In-situ Characterization of Light-Matter Coupling in Multimode Circuit-QED Systems

This paper presents a general measurement protocol that enables the in-situ characterization of light-matter coupling to individual photonic modes in complex multimode circuit-QED systems by leveraging AC-Stark and Kerr effects, thereby eliminating the need for single-photon resolution or calibration, and validates the method using a superconducting transmon qubit coupled to a microwave resonator lattice.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: Finding the "Volume" of Invisible Connections

Imagine you are in a giant, echoey concert hall (the multimode cavity) filled with hundreds of different musical instruments (the photon modes). You have a single, very sensitive microphone (the qubit) that can hear these instruments.

The scientists want to know exactly how loud each instrument is when it talks to the microphone. In physics terms, they want to measure the coupling strength ($g$).

The Problem:
Usually, to measure how loud an instrument is, you need to count exactly how many sound waves hit the microphone. But in this quantum concert hall, the sound is so faint and the room so complex that counting individual "sound packets" (photons) is incredibly difficult. It's like trying to count individual raindrops in a hurricane while wearing blindfolded goggles. You also don't know how much the sound gets muffled by the walls (insertion loss) or how sensitive your microphone really is.

The Solution:
The team developed a clever trick. Instead of trying to count the raindrops, they listen to how the rain changes the shape of the microphone and how the rain changes the shape of the other instruments. By comparing these two changes, they can figure out the volume of the connection without ever needing to count a single drop.

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The Analogy: The "Rubber Band" and the "Echo"

To understand their method, let's use an analogy involving a trampoline (the qubit) and two springs (the photon modes).

1. The Setup

• The Trampoline (Qubit): A superconducting qubit that bounces up and down at a specific rhythm.
• Spring A (Drive Mode): A spring you can push and pull with your hand.
• Spring B (Monitor Mode): A second spring nearby that you can't touch directly, but you can watch.
• The Connection: Both springs are connected to the trampoline by invisible rubber bands. The strength of these bands is what the scientists want to measure.

2. The Two Effects

When you push Spring A (the Drive), two things happen to the trampoline and the springs:

• Effect 1: The AC-Stark Shift (The "Heavy Blanket")
  Imagine you put a heavy blanket on the trampoline. It doesn't change the blanket's shape, but it makes the trampoline bounce slower.

  • In the lab: Pushing Spring A changes the frequency (pitch) of the qubit. The harder you push (more power), the slower the qubit bounces.
  • The Catch: You don't know exactly how much "blanket weight" (photon number) you added because you don't know how efficiently your hand pushes the spring.

• Effect 2: The Kerr Shift (The "Stiffening Spring")
  Imagine that when you push Spring A, the spring itself gets stiffer or looser, changing its own rhythm. Also, because Spring A is connected to Spring B through the trampoline, Spring B also changes its rhythm slightly.

  • In the lab: Pushing Spring A changes the frequency of Spring A itself (Self-Kerr) and the frequency of Spring B (Cross-Kerr).

3. The Magic Trick: Canceling the Unknown

Here is the genius part of the paper.

Usually, to find the strength of the rubber band, you need to know:
$$\text{Shift} = \text{Band Strength} \times \text{How Hard You Pushed}$$

You know the Shift (you measured the frequency change).
You don't know How Hard You Pushed (the exact number of photons).

But wait! You measured two shifts caused by the same push:

1. The shift on the Trampoline (AC-Stark).
2. The shift on the Springs (Kerr).

If you take the ratio of these two shifts, the "How Hard You Pushed" part cancels out!
$$\frac{\text{Shift on Trampoline}}{\text{Shift on Spring}} = \frac{\text{Band Strength} \times \text{Push}}{\text{Band Strength} \times \text{Push}} = \text{Just the Band Strength}$$

It's like if you didn't know how heavy a bag of apples was, but you knew that:

• The bag made a scale tip 10 degrees.
• The bag made a spring stretch 2 inches.
• You knew the physics of the scale and the spring perfectly.
• By comparing the 10 degrees to the 2 inches, you could calculate the weight of the apples without ever needing to weigh them directly.

Why This Matters

1. No "Single-Photon" Glasses Needed:
Old methods required the scientists to see individual photons (like needing night-vision goggles to count fireflies). This new method works even if the light is too dim to see individual particles.

2. It Works in a Crowded Room:
In a multimode system, there are dozens of frequencies overlapping. It's like trying to hear one violin in an orchestra. This method allows them to pick out the specific "violin" (mode) they are interested in, even if it's a "flat" mode that doesn't transmit sound well to the outside world. They use a "loud" mode as a microphone to listen to the "quiet" mode.

3. It's Universal:
The paper shows this works for superconducting circuits (artificial atoms), but the math applies to real atoms, phonons (vibrations), and even light in optical fibers. It's a universal translator for light-matter interactions.

The Result

The team tested this on a "lattice" of 54 different modes. They picked three modes, measured them in every possible combination (A driving B, B driving A, etc.), and found that the calculated connection strengths were identical regardless of which pair they used. They also proved the method works even when they changed the "tuning" of the qubit.

In short: They invented a way to measure how strongly light and matter talk to each other by listening to how they change each other's "voices," completely bypassing the need to count the individual words (photons) being spoken.

Technical Summary

Here is a detailed technical summary of the paper "In-situ Characterization of Light-Matter Coupling in Multimode Circuit-QED Systems."

1. Problem Statement

Multimode cavity Quantum Electrodynamics (cQED) systems are powerful platforms for exploring complex quantum phenomena, such as phase transitions and quantum spin models. However, a significant bottleneck in utilizing these systems is the systematic characterization of light-matter coupling strengths ($g$) between individual atom-like degrees of freedom (e.g., transmon qubits) and specific photonic modes.

Existing methods for determining $g$ typically rely on:

• Single-photon resolution: Requiring the detection of individual photons, which is experimentally demanding.
• Independent photon-number calibration: Necessitating precise knowledge of the intracavity photon number, which depends on hard-to-calibrate factors like input attenuation and mode matching.
• Vacuum Rabi splitting: Only applicable when light-matter coupling is extremely strong or modes are well-separated.

In highly multimode systems (e.g., resonator lattices with dozens of modes), modes often have varying coupling strengths, localization properties, and frequency splittings, making standard characterization techniques inefficient or inapplicable, particularly for localized or under-coupled modes.

2. Methodology

The authors propose a general measurement protocol that determines the coupling strength $g$ to individual modes without requiring single-photon resolution or independent photon-number calibration. The method relies on comparing two distinct nonlinear effects induced by driving a specific photonic mode: the AC-Stark shift and the Kerr shifts (self-Kerr and cross-Kerr).

Theoretical Framework

The protocol utilizes a transmon qubit coupled to two photonic modes: a Drive mode (D) and a Monitor mode (M).

1. Hamiltonian: The system is modeled using a dispersive Hamiltonian where the qubit is far detuned from the modes. The effective interaction includes AC-Stark shifts ($\tilde{\chi}$) and Kerr shifts ($\chi$).

2. Key Insight: The frequency shifts per unit of drive power ($\partial_P \omega'$) are directly measurable. While the absolute intracavity photon number ($\bar{n} = \beta P$) depends on an unknown conversion factor $\beta$ (due to losses), the ratio of frequency shifts eliminates $\beta$.

3. Extraction Equations:

   • AC-Stark Shift: The drive mode shifts the qubit frequency: $\partial_P \omega'_q \propto \beta \tilde{\chi}_D$.
   • Self-Kerr Shift: The drive mode shifts its own frequency: $\partial_P \omega'_D \propto \beta \chi_{DD}$.
   • Cross-Kerr Shift: The drive mode shifts the monitor mode frequency: $\partial_P \omega'_M \propto \beta \chi_{DM}$.

   By taking the ratios of these derivatives (e.g., $\frac{\partial_P \omega'_D}{\partial_P \omega'_q}$), the unknown $\beta$ cancels out. The remaining terms depend only on known detunings ($\Delta$), transmon anharmonicity ($\alpha$), and the unknown coupling strengths $g_D$ and $g_M$. This allows for the direct algebraic extraction of $g$.

Experimental Implementation

• Platform: A superconducting transmon qubit coupled to a 1D coplanar waveguide (CPW) resonator lattice with 54 modes.
• Technique: Two-tone spectroscopy (pump-probe).
  • Case 1 (AC-Stark): A weak drive is applied to Mode D. The shift in the qubit frequency is measured via the transmission of a probe tone through Mode D.
  • Case 2 (Kerr): A stronger drive is applied to Mode D. The self-Kerr shift of Mode D and the cross-Kerr shift of Mode M are measured simultaneously by tracking the resonance frequencies of both modes using two-tone spectroscopy.
• Validation: The protocol was tested on three different lattice modes using all six possible drive/monitor pairings. Additionally, the qubit frequency was swept over a range of detunings (400–800 MHz) to verify consistency.

3. Key Contributions

• Calibration-Free Protocol: The method eliminates the need for single-photon resolution, independent photon-number calibration, or insertion-loss calibration. It relies solely on the relative power dependence of nonlinear frequency shifts.
• Applicability to Multimode Systems: The protocol works effectively in complex environments with many modes, including those with weak transmission or strong localization (e.g., flat bands), by using an easily measurable "auxiliary" mode to characterize a difficult-to-measure mode.
• Generalizability: While demonstrated with a transmon, the framework is applicable to any atom-like system coupled to bosonic modes, provided the appropriate AC-Stark and Kerr coefficients are used.
• Theoretical Derivation: The authors provide a rigorous fourth-order Schrieffer-Wolff perturbation theory derivation for the Kerr coefficients of a transmon (including higher excited states), offering closed-form expressions for self- and cross-Kerr shifts that go beyond the standard two-level approximation.

4. Results

• Consistency Across Pairs: When measuring the coupling strengths ($g$) for three selected modes (A, B, C) using all possible drive/monitor combinations, the extracted values were highly consistent. For example, $g_A$ was measured to be $\approx 12.6 - 13.4$ MHz depending on the pair, with systematic errors dominating over statistical noise.
• Detuning Independence: By sweeping the qubit frequency from $\sim 4.2$ GHz to $\sim 4.6$ GHz, the authors demonstrated that the extracted coupling strengths remained stable (varying by only $\sim 5\%$), confirming the model's accuracy across a wide range of detunings.
• Quantitative Accuracy:
  • Drive mode coupling ($g_D$): $\approx 14.2 \pm 0.6$ MHz.
  • Monitor mode coupling ($g_M$): $\approx 13.4 \pm 0.5$ MHz.
  • The method successfully extracted these values without prior knowledge of the input attenuation or photon conversion efficiency.

5. Significance

This work provides a robust, scalable tool for the characterization of multimode quantum systems. Its significance lies in:

1. Enabling Analog Quantum Simulation: It allows researchers to accurately map the parameters of complex spin-boson models and exotic band structures in resonator lattices, which is a prerequisite for reliable quantum simulation.
2. Characterizing "Dark" Modes: It enables the measurement of coupling strengths for modes that are difficult to access directly (e.g., localized flat-band modes or phononic modes in hybrid systems) by using a well-coupled auxiliary mode as a probe.
3. Simplifying Experimental Workflows: By removing the need for complex photon-number calibration, the protocol lowers the barrier to entry for characterizing high-dimensional quantum systems, accelerating the development of multimode circuit-QED architectures.

In summary, the paper presents a practical and theoretically grounded method to "in-situ" determine light-matter coupling in complex multimode environments, overcoming the limitations of traditional calibration-heavy techniques.