⚛️ quantum physics

Bose condensation and Bogoliubov excitation in resonator-embedded superconducting qubit network

This paper reports a two-tone spectroscopy experiment on a network of 10 superconducting flux qubits coupled to a resonator, demonstrating the formation of a macroscopic Bose-Einstein condensate of microwave photons and the observation of Bogoliubov-like excitations that exhibit a sharp, tunable frequency shift indicative of photon-number bistability when the pump power exceeds a critical threshold.

Original authors: Patrick Navez, Valentina Di Meo, Berardo Ruggiero, Claudio Gatti, Fabio Chiarello, Alessandro D'Elia, Alessio Rettaroli, Emanuele Enrico, Luca Fasolo, Mikhail Fistul, Ilya Eremin, Alexandre Zagoskin

Published 2026-01-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Patrick Navez, Valentina Di Meo, Berardo Ruggiero, Claudio Gatti, Fabio Chiarello, Alessandro D'Elia, Alessio Rettaroli, Emanuele Enrico, Luca Fasolo, Mikhail Fistul, Ilya Eremin, Alexandre Zagoskin, Paolo Vanacore, Paolo Silvestrini, Mikhail Lisitskiy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a superconducting qubit network (a tiny circuit made of superconducting loops) as a large choir of 10 singers standing inside a very quiet, echo-prone room (a resonator). Normally, these singers are quiet and independent. But in this experiment, the researchers decided to turn up the volume on a specific "pump" microphone, blasting a strong tone into the room.

Here is what happened, broken down into simple concepts:

1. The "Condensate" (The Choir Singing in Unison)

When the researchers blasted a strong microwave signal (the pump) into the room at just the right frequency, something magical happened. Instead of the singers acting individually, they all suddenly locked into step. The room filled up with a massive, synchronized wave of energy. The paper calls this a Bose-Einstein condensate.

Think of it like a crowd of people in a stadium doing "the wave." At first, everyone is just sitting or standing randomly. But once the "wave" starts, everyone moves together as one giant, single entity. In this experiment, the microwave photons (particles of light) inside the resonator behaved like that single, giant wave.

2. The "Probe" (The Second Microphone)

While the "choir" was singing loudly (the pump), the researchers used a second, much quieter microphone (the probe) to listen to the room. They swept this second microphone through different frequencies to see how the room responded.

In a normal room, you would expect the sound to change smoothly as you turn the volume up. But here, the room acted strangely.

3. The "Switch" (Bistability)

As the researchers turned up the volume of the main "pump" signal, they hit a critical threshold (a specific power level). Suddenly, the room didn't just get louder; it snapped into a completely different state.

Before the snap: The room resonated at one specific pitch.
After the snap: The room suddenly resonated at a lower pitch.

This is called bistability. It's like a light switch that has two stable positions: ON and OFF. You can wiggle the switch back and forth a little bit, but it stays in one position until you push it hard enough to make it "click" to the other side. The researchers found that once the pump power crossed a critical line, the system "clicked" from one state to another.

4. The "Bogoliubov Excitation" (The Ripple Effect)

When the researchers listened with their second microphone, they didn't just hear the main note. They heard a new, specific "ripple" or vibration that only appeared because the choir was singing in unison.

The paper calls this a Bogoliubov excitation. Imagine a calm pond (the resonator). If you throw a single pebble, you get a small ripple. But if the whole pond suddenly starts vibrating in a synchronized way (the condensate), a new type of ripple appears that behaves differently than a normal ripple. This special ripple is what the researchers detected, proving that the photons were interacting with each other as a collective group, not just as individual particles.

5. The Magnetic "Tuner"

The researchers also tried turning a knob (applying a magnetic field) to see if they could change the behavior. They found that applying a magnetic field made it easier to trigger the "snap" (the switch to the new state). It was as if the magnetic field loosened the switch, requiring less force to flip it.

The Big Picture

The paper demonstrates that by connecting a network of superconducting qubits to a resonator, they created a system where light (microwaves) behaves like a fluid or a collective group of atoms.

The Discovery: They proved that these artificial atoms can force microwave photons to interact strongly, creating a "condensate" that can switch between two distinct states abruptly.
The Proof: They used a two-tone experiment (a loud pump and a quiet probe) to map out exactly where this switch happens and confirmed their findings with a mathematical model that matched their data perfectly.

In short, they built a tiny, super-cooled "switch" where light can be forced to act like a synchronized crowd, and they figured out exactly how to flip that switch.

Technical Summary: Bose Condensation and Bogoliubov Excitation in Resonator-Embedded Superconducting Qubit Networks

Problem and Context
Superconducting qubit networks (SQNs) embedded in low-dissipative resonators offer a platform for studying collective quantum dynamics on a macroscopic scale and enhancing microwave photon detector sensitivity. While previous research has explored multistability in single Josephson junctions, trapped fluxons, and arrays of RF-SQUIDs, and while the ac Stark effect has been used to resolve Fock states in single-qubit systems, the behavior of a large set of qubits coupled to a high-quality resonator under strong microwave pumping remains less explored. Specifically, the transition to collective non-classical states of the microwave photon field and the observation of Bogoliubov-like excitations in such a regime have been predicted but not yet experimentally demonstrated in a network of flux qubits.

Methodology
The authors constructed a device comprising a network of 10 capacitively-shunted superconducting flux qubits (each containing three Josephson junctions) coupled inductively to a transmission line (T-resonator) and capacitively to a readout resonator (R-resonator). The system was operated at 15 mK in a dilution refrigerator.

The experimental approach utilized two-tone spectroscopy:

Pump Tone: A strong microwave field ( $P_{pump}$ ) was injected into the R-resonator at a frequency ( $f_{pump}$ ) close to the resonant frequency ( $f_r \approx 7.7$ GHz). This pump field macroscopically populates the resonator mode, creating a state analogous to a Bose-Einstein condensate (BEC) of photons.
Probe Tone: A second, weaker signal ( $f_{probe}$ ) was used to scan the transmission coefficient ( $|S_{21}|$ ) of the T-resonator.
Variable Parameters: The pump power was swept from –110 dBm to –60 dBm, and the pump frequency was varied within a narrow window around resonance. Experiments were conducted both with and without an external static magnetic field ( $B = 0.489$ G).

Theoretical modeling involved deriving a time-dependent, nonlinear Schrödinger-like equation (analogous to the Gross-Pitaevskii equation but with distinct nonlinearity) for the photon field $\psi(t)$ inside the hybrid R+T resonator. This model accounted for the ac Stark shift induced by the coupling between the SQN and the microwave photons.

Key Results

Resonant Dip Shift and Bistability: As the pump power exceeded a critical threshold ( $P_{cr}$ ), the resonant dip in the transmission coefficient $|S_{21}|$ exhibited an abrupt shift toward lower frequencies. This shift occurred within a narrow window of pump frequencies ( $7.743 < f_{pump} < 7.746$ GHz).
Hysteresis: The system displayed bistability. When the pump power was swept up and down near $P_{cr}$ , the position of the resonant dip followed a hysteretic loop, indicating the existence of two stable metastable configurations for the photon number inside the resonator.
Bogoliubov-like Excitations: The appearance of a secondary resonant feature (an idler mode) near the main dip, which becomes dominant once $P_{pump} \geq P_{cr}$ , was identified as a signature of Bogoliubov-like collective excitations. This confirms the presence of interacting photon modes.
Magnetic Field Tuning: Applying a static magnetic field shifted the critical pump power threshold $P_{cr}$ to lower values. Theoretical analysis estimated the qubit frequency shift per Gauss to be approximately 200 MHz/G, consistent with expectations.
Quantitative Agreement: The experimental data was quantitatively reproduced by the derived theoretical model. The fit yielded a coupling quality factor $Q_c \approx 25,000$ and a collective frequency shift of $2\pi \times 3.4$ MHz. The model successfully predicted the transition from a single-valued regime to a bistable regime based on cavity detuning and leakage parameters.

Significance and Claims
The paper claims to provide the first experimental evidence of collective states of interacting microwave photons in a resonator-embedded SQN, characterized by a sharp frequency shift and bistability. The authors identify the observation of Bogoliubov-like excitations as a fingerprint of these collective states.

The significance of this work lies in:

Demonstrating Macroscopic Quantum Dynamics: Showing that a network of 10 qubits coupled to a resonator can support collective quantum states analogous to a photon BEC.
Theoretical Validation: Providing a tractable nonlinear equation that accurately describes the spectroscopy of SQNs in a resonator, bridging the gap between single-qubit ac Stark effects and many-body photon interactions.
Potential Applications: The authors suggest that the observed bistability and hysteretic behavior could be exploited for realizing quantum threshold detectors or bistable memory elements in circuit quantum electrodynamics (cQED) architectures.

The authors remain modest regarding the interpretation of the magnetic field's specific influence on the bistability region, noting that while the threshold shifts, the fundamental bistability region did not show significant changes upon tuning qubit frequencies, a point requiring further investigation.