Present address: ]Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg,
Germany

Present address: ]WSL Institute for Snow and Avalanche Research SLF, 7260
Davos Dorf, Switzerland

Present address: ]Present address: Institut Laue-Langevin, CS 20156, 38042
Grenoble Cedex 9, France.

# Resonant Cancellation Effect in Ramsey Experiments

Ivo Schulthess<EMAIL_ADDRESS>[  Laboratory for High Energy Physics
and Albert Einstein Center for Fundamental Physics, University of Bern, 3012
Bern, Switzerland  Ivan Calic  [  Laboratory for High Energy Physics and
Albert Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Estelle Chanel  [  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Anastasio Fratangelo  Laboratory for High Energy Physics and
Albert Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Philipp Heil  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Christine Klauser  Paul Scherrer Institut, 5232 Villigen PSI,
Switzerland  Gjon Markaj  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Marc Persoz  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Ciro Pistillo  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Jacob Thorne  Laboratory for High Energy Physics and Albert
Einstein Center for Fundamental Physics, University of Bern, 3012 Bern,
Switzerland  Florian M. Piegsa<EMAIL_ADDRESS>Laboratory for High
Energy Physics and Albert Einstein Center for Fundamental Physics, University
of Bern, 3012 Bern, Switzerland

(August 30, 2024)

###### Abstract

We investigate the response of a Ramsey-type experiment on an additional
oscillating magnetic field. This superimposed field is oriented in the same
direction as the static main magnetic field and causes a modulation of the
original Larmor spin precession frequency. The observable magnitude of this
modulation reduces at higher frequencies of the oscillating field. It
disappears completely if the interaction time of the particles matches the
oscillation period, which we call resonant cancellation. We present an
analytical approach that describes the effect and compare it to a measurement
using a monochromatic cold neutron beam.

##  I Introduction

Rabi’s nuclear magnetic resonance method [1, 2] and Ramsey’s techniques of
separated oscillatory fields [3, 4, 5] have been used very effectively in
various scientific experiments. These methods utilize both constant and time-
varying magnetic fields to manipulate the spin of probe particles.
Specifically, Ramsey’s technique allows for precisely determining of the
Larmor precession frequency of a spin in a magnetic field $B_{0}$. This is
achieved by initially flipping the spin-polarized particles into the plane
orthogonal to $B_{0}$ using an oscillating field $B_{1}$, allowing them to
precess for a defined period of time, and then flipping them again using a
second oscillating $B_{1}$ field. By scanning the frequency of the oscillating
fields close to the resonance and keeping the phase between the two signals
locked, an interference pattern of the spin polarization in the frequency
domain is obtained. Another option is to scan the relative phase between the
two oscillating spin-flip signals while keeping their frequencies on
resonance. Ramsey’s technique is highly versatile and can be applied for
precise measurements of changes in magnetic and pseudo-magnetic fields. For
instance, it has been utilized in a variety of applications such as atomic
clocks [6, 7], the measurement of the Newtonian gravitational constant [8],
the measurement of the neutron magnetic moment [9], the search for the neutron
electric dipole moment [10, 11, 12], probing for dark matter [13, 14], and the
search for new particles and interactions [15].

The main magnetic field $B_{0}$ that causes the Larmor precession in a Ramsey-
type experiment is usually constant over time. In this manuscript, we describe
the effect of a superimposed oscillating (pseudo-)magnetic field on the Ramsey
signal, called resonant cancellation. Such a signal is potentially present in
an axionlike dark-matter field that couples to the electric dipole moment of
particles. Two experiments searching for an oscillating electric dipole moment
of neutrons [14, 16] and electrons [17] showed how this systematic effect
attenuates the signal and reduces their sensitivity. Moreover, ultralight
scalar or pseudo-scalar dark matter could lead to oscillating nuclear charge
radii that could be detected with Ramsey’s technique in optical atomic clocks
[18]. Such measurements would also potentially suffer from the effect.

In the following, we provide a theoretical derivation of the resonant
cancellation effect for a monochromatic particle beam with a velocity $v$. In
a Ramsey-type experiment, the spins precess in the main magnetic field $B_{0}$
due to the Larmor precession

$\varphi=\omega T=\gamma B_{0}T\ ,$ (1)

where $\gamma$ is the gyromagnetic ratio and $T$ is the interaction time. The
phase shift that the spins acquire when interacting with an oscillating
magnetic field ${B_{a}(t)=B_{a}\cos{(2\pi f_{a}t)}}$ parallel to $B_{0}$ is

$\Delta\varphi=\gamma\int B_{a}\cos{(2\pi f_{a}t)}\,\mathrm{d}t\ ,$ (2)

where $B_{a}$ is the amplitude and $f_{a}$ the frequency of the oscillating
field. This becomes maximal when the integral is evaluated symmetrically
around zero which leads to a maximum value of the phase

$\displaystyle\Delta\varphi_{\text{max}}$ $\displaystyle=\left|\gamma
B_{a}\int_{-T/2}^{T/2}\cos{(2\pi f_{a}t)}\,\mathrm{d}t\right|$ (3)
$\displaystyle=\left|\frac{\gamma B_{a}}{\pi f_{a}}\sin{\left(\pi
f_{a}\frac{L}{v}\right)}\right|\ ,$ (4)

where the interaction length $L=vT$. Equation (4) has the shape of a sinc-
function with roots at ${f_{a}=k\cdot v/L}$, $k\in\mathbb{N}$. Hence, when the
interaction time with the oscillating field matches one or multiple periods of
the oscillating magnetic field, the integral becomes zero, yielding no phase
shift. Therefore, a Ramsey-type experiment is insensitive to magnetic field
effects of specific frequencies. In the case of a white beam, e.g., with a
Maxwell-Boltzmann-like velocity distribution, this effect is suppressed, and
there is no complete cancellation. It still results in a reduced phase shift
and, therefore, a decreased sensitivity to oscillating magnetic field effects
at higher frequencies.

##  II Setup

To investigate the resonant cancellation effect we conducted measurements
during a dedicated beamtime in May 2021 at the Narziss instrument at the Paul
Scherrer Institute in Switzerland. The Narziss beamline is usually used for
neutron reflectivity measurements. It provides neutrons with a de Broglie
wavelength of $4.96\leavevmode\nobreak\ \text{\r{A}}$ and can be regarded as
monochromatic for the purpose of this measurement. A schematic of the
experiment is presented in Fig. 1.

Figure 1: Schematic of the experimental apparatus. Neutrons enter the setup
from the left. They first pass a polarizer that transmits only one spin state.
The neutrons then enter the magnetic field region where a constant magnetic
field $B_{0}$ and an oscillating magnetic field $B_{a}(t)$ are applied in the
vertical direction. Two $6\leavevmode\nobreak\ \mathrm{cm}$-long spin-flip
coils with a center-to-center separation of $50\leavevmode\nobreak\
\mathrm{cm}$ allow to flip the spins. After the second spin-flip coil the
neutron spin states are analyzed and counted with a ${}^{\text{3}}$He
detector. There are four apertures to define the beam cross-section and
divergence. They also block the reflected beams from the polarizer and
analyzer.

The neutrons pass a beam monitor after exiting from the neutron guide. This
monitor signal can be used to normalize all measurements to the same number of
incoming neutrons and accounts for potential fluctuations in the neutron flux
of the source. There are four apertures with an opening of $30\times
2\leavevmode\nobreak\ \mathrm{mm^{2}}$ that define the beam cross-section and
divergence along the beam path. Their position is indicated in Fig. 1. The
neutrons are polarized with a polarizing supermirror that transmits only one
spin state. The other spin state is reflected and stopped in an aperture. To
maintain the polarization, there is a small magnetic guiding field between the
polarizer and the first spin-flip coil. The main magnetic field $B_{0}\approx
3\leavevmode\nobreak\ \mathrm{mT}$ is created by an electromagnet surrounding
the back of a C-shaped iron yoke. The yoke guides the field lines and creates
a field between its flanks in the vertical direction. The oscillating magnetic
field $B_{a}(t)$ is applied over the same region as the $B_{0}$ field. It is
created by a Keysight waveform generator [19] that is connected via a Kepco
bipolar power amplifier [20] to an auxiliary coil. This coil can also be used
to apply DC offset fields to test the functionality of the setup. Two
rectangular solenoid-type spin-flip coils induce resonant $\pi/2$-flips. They
have a length of $6\leavevmode\nobreak\ \mathrm{cm}$ and a center-to-center
separation of $50\leavevmode\nobreak\ \mathrm{cm}$. Their axis is aligned with
the neutron beam and their cross-section is much bigger than the size of the
beam. Their signals are produced by a waveform generator and amplified by two
Mini-Circuits LYZ-22+ power amplifiers [21]. They are connected to the spin-
flip coils via resonance circuits that match $50\leavevmode\nobreak\ \Omega$
and are tuned to the resonance frequency of about 90 kHz. The two sides are
covered with $2\leavevmode\nobreak\ \mathrm{mm}$ thick aluminum plates to
minimize their fringe fields. The spins are analyzed with a second polarizing
supermirror. The transmitted neutrons are counted using a ${}^{\text{3}}$He
detector. Each neutron creates an electronic pulse. The timing of the pulse is
processed and recorded by a custom-made data-acquisition system using an
Arduino.

##  III Measurements

The spin-flip signals were optimized for Ramsey-type measurements. We
individually measured a Rabi resonance frequency of ${(91.94\pm
0.01)\leavevmode\nobreak\ \mathrm{kHz}}$ and ${(91.46\pm
0.01)\leavevmode\nobreak\ \mathrm{kHz}}$ for the first and the second spin-
flip coil, respectively. For further measurements we applied a signal at a
frequency of $91.7\leavevmode\nobreak\ \mathrm{kHz}$. The amplitudes of both
spin-flip signals were optimized for the highest signal visibility in a Ramsey
measurement, corresponding to a $\pi/2$-flip.

To test the functionality and characterize the setup we performed various
Ramsey frequency and phase scans. Two such measurements are presented in Fig.
2. For each data point, the neutron counts were integrated over roughly $10$
seconds, corresponding to ${2\times 10^{5}}$ monitor counts. The Ramsey
frequency scan presented in Fig. 2(a) shows an overall envelope which arises
from the Rabi resonance. The fringes are the interference pattern produced by
the two spatially separated spin-flip coils. All fringe maxima are close to
the maximum neutron counts of about $2.8\times 10^{4}$ neutrons, confirming
the small wavelength spread. We measured the shift of the resonance frequency
by scanning only the central fringe while applying various DC offset currents
through the auxiliary coil. A linear fit through all resonance frequencies
results in a value for the shift of $(-1.096\pm 0.002)\leavevmode\nobreak\
\mathrm{kHz/A}$ which corresponds to roughly $38\leavevmode\nobreak\
\mathrm{\mu T/A}$. The Ramsey phase scan in Fig. 2(b) shows a sinusoidal
behavior whose phase shifts if an additional offset current is applied, as in
the case of the central resonance fringe. We performed several more
measurements to get the phase shift as a function of the applied current. A
linear least-squares fit through all phases results in a value of $(-257.0\pm
0.4)\mathrm{\degree/A}$. The measurements presented in Fig. 2 show that the
apparatus works as expected and that an offset current through the auxiliary
coil indeed changes the precession frequency of the neutrons.

(a)

(b)

Figure 2: (a) Ramsey frequency scan over the full resonance. The data shows
the neutron counts as a function of the spin-flip signal frequency. The solid
line serves only as a guide for the eyes. (b) Ramsey phase scans were the
neutron counts are shown as a function of the relative phase between the two
spin-flip signals. The frequency was fixed on resonance at
$91.7\leavevmode\nobreak\ \mathrm{kHz}$. Besides a reference measurement
(black ⚫), we applied various DC offset currents via the auxiliary coil. The
measurements with a current of $-100\leavevmode\nobreak\ \mathrm{mA}$ (blue
★), $+100\leavevmode\nobreak\ \mathrm{mA}$ (orange ▼), and
$+200\leavevmode\nobreak\ \mathrm{mA}$ (green ✚) are shown. The solid lines
correspond to least-squares fits of a sinusoidal function.

The results of the Ramsey frequency and phase scans can be used to calculate
the effective interaction length with the use of Eq. (1) and ${v\approx
798\leavevmode\nobreak\ \mathrm{m/s}}$. The resulting value of ${(51.9\pm
0.1)\leavevmode\nobreak\ \mathrm{cm}}$ is slightly longer than the separation
of the two spin-flip coils. The reason is that the spins already start to
precess within the spin-flip coil when partially flipped.

To investigate the resonant cancellation effect, we measured the neutron
counts continuously and saved their timing information. Because of limitations
from the data-acquisition system, we phased/mapped the data into two periods
of the oscillating magnetic field $B_{a}(t)$. This mapping was triggered by
the waveform generator which created the oscillating signal. We chose a time
bin size of $20\leavevmode\nobreak\ \mathrm{\mu s}$. This size allowed us to
have more than ten bins for the highest frequency of $3500\leavevmode\nobreak\
\mathrm{Hz}$. The relative phase between the spin-flip signals was set to
$105\degree$ which corresponds to the point of steepest slope in the reference
measurement of the Ramsey phase scan shown in Fig. 2(b). At this point, the
measurement is most sensitive to magnetic field changes, and the relation
between the neutron counts and the phase is approximately linear.

Figure 3: Neutron signals for oscillating magnetic fields at 252 Hz, 1020 Hz, 1477 Hz, and 2227 Hz. They show the neutron counts as a function of time. The counts are phased/mapped into two periods of the signal. The data (black) are fitted with a sinusoidal function with a fixed frequency (orange). The fit results are presented in Tab. 1. The measurement time was roughly $100$ seconds for each setting. | amplitude | offset | amplitude/offset |  $\chi^{2}$/NDF
---|---|---|---|---
252 Hz | $37\pm 1$ | $368\pm 1$ | $0.100\pm 0.004$ | 329/391
1020 Hz | $59\pm 6$ | $1463\pm 4$ | $0.040\pm 0.004$ | 120/92
1477 Hz | $5\pm 8$ | $2093\pm 6$ | $0.003\pm 0.004$ | 52/62
2227 Hz | $67\pm 12$ | $3108\pm 9$ | $0.022\pm 0.004$ | 47/39
Table 1: Results of the sinusoidal fit of the data presented in Fig. 3. The
amplitude is normalized by the offset to make the measurements comparable. The
phase of the sinusoidal fit is not relevant for the analysis of the resonant
cancellation effect. The $\chi^{2}$ of each fit and the corresponding number
of degrees of freedom are also given.

We measured the oscillating neutron amplitude for $100$ frequencies between
$60\leavevmode\nobreak\ \mathrm{Hz}$ and $3500\leavevmode\nobreak\
\mathrm{Hz}$. Four examples of neutron signals are shown in Fig. 3. The
results of their sinusoidal fits are presented in Tab. 1. All measurements
have different periods but the same time bin sizes. Therefore, the number of
time bins is different. Since the total number of neutrons is the same for all
measurements, the amplitude and offset of the sinusoidal neutron signal scale
with the frequency of the oscillating field. To account for this, we
normalized the amplitude of the neutron signal by its offset. To improve the
signal-to-noise ratio we repeated this sequence three times and averaged the
data. The frequency dependence of the oscillating magnetic field amplitude
$B_{a}$ was investigated in an auxiliary measurement with a fluxgate sensor.
It features a slight decrease for higher frequencies which is due to the
experimental setup. We used the fluxgate data to account for this
characteristics. Additionally, we normalized the amplitude to be one at DC.
The result of this measurement and analysis is shown in Fig. 4. A fit of Eq.
(4) yields the values of the roots at $(1529\pm 7)\leavevmode\nobreak\
\mathrm{Hz}$ and $(3057\pm 14)\leavevmode\nobreak\ \mathrm{Hz}$. The reduced
chi-square of the fit is $\chi_{r}^{2}=1.04$ for 98 degrees of freedom.

Figure 4: Measurement of the resonant cancellation effect at the Narziss
beamline. The normalized amplitude as a function of the frequency of the
oscillating signal is shown. The data (black) are fitted with Eq. (4) (red).
The lower subfigure shows the residuals.

Figure 4 shows that the effect of the resonant cancellation behaves as
expected for a monochromatic neutron beam. Given a neutron wavelength
(velocity) of $4.96\leavevmode\nobreak\ \text{\r{A}}$
($798\leavevmode\nobreak\ \mathrm{m/s}$), the fit values of the root leads to
an interaction length of ${(52.2\pm 0.2)\leavevmode\nobreak\ \mathrm{cm}}$.
This value is in agreement with the aformentioned effective interaction length
of ${(51.9\pm 0.1)\leavevmode\nobreak\ \mathrm{cm}}$ which was determined from
the Ramsey data.

##  IV  Conclusion

In conclusion, we investigated the resonant cancellation effect. This effect
is present in Ramsey-type experiments where an oscillating magnetic field is
superimposed to the main magnetic field that causes the Larmor precession. We
showed that the neutron oscillation amplitude follows the expected theoretical
behavior in the case of a monochromatic neutron beam. This results in a
reduced sensitivity of the experiment for higher frequencies to the
superimposed field. In particular, the sensitivity becomes zero if the
interaction time of the neutrons matches the period of the oscillating field.
This approach can be used to estimate the attenuation effect for experiments
searching for time dependent signals. For instance, this systematic effect is
important in axionlike dark-matter searches using electric dipole moments of
particles.

###### Acknowledgements.

We gratefully acknowledge the excellent technical support by R. Hänni, J.
Christen, and L. Meier from the University of Bern. The experiment has been
performed at the Swiss Spallation Neutron Source SINQ at the Paul Scherrer
Institute in Villigen, Switzerland. This work was supported via the European
Research Council under the ERC Grant Agreement no. 715031 (BEAM-EDM) and via
the Swiss National Science Foundation under grants no. PP00P2-163663 and
200021-181996.

## Conflict of Interest

The authors have no conflicts to disclose.

## Author Contributions

I. Schulthess: conceptualization (equal); data curation (lead); formal
analysis (equal); investigation (equal); methodology (lead); software (lead);
validation (equal); visualization (lead); writing - original draft (lead);
writing – review & editing (equal). I. Calic: formal analysis (equal);
investigation (equal); software (supporting); validation (equal); writing –
review & editing (supporting). A. Fratangelo: investigation (equal); writing –
review & editing (supporting). P. Heil: writing – review & editing
(supporting). Ch. Klauser: resources (supporting); writing – review & editing
(supporting). G. Markaj: writing – review & editing (supporting). M. Persoz:
writing – review & editing (supporting). C. Pistillo: writing – review &
editing (supporting). J. Thorne: writing – review & editing (supporting). F.
M. Piegsa: conceptualization (equal); funding acquisition (lead);
investigation (equal); project administration (lead); resources (lead);
supervision (lead); validation (equal); writing – review & editing (equal).

## Data Availability

The data and analysis that support the findings of this study are openly
available in a Github repository [22].

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