Landé $g$ factor measurement of $^{48}$Ti$^+$ using simultaneous co-magnetometry and quantum logic spectroscopy

This paper presents a quantum logic spectroscopy scheme using simultaneous co-magnetometry to measure the ground state $g$ factors of a single $^{48}$Ti$^+$ ion with $10^{-6}$ uncertainty, achieving agreement between experimental results and new theoretical predictions while offering a method applicable to various atomic species, including those not directly laser coolable.

The Gist

Imagine you are trying to measure the speed of a car, but the road you are driving on is constantly shaking and vibrating. If you try to time the car's speed while the road is jiggling, your measurement will be messy and inaccurate.

This is essentially the problem scientists faced when trying to measure a fundamental property of atoms called the Landé g-factor. This number tells us how strongly an atom reacts to a magnetic field, acting like a "magnetic personality" for the particle. To measure it precisely, scientists usually need a very stable magnetic environment. But in the real world, magnetic fields fluctuate due to power lines, elevators, and even the Earth's own shifting fields.

Here is how the team at the Physikalisch-Technische Bundesanstalt (PTB) in Germany solved this, using a clever trick involving Quantum Logic and Co-Magnetometry.

The Problem: The Shaky Road

In the past, to measure these magnetic properties, scientists used massive machines called Penning traps. These traps use incredibly strong magnetic fields (thousands of times stronger than a fridge magnet) to hold the atoms still.

• The Catch: When the magnetic field is too strong, it messes up the delicate internal structure of the atom (specifically for Titanium ions), making it impossible to measure the specific property they wanted.
• The Alternative: They tried using weaker magnetic fields, but then the "shaky road" problem returned. The magnetic field would drift slightly during the experiment, ruining the precision.

The Solution: The "Tandem Bike" Analogy

The researchers came up with a brilliant solution: Don't measure the road; measure the difference between two riders on the same bike.

They trapped two different ions together in a single line:

1. The "Logic" Ion (Calcium): This is the expert rider. We already know its "magnetic personality" (g-factor) perfectly. It acts as a built-in reference.
2. The "Spectroscopy" Ion (Titanium): This is the mystery rider. We want to find its magnetic personality.

Instead of measuring them one after the other (which would let the magnetic field drift between measurements), they measured both at the exact same time.

Think of it like a belt drive system (as visualized in their Figure 1):

• Imagine a belt connecting two pulleys.
• The speed of the belt is the fluctuating magnetic field (the noise).
• The size of the pulleys represents the g-factors of the two ions.
• The rotation speed of each pulley is how the atom reacts to the field.

If the belt speeds up (magnetic field gets stronger), both pulleys spin faster. If the belt slows down, both spin slower. By comparing the rotation speed of the Titanium pulley to the Calcium pulley simultaneously, the speed of the belt cancels out! You don't need to know how fast the belt is moving; you only need to know the ratio of the pulley sizes.

The Magic Tool: Quantum Logic Spectroscopy

Now, here is the tricky part: Titanium ions are "grumpy." You can't easily shine a laser on them to see what state they are in (like you can with Calcium). They are invisible to our standard lasers.

So, how do they read the Titanium?
They use the Calcium ion as a messenger.

1. They use a special "quantum handshake" (entangling the two ions via their shared motion) to transfer the Titanium's secret information to the Calcium ion.
2. Once the information is on the Calcium, they shine a laser on the Calcium. The Calcium lights up or stays dark depending on what the Titanium was doing.
3. It's like asking a friend (Calcium) to whisper a secret to you, because the person you actually want to talk to (Titanium) is wearing a blindfold and can't speak.

The Results: Why Does This Matter?

Using this "Tandem Bike" method, they measured the magnetic personality of the Titanium-48 ion with incredible precision (uncertainty of 1 in a million).

Why should we care?

1. Stellar Detective Work: Titanium is everywhere in the universe. It's in stars and ancient gas clouds. By knowing exactly how Titanium behaves in magnetic fields, astronomers can better understand the composition of stars and even test if the fundamental laws of physics have changed over billions of years.
2. Testing the Universe's Rulebook: They compared their experimental results with complex computer simulations based on Quantum Electrodynamics (QED). The results matched almost perfectly, but with tiny, exciting differences. These tiny differences might be the key to understanding "negative energy states" and other weird quantum effects that we don't fully understand yet.

Summary

The scientists didn't just build a better ruler; they built a self-correcting system. By pairing a known "control" atom with a "mystery" atom and measuring them simultaneously, they canceled out the noise of the universe. They turned a shaky, noisy measurement into a crystal-clear window into the quantum world, proving that even the most elusive atoms can be understood if you ask the right questions in the right way.

Technical Summary

1. Problem Statement

Precise determination of atomic Landé $g$ factors is essential for using atomic systems as magnetic field probes in laboratory and astrophysical contexts (e.g., stellar spectra analysis and testing fundamental constants). However, current measurement techniques face significant limitations:

• Penning Traps: While offering high precision ($<10^{-9}$), they require multi-Tesla magnetic fields. In such strong fields, weak $LS$-coupling ions (like singly charged transition metals) enter the Paschen-Back regime where orbital angular momentum ($L$) and spin ($S$) decouple, preventing the measurement of $g_J$ factors. Furthermore, long averaging times required in Penning traps hinder the study of short-lived metastable states.
• Weak Field Regime: In linear Paul traps (weak fields), measurements are typically limited by temporal magnetic field fluctuations. Traditional co-magnetometry schemes often use interleaved measurements, which fail to fully suppress noise if the field fluctuates faster than the measurement cycle.
• Theoretical Gaps: Many atomic states, particularly in complex transition metal ions like Titanium ($Ti^+$), lack precise experimental $g$-factor data, relying solely on theoretical calculations that may miss higher-order Quantum Electrodynamics (QED) effects or negative-energy state contributions.

2. Methodology

The authors present a novel measurement scheme combining Quantum Logic Spectroscopy (QLS) with Simultaneous Co-magnetometry in a linear segmented Paul trap.

• Experimental Setup:

  • Ion Species: A two-ion crystal consisting of a spectroscopy ion ($^{48}\text{Ti}^+$) and a logic ion ($^{40}\text{Ca}^+$).
  • Logic Ion: $^{40}\text{Ca}^+$ serves as the co-magnetometer. Its ground state $g$-factor ($g_{Ca}$) is precisely known from Penning trap experiments.
  • Spectroscopy Ion: $^{48}\text{Ti}^+$ is the target. Its ground state is the $a^4F$ manifold with four fine-structure levels ($J = 3/2, 5/2, 7/2, 9/2$).
  • Cooling & Readout: The system uses Doppler, EIT, and sideband cooling. State information from the $Ti^+$ ion is transferred to the shared motional mode of the crystal and read out via the $Ca^+$ ion using quantum logic techniques (Raman transitions and Rapid Adiabatic Passage).

• Simultaneous Ramsey Interferometry:

  • Unlike previous interleaved methods, the experiment probes Zeeman transitions in both ions simultaneously using a single radio-frequency (RF) antenna.
  • A Ramsey sequence is applied to both the $^{40}\text{Ca}^+$ $2S_{1/2}$ ground state and the $^{48}\text{Ti}^+$ $a^4F_J$ states.
  • Principle: The Zeeman splitting energy $\Delta E$ is proportional to $g \cdot B$. By measuring the ratio of the Zeeman splittings ($\Delta E_{Ti} / \Delta E_{Ca}$) simultaneously, the magnetic field $B$ cancels out:
    $$g_{Ti} = g_{Ca} \frac{\Delta E_{Ti}}{\Delta E_{Ca}}$$
  • This simultaneous interrogation suppresses systematic errors caused by temporal magnetic field fluctuations (e.g., 50 Hz AC line noise), as both ions experience the same field variation at the exact same time.

• Theoretical Framework:

  • Theoretical $g$-factors were calculated using a combination of Configuration Interaction (CI) and Second-Order Many-Body Perturbation Theory (MBPT).
  • This approach accounts for both valence-valence and core-valence correlations.
  • The study also evaluated contributions from QED corrections and negative-energy states.

3. Key Contributions

1. Simultaneous Co-magnetometry Scheme: The primary innovation is the implementation of a simultaneous Ramsey interrogation on two different species in a Paul trap. This achieves a suppression of magnetic field noise that allows for $g$-factor measurements at the $10^{-6}$ level without the need for high magnetic fields.
2. First Measurement of $^{48}\text{Ti}^+$ Ground State $g$-factors: This work provides the first experimental determination of the Landé $g$-factors for all four $|J\rangle$ states of the $a^4F$ ground state of $^{48}\text{Ti}^+$.
3. Validation of Advanced Theory: The study provides a rigorous benchmark for CI+MBPT theoretical methods in complex multi-electron systems, specifically testing the sensitivity of $g$-factors to core-valence correlations and QED effects.
4. Scalability: The method is species-independent and applicable to ions that cannot be directly laser-cooled, opening avenues for precision spectroscopy of other astrophysically relevant elements.

4. Results

• Experimental Precision: The experiment achieved a relative statistical uncertainty of approximately $10^{-6}$ for all four measured $g$-factors. The Allan deviation analysis confirmed that the scheme successfully recovers white noise scaling, indicating robustness against magnetic field drifts.
• Measured Values: The experimentally determined $g$-factors (e.g., $g_{3/2} = 0.3984617(5)_{stat}(244)_{sys}$) show excellent agreement with the new theoretical predictions.
• Systematic Uncertainties: The dominant systematic uncertainties arise from:
  • Magnetic field gradients between the two ion positions (estimated $\sim 1.7 \times 10^{-6}$ relative uncertainty).
  • AC Zeeman shifts induced by the trap drive and RF pulses.
  • Quadratic Zeeman shifts (second-order effects) were calculated and found to be below the statistical uncertainty but were included in the theoretical model.
• Theory vs. Experiment:
  • The CI+MBPT theoretical values agreed with the experimental data within the expected accuracy.
  • The study found that core-valence correlations contribute at the level of $(3-5) \times 10^{-5}$.
  • A small residual discrepancy between theory and experiment suggests that higher-order QED contributions and negative-energy state effects (not fully included in the current theory) become significant at this level of precision.

5. Significance

• Astrophysics: Titanium is abundant in cosmic spectra (stars, quasars). Precise $g$-factors are critical for interpreting these spectra, particularly for studies on the variation of fundamental constants and stellar composition.
• Fundamental Physics: The agreement between experiment and theory validates the CI+MBPT approach for transition metal ions. The remaining discrepancies highlight the necessity of including negative-energy states and higher-order QED effects in atomic structure calculations for heavy, multi-electron systems.
• Metrology: The demonstrated technique offers a pathway to high-precision magnetic field sensing and $g$-factor measurements for a wide range of atomic species, including those with short-lived states or those lacking direct cooling transitions, which were previously inaccessible to high-precision Penning trap measurements.

In conclusion, this paper establishes a robust method for high-precision $g$-factor measurements in weak magnetic fields, successfully bridging the gap between experimental capabilities and complex theoretical predictions for transition metal ions.