††thanks: Current address: Exponent, Inc., Electrical Engineering and Computer
Science Practice, Warrenville, IL

# Systematic study of rotational decoherence with a trapped-ion planar rotor

Neil Glikin  Department of Physics, University of California, Berkeley,
Berkeley, California 94720, USA  Challenge Institute for Quantum Computation,
University of California, Berkeley, Berkeley, California 94720, USA  Benjamin
A. Stickler  Institute for Complex Quantum Systems, Ulm University, Albert-
Einstein-Allee 11, 89069 Ulm, Germany  Ryan Tollefsen  Department of Physics,
University of California, Berkeley, Berkeley, California 94720, USA  Challenge
Institute for Quantum Computation, University of California, Berkeley,
Berkeley, California 94720, USA  Sara Mouradian  Department of Electrical and
Computer Engineering, University of Washington, Seattle, Washington 98195, USA
Neha Yadav  Department of Physics, University of California, Berkeley,
Berkeley, California 94720, USA  Challenge Institute for Quantum Computation,
University of California, Berkeley, Berkeley, California 94720, USA  Erik
Urban  Department of Physics, University of California, Berkeley, Berkeley,
California 94720, USA  Klaus Hornberger  University of Duisburg-Essen, Faculty
of Physics, Lotharstraße 1, 47057 Duisburg, Germany  Hartmut Haeffner
Department of Physics, University of California, Berkeley, Berkeley,
California 94720, USA  Challenge Institute for Quantum Computation, University
of California, Berkeley, Berkeley, California 94720, USA

###### Abstract

Quantum rotors promise unique advantages for quantum sensing, quantum
simulation, and quantum information processing. At present, a variety of
systems ranging from nanoparticles to single molecules and trapped ions have
demonstrated detection and control of rotational motion in and near the
quantum regime. For future applications of quantum rotors, understanding their
dynamics in the presence of ambient environments and decoherence will be
critical. While other model quantum systems such as the harmonic oscillator
have seen extensive experimental study of their decoherence dynamics, such
experiments remain an open task for the rigid rotor. We present measurements
of fundamental scaling relationships for decoherence of a quantum planar rotor
realized with two trapped ions, and find excellent agreement with recent
theoretical work.

The quantum rigid rotor is among the simplest model quantum systems, on par
with the quantum harmonic oscillator and the qubit. In contrast to these, the
technological potential of quantum rotor dynamics has not yet been fully
tapped. There have recently been remarkable advances in the preparation and
control of molecular rotational states Koch  _et al._ (2019); Chou  _et al._
(2020); Lin  _et al._ (2020); Sinhal  _et al._ (2020); Voges  _et al._ (2020);
Schindewolf  _et al._ (2022); Stevenson  _et al._ (2023), in using such states
for coherent interactions Yan  _et al._ (2013); Bao  _et al._ (2022);
Christakis  _et al._ (2023); Li  _et al._ (2023); Gregory  _et al._ (2023),
and in the manipulation of nanoscale rotors Delord  _et al._ (2020); Ahn  _et
al._ (2020); van der Laan  _et al._ (2021); Stickler  _et al._ (2021); Pontin
_et al._ (2023); Kamba  _et al._ (2023). These developments are motivated by
prospects of using the unique properties of rotors for encoding quantum
information DeMille (2002); Grimsmo  _et al._ (2020); Albert  _et al._ (2020),
performing quantum simulations using tunable dipole-dipole interactions Zhou
_et al._ (2011); Lechner and Zoller (2013); Sundar  _et al._ (2018), torque
sensing Ahn  _et al._ (2020), and tests of the quantum superposition principle
at high masses Stickler  _et al._ (2018a); Ma  _et al._ (2020).

Experiments which aim to exploit rotational quantum coherence will also need
to contend with interaction with their environment and the resulting
decoherence. In rotor systems, decoherence has been experimentally observed
only in certain limited contexts. These include the decay of rotational
coherences in gas-phase ensembles of molecules due to collisions or radiation
Vieillard  _et al._ (2008); Owschimikow  _et al._ (2010); Milner  _et al._
(2014); Tenney  _et al._ (2016); Damari  _et al._ (2017), and due to light
scattering or uncontrolled differential shifts in optical traps Neyenhuis  _et
al._ (2012); Yan  _et al._ (2013); Seeßelberg  _et al._ (2018); Bause  _et
al._ (2020); Burchesky  _et al._ (2021); Gregory  _et al._ (2023). To push
quantum rotors into a useful coherent regime, a comprehensive understanding of
their decoherence dynamics will be crucial.

The theory of rotor decoherence, which fully accounts for orientational
periodicity, has been developed only within the last decade Stickler  _et al._
(2016); Zhong and Robicheaux (2016); Papendell  _et al._ (2017); Stickler  _et
al._ (2018b); Schlosshauer (2019). An important case is interaction with an
isotropic environment, which leads to diffusion of a rotor’s angular momentum
state. In this case, a superposition of two orientation states
$\ket{\phi}+\ket{\phi^{\prime}}$ is predicted to decay into a statistical
mixture at a rate proportional to two important parameters: (i) the angular
momentum diffusion coefficient, and (ii) for inversion-symmetric planar
rotors, the sine squared of their relative angle,
$\sin^{2}(\phi-\phi^{\prime})$. This can be compared with the decoherence rate
$\absolutevalue{\alpha-\alpha^{\prime}}^{2}$ for a superposition of harmonic
oscillator coherent states $\ket{\alpha}+\ket{\alpha^{\prime}}$ Walls and
Milburn (1985). Experimentally verifying harmonic oscillator decoherence
scalings Brune  _et al._ (1996); Myatt  _et al._ (2000); Auffeves  _et al._
(2003) proved crucial for developing oscillators into building blocks for
quantum technologies. For rotors, the analogous relations have not yet been
experimentally probed.

Here we report on a systematic study of the decoherence dynamics of a planar
rigid rotor. The rotor is composed of two trapped atomic ions separated by
several micrometers and rotating with a frequency on the order of one hundred
kilohertz. We prepare this system in a superposition of different rotation
rates, which turns into a superposition of orientation states, and engineer a
noisy electric-field environment to induce rotational decoherence on the
millisecond timescale. Our results demonstrate a sine-squared scaling of the
decoherence rate for orientational superpositions in the small-angle limit and
the expected scaling with the angular momentum diffusion coefficient. Both
findings are in excellent agreement with theory.

This work will be relevant for future studies of the prospects and practical
challenges of fault-tolerant quantum information processing with rotors
Grimsmo  _et al._ (2020); Albert  _et al._ (2020), of decoherence-free
subspaces of orientation states Pedernales  _et al._ (2020), and of
macroscopic quantum tests with nanorotors Stickler  _et al._ (2018a); Ma  _et
al._ (2020). Moreover, it constitutes a first step towards systematically
probing surface-induced heating and decoherence of rotors that are
characterized by charge multipoles Martinetz  _et al._ (2022).

Fig. 1: Experimental setup and sequence. a, Two ${}^{40}{\rm Ca}^{+}\,$ions
(not to scale) rotate above a circular surface ion trap in the plane parallel
to the surface. A tilted vertical beam of 729 nm laser light addresses the ion
crystal on rotational sidebands of the electronic
$\ket{S}\leftrightarrow\ket{D}$ transition, labeled by their order
$\Delta\ell$, to create angular momentum superpositions. Sidebands of
different orders $\Delta\ell$ are spectrally resolved from one another.
$\ket{SS,\ell}$ denotes a state with both ions in the electronic $\ket{S}$
state and with rotor angular momentum $\hbar\ell$. b, Circuit schematic of the
setup for creating the electric-field gradient noise environment. A narrowband
noisy voltage is applied to the outer control electrodes in an alternating
pattern, giving rise to a quadrupole electric field at the location of the ion
rotor. The zoom-in shows the ion rotor and the quadrupole field (blue). The
noisy gradient of this field exerts random torques on the rotor when on
resonance with the ions’ rotation, resulting in angular momentum diffusion and
decoherence. c, Simplified diagram of the Ramsey sequence used to measure
rotational decoherence. Ions are shown in a co-rotating frame. A $\pi/2$ pulse
from the 729 nm laser on a rotational sideband of order $\Delta\ell$ creates a
superposition of angular momenta. This may be equated to a superposition of
two orientations of the rotor which are freely evolving at different angular
velocities and have a linearly increasing angular separation $\Delta\phi$. A
$\pi$ pulse reverses the separation, and a $\pi/2$ pulse closes the
interferometer. The electric field gradient noise environment is turned on
while the interferometer is open. The remaining orientational coherence is
mapped to the electronic state, and is quantified as the amplitude of the
interference fringe.

We realize a quantum rigid rotor as a pair of ${}^{40}{\rm Ca}^{+}\,$ions
Paul-trapped in a cylindrically symmetric potential, created by a surface ion
trap with circular radio-frequency (rf) electrodes. The electrodes produce a
three-dimensional harmonic potential centered 180 µm above the trap surface,
whose confinement strength is characterized by secular trap frequencies
$\omega_{z}={2\pi\times 2.87}$ MHz normal to the trap surface and
$\omega_{x}=\omega_{y}={2\pi\times 1.44}$ MHz in the transverse plane. Since
$\omega_{z}>\omega_{x,y}$, the ions are confined to the tranverse plane, while
the cylindrical symmetry allows for a free rotational motion of the ion
crystal within the plane Li  _et al._ (2017); Urban  _et al._ (2019). The
ions’ mutual Coulomb repulsion balanced with the trap potential creates a
rigid rotor with a calculated radius $r=2.2$ µm. Non-rigid corrections are of
order $10^{-2}$ and are negligible (see Methods).

The Hamiltonian describing the rotational motion of the ion crystal is that of
a planar rotor $H_{0}=L_{z}^{2}/2I$, where $L_{z}$ is the angular momentum
operator component normal to the rotor plane and $I=2mr^{2}$ is the moment of
inertia. The angular momentum eigenstates $\ket{\ell}$ are simultaneous
eigenstates of the Hamiltonian $H_{0}$ and the angular momentum operator
$L_{z}$, with angular momentum eigenvalues $\hbar\ell$ where $\ell$ takes
integer values. The energy scale is set by the rotational constant
$\omega_{\rm r}=\hbar/2I=2\pi\times 13$ Hz, equal to the transition frequency
between the ground and first excited rotational states.

Each ion contains an independent electronic degree of freedom, of which only
two states are of interest: the ${}^{2}S_{1/2}\,(m=-1/2)$ sublevel of the
ground electronic state (labelled $\ket{S}$) and the metastable
${}^{2}D_{5/2}\,(m=-1/2)$ state (labelled $\ket{D}$). The ions’ internal
electronic states are coherently manipulated by a narrow-linewidth laser near
the $\ket{S}\leftrightarrow\ket{D}$ resonance at 729 nm. Sidebands of this
resonance are used to couple to the rotational degree of freedom. The 729 nm
laser addresses the full ion crystal nearly normal to the rotor, at a small
angle of 5.6$\degree$ with respect to the normal of the rotor plane, in order
to maximize coupling to the desired sideband orders Urban  _et al._ (2019). A
uniform magnetic field of 409 µT separates the Zeeman sublevels of the
electronic states to allow frequency selectivity of the sublevels of interest.
State measurement is performed by applying cycling transition laser light at
397 nm, which is resonant with a dipole-allowed transition from the $\ket{S}$
state but off-resonant from any transition from the $\ket{D}$ state. This
allows state discrimination by counting scattered photons. All measurements
record the $\ket{D}$ state population averaged between the two ions, $P(D)$.

We experimentally realize a controllable environment by applying a noisy
voltage to the outer control electrodes of the trap, giving rise to a noisy
electric field that couples to the rotor via the ions’ charge. Since the two-
ion crystal has zero dipole moment, the interaction with the noise field is
via its quadrupole component and thus is proportional to the local field
gradients. We inject voltage noise onto four of the eight control electrodes
which surround the rf electrodes with alternating sign, as shown in Fig. 1b.
The resulting quadrupole field is proportional to the applied voltage, which
is generated as white noise from an arbitrary waveform generator and passed
through a bandpass filter with a 19 kHz bandwidth and tunable center
frequency. This allows us to directly control the noise spectral density of
the electric field gradient. In the absence of injected voltage noise,
uncontrolled ambient electric field gradient noise is still present at the
location of the ions due to stray voltage noise and thermally induced field
fluctuations originating from the electrodes Brownnutt  _et al._ (2015);
Martinetz  _et al._ (2022).

Averaging the rotor-field interaction over many realizations of the noisy
electric field gradient, the dynamics of the rotor’s quantum state $\rho$ are
described by the Lindblad master equation of orientational decoherence for
quadrupolar interactions
$\partial_{t}\rho=-i[H_{0},\rho]/\hbar+\mathcal{L}\rho$ with (see Methods),

$\mathcal{L}\rho=\frac{D}{4\hbar^{2}}\left(L_{-}^{2}\rho
L_{+}^{2}+L_{+}^{2}\rho L_{-}^{2}-2\rho\right).$ (1)

Here, $L_{\pm}$ are dimensionless angular momentum raising and lowering
operators, $L_{\pm}\ket{\ell}=\ket{\ell\pm 1}$. The master equation describes
angular momentum diffusion at a rate $D$, i.e.
$\partial_{t}\expectationvalue{L_{z}^{2}}=2D$, and decoherence in the rotor
orientation $\phi$ Papendell  _et al._ (2017),

$\bra{\phi}\mathcal{L}\rho\ket{\phi^{\prime}}=-\frac{D}{\hbar^{2}}\sin^{2}(\phi-\phi^{\prime})\,\bra{\phi}\rho\ket{\phi^{\prime}}.$
(2)

The scaling with $\sin^{2}(\phi-\phi^{\prime})$ is $\pi$-periodic rather than
$2\pi$-periodic due to the inversion symmetry of the quadrupole interaction
Papendell  _et al._ (2017); Pedernales  _et al._ (2020). For a rapidly
spinning rotor with mean angular frequency $\omega_{\rm
rot}=\expectationvalue{L_{z}}/I$, the diffusion coefficient is determined by
the power spectral density of the field gradient at twice the rotation
frequency,

$D=\left(\frac{er^{2}}{2}\right)^{2}S_{\varepsilon}(2\omega_{\text{rot}}).$
(3)

Here, $S_{\varepsilon}(\omega)$ is the power spectral density of
$\varepsilon_{t}=-\partial_{x}E_{x}+\partial_{y}E_{y}+2i\partial_{x}E_{y}$,
which is the relevant field gradient for quadrupolar interaction in the plane
parallel to the surface (see Methods). The environment does not induce
rotational friction within measurement limits (see Methods), and thus friction
terms Stickler  _et al._ (2018b) are omitted from the master equation (1).

We present two types of measurements, (i) angular momentum diffusion of the
quantum rotor, and (ii) decoherence of angular momentum superpositions. In
both cases, we begin with the ions trapped in an additional pinning potential
which fixes their locations. We first cool all motional degrees of freedom
with Doppler cooling, then sideband cool the transverse librational motion and
the motion normal to the rotor plane. We then initialize the ions’ internal
states to $\ket{S}$ via optical pumping, and spin up the rotor by rotating and
then releasing the pinning potential via a protocol described in Ref. Urban
_et al._ (2019). This approximately produces a coherent rotor state
$\ket{\Psi_{\overline{\ell}}}$ that is localized in angular momentum space at
$\bar{\ell}=\expectationvalue{L_{z}}/\hbar\gg\sigma_{\ell}$, where
$\sigma_{\ell}$ is the standard deviation in units of $\hbar$. In the present
experiment, $\bar{\ell}\approx 6\times 10^{3}$ and $\sigma_{\ell}\approx 20$.
The corresponding mean rotation frequency, $\omega_{\text{rot}}=2\omega_{\rm
r}\bar{\ell}$, is chosen to be between 142 and 149 kHz. This procedure yields
resolved sidebands on the $\ket{S}\leftrightarrow\ket{D}$ 729 nm transition at
frequencies $\Delta\ell\omega_{\text{rot}}$ for integer $\Delta\ell$,
corresponding to rotational state transitions
$\ket{\ell}\to\ket{\ell+\Delta\ell}$. This allows selection of well-defined
angular momentum changes $\Delta\ell$ by tuning the frequency of the 729 nm
laser to the corresponding sideband frequency.

Fig. 2: Measurements of the angular momentum diffusion coefficient. a, Diagram
of the measurement protocol for the angular momentum spread. A rotating
coherent state is prepared, and noise is injected for a time $t_{\rm diff}$.
Afterward, the linewidth of the $\Delta\ell=1$ sideband is measured with a
Ramsey experiment. b, A sample diffusion coefficient measurement. Angular
momentum distribution width $\sigma_{\ell}$ is measured as a function of time
after rotational state preparation, with a constant level of noise injected
during the diffusion time. The solid line is a fit giving
$D=156(20)\,\hbar^{2}$/ms. Inset: Selected individual Ramsey measurements of
$\sigma_{\ell}$ with a detuning of 6 kHz from the $\Delta\ell=1$ sideband
transition. The $y$-axis is the average probability of an ion to be found in
the $\ket{D}$ state. The shaded region shows the inferred phase contrast from
the data, from which $\sigma_{\ell}$ is inferred. c, Demonstration of the
$2\omega_{\text{rot}}$ resonance. Diffusion is measured at a rotation
frequency $\omega_{\text{rot}}=144(2)$ kHz. For each data point, the center
frequency of the injected noise spectrum is shifted, while the amplitude and
bandwidth are preserved, and the rotation frequency is kept constant. A
Lorentzian lineshape, centered at $2\omega_{\text{rot}}$ and with bandwidth
equal to the injected noise bandwidth of 19 kHz, is shown for reference.
Vertical bands are at $\omega_{\text{rot}}$ and $2\omega_{\text{rot}}$
including variation.

To determine the angular momentum diffusion coefficient, we measure the width
of the rotor state $\sigma_{\ell}$ over time as it interacts with the
environment. This is done with a Ramsey experiment to measure the linewidth of
the $\Delta\ell=1$ sideband, which is equal to $4\omega_{\rm r}\sigma_{\ell}$
Urban  _et al._ (2019). This relation expresses the fact that a spread in
angular momentum corresponds to an uncertainty in mean angular velocity, and
thus also to a spread of transition frequencies on a given rotational
sideband. The rotational sideband linewidth is thus a direct measure of
$\sigma_{\ell}$. One expects this width to increase with time as
$\sigma_{\ell}(t_{\text{diff}})=\sigma_{\ell}(0)+\sqrt{2Dt_{\text{diff}}}/\hbar$.
The measurement protocol is shown in Fig. 2a. After letting the rotor interact
with the environment for time $t_{\rm diff}$, we perform a Ramsey experiment
on the $\Delta\ell=1$ sideband. We add a small detuning of a few kHz to see
Ramsey fringes, and extract the sideband linewidth directly from the fringe
decay rate. For interaction with the engineered environment, we inject noise
during the time between rotational state preparation and the Ramsey
measurement. We find that the uncontrolled ambient environment induces
diffusion coefficients between 2 and 20 $\hbar^{2}$/ms. Figure 2b shows a
sample diffusion measurement of the engineered environment. In Fig. 2c, we
plot the measured diffusion coefficient for noise spectral densities with
varying center frequency. We find that the measured diffusion coefficient is
maximized when the injected noise resonance is equal to
$2\omega_{\text{rot}}$, and is consistent with a Lorentzian lineshape with
bandwidth of 19 kHz as the center frequency of the noise is swept. Measured
diffusion coefficients below 20 $\hbar^{2}$/ms are likely prevented here by
ambient noise.

To verify that the angular momentum diffusion is accompanied by decoherence, a
direct coherence measurement is required. For this, we perform a Ramsey
experiment on a chosen $\Delta\ell$ sideband in the presence of the noise
bath. The initial $\pi/2$ pulse applies the operation
$\ket{SS,\ell}\to\left(\ket{SS,\ell}\pm\ket{SD,\ell\\!+\\!\Delta\ell}-\ket{DS,\ell\\!+\\!\Delta\ell}\mp\ket{DD,\ell\\!+\\!2\Delta\ell}\right)/2$,
where the signs depend on whether $\Delta\ell$ is odd (top) or even (bottom).
Subsequent coherent evolution results in a dephasing of the electronic state
on the 100 µs timescale due to the spread in angular momentum space of the
initial state and the nonlinearity of the rotor spectrum Urban  _et al._
(2019). We therefore add a Hahn echo pulse to the center of the Ramsey scheme
to compensate for this by reversing the free evolution. The interferometer is
closed by a final $\pi/2$ pulse after time $\tau$. With the echo pulse
included, any measured loss in phase contrast can be attributed to nonunitary
dynamics. For interaction with the engineered environment, we inject noise
during the Ramsey evolution. The remaining coherence is quantified by the
amplitude of the Ramsey fringe when the phase of the final pulse is scanned
from 0 to 2$\pi$. We emphasize that this measurement is sensitive only to
coherences between angular momentum states whose separation is exactly
$\Delta\ell$, even in the presence of an initial state with $\sigma_{\ell}\gg
1$, due to the resolved $\Delta\ell$ sidebands of the electronic transition
being addressed by the laser. We rule out other known potential decoherence
mechanisms which could otherwise contribute to decay in fringe contrast by
independent measurements (see Methods).

Fig. 3: Measurements of rotational decoherence. a, Sample measurements of the
decay of phase contrast for superposition states with $\Delta\ell=$1, 2, 3 at
a constant level of noise injected such that $D=70\,\hbar^{2}$/ms. Inset:
Selected individual phase scans for $\Delta\ell=1$ using the pulse sequence
shown in Fig. 1c. b, Decoherence rate measured against angular momentum
diffusion coefficient. The leftmost datapoints are derived from an ambient
environment where $D=2.1\,\hbar^{2}$/ms, and the others are obtained from use
of the engineered environment, with varying amplitudes of injected noise.
Solid curves are produced from Eq. (6), with no free fitting parameters.
Inset: Decoherence rates, from the data points at $D=70\,\hbar^{2}$/ms in the
main plot, plotted against $\Delta\ell$ on a log-log scale. The shaded area
shows a power law fit including uncertainty. The black line shows the
theoretically expected power law of $\Delta\ell^{2/3}$ from Eq. (6).

A sample of rotational decoherence measurements is presented in Fig. 3a, which
shows the Ramsey phase contrast $C$ as a function of total Ramsey time $\tau$
for angular momentum superposition orders $\Delta\ell=$ 1, 2, and 3. These
measurements are each made with the same level of noise injection such that
$D=70\,\hbar^{2}$/ms, as determined by an independent measurement of
diffusion. Each is fit to the theoretical prediction
$C(\tau)=C_{0}\exp\left[-(\gamma\tau)^{3}\right]$ (see below), where $C_{0}$
is the phase contrast without rotational decoherence, which may be less than
unity due to imperfect operations. We interpret the fitted parameter $\gamma$
as the measured decoherence rate. The fitted decoherence rates from Fig. 3a
are plotted against superposition order $\Delta\ell$ in the inset of Fig. 3b,
and are fit to a power law. In Fig. 3b, we present similar triplets of
decoherence rates for superpositions $\Delta\ell=1,2,3$ across various
strengths of the system-environment interaction, which are measured by the
angular momentum diffusion coefficient. By varying the amplitude of injected
noise, we tune the diffusion coefficient from 2.1 $\hbar^{2}$/ms (ambient
environment, no noise injected) to 1000 $\hbar^{2}$/ms. The solid curves are
derived directly from the solution to the master equation with no free fitting
parameters, as described next.

To understand our decoherence measurements theoretically, we approximately
solve the master equation (1) between the initial $\pi/2$ and the Hahn echo
pulse and again between the Hahn echo pulse and the final $\pi/2$ pulse with
the initial state $\ket{SS,\Psi_{\overline{\ell}}}$. The first $\pi/2$ pulse
entangles the ion crystal’s electronic state with the rotational state,
creating superpositions of the ion crystal revolving at different angular
velocities as determined by $\Delta\ell$. The resulting angular separation
increases linearly in time until $\tau/2$, leading to a maximal instantaneous
decoherence rate of $\sin^{2}\left(\omega_{\rm r}\Delta\ell\,\tau\right)$.
After the Hahn echo pulse, the angular separation decreases again until the
interferometer is closed by the final $\pi/2$ pulse. We measure the
decoherence rate via the decay of the Ramsey fringe contrast, which can be
shown to be given by (see Methods)

$C(\tau)=C_{0}\,\exp\left(-\frac{D\tau}{2\hbar^{2}}\big{[}1-\text{sinc}(2\omega_{\rm
r}\Delta\ell\,\tau)\big{]}\right).$ (4)

This describes an exponential decay modulated with the rate of angular
separation $2\omega_{\rm r}\Delta\ell$. The experiment takes place in a regime
where the angular frequency difference between the states $2\omega_{\rm
r}\Delta\ell$ is small compared to the diffusion coefficient: $2\omega_{\rm
r}\Delta\ell\ll D/\hbar^{2}$. In this regime, decoherence occurs well before
the completion of the first oscillation at time $\tau=\pi/\omega_{\rm
r}\Delta\ell$. This restricts the angular separation to small angles. The
decay profile (4) then reduces to

$C(\tau)=C_{0}\,\exp\left[-(\gamma\tau)^{3}\right],$ (5)

with the rate

$\gamma=\left(\frac{\omega_{\rm
r}^{2}\Delta\ell^{2}D}{3\hbar^{2}}\right)^{1/3}.$ (6)

We experimentally confirm three important predictions from (5) and (6): (i)
The shape of the contrast decay profile (5) matches the measured decay profile
in Fig. 3a. In the picture of superposed time-dependent orientations, the
$\tau^{3}$-dependence may be understood as the integral of an instantaneous
decoherence rate that is quadratic in the angular separation. This is a
signature of the sine-squared orientational scaling of rotational decoherence
in the small-angle limit, since for small angular separation one obtains
$\sin^{2}[\phi(t)-\phi^{\prime}(t)]=\sin^{2}(2\omega_{\rm r}\Delta\ell
t)\approx(2\omega_{\rm r}\Delta\ell t)^{2}$. (ii) The power-law scaling for
the decoherence rate (6) with $\Delta\ell$, $\gamma\sim\Delta\ell^{2/3}$, is
confirmed by the fit in the inset of Fig. 3b. This yields
$\gamma\sim\Delta\ell^{0.71(5)}$, which is consistent with the expected
scaling. Repeating this analysis for the four diffusion coefficients in the
main plot of Fig. 3b and averaging the exponents, we find
$\gamma\sim\Delta\ell^{0.65(4)}$. As with the shape of the coherence profile
in time, the scaling $\gamma\sim\Delta\ell^{2/3}$ also results directly from
the sine-squared orientational scaling, since $\Delta\ell$ sets the speed of
angular separation. (iii) The power-law scaling of the decoherence rate (6)
with diffusion coefficient, $\gamma\sim D^{1/3}$, can be seen in our
measurements in Fig. 3b, where the slope of the solid curves follows this
$1/3$ power law. We emphasize that the solid curves come directly from (6)
with no free parameters. Thus in addition to finding agreement with all
expected scalings of the rotational decoherence process, we also find that the
absolute values of the measured decoherence rates agree with predictions from
the master equation. We see agreement over the full range of diffusion
coefficients, spanning nearly three orders of magnitude.

In summary, we have presented a systematic experimental study of the
decoherence dynamics of a trapped-ion planar rotor due to a noisy electric
field. We have verified the theoretical prediction that the decoherence rate
of a superposition of two orientations is proportional to
$\sin^{2}(\phi-\phi^{\prime})$. In addition, by independently engineering and
measuring the rate of angular momentum diffusion, we have confirmed the
dependence of rotational decoherence on angular momentum diffusion. These
results serve as a controlled, systematic test of the theory of rotor
decoherence.

There are several ways to extend what we have demonstrated in the near term
and probe further aspects of rotor decoherence. Extending the present work to
more complicated rotors, containing for instance three or more ions revolving
at fixed distances in a plane, would enable the observation of decoherence-
free subspaces in the rotational degree of freedom Pedernales  _et al._
(2020). In addition, correlating the observed rotational heating with heating
of the rotor’s center-of-mass motion and rocking motion may enable systematic
studies of surface noise that go beyond heating measurements with single ions
Martinetz  _et al._ (2022). In the long term, this work constitutes a
necessary step towards utilizing rotational quantum coherence for sensing,
quantum information processing, and fundamental tests DeMille (2002); Grimsmo
_et al._ (2020); Albert  _et al._ (2020); Zhou  _et al._ (2011); Lechner and
Zoller (2013); Sundar  _et al._ (2018); Ahn  _et al._ (2020); Stickler  _et
al._ (2018a); Ma  _et al._ (2020).

## References

  * Koch  _et al._ (2019) C. P. Koch, M. Lemeshko, and D. Sugny, Reviews of Modern Physics 91, 035005 (2019).
  * Chou  _et al._ (2020) C. W. Chou, A. L. Collopy, C. Kurz, Y. Lin, M. E. Harding, P. N. Plessow, T. Fortier, S. Diddams, D. Leibfried, and D. R. Leibrandt, Science 367, 1458 (2020).
  * Lin  _et al._ (2020) Y. Lin, D. R. Leibrandt, D. Leibfried, and C.-w. Chou, Nature 581, 273 (2020).
  * Sinhal  _et al._ (2020) M. Sinhal, Z. Meir, K. Najafian, G. Hegi, and S. Willitsch, Science 367, 1213 (2020).
  * Voges  _et al._ (2020) K. K. Voges, P. Gersema, M. Meyer zum Alten Borgloh, T. A. Schulze, T. Hartmann, A. Zenesini, and S. Ospelkaus, Physical Review Letters 125, 083401 (2020).
  * Schindewolf  _et al._ (2022) A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda, T. Karman, I. Bloch, and X.-Y. Luo, Nature 607, 677 (2022).
  * Stevenson  _et al._ (2023) I. Stevenson, A. Z. Lam, N. Bigagli, C. Warner, W. Yuan, S. Zhang, and S. Will, Physical Review Letters 130, 113002 (2023).
  * Yan  _et al._ (2013) B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature 501, 521 (2013).
  * Bao  _et al._ (2022) Y. Bao, S. S. Yu, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, “Dipolar spin-exchange and entanglement between molecules in an optical tweezer array,” (2022), arxiv:2211.09780 [physics, physics:quant-ph] .
  * Christakis  _et al._ (2023) L. Christakis, J. S. Rosenberg, R. Raj, S. Chi, A. Morningstar, D. A. Huse, Z. Z. Yan, and W. S. Bakr, Nature 614, 64 (2023).
  * Li  _et al._ (2023) J.-R. Li, K. Matsuda, C. Miller, A. N. Carroll, W. G. Tobias, J. S. Higgins, and J. Ye, Nature 614, 70 (2023).
  * Gregory  _et al._ (2023) P. D. Gregory, L. M. Fernley, A. L. Tao, S. L. Bromley, J. Stepp, Z. Zhang, S. Kotochigova, K. R. A. Hazzard, and S. L. Cornish, “Second-scale rotational coherence and dipolar interactions in a gas of ultracold polar molecules,”  (2023), arxiv:2306.02991 [cond-mat, physics:physics] .
  * Delord  _et al._ (2020) T. Delord, P. Huillery, L. Nicolas, and G. Hétet, Nature 580, 56 (2020).
  * Ahn  _et al._ (2020) J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, and T. Li, Nature Nanotechnology 15, 89 (2020).
  * van der Laan  _et al._ (2021) F. van der Laan, F. Tebbenjohanns, R. Reimann, J. Vijayan, L. Novotny, and M. Frimmer, Physical Review Letters 127, 123605 (2021).
  * Stickler  _et al._ (2021) B. A. Stickler, K. Hornberger, and M. S. Kim, Nature Reviews Physics 3, 589 (2021).
  * Pontin  _et al._ (2023) A. Pontin, H. Fu, M. Toroš, T. S. Monteiro, and P. F. Barker, Nature Physics 19, 1003 (2023).
  * Kamba  _et al._ (2023) M. Kamba, R. Shimizu, and K. Aikawa, “Nanoscale feedback control of six degrees of freedom of a near-sphere,”  (2023), arxiv:2303.02831 [physics, physics:quant-ph] .
  * DeMille (2002) D. DeMille, Physical Review Letters 88, 067901 (2002).
  * Grimsmo  _et al._ (2020) A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Physical Review X 10, 011058 (2020).
  * Albert  _et al._ (2020) V. V. Albert, J. P. Covey, and J. Preskill, Physical Review X 10, 031050 (2020).
  * Zhou  _et al._ (2011) Y. L. Zhou, M. Ortner, and P. Rabl, Physical Review A 84, 052332 (2011).
  * Lechner and Zoller (2013) W. Lechner and P. Zoller, Physical Review Letters 111, 185306 (2013).
  * Sundar  _et al._ (2018) B. Sundar, B. Gadway, and K. R. A. Hazzard, Scientific Reports 8, 3422 (2018).
  * Stickler  _et al._ (2018a) B. A. Stickler, B. Papendell, S. Kuhn, B. Schrinski, J. Millen, M. Arndt, and K. Hornberger, New Journal of Physics 20, 122001 (2018a).
  * Ma  _et al._ (2020) Y. Ma, K. E. Khosla, B. A. Stickler, and M. S. Kim, Physical Review Letters 125, 053604 (2020).
  * Vieillard  _et al._ (2008) T. Vieillard, F. Chaussard, D. Sugny, B. Lavorel, and O. Faucher, Journal of Raman Spectroscopy 39, 694 (2008).
  * Owschimikow  _et al._ (2010) N. Owschimikow, F. Königsmann, J. Maurer, P. Giese, A. Ott, B. Schmidt, and N. Schwentner, Journal of Chemical Physics 133, 184302 (2010).
  * Milner  _et al._ (2014) A. A. Milner, A. Korobenko, J. W. Hepburn, and V. Milner, Physical Review Letters 113, (2014).
  * Tenney  _et al._ (2016) I. F. Tenney, M. Artamonov, T. Seideman, and P. H. Bucksbaum, Physical Review A 93, 13421 (2016).
  * Damari  _et al._ (2017) R. Damari, D. Rosenberg, and S. Fleischer, Physical Review Letters 119, 033002 (2017).
  * Neyenhuis  _et al._ (2012) B. Neyenhuis, B. Yan, S. A. Moses, J. P. Covey, A. Chotia, A. Petrov, S. Kotochigova, J. Ye, and D. S. Jin, Physical Review Letters 109, 230403 (2012).
  * Seeßelberg  _et al._ (2018) F. Seeßelberg, X.-Y. Luo, M. Li, R. Bause, S. Kotochigova, I. Bloch, and C. Gohle, Physical Review Letters 121, 253401 (2018).
  * Bause  _et al._ (2020) R. Bause, M. Li, A. Schindewolf, X.-Y. Chen, M. Duda, S. Kotochigova, I. Bloch, and X.-Y. Luo, Physical Review Letters 125, 023201 (2020).
  * Burchesky  _et al._ (2021) S. Burchesky, L. Anderegg, Y. Bao, S. S. Yu, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Physical Review Letters 127, 123202 (2021).
  * Stickler  _et al._ (2016) B. A. Stickler, B. Papendell, and K. Hornberger, Physical Review A 94, (2016).
  * Zhong and Robicheaux (2016) C. Zhong and F. Robicheaux, Physical Review A 94, 52109 (2016).
  * Papendell  _et al._ (2017) B. Papendell, B. A. Stickler, and K. Hornberger, New Journal of Physics 19, (2017).
  * Stickler  _et al._ (2018b) B. A. Stickler, B. Schrinski, and K. Hornberger, Physical Review Letters 121, (2018b).
  * Schlosshauer (2019) M. Schlosshauer, Physics Reports Quantum Decoherence, 831, 1 (2019).
  * Walls and Milburn (1985) D. Walls and G. Milburn, Physical Review A 31, 2403 (1985).
  * Brune  _et al._ (1996) M. Brune, E. Hagley, J. Dreyer, X. MaÎtre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Physical Review Letters 77, 4887 (1996).
  * Myatt  _et al._ (2000) C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinskl, W. M. Itano, C. Monroe, and D. J. Wineland, Nature 403, 269 (2000).
  * Auffeves  _et al._ (2003) A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, Physical Review Letters 91, 230405 (2003).
  * Pedernales  _et al._ (2020) J. S. Pedernales, F. Cosco, and M. B. Plenio, Physical Review Letters 125, 090501 (2020).
  * Martinetz  _et al._ (2022) L. Martinetz, K. Hornberger, and B. A. Stickler, PRX Quantum 3, 030327 (2022).
  * Li  _et al._ (2017) H.-K. Li, E. Urban, C. Noel, A. Chuang, Y. Xia, A. Ransford, B. Hemmerling, Y. Wang, T. Li, H. Häffner, and X. Zhang, Physical Review Letters 118, 053001 (2017).
  * Urban  _et al._ (2019) E. Urban, N. Glikin, S. Mouradian, K. Krimmel, B. Hemmerling, and H. Haeffner, Physical Review Letters 123, 133202 (2019).
  * Brownnutt  _et al._ (2015) M. Brownnutt, M. Kumph, P. Rabl, and R. Blatt, Reviews of Modern Physics 87, 1419 (2015).

## Methods

### Derivation of the master equation

We consider a planar rigid rotor formed by two ions with charges $e$ at
positions $r_{1,2}=z{\bf e}_{z}\pm r{\bf e}_{\rho}(\phi)$, where $z$ is the
height above the surface trap, $2r$ is the ion separation, and the angle
$\phi$ describes the rotor orientation. The potential energy of the rotor in
the quadrupole potential follows as

$V_{t}(\phi)=-er^{2}{\bf e}_{\rho}(\phi)\cdot\left[\nabla\otimes{\bf E}(z{\bf
e}_{z},t)\right]{\bf e}_{\rho}(\phi),$ (7)

where ${\bf E}({\bf r},t)$ is the local field strength at time $t$. The vector
gradient $\nabla\otimes{\bf E}(z{\bf e}_{z},t)$ is symmetric, traceless, and
determined by the trap geometry. Using $L_{\pm}=e^{\pm i\phi}$ the potential
energy can be rewritten, up to an additive constant, as

$V_{t}(\phi)=\frac{er^{2}}{4}\left(\varepsilon_{t}L_{+}^{2}+\varepsilon_{t}^{*}L_{-}^{2}\right).$
(8)

Here, we defined the complex-valued combination of field gradients

$\varepsilon_{t}=-\partial_{x}E_{x}+\partial_{y}E_{y}+2i\partial_{x}E_{y},$
(9)

whose magnitude is determined by the trapping geometry and the time-dependent
voltage at the control electrodes.

The rotor quantum state $\ket{\psi_{t}}$ evolves according to the Schrödinger
equation with Hamiltonian $H_{0}+V_{t}(\phi)$, involving the kinetic energy
$H_{0}=L_{z}^{2}/2I$. Switching to the interaction picture
$\ket{\chi_{t}}=\exp(iH_{0}t/\hbar)\ket{\psi_{t}}$ introduces the interaction
picture potential $\exp(iH_{0}t/\hbar)V\exp(-iH_{0}t/\hbar)$, which can be
evaluated by using that

$e^{iH_{0}t/\hbar}L_{\pm}^{2}e^{-iH_{0}t/\hbar}=e^{2i\hbar
t/I}L_{\pm}^{2}e^{\pm 2itL_{z}/I},$ (10)

as follows from the commutation relations $[L_{z},L_{\pm}]=\pm\hbar L_{\pm}$
and unitarity, $L_{\pm}L_{\mp}=\mathbb{I}$. We consider a rapidly revolving
rotor state $\ket{\chi_{t}}$ with mean angular momentum quantum number
$\overline{\ell}=I\omega_{\rm rot}/\hbar\gg 1$ much greater than the spread
$\sigma_{\ell}$. We can thus approximate $\exp(\pm
2itL_{z}/I)\ket{\chi_{t}}\simeq e^{\pm 2i\omega_{\rm rot}t}\ket{\chi_{t}}$,
yielding the interaction-picture Schrödinger equation
$i\hbar\partial_{t}\ket{\chi_{t}}=\widetilde{V}_{t}\ket{\chi_{t}}$ with

$\widetilde{V}_{t}=\frac{er^{2}}{4}\left(\varepsilon_{t}e^{2i\omega_{\rm
rot}t}L_{+}^{2}+\varepsilon_{t}^{*}e^{-2i\omega_{\rm rot}t}L_{-}^{2}\right).$
(11)

Integrating this Schrödinger equation interatively over the time interval
$\Delta t\gg 1/\omega_{\rm rot}$ yields an integral equation

$\displaystyle\ket{\chi_{t+\Delta t}}=$
$\displaystyle\ket{\chi_{t}}-\frac{i}{\hbar}\int_{t}^{t+\Delta
t}dt^{\prime}\widetilde{V}_{t^{\prime}}\ket{\chi_{t}}$
$\displaystyle-\frac{1}{\hbar^{2}}\int_{t}^{t+\Delta
t}dt^{\prime}\int_{t}^{t^{\prime}}dt^{\prime\prime}\widetilde{V}_{t^{\prime}}\widetilde{V}_{t^{\prime\prime}}\ket{\chi_{t^{\prime\prime}}}.$
(12)

where we may approximate $\ket{\chi_{t^{\prime\prime}}}\simeq\ket{\chi_{t}}$
to express that $\ket{\chi_{t}}$ evolves little in $\Delta t$.

This equation describes how the dynamics of $\ket{\chi_{t}}$ is influenced by
the random voltage noise applied to the control electrodes. Given that the
rotor-surface distance is so short that field retardation can be neglected,
the statistical properties the voltage noise directly translate to those of
$\varepsilon_{t}$. In the ensemble average, the rotor state is then described
by a Lindblad quantum master equation for the density operator
$\widetilde{\rho}$ (see Supplement),

$\displaystyle\partial_{t}\widetilde{\rho}=$ $\displaystyle\frac{1}{\Delta
t}\left[\mathbb{E}(\ket{\chi_{t+\Delta t}}\bra{\chi_{t+\Delta
t}})-\ket{\chi_{t}}\bra{\chi_{t}}\right]$ $\displaystyle\simeq$
$\displaystyle\frac{D}{4\hbar^{2}}\left(L_{+}^{2}\widetilde{\rho}L_{-}^{2}+L_{-}^{2}\widetilde{\rho}L_{+}^{2}-2\widetilde{\rho}\right).$
(13)

Here, $\mathbb{E}$ denotes the ensemble average over the stationary stochastic
voltage process, and we defined the angular momentum diffusion coefficient (3)
with the power spectral density

$S_{\varepsilon}(\omega)=\int_{-\infty}^{\infty}d\tau\mathbb{E}(\varepsilon_{t}\varepsilon^{*}_{t-\tau})e^{i\omega\tau}.$
(14)

In Eq. (Derivation of the master equation) we used that $\Delta t$ is large in
comparsion to the noise correlation time as determined by the noise filter
bandwidth of 19 kHz. The latter is indeed much greater than the width of
rotation frequencies $\sigma_{\ell}\hbar/I\simeq 1$ kHz. Transforming back to
the Schrödinger picture yields (under the same assumptions as above) the
master equation (1).

### Decoherence of angular momentum superpositions

Driving the rotor state at a rotational sideband of angular momentum offset
$\Delta\ell$ allows applying the $\pi/2$ pulse (see Supplement)

$\displaystyle X_{\pi/2}=$
$\displaystyle\frac{\mathbb{I}}{2}-\frac{\sigma_{1}^{+}+(-1)^{\Delta\ell}\sigma_{2}^{+}}{2}L_{+}^{\Delta\ell}+\frac{\sigma_{1}^{-}+(-1)^{\Delta\ell}\sigma_{2}^{-}}{2}L_{-}^{\Delta\ell}$
$\displaystyle+\frac{(-1)^{\Delta\ell}}{2}\left(\sigma_{1}^{+}\sigma_{2}^{+}L_{+}^{2\Delta\ell}+\sigma_{1}^{-}\sigma_{2}^{-}L_{-}^{2\Delta\ell}\right.$
$\displaystyle\left.-\sigma_{1}^{+}\sigma_{2}^{-}-\sigma_{1}^{-}\sigma_{2}^{+}\right).$
(15)

and the $\pi$ pulse

$\displaystyle X_{\pi}=$
$\displaystyle(-1)^{\Delta\ell}\left(\sigma_{1}^{+}\sigma_{2}^{+}L_{+}^{2\Delta\ell}+\sigma_{1}^{-}\sigma_{2}^{-}L_{-}^{2\Delta\ell}\right.$
$\displaystyle\left.-\sigma_{1}^{+}\sigma_{2}^{-}-\sigma_{1}^{-}\sigma_{2}^{+}\right).$
(16)

Here, $\sigma_{1,2}^{\pm}$ are the raising/lowering operators for the
electronic levels of the two ions. The pulses can be used to implement a
Ramsey interference scheme involving the steps:

  1. (0)

Initialization—The rotor is prepared in the coherent rotor state
$\ket{\Phi_{0}}=\ket{SS,\Psi_{\overline{\ell}}}$ with mean angular momentum
$\overline{\ell}\gg\sigma_{\ell}$.

  2. (1)

Creation of the superposition—At $t=0$, a $\pi/2$ pulse creates a
superposition of angular momentum states
$\ket{\Phi_{1}}=X_{\pi/2}\ket{\Phi_{0}}$. Direct application of Eq.
(Decoherence of angular momentum superpositions) yields the threefold
superposition

$\displaystyle\ket{\Phi_{1}}=$
$\displaystyle\frac{1}{2}\left(\ket{1}-\sqrt{2}\ket{2}+\ket{3}\right),$ (17)

of the orthonormal states

$\displaystyle\ket{1}=$ $\displaystyle\ket{SS,\Psi_{\overline{\ell}}}$ (18a)
$\displaystyle\ket{2}=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{DS}+(-1)^{\Delta\ell}\ket{SD}\right)\ket{\Psi_{\overline{\ell}+\Delta\ell}}$
(18b) $\displaystyle\ket{3}=$
$\displaystyle(-1)^{\Delta\ell}\ket{DD,\Psi_{\overline{\ell}+2\Delta\ell}}.$
(18c)

The laser pulse thus entangles the two ion’s internal state with that of the
rotational degree of freedom. Whether $\Delta\ell$ is even or odd determines
whether the $\pi/2$ pulse creates the even or odd superposition of $\ket{DS}$
and $\ket{SD}$.

  3. (2)

Time evolution—This is followed by a evolution in presence of the stochastic
potential (8) until $t=\tau/2$, $\ket{\Phi_{2}}=e^{-iH_{\rm
rot}\tau/2\hbar}U_{\tau/2}\ket{\Phi_{1}}$, where $H_{\rm
rot}=H_{0}+\hbar\Delta\omega(\sigma_{1}^{+}\sigma_{1}^{-}+\sigma_{2}^{+}\sigma_{2}^{-})$
and $U_{\rm\tau/2}$ describes the interaction picture time evolution in
presence of a realisation of the stochastic potential with the laser-atom
detuning $\Delta\omega$ (see Supplement).

  4. (3)

Hahn echo pulse—At $t=\tau/2$, a $\pi$ pulse implements a Hahn echo
$\ket{\Phi_{3}}=X_{\pi}\ket{\Phi_{2}}$.

  5. (4)

Time evolution—This is followed by a second time evolution in presence of the
environment until $t=\tau$, $\ket{\Phi_{4}}=e^{-iH_{\rm
rot}\tau/2\hbar}U_{\tau/2}\ket{\Phi_{3}}$. Note that all states in this
superposition are now translated by the same angle
$2(\overline{\ell}+\Delta\ell)\omega_{\rm r}\tau$, meaning that the
interferometer can be closed.

  6. (5)

Closing of the interferometer—A second $\pi/2$ pulse
$\ket{\Phi_{5}}=X_{\pi/2}\ket{\Phi_{4}}$ closes the interferometer.

  7. (6)

Measurement—Measuring the mean flourescence light of the two atoms yields the
mean probability of being in the excited state,

$\displaystyle P(D)=$ $\displaystyle\frac{1}{2}+\frac{1}{2}\left[{\rm
Prob}(DD)-{\rm Prob}(SS)\right].$ (19)

Using the final state $\ket{\Phi_{5}}$ and averaging over many repetitions of
the experiment yields (see Supplement)

$\displaystyle P(D)=$
$\displaystyle\frac{1}{2}-\frac{1}{2}\cos\left(\omega_{\rm
r}\Delta\ell^{2}\tau\right)C(\tau),$ (20)

with the reduction factor of the Ramsey fringe contrast (4).

### Coherent effects on the contrast decay profile

From (20) we see that the phase contrast, which decays from decoherence as
$C(\tau)$ given by (4), is additionally modulated by a factor

$C_{\text{mod}}(\tau)=\cos(\omega_{\text{r}}\Delta\ell^{2}\tau).$ (21)

This arises from interference between the two relevant coherences of the rotor
state: $\ket{\ell}\leftrightarrow\ket{\ell+\Delta\ell}$ and
$\ket{\ell+\Delta\ell}\leftrightarrow\ket{\ell+2\Delta\ell}$, whose respective
transition frequencies differ by $2\Delta\ell^{2}\omega_{\rm r}=2\pi\times 26$
Hz $\times\,\Delta\ell^{2}$. This modulation is necessarily present in the
coherence measurements presented in this work, which otherwise yield a
contrast profile of the form $\exp[-(\gamma\tau)^{3}]$. Equation (21) predicts
a node in the fringe contrast at $\tau=\pi/(2\Delta\ell^{2}\omega_{\rm r})$.
The modulation may therefore be safely neglected only if the coherence time is
much shorter than the node time, $1/\gamma\ll\pi/(2\Delta\ell^{2}\omega_{\rm
r})$. Otherwise, the effect of the modulation may be measurable before
decoherence has occurred, and thus this effect must be accounted for in order
to accurately estimate $\gamma$ from the measurements. We find from numerical
simulations that the modulation profile may be appreciably influenced by
imperfect operations, where only a partial initial coherence is created, which
we therefore must also take into account. Including corrections due to
imperfect operations, the contrast node is expected to occur at $\tau=19$ ms
for $\Delta\ell=1$ superpositions, at 4.8 ms for $\Delta\ell=2$, and at 2.4 ms
for $\Delta\ell=3$. For most of our measurements, this effect is negligible.
In estimating the decoherence rate $\gamma$ in Fig. 3b, we explicitly account
for this effect only when $\Delta\ell^{2}\omega_{\rm r}/2\pi\gamma>0.1$. See
Extended Data Fig. 1 for a demonstration of this effect.

### Non-rigid effects

In the main text, we approximate the ion crystal as a rigid rotor. The
leading-order non-rigid effect is centrifugal distortion: the moment of
inertia increases as the square of the rotation frequency due to centrifugal
force, effectively reducing the rotational constant $\omega_{\rm r}$ with
increasing angular momentum. For an in-plane confinement strength such that
the center-of-mass vibrational frequency is $\omega_{x}$, the fractional
change to $\omega_{\rm r}$ due to finite rotation frequency
$\omega_{\text{rot}}$ is given by
$2\omega_{\text{rot}}^{2}/\omega_{x}^{2}=2.0\times 10^{-2}$ for parameters
used in this work. This quantity is smaller than the fractional uncertainty of
any measurement presented, so we neglect this correction.

### Measurements of rotational friction

A finite-temperature environment interacting with a quantum rotor will have a
frictional effect in addition to angular momentum diffusion, which in general
may affect the decoherence dynamics Stickler  _et al._ (2018b). We thus verify
that the environment is well-approximated as infinite temperature by
ascertaining that rotational friction is negligible. We measure rotational
friction by recording the center frequency of a rotational sideband at various
wait times after rotational state preparation. For increased sensitivity, the
third rotational sideband frequency $\omega_{sb}^{(\Delta\ell=3)}$ is chosen,
and the rotation frequency inferred as
$\omega_{\text{rot}}=\omega_{sb}^{(\Delta\ell=3)}/3$. We do this for both an
ambient environment ($D=19\,\hbar^{2}$/ms) and an engineered environment
($D=110\,\hbar^{2}$/ms). In both cases, we find the rate of slowdown to be
consistent with zero. In the case of the ambient environment, we bound the
rate of slowdown (2$\sigma$ uncertainty of the slope of the fit) at
$<0.2\,\hbar$/ms in the case of the ambient environment, and $<0.3\,\hbar$/ms
in the case of the engineered environment. We thus neglect rotational friction
in the theoretical treatment in the main text. See Extended Data Fig. 2 for
these measurements.

### Additional decoherence sources

Two other potentially relevant effects are decoherence of the ions’ electronic
state and changes of the rotor’s moment of inertia due to fluctuations of the
trap frequency. We rule out both of these as significant contributions.

Since the Ramsey experiment entangles the angular momentum state of the rotor
with the electronic state of the ions, decoherence of the electronic states
will also result in a loss of measured phase contrast. To measure the
electronic coherence time, we trap a single ion and prepare it in the
superposition $\ket{S}+\ket{D}$ with a Ramsey experiment including a Hahn echo
pulse. We find the $1/e$ coherence time to be 38 ms (see Extended Data Fig.
3), much longer than any coherence measurement presented in this work.

Changes in the rotor’s moment of inertia conserve its angular momentum but
change the rotation frequency, thereby also decohering a superposition of
angular displacements. The stability of the moment of inertia $I$ is
determined by the stability of the transverse trap frequency $\omega_{x}$,
related by $I\propto\omega_{x}^{-4/3}$. We measure the stability of the trap
frequency by trapping a single ion and preparing it in the Fock-state
superposition $\ket{0}+\ket{1}$ with a Ramsey experiment including a Hahn echo
pulse. We find a $1/e$ coherence time of 32 ms (see Extended Data Fig. 3).
When propagated to the case of angular momentum superpositions in the rotor
for $\omega_{\text{rot}}=145$ kHz, this yields an inferred rotational
coherence time of $240$ ms$/\Delta\ell$, again much longer than the coherence
measurements presented.

### Acknowledgements

Funding provided by the NSF Grant No. PHY 2011973 and by AFOSR Grant No.
FA9550-23-1-0546.

B.A.S. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – 510794108.

## Extended Data Figures

Extended Data Fig. 1: Coherent oscillations of phase contrast amplitude. All
contrast decay measurements for $\Delta\ell=3$ measurements of decoherence
presented in Fig. 3b are shown here. The shaded region is the contrast profile
from coherent 4-level dynamics, modified from (21) to account for imperfect
operations (sub-unity initial contrast) such that it yields agreement with
numerical simulations (see Methods). Solid lines are fits of the form (5),
additionally multiplied by the shaded profile in cases where
$\Delta\ell^{2}\omega_{\rm r}/2\pi\gamma>0.1$, which here are
$D=2.1\,\hbar^{2}$/ms and $D=13\,\hbar^{2}$/ms.

Extended Data Fig. 2: Measurements of rotational friction. The rotation
frequency is measured as a function of wait time after preparation of a
rapidly rotating state, both without without ($D=19\,\hbar^{2}$/ms) and with
($D=110\,\hbar^{2}$/ms) voltage noise injection. Over 100 ms, no measurable
slowdown is observed.

Extended Data Fig. 3: Other coherence measurements. Measurements of the
electronic degree of freedom and the single-ion transverse vibrational degree
of freedom are shown. Electronic coherence is measured using a single ion
placed in the superposition $\ket{S}+\ket{D}$. A vibrational coherence
measurement of the $\ket{0}+\ket{1}$ Fock state superposition probes the
coherence of the transverse vibrational motion of a single ion, placing a
limit on the stability of the rotor’s moment of inertia. A Hahn echo pulse is
included in both of these measurements to match the conditions of the
rotational decoherence measurements.