Flux-modulated tunable interaction regimes in two strongly nonlinear oscillators

This paper demonstrates a flux-modulated scheme for two strongly nonlinear superconducting oscillators that enables the selective activation of distinct interaction regimes—including photon-hopping, two-mode squeezing, and cross-Kerr interactions—to facilitate the analog simulation of arbitrary spin systems and the exploration of driven-dissipative dynamics in previously unexplored parameter spaces.

The Gist

Imagine you have two tiny, super-fast pendulums (let's call them "Quantum Pendulums") hanging in a vacuum. In the world of quantum physics, these aren't just swinging back and forth; they are vibrating at incredibly high speeds, and they are made of superconducting materials that act like magic.

Usually, if you want these two pendulums to talk to each other, you just tie a string between them. If one swings, it pulls the other. This is simple, but it's limited. You can only get them to do one thing: swap energy back and forth.

The Big Idea of This Paper
The scientists in this paper wanted to build a "universal translator" for these pendulums. They wanted a way to make the pendulums talk to each other in different languages (different types of interactions) just by changing the tune of a signal they send through a special control knob.

They built a device using two superconducting circuits (the pendulums) connected by a special bridge (a tunable coupler). This bridge is the star of the show. It's like a magical door that can change its shape depending on how you knock on it.

The Magic Trick: The "Flux Knob"

Imagine the bridge between the pendulums is a door made of a special material that reacts to magnetic fields. The scientists have a remote control (a magnetic field) that they can wiggle back and forth very quickly.

• If they wiggle the remote at a slow, steady pace: The door stays mostly closed, and the pendulums just feel a gentle, static push from each other. This is like them just sitting next to each other, aware of one another but not really interacting much.
• If they wiggle the remote at a specific "Red" rhythm: The door opens just enough to let the pendulums swap energy. If Pendulum A has a "kick" of energy, it can pass it to Pendulum B. It's like a game of hot potato where they toss the energy back and forth perfectly. This is called Photon Hopping.
• If they wiggle the remote at a specific "Blue" rhythm: The door opens in a weird way. Now, if Pendulum A gets a kick, Pendulum B also gets a kick at the exact same time. They don't swap; they grow together. It's like if you pushed one swing, and the other swing magically started swinging higher too, even though no one touched it. This is called Two-Mode Squeezing.

The "Strongly Nonlinear" Twist

Here is the tricky part. Most pendulums in physics are "linear," meaning if you push them twice as hard, they swing twice as high. But these "Quantum Pendulums" are strongly nonlinear.

Think of a normal swing: it's predictable.
Think of these quantum swings: they are like a swing that changes its own rules. If you push it hard, it doesn't just go higher; it changes its rhythm, its shape, and how it reacts to the other swing. They are "stubborn" pendulums.

Usually, when you have two stubborn pendulums, they are hard to control. Their own stubbornness (nonlinearity) usually overpowers any attempt to make them talk to each other.

The Discovery:
The scientists found that even though these pendulums are very stubborn, they could still force them to talk in these different "languages" (hopping or squeezing).

• They saw Level Repulsion: When they tried to make the pendulums swap energy, the pendulums pushed each other away, like two magnets with the same pole facing each other.
• They saw Level Attraction: When they tried the "Blue" rhythm, the pendulums seemed to pull toward each other, almost like they wanted to merge. This is rare and usually only happens in very specific, unstable conditions, but they managed to see it even with these stubborn pendulums.

Why Does This Matter?

Why should you care about two wiggly pendulums?

1. Building a Quantum Simulator: Imagine you want to simulate how a complex molecule works, or how a new material conducts electricity. These are hard to calculate on a normal computer. But if you can program these pendulums to act exactly like the atoms in that molecule, you can use the pendulums to "solve" the problem for you. This paper gives us a "universal remote" to program the pendulums to act like almost any system we want.
2. New Types of Computers: This technology helps build better quantum computers. By being able to switch between different types of interactions instantly, we can perform calculations much faster and with fewer errors.
3. Exploring the Unknown: The scientists found that even when the pendulums are very "stubborn" (strongly nonlinear), they can still exhibit strange behaviors like "Level Attraction" without falling apart. This opens up a whole new playground for physicists to study how energy moves in complex systems.

The Bottom Line

Think of this paper as the invention of a universal translator for quantum machines. Before, scientists could only make quantum machines talk in one or two ways. Now, by wiggling a magnetic knob at the right rhythm, they can make two stubborn, complex quantum pendulums swap energy, grow together, or push each other apart at will. This gives us the power to build custom quantum machines that can simulate almost any physical phenomenon we can imagine.

Technical Summary

1. Problem Statement

The primary goal of analog quantum simulation is to emulate complex quantum systems (such as arbitrary spin models or lattice gauge theories) that are difficult to study directly. While superconducting circuit quantum electrodynamics (cQED) using transmon qubits has been successful for digital quantum computing, there is a need for greater flexibility in analog simulations.

Current challenges include:

• Limited Interaction Regimes: Most tunable couplers in cQED are optimized for specific interactions (e.g., single-photon exchange or cross-Kerr) but struggle to access regimes where nonlinear interactions dominate or where specific entangling terms (like two-mode squeezing) are selectively activated.
• Strong Nonlinearity: Previous studies on "level attraction" (a phenomenon where energy levels merge rather than repel) were largely confined to linear or weakly nonlinear systems. The behavior of strongly nonlinear oscillators (like transmons with high anharmonicity) under parametric driving, particularly regarding level attraction and instability, remains underexplored.
• Control: There is a need for a single device capable of switching between different interaction Hamiltonians (e.g., beam-splitter, squeezing, and cross-Kerr) to simulate diverse physical models (Ising, Heisenberg, Bose-Hubbard).

2. Methodology

The authors designed and characterized a superconducting circuit comprising two flux-tunable transmon qubits coupled via a tunable nonlinear inductive coupler (a SQUID) and a fixed capacitive coupling.

• Device Architecture:
  • Two transmon qubits ($A$ and $B$) connected to a central coupler.
  • The coupler consists of a symmetric SQUID loop, allowing its inductance to be tuned by an external magnetic flux ($\Phi_{DC}$).
  • The system is driven and probed via coplanar waveguide resonators.
• Theoretical Framework:
  • The system is modeled as two coupled Kerr-nonlinear oscillators. The Hamiltonian includes single-photon hopping ($J_1$), two-mode squeezing ($J_2$), and cross-Kerr ($V$) interactions.
  • Parametric Modulation: The authors apply an AC flux modulation ($\Phi_{AC} \cos(\omega_m t)$) to the coupler's SQUID loop.
  • Resonance Conditions:
    • Red Sideband (RSB): Modulation at the difference frequency $\omega_m = |\omega_A - \omega_B|$ selectively activates the single-photon hopping (beam-splitter) interaction ($J_1$).
    • Blue Sideband (BSB): Modulation at the sum frequency $\omega_m = \omega_A + \omega_B$ selectively activates the two-mode squeezing interaction ($J_2$).
  • The modulation effectively converts the static, off-resonant interactions into resonant, tunable couplings while the cross-Kerr term ($V$) remains as a static background interaction.
• Experimental Protocol:
  • Two-tone spectroscopy was performed on one transmon while sweeping the modulation frequency $\omega_m$.
  • The transmission spectrum was analyzed to observe avoided crossings (level repulsion) or merging of levels (level attraction).
  • Numerical simulations using the Quantum Master Equation (QuTiP) were employed to validate the analytical models and extract interaction strengths.

3. Key Contributions

1. Selective Activation of Interaction Regimes: The work demonstrates a method to dynamically switch between photon-hopping, two-mode squeezing, and cross-Kerr dominated regimes on the same hardware simply by changing the modulation frequency and amplitude.
2. Observation of Level Attraction in Strongly Nonlinear Systems: The authors observed the phenomenon of level attraction (associated with dissipative coupling and two-mode squeezing) in a system of strongly nonlinear oscillators. This contrasts with previous observations in linear systems where level attraction often leads to parametric instability.
3. Characterization of Non-Instability Regimes: They showed that in the presence of strong self-Kerr nonlinearity (characteristic of transmons), the system does not enter a parametric instability region (where photon numbers diverge) even under blue-sideband driving. Instead, it generates low-photon-number entangled states.
4. Quantitative Extraction: The paper provides precise extraction of interaction strengths ($J_{AC}$ and $V$) and demonstrates that these can be tuned to regimes where $J > V$, $J \approx V$, and $J < V$.

4. Key Results

• Red Sideband (Level Repulsion):
  • Modulating at the difference frequency induced a single-photon hopping interaction.
  • Result: An avoided crossing (level repulsion) was observed in the transmission spectrum.
  • Extracted Parameters: Hopping strength $J_{AC}/2\pi = 7.462$ MHz and Cross-Kerr strength $V/2\pi = -6.543$ MHz.
• Blue Sideband (Level Attraction):
  • Modulating at the sum frequency induced a two-mode squeezing interaction.
  • Result: A distinct level attraction feature was observed, characterized by the merging of eigenfrequencies and a dip in transmission (loss of excited state population).
  • Extracted Parameters: Squeezing strength $J_{AC}/2\pi = 1.131$ MHz and Cross-Kerr strength $V/2\pi = -9.158$ MHz.
  • Shifted Features: Due to the strong cross-Kerr coupling, a secondary spectral feature shifted by $V$ was observed, confirming the theoretical prediction of frequency shifts due to nonlinearity.
• Nonlinear Dynamics:
  • Numerical simulations confirmed that for strongly nonlinear oscillators ($\alpha \gg \kappa$), the two-mode squeezing interaction acts effectively within the qubit subspace, generating an $XX-YY$ interaction without triggering the parametric instability seen in linear systems.
  • The system remained stable (photon number $< 1$) even under conditions that would cause instability in linear oscillators.

5. Significance and Outlook

• Analog Quantum Simulation: This work provides a versatile platform for simulating arbitrary XYZ spin-model Hamiltonians and Ising models. By tuning the ratio of hopping ($J$) to cross-Kerr ($V$) and utilizing parametric driving, researchers can emulate complex many-body physics.
• Exotic Physics: The ability to access regimes where nonlinearities exceed coupling strengths allows for the study of driven-dissipative dynamics and non-Hermitian physics in previously unexplored parameter spaces.
• Quantum Control: The demonstration of controllable level attraction and repulsion in strongly nonlinear systems opens new avenues for quantum-limited parametric amplification, entanglement generation, and topological energy transfer.
• Scalability: The scheme relies on standard superconducting circuit components (transmons and SQUIDs), making it highly compatible with scalable quantum processor architectures. Future work could extend this to multi-qubit arrays or utilize asymmetric elements (like SNAILs) to access even more exotic interactions (e.g., photon-pair tunneling).

In conclusion, this paper establishes a robust method for flux-modulated tunable interactions in strongly nonlinear oscillators, bridging the gap between digital gate-based control and analog simulation of complex quantum phenomena.