🔬 mesoscale physics

Measuring high field gradients of cobalt nanomagnets in a spin-mechanical setup

This paper presents a spin-mechanical setup utilizing a focused electron beam-induced deposition cobalt nanomagnet to achieve a directly measured high magnetic field gradient of 170 kT/m while preserving spin coherence, thereby demonstrating viable spin-mechanics coupling for future quantum information and sensing applications.

Original authors: Felix Hahne, Teresa Klara Pfau, Liza Žaper, Lucio Stefan, Thibault Capelle, Andrea Ranfagni, Martino Poggio, Albert Schliesser

Published 2026-02-04

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CC BY 4.0

Original authors: Felix Hahne, Teresa Klara Pfau, Liza Žaper, Lucio Stefan, Thibault Capelle, Andrea Ranfagni, Martino Poggio, Albert Schliesser

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 you are trying to build a super-sensitive scale that can weigh the tiniest things in the universe, like a single atom. To do this, you need two things to talk to each other: a tiny, invisible "spin" (a magnetic property of an atom) and a tiny, vibrating "swing" (a mechanical object). The problem is, they are very shy and don't like to interact unless you get them very close and very loud.

This paper is about building a special "megaphone" to help these two talk to each other. Here is the story of how they did it, explained simply:

1. The Goal: Making a Magnetic "Whisper" Loud

The scientists wanted to create a setup where a single atom (specifically, a defect in a diamond called an NV center) could feel the movement of a tiny mechanical swing. To make this happen, they needed a magnetic gradient.

Think of a magnetic gradient like a steep hill. If you roll a ball down a gentle slope, it moves slowly. If you roll it down a steep cliff, it accelerates fast. In this experiment, the "ball" is the magnetic field, and the "cliff" is the gradient. The steeper the cliff, the more the atom feels the movement of the swing. The scientists wanted to build the steepest magnetic cliff possible without breaking the delicate atom or the swing.

2. The Tool: A "Magnetic Pen" (FEBID)

To build this cliff, they used a technique called Focused Electron Beam Induced Deposition (FEBID).

The Analogy: Imagine you have a magic pen that shoots tiny, invisible beams of electrons. When this pen touches a special "ink" (a gas), it turns the ink into solid metal instantly, right where the pen is pointing.
What they did: They used this "pen" to draw a tiny, 3D tower out of cobalt metal on a silicon chip. This tower is the "magnet" in their experiment. Because they drew it with a pen, they could make it the exact shape and size they needed (about the width of a virus).

3. The Test: Measuring the "Steepness"

Once they built their cobalt tower, they needed to see how steep the magnetic "hill" was.

They brought their diamond atom (the sensor) very close to the tower—just a few hundred nanometers away (that's like being a few steps away from a house if you were shrunk to the size of an ant).
They measured how much the atom's magnetic "tuning" changed as it moved up and down.
The Result: They found a spot where the magnetic field changed incredibly fast. They measured a gradient of 170,000 Tesla per meter.
- To visualize this: If you were standing on this magnetic hill, the field would change so drastically over a tiny distance that it's like going from a gentle breeze to a hurricane in the blink of an eye.

4. The Catch: Keeping the Atom Calm

There was a risk: being this close to a strong magnet might make the atom "nervous" and lose its ability to hold information (a problem called losing "coherence").

They tested this by checking how long the atom could stay calm (coherent) while sitting near the magnet.
The Result: Even with a very steep magnetic hill (up to 25,000 Tesla per meter), the atom stayed calm for 20 microseconds. That's a very long time in the world of quantum physics! This proved that their "cobalt tower" was strong but didn't ruin the atom.

5. The Big Moment: The Swing and the Atom Dance

Finally, they wanted to see if the mechanical swing could actually push the atom.

They attached their cobalt tower to a tiny tuning fork (the swing) and made it vibrate.
As the tuning fork wiggled back and forth, it moved the magnetic field up and down.
The Result: The atom felt this wiggling! The scientists saw the atom's signal change in a rhythmic pattern that matched the vibration of the fork. This proved that the "swing" and the "atom" were finally holding hands and dancing together.

Why This Matters (According to the Paper)

The scientists say this method is special because:

It's gentle: They built the magnet directly on the chip without damaging it (non-invasive).
It's precise: They can draw the magnet exactly where they want.
It works: They proved that you can have a super-strong magnetic gradient that still lets the quantum atom stay calm.

They conclude that this setup is a promising step toward future "quantum machines" where tiny magnets and mechanical swings work together to sense the world or process information, but they specifically note that this is a foundational step for hybrid quantum systems and quantum sensing, not for medical use or other applications yet.

Technical Summary: Measuring High Field Gradients of Cobalt Nanomagnets in a Spin-Mechanical Setup

Problem Statement
The realization of quantum information protocols and advanced quantum sensing applications relies on hybrid systems where a single electron spin is magnetically coupled to a macroscopic mechanical resonator. A critical bottleneck in this field is the identification and implementation of a magnetic structure that can generate sufficiently high magnetic field gradients to enable strong coupling, while simultaneously preserving the coherence of the spin and the mechanical properties of the resonator. Previous approaches have utilized various magnetic materials and deposition methods, including NdFeB structures, evaporated CoFe films, and deposited CoCr layers, but challenges remain in balancing gradient magnitude with system coherence and non-invasive integration.

Methodology
To address these challenges, the authors employed Focused Electron Beam Induced Deposition (FEBID) to grow soft cobalt nanomagnets directly onto a silicon chip. The fabrication utilized a circular pattern with specific parameters: 5 kV acceleration voltage, 100 pA beam current, 1 µs dwell time, and a precursor flux corresponding to vacuum pressures between $1.1 \times 10^{-6}$ and $1.2 \times 10^{-6}$ mbar. The resulting nanomagnets possessed a cylindrical geometry with approximately 830 nm height and 400 nm diameter.

The experimental setup was a scanning Nitrogen-Vacancy (NV) microscopy configuration. A diamond scanning probe hosting an NV center was mounted on a quartz tuning fork and positioned relative to the nanomagnet using x,y,z piezoelectric nanopositioners. The system operated under an external bias magnetic field (up to 140 mT) aligned with the NV spin quantization axis. Characterization involved:

Magnetometry: Optically Detected Magnetic Resonance (ODMR) was performed via continuous-wave microwave sweeps while raster scanning the NV probe above the nanomagnet to map the stray field ( $B_{NV}$ ).
Modeling: The experimental data was fitted using a magnetic dipole model to extract the magnetic moment ( $m_{dip}$ ). These results were cross-validated against micromagnetic simulations based on AFM topography.
Gradient Measurement: The magnetic field gradient ( $G_z$ ) was directly measured by analyzing the Zeeman shift of the ODMR spectra as the NV center approached the nanomagnet surface.
Coherence and Dynamics: Spin coherence ( $T_2^E$ ) was measured using spin-echo sequences at various gradient strengths. The coupling between mechanical motion and spin dynamics was probed by driving the tuning fork and observing the modulation of the spin-echo visibility.

Key Contributions and Results

Direct Gradient Measurement: The study achieved a direct measurement of a maximum magnetic field gradient of 170 kT m $^{-1}$ at a distance of approximately 250 nm from the cobalt nanomagnet surface. This was measured at a bias field of $\sim$ 120 mT.
Magnetic Characterization: The nanomagnets were characterized as having a soft magnetic nature, with the dipole moment increasing linearly with the external bias field up to the tested limit. The experimental dipole fits, simulation results, and net magnetization calculations showed mutual agreement, validating the magnetic model.
Spin Coherence Preservation: Spin coherence was preserved up to a gradient of 25 kT m $^{-1}$ , with a measured $T_2^E$ of at least 20 µs. While coherence decreased at higher gradients (attributed to magnetic noise or thermal drifts), the system maintained functional spin control.
Spin-Mechanical Coupling: The authors observed the effect of mechanical motion on spin dynamics. When the tuning fork was driven at its resonance frequency ( $\approx$ 32 kHz), the spin-echo visibility exhibited a modulation fitting a zeroth-order Bessel function. This confirmed the presence of spin-mechanical coupling, with extracted mechanical amplitudes of 3 nm and 11 nm corresponding to drive amplitudes of 3 mV and 9 mV, respectively.

Significance and Claims
The paper claims that the non-invasive nature of FEBID allows for the successful integration of magnetic structures onto high-quality factor resonators, such as silicon nitride membranes, without the need for invasive transfer processes. The authors posit that this "magnet-on-oscillator" configuration is a promising platform for future hybrid quantum systems.

Specifically, the authors project that if such structures are integrated with ultra-high quality factor resonators ( $Q=10^9$ ) and operated with improved spin coherence times (e.g., 1 ms) and higher gradients (1 MT m $^{-1}$ ), the system could achieve a single-phonon coupling strength of $g_0/2\pi = 8$ Hz. This would result in a quantum cooperativity ( $C$ ) four orders of magnitude higher than current state-of-the-art setups, potentially enabling the detection of the force exerted by a single spin within seconds. The work establishes a foundational step toward realizing quantum gates and entanglement generation in hybrid nano-electromechanical systems by demonstrating the feasibility of creating strong magnetic gradients while maintaining spin coherence.