Characterization of rf field-induced a.c. Zeeman shift in multi-level highly charged ions

This paper experimentally characterizes the trap radio-frequency-induced a.c. Zeeman shift in highly charged $\mathrm{Ca}^{14+}$ ions using quantum logic spectroscopy with a co-trapped $\mathrm{Be}^{+}$ ion, confirming the shift's negligible impact on optical clock accuracy while demonstrating techniques applicable to other multi-level atomic systems.

The Gist

The Big Picture: Building a Perfect Clock

Imagine scientists are trying to build the most perfect clock in the universe. They aren't using gears or springs; they are using ions (electrically charged atoms) that vibrate at incredibly precise frequencies. These "atomic clocks" are so accurate they could measure the age of the universe without losing a single second.

However, to make these clocks perfect, the scientists need to eliminate every tiny source of error. One of the sneaky culprits is the trap holding the ion.

The Problem: The "Shaking Cage"

To keep an ion from flying away, scientists use a device called a Paul Trap. Think of this trap as an invisible, electromagnetic cage. It uses rapidly oscillating radio waves (RF) to keep the ion suspended in mid-air, like a juggling ball kept in the air by a fan blowing up and down.

But here's the catch: When you wiggle an electric charge with radio waves, it creates a tiny, invisible magnetic field that shakes along with it.

• The Analogy: Imagine you are trying to listen to a whisper (the ion's clock tick) while standing next to a loud, vibrating speaker (the trap's radio waves). That vibration creates a "magnetic hum" that can slightly distort the whisper. In physics, this is called the AC Zeeman shift. If you don't measure and correct for this hum, your clock will be slightly off.

The Challenge: The "Super-Heavy" Ion

Usually, scientists use simple atoms (like Calcium or Barium) for these clocks. But for the next generation of ultra-precise clocks, they want to use Highly Charged Ions (HCIs).

• The Analogy: Think of a regular atom as a light, fluffy dandelion seed. It's easily blown around by the wind (external magnetic fields). A Highly Charged Ion is like a heavy, dense bowling ball. Because it has so much charge packed into a tiny space, it is incredibly stubborn. It barely notices the wind.

The problem? Because the "bowling ball" is so stubborn, the usual tricks to measure the magnetic hum don't work well. The ion doesn't react strongly enough to the magnetic shake to give a clear signal.

The Solution: The "Translator" and the "Dance"

The team (led by researchers in Germany) came up with a clever two-part strategy to measure this magnetic hum using a Highly Charged Calcium ion (Ca¹⁴⁺).

1. The "Translator" (The Logic Ion)

Since the Calcium ion is so tough to read directly, they trapped a second, lighter ion (Beryllium) right next to it.

• The Analogy: Imagine the Calcium ion is a mute, heavy rock. The Beryllium ion is a chatty bird sitting on the rock. You can't talk to the rock, but you can talk to the bird. If the rock shakes, the bird shakes with it. By watching the bird, you know exactly what the rock is doing.
• In the lab: They use the Beryllium ion to "cool" the Calcium ion down to a standstill and then "read" its state. This is called Quantum Logic Spectroscopy.

2. Measuring the "Shake" (The Transverse Component)

To measure the magnetic field shaking side-to-side (transverse), they used a phenomenon called Autler-Townes splitting.

• The Analogy: Imagine the Calcium ion has three specific dance steps (energy levels) it can do. Usually, these steps are spaced out evenly. But when the trap's radio wave shakes at just the right frequency, it forces the ion to "dance" between these steps.
• The Result: This interaction splits the ion's signal into two distinct peaks, like a single note splitting into a harmony. By measuring how wide this split is, they can calculate exactly how strong the magnetic "shake" is. They found that because the Calcium ion is so heavy, the shake is actually quite small compared to lighter ions.

3. Measuring the "Push" (The Longitudinal Component)

To measure the magnetic field pushing up and down (longitudinal), they used the Beryllium bird again.

• The Analogy: They asked the Beryllium bird to listen to a specific radio frequency that is supposed to be immune to magnetic fields. But, because the "shake" from the trap is so strong, even this immune frequency gets slightly distorted. By measuring that tiny distortion, they could calculate the strength of the vertical magnetic push.

The Verdict: The Clock is Safe!

After all this measuring, the team found something exciting:

1. The Shake is Tiny: Because the Highly Charged Ion is so "heavy" and stable, the magnetic noise from the trap is incredibly weak.
2. The Impact is Negligible: The error this noise causes in the clock is so small (less than 1 part in 10²²) that it doesn't matter for practical purposes.

Why This Matters

This paper proves that Highly Charged Ions are ready to be the stars of the next generation of atomic clocks. They are so robust that they don't need the ultra-perfect, expensive magnetic shielding that current clocks require.

The Takeaway:
Scientists successfully used a "chatty bird" (Beryllium) to listen to a "mute bowling ball" (Calcium) to prove that the "shaking cage" holding them doesn't actually bother the ball much. This means we can build clocks that are even more accurate, helping us test the fundamental laws of the universe and keep time better than ever before.

Technical Summary

1. Problem Statement

Optical ion clocks based on Highly Charged Ions (HCIs) are promising candidates for next-generation frequency standards and tests of fundamental physics due to their extreme suppression of sensitivity to external perturbations. However, achieving high accuracy requires a precise characterization of systematic shifts, specifically the second-order Zeeman shift induced by the radio-frequency (rf) magnetic field ($B_{rf}$) inherent to Paul traps.

While the static (d.c.) magnetic field can be measured via Zeeman splitting, the oscillating (a.c.) magnetic field generated by the trap drive is more complex. Previous methods for measuring the transverse component of this field relied on Autler-Townes (AT) splitting in effectively two-level systems. This approach faces a significant challenge in HCIs (and some singly charged ions like $B^+$, $Al^+$, $In^+$):

• HCIs have extremely small second-order Zeeman sensitivity coefficients.
• Consequently, the Zeeman sublevels within a fine-structure manifold (e.g., $J>0$) remain nearly equally spaced under typical static fields.
• This creates a multi-level interacting system rather than an isolated two-level system, rendering standard AT splitting analysis for two-level systems insufficient or inapplicable.

2. Methodology

The authors developed a comprehensive experimental framework to characterize the vector components of the trap-induced a.c. magnetic field using a $^{9}\text{Be}^+ - ^{40}\text{Ca}^{14+}$ two-ion crystal confined in a cryogenic linear Paul trap.

Experimental Setup

• Ion Species: $^{40}\text{Ca}^{14+}$ serves as the spectroscopy ion (HCI), while $^{9}\text{Be}^+$ acts as the logic ion for sympathetic cooling and quantum logic readout.
• Magnetic Field Configuration: A static magnetic field ($B_{dc}$) is applied at $\approx 30^\circ$ relative to the trap axis to define the quantization axis.
• Trap Drive: The Paul trap operates at $\Omega_{rf}/(2\pi) \approx 24.2$ MHz.

Measurement Techniques

The study decomposes the a.c. magnetic field $\mathbf{B}_{rf}$ into spherical components ($B_+, B_-$) and a longitudinal component ($B_z$), measuring them via two distinct methods:

A. Transverse Components ($B_+$ and $B_-$) via Multi-Level AT Splitting

• Principle: The authors exploit the resonant coupling between the three Zeeman sublevels ($m_J = -1, 0, +1$) of the $^{40}\text{Ca}^{14+}$ $^3P_1$ state when the Zeeman splitting ($\delta$) matches the trap drive frequency ($\Omega_{rf}$).
• Process:
  1. Adjust $B_{dc}$ to satisfy the resonance condition ($\delta \approx \Omega_{rf}$).
  2. Probe the optical transition $|^3P_0, m_J=0\rangle \to |^3P_1, m_J=0\rangle$ using a 570 nm laser.
  3. Observe the Autler-Townes splitting in the excitation spectrum caused by the coupling of the $m_J=0$ state to $m_J=\pm 1$ states via the a.c. field.
  4. Directional Control: By aligning $B_{dc}$ parallel and anti-parallel to the laser propagation direction, the authors isolate the coupling strengths for $B_+$ and $B_-$ separately.
• Analysis: The experimental spectra are fitted to a three-level interaction model (solving the Liouville equation numerically) to extract the Rabi frequencies ($\Omega_+$), which are directly proportional to the magnetic field amplitudes.

B. Longitudinal Component ($B_z$) via $^{9}\text{Be}^+$ Hyperfine Shift

• Principle: The longitudinal component is inferred by measuring the second-order Zeeman shift of the magnetic-field-insensitive "clock" transition in $^{9}\text{Be}^+$ ($|F=2, m_F=0\rangle \to |F=1, m_F=0\rangle$).
• Process:
  1. Measure the frequency shift of the $^{9}\text{Be}^+$ clock transition as a function of the trap rf power (characterized by the secular motional frequency $\omega_r$).
  2. The shift is proportional to the mean square of the total a.c. field: $\langle B_{rf}^2 \rangle = \langle B_z^2 \rangle + \frac{1}{4}(\langle B_+^2 \rangle + \langle B_-^2 \rangle)$.
  3. Since $B_+$ and $B_-$ were already measured, $B_z$ is derived by subtracting the known transverse contributions from the total measured shift.
  4. Statistical Handling: Due to experimental uncertainties potentially yielding non-physical results ($B_z^2 < 0$), a Monte-Carlo sampling method was employed to determine the probability distribution of $B_z$.

3. Key Contributions

1. Extension of AT Splitting to Multi-Level Systems: The paper successfully adapts Autler-Townes spectroscopy for systems where Zeeman sublevels are equally spaced, treating the $^3P_1$ manifold of $Ca^{14+}$ as a driven three-level system rather than a two-level approximation.
2. Vector Characterization of $B_{rf}$: It provides the first complete vector characterization (transverse and longitudinal components) of the trap-induced a.c. magnetic field in a highly charged ion system.
3. Quantum Logic Spectroscopy Application: Demonstrates the robust application of quantum logic spectroscopy with $Be^+$ to read out the state of a highly charged ion ($Ca^{14+}$) and measure magnetic field effects with high precision.
4. Validation of HCI Clock Robustness: Confirms that the a.c. Zeeman shift in HCIs is negligible for clock operation, validating theoretical predictions.

4. Results

The team quantified the a.c. magnetic field amplitudes normalized by the secular motional frequency ($\omega_r$), denoted as $B_{norm}$:

• Transverse Components:
  • $B_{+,norm} = 288(31) \text{ nT MHz}^{-1}$
  • $B_{-,norm} = 261(95) \text{ nT MHz}^{-1}$
• Longitudinal Component:
  • $B_{z,norm} = 80(49) \text{ nT MHz}^{-1}$ (derived via Monte-Carlo analysis).
• Total Field Impact:
  • When scaled to the trapping voltages typically used for singly charged ions, the field strength is comparable to other optical clock systems.
  • However, under the specific operating conditions of the $Ca^{14+}$ clock (which requires lower rf voltages due to the high charge-to-mass ratio), the resulting fractional second-order Zeeman shift is calculated to be below $10^{-22}$.

5. Significance

• Validation of HCI Clocks: The results confirm that the a.c. Zeeman shift is a negligible systematic error for HCI-based optical clocks, reinforcing their potential for ultra-high accuracy (beyond $10^{-19}$).
• Methodological Transferability: The technique for measuring transverse a.c. fields in multi-level systems is not limited to HCIs. It is directly applicable to other singly charged ions with low second-order magnetic sensitivity (e.g., $B^+$, $Al^+$, $In^+$), which previously posed challenges for standard AT splitting analysis.
• Relaxed Requirements: The study suggests that the requirements for characterizing trap-induced magnetic fields are less stringent for HCI clocks compared to other species, simplifying the path toward realizing next-generation optical clocks.