On-chip Dicke-type magnon polaritons in the ultrastrong coupling regime via spatially separated nanomagnets

This paper reports the on-chip realization of a Dicke-type magnon polariton system in the ultrastrong coupling regime using spatially separated nanomagnets, which successfully suppresses detrimental self-interaction terms to enable the observation of critical quantum phenomena like the Bloch-Siegert shift.

The Gist

The Big Idea: A Quantum Dance Floor with a Catch

Imagine you are trying to get a huge crowd of people (atoms or spins) to dance in perfect unison with a single DJ (a photon). In the world of quantum physics, this is called the Dicke model. If you can get them to dance perfectly together, you unlock superpowers like "superradiance" (where the crowd shouts so loud it changes the room's physics) and "squeezed states" (a special kind of quantum magic useful for computers).

However, there's a massive problem. In the real world, when you try to get these people to dance, they don't just listen to the DJ; they also start bumping into each other and arguing. In physics terms, this is called self-interaction. These arguments (self-interactions) ruin the perfect dance, making the "superradiant" magic impossible to achieve. It's like a "No-Go" sign on the dance floor.

The Solution: The "Social Distancing" Strategy

The researchers in this paper found a clever way to break the "No-Go" sign. They realized that if the dancers are too close together, they fight. But if you spread them out, they can still hear the DJ perfectly without bumping into each other.

Here is how they did it:

1. The DJ (The Resonator): They built a tiny, super-conducting microwave circuit (like a super-fast DJ booth) made of a special material called YBCO. This DJ creates a very strong, focused magnetic field.
2. The Dancers (The Magnons): Instead of one big block of magnetic material, they used 26 tiny, separate stripes of a magnetic metal called Permalloy.
3. The Trick (Spatial Separation): By placing these 26 stripes in a line but keeping them physically separated (like dancers standing in a circle with space between them), they achieved two things:
   • Cooperative Power: Because they are all listening to the same DJ at the same time, they act as one giant team. Their combined strength grows with the square root of the number of stripes (26 stripes = much stronger dance than 1 stripe).
   • No Fighting: Because they are separated, they can't bump into each other. The "self-interaction" (the arguing) stays small and doesn't grow with the team size.

The Result: Ultrastrong Coupling and the "Ghost" Shift

Because of this setup, the team achieved something called Ultrastrong Coupling. This is a state where the energy exchange between the DJ and the dancers is so fast and intense that it breaks the usual rules of physics.

Usually, physicists ignore certain "backwards" movements in the dance (called counter-rotating terms) because they are too weak to matter. But in this ultrastrong regime, those backwards moves become huge.

The Proof: The Bloch-Siegert Shift
The researchers found a specific "glitch" in the music, called the Bloch-Siegert shift.

• Analogy: Imagine you are pushing a child on a swing. Usually, you push when they come toward you. But in this ultrastrong regime, the swing is so energetic that it starts reacting to your push before you even make it, or even pushes back when you aren't touching it. This creates a slight change in the rhythm (frequency) of the swing.
• The Finding: They measured this shift and found it was huge (up to 60 MHz). This proved that the "backwards" quantum terms were active and that the system was behaving exactly like the theoretical Dicke model they wanted to build.

Why This Matters

This isn't just a cool physics trick; it's a blueprint for the future.

• Quantum Computers: This setup creates a "playground" where we can study exotic quantum states that were previously impossible to see.
• Scalability: Because they used tiny stripes on a chip, this can be mass-produced. It's like moving from a one-off art project to a factory line for quantum devices.
• Breaking the Rules: They successfully bypassed the "No-Go" theorem that has blocked scientists for decades. They proved that by simply changing the geometry (spacing the magnets out), you can suppress the bad interactions while keeping the good ones.

Summary in One Sentence

The researchers built a quantum dance floor where they spaced out the dancers just enough so they could all dance in perfect, super-powered unison with a DJ without tripping over each other, finally allowing them to observe rare quantum phenomena that were thought to be impossible.

Technical Summary

1. Problem Statement

The research addresses a fundamental challenge in quantum electrodynamics (QED) and the realization of the Dicke model, a theoretical framework describing the interaction between $N$ two-level systems and a quantized radiation field.

• The No-Go Theorem: In the thermodynamic limit ($N \to \infty$), the Dicke model predicts exotic phenomena like equilibrium superradiant phase transitions and ground-state two-mode squeezing. However, experimental realization has been elusive due to the no-go theorem. This theorem arises from gauge invariance, which mandates the existence of self-interaction terms (specifically the $A^2$ term for photons and magnetic self-interaction for magnons). These terms scale with the coupling strength, effectively forbidding the superradiant phase transition in continuous bulk media.
• The Limitation of Existing Systems: While ultrastrong coupling (where coupling strength $g > 10\%$ of the resonance frequency) has been achieved in circuit QED and semiconductor microcavities, the simultaneous enhancement of coupling strength and self-interaction terms prevents the observation of genuine Dicke physics. Spin-based systems (magnons) were thought to avoid the photon $A^2$ term, but they suffer from a magnonic counterpart: magnetic self-interaction.

2. Methodology

The authors propose and experimentally realize a novel architecture that decouples the cooperative enhancement of coupling strength from the enhancement of self-interaction energy.

• Device Architecture:

  • Resonator: A planar, lumped-element superconducting microwave resonator fabricated from high-$T_c$ Yttrium Barium Copper Oxide (YBCO). It features a meander-shaped inductor to confine the microwave magnetic field.
  • Magnets: Instead of a single continuous ferromagnetic film, the device utilizes an array of spatially separated Permalloy (NiFe) nanostripes (30 nm thick) placed on the resonator's inductor.
  • Mechanism: The spatial separation ensures that while the individual magnon modes in each stripe couple coherently to the single photon mode (forming a "bright mode"), the direct dipole-dipole interactions between the separated stripes are negligible.

• Theoretical Framework:

  • The system is modeled using a gauge-invariant Hopfield Hamiltonian.
  • The authors derive that the effective coupling strength ($G_{eff}$) scales with the square root of the number of elements ($\sqrt{n}$), while the magnon self-interaction term ($D_m$) remains independent of $n$ because the interaction is local to each isolated element.
  • This breaks the standard scaling law ($D \propto G^2$) that enforces the no-go theorem.

• Experimental Setup:

  • Measurements were conducted at 10 K to minimize thermal noise while maintaining the superconducting state of YBCO.
  • Transmission spectra ($S_{21}$) were measured using a vector network analyzer under varying external magnetic fields (up to $\pm 145$ mT).
  • Devices with varying numbers of permalloy stripes ($n = 1, 8, 16, 26$) were fabricated to verify scaling laws.

3. Key Contributions

1. Spatial Separation Strategy: The primary contribution is the demonstration that spatially separating ferromagnetic elements allows for the cooperative enhancement of coupling strength without the concomitant enhancement of self-interaction terms. This physically realizes the "Dicke limit" where emitters are macroscopic but non-interacting.
2. Ultrastrong Coupling (USC): The system achieves a coupling ratio $G_{eff}/\omega_p \approx 0.102$, exceeding the 10% threshold for the ultrastrong coupling regime.
3. Suppression of Self-Interaction: The authors experimentally quantify the suppression of the magnon self-interaction term, showing a suppression factor ($\xi_m$) of $\approx 0.038$, consistent with the theoretical prediction of $1/n$.
4. Observation of Bloch-Siegert Shift: The system exhibits a clear Bloch-Siegert shift (up to 60 MHz), providing direct evidence of significant counter-rotating terms in the Hamiltonian, a hallmark of the ultrastrong coupling regime.

4. Key Results

• Spectral Analysis: The transmission spectra show clear anticrossing between the photon and magnon modes. Fitting the data to the quartic equation derived from the full Hamiltonian yielded:
  • Effective coupling strength: $G_{eff}/2\pi = 512.7$ MHz.
  • Coupling ratio: $G_{eff}/\omega_p = 0.102$ (Ultrastrong regime).
• Bloch-Siegert Shift: By comparing experimental data with simulations that exclude counter-rotating terms, the authors isolated a frequency shift ($\Delta f_{BS}$) of up to 60 MHz. This shift scales with $G_{eff}^2$, confirming the dominance of counter-rotating terms.
• Cooperativity Verification: The effective coupling strength was shown to scale linearly with $\sqrt{n}$ (where $n$ is the number of stripes), confirming the formation of a collective magnon bright mode (Dicke cooperativity).
• Self-Interaction Suppression: The extracted scaling parameters for self-interaction ($\xi_p \approx 0.048$ for photons, $\xi_m \approx 0.038$ for magnons) satisfy the condition $\xi_p + \xi_m < 1$. This is the necessary condition to bypass the no-go theorem and allows for a finite critical coupling strength for a superradiant phase transition.
• Critical Coupling: The calculated critical coupling strength required for the phase transition is approximately 2.64 GHz, which is roughly 5 times the current experimental coupling strength, indicating the system is approaching the regime where ground-state squeezing and phase transitions could be observed.

5. Significance

• Platform for Quantum Physics: This work establishes a scalable, on-chip platform for exploring Dicke physics in the ultrastrong coupling regime. It provides a "playground" to study phenomena previously thought impossible in equilibrium, such as ground-state two-mode squeezing and superradiant phase transitions.
• Bypassing the No-Go Theorem: By physically decoupling the coupling enhancement from self-interaction via spatial separation, the authors offer a robust solution to the long-standing no-go theorem problem in cavity QED.
• Future Applications: The platform paves the way for:
  • Noise-tolerant quantum technologies: Utilizing the robustness of Dicke states.
  • Quantum Information: Generating entangled magnon-photon states and virtual photon generation.
  • Spintronics-Quantum Optics Integration: Merging magnetic spin dynamics with superconducting circuits.
• Material Insights: The observation of a finite photon self-interaction ($A^2$ term) in metallic permalloy (unlike insulating YIG) suggests significant electronic contributions (electron-magnon/photon coupling), offering new avenues to study orbital dynamics in ferromagnets.

In summary, this paper successfully demonstrates a method to engineer a "genuine" Dicke system on a chip by spatially separating nanomagnets, achieving ultrastrong coupling while suppressing the self-interaction terms that typically prevent the observation of exotic quantum collective phenomena.