Ramsey correlation spectroscopy with phase cycling using a single quantum sensor

This paper introduces RESOLUTE, a novel Ramsey correlation spectroscopy protocol utilizing phase cycling and population imbalance storage that extends effective coherence time beyond the standard $T_2^*$ limit, enabling the detection of low-frequency signals such as $^{13}$C nuclear spin precession in regimes previously inaccessible to single quantum sensors.

The Gist

Imagine you are trying to listen to a very quiet, slow song playing in a noisy room. You have a microphone (the quantum sensor, specifically a tiny defect in a diamond called an NV center), but the microphone has a flaw: it gets "tired" very quickly. If the song is too slow, the microphone forgets the melody before it can record enough of it to understand what's being played. This is the main problem scientists face when trying to detect slow magnetic signals at the nanoscale.

This paper introduces a clever new trick called RESOLUTE (Ramsey corrElation SpectroscOpy puLse seqUence wiTh phasE cycling) to solve this problem.

Here is the explanation using simple analogies:

1. The Problem: The "Short Attention Span" Microphone

Think of the quantum sensor like a student trying to memorize a long, slow poem.

• The Poem: The magnetic signal you want to detect (like the spin of a carbon atom).
• The Student's Attention Span: The sensor's "coherence time" ($T_2^*$).
• The Issue: If the poem is very slow, the student gets distracted (noise) and forgets the beginning before they reach the end. They can't tell you what the poem was about. Traditional methods (like a standard "Ramsey" measurement) just give up if the signal is too slow because the student's memory fades too fast.

2. The Solution: The "Note-Taking" Strategy (RESOLUTE)

Instead of trying to memorize the whole slow poem in one go, RESOLUTE changes the strategy. It breaks the listening session into two parts with a "break" in the middle.

• Step 1: The First Listen. The student listens to the first half of the poem and writes down a quick note about the rhythm they heard.
• Step 2: The Break (Correlation Time). The student stops listening but keeps the note. Crucially, this note is written in a way that is immune to the noise that usually makes them forget things. It's like writing the note in permanent ink while the rest of their brain is still fuzzy.
• Step 3: The Second Listen. The student listens to the second half of the poem.
• Step 4: The Comparison. Now, they compare the new rhythm they just heard with the old note from the break.

Why this works:
If the noise in the room changes randomly between the two listening sessions, it cancels out when you compare the two. But if the poem itself (the signal) is consistent, the connection between the note and the new sound remains. This allows the student to "remember" the slow song for much longer than their natural attention span would allow.

3. The "Phase Cycling" Trick: Tuning Out the Static

The researchers added a special ingredient called Phase Cycling. Imagine the student wearing noise-canceling headphones that can be flipped upside down.

• They run the experiment twice: once with the headphones "normal" and once with them "flipped."
• When they add the results together, the constant background noise (like the hum of a fridge) cancels out because it sounds the same in both runs.
• However, the specific signal they are looking for (which changes in a specific way during the break) adds up and becomes louder.
• Result: They can now hear the "AC" (changing) signal clearly, while ignoring the "DC" (static) noise.

4. The Real-World Win: Hearing the "Whisper"

Using this new method, the team achieved something amazing:

• Extended Memory: They stretched the sensor's memory from 0.38 microseconds to 5.1 microseconds. That's a 13x improvement!
• Low Frequency Detection: They successfully detected the "Larmor precession" (a wobble) of Carbon-13 atoms in a diamond. These atoms were wobbling very slowly, at a frequency of about 50 kHz, under a very weak magnetic field (49 Gauss).
• Why it matters: Previous methods couldn't hear these slow wobbles because the sensor would "forget" them before the measurement was done. RESOLUTE allowed them to hear a whisper that was previously inaudible.

5. The "Chirped Pulse" Bonus: The Universal Remote

Finally, they combined this with Adiabatic (Chirped) Pulses.

• Imagine trying to flip a switch on a device that is spinning or tilted in a weird direction. A standard "hard" button press (a standard pulse) might miss the switch if the angle is wrong.
• A Chirped Pulse is like a "Universal Remote" that slowly sweeps through all frequencies. It doesn't matter how the target is tilted; the remote eventually finds the right frequency and flips the switch perfectly.
• By combining this "Universal Remote" with the "Note-Taking" strategy (RESOLUTE), they could detect the magnetic signature of a single electron spin with much higher clarity than before.

Summary

RESOLUTE is like a smart recording technique for quantum sensors. Instead of trying to hold a long, slow memory in your head (which fails due to noise), you write a quick note, take a break, and then compare the new sound to the note. This cancels out the noise, extends your memory, and lets you hear the faint, slow signals that were previously impossible to detect. This opens the door to seeing the magnetic "fingerprints" of single molecules and atoms with incredible precision.

Technical Summary

Here is a detailed technical summary of the paper "Ramsey correlation spectroscopy with phase cycling using a single quantum sensor" (RESOLUTE).

1. Problem Statement

Nanoscale magnetic spectroscopy using quantum sensors, such as Nitrogen-Vacancy (NV) centers in diamond, is limited by the sensor's finite coherence time ($T_2^*$).

• The Low-Frequency Gap: Conventional Ramsey spectroscopy is sensitive to DC fields but loses sensitivity to oscillating signals with frequencies lower than $1/T_2^*$ because the sensor's coherence decays before sufficient phase accumulation occurs.
• Limitations of Existing Protocols:
  • Dynamical Decoupling (DD): While sequences like Hahn Echo extend coherence ($T_2$), they rely on frequency matching where the pulse separation must match the signal frequency. This makes detecting very low-frequency signals difficult, as the sensor loses coherence between pulses.
  • Phase-Sensitive Correlation: Existing methods to detect low frequencies often require complex phase-sensitive measurements, cumbersome post-processing, and lack intrinsic frequency selectivity or coherence time extension.
• Goal: Develop a protocol that extends the effective coherence time, filters out noise, and enables the detection of low-frequency AC signals (specifically in the range between $1/T_1$and$1/T_2^*$) with high sensitivity and selectivity.

2. Methodology: The RESOLUTE Protocol

The authors introduce RESOLUTE (Ramsey corrElation SpectroscOpy puLse seqUence wiTh phasE cycling). The protocol combines Ramsey interferometry, correlation spectroscopy, and phase cycling.

• Pulse Sequence Structure:

  1. First Sensing Period: A $\pi/2$ pulse creates a superposition state. The sensor evolves for time $\tau/2$, accumulating a phase $\Phi_1$ dependent on environmental fields.
  2. Correlation Period ($T_{corr}$): A second $\pi/2$ pulse projects the accumulated phase into a population imbalance (z-axis). This state is stored for a duration $T_{corr}$ (where $T_{corr} < T_1$). Crucially, this population state is immune to dephasing ($T_2$) effects, limited only by relaxation ($T_1$).
  3. Second Sensing Period: A third $\pi/2$ pulse re-creates a superposition based on the stored population. The sensor evolves for another $\tau/2$, accumulating a second phase $\Phi_2$.
  4. Readout: A final $\pi/2$ pulse projects the state for optical readout.

• Phase Cycling:

  • The sequence is repeated four times with specific phase combinations for the middle pulses ($\phi = x$ or $y$) and the final readout pulse ($RO = \pm x$).
  • Signal Extraction: By adding and subtracting the signals from different phase cycles ($S(x) \pm S(y)$), the protocol isolates specific field components:
    • Subtraction ($S(x) - S(y)$): Eliminates static DC fields and correlated noise that remains constant between the two sensing periods. It isolates signals that change during $T_{corr}$ (e.g., dipolar coupling flips).
    • Addition ($S(x) + S(y)$): Isolates correlated fields (e.g., AC fields oscillating at a frequency matching $T_{corr}$).

• Frequency Filtering Mechanism:

  • The protocol acts as a frequency filter where the matching condition is determined by the correlation time $T_{corr}$ rather than the coherence time.
  • If $T_{corr}$ matches the period of an oscillating signal ($T_{corr} = n \cdot T_{signal}$), the phase shifts accumulate constructively, while uncorrelated noise averages out.

• Enhanced Sensing (Chirped Pulses):

  • To detect electron spins (DC dipolar interactions), the authors combine RESOLUTE with adiabatic (chirped) pulses applied during $T_{corr}$.
  • These pulses flip the target electron spin state, converting the static DC dipolar interaction into a correlated AC signal that survives the subtraction process. Chirped pulses are used to overcome orientation dependence and bandwidth limitations of hard $\pi$-pulses.

3. Key Contributions

1. Extension of Effective Coherence Time: RESOLUTE creates a new effective coherence time $T_2^p$ that is significantly longer than the intrinsic Ramsey $T_2^*$. It shifts the frequency-matching condition from the coherence time to the correlation time ($T_{corr}$).
2. Access to the "Low-Frequency Gap": The protocol enables detection of signals in the spectral region between $1/T_1$and$1/T_2^*$, a regime previously inaccessible to standard Ramsey or Hahn Echo sequences.
3. Noise Filtering via Correlation: By storing phase as population imbalance, the protocol filters out noise sources that fluctuate slower than $T_{corr}$, effectively creating a "quiet" window for sensing.
4. Theoretical Framework: The authors provide a rigorous analysis using Fisher Information to quantify the frequency estimation capabilities, comparing RESOLUTE against Ramsey, Hahn Echo, and Correlated Ramsey protocols.
5. Robust Spin Control: Integration of adiabatic pulses allows for robust detection of electron spins regardless of their orientation, overcoming limitations of standard DEER sequences.

4. Experimental Results

• Coherence Time Extension:
  • Using a single NV center, the authors measured $T_2^* = 0.38 \pm 0.2$ $\mu$s (Ramsey) and $T_2 = 4.36 \pm 0.01$ $\mu$s (Hahn Echo).
  • RESOLUTE achieved $T_2^p = 5.1 \pm 0.2$ $\mu$s, surpassing even the Hahn Echo measurement. This indicates that RESOLUTE successfully refocuses noise sources that Hahn Echo cannot.
• Detection of Low-Frequency Nuclear Spins:
  • The protocol successfully detected the Larmor precession of naturally abundant $^{13}$C nuclear spins.
  • Low-Field Detection: Signals were detected at magnetic fields as low as 49 G (corresponding to $\sim 50$ kHz), a regime where previous correlation methods (e.g., standard Hahn Echo correlation) failed (which typically required $>110$ G).
  • Oscillations were observed at frequencies matching the $^{13}$C Larmor frequency for fields of 49.5 G, 64.3 G, and 107.2 G.
• Electron Spin Detection:
  • By combining RESOLUTE with chirped pulses, the authors detected a target electron spin via dipolar coupling.
  • Contrast Improvement: The RESOLUTE sequence with chirped pulses yielded a signal contrast of 6%, compared to 1.4% for a standard DEER sequence with a hard $\pi$-pulse.
  • The chirped pulse (adiabaticity factor $Q=5$) significantly improved the flip efficiency and robustness against detuning compared to rectangular pulses.

5. Significance and Impact

• Single-Molecule Imaging: The ability to detect weak dipolar interactions and low-frequency nuclear spins at low magnetic fields is crucial for imaging single molecules and resolving hyperfine interactions in complex environments.
• Quantum Sensing Applications: RESOLUTE provides a versatile tool for nanoscale magnetometry, offering a simple implementation (requiring only least-squares fitting for post-processing) and superior performance in noisy environments.
• Theoretical Validation: The Fisher information analysis confirms that RESOLUTE is the optimal protocol for frequency estimation in the $1/T_2^p > \omega > 1/T_1$ range, offering linear information accumulation over time.
• Future Potential: The authors suggest that by using external memory qubits to extend $T_{corr}$ beyond $T_1$, the technique could potentially access even lower frequencies (sub-kHz), further expanding the dynamic range of quantum sensors.

In summary, RESOLUTE represents a significant advancement in quantum sensing by decoupling the frequency detection limit from the sensor's coherence time, enabling high-sensitivity detection of low-frequency magnetic signals that were previously undetectable.