Optical pumping simulations and optical Rabi frequency measurements in $^{151}\mathrm{Eu}^{3+}\!:\mathrm{Y}_2\mathrm{SiO}_5$ under magnetic field

This paper presents a numerical simulator for optical pumping and develops experimental methodologies to precisely characterize the 36 optical-hyperfine transitions in $^{151}\mathrm{Eu}^{3+}\!:\mathrm{Y}_2\mathrm{SiO}_5$ under a magnetic field, ultimately deriving the optical dipole moment for the $^7\!F_0 \leftrightarrow {}^5\!D_0$ transition.

The Gist

Imagine you are trying to run a high-tech library, but instead of books, you are storing quantum information (the "secret codes" of the universe). To keep these codes safe for a long time, you use special crystals called Europium-doped Yttrium Orthosilicate (Y2SiO5).

However, there is a massive problem: this library is incredibly messy.

The Problem: The "Noisy Library"

Imagine this library has millions of books, but they aren't organized. Every book is slightly different—some are a bit thicker, some are a bit thinner, and some are slightly different colors. In science, we call this "inhomogeneous broadening."

Even worse, each book has 36 different "chapters" (called hyperfine transitions) that you can use to store information. If you try to shout a command to the library, the command hits all the books at once, creating a chaotic mess of noise. You can't pick just one specific chapter to write your secret code in because everything is overlapping.

The Solution: The "Laser Librarian"

The researchers in this paper, Jingjing Chen and Mikael Afzelius, have developed a way to become "Master Librarians." They use three main "tools" to clean up the mess:

1. The "Class Cleaning" Simulator (The Sorting Machine)

Before they even touch the crystal, they built a computer simulator. Think of this as a virtual sorting machine. It predicts exactly how much "noise" they will get if they use certain laser frequencies. It helps them figure out how to "pump" the unwanted books away, leaving only one specific "class" of books (one specific frequency) standing in the middle of the room. This is like clearing a single aisle in a crowded warehouse so you can work in peace.

2. The "Magnetic Compass" (The Precision Tuner)

To make the library work, they need to apply a magnetic field. But this isn't just any field; it’s like trying to balance a needle on its tip while driving a car. If the magnetic field is slightly tilted, the "chapters" in the books move to the wrong places.

The researchers developed a way to measure the magnetic field vector with incredible precision. They use techniques called RHS and SHB (which you can think of as "listening to the echoes" of the atoms) to figure out exactly which way the magnetic field is pointing. This allows them to "tune" the crystal perfectly, ensuring the chapters are exactly where they need to be.

3. The "Rabi Frequency" Test (The Speedometer)

Once the library is clean and the magnetic field is set, they need to know how fast they can actually "write" and "read" the information. They use something called the Rabi frequency.

Think of this as a speedometer for light. By hitting the atoms with laser pulses, they watch how the atoms "dance" (oscillate) between states. By measuring this dance, they can calculate the optical dipole moment—essentially, how "sensitive" the atoms are to the light. It’s like figuring out exactly how much pressure you need to apply to a pen to write clearly on a specific type of paper.

Why does this matter?

By mastering this "messy library," the researchers have proven that we can control these crystals with extreme precision.

They successfully mapped out the "branching ratios"—a master map of which chapters connect to which. This is the "instruction manual" for future quantum computers. Because they can now isolate single frequencies and predict exactly how the atoms will behave, they have paved the way for Quantum Memories: devices that can catch a quantum signal, hold onto it without it fading, and release it exactly when we need it.

In short: They turned a chaotic, noisy room of overlapping signals into a perfectly organized, high-speed filing cabinet for the quantum age.

Technical Summary

Technical Summary: Optical Pumping Simulations and Optical Rabi Frequency Measurements in $^{151}\text{Eu}^{3+}:\text{Y}_2\text{SiO}_5$ under Magnetic Field

Problem Statement

Europium-doped $\text{Y}_2\text{SiO}_5$ is a premier candidate for optical quantum memories due to its exceptionally long optical and spin coherence times. However, operating these memories at high magnetic fields—necessary to resolve hyperfine levels and reach Zero First-Order Zeeman (ZEFOZ) transitions—introduces significant complexity. In $^{151}\text{Eu}^{3+}:\text{Y}_2\text{SiO}_5$, an external magnetic field splits the hyperfine structure into 36 distinct optical-hyperfine transitions. The frequencies and transition strengths (branching ratios) of these transitions are highly sensitive to the magnetic field's magnitude and orientation due to the anisotropy of the doping site. Precise knowledge of these parameters is essential for designing efficient quantum memory protocols, such as Atomic Frequency Comb (AFC) storage.

Methodology

The authors employ a multi-faceted approach combining numerical modeling, advanced spectroscopy, and coherent control:

1. Numerical Simulation: The researchers developed a generalized numerical simulator to model optical pumping in inhomogeneously broadened multi-level systems. The model simulates population redistribution via optical excitation and spontaneous emission, allowing them to design "class cleaning" (CC) and "spin polarization" (SP) sequences to isolate a single frequency class of ions from the inhomogeneous broadening.
2. Magnetic Field Vector Estimation: To accurately predict transition frequencies, the magnetic field vector $\mathbf{B}$ must be known. The authors used two complementary techniques:
   • Raman Heterodyne Scattering (RHS): Measuring radio-frequency (RF) spin transitions between ground-state hyperfine doublets to determine the field orientation and magnitude.
   • Spectral Hole Burning (SHB): Measuring optical hole and anti-hole frequencies to extract information from both ground and excited state manifolds.
3. Rabi Frequency Measurements: To determine transition strengths, the authors performed coherent driving of 21 out of the 36 possible optical-hyperfine transitions. By measuring the optical depth oscillations (Rabi oscillations) as a function of pulse duration, they extracted the Rabi frequencies.
4. Branching Ratio Reconstruction: Using the measured Rabi frequencies and applying non-linear least-squares optimization subject to conservation of population constraints (row and column sums of the $6 \times 6$ matrix), they reconstructed the relative transition strength matrix ($\gamma$).

Key Contributions

• A Robust Simulator: A versatile tool for predicting the effect of optical pumping on inhomogeneous spectra, useful for any multi-level atom.
• High-Precision Field Mapping: Demonstrated methodologies to estimate the magnetic field vector with high accuracy using both RF and optical spectroscopy.
• Experimental Validation of Spin Hamiltonians: Provided a rigorous test of existing spin Hamiltonian models by comparing predicted branching ratios with experimentally derived ones.
• Determination of Fundamental Constants: Derived the absolute optical dipole moment for the $^7\text{F}_0 \leftrightarrow {}^5\text{D}_0$ transition.

Results

• Branching Ratio Matrix: The reconstructed $6 \times 6$ matrix of relative transition strengths showed excellent agreement with theoretical predictions derived from spin Hamiltonian models. This confirmed that the specific tensor solutions proposed in previous literature (Ref [32]) are correct.
• Optical Dipole Moment: The study yielded an absolute optical dipole moment for the $^7\text{F}_0 \leftrightarrow {}^5\text{D}_0$ transition of $(6.94 \pm 0.09) \times 10^{-33} \text{ C}\cdot\text{m}$.
• Field Precision: The magnetic field vector estimation was highly precise, with residuals in the RF frequency fits as low as $12 \text{ kHz}$ and SHB deviations within the $5\text{--}12 \text{ kHz}$ regime.
• Bandwidth Limits: The simulation identified that the effective bandwidth for preparing a single frequency class is limited by the Zeeman splitting of the excited state (approx. $2.36 \text{ MHz}$ at $230 \text{ mT}$).

Significance

This work provides the "instruction manual" for operating $^{151}\text{Eu}^{3+}:\text{Y}_2\text{SiO}_5$ in high-field regimes. By proving that spin Hamiltonian models can reliably predict energy levels and transition strengths, the authors enable the precise engineering of quantum memory protocols. The ability to isolate specific frequency classes and accurately predict transition strengths is critical for maximizing the efficiency and bandwidth of quantum repeaters and long-distance quantum communication networks.