximation used in the spectral analysis.

5. Significance

• Enabling Open Quantum Systems: This work provides a critical toolkit for characterizing superconducting qubits in open waveguides, which is a prerequisite for developing quantum routers, single-photon sources, and distributed quantum networks.
• Flexibility and Control: By decoupling the drive and readout frequencies, the method offers greater flexibility. It allows researchers to manipulate the qubit state under weak-drive conditions (minimizing unwanted heating or AC-Stark shifts) while using optimized, potentially stronger pulses for readout.
• Scalability: The technique relies on standard microwave components and does not require complex resonator structures, making it highly scalable for future quantum processors and networks that operate in the itinerant photon regime.
• Fundamental Insight: The successful separation of radiative and non-radiative loss mechanisms in an open system deepens the understanding of how qubits interact with their electromagnetic environment, which is vital for improving qubit coherence times.

In conclusion, Sultanov et al. demonstrate a robust, versatile protocol for measuring the coherent dynamics of transmon qubits in resonator-free environments, bridging the gap between theoretical models of open quantum systems and experimental reality.