Phonon-bottlenecked spin relaxation of Er$^{3+}$:CaWO$_4$ at milliKelvin temperatures

This study demonstrates that spin-lattice relaxation times in Er$^{3+}$:CaWO$_4$ at millikelvin temperatures are governed by a phonon bottleneck, evidenced by a unique $[\tanh (\hbar \omega_0/k_\text{B} T)]^2$ temperature dependence and an increase in relaxation times with spin excitation, which has significant implications for quantum technologies utilizing rare-earth spin ensembles.

The Gist

The Big Picture: A Traffic Jam for Energy

Imagine you have a room full of people (these are the electrons in the crystal) who are all dancing. When you turn on a loud speaker (a microwave pulse), they all start jumping up and down excitedly. Eventually, they get tired and want to sit down to rest.

In a normal room, they would just walk over to a wall and lean against it to cool off. In physics, this "wall" is the crystal lattice (the solid structure of the material), and the "cooling off" is called spin-lattice relaxation.

However, in this specific experiment, the researchers found that at extremely cold temperatures (colder than outer space), the people couldn't just walk to the wall. The "exit" was clogged. This is called a Phonon Bottleneck.

The Cast of Characters

• The Dancers (Er³⁺ Ions): These are tiny magnetic particles (electrons) trapped inside a crystal made of Calcium Tungstate (CaWO₄). They are the "stars" of the show.
• The Heat Carriers (Phonons): When the dancers get tired, they need to dump their energy. They do this by throwing little packets of energy called "phonons" (vibrations) into the crystal structure. Think of phonons as messengers carrying the "I'm tired" message to the rest of the building.
• The Super-Cold Room (MilliKelvin): The experiment happens at temperatures near absolute zero. At this temperature, the building is so quiet that there are very few empty seats (phonons) available for the dancers to sit in.

The Problem: The Messengers Get Stuck

Usually, when a dancer gets tired, they throw a messenger (phonon) to the wall, and the wall absorbs it instantly.

But in this experiment, the researchers cranked up the number of excited dancers. Because the room is so cold, there aren't enough "empty seats" (phonons) in the building to receive the messages.

1. The dancers throw their messengers.
2. The messengers hit the wall, but the wall is already full of other messengers from other dancers.
3. The messengers get stuck in the hallway.
4. Because the messengers can't leave, the dancers can't sit down. They stay excited for a much longer time than expected.

This traffic jam is the Phonon Bottleneck. It makes the "cooling down" process (relaxation) take much longer.

The "Traffic Jam" Analogy in Action

The researchers noticed something very specific about how long the dancers stayed excited:

• The Temperature Rule: They found that the time it took to cool down followed a very specific mathematical pattern related to the temperature, described as [tanh(ℏω0/kBT)]².
  • Simple translation: As the room gets colder, the traffic jam gets worse, and the dancers stay excited much longer. The relationship isn't a straight line; it's a curve that gets steep very quickly.
• The Magnetic Field: They also found that if they changed the magnetic field (like changing the direction the dancers are facing), the traffic jam got worse or better depending on how hard the dancers had to "throw" their messengers.

Why This Matters (According to the Paper)

The paper explains that this isn't just a weird quirk of physics; it's a real phenomenon that happens when you have a lot of these "dancers" packed together in a super-cold environment.

• The "Avalanche" Risk: The paper mentions that if too many messengers get stuck, they might suddenly all get released at once, causing a "phonon avalanche." Imagine a crowd of people all trying to leave a room at once, causing a stampede. This is bad for keeping the system stable.
• The Good News: The researchers found that if you have fewer dancers (lower concentration) or if the "hallway" is wider (different magnetic angles), the traffic jam clears up.

The Takeaway

The scientists successfully watched this "traffic jam" happen in a crystal at temperatures colder than almost anywhere else in the universe. They proved that when you try to cool down a lot of excited particles at once in a super-cold room, the energy gets stuck in the building's walls before it can escape.

This is important because if we want to use these crystals for future quantum computers (which need to keep information stable), we need to understand exactly how long it takes for the energy to clear out, so we don't accidentally cause a "stampede" that ruins the information.

Technical Summary

Technical Summary: Phonon-bottlenecked spin relaxation of Er3+:CaWO4 at milliKelvin temperatures

Problem Statement
Rare-earth ion-doped crystals, particularly Erbium-doped Calcium Tungstate (Er³⁺:CaWO₄), are promising candidates for quantum information processing due to their optical transitions in the telecom C-band and compatibility with superconducting qubit architectures. While spin coherence times ($T_2$) up to 23 ms have been reported, the ultimate limit on coherence is set by the spin-lattice relaxation time ($T_1$). At millikelvin (mK) temperatures, standard relaxation mechanisms face complications. Specifically, the scarcity of resonant phonons required to transport energy from spins to the thermal bath can lead to a "phonon bottleneck" (PB). In this regime, excitations accumulate in resonant lattice modes, potentially causing unintended re-excitation of spins or sudden energy release via phonon avalanches. While PB has been predicted and observed in other magnetic systems, characterizing it in highly doped rare-earth crystals at mK temperatures remains challenging due to spin-spin interactions and cross-relaxation. Understanding these dynamics is critical for utilizing Er³⁺ ensembles as coherent resources, especially for protocols involving high-power control pulses.

Methodology
The authors investigated spin-lattice relaxation in a nominally 50 ppm Er³⁺:CaWO₄ crystal coupled to a low-mode-volume NbN superconducting resonator (operating at 4.44 GHz). The experiment was conducted in a dilution refrigerator with a base temperature of 20 mK.

• Measurement Technique: To access slow spin-lattice relaxation processes distinct from fast spin-spin flip-flops, the authors employed a pump-probe sequence. A strong saturation pump (power ~2.5 nW, duration $t_{pump}$) was applied to depolarize the spins over a large mode volume. Subsequently, the system was weakly probed (power ~1 pW) to monitor the resonator's transmission response as spins repopulated the ground state.
• Variables: The study systematically varied:
  • Magnetic Field ($B_{mag}$): Oriented in the $ac$ plane of the crystal at angles $\theta$ relative to the $c$-axis, covering fields from ~38 mT to ~254 mT.
  • Temperature: Ranging from 20 mK to ~400 mK.
  • Pump Duration: Varied from 10 ms to 3 s to control the density of spin excitations.
  • Inhomogeneous Linewidth ($\Gamma_{inh}$): Exploited natural misalignments between cooldowns to study samples with different effective spin densities (28 MHz vs. 14 MHz).
• Data Analysis: The transmitted signal decay was fitted to triple-exponential functions. The authors focused on the slowest time constant ($T_1^{sl}$), identified as the spin-to-bath relaxation time, while analyzing the temperature and field dependencies against theoretical models for direct phonon processes and phonon-bottlenecked relaxation.

Key Results

1. Observation of Phonon Bottleneck: The study observed a significant increase in spin relaxation times ($T_1^{sl}$) with increasing spin excitations, a hallmark of the phonon bottleneck. This effect was most pronounced at lower magnetic fields and narrower spin inhomogeneous linewidths.
2. Unique Temperature Dependence: The relaxation dynamics exhibited a distinct temperature dependence following a $[\tanh(\hbar\omega_0/k_B T)]^2$ law. This deviates from the standard direct phonon process prediction and aligns with the theoretical prediction for a phonon-bottlenecked regime where the relaxation time is modified by a bottleneck term ($T_1^{sl} \approx T_{1D} + T_{1b}$).
3. Magnetic Field Dependence: At 20 mK, the relaxation rate followed an approximate $B_0^2$ dependence, consistent with spontaneous phonon emission for Kramers ions. However, deviations at higher fields and the strong dependence on pump duration confirmed the non-linear nature of the bottleneck.
4. Role of Spin Density: The bottleneck effect was suppressed in conditions with lower effective spin density (achieved via narrower linewidths or shorter pump durations). Specifically, halving the inhomogeneous linewidth ($\Gamma_{inh}$) resulted in a doubling of the bottleneck correction term ($T_{1b}$), confirming that the effect scales with the density of excited spins relative to the available phonon modes.
5. Multi-exponential Decay: The relaxation curves consistently required a triple-exponential fit. The slowest component ($T_1^{sl}$) was attributed to the spin-to-bath relaxation, while the faster components were linked to non-spin origins or potentially phonon-avalanche mediated fast relaxation preceding the bottlenecked slow phase.

Significance and Claims
The paper claims the observation of phonon-bottleneck effects in Er³⁺:CaWO₄ electron spin ensembles at millikelvin temperatures and nanowatt power excitation regimes. This is significant because:

• Sensitivity: The use of a superconducting resonator with a small mode volume allows the detection of PB effects at extremely low power levels (nW), contrasting with previous observations in electron spin ensembles that required watt-level saturation powers.
• Quantitative Characterization: The authors provide a quantitative description of the PB effect, demonstrating its dependence on spin density, magnetic field, and temperature, and validating the $[\tanh(\hbar\omega_0/k_B T)]^2$ temperature scaling.
• Implications for Quantum Technology: The findings highlight that while PB can enhance $T_1$ (potentially beneficial for coherence), it introduces non-linear thermalization dynamics. The authors note that understanding these mechanisms is vital for mitigating unfavorable effects in quantum technologies that rely on Er³⁺ ensembles, particularly for protocols involving high-power control pulses (e.g., quantum memory retrieval or microwave-to-optical transduction) where rapid phonon thermalization with the bath is necessary.

The authors conclude that while simple theoretical descriptions explain the strong dependencies observed, further study is required to fully understand the relative weights and origins of the multi-exponential decay components and the precise mechanisms of phonon avalanches in this system.