On the interpretation of Hahn echo measurements in electron spin resonance scanning tunneling microscopy

This paper demonstrates that Hahn echo measurements in electron spin resonance scanning tunneling microscopy (ESR-STM) are often misinterpreted as intrinsic spin coherence times because they are dominated by RF-voltage-induced relaxation, and it proposes a modified pulse protocol to reliably extract the true, significantly shorter coherence time of approximately 30 ns.

The Gist

The Big Picture: A Misunderstanding in the Lab

Imagine you are trying to listen to a very faint, rhythmic drumbeat (the spin of an atom) in a noisy room. You have a special microphone (the Scanning Tunneling Microscope, or STM) that can hear individual atoms. To keep the drumbeat going, you tap a stick against the drum (using radio waves).

For years, scientists have been using a specific trick called a "Hahn Echo" to measure how long that drumbeat stays in rhythm before it gets messy. They thought they were measuring the pure rhythm of the drum.

This paper says: "Wait a minute. You aren't measuring the rhythm; you're measuring how fast the drum gets tired."

The authors discovered that in their specific setup, the "tap" they use to listen to the drum is actually making the drum stop beating. They realized that previous measurements of how long these atoms stay "coherent" (in sync) were likely wrong because they were confusing tiredness (relaxation) with forgetting the rhythm (decoherence).

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The Characters and the Setup

• The Atom (FePc): Think of this as a tiny, spinning top sitting on a surface. It has a "spin" (like a magnetic direction).
• The Microscope Tip: A super-sharp needle hovering just above the atom.
• The Radio Frequency (RF) Voltage: This is the "tap." It's an electrical signal sent down the needle to spin the top and make it dance.
• The Tunneling Electrons: Imagine a stream of tiny raindrops falling from the needle onto the atom. These are electrons jumping across the gap.

The Problem: The Tap is Too Loud

In a perfect world, you tap the drum to make it spin, wait a moment, and then tap it again to see if it's still spinning in sync.

But in this experiment, the "tap" (the RF voltage) does three things at once:

1. It drives the spin: It makes the atom spin.
2. It probes the spin: It creates a stream of electrons (raindrops) that tell us where the spin is pointing.
3. It relaxes the spin: The act of hitting the atom with these electrons knocks it out of its rhythm.

The Analogy:
Imagine you are trying to measure how long a spinning top stays upright.

• The Old Way: You spin it, wait, and check. If it falls after 10 seconds, you say, "It has a 10-second lifespan."
• The Problem: In this experiment, every time you check the top, you have to blow on it with a straw to see if it's moving. But blowing on it actually knocks it over faster!
• The Result: You think the top is very fragile and falls quickly. But actually, you are the one knocking it over every time you look.

The authors found that the "echo" signal they were seeing wasn't the top remembering its rhythm; it was just the top getting knocked over by the electrons they were using to look at it.

The "Fake" Echo vs. The Real Echo

The paper shows that when scientists used the standard "Hahn Echo" test (a specific sequence of taps), they saw a signal that looked like a perfect, smooth decay. They thought, "Wow, the spin is staying coherent for 200 nanoseconds!"

But the authors ran a Control Test:
They messed up the sequence. They changed the timing, used the wrong pulse lengths, or even used a sequence that shouldn't create an echo at all.

• Surprise: The signal was still there! It still looked like a smooth decay.
• Conclusion: If the signal appears even when you do the "wrong" thing, it's not a real echo. It's just the atom getting tired (relaxing) because of the electrons hitting it.

The Solution: The "Two-Stop" Test

How do you prove it's a real echo and not just the atom getting tired? You need a test that only works if the atom is actually remembering its rhythm.

The authors used a "Two-Delay" test (like a two-step dance):

1. Wait for a while (Time A).
2. Tap the atom.
3. Wait for a different amount of time (Time B).
4. Tap again.

The Analogy:
Imagine a musician playing a note, waiting, then playing another note.

• Real Echo (Coherent): If the musician is in perfect time, the sound only comes back clearly if the two waiting periods are perfectly matched. If you change the timing, the music becomes a mess.
• Fake Echo (Relaxation): If the musician is just getting tired and stopping, it doesn't matter how you time the pauses. The sound will just fade away regardless of the timing.

When the authors did this "Two-Stop" test on a slightly different molecule (Fe-FePc), they saw the signal only appear when the timing was perfect. This proved that the "real" coherence time is actually much shorter (about 30 nanoseconds) than the "fake" echo time (200+ nanoseconds) they had been seeing before.

The Takeaway

1. Be Careful: Just because you see a smooth, decaying signal in these experiments doesn't mean you are seeing "quantum coherence." It might just be the measurement tool disturbing the system.
2. The Measurement is the Disturbance: In this specific type of microscope, the act of looking at the atom is what kills its rhythm.
3. New Rules: To be sure you are seeing a true quantum echo, you must use more complex tests (like the two-delay sweep) that rule out simple "tiredness" or relaxation.

In short: The paper is a "correction notice" to the scientific community, saying, "We thought we were measuring how long the quantum memory lasts, but we were actually measuring how fast the measurement tool breaks it. Here is how to tell the difference."

Technical Summary

1. Problem Statement

Electron Spin Resonance Scanning Tunneling Microscopy (ESR-STM) is a powerful technique for probing spin dynamics at the atomic scale. A standard method for measuring spin coherence times ($T_2$) in this field involves Hahn echo and Carr-Purcell (CP) pulse sequences.

• The Issue: Previous studies on systems like Ti adatoms and FePc molecules reported Hahn echo coherence times ($T_2^{Hahn}$) significantly longer (e.g., ~200 ns) than Rabi oscillation coherence times ($T_2^{Rabi}$, ~40 ns). These long values were interpreted as successful suppression of inhomogeneous dephasing.
• The Hypothesis: The authors argue that in ESR-STM, the applied radio-frequency (RF) voltage serves a dual role: it drives the spin coherently but also generates tunneling electrons that probe and relax the spin state. Consequently, the exponential decay observed in standard Hahn echo measurements may not reflect intrinsic phase coherence ($T_2$) but rather spin relaxation ($T_1$) induced by tunneling electrons. This leads to a systematic overestimation of $T_2$.

2. Methodology

The study utilizes Iron Phthalocyanine (FePc) molecules adsorbed on a bilayer of Magnesium Oxide (MgO) on Ag(001) as a model system. Experiments were conducted in a low-temperature (50 mK) ESR-STM setup.

Key Experimental Protocols:

1. Rabi Oscillations: Used to establish baseline coherence times and determine pulse durations for $\pi/2$ and $\pi$ rotations.
2. Standard Hahn Echo: A sequence of $\pi/2$ – $\tau$ – $\pi$ – $\tau$ – $\pi/2$. The delay time $\tau$ was varied to measure decay.
3. Carr-Purcell (CP) Sequences: Extended sequences with $N$ refocusing $\pi$ pulses to test for dynamical decoupling saturation.
4. Control Experiments:
   • Frequency Detuning: Measuring off-resonance to rule out electrical artifacts.
   • Sequence Imperfections: Deliberately introducing errors (unequal spacing, wrong pulse lengths, non-echo sequences) to test if the decay signal persists without coherent refocusing conditions.
5. Two-Delay Hahn Echo: A rigorous test where the first delay ($\tau_1$) and second delay ($\tau_2$) are swept independently. This was performed on Fe–FePc molecular complexes, which are less susceptible to tunneling-induced scattering, to isolate true coherent echoes.

3. Key Contributions & Findings

A. The "Relaxometry" Mechanism

The authors demonstrate that the RF voltage in ESR-STM generates tunneling currents that act as a homodyne-like detection mechanism.

• The signal depends on the spin population (longitudinal magnetization) rather than just phase coherence.
• In a standard Hahn echo, the $\pi$ and second $\pi/2$ pulses not only refocus the spin but also partially measure the remaining population.
• If the spin relaxes ($T_1$) during the delay $\tau$, the subsequent pulses detect a reduced population, creating an exponential decay that mimics a coherent echo but is actually driven by incoherent relaxation.

B. Re-evaluation of Decay Times

• Current Dependence: The extracted decay rate ($1/T_2$) scales linearly with tunneling current, consistent with $T_1$-limited relaxation.
• Carr-Purcell Anomaly: In standard CP measurements, increasing the number of pulses ($N$) led to a linear increase in apparent $T_2$ (up to ~650 ns for $N=16$) without saturation. In true dynamical decoupling, $T_2$ should saturate at $2T_1$. The lack of saturation confirms the signal is not a true echo but a cumulative measurement of $T_1$ relaxation.
• Control Experiments: Exponential decay signals persisted even when the pulse sequence was deliberately broken (e.g., unequal spacing, wrong pulse angles). This proves the decay does not rely on phase-coherent refocusing.

C. The Two-Delay Benchmark

To distinguish true coherence from relaxometry, the authors employed two-delay Hahn echo measurements on Fe–FePc complexes.

• Result: A genuine echo signal (interference dip) was observed only when $\tau_1 = \tau_2$.
• Decay: The coherent signal vanished rapidly for total evolution times >40 ns.
• Conclusion: The intrinsic coherence time ($T_2$) for these systems is actually ~30 ns, comparable to Rabi oscillation measurements and significantly shorter than previously reported Hahn echo values (~200 ns).

4. Significance

• Correction of Literature: The paper challenges the interpretation of previous ESR-STM coherence measurements, suggesting that reported $T_2$ values derived from one-delay Hahn echoes are likely overestimates reflecting $T_1$ relaxation.
• New Diagnostic Criteria: It establishes that an exponentially decaying signal alone is insufficient evidence for phase coherence in ESR-STM.
  • True Echo: Requires independent variation of $\tau_1$ and $\tau_2$ to observe interference patterns (echo condition).
  • False Echo: Signals that persist despite pulse sequence imperfections or show linear $T_2$ growth with pulse number in CP sequences are likely relaxometry artifacts.
• Practical Guidelines: The authors provide a framework for future ESR-STM experiments to distinguish between genuine quantum coherence and tunneling-induced relaxometry, urging caution in interpreting $T_2$ times from standard echo protocols.

Summary of Quantitative Results

Metric	Previous Interpretation (Hahn Echo)	New Interpretation (This Work)
FePc $T_2$	~200 ns	~30 ns (consistent with Rabi $T_2$)
Mechanism	Phase-coherent refocusing ($T_2$)	Population relaxation ($T_1$) via tunneling electrons
CP Sequence	Saturation expected	Linear increase observed (artifact)
Verification	Single delay sweep	Two-delay sweep ($\tau_1 \neq \tau_2$) required

In conclusion, the paper redefines the understanding of spin dynamics in ESR-STM, highlighting that the measurement apparatus (tunneling electrons driven by RF) fundamentally alters the observed dynamics, often masking true coherence times with relaxation-dominated signals.