Spectroscopy and Coherence of an Excited-State Transition in Tm$^{3+}$:YAlO$_3$ at Telecommunication Wavelength

This paper reports the first demonstration of optical coherence in an excited-state transition of a rare-earth crystal, characterizing the spectroscopic and coherence properties of the 1451.37 nm transition in Tm$^{3+}$:YAlO$_3$ at telecommunication wavelengths and achieving a coherence time of 4.75 $\mu$s, thereby suggesting its potential for quantum technology applications.

The Gist

Imagine a crystal as a vast, quiet library filled with millions of tiny, invisible librarians. In this specific story, the library is made of Yttrium Aluminum Perovskite, and the librarians are Thulium ions (a type of rare-earth element).

Usually, scientists study these librarians when they are sitting in their "ground state"—essentially, when they are resting in their chairs at the bottom of the library. But this paper is special because the researchers decided to study the librarians while they were standing up and working in a higher, more active part of the library.

Here is a breakdown of what they did, using simple analogies:

1. The Special Wavelength (The "Telecom" Connection)

Most of these crystal libraries are studied using light that travels at a wavelength of about 1532 nanometers (like a specific shade of infrared). However, the researchers found a different "aisle" in the library where the light travels at 1451 nanometers.

Why does this matter? Think of the internet's fiber-optic cables as a highway. The 1532 nm light is like a car driving on a highway that has a few speed bumps. The 1451 nm light found in this paper is like a car driving on a highway that is almost perfectly smooth, with very little friction (loss). This makes it a potential "super-highway" for future quantum internet, allowing information to travel further without degrading.

2. The "Excited State" Challenge

Usually, when a librarian (an ion) stands up (gets excited), they are very wobbly and lose their balance quickly. It's hard to get them to hold a steady pose long enough to do complex tasks.

In this experiment, the researchers managed to get these ions to stand up and hold a steady, coherent pose for a surprisingly long time: 4.75 microseconds.

• The Analogy: Imagine trying to balance a spinning top on a table. Usually, it falls over in a split second. These researchers managed to keep the top spinning steadily for a tiny fraction of a second longer than anyone has ever managed for this specific type of "standing up" (excited-state) transition in a rare-earth crystal.

3. The Magnetic "Tuning Fork"

To keep these wobbling ions steady, the researchers used a magnetic field (like a giant, invisible tuning fork).

• They found that as they increased the magnetic field strength, the ions became more stable and less likely to wobble.
• They also discovered that the ions' "voices" (their energy levels) shifted slightly depending on the magnetic field, similar to how a guitar string changes pitch when you tighten it. This shift followed a specific mathematical rule (the quadratic Zeeman effect), which helped them understand the internal structure of the ions.

4. The "Spectral Hole" Game

To measure how steady the ions were, the researchers played a game called Spectral Hole Burning.

• The Analogy: Imagine a crowded room where everyone is humming at slightly different pitches. If you shout a specific note, the people humming that exact note stop and go silent, creating a "hole" in the noise.
• By shouting a specific laser note, they created a quiet spot (a hole) in the crowd's noise. They then watched how fast that hole got filled back in by the "wobbly" neighbors.
• They found that if they reduced the number of ions in the room (lower concentration) and used a stronger magnetic field, the hole stayed open longer. This proved that the ions were holding their "coherence" (staying in sync) for that record-breaking 4.75 microseconds.

5. Why This is a Big Deal (According to the Paper)

The paper claims this is the first time anyone has successfully measured this kind of stability (coherence) for an "excited-state" transition in a rare-earth crystal.

• The Metaphor: Previously, scientists could only study the librarians when they were sitting down (ground state). This paper proves you can study them while they are standing up and working, and they can still stay focused long enough to be useful.
• The Potential: Because this light travels so well in standard fiber-optic cables (the "smooth highway"), the authors suggest this could be a new way to build quantum memories (storage for quantum information) or single-photon sources (generators for single particles of light) that work directly with existing internet infrastructure.

In Summary:
The researchers took a crystal, cooled it to near absolute zero, and used magnets to help a specific group of atoms stand up and stay steady. They proved that these "standing" atoms can hold a quantum state for a tiny but record-breaking amount of time, using a color of light that is perfect for traveling through the world's existing internet cables.

Technical Summary

Technical Summary: Spectroscopy and Coherence of an Excited-State Transition in Tm³⁺:YAlO₃ at Telecommunication Wavelength

Problem and Motivation
Rare-earth ions doped in cryogenically cooled inorganic crystals have become central to quantum technology, particularly for quantum memories and single-photon sources. Historically, research has focused on optical coherence for transitions originating from the electronic ground state, where coherence times can exceed milliseconds or even hours. However, transitions between long-lived excited states, while crucial for solid-state lasers (e.g., Nd:YAG), have not been investigated for optical coherence at cryogenic temperatures. This gap is significant because the long excited-state lifetimes of rare-earth 4f levels theoretically permit similarly long coherence times for excited-state transitions. Furthermore, exploiting such transitions could extend rare-earth crystal applications to new wavelengths, specifically the telecommunication band, offering alternatives to existing erbium-doped systems.

Methodology
The authors characterized the spectroscopic and coherence properties of the 1451.37 nm zero-phonon line (ZPL) corresponding to the excited-state transition between the $^3F_4$ and $^3H_4$ manifolds in a thulium-doped yttrium aluminum perovskite (Tm³⁺:YAlO₃) crystal at approximately 1.5 K.

• Experimental Setup: Experiments were conducted on a 0.1% doped YAlO₃ crystal cooled to ~1.5 K using a pulse-tube cooler with an adiabatic demagnetization stage. A superconducting solenoid applied homogeneous magnetic fields up to 2 T along the crystal's b-axis.
• Optical Pumping: To study the $^3F_4 \to ^3H_4$ transition, the authors first populated the lowest crystal field level of the $^3F_4$ manifold (Y1). This was achieved by optically pumping ions from the ground state ($Z_1$ in $^3H_6$) to higher levels within the $^3F_4$ manifold using a 1686 nm laser. Rapid intra-manifold relaxation (nanosecond scale) funnels population into the Y1 level.
• Spectroscopy: The team measured absorption spectra between the $^3H_6$ and $^3F_4$ manifolds, and subsequently between the $^3F_4$ and $^3H_4$ manifolds using a tunable continuous-wave laser around 1450 nm. Photoluminescence detection at ~800 nm was used to identify resonances and optimize polarization.
• Coherence Measurements: Optical coherence times ($T_2$) were assessed using two primary techniques:
  1. Spectral Hole Burning (SHB): Narrow-bandwidth laser pulses created spectral holes. The width of these holes, measured as a function of wait time and magnetic field, provided bounds on the homogeneous linewidth.
  2. Free Induction Decay (FID): Ions were excited with pulsed lasers, and the subsequent exponential decay of the emitted light intensity was measured to determine dephasing rates.
• Variable Parameters: The study systematically varied magnetic field strength (0–2 T), ion concentration (via optical depth), and wait times between excitation and readout to isolate sources of decoherence, such as spectral diffusion and spin-flip processes.

Key Results

• Spectroscopic Characterization: The authors identified a well-isolated ZPL at 1451.37 nm with an inhomogeneous linewidth of 1.29 GHz. They measured radiative lifetimes of 6.21 ± 0.01 ms for the $^3F_4$ (Y1) level and 845.4 ± 2.4 µs for the $^3H_4$ (X1) level.
• Magnetic Interactions: The study quantified a quadratic Zeeman shift for the transition of $\Delta\nu = -101 \pm 3$ MHz/T². Spectral hole burning revealed hyperfine splitting of the energy levels due to the enhanced nuclear Zeeman effect, with splitting rates of $\pm 25.6$ MHz/T and $\pm 41.6$ MHz/T attributed to the splitting of the Y1 and X1 doublets, respectively.
• Coherence Times: The optical coherence time ($T_2$) was found to be dependent on magnetic field, wait time, and ion concentration.
  • Spectral diffusion was observed, characterized by an average bath spin-flip rate of $R = 7.82 \pm 1.75$ kHz.
  • By reducing the optical depth (lowering the concentration of ions in the Y1 level) and applying a 2 T magnetic field, the authors achieved a maximum optical coherence time of 4.75 ± 0.07 µs.
  • Results from SHB and FID measurements showed good agreement.
• Decoherence Mechanisms: The data suggests that spin flip-flops between neighboring Tm ions (or instantaneous spectral diffusion caused by exciting ions to the X1 level) are the primary sources of spectral diffusion in the measured regime. The impact of phonon-driven spin flips was found to be insignificant.

Significance and Claims
The paper claims to present the first demonstration of optical coherence for an excited-state transition in any rare-earth crystal. While the measured coherence time of ~4.75 µs is currently shorter than those achieved for ground-state transitions in similar materials, the authors argue that this result proves the fundamental feasibility of using excited-state transitions for quantum applications.

The significance of this work lies in:

1. New Wavelength Access: The 1451 nm transition operates in a telecommunication window with low transmission loss (comparable to the C-band), offering a potential alternative to erbium-doped crystals (1532 nm) for quantum networks.
2. Novel Protocols: The specific energy-level structure of Tm³⁺:YAlO₃, featuring a ring of three ZPLs (two ground-state and one excited-state), enables potential novel protocols. These include the generation of energy-time entangled photon pairs via cascading transitions and quantum computing schemes utilizing one transition for readout and another for dipole-blockade gates.
3. Future Potential: The authors note that coherence times are expected to increase with further optimization, such as lower temperatures (<0.8 K), reduced ion concentrations, and higher magnetic fields, potentially reaching tens of microseconds. This suggests that excited-state transitions in rare-earth crystals could become viable for quantum information processing applications.