Long nuclear spin coherence times for molecules trapped in high-purity solid parahydrogen

By utilizing high-purity solid parahydrogen matrices, researchers achieved significantly extended proton spin coherence times (T2 and T2*) and measured the longitudinal relaxation time (T1) for trapped HD molecules, establishing the intrinsic limits of coherence imposed by the matrix itself.

The Gist

Imagine you are trying to listen to a single violinist playing a perfect, sustained note in a crowded, noisy concert hall. If the crowd is rowdy, the violinist gets distracted, the note wavers, and the sound fades away quickly. This is exactly what happens to tiny particles called protons inside molecules when they are trapped in a solid block of frozen hydrogen.

This paper is about a team of scientists who managed to build the quietest possible concert hall in the universe to let these protons play their notes for a record-breaking amount of time.

Here is the story of how they did it, explained simply:

1. The Problem: A Noisy Crowd

In the world of quantum physics, scientists want to study the "spin" of atomic nuclei (like protons) to look for new laws of physics. Think of spin like a tiny, spinning top. To measure it accurately, all the tops need to spin in perfect sync for as long as possible.

However, in most solid materials, the environment is chaotic. The molecules are stuck in weird angles, and they bump into magnetic neighbors. It's like trying to get a choir to sing in perfect harmony while everyone is wearing different shoes, standing in different spots, and shouting at each other. The result? The "song" (the signal) gets messy and dies out in a fraction of a second.

2. The Solution: The "Silent" Ice

The scientists used a special type of frozen hydrogen called parahydrogen.

• Normal Hydrogen is like a crowd of people where some are wearing red shirts and some are wearing blue. The "red shirts" (called orthohydrogen) are magnetic and act like noisy neighbors, disrupting the signal.
• Parahydrogen is like a crowd where everyone is wearing white. They are magnetically "invisible" to each other. They don't shout or interfere.

By freezing this "white-shirted" hydrogen, they created a perfectly quiet, magnetic vacuum. They trapped a specific molecule called HD (a mix of hydrogen and deuterium) inside this ice.

3. The Magic Trick: The Spinning Top

Usually, when you trap a molecule in a solid, it gets stuck and can't move. But in this special "white-shirt" ice, the HD molecules are like ice skaters on a frictionless rink. They are trapped in the ice, but they can still spin freely!

Because they are spinning freely, they average out all the tiny magnetic jitters. It's like if a noisy fan was spinning so fast that the sound of the blades blurred into a single, smooth hum. This "spinning" allowed the protons to keep their perfect sync for much longer than anyone thought possible.

4. The Results: Holding the Note

The team measured how long the protons could stay in sync (this is called coherence time).

• The Old Way: In previous experiments with "noisier" hydrogen, the signal faded away almost instantly.
• The New Way: By cleaning up the hydrogen to remove almost all the "red shirts" (impurities), they extended the time the signal lasted.
  • They found that as they made the hydrogen cleaner, the signal lasted longer and longer.
  • Eventually, they hit a limit where the signal lasted about 0.3 seconds. In the world of quantum particles, this is an eternity!

5. Why Does This Matter?

Why do we care about a spinning proton in frozen gas?

• Super-Sensitive Detectors: These long-lasting signals act like incredibly sensitive microphones. If there is a tiny, unknown force in the universe (like a new particle or a violation of symmetry), it would make the proton's "note" wobble slightly.
• The Future: Because the signal lasts so long, scientists can detect these tiny wobbles that were previously impossible to hear. This could help us discover new physics beyond what we currently know.

The Catch and The Fix

There was one small problem: The "ice" was so quiet that the protons were too relaxed. They stopped spinning so fast that it took forever to get them to start again (a problem called long T1 time). It's like a swing that, once stopped, takes a century to start moving again.

The Fix: The scientists suggest a clever trick. They plan to sneak in a different molecule that acts like a "battery charger." They would use light to wake up this new molecule, which would then transfer its energy to the protons, getting them spinning again quickly without making the environment noisy.

In a Nutshell

The scientists built a magnetically silent, frozen stage where molecules can spin freely without being disturbed. This allowed them to keep a quantum signal alive for a record-breaking time, opening the door to listening for the faintest whispers of new physics in the universe.

Technical Summary

1. Problem Statement

Molecules trapped in solid matrices are promising candidates for precision measurements of physics beyond the Standard Model, particularly searches for symmetry-violating new physics (e.g., electron or nuclear electric dipole moments). However, a critical limitation for these experiments is the nuclear spin transverse relaxation time ($T_2^*$).

• The Challenge: In typical solid matrices, $T_2^*$ is extremely short due to inhomogeneous molecular orientations, conformational variations, and magnetic dipolar interactions with other spins in the matrix.
• The Goal: To achieve long coherence times necessary for high statistical sensitivity, researchers need a matrix that allows molecules to rotate freely (averaging out dipolar interactions) and possesses magnetic purity to minimize external spin noise.
• Previous Limitations: Prior work using solid parahydrogen ($p$-H$_2$) as a matrix achieved orthohydrogen ($o$-H$_2$) fractions ($X$) down to $2 \times 10^{-3}$. The authors aimed to extend this purity to $X = 10^{-6}$ to investigate the fundamental limits of coherence imposed by the matrix itself.

2. Methodology

The authors developed an experimental apparatus to grow and characterize HD (hydrogen-deuteride) molecules trapped in high-purity solid parahydrogen.

• Apparatus:

  • Cryogenics: A single vacuum chamber cooled by a closed-cycle pulse-tube refrigerator to 4 K.
  • Sample Growth: Samples are grown via vapor deposition of hydrogen gas onto a sapphire rod attached to a copper cold finger. Sapphire was chosen for high thermal conductivity and low magnetic susceptibility.
  • Purification: An "in-line" cryogenic catalyst purifies the hydrogen gas. By controlling the catalyst temperature, the orthohydrogen fraction ($X$) is reduced to as low as $1 \times 10^{-6}$. The catalyst also performs isotopic separation, suppressing the HD component in natural hydrogen, allowing for controlled doping of HD into the matrix via a separate gas line.
  • Magnetic Field: A 1.4 T superconducting magnet (proton Larmor frequency $\sim$60 MHz) with shim coils to ensure field uniformity.
  • Detection: A resonant "saddle coil" is used for both RF excitation (Rabi frequencies $\sim$3 kHz) and signal pickup (Free Induction Decay - FID).

• Measurement Techniques:

  • $T_2^*$ (Ensemble Transverse Relaxation): Measured via FID. The power spectrum is Fourier transformed, and the Full Width at Half Maximum (FWHM) is used to calculate $T_2^* = (\pi \cdot \text{FWHM})^{-1}$.
  • $T_2$ (Spin-Echo Coherence): Measured using Hahn echo sequences ($\pi/2 - \tau - \pi - \tau$) to refocus static field inhomogeneities. The decay of the echo amplitude yields $T_2$.
  • $T_1$ (Longitudinal Relaxation): Measured using a saturation-recovery sequence.
  • Variable Parameters: The orthohydrogen fraction ($X$) and the relative HD flow rate (dopant density) were systematically varied.

3. Key Contributions

• Record Purity: Successfully extended the orthohydrogen fraction purity in solid parahydrogen matrices from $X \approx 2 \times 10^{-3}$ to $X \approx 10^{-6}$.
• Resolution of Triplet Structure: For the first time in a solid matrix, the authors resolved the proton triplet structure of the HD molecule. This confirmed that HD molecules rotate freely within the solid $p$-H$_2$ lattice, effectively averaging intramolecular dipolar interactions.
• Scaling Laws: Established distinct scaling behaviors for coherence times ($T_2, T_2^*$) and longitudinal relaxation ($T_1$) as a function of impurity concentration ($X$).
• Identification of Fundamental Limits: Demonstrated that while $T_2$ scales with impurity concentration, there is a "zero-dopant" limit imposed by the matrix itself that cannot be overcome simply by further purification.

4. Key Results

• Free Rotation Confirmation:

  • At low $X$, the FID spectrum revealed a resolved triplet splitting of $J(H, D) = 47.2 \pm 1.1$ Hz.
  • This splitting is significantly different from the free gas phase value (43.1 Hz) but confirms that the molecular axis is not fixed, eliminating broadening from inhomogeneous orientations.

• Coherence Times ($T_2^*$ and $T_2$):

  • High Purity Regime ($X \lesssim 3 \times 10^{-3}$): Both $T_2^*$ and $T_2$ increase as $X$ decreases, scaling approximately as $1/X$. This confirms that magnetic orthohydrogen impurities are the dominant source of decoherence in this range.
  • The Plateau: Below $X \approx 10^{-4}$, $T_2$ plateaus at approximately 0.3 seconds.
  • $T_2^*$ Limit: Due to extremely long $T_1$ times at low $X$, shimming the magnetic field became impractical, preventing direct $T_2^*$ measurement. However, the measured $T_2$ provides an upper bound, suggesting $T_2^*$ plateaus between 0.02 s and 0.3 s.
  • Mechanism: The plateau suggests a fundamental limit imposed by the parahydrogen matrix itself (e.g., lattice defects or residual interactions), distinct from the magnetic impurities.

• Longitudinal Relaxation ($T_1$):

  • In the range $10^{-4} \lesssim X \lesssim 3 \times 10^{-3}$, $T_1$ scales roughly as $X^{-2}$.
  • This scaling is attributed to Fermi's Golden Rule, where the transition rate depends on the square of the magnetic field fluctuation amplitude (which scales linearly with $X$).
  • At very low $X$ ($<10^{-4}$), $T_1$ continues to increase but deviates from the $X^{-2}$ scaling, becoming strongly dependent on the HD dopant density.

5. Significance and Future Outlook

• Sensitivity for New Physics: The achieved coherence times ($T_2 \sim 0.3$ s) are significantly longer than those in molecular beams (limited by transit time) and comparable to state-of-the-art solid-state NMR using magic-angle spinning. This offers a high-density, low-temperature platform for searching for symmetry-violating physics.
• Overcoming $T_1$ Limitations: The extremely long $T_1$ at high purity makes traditional thermal polarization inefficient. The authors propose using Dynamic Nuclear Polarization (DNP) with optically excited metastable paramagnetic molecules to rapidly polarize nuclear spins, bypassing the slow $T_1$ recovery.
• Generalizability: The results suggest that the limits on coherence are general for diamagnetic molecules in this matrix, scaling with the gyromagnetic ratio. Future work will explore different molecular species to understand how rotational constants and spin-rotation couplings affect the matrix-imposed limits.

In summary, this work establishes high-purity solid parahydrogen as a viable, high-sensitivity host for molecular quantum sensors, identifying both the magnetic impurity limits and the intrinsic matrix limits on nuclear spin coherence.