Long-Lived Circular Rydberg Qubits of Alkaline-Earth Atoms in Optical Tweezers

C. Hölzl , A. Götzelmann , E. Pultinevicius , M. Wirth , and F. Meinert
5. Physikalisches Institut and Center for Integrated Quantum Science and Technology,

Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

(Received 19 January 2024; accepted 28 March 2024; published 3 May 2024)

Coherence time and gate fidelities in Rydberg atom quantum simulators and computers are
fundamentally limited by the Rydberg state lifetime. Circular Rydberg states are highly promising
candidates to overcome this limitation by orders of magnitude, as they can be effectively protected from
decay due to their maximum angular momentum. We report the first realization of alkaline-earth circular
Rydberg atoms trapped in optical tweezers, which provide unique and novel control possibilities due to the
optically active ionic core. Specifically, we demonstrate creation of very high-n (n ¼ 79) circular states of
88Sr. We measure lifetimes as long as 2.55 ms at room temperature, which are achieved via cavity-assisted
suppression of black-body radiation. We show coherent control of a microwave qubit encoded in circular
states of nearby manifolds, and characterize the qubit coherence time via Ramsey and spin-echo
spectroscopy. Finally, circular-state tweezer trapping exploiting the Srþ core polarizability is quantified
via measurements of the trap-induced light shift on the qubit. Our work opens routes for quantum
simulations with circular Rydberg states of divalent atoms, exploiting the emergent toolbox associated with
the optically active core ion.

DOI: 10.1103/PhysRevX.14.021024 Subject Areas: Atomic and Molecular Physics,
Quantum Physics, Quantum Information

I. INTRODUCTION

Arrays of individually controlled and interacting
Rydberg atoms based on optical tweezer technology have
recently enabled rapid advances in the development of
neutral-atom quantum simulators and computers [1].
Prominent examples range from large-scale simulation of
quantum spin models [2,3], over the implementation of
optimization problems [4,5], to high-fidelity gate opera-
tions in quantum circuits [6,7], and even demonstrations of
key steps toward quantum error correction [8–12].
For all of these applications, the lifetime of the highly

excited Rydberg levels sets a fundamental limit for achiev-
able coherence times or gate fidelities. In this context, the
use of circular Rydberg states has recently attracted
increasing attention to overcome this constraint, both for
analog quantum simulators and gate-based quantum com-
puters [13,14]. Circular Rydberg states have maximum
allowed angular momentum (i.e., jmj ¼ n − 1, where m
and n denote the orbital magnetic and principal quantum
number), which inhibits optical decay to low-lying orbitals
by selection rules [15]. This opens up exciting prospects to

increase the coherence time of Rydberg atom arrays by
orders of magnitude either in cryogenic or room-temper-
ature setups [16–18]. Only very recently, first tweezer
arrays with circular Rydberg states of rubidium atoms have
been demonstrated using optical bottle beam traps [19].
In this article, we demonstrate the first tweezer-trapped

circular states of alkaline-earth atoms, which in contrast to
alkali atoms, provide a second optically active electron.
Combining the richer low-lying electronic structure of
divalent atoms with atom arrays already gave rise to
powerful new tools [20–22], for example, for optical clock
metrology or neutral-atom quantum computing [23–26]. In
contrast to low-angular-momentum Rydberg states (e.g., S
or D orbitals), circular states of alkaline-earth atoms allow
for ionic-core excitation in the absence of rapid autoioniza-
tion, providing a plethora of unique possibilities. First, the
ion core enables conservative trapping in standard Gaussian
beam tweezers, which brings scaling advantages in view of
power requirements when compared to the bottle traps
needed for alkali atoms. Second, photon scattering at the
broad and narrow core transitions can be exploited for direct
laser cooling and imaging of the trapped Rydberg atoms,
making use of central manipulation techniques developed
for trapped ions. Third, combiningmicrowave control of the
Rydberg electron with narrow-line optical core spectros-
copy involving the ion’s D level enables local control and
readout of the circular Rydberg qubit via the quadrupole
interaction between the two electrons (the interaction was
recently demonstrated in an atomic beam experiment [27]).

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Here, we create very-high-n (n ¼ 79) circular Rydberg
states of 88Sr atoms from an array of optical tweezers and
demonstrate coherent control of a qubit encoded in circular
states separated by two principal quantum numbers which
is driven by a two-photon microwave (MW) transition. We
demonstrate trapping of the circular Rydberg atom in the
optical tweezer exploiting the dominating Srþ core polar-
izability and analyze the effect of the trapping light on the
qubit coherence. We observe lifetimes of the circular
Rydberg states as long as 2.55 ms, which is about an
order of magnitude longer than the free-space black-body
decay at room temperature. The long lifetime is achieved
by placing the atoms inside a pair of optically transparent
capacitor plates, which suppress the black-body field at
microwave frequencies.

II. PREPARING CIRCULAR RYDBERG STATES

Our experiments start with an array of ten optical
tweezers at a wavelength of λ ¼ 539.91 nm and a waist

of 564(5) nm, which are stochastically loaded with single
88Sr atoms, cooled close to the motional ground state [see
Figs. 1(a) and 1(b); for details on tweezer loading, in-trap
laser cooling, and parity projection in our setup, see
Ref. [28] ]. The atoms are prepared inside a structure
consisting of six electrodes. Four of them form a ring
structure and allow us to apply electric fields in the x-y
plane of the tweezer array. The remaining two plate
electrodes are placed below and above this ring structure
for controlling the electric field along the z direction
(direction of axial tweezer confinement). They are fabri-
cated from glass coated with a thin film (approximately
700-nm thickness) of indium tin oxide (ITO) [17], which
grants excellent optical access for laser cooling, high-NA
tweezer generation, and single-atom imaging.
We prepare circular Rydberg states (CRSs) by first

exciting the atoms from the 1S0 ground state to the n ¼
79; 1F3; m ¼ 2 Rydberg level in the presence of a magnetic
field B ¼ 0.40ð5Þ G pointing along the z direction [see
Fig. 1(a)] [29]. During this tRyd ¼ 20-μs-long off-resonant

(a)

(b)

(c) (d) (e)

(f)

MW horn

f

p

FIG. 1. (a) Schematic drawing of the experiment. Single 88Sr atoms are trapped in optical tweezers (green) inside an electrode structure
consisting of two transparent electrodes (red) and four circularly shaped electrodes for applying σþ-polarized radio-frequency (rf) fields
(indicated with orange arrows). Magnetic and electric control fields E and B are aligned with the tweezer axis. (b) Averaged fluorescence
image of single atoms in the ten-tweezer array used throughout this work obtained through the transparent electrodes. (c) The atoms are
excited to the j79F;m ¼ 2i state via an off-resonant three-photon transition, from where they are transferred to the j79Ci circular
Rydberg state by an adiabatic rapid passage (ARP). The qubit is implemented by coherently coupling j79Ci to j77Ci by two microwave
photons at approximately 13.9 GHz. (d) State-selective field-ionization profile of the j79Fi, j79Ci, and j77Ci states ionized by a 10-μs
linear field ramp to 27ð5Þ Vcm−1 before (blue) and after circularization and a partial transfer to j77Ci (red). The three states are well
distinguishable, and by counting the events within the gray integration windows, the state populations pn are extracted. (e) Sketch of the
experimental sequence showing tweezer trap depth (green), electric control field strength (red), Rydberg laser light (purple), rf field
(orange), microwave for qubit control (blue), and state readout via state-selective ramped field ionization (SSFI) (brown). The employed
technical components in (a) are colored accordingly. (f) Rabi oscillations on the j79Ci ↔ j77Ci microwave transition. The population
transfer p77=ðp77 þ p79Þ to find the atom in j77Ci as a function of the microwave pulse length tMW is shown. The solid blue line is a
sinusoidal fit with a Gaussian decay envelope. Note that for the three areas separated by the gray dashed lines, the frequency is varied
independently due to microwave power fluctuations between measurements. Error bars represent 1 standard deviation.

C. HÖLZL et al. PHYS. REV. X 14, 021024 (2024)

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three-photon excitation [cf. Fig. 1(e)], the tweezer light is
turned off to prevent light shifts. Subsequently, the
electron is transferred into the circular Rydberg orbit
j79Ci (n ¼ 79; l ¼ m ¼ 78) via an adiabatic rapid passage
protocol. To this end, we first ramp up the electric
field along z from zero to E ¼ 478ð1Þ mVcm−1 within
tS ¼ 5 μs. At this field, the initial Rydberg F state
smoothly attaches to the Stark-shifted manifold of high-
l levels, effectively forming an equidistant ladder of states
with increasing m up to the circular state. The equidistant
spacing allows for resonant coupling of all levels in that
ladder with a single σþ-polarized rf field. In our setup, we
generate this field by applying sinusoidal voltages with
frequency frf ¼ 70 MHz and phase shifts of about 90 deg
between pairs of neighboring electrodes on the four ring
electrodes. The adiabatic passage is then driven by slowly
(within tARP ¼ 20 μs) sweeping the electric field along z
further up to 597ð1Þ mVcm−1 through the rf-induced
multilevel avoided crossing (see Appendix A for details).
Note that prior to this work, circularization via rapid
adiabatic transfer has been explored for Rydberg states of
much lower principal quantum number (n ≤ 52) [30].
We exploit SSFI for ensemble-averaged state readout.

The ionization field is applied along the x direction via two
of the ring electrodes and guides the produced ions toward
a microchannel plate (MCP) mounted outside the electrode
cage for detection. Figure 1(d) shows histograms of the
ionization fields derived from the time of flight to the
detector before and after the adiabatic rapid passage. For
the data after transfer, we apply a resonant two-photon MW
pulse which partially transfers j79Ci to j77Ci, as discussed
in more detail in the next paragraph. SSFI provides
powerful means to tell apart low-l and high-l states, but
does not give enough resolution to distinguish the target
circular Rydberg level from not fully stretched “elliptical
states” with l < n − 1. In order to probe the fidelity for
preparing j79Ci, we implement coherent driving of a
microwave qubit encoded in the circular Rydberg levels
j79Ci and j77Ci, which are well separated in the SSFI
signal [cf. Fig. 1(d)], from which the population p77 (p79)
in j77Ci (j79Ci) is extracted. Coupling the qubit states is
achieved by an off-resonant two-photon microwave drive at
fMW ¼ 13.86 GHz [see Fig. 1(c)]. Moreover, after the
adiabatic passage and prior to the MW pulse, the electric
field is ramped up to approximately 2 Vcm−1, which shifts
transitions from j79Ci to states other than j77Ci out of
resonance. Recording the population transfer P ¼
p77=ðp79 þ p77Þ from j79Ci to j77Ci as a function of
the MW pulse length tMW reveals coherent Rabi oscilla-
tions as depicted in Fig. 1(f). From the maximum pop-
ulation transfer into j77Ci after a π pulse [i.e., after
tMW ¼ 870ð5Þ ns], we infer about ϵCRS ≈ 70% preparation
efficiency of the j79Ci circular Rydberg level. Damping of
the Rabi oscillations is attributed to small shot-to-shot
fluctuations of the microwave amplitude at the position of

the atoms. It is well modeled by a Gaussian envelope [6],
from which we extract a 1=e time of 60ð3Þ μs. Note that the
Rabi frequency also changed by approximately 10%
between measurement sets due to fluctuations of the output
power of the microwave generator. This is accounted for by
fitting a function with piecewise-independent frequencies
in the areas separated by the gray dashed lines.

III. CIRCULAR-STATE QUBIT COHERENCE

In the next step, we probe the transverse coherence time
of the circular-state qubit j77Ci ↔ j79Ci via its free
induction decay. An exemplary dataset obtained from a
standard Ramsey measurement (π=2 pulse, wait time tR,
π=2 pulse) with a π=2-pulse time of tπ=2 ¼ 227ð2Þ ns is
shown in Fig. 2(a) (blue circles). A sinusoidal fit of the
Ramsey signal with a Gaussian envelope ∝ expð−t2R=2T�

2
2Þ

reflecting a stochastic dephasing process from fluctuations
of the qubit resonance frequency reveals a reversible
coherence time T�

2 ¼ 43ð2Þ μs. In fact, during this time,
longitudinal decay out of the qubit subspace is compara-
tively small [gray diamonds in Fig. 2(a)], and decoherence
is primarily attributed to the first-order magnetic and
second-order electric sensitivity of the transition between
the circular states (see Appendix B for details). From the

(a)

(b)

FIG. 2. (a) Measurement of the circular-state qubit coherence
via Ramsey (blue) and spin-echo (red) experiments. The pop-
ulation transfer P into the j77Ci state is plotted against the
cumulative Ramsey time and echo time tR þ tE. The microwave
pulse sequences for the Ramsey and spin-echo experiments are
shown schematically above the data. Solid lines are Gaussian-
damped sine functions fitted to the data. The gray diamonds show
the combined ion detection probability in j79Ci and j77Ci
(p77 þ p79) scaled by the first data point. An exponential fit
(gray dashed line) to the data reveals a population lifetime in the
qubit subspace of 1.3(1) ms. (b) Maximum fringe contrast of the
spin-echo signal scaled by the preparation efficiency ϵCRS, as a
function of the revival time tC. The solid line shows a fit of Eq. (1)
to the data. Error bars show 1 standard deviation.

LONG-LIVED CIRCULAR RYDBERG QUBITS OF ALKALINE- … PHYS. REV. X 14, 021024 (2024)

021024-3



oscillation frequency of the Ramsey fringes, we deduce an
MW (red) detuning of Δ ¼ 66.5ð2Þ kHz.
Extending the Ramsey measurement to spin-echo inter-

ferometry (π=2 pulse, wait time tR, π pulse, wait time tE,
π=2 pulse) allows us to probe reversibility of the qubit
dephasing. An exemplary dataset, for which we apply the
echo π pulse after tR ¼ 160 μs and scan tE, is shown in
Fig. 2(a) (red circles). A clear reappearance of fringes in the
coherent qubit evolution is observed near tE ≈ tR. We
define the revival time tC as the time where the contrast
is maximal. Notably, tC of the spin echo is shifted from 2tR
to earlier times (see Appendix B). From such measure-
ments, we quantify the degree of reversibility by extracting
the contrast at tC from a sinusoidal fit to the data with a
Gaussian envelope function. Figure 2(b) depicts this con-
trast scaled by the CRS preparation efficiency ϵCRS as a
function of tC. These measurements reveal coherent qubit
evolution up to several hundred microseconds, a timescale
for which population loss out of the qubit subspace cannot
be fully ignored anymore. For this reason, we model the
decay of the echo contrast with a functional form which
also includes longitudinal decay (see Appendix B),

CðtCÞ ¼ exp

�
−D

�
tC
2
−
T�
2
2D
2

tanh
tC

T�
2
2D

�
−
tC
τl

�
; ð1Þ

where the first part describes irreversible dephasing,
which is modeled in a similar manner as in Refs. [16,31],
assuming a realistic Lorentzian noise spectrum. It is
quantified by a noise amplitude D and the reversible
coherence time T�

2 introduced above. The second term
accounts for the effective circular-state lifetime τl. A fit to
the data over the free parameters D and τl reveals
τl ¼ 1.4ð6Þ ms, which is in good agreement with results
obtained from detailed measurements of the qubit lifetime
below (see Sec. V). The irreversible coherence time
[CðT2Þ ¼ Cð0Þ=2] is derived from the fit function to
T2 ¼ 278ð10Þ μs.
For all of the measurements so far, the optical tweezer

depth has been set to zero during circular-state preparation
and probing; i.e., the circular Rydberg qubit was not
trapped during the measurements but was slowly dispersing
freely in the trapping region. In the following, we keep the
tweezer light on to demonstrate trapping of the circular
Rydberg states and to quantify the degree of trap-light-
induced qubit decoherence.

IV. TWEEZER TRAPPING

The trapping potential seen by the circular Rydberg atom
is a sum of two contributions, as depicted in Fig. 3(a). First,
the driven motion of the Rydberg electron in the oscillating
laser field results in a spatially dependent ponderomotive
energy shift [32]. For circular Rydberg orbitals similar or
smaller in size than the waist of the optical tweezer, this
yields a repulsive potential expelling the Rydberg atom

from the trap focus [blue dashed line in Fig. 3(a)]. As a
consequence, optical trapping of alkali (circular) Rydberg
atoms, i.e., atoms with a single valence electron, requires
tailored bottle beam potentials with a light intensity
minimum at the center [19,33]. Notably, for very high-n
circular Rydberg states, with a radius that exceeds the trap
waist, the ponderomotive potential develops a central
minimum even for Gaussian beam tweezers, allowing to
pin circular orbits via the ponderomotive force alone. For
our trap waist, this “needle-trap” effect is expected for
circular states with n ≳ 100 [red dashed line in Fig. 3(a)]
[34]. Alkaline-earth Rydberg atoms feature additional ac
polarizability due to the optically active ionic core, result-
ing in the second contribution to the net trapping potential.
For our tweezer wavelength, this results in a strong
attractive potential [dotted line in Fig. 3(a)], which

(a)

(c) (d)

(b)

FIG. 3. (a) Radial cut through the calculated tweezer potential
for CRS with n ¼ 77 (blue) and n ¼ 100 (red) at the trap center.
The dashed lines show the contribution from the ponderomotive
potential of the electron in the CRS, and the gray dotted line
depicts the contribution from the ionic-core polarizability. The
size of the electron wave function (shaded areas) is comparable to
the tweezer waist. The total trap potential (solid lines) depends on
n, leading to a differential light shift δLS. (b) Differential light
shift δLS for the two-photon MW transition jnCi → jðnþ 2ÞCi as
a function of n. (c) Measured microwave spectrum j79Ci →
j77Ci for tweezer depth U0 ¼ 2.39 MHz (red diamonds) and
U0 ¼ 5.13 MHz (green squares) compared to the free-space
resonance (blue circles). Solid lines are Gaussian fits to the data
to extract the light shift. Linewidths are dominated by power
broadening from different microwave powers set for the three
measurements. (d) Light shift δLS as a function of the tweezer
depth U0 (tweezer power PCRS) obtained from microwave
spectroscopy data as in (c) (red circles) and from Ramsey
oscillations as in Fig. 4 (green triangles). The blue shaded area
is an ab initio calculation of the expected light shift including
experimental uncertainties in the tweezer power, while the gray
line is a linear fit to the data. In all panels, vertical error bars
represent 1 standard deviation.

C. HÖLZL et al. PHYS. REV. X 14, 021024 (2024)

021024-4



overcomes the repulsive ponderomotive force and readily
allows for trapping in a standard Gaussian beam tweezer.
Specifically, the net potential depth U0 for our j79Ci is

about 1=5 of the potential seen by the 1S0 electronic ground
state. To ensure qubit trapping in our experiment, we now
switch the tweezer light back on immediately after the
optical F-state excitation and set the power PCRS in each
tweezer spot to a value at least 6 times larger (approx-
imately 350 μW) than the value set for trapping 1S0 [35].
Note that we do not observe any effect of the trapping
potential on the circular-state preparation efficiency in our
measurements.
In a first set of experiments, we measure the differential

light shift of our trapped circular Rydberg qubit over a
range of tweezer depths. To this end, we slowly (within
50 μs) ramp the tweezer power to larger values up to
PCRS ≈ 5.4 mW after preparing j79Ci, and then perform
in-trap microwave spectroscopy on the j77Ci ↔ j79Ci
qubit transition. Exemplary spectra are depicted in
Fig. 3(c), revealing a clear trap-induced light shift on the
microwave resonance δLS < 0. Specifically, the tweezer
potential is slightly deeper for j79Ci due to the reduced
ponderomotive potential [cf. solid lines in Fig. 3(a)],
causing a redshift of the qubit resonance in the trap center
with respect to the free-space microwave transition
(δLS < 0). In Fig. 3(d), the light shift, which is extracted
from the center value of Gaussian fits to the data, is shown
as a function of PCRS (red circles). We find good agreement
with ab initio n-dependent trap depth calculations (blue
shaded area). Residual discrepancies are attributed to
thermal motion in the trap, for which the atom does not
ideally probe the trap bottom.
We also record that the differential light shift is signifi-

cantly n dependent and for our tweezer waist maximal
around the principal quantum numbers used throughout
this work [see Fig. 3(b)]. This can be used for local
microwave addressing, at the expense of enhanced
motional dephasing.
In a second set of experiments, we repeat the Ramsey

and spin-echo experiments from above, but now with
trapped circular Rydberg atoms. Exemplary Ramsey
fringes for three different values of PCRS are shown in
Figs. 4(a)–4(c). We observe that the oscillation frequency
of the free induction decay decreases for increasing trap
depth. This is attributed to the differential light shift δLS
between the two qubit states j79Ci and j77Ci, causing a
redshift and thereby reducing the effective detuning during
tR. The extracted light shifts are added to Fig. 3(d) (green
circles), where we find good agreement with the expected
values as well as with light shifts extracted from the
spectroscopy measurements.
We also identify a reduction in T�

2 with increasing
tweezer depth [Fig. 4(d)]. A comparison with results from
a semiclassical simulation of the dephasing dynamics
suggests that the decrease in T�

2 is due to (thermal) motion

in the trap (see Appendix C). This motion is dominated by
the short trap release in combination with the photon recoil
during the 20-μs-long Rydberg F-state excitation.
Increasing the currently limited Rydberg laser power and
thereby shortening the release duration should allow us to
significantly reduce this source of dephasing. Perspectively,
one may also compensate the differential light shift on the
qubit by combining the Gaussian trap with a second
Laguerre-Gaussian beam with radial index p ¼ 0, which
counteracts the light intensity gradient around the circular
electron orbit. Interestingly, using a pair of Laguerre-
Gaussian modes with azimuthal index opposite in sign
would allow for locally driving transitions between distant
circular states [14].
Notably, also the trap-induced dephasing can be rephased

via spin echo [see Fig. 4(e)], though with a slight reduction
of the irreversible transverse coherence time compared
to the free-space scenario [dashed line in Fig. 4(e)].

V. LIFETIME

Finally, we investigate the lifetime of our circular
Rydberg qubit in more detail. To this end, we initialize
the qubit either in j79Ci or j77Ci, the latter via a microwave
π pulse applied after preparing j79Ci. For both scenarios,
we hold the atoms in shallow tweezers (approximately
400 μW) for a variable time t and subsequently perform
state-selective field ionization. This allows us to identify
black-body-induced population transfer into neighboring n
manifolds. Representative histograms of the field-ioniza-
tion signal are depicted in Figs. 5(c) and 5(d) at t ≈ 630 μs
(indicated by dashed vertical lines) when starting in either
of the qubit states. Evidently, decay from the circular state

(a) (d) (e)

(b)

(c)

FIG. 4. (a)–(c) Ramsey signal for three different values of the
tweezer power PCRS ¼ ð1.15; 0.73; 0.32Þ mW. For increasing
power, the frequency decreases by the tweezer-induced light
shift. (d) Reversible coherence time T�

2 extracted from Ramsey
measurements as a function of the tweezer power PCRS (green
circles) compared to expectations from a semiclassical dephasing
model (green line). The gray dashed lines mark the correspon-
dence to the Ramsey data in (a)–(c). (e) Spin-echo contrast
obtained with the same method as in Fig. 2(b) but for atoms
trapped in tweezers with PCRS ¼ 0.32 mW (red circles). The
red dashed line shows the model fitted to the nontrapped case
[Fig. 2(b)] for comparison. Error bars show 1 standard deviation.

LONG-LIVED CIRCULAR RYDBERG QUBITS OF ALKALINE- … PHYS. REV. X 14, 021024 (2024)

021024-5



appears to be significantly slower for j79Ci. For a
quantitative analysis, we fit the histograms for each hold
time with a sum over multiple skewed Gaussians with the
individual amplitudes as only free parameters and extract
the time-dependent state population pn from the areas
under the individual curves. The results are depicted in
Figs. 5(a) and 5(b), and allow for extracting lifetimes τjnCi
by fitting the data with a rate model (see Appendix E).
Specifically, we obtain τj79Ci ¼ 2.55ð10Þ ms and τj77Ci ¼
0.53ð8Þ ms. Comparing these observations to predictions
in free space at room temperature (τj79Ci;FS ¼ 303 μs and
τj77Ci;FS ¼ 297 μs), we find an enhancement in lifetime by
factors of 8.4 and 1.8, respectively. The long circular-state
lifetimes are attributed to the presence of the ITO electrodes
forming a plate capacitor in the x-y plane, which coincides
with the orbital plane of the Rydberg electron. The
capacitor plates are spaced by d ¼ 10.0ð2Þ mm, which is
slightly smaller than the half wavelength of black-body
transitions into the neighboring n manifolds λn→n−1

BB =2 ¼
10.6 (10.2) mm for j79Ci (j77Ci). This leads to suppres-
sion of the most detrimental circularly polarized black-
body modes inside the electrode structure and allows
us to create and control long-lived circular states without

cryogenic cooling. The fact that the capacitor spacing is
only slightly larger than the relevant black-body wave-
length leads to the large difference in the suppression
factors that we find for the two circular qubit states. We find
good agreement with calculations for a infinite plate
capacitor along the lines of Ref. [17], yielding τj77Ci ¼
0.56 ms and τj79Ci ¼ 2.37 ms when assuming a capacitor
spacing of d ¼ 10.15 mm and plate reflectivity of
R ¼ 0.96. As the lifetime is extremely sensitive to the
capacitor spacing in this regime, measurements at higher n
and a simulation of our finite electrode geometry is
necessary to fully characterize the suppression effect.
Nevertheless, our calculations suggest that lifetimes
>10 ms should be reached when increasing n to ≥96.

VI. CONCLUSION AND OUTLOOK

We have demonstrated creation and trapping of alkaline-
earth atoms in very high-n circular Rydberg states by
applying radio-frequency-driven adiabatic state preparation
to a record-breaking principal quantum number of n ¼ 79.
Trapping in a standard Gaussian tweezer beam is possible
due to the core polarization of the divalent Rydberg atom.
We have implemented microwave control of a circular
Rydberg atom qubit and characterized qubit coherence via
Ramsey and spin-echo spectroscopy, including analysis of
trap-induced dephasing. Our results have shown coherence
times up to 278ð10Þ μs limited by residual magnetic-field
noise and electric-field gradients in our current setup.
Assisted by an in-vacuum suppression capacitor for
black-body modes, we have found that the produced
circular states live for up to 2.5 ms, which to the best of
our knowledge is the longest-lived Rydberg atom ever
observed in a room-temperature environment.
Our work opens the door to quantum simulations with

alkaline-earth circular Rydberg states, and provides pros-
pects for novel qubit concepts in gate-based Rydberg
quantum computers [13,14,17]. Higher preparation fidel-
ities or partial-ionization methods to purify the created
circular Rydberg states [36] will allow us to decrease the
electric field we applied here to separate transitions of
nearby elliptical states, which drastically decreases electric-
field sensitivity. Together with improved magnetic-field
noise cancellation, this should allow for reaching qubit
coherence in the millisecond range and beyond [17], which
provides exciting opportunities to overcome fundamental
lifetime limitations in state-of-the-art quantum simulators
using low-l Rydberg states by orders of magnitude.
Specifically, the high-n circular states we have demon-
strated here give >10 MHz exchange coupling between
neighboring n, and thus on the order of 104–105 coherent
flip-flops within their lifetime even at room temperature.
Perspectively, reaching two-body gate fidelities beyond
99.9% would also profit from increased Rydberg state
lifetime, at the expense of additional challenges associated

(a)

(b)
(c)

(d)

FIG. 5. Decay dynamics of j79Ci (a) and j77Ci (b) into
neighboring circular Rydberg states. Symbols depict the pop-
ulations pn in different n manifolds as a function of the hold time
t. The populations in each state are recorded via SSFI with a
24-μs-long linear ionization ramp to 27ð5Þ Vcm−1 at various
values of t and are extracted by fitting multiple skewed Gaussians
to the obtained ion histograms. The initial population p77 and p79

in (b) arises from the preparation efficiency of j77Ci via an MW π
pulse from j79Ci. Solid lines depict a fit of a rate model to the
data, which yields lifetimes for j79Ci (j77Ci) of 2.55(10) ms
[0.53(8) ms]. Insets (c) and (d) show representative ion histo-
grams for the datasets in (a) and (b), respectively, recorded at
values of t marked by the dashed vertical lines. The colors
indicate the principal quantum numbers labeled in (a) and apply
to all subfigures.

C. HÖLZL et al. PHYS. REV. X 14, 021024 (2024)

021024-6



with using circular states for digital mode operation [14].
The available ionic-core transitions can be exploited in
future experiments for optical readout and laser cooling of
circular Rydberg atoms [37–39]. Narrow-line spectroscopy
on the Srþ core also enables coherent and local optical
manipulation of the circular-state qubit mediated by the
electrostatic coupling between the two electrons [27]. Fully
utilizing this coupling requires trapping of the circular atom
also when the second electron is in the excited metastable
Srþ core levels. This is not the case for the traps used here
but can be achieved by overlapping a near-infrared tweezer
to form a bichromatic array.
The next obvious steps include the creation and control

of interacting circular Rydberg atoms, scaling the system to
large qubit arrays, possibly assisted by rapid Rabi coupling
to circular states [40], and optimal control techniques for
improving state preparation efficiencies [41,42]. Moreover,
dynamical rearrangement of the Rydberg atoms within their
long lifetime is in reach, and together with nondestructive
state detection [27], e.g., via low-l ancilla atoms [14], could
be used for filling defects in the array. Residual F states
could be efficiently removed by autoionization using the
inner core transitions. The long coherence times may then
be used to extend simulations of quantum magnets to
situations with strong spin-phonon coupling [43–45], and
additional control of radio-frequency coupling to other
high-l states in the giant Rydberg manifold provides access
to large-spin Heisenberg models [46].

ACKNOWLEDGMENTS

We thank the Quantum Länd team for fruitful discussions
and Jennifer Krauter for proofreading. We acknowledge
funding from the Federal Ministry of Education and
Research under the Grants CiRQus and QRydDemo, the
Carl Zeiss Foundation via IQST, the Horizon Europe
ProgrammeHORIZON-CL4-2021-DIGITAL-EMERGING-
01-30 via Project No. 101070144 (EuRyQa), and the Vector
Foundation.

APPENDIX A: CIRCULARIZATION

In this section, we provide additional details on the
adiabatic circularization method described in the main text.
By coupling the lowest-lying states of each m, the initial
optically accessible F state can be coupled to the circular
Rydberg state. To optimize the adiabatic multicrossing
efficiency when sweeping the electric field, the frequency
of all involved transitions must be approximately equal
fm;mþ1 ≈ frf . Figure 6(a) shows those frequencies calcu-
lated (using Ref. [47]) with the quantum defects reported in
Ref. [48] as a function of the electric field and magnetic
field of B ¼ 0.4 G. The m ≤ 3 states are split off from the
equally spaced hydrogenic manifold with m ≥ 4 by the
quantum defect. While the transition frequencies f2;3 and
f3;4 attach smoothly to the transition frequency of the

hydrogenic manifold fHM with increasing electric field, the
lower-m states get further shifted by states of the neighbor-
ing n ¼ 78 manifold, decreasing the transition frequency
below fHM. The electric field during the ARP must be
chosen close to the point where the transition frequency
from the initial m ¼ 2 state f2;3 is close to fHM, while
keeping f1;2 out of resonance to prevent malicious pop-
ulation transfer to states with m < 2. Note that the
calculated crossing point is at E ¼ 0.43 Vcm−1, whereas
we find it to be between E ¼ 0.46 Vcm−1 and E ¼
0.5 Vcm−1 in the experiment. We attribute this to uncer-
tainties in the quantum defects which are measured at
n ¼ 50. In Fig. 6(b), SSFI traces after the tARP ¼ 20-(μ)s-
long ARP and a subsequent microwave transfer from j79Ci
to j77Ci for varying rf voltage amplitudes applied to the
circular electric-field electrodes is shown. For low ampli-
tudes <6 mV, the emergent avoided crossing is too weak
and (partial) diabatic crossing to states with m < n − 1
occurs. For amplitudes between 6 and 12 mVa clear, robust
transfer to the j77Ci is visible, indicating a high success
rate of the ARP. If the amplitude is chosen even higher, the
transfer efficiency suddenly drops drastically. This can be
explained by a coupling to the m ¼ 0, 1 states if the split-
off frequency jf1;2 − frf j from the m ≥ 4 states becomes
comparable to the Rabi frequency of the rf drive, leading to
population transfer to those states.
Since the circularization parameters are very sensitive to

the F state attaching to the manifold, the quantum defects
are likely not accurate enough to find the optimal param-
eters. We therefore propose that the preparation fidelity can
be strongly increased from the currently 70% by an in-
depth optimization, preferably guided by optimal control
methods. Since the parameter space including all electric,
magnetic, and radio-frequency fields is very high dimen-
sional, we leave this for future work. Perspectively, we also
expect to reach preparation times on the order of 100 ns and

(a) (b)

FIG. 6. (a) Calculated transition frequency fm;mþ1 between the
lowestm states of the n ¼ 79 hydrogenic manifold of 88Sr used in
the circularization process. The gray dashed line indicates the rf
frequency frf ¼ 70 MHz used in the experiment. (b) rf voltage
dependence of the transfer from j79F;m ¼ 2i to j77Ci. The SSFI
traces are recorded with the same linear ionization ramp as in
Fig. 1(d), after the ARP to j79Ci and a subsequent MW π pulse
resonant with the j79Ci-to-j77Ci transition.

LONG-LIVED CIRCULAR RYDBERG QUBITS OF ALKALINE- … PHYS. REV. X 14, 021024 (2024)

021024-7



fidelities of 99% by optimal-control-assisted coherent
excitation as reported in Ref. [42] for lower n.

APPENDIX B: MODELING QUBIT
DECOHERENCE

This section details the decoherence model we use to
evaluate the spin-echo measurements of Fig. 2. Our method
is similar to the approach in Refs. [16,31] with an addi-
tional term to include the longitudinal, finite-lifetime-
induced decay. Specifically, we consider the superposition
state

j77Ci þ ei½ϕ0þϕiðtÞ�j79Ci ðB1Þ

being affected by an acquired nondeterministic phase
contribution ϕiðtÞ, which fluctuates differently for each
experimental run i. For a spin-echo experiment, ϕiðtÞ arises
from time-dependent fluctuations of the qubit resonance
frequency and is connected to associated fluctuations of the
microwave detuning Δ0 þ ΔiðtÞ via

ϕiðtÞ ¼ −
Z

tR

0

Δiðt0Þdt0 þ π þ
Z

t

tR

Δiðt0Þdt0; ðB2Þ

assuming of a perfect π pulse at t ¼ tR.
We model the detuning noise ΔiðtÞ assuming Gaussian

white noise WðtÞ with amplitude D and filtered by an
exponential decay

ΔiðtÞ ¼
Z

t

−∞
Wiðt0Þ

exp
�
t0−t
τ

�
τ

dt0; ðB3Þ

where τ denotes the correlation time of the resulting noise
function. Note that the exponential decay results in a
Lorentzian noise spectrum reflecting the low-pass character
of magnetic-field noise in the experimental setup.
Under these assumptions, the phase variable ϕiðtÞ is

normally distributed over the ensemble of experimental
realizations with variance σ2ðtÞ ¼ hϕ2

i ðtÞi, and results in a
Gaussian-like decay of the echo contrast

CðtÞ ¼ e−t=τl
D
ℜ
�
e−iϕiðtÞ

�E
¼ e−

1
2
σ2ðtÞ−t=τl ; ðB4Þ

where we now also include the effective qubit lifetime τl.
Expressing σ in terms of the noise parameters D and τ, it

is straightforward to show that the maximum of the contrast
∂tCðtÞjtC ¼ 0 is found at

tC ¼ τ log
�
Dτlð2 expðtR=τÞ − 1Þ

2þDτl

�
: ðB5Þ

Notably, this expression is always smaller than 2tR, shifting
the maximum contrast to times earlier than given by
tE ¼ tR. This shift is directly evident in the experimental

data of Fig. 2. If we assume that the reversible decay is
much faster than the effective qubit lifetime T�

2 ≪ τl, and
therewith the contrast time shift is dominated by the noise,
we can approximate Eq. (B5) to

tC ≈ τ log ½2 expðtR=τÞ − 1�: ðB6Þ

Here we use that the noise parameters and the reversible
coherence time are intimately connected via τ ¼ 1=2DT�

2
2,

which is found from a similar analysis of the Ramsey signal
in absence of the echo. Finally, evaluating Eq. (B4) at the
time of maximal contrast t ¼ tC results in

CðtCÞ ¼ exp

�
−D(

tC
2
− τ tanh

�
tC
2τ

�
) −

tC
τl

�
: ðB7Þ

This allows us to fit the experimental data in Fig. 2(b) with
Eq. (1), using T�

2 as obtained from the Ramsey measure-
ment [cf. Fig. 2(a)].
With the qubit transition sensitive to magnetic and

electric fields, we identify four possible decoherence
sources to the T�

2 time: temporal fluctuations and spatial
gradients, both in the electric and magnetic field. From
independent Rydberg F-state spectroscopy at varying
electric fields in single tweezers, we find an upper bound
for electric-field fluctuations of δE ¼ 50 μV=cm. With a
sensitivity of the j79Ci → j77Ci transition of ξE ¼ 2π ×
8.8 kHz ðmV=cmÞ−1 at the electric control field of E ¼
2 V=cm used throughout this work, we find a variance of
the effective noise-induced detuning of

σE ¼ ξEδE ≈ 2π × 0.44 kHz: ðB8Þ

From similar spectroscopy measurements, we find that the
electric-field gradient across the tweezer array is
ΔE ¼ 1.1 mV=mm2. The gradient causes a discrete, con-
stant detuning on each tweezer, for which the variance can
be obtained from the sum over all tweezer sites k,

σ2ΔE
¼ 1

10

X9
k¼0

ðkdΔEξE − μEÞ2 ¼ ð2π × 2.8 kHzÞ2; ðB9Þ

with the mean μE ¼ 1
10

P
9
k¼0 ðkdΔEξEÞ and the distance

between tweezers d ¼ 10 μm. Analogously, we obtain
σΔB

¼ 2π × 20 mHz for a magnetic-field gradient of ΔB ¼
26 mG=cm we find in our experiment and a magnetic
sensitivity of the transition of ξB ¼ 2π × δmμB with δm ¼ 2
the difference in magnetic quantum number m, and
μB ¼ 1.4 MHz=G. Evidently, decoherence by magnetic-
field gradients and electric-field fluctuations are orders of
magnitude smaller than the influence of the electric-field
gradient and can be neglected. Assuming uncorrelated
noise sources, the T�

2 time is directly connected to the
summed variances via

C. HÖLZL et al. PHYS. REV. X 14, 021024 (2024)

021024-8



1

T�
2
2
¼ σ2ΔE þ σ2B: ðB10Þ

With the measured T�
2 time, we can give an approximate

value for the magnetic-field noise by solving Eq. (B10) for
σB. We find σB ¼ 2π × 2.4 kHz on the same order as the
electric-field-gradient-induced noise. This corresponds to
magnetic-field fluctuations of δB ≈ 0.9 mG comparable
with values we find from independent measurements of
the microwave transition resonance.

APPENDIX C: SEMICLASSICAL SIMULATION
OF TRAP-INDUCED DEPHASING

In this section, we provide details on the semiclassical
analysis of the dephasing induced by motion of the atom in
the tweezer potential shown in Fig. 4. The dominating
source of motion is the heating due to the short switch off of
the tweezer light for Rydberg laser excitation during which
461-nm photons are scattered. The scattering rate of the two
upper Rydberg photons (768 and 893 nm) can be neglected
since the transferred momentum and the scattering rate is
much lower in our setup. We start the simulation by
drawing a Monte Carlo sample of the ground-state atoms
position and velocity vectors in a harmonic potential from a
classical thermal distribution. Guided by earlier measure-
ments in Ref. [28], we approximate the initial temperature
to T ≈ 15 μK. We then calculate the atoms position and
velocity after evolution in free space for the 20-μs-long
Rydberg excitation, during which it absorbs between one
and two 461-nm photons at random times. After the
Rydberg excitation, quenching on the trapping potential
to a variable depthU0 leads to a kick of the atom dependent
on its position and velocity. The resulting motion leads to a
time-dependent light shift experienced by the atom, altering
the detuning during the Ramsey sequence. This effect is
included in the two-level atom Hamiltonian describing the
qubit evolution during the Ramsey sequence by a time-
dependent detuning. By solving the von Neumann equation
numerically for each Monte Carlo sample, we obtain the
temporal evolution of the qubit states. By fitting a Gaussian
envelope to the result, the T�

2 time is extracted for different
tweezer powers PCRS which is plotted in Fig. 4.

APPENDIX D: TRAPPING POTENTIAL
CALCULATION

1. Sr+ ionic-core potential

To calculate the trapping potential for the ionic core, we
use a sum-over-states method [49]

Uðr⃗Þ ¼ −
3πc2

2ω3
0

X
k

�
Γk

ω0 − ωk
þ Γk

ω0 þ ωk

�
2

Iðr⃗Þ; ðD1Þ

where ω0 is the frequency of the trapping laser, and Iðr⃗Þ ¼
Iðr; zÞ the position-dependent tweezer intensity. The index

k iterates over all optical dipole transitions with linewidth
Γk and frequency ωk connected to the electronic ground
state 5s2S1=2 listed in Ref. [50]. We assume the tweezer to
be of Gaussian form at the position of the focus with

Iðr; zÞ ¼ 2P0

πwðzÞ2 exp
�
−

2r2

wðzÞ2
�
; ðD2Þ

where wðzÞ is the spot-size parameter and P0 the
laser power.

2. Ponderomotive potential

The ponderomotive potential for the Rydberg electron
shown in Fig. 3(a) is that of a free electron [51] defined as

Upðr⃗Þ ¼
e2

2mecϵ0ω2
0

Z
Ið r0!þ r⃗ÞjΨð r0!Þj2d3r0; ðD3Þ

where Ψðr⃗Þ is the circular-state wave function. The
integration is then carried out numerically.

APPENDIX E: CRS DECAY-RATE MODEL

To measure the lifetimes of the CRS we perform SSFI
with a 24-μs-long linear electric-field ramp to approxi-
mately 28 Vcm−1 after a variable hold time t. The so-
obtained histograms [see Figs. 5(c) and 5(d)] have a
high resolution in the ionization field range between 20
and 27 Vcm−1, allowing for the distinction of CRS with
neighboring principal quantum number nwith 76≤ n≤ 80.
Noncircular states with the same n are counted to the
corresponding CRS since they have very similar ionization
threshold fields and cannot be distinguished with SSFI.
This effect will not influence the prime results of this
analysis since the decay rates are orders of magnitude
smaller than the black-body-induced transfer between
CRSs [13].
To obtain the state populations pn from the histograms,

we find that a sum of skewed Gaussians of the form

X80
n¼76

An
2

σn2π
e
−ðE−E0nÞ2

2σ2n

Z ðE−E0
nÞα=σn

−∞
e−

E02
2 dE0; ðE1Þ

with the skew factor α approximates the form of the
histograms reasonably well. Here, E0

n (σn) is the center
field (width) of each Gaussian. The individual amplitudes
An are normalized to obtain the state populations
pn ¼ An=

P
m Am. Since the different CRSs have a finite

overlap in the ion histograms, it is not feasible to fit
Eq. (E1) over all parameters. Instead, a stepwise procedure
is necessary. First, for j79Ci and j77Ci at t ¼ 0, a
histogram where the corresponding state is dominant is
measured and fitted with a single skewed Gaussian. Then,
the width σn and position E0

n are fixed, and only the

LONG-LIVED CIRCULAR RYDBERG QUBITS OF ALKALINE- … PHYS. REV. X 14, 021024 (2024)

021024-9



population pn is used as free parameter in the following fits.
With the same procedure, one state after the other is added
to the sum, fixing all σn and E0

n, leaving the populations pn
the only free parameters.
The black-body-induced transfer between adjacent CRSs

can be described by a classical rate model [18]. Each atom
in jnCi has a certain chance to be transferred either up or
down by one n to jðn� 1ÞCi with rate γn→n�1. Since the
amount of black-body photons with f ≈ 13.9 GHz is large
at room temperature, we can assume γn→m ¼ γm→n. The
change in population ṗn is then governed by the transfer
out of the state but also by the transfer from neighboring
states into it, yielding

ṗn ¼ −ðγn→nþ1 þ γn→n−1Þpn

þ γn−1→npn−1 þ γnþ1→npnþ1:

We limit the differential equation to the experimentally
accessible states 76 ≤ n ≤ 80 by setting γ81→80 ¼ γ75→76 ¼
0. This system of differential equations is then solved
analytically and fitted over the decay rates γn→m to the data
shown in Fig. 5 to obtain lifetimes τn ¼ 1=ðγn→nþ1þ
γn→n−1Þ.

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