# Cavity QED in a High NA Resonator

Danial Shadmany∗ Department of Physics, Stanford University, Stanford, CA
Aishwarya Kumar∗ Department of Physics, Stanford University, Stanford, CA
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
Anna Soper  Department of Applied Physics, Stanford University, Stanford, CA
Lukas Palm  Department of Physics, The University of Chicago and the James
Franck Institute, , Chicago, IL  Chuan Yin  Department of Physics, The
University of Chicago and the James Franck Institute, , Chicago, IL  Henry
Ando  Department of Physics, The University of Chicago and the James Franck
Institute, , Chicago, IL  Bowen Li  Department of Physics, Stanford
University, Stanford, CA  Lavanya Taneja  Department of Physics, The
University of Chicago and the James Franck Institute, , Chicago, IL
Department of Physics, Stanford University, Stanford, CA  Matt Jaffe
Department of Physics, The University of Chicago and the James Franck
Institute, , Chicago, IL  Department of Physics, Montana State University,
Bozeman, MT  David I Schuster  Department of Applied Physics, Stanford
University, Stanford, CA  Jon Simon  Department of Physics, Stanford
University, Stanford, CA  Department of Applied Physics, Stanford University,
Stanford, CA

(August 28, 2024)

###### Abstract

From fundamental studies of light-matter interaction to applications in
quantum networking & sensing, cavity quantum electrodynamics (QED) provides a
platform-crossing toolbox to control interactions between atoms and photons.
The coherence of such interactions is determined by the product of the single-
pass atomic absorption and the number of photon round-trips. Reducing the
cavity loss has enabled resonators supporting nearly 1-million optical
roundtrips at the expense of severely limited optical material choices and
increased alignment sensitivity. The single-pass absorption probability can be
increased through the use of near-concentric, fiber or nanophotonic cavities,
which reduce the mode waists at the expense of constrained optical access and
exposure to surface fields. Here we present a new high numerical-aperture,
lens-based resonator that pushes the single-atom-single-photon absorption
probability per round trip close to its fundamental limit by reducing the mode
size at the atom below a micron while keeping the atom mm-to-cm away from all
optics. This resonator provides strong light-matter coupling in a cavity where
the light circulates only $\sim 10$ times. We load a single 87Rb atom into
such a cavity, observe strong coupling, demonstrate cavity-enhanced atom
detection with imaging fidelity of $99.55(6)\%$ and survival probability of
$99.89(4)\%$ in $130$ $\mu$s, and leverage this new platform for a time-
resolved exploration of cavity cooling. The resonator’s loss-resilience paves
the way to coupling of atoms to nonlinear & adaptive optical elements and
provides a minimally invasive route to readout of defect centers. Introduction
of intra-cavity imaging systems will enable the creation of cavity arrays
compatible with Rydberg atom array computing technologies, vastly expanding
the applicability of the cavity QED toolbox.

##  I Introduction

Cavity quantum electrodynamics, the study of light-matter interaction through
coupling of emitters to the field of an optical resonator, enables coherent
information exchange between material and photonic degrees of freedom, with
wide-ranging applications from qubit state detection teper2006resonator;
bochmann2010lossless; dhordjevic2021entanglement; deist2022mid, to sensing
leroux2010implementation; cox2016deterministic; hosten2016measurement;
pedrozo2020entanglement, and networking ritter2012elementary;
reiserer2015cavity; kimble2008quantum. Such tools apply not only to coupling
of light to laser-cooled atoms and ions birnbaum2005photon; steiner2013single;
sterk2012photon; keller2004continuous, but also transmon qubits, rare earth
ionic dopants raha2020optical; kindem2020control, color centers
calusine2014silicon; evans2018photon; zhang2018strongly, quantum dots
englund2007controlling; press2007photon, and other optically active emitters.

The fundamental figure of merit that controls the coherence of light-matter
interactions is the cooperativity, given by $C=\frac{4g^{2}}{\kappa\Gamma}$,
where $g$ is the coherent information exchange rate, $\kappa$ is the decay
rate of the cavity field and $\Gamma$ is the decay rate of the material
excitation tanji2011interaction. For applications in quantum information
science, the number of coherent information exchanges is approximately
$\sqrt{C}$ (SI B), and the infidelity of a cavity mediated gate is
$\epsilon\approx\frac{2\pi}{\sqrt{C}}$(sorensen2003measurement;
benito2019optimized and SI B). Cavities can also be harnessed for Purcell
enhanced atomic state detection, where the ratio of atomic emission into the
cavity vs free space is $C$ tanji2011interaction; equivalently, the Purcell
enhanced collection solid angle is $4\pi\times C$.

For a closed optical transition such as those available in alkali atoms, the
cooperativity of a macroscopic resonator may be expressed in terms of the
resonator geometry
$C=\frac{12}{\pi^{2}}\frac{F}{2\pi}\frac{\lambda^{2}}{w^{2}}$
(tanji2011interaction and SI P), where $F$ is the resonator finesse defined so
that $F/(2\pi)$ is the mean number of times the light passes the atom within
the cavity, $w$ is the optical mode size at the atom, and $\lambda$ is the
wavelength of the optical transition. At fixed $\lambda$, this expression
suggests two routes to strong coupling ($C>1$)– large finesse or small mode
waist.

Finesses above $10^{5}$ have been achieved using ion-beam sputtered dielectric
coatings on super polished hood2001characterization or laser ablated
hunger2010fiber substrates, and recently finesses above $10^{6}$ have been
achieved using reactive ion etching jin2022micro to reduce surface roughness.
Further gains would necessitate breakthroughs in atom-scale surface polishing
and part-per-million-level coating absorption.

The decades-long effort to reduce the mode size $w$ has spanned numerous
approaches: in two-mirror cavities, the near-confocal geometry
siegman1986lasers with length $L$ equal to the mirror radius of curvature $R$
is a compromise between resonator alignment sensitivity and optical access
that results in a mode-waist of $w=\sqrt{\frac{R\lambda}{2\pi}}$. Efforts to
reduce the mode waist in two-mirror cavities beyond this limit have focused on
either (a) small mirror separation $L\ll R$ (near-planar resonators),
providing a substantial reduction in mode waist birnbaum2005photon;
hunger2010fiber at the expense of reduced optical access and sensitivity to
E-fields from proximal surfaces; or (b) near-concentric resonator geometry
$L\approx 2R$, providing a few-fold waist reduction compared to the confocal
geometry at the expense of increased alignment sensitivity haase2006detecting;
periwal2021programmable; deist2022mid.

Figure 1:  Resonator Design. The heart of the apparatus, shown in (a), is a
cavity consisting of a spherical mirror (radius $R$) that focuses light down
to a sub-micron spot, a high-NA aspheric lens (focal length $f$) that re-
collimates it, and finally a long propagation (length $L_{long}$). This long
propagation distance necessitates that the vacuum chamber window (gray) reside
inside of the resonator. An optical conveyor belt vertically transports 87Rb
atoms to the resonator from a magneto-optical trap. (inset) A cavity
transmission spectrum reveals a (fitted) finesse $F=40(2)$; for a 480 MHz free
spectral range (FSR), the linewidth is thus $\kappa=2\pi\times 12.4(7)$ MHz.
(b) The paraxially computed mode waist (w0) is plotted vs the distance between
the aspheric lens and spherical mirror, quantified as the deviation from
overlapping lens and mirror foci. Introduction of the asphere within the
cavity allows for the exploration of resonators that are asymmetric,
consisting of two propagation arms of unequal lengths. For $L_{long}\gg
L_{short}$, the width of the stability region ($f^{2}/L_{long}$) compresses,
leading in the extreme case ($L_{long}\approx 30$ cm) to a sub-micron waist
that persists even in the middle of the stable range, rather than exclusively
near the (highly-sensitive) edges as is common in macroscopic two mirror
cavities. (inset) A direct measurement of the mode waist (via scanning a nano-
pore across the mode, see SI L) bounds the mode size at a maximum $1/e^{2}$
radius of $1000$ nm.  Figure 2: Single Atom Loading (a) Early time dynamics of
the Rb cloud emission into the cavity. First (dark green), the signal
increases as atoms are cooled into the cavity-enhanced 785 nm dipole trap.
After reaching a maximum (yellow) the atoms at the center begin to parity
project, leading to loss. (inset) Finally (orange) the signal settles to
quantized levels, indicating the presence of 0 or 1 atoms. (b) Full schematic
of single-atom loading and localization schemes. First, the atoms are released
from the transport lattice and cooled and compressed into the intracavity-
dipole trap (red) by the 3D molasses beams (dark blue), where the intracavity-
dipole trap is created by the addition of off-resonant sidebands exactly 10
FSRs away (see SI D). Near the small waist, the atoms in the dipole trap
undergo light-assisted collisions, parity projecting until only 1 atom is
left. In light blue, we show the histogram of scattering statistics with a 1
ms bins for a single atom in a dipole trap. This corresponds to a scattering
rate of $\approx{20}$ kHz. The single atom in the dipole trap samples the
nodes and anti-nodes readout cavity lattice (light green) equally, cutting the
effective cooperativity by a factor of two. To improve on this, we alter the
profile of the trapping light from a dipole trap to a lattice by removing the
far sidebands. We take care to choose a 785 nm FSR which is phase-matched with
the 780 nm readout lattice. Cooling is performed during this process, leading
to a single atom which is localized to a single cavity anti-node. We plot a
histogram of statistics for the localized single atom with a 1 ms bin size
(pink) and find a scattering rate of $\approx{50}$ kHz.

Achieving yet-smaller mode waists requires exploring more sophisticated
resonator geometries. For example, by employing an additional long propagation
arm and/or a pair of _convex_ mirrors, the bowtie resonator allows for mode-
waists down to a limit imposed by astigmatism induced by off-axis incidence on
the focusing mirrors ningyuan2016observation; clark2020observation. Such
resonators have demonstrated waists down to $\sim\\!7$ $\mu$m chen2022high,
the limit thus far achieved for high-finesse macroscopic Fabry-Pérot cavities.
Wavelength-scale resolutions have been observed in a multi-pass imaging system
for biological samples juffmann2016multi, but never in a macroscopic device
compatible with atomic cavity QED.

In this work we present, for the first time, a macroscopic resonator with a
$\lambda$-scale mode waist. This enables us to enter the strong coupling
regime of cavity QED in a resonator with finesse below $50$. In Section II we
introduce this first-of-its-kind resonator geometry and describe its
construction and stability. In Section III we describe the process of
loading/detecting a single atom and use it to characterize the resonator with
a measurement of the atom-cavity coupling. In Section IV we demonstrate fast
atom detection and in Section V we conclude by describing opportunities to
leverage this new approach for emerging quantum science and technology.

##  II Small Waist Lens Cavity

The principal innovation in our resonator design is the integration of a high
numerical aperture (NA) aspheric lens (focal length $f$) within a two mirror
Fabry-Perot cavity. This lens divides the resonator into a short arm (length
$L_{short}$) between the lens and a spherical mirror (radius of curvature
$R$), and a long arm (length $L_{long}$) between the lens and a planar mirror
(see Fig. 1a). When $L_{long},R\gg f$, a small waist appears in the short arm
near the center of curvature of the spherical mirror and focus of the aspheric
lens. The size of this waist,
$w_{0}\approx\sqrt{\frac{f}{L_{long}}}\times\sqrt{\frac{f\lambda}{2\pi}}$, is
very similar to that of a confocal cavity, but with additional demagnification
induced by the long arm.

There are several important features of note: (i) if the long arm is _long_
enough, sub-micron waists are possible at the center of the stability diagram,
limited by clipping and aberration due to finite lens NA (see SI M. (ii) using
an asphere allows us to minimize aberrations while maintaining a high
numerical aperture (NA), unlike a bowtie resonator with spherical mirrors
where astigmatism limits the achievable waist ningyuan2016observation (iii)
with a round trip Gouy phase of $\pi/2$, the transverse mode structure of the
cavity is neither that of a concentric or a confocal resonator, suppressing
mode mixing due to lower order aberrations jaffe2021aberrated. For our
parameters ($L_{long}=30$ cm, $f=1.45$ mm), we expect a waist of $930$ nm (see
Fig. 1b). See SI H

In our implementation of the resonator geometry described above, we mount the
asphere and the curved mirror in an ultra-high vacuum load-lock system (see SI
Fig. S8 and yin2023cavity). The mount is a custom designed 3-axis piezo-driven
and mechanically multiplied flexure-stage, providing precise control of the
relative position of the asphere and mirror, crucial to aligning the cavity in
the presence of drifts (see SI N). We place the in-coupling (R=98$\%$) flat
end mirror of the cavity outside vacuum and use it primarily to in-couple our
probe and trapping light. We insert a pellicle right after the flat mirror to
pick-off a controllable (angle-dependent) fraction of the light circulating in
the cavity to either measure cavity transmission or atomic fluorescence.
Furthermore, we employ an electro-optic modulator between the pellicle and
vacuum window for high bandwidth cavity locking. We directly verify a micron
scale waist in an out-of-vacuum test setup by using a gold coated film
containing a $200$ nm diameter aperture (see SI L).

##  III Atom-Cavity Coupling

To conclusively demonstrate high cooperativity and the utility of the platform
in atom-based quantum information protocols, we trap a single atom at the
waist of the resonator and use it to probe the cavity. Our experiments begin
with a cloud of laser cooled 87Rb atoms from a magneto optical trap (MOT)
which is transported to the small cavity waist location in a one-dimensional
optical conveyor belt (see Fig. 1a). Additional polarization gradient cooling
(PGC) at the cavity waist location loads the atoms into an intra-cavity dipole
trap at $785$ nm with a peak depth of $U_{0}/k_{B}=2.0(5)$ mK (measured as
half the trap-induced shift of the atom-cavity resonance condition).

Figure 3:  Trapped Atom Characterization. Once an atom is detected, we probe
its cavity coupling: (a) Cavity transmission vs probe laser frequency,
with/without atom in orange/purple; the mode splitting apparent in the
presence of the atom indicates strong light-matter coupling with $C=1.6(2)$.
(b) Two-time correlator $g^{(2)}(\tau)$ of cavity-fluorescence from a dipole-
trapped atom vs time difference $\tau$. The fast ($\sim 3\leavevmode\nobreak\
\mu$s) oscillations (inset) come from radial oscillations away from the cavity
axis where coupling is strongest, rapidly damped by PGC. The slow decay over
$\sim 50\leavevmode\nobreak\ \mu$s comes from over-damping the weak axial
dipole trap. A fit (solid, see SI J), indicates axial & radial temperatures
differ by a factor $\sim 100$ ($T_{z}\approx 5\leavevmode\nobreak\ \mu$K,
$T_{x,y}\approx 600\leavevmode\nobreak\ \mu$K), while axial & radial PGC-
damping rates differ by a factor $\sim 10$ ($\gamma_{x,y}=500$ ms-1,
$\gamma_{z}=5000$ ms-1). (c) $g^{(2)}(\tau)$ of lattice trapped atom; absence
of slow decay indicates stronger axial trapping of the lattice. The fit
(solid, see SI J), reveals equal axial & radial temperatures
$=200\leavevmode\nobreak\ \mu$K, with axial & radial damping rates differing
by a factor of four ($\gamma_{x,y}=112$ ms-1, $\gamma_{z}=400$ ms-1). b & c,
insets depict trapping potential in red, light-matter coupling in white, and
extent of atomic motion in purple.

It is essential that atoms within the cavity be able to collide to ensure that
the PGC process drives parity projection ueberholz2002cold. As such, the atoms
are first loaded into a dipole trap rather than a lattice. We wash out the
cavity standing wave by applying sidebands to the $785$ nm cavity lattice
laser, to excite cavity modes 10 free spectral ranges away, creating a dipole
trap at the small waist location (see SI D). Once a single atom is loaded into
the dipole trap, we remove the sidebands while cooling over $2$ ms, reloading
the atom into a single cavity lattice well. The trapping wavelength must thus
be chosen to align the $780$ nm and $785$ nm cavity standing-waves at the
location of the small cavity waist (see SI E). To probe the system we monitor
the PGC-induced fluorescence scattered into the cavity, picked off by the
pellicle and directed into a pair of single photon counting modules (SPCMs).

We first study the atom-cavity interaction with outcoupling $T=4\%$ well below
the internal cavity round trip loss $L_{rt}=11.7\%$ to achieve a near-maximal
finesse of $\mathcal{F}=40.0(1)$, and hence a predicted single-atom
cooperativity $C_{max}=6$. Fig. 2a shows averaged fluorescence as an atomic
ensemble is cooled into the intra-cavity dipole trap. An initial compression
leads to an increase in the fluorescence, followed by a sharp decrease during
parity-projection, after which the signal settles to quantized levels (Fig.
2a). This quantized signal arises from discrete atomic occupancy of the cavity
trap, dominated by either 0 or 1 atom loaded schlosser2001sub. Fig. 2b shows
the histogram of collected fluorescence photon numbers from dipole trapped
atoms, with a clear separation between the background and a single atom
loaded. After transferring to an intracavity lattice with depth
$800(80)\leavevmode\nobreak\ \mu$K, the signal is enhanced 2-fold because the
atom is better localized to an antinode of the $780$ nm cavity field (see SI
E).

Armed with the ability to condition measurements on the presence of an atom,
we extract the single-atom cavity coupling strength by measuring the vacuum
Rabi splitting. In Fig. 3a, we plot the cavity transmission as we scan a $780$
nm probe laser across the cavity resonance, both with (purple) and without
(orange) an atom. A single-atom vacuum Rabi splitting is clearly visible with
a fitted coupling strength $g=2\pi\times 5.6(3)$ MHz. Combined with the atomic
linewidth $\Gamma=2\pi\times 6.065$ MHz and our measured $\kappa=2\pi\times
13.3(1)$ MHz, this yields a cooperativity, $C=1.6$(2), placing us in the
regime where emission into the cavity is more likely than into free space.
This cooperativity is about a factor of $\sim 4$ lower than the maximum
achievable due to a combination of cavity birefringence ($\approx 2\times$
reduction) and atomic motion ($\approx 2\times$ reduction) (see SI K). We also
leverage our ability to perform atom-conditioned measurements to extract the
second order correlation function, $g^{(2)}(\tau)$ of the atomic fluorescence,
taken both in the dipole trap (Fig. 3b) and lattice (Fig. 3c) (see Supplement
G for parameters). Both plots exhibit $\mu$s-scale oscillations indicative of
radial trapping that is damped out over a few periods by the optical-molasses
(see SI J for derivation of the solid fits); this oscillation reflects the
fact that atoms are more likely to scatter at times when they are near cavity
axis (large $g_{2}(\tau=0)$), and less likely to scatter a half trap period
later when the atom has oscillated away from the cavity axis (small
$g_{2}(\tau=T_{trap}/4)$). Data from the dipole-trapped atom exhibits an
additional slow decay indicative of over-damping of the weakly-trapped axial
motion. By contrast, the lattice-trapped atom is strongly confined in the
axial direction, and so exhibits an additional, even faster (axial)
oscillation in the $g_{2}$ instead place of over-damped decay apparent for the
dipole-trapped atom.

##  IV Optimized Atom Detection

The low outcoupling probability employed for the aforementioned experiments is
not optimal for atomic state detection in the presence of fixed internal
cavity losses. The optimum outcoupling is a tradeoff: increased outcoupling
improves the fraction of cavity photons leak out of the cavity before they are
lost to scattering/absorption of cavity optics, while reduced outcoupling
improves the cooperativity and thus the probability that the atom scatters
light into the cavity in the first place. As we show in SI A, the optimal
operating point occurs when the outcoupling probability is
$L_{int}\sqrt{1+C_{max}}$, where $C_{max}$ is the cooperativity with no
outcoupling and $L_{int}$ is the internal cavity loss per round trip. To
further demonstrate the flexibility of our platform, we enter this optimal
regime by tuning the angle of the intra-cavity pellicle to outcouple more
light (optimized at 20 $\%$, with an estimated new cooperativity of
$\approx{0.8}$).

Figure 4: Characterizing Atom Detection Fidelity. (a) Histogram of cavity
fluorescence photons collected in $1\leavevmode\nobreak\ $ms bins from a
single lattice-trapped atom at larger $T=20\%$ cavity outcoupling. The
measured scattering rate is $\sim 75$ kHz, $1.7\times$ higher than that
measured in the same configuration at $T=4\%$ outcoupling. See Supplement A
for derivation of relationship between total cavity collection rate and
outcoupling. (inset) Correlations between subsequent measurements of a single
atom: upper-right quartile reflects persistence of an atom through adjacent
measurements; lower-right quartile reflects an atom lost between the
measurements; lower-left quantile reflects measurements both with no atom;
upper-left quantile reflects appearance of an atom between measurements,
indicating imperfect dispersal of the transported cloud. The elliptical shape
of the data in the upper-right quadrant indicates correlations in scattering
rate between subsequent measurements, likely due to site to site variation in
cavity coupling strength. (b) The model-independent fidelity estimates are
plotted vs measurement time, showing a fidelity of $99.55(6)\%$ reached at a
measurement time of $130\leavevmode\nobreak\ \mu$s. The (inset) depicts the
corresponding estimated loss probability vs the imaging time and with a model
(solid curve) for a survival rate of 99.89(4)$\%$ and a$112(8)$ ms lifetime
(see SI F).

Figure 4a shows atom detection for optimized outcoupling. Compared to the low-
outcoupling configuration, we observe an increase in the detected photon rate
by a factor of $\sim$1.7 (Fig. 4a), consistent with expectations (see SI A). A
systematic, model-independent analysis of atom detection (see Ref.
norcia2018microscopic and SI O) reveals that, within 130 $\mu$s, we are able
to detect an atom with fidelity of $99.55(6)\%$ (see Fig. 4b), while
maintaining an atom survival probability of 99.89(4) $\%$ (see Fig. 4b inset),
consistent with the atom lifetime in the presence of the molasses light. These
numbers are limited by the internal losses of our resonator, without which we
could increase the resonator finesse, and thereby the Purcell enhancement
factor of the resonator, while maintaining efficient resonator outcoupling.
Reducing these losses, through a combination of better anti-reflection
coatings and lower surface roughness, will immediately reduce the readout
time.

##  V Outlook

In this paper we have introduced an approach to cavity QED that leverages high
NA, low-finesse cavities to enter the strong coupling regime. We load a single
atom into such a cavity, with a finesse $F=40$, harnessing cavity enhancement
to create first a dipole trap and then a lattice, which efficiently confines
the atom. We validate that we are in the strong coupling regime via an atom-
conditioned vacuum Rabi splitting measurement, proving the presence of a
single atom via a quantized signal. Temporal correlations of the light
scattered by the atom into the cavity enable precise measurements of both atom
temperature and PGC damping coefficients. When we optimize the cavity
outcoupling to maximize total atomic light scattering, we are able to detect a
single atom with a fidelity of 99.55(6)% in a time of $130\leavevmode\nobreak\
\mu$s, with a survival rate of $99.9\%$.

The parameters of our platform are far from optimized: a non-birefringent
cavity will provide $2\times$ the collection efficiency, while lower atom
temperature will provide an additional factor of $1.8$; free-space photon
counters provide an extra factor $2$ vs their fiber-coupled counterparts, for
a projected detection time of $20\leavevmode\nobreak\ \mu$s. Reduced
intracavity loss will yield further improvements in collection efficiency and
cooperativity, enabling non-destructive atom detection in less than $\sim
10\leavevmode\nobreak\ \mu$s. See SI Q. The minimal finesse requirements mean
that it will be possible to strongly couple the atom to more sophisticated
intracavity optics: nonlinear crystals will enable in-situ wavelength
conversion for integration with telecom infrastructure chen2005intracavity;
electro-optics will provide rapid tunability hudson2005mode; adaptive optics
will enable yet-higher NA operation benea2021electro; and nanophotonics in the
end-mirror will enable direct integration with waveguide devices.

Equally exciting is the possibility of integrating these cavities with solid-
state emitters that otherwise require nanoscale cavities to achieve large
cooperativity due to inherent material-induced scattering/absorption: coupling
to rare-earth ions and silicon vacancies, with their nearly-closed
transitions, would greatly increase the flexibility of those platforms.
Finally, in combination with microlens arrays to stabilize an array of waists,
it should be possible to extend the small-waist resonator technique
demonstrated in this paper to _arrays of resonators spaced by a few microns_ ,
for immediate integration with Rydberg atom arrays endres2016atom;
barredo2016atom. Crucially, because atoms can be mm-to-cm distances from the
nearest optic, the sensitivity of Rydberg atoms to surface potentials is
mitigated in these cavities compared to their nanophotonic counterparts. In
short, this work heralds a new era of optical cavity QED where strong light-
matter coupling is widely available, rapidly extensible, and compatible with a
much broader array of experimental platforms and technologies.

##  VI Acknowledgments

This work was supported by AFOSR grant FA9550-22-1-0279, ARO grant
W911NF-23-1-0053, AFOSR MURI Grant FA9550-19-1-0399, and AFOSR DURIP
FA9550-19-1-0140. D.S. and H.A. acknowledge support from the NSF GRFP. A.S.
acknowledges support from the Hertz Foundation and the DoD NDSEG Fellowship.
We acknowledge Mark Stone for assistance in construction of the load-lock
vacuum system, and Adam Shaw for discussions of model-free estimates of atom
detection fidelity.

##  VII Author Contributions

The experiments were designed by A.K., D.S., A.S., M.J., L.P., D.I.S., and
J.S. The apparatus was built by C.Y., H.A., M.J., D.S., A.S., A.K., L.T., and
J.S. The collection of data was handled by A.K., D.S. and A.S. L.P. and B.L.
performed the waist measurements in a test setup. All authors analyzed the
data and contributed to the manuscript.

∗ These authors contributed equally.

##  VIII Competing Interests

The authors declare no competing financial or non-financial interests.

## References

###  VIII.1 Data Availability

The experimental data presented in this manuscript are available from the
corresponding author upon request, due to the proprietary file formats
employed in the data collection process.

###  VIII.2 Code Availability

The source code for simulations throughout are available from the
corresponding author upon request.

###  VIII.3 Additional Information

Correspondence and requests for materials should be addressed to J.S.
(jonsimon@stanford.edu). Supplementary information is available for this
paper.

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## Supplementary Information

##  Supplement A Derivation of Out-scattering Efficiency

For a cavity with cooperativity $C_{max}$ in the absence of any outcoupling
(just intracavity losses), the question we want to address here is how much
outcoupling to _add_ to maximize the fraction of photons scattered by the atom
that leak out of the cavity: if we outcouple too little, more of the photons
are lost to intracavity losses; if we outcouple too much we reduce the
cooperativity and more photons are lost to free-space scattering.

Suppose we outcouple a factor $\beta$ more photons than are lost intracavity.
This provides a cavity outcoupling fraction $\frac{\beta}{1+\beta}$ and a
cavity cooperativity $C=\frac{C_{max}}{1+\beta}$ The fraction of photons
scattered by the atom that leak out of the cavity for detection is then
$\chi=\frac{\beta}{1+\beta}\times\frac{C_{max}/(1+\beta)}{1+C_{max}/(1+\beta)}$.
Maximizing this over $\beta$ yields $\beta=\sqrt{1+C_{max}}$, and
$\chi=1-2\frac{\sqrt{1+C_{max}}-1}{C_{max}}$.

In the limit of large maximum cooperativity $C_{max}\gg 1$, we then find that
$C\approx\sqrt{C_{max}}$ and $\chi\approx 1-\frac{2}{\sqrt{C_{max}}}$; on the
other hand, in the limit of small cooperativity $C_{max}\ll 1$, we find that
$C\approx\frac{C_{max}}{2}$ and $\chi\approx\frac{C_{max}}{4}$.

That is, in the limit of large $C_{max}$ we should outcouple enough to reduce
the cooperativity to the square root of its maximum value, resulting in
outcoupling which is $\approx\sqrt{C_{max}}$ times larger than the intracavity
losses. By contrast small $C_{max}$ we should set our outcoupling equal to our
intracavity losses, halving the cooperativity.

These results are interesting because common knowledge (see SI B) was that
coherent transfer of photons between atoms has an infidelity (for $C\gg 1$) of
approximately $\frac{2\pi}{\sqrt{C}}$, whereas scattering light into a cavity
(thought of as an incoherent process) has an infidelity of approximately
$\frac{1}{C}$; rather than attributing this difference to coherent vs
incoherent scattering, it is now clear that the difference arises from
_neglecting cavity outcoupling efficiency_ in the latter case. This is
valuable to know, because otherwise it would have appeared more efficient to
pitch and catch quantum information between atoms in separate cavities rather
than coherently transferring between atoms in the same cavity - it is now
apparent that these two processes have the same (unfavorable) scaling with the
maximum cooperativity of the resonator.

Supplemental Information Fig. S1: Cavity Collection Efficiency. (a) Photon
collection efficiency as function of outcoupling fraction for measured
cooperativity (b) histogram showing improvement between increase in
outcoupling from 4$\%$ (pink) to 20$\%$ (red)

##  Supplement B Cavity-Mediated Coupling Between Ensembles

This section follows the approach outlined in Jon Simon’s thesis
simon2010cavity. Suppose we have two atoms ($a$ & $b$) in the same cavity and
want to coherently couple a hyperfine excitation (between states $g$ & $f$)
from one to the other via a cavity-stimulated Raman transition. The
Hamiltonian is:

$H=g\left[c^{\dagger}(\sigma^{fe}_{a}+\sigma^{fe}_{b})+c(\sigma^{ef}_{a}+\sigma^{ef}_{b})\right]+\Omega(\sigma^{eg}_{a}+\sigma^{eg}_{b}+\sigma^{ge}_{a}+\sigma^{ge}_{b})+(\delta_{c}+i\frac{\kappa}{2})c^{\dagger}c+(\Delta+i\frac{\Gamma}{2})(\sigma^{ee}_{a}+\sigma^{ee}_{b})$
(B.1)

Here the cavity is detuned from Raman resonance by an amount $\delta_{c}$, the
Raman transition is detuned from the excited state by an amount $\Delta$, $g$
is the single photon/single atom light-matter coupling strength on the
$f\leftrightarrow e$ transition, $\Omega$ is the Rabi frequency for the laser
driving the $g\leftrightarrow e$ transition, $\Gamma$ & $\kappa$ are the
excited atom and cavity decay rates, $c\dagger$/$c$ are the field creation and
annihilation operators, and $\sigma^{pq}_{l}$ is the transition operator from
state $q$ to state $p$ in atom $l$.

There are, of course many pulse protocols to convert
$|fg;0\rangle\leftrightarrow|gf;0\rangle$. In the presence of dominant cavity
and excited atom loss, they all have the same optimal performance up to
factors of order unity. For simplicity we will skip counter-intuitive driving
and all other adiabatic approaches (though they are covered in Simon’s
thesis), and consider only a 4-photon Rabi process.

In that case we can adiabatically eliminate the $e$-states, resulting in an
effective 2-photon Hamiltonian (neglecting second order shifts that can be
compensated with cavity & laser detunings):

$H^{eff}_{2}=\frac{g\Omega}{\Delta}\left[c^{\dagger}(\sigma^{fg}_{a}+\sigma^{fg}_{b})+c(\sigma^{gf}_{a}+\sigma^{gf}_{b})\right]+i\frac{\Omega^{2}}{\Delta^{2}}\Gamma(\sigma^{ff}_{a}+\sigma^{ff}_{b})+(\delta_{c}+i\kappa/2+i\frac{g^{2}}{\Delta^{2}}\Gamma)c^{\dagger}c$
(B.2)

Further eliminating the cavity mode results in an effective 4-photon
Hamiltonian (neglecting fourth order shifts that can be tuned out):

$H^{eff}_{4}=\frac{g^{2}\Omega^{2}}{\Delta^{2}\delta_{c}}\left[\sigma^{fg}_{a}\sigma^{gf}_{b}+\sigma^{gf}_{a}\sigma^{fg}_{b}\right]+i\left[\frac{\Omega^{2}}{\Delta^{2}}\Gamma+\frac{g^{2}\Omega^{2}}{\Delta^{2}\delta_{c}^{2}}(\kappa/2+\frac{g^{2}}{\Delta^{2}}\Gamma)\right](\sigma^{ff}_{a}+\sigma^{ff}_{b})$
(B.3)

A full flip-flop of an excitation between atoms $a$ and $b$ thus requires a
time $\tau=\pi/\left(\frac{g^{2}\Omega^{2}}{\Delta^{2}\delta_{c}}\right)$,
resulting in a total decoherence induced loss
$\epsilon=\tau\times\left[\frac{\Omega^{2}}{\Delta^{2}}\Gamma+\frac{g^{2}\Omega^{2}}{\Delta^{2}\delta_{c}^{2}}(\kappa/2+\frac{g^{2}}{\Delta^{2}}\Gamma)\right]$.
In the limit $\Delta\rightarrow\infty$, this becomes
$\epsilon=\pi\frac{2\Gamma\delta_{c}^{2}+g^{2}\kappa}{2g^{2}\delta_{c}}$. This
loss is minimized for
$\delta_{c}=\frac{g\sqrt{\kappa}}{\sqrt{2\Gamma}}=\kappa\sqrt{\frac{C}{8}}$,
with a value $\epsilon=\frac{2\pi}{\sqrt{2C}}$.

Because any two-atom gate can be implemented by pellizzari1995decoherence
mapping two physically separate qubits into one four-level atom via the cavity
field, performing internal four-level operations on the atom, and then mapping
back to separate physical qubits via the cavity field, $\epsilon$ sets a lower
limit on the infidelity of such a gate.

Similarly, the number of information exchanges between atoms, or between atom
and cavity field, is $1/\epsilon\propto\sqrt{C}$.

Finally, a bit of interpretation: if we go to too large a detuning from the
cavity, the 4-photon Rabi oscillation slows down too much and we are dominated
by excited-state loss; if we go to too small a detuning from the cavity, the
4-photon Rabi oscillation speeds up, but not as much as the cavity leakage,
and we are dominated by that. The sweet spot is in the middle. The excited
state detuning, as long as it is large enough, does not contribute at all.
Indeed one can show that even on resonance with excited state a counter-
intuitive pulse sequence gives the same performance up to a numerical factor
of order unity (see ref. simon2010cavity).

##  Supplement C Cavity Optics Setup

See Figure S2 for a full schematic, including intra-cavity optics, photo-
detectors, and in-coupling paths. The setup includes an EOM and a pellicle
beam splitter inserted into the long propagation arm of the cavity. These are
used for high-bandwidth cavity stabilization and tunable outcoupling
respectively. During operation, we tune the angle of the pellicle to switch
between 4$\%$ and 20$\%$ pick-off, depending on if we are optimizing for atom-
light coupling (as for a VRS) or overall photon collection efficiency (as for
the readout histograms). The pellicle reflects light out on both sides,
requiring the use of two SPCMs to maximize collection. Using the same
pellicle, we also pick off a fraction the 785 nm trap light for a low-
bandwidth, slow locking path, from which we generate an error signal and
feedback to the piezo and EOM.

Supplemental Information Fig. S2: Detailed Cavity Setup. The resonator with
all in-cavity optics is highlighted in gray. In addition to the aspheric lens,
spherical mirror, and planar mirror, we also depict the pellicle used for
outcoupling, and the intra-cavity electro-optic modulator that we use to lock
the resonator with $\sim 100$ kHz of bandwidth. Light outcoupled from both
sides of the pellicle is coupled to single photon counting modules (SPCMs); a
dichroic on one side of the outcoupler picks off the 785 nm locking light.

##  Supplement D Intracavity Dipole Trap from Multichromatic Driving

We form our intra-cavity trap by driving the cavity with a far detuned $785$
nm laser. Driving a single longitudinal mode of the cavity results in a one-
dimensional optical lattice with a small waist, which after parity projection,
typically leads to single atoms trapped at multiple sites of the lattice.
Deterministic trapping of only one atom coupled to the cavity mode requires an
optical tweezer potential – a dipole trap with a small waist. To create this
dipole trap, we drive multiple longitudinal modes of the cavity by phase
modulation of the trapping light. Since different longitudinal modes are phase
shifted (in space) with respect to each other, choosing appropriate phase
modulation frequency and strength can result in significant suppression of the
sinusoidal intensity variation across a finite region cox2016spatially. For
our cavity, we require this region to be centred at $d=1$ cm from the curved
mirror – the location of the small waist. Phase modulation with an EOM at $n$
times the free spectral range (FSR) results in a trapping potential given by
(to the first order):

$U=U_{0}\left(J_{-1}(\beta)^{2}\sin^{2}(kz-\frac{nd}{L}\pi)+J_{0}(\beta)^{2}\sin^{2}(kz)+J_{1}(\beta)^{2}\sin^{2}(kz+\frac{nd}{L}\pi)\right)$
(D.1)

where $U_{0}$ is the depth of the initial optical lattice, $k$ is the wave-
vector of the carrier, $\beta$ is the modulation depth, $L\approx 30$ cm is
the length of our cavity, and $J_{\alpha}$ is the Bessel function of first
kind. We note the following identity :

$\sin^{2}(kz-\frac{10}{30}\pi)+\sin^{2}(kz)+\sin^{2}(kz+\frac{10}{30}\pi)=\frac{3}{2}$
(D.2)

For our cavity $L/d\approx 30$, thus driving the $n=10$ sidebands at equal
intensity as the carrier (i.e.
$J_{-1}(\beta)^{2}=J_{0}(\beta)^{2}=J_{1}(\beta)^{2}$), results in a
cancellation of the sinusoidal potential at the cavity waist resulting in a
dipole trap with depth $U_{0}/2$ (see Fig. S3). While this seems to imply that
such a cancellation is only achievable with a fine tuning of the length and
the waist location, numerical calculations suggest that it is possible for any
location by changing the modulation depth. Note that the effect of higher
order sidebands can be similarly compensated for by slightly changing the
modulation depth.

Supplemental Information Fig. S3:  left the intensity profile of a single
longitudinal mode of the cavity, which takes the form of standing wave
alternating with nodes and anti-nodes right a smooth intensity profile is
required to trap a single atom using light assisted collisions at the small
waist. To realize this in the cavity, we drive sidebands 10 FSRs from the
carrier, tuning the relative power in the peaks until the multiple
longitudinal modes interfere to create a dipole trap.

##  Supplement E Aligning the 780 nm cavity mode and 785 nm intra-cavity
lattice

Getting the strongest possible coupling between the atom and the cavity
requires that the atom be trapped at the $780$ nm cavity mode anti-node. We
achieve this by using an intra-cavity lattice for trapping the atom, and
choosing the frequency of the lattice laser such that the anti-nodes of
lattice standing wave and the cavity mode are approximately aligned at $z=d$,
where $d\approx 1$ cm is the distance of the mode anti-node closest to the
cavity waist, from the curved mirror 111Note that we can’t guarantee that
there is an anti-node at the small waist location. At worst, the anti-node is
$\lambda/4$ away from the waist, which only changes the coupling by $0.5\%$.

To see how we find the correct lattice frequency, note that the lattice
standing wave is given by the function $\sin^{2}(\frac{nd}{L})$, where $n$ is
the order of the mode at which the lattice laser is resonant with the cavity,
i.e. $L=n\lambda/2$, $\lambda\approx 785$ nm. Moving this standing wave one
period requires changing the phase the frequency by $\Delta n\times$FSR, where
$\Delta n=\frac{L}{d}$. Our measured FSR gives $L=31.2$ cm, so $L/d$ is not an
integer and therefore perfect alignment is not generally possible. But this
guarantees that moving the frequencies by 16 FSRs would either pass through
either a minimum or maximum of coupling.

One of the ways we keep track of this coupling is by measuring the VRS.
Another, faster way to keep track of this coupling, is to trap the particle in
the dipole trap with a small residual standing wave, which still leads to
parity projection, but slightly increases the probability of finding an atom
at the residual lattice intensity anti-nodes. This leads to change in
scattering rate in the dipole trap as the lattice laser is moved through the
different longitudinal modes. We have used both methods at different times.

We then use a binary search like procedure to find the minimum of the
coupling, since that is a clearer signal than a maximum and then change the
frequency by 16 FSRs to find the maximum. Finally, we note that since smallest
step that we can change the phase of the standing wave is $\frac{d}{L}\pi$,
theoretically the worst possible phase offset (if we perform our alignment
procedure perfectly) we can have is $\frac{d}{2L}\pi=0.016\pi$, which
corresponds to a misalignment of $\approx 6$ nm.

##  Supplement F Single Atom Lifetime

To characterize the lifetime of our trapped single atoms, we observe the atom
while under molasses cooling light for 200 ms and plot average fluorescence
level vs hold time. An exponential fit then yields a lifetime of 146(6) ms.
See Figure ]S4.

The atom lifetime without the molasses light (in the dark) fits to a lifetime
of 50 ms. The nearly threefold increase in lifetime in the presence of
molasses light is likely indicative of intensity-noise-induced parametric
heating in the cavity dipole trap/lattice.

Supplemental Information Fig. S4: Survival probability of single atoms in the
trap vs hold time, characterized by average fluorescence level over 700 shots,
with lifetime fitted to 146(6) ms.

##  Supplement G Experimental Conditions for $g_{2}(\tau)$ dataset

The $g_{2}(\tau)$ dataset was collected at high outcoupling ($T=20\%$ per
round-trip) to maximize the data rate, using a pair of single photon counters
(SPCMs) to minimize the impact of detector afterpulsing mckeever2004trapped,
and time-tagged using custom firmware running on a RedPitaya built on top of
the Zynq Time-to-Digital package (adamivc2019fast and
https://github.com/madamic/zynq_tdc).

##  Supplement H Cavity Stability

We calculate the waist size and Guoy phase of the small-waist resonator in the
usual way, by taking the eigenvalues and vectors of the round trip ABCD
matrix. Expressed as (1,q), the eigenvector provides the complex beam
parameter q from which the waist size at a given cavity length can be
extracted. The stability condition $\frac{f^{2}}{L_{long}-f}>L_{short}-f-R>0$
is extracted by setting the norm of the eigenvalue equal to 1. The argument of
this round trip eigenvalue provides the Guoy phase, or transverse mode
splitting. We show plots of the resonator waist size and Guoy phase at the
operating point of our cavity, where we set the long arm $L_{long}=30$ cm and
the the focal length $f=1.45$ mm for the C140TMD-B asphere. The only varying
parameter then is the length of the short arm (distance between C140 and
curved mirror). We plot the waist size against the perturbation in this short
arm length from $f$ (with $f$ chosen as the edge of the stability diagram and
the “design” distance of a normal lens system). See Figure S5.

It is worth noting that the small waist in a occurs at the center of the
stability diagram. This is the salient feature of our new resonator geometry,
indicating that a sub-micron mode waist can be achieved at the location of the
atom without operating at the edge of the stability range, as is required in
concentric cavities. It is also worth noting at this point that the Guoy phase
for this resonator does not follow the usual bounds for a two mirror cavity.
There, the Guoy phase and consequently the transverse mode splitting runs from
0 to $2\pi$ across the edges of the stability region. In the small waist
geometry sketched here, the phase range is halved, running only from $\pi$ on
one end to 0. This is due to the asymmetry of the cavity. At the operation
point of our cavity - the center of the stability diagram - the Guoy phase of
the resonator is $\frac{\pi}{2}$. This has the favorable property that neither
the first or second order modes are overlapped with the fundamental mode of
the cavity, circumventing the mode-mixing issues of the confocal and
concentric designs.

Supplemental Information Fig. S5:  left the small waist cavity stability
diagram with a long arm $L_{long}=30$ cm. A waist of 930 nm is achieved at the
center of the stability diagram right the Guoy phase of the cavity across the
stability range, sweeping from $\pi$ to 0.

##  Supplement I Error and Uncertainty Calculations

Error propagation for quoted parameters is carried out using Python’s
uncertainties package, with error for fitted parameters provided by the
covariance matrix. For uncertainty bars, two methods are used: (1) chi-
squared, fit based for the lifetime data, and (2) bootstrapping for the
survival rate and fidelity data. For bootstrapping, we split the data into
three sets, calculate relevant quantities individually, then calculate the
average and standard deviation.

##  Supplement J $g^{(2)}(\tau)$ Calculation

There are three relevant timescales in this system : (i) the fastest one set
by the Rabi frequency in PGC, detuning, atomic decay rates and the cavity
lifetime, (ii) the time scale set by the scattering rate in the PGC, and (iii)
the slowest timescale, set by the motion in the trap. While it might seem that
the fastest, quantum mechanical timescale would be most interesting, it has
been thoroughly explored in the context of resonance fluorescence. With time-
resolved explorations of laser-cooling in mind, we are interested in the
slowest of these timescales. Here the dynamics of the atom are entirely
described by the classical Langevin equation of a harmonic oscillator. The 1D
Langevin equation is of the form :

$\frac{d^{2}x(t)}{dt^{2}}+\gamma\frac{dx(t)}{dt}+\omega^{2}x(t)=\eta(t)$ (J.1)

Here $\eta(t)$ is the random force (noise term) that obeys
$\langle{\eta(t)\eta(t^{\prime})}\rangle=\frac{2\gamma
k_{B}T}{m}\delta(t-t^{\prime})$, $\gamma$ is the damping rate, $m$ is the
mass, $\omega$ is the oscillator angular frequency, and $T$ is the
temperature.

Our program for analyzing these dynamics is as follows:

  1. 1.

Treat the atom as moving in a (separable) intra-cavity potential
$U(x,y,z)=\frac{U_{0}}{1+\left(\frac{z}{z_{r}}\right)^{2}}\times\exp\left(-\frac{2(x^{2}+y^{2})}{w_{0}^{2}(1+(z/z_{r})^{2})}\right)\approx
U_{0}\times\left(1-(\frac{z}{z_{r}})^{2}\right)\times\left(1-2\frac{x^{2}+y^{2}}{w_{0}^{2}}\right)\approx
U_{0}\times\left(1-(\frac{z}{z_{r}})^{2}-2\frac{x^{2}+y^{2}}{w_{0}^{2}}\right)$,
where $z_{r}\equiv\frac{\pi w_{0}^{2}}{\lambda}\approx 3$ $\mu$m is the
Rayleigh range of the cavity mode, $w_{0}\approx 930$ nm is the mode waist,
and $\lambda\approx 780$ nm is the wavelength of the trapping light.

  2. 2.

Treat the atomic scattering rate as separable in the Cartesian atom
coordinates $I_{sc}\approx I_{pk}\times\cos^{2}{kz}\times
e^{-2(x^{2}+y^{2})/w_{0}^{2}}$. Note that where the _trap_ varies in $z$ on a
length scale of $z_{r}$, the scattering rate varies in $z$ on a much shorter
length-scale of $k^{-1}=\frac{\lambda}{2\pi}$ – by contrast, the radial
trapping and radial scattering rates both vary on the length scale of $w_{0}$
\- it is for this reason that we claim that the variations in fluorescence
rate in the dipole trap arise predominantly from axial motion222At least
insofar as axial and radial temperatures are equal.

  3. 3.

Generate the equations of motion for each of the spatial degrees of freedom of
the atom, including velocity dependent damping and Langevin noise term arising
from the cooling beams (itself normalized based upon the axis-dependent
temperature).

  4. 4.

For the radial direction, solve these equations for the two-time correlators
of each degree of freedom of the atom. For the axial direction, the fast
$\cos^{2}(kz)$ variation of the scattering rate requires calculation of the
the marginal probability distribution $P(z,t)$

  5. 5.

Employ these two-time correlators and the marginal distribution of the atomic
position, in conjunction with the knowledge that since the equations of motion
are linear the motional degrees of freedom act as gaussian processes, to
compute two-time correlators of the intensity. We will find that the two-time
intensity correlator factorizes into independent correlators in $x$,$y$, and
$z$.

  6. 6.

Normalize the two-time correlator using the expected value of the intensity.

We first note that since motion in the three directions is approximately
independent, the variations of scattering rates due to motion in the three
directions are uncorrelated. Writing scattered intensity as
$I(t)=I_{pk}I_{x}(x(t))I_{y}(y(t))I_{z}(z(t))$, we find :

$\displaystyle g^{(2)}(\tau)$
$\displaystyle=\frac{\langle{I_{x}(x(\tau))I_{y}(y(\tau))I_{z}(z(\tau))I_{x}(x(0))I_{y}(y(0))I_{z}(z(0)}\rangle}{\langle{I_{x}(x(0))I_{y}(y(0))I_{z}(z(0)}\rangle^{2}}$
(J.2)
$\displaystyle=\frac{\langle{I_{x}(x(\tau))I_{x}(x(0))}\rangle}{\langle{I_{x}(x(0))}\rangle^{2}}\frac{\langle{I_{y}(y(\tau))I_{y}(y(0))}\rangle}{\langle{I_{y}(y(0))}\rangle^{2}}\frac{\langle{I_{z}(z(\tau))I_{z}(z(0))}\rangle}{\langle{I_{z}(z(0))}\rangle^{2}}$
$\displaystyle=g^{(2)}_{x}(\tau)g^{(2)}_{y}(\tau)g^{(2)}_{z}(\tau)$

###  J.1 The radial part

In one of the radial directions, say $x$, the scattered intensity varies as
$I_{x}(x)=e^{-\frac{2x^{2}}{w_{0}^{2}}}\approx 1-\frac{2x^{2}}{w_{0}^{2}}$.
This implies :

$g_{x}^{(2)}(\tau)=\frac{\biggl{<}\left(1-\frac{2x(\tau)^{2}}{w_{0}^{2}}\right)\left(1-\frac{2x(0)^{2}}{w_{0}^{2}}\right)\biggr{>}}{\biggl{<}1-\frac{2x(0)^{2}}{w_{0}^{2}}\biggr{>}^{2}}$
(J.3)

We assume a gaussian process and use Wick’s theorem to simplify the fourth
order correlator and obtain:

$g_{x}^{(2)}(\tau)=1+\frac{8}{w_{0}^{4}}\frac{\langle{x(\tau)x(0)}\rangle^{2}}{\left(1-\frac{2\langle{x(0)^{2}}\rangle}{w_{0}^{2}}\right)^{2}}$
(J.4)

We further note that for a particle governed by equation J.1, the second order
position correlator is well known and given by Ref wang1945theory:

$\langle{x(\tau)x(0)}\rangle=\frac{k_{B}T_{r}}{m\omega_{r}^{2}}e^{-\frac{\gamma_{r}}{2}\lvert\tau\rvert}\left(\cos(\omega^{\prime}\tau)+\frac{\gamma_{r}}{2\omega^{\prime}}\sin(\omega^{\prime}\lvert\tau\rvert)\right),$
(J.5)

where $T_{r}$ is the radial temperature, $\omega_{r}$ is the radial trap
angular frequency, $\gamma_{r}$ is the radial damping rate, and
$\omega^{\prime}=\sqrt{\omega_{r}^{2}-\gamma_{r}^{2}/4}$. This completes the
calculation for $g^{(2)}_{x}(\tau)$. We assume that our system has cylindrical
symmetry, so $g^{(2)}_{y}(\tau)=g^{(2)}_{x}(\tau)$.

###  J.2 The axial part

In the axial direction $z$, the variation in scattered intensity is due to the
standing wave nature of the mode, i.e. $I_{z}(x)=\cos^{2}(kz)$, leading to:

$g^{(2)}_{z}(\tau)=\frac{\langle{\cos^{2}(kz(\tau))\cos^{2}(kz(0))}\rangle}{\langle{\cos^{2}(kz(0))}\rangle^{2}}$
(J.6)

####  J.2.1 Intra-cavity lattice

For an atom trapped in tashe intra-cavity lattice, we can follow the same
procedure as the radial calculation i.e. Taylor expand the cosine, and apply
Wick’s theorem, yielding :

$g_{x}^{(2)}(\tau)=1+2k^{4}\frac{\langle{z(\tau)z(0)}\rangle^{2}}{\left(1-k^{2}\langle{z(0)^{2}}\rangle\right)^{2}}$
(J.7)

####  J.2.2 Dipole trap

This is not possible for an atom in the dipole trap, which has a length scale
given by $z_{r}$, the Rayleigh range. An atom could possibly sample even
multiple longitudinal mode anti-nodes while staying in the trap.

Here we instead use the fact that the trap frequency in dipole trap is much
lower, and assume that the motion of the particle is overdamped. This allows
us to simplify the Fokker-Planck equation and solve for the marginal
probability distribution, $P(z,t;z_{0})$, the probability of finding a
particle at position $z$ at time $t$ given that the particle was at position
$z_{0}$ at time $t=0$. The Fokker-Planck equation simplifies to
wang1945theory:

$\frac{\partial P(z,t)}{\partial t}=\frac{\partial}{\partial
z}\left(\frac{\omega_{ax}^{2}}{\gamma_{ax}}zP(z,t)\right)+\frac{k_{B}T_{ax}}{m\gamma_{ax}}\frac{\partial^{2}P(z,t)}{\partial
z^{2}}$ (J.8)

Here $T_{ax}$ is the axial temperature, $\omega_{ax}$ is the axial trap
angular frequency, $\gamma_{ax}$ is the radial damping rate. Such a
simplification and marginalization of the full distribution including velocity
is possible because the particle attains terminal velocity at every position,
giving $\langle{v}\rangle=\frac{\omega_{ax}^{2}}{\gamma_{ax}}z$. This allows
the inertial term in the Langevin equation to be neglected. For the initial
condition $P(z,0)=\delta(z-z_{0})$, the solution to this equation is a
gaussian given by:

$P(z,t;z_{0})=\frac{1}{\sigma(t)\sqrt{2\pi}}\exp\left(-\frac{(z-z_{0}\alpha(t))^{2}}{2\sigma(t)^{2}}\right),$
(J.9)

with $\alpha(t)=e^{-\frac{\omega_{ax}^{2}}{\gamma_{ax}}t}$,
$\sigma(t)=\sigma_{0}(1-\alpha(t)^{2})$ and
$\sigma_{0}=\frac{k_{B}T_{ax}}{m\omega_{ax}^{2}}$. $P(z,t;z_{0})$ is thus the
conditional probability of finding the atom at position $z$ at time $t$, given
that it was at position $z_{0}$ at time $t=0$, exactly the distribution
required to calculate two-time correlators. Furthermore,
$P_{\infty}(z)=P(z,\infty;z_{0})$ gives the probability of finding an atom at
position $z$ in steady state.

For the denominator of $g^{(2)}_{z}(\tau)$ we have:

$\displaystyle\langle{\cos^{2}(kz)}\rangle^{2}$
$\displaystyle=\left(\int_{-\infty}^{\infty}dzP_{\infty}(z)\cos^{2}(kz)\right)^{2}$
(J.10) $\displaystyle=\frac{1}{2}\left(1+e^{-2k^{2}\sigma_{0}^{2}}\right)^{2}$

For the numerator we get:

$\displaystyle\langle{\cos^{2}(kz(\tau)\cos^{2}(kz_{0})}\rangle$
$\displaystyle=\int_{-\infty}^{\infty}\left(dz_{0}P_{\infty}(z_{0})\cos^{2}(kz_{0})\int_{-\infty}^{\infty}dzP(z,t;z_{0})\cos^{2}(kz(\tau))\right)$
(J.11)
$\displaystyle=\frac{1}{4}\left(e^{-4k^{2}\sigma_{0}^{2}}\cosh\left(4\alpha(\tau)k^{2}\sigma_{0}^{2}\right)+2e^{-2k^{2}\sigma_{0}^{2}}+1\right)$

Putting it all together and simplifying with a bit of algebra, we get for the
dipole trap:

$g^{(2)}_{z}(\tau)=1+\frac{2\sinh^{2}\left(2k^{2}\sigma_{0}^{2}e^{-\left(\frac{\omega_{ax}^{2}}{\gamma_{ax}}\lvert\tau\rvert\right)}\right)}{\left(e^{2k^{2}\sigma_{0}^{2}}+1\right)^{2}}$
(J.12)

##  Supplement K Effect of temperature on average atom-cavity coupling

The geometric expression for the cooperativity mentioned in the main text is
valid for a standing wave cavity with the atom perfectly localized to a cavity
mode anti-node at the cavity waist. In reality, due to its finite temperature,
the atom samples a range of different positions and therefore different
couplings to the mode. This leads to an average observed cooperativity lower
than the peak value. We note that we are neglecting the effect of the atom
sampling different velocities, Doppler broadening the atomic transition. We
expect this effect to be much smaller than the lowering of effective coupling,
even at the highest expected temperatures.

The functional dependence of cooperativity on the position is exactly the same
as that of scattered intensity discussed in SI J. In fact, if the average
value of the cooperativity is reduced from the peak cooperativity ($C_{peak}$
by a factor f), i.e. $\langle{C}\rangle=fC_{peak}$, the factor f is given by
exactly the kind of integrals we have already calculated for the denominator
of $g^{(2)}$ in SI J. We are only interested in the cooperativity when the
atom is trapped in the lattice. In this case the reduction factor is given by:

$f=\biggl{(}1-\frac{2\langle{x^{2}}\rangle}{w_{0}^{2}}\biggr{)}\biggl{(}1-\frac{2\langle{y^{2}}\rangle}{w_{0}^{2}}\biggr{)}\left(1-k^{2}\langle{z^{2}}\rangle\right)$
(K.1)

As before, $x$ and $y$ are the radial directions and $z$ is the axial
direction,

$\displaystyle\langle{x^{2}}\rangle$
$\displaystyle=\langle{y^{2}}\rangle=\frac{k_{B}T_{r}}{m\omega_{r}^{2}}$ (K.2)
$\displaystyle\langle{z^{2}}\rangle$
$\displaystyle=\frac{k_{B}T_{ax}}{m\omega_{ax}^{2}}$

This gives,

$f=\biggl{(}1-\frac{k_{B}T_{r}}{2U_{0}}\biggr{)}^{2}\left(1-\frac{k_{B}T_{ax}}{2U_{0}}\right)$
(K.3)

Here $U_{0}$ is the depth of the trap. In this expression we have neglected
the slight difference between $785$ nm trap profile and the $780$ nm mode
profile and also assumed perfect alignment between both of them. For our
measured values of $U_{0}/k_{B}\approx 800$ $\mu$K and $T_{r}\approx
T_{ax}\approx 200$ $\mu$K, we get $f\approx 0.67$. Based on our waist,
$w_{0}=930$ nm and finesse, $F=40$, $C_{peak}=5.6$. Accounting for another
factor of 2 from lack of optical pumping and probing with a linear
polarization, we predict $\langle{C}\rangle=1.87$, close to the measured value
of $1.6$. This difference could potentially be explained by the effect of
stochastic loading of the atom in different sites along the $z$ direction on
the scale of a Rayleigh range, which would lead to additional variation in
coupling, which we have neglected.

##  Supplement L Mode Waist Measurement

Approaching the diffraction limit $1/kw_{0}\rightarrow 1$, the smallest
achievable cavity waist is influenced by several non-quadratic optical
effects: non-paraxial propagation & aberrations of the asphere
jaffe2021aberrated and vector polarization effects chaumet2006fully that are
small in free space are enhanced by numerous cavity round-trips. Rather than
relying exclusively on modeling to estimate the beam-waist, we prefer to
measure it directly.

A traditional knife edge measurement of the waist is not possible, because
once the losses from the edge become comparable to the external and internal
losses of the cavity (long before it actually approaches the waist), the
finesse drops precipitously. In addition, the short ($z_{r}\sim
3\leavevmode\nobreak\ \mu$m) Rayleigh range of the sub-micron waist
necessitates measurement with an object that is thin compared to $z_{r}$,
impractical for a razor blade. Indeed, even SNOM-based mode-mapping techniques
ferri2020mapping result in too much loss near the waist of a sub-micron waist
resonator.

We therefore introduce a new way of measuring the mode waist of a cavity in
situ using an end mirror with a small, sub-wavelength hole providing local
outcoupling of the cavity field. The size of the hole is chosen such that the
additional loss introduced is small compared to the cavity round trip loss and
the finesse does not drop significantly for all measurement locations, while
still providing enough outcoupling to detect the transmission. We use a gold
coated SiN membrane (Norcada) with two holes of nanopores (of radius $100$,
$200$, or $350$ nm) separated by $7\leavevmode\nobreak\ \mu$m. The gold coated
membrane acts as the flat end mirror of the cavity which guarantees the focus
to be right on the membrane because of the flat wavefront set by the boundary
conditions. We carried out our test by using aspheric lens with $f=1.5$ mm
(Thorlabs C140 TMD-B), identical to what we used in the main text.

The transmission through a sub-wavelength nanopore is approximated by the
Bethe formula Bethe1944 $T\propto(\lambda/r)^{4}$ for a circular hole with
radius $r$ at wavelength $\lambda$ in a thin and transversely infinite
conductor. In practice the finite thickness of the gold deposition as well as
the existence of surface plasmons and plasmon resonances around the hole
complicate the calculation of the hole transmission; because of the $r^{-4}$
dependence of transmission on hole size, the size must be carefully chosen to
ensure that the transmission is smaller than the mirror transmission
$T=1-98.5\%=0.015$ to avoid spoiling the cavity finesse, but large enough to
yield a detectable signal. Experimentally we found a diameter of $200$ nm gave
a clear transmission signal while only reducing the cavity finesse by $15\%$
from 52 to 42, whereas $350$ nm introduced excess loss destroying the cavity
and $100$ nm yielded no transmission detectable above the noise floor. We
expect that the impact of the $200$ nm pinhole’s size on our waist measurement
to be ignorable. The hole is then scanned by a homebuilt, mechanically
multiplied 2D piezo stage transverse to the cavity axis to determine the
profile of the cavity waist. The well-defined separation of the two holes is
used as an absolute calibration of the scanning setup.

In figure S6 the transmission through the nanopore is shown as a function of
transverse position. The waist is determined by a gaussian fit to both holes.

Supplemental Information Fig. S6: Measurement on Mode Waist. (a) Schematic of
measurement setup. We built a similar cavity setup as in the main text and
placed a gold membrane with pinhole with 200 nm diameter and pitch of
$7\leavevmode\nobreak\ \mu$m on a piezo scanning stage behind the aspherical
lens. The extra distance $\epsilon$ is tunable via the translation stage
holding the asphere. The piezo stage scans the 2D transverse plane as the
transmission signal is recorded by a photodiode. (b) 2D Plot of transmission.
As cavity mode crosses the pinhole, the finesse drops only around 15% from 52
to 42, which allows the transmission measurement to reflect the point spread
function (PSF) without a substantial impact to finesse variation/impedance
matching. Note that the duplication of pinhole pairs on the upper and lower
regions is due to the triangle scanning waveform in the $y$ direction of piezo
stage. (c)-(d) Comparison of experiment and paraxial model when cavity length
is $29$ cm in (c) and $50$ cm in (d). Note that the $x$ axis of experiment
plots $\Delta\epsilon$ refers to relative displacements in $\epsilon$ while
the $x$ axis $\epsilon$ in theory plots refers to absolute displacements. We
expect the absolute $\epsilon$ values in the experiment overlap with the
central parts of theory $\epsilon$ values. The scale in x direction was
calibrated by using $7\leavevmode\nobreak\ \mu$m pitch between the pinholes
and we only took the gaussian fit in the $x$ direction as our data. The error
bar is the standard deviation of the average of 4 PSF features in each
transmission image.

We extract the waist size for several different positions in the stability
diagram, as a function of mirror lens separation. We measure the finesse and
amplitude by sweeping the laser over multiple FSRs at every transverse
position. From the higher order modes in the transmission spectrum we infer
the transverse mode spacing and the position in the stability diagram.

##  Supplement M Comparing Lens and Cavity Light Collection Rates

We would like to compare the collection efficiency of a high numerical
aperture lens (parameterized here by $\mathrm{NA}_{lens}$) to that of a
standing wave cavity with waist size also limited by $\mathrm{NA}_{lens}$.

The first thing to note is that if all we care about is maximizing the cavity
cooperativity and our loss comes entirely from clipping, it is _always_ better
to reduce the cavity NA by increasing the mode waist to improve the finesse.
The idea is that the finesse limit is set by clipping four per round-trip to
$F_{cav}=\frac{2\pi}{4e^{-2\mathrm{NA}_{lens}^{2}/\mathrm{NA}_{cav}^{2}}}$,
where $\mathrm{NA}_{cav}=\frac{\lambda}{\pi w_{0}}$ and $w_{0}$ is the cavity
mode waist. The cavity cooperativity is then:
$C=\frac{24F}{\pi}\frac{1}{(kw)^{2}}=\frac{6F}{\pi}\mathrm{NA}_{cav}^{2}=3\mathrm{NA}_{cav}^{2}e^{2\mathrm{NA}_{lens}^{2}/\mathrm{NA}_{cav}^{2}}$

Because the gaussian factor grows faster than the quadratic prefactor shrinks,
we should go to low $\mathrm{NA}_{cav}$ (large cavity waist) to maximize the
cooperativity!

What went wrong? Well, for starters we might _want_ a small mode waist for
other reasons, like coupling to atoms in an array, or because achieving high
cooperativity at small mode waist relaxes finesse requirements and speeds up
readout. The upshot is that, if we do not expect to be able to get our finesse
higher than a few thousand (for example due to material or linewidth
constraints), choosing $\mathrm{NA}_{cav}\approx\frac{1}{2}\mathrm{NA}_{lens}$
is plenty, as it leads to a finesse of approximately 4600; increasing the
cavity NA rapidly reduces the finesse – choosing
$\mathrm{NA}_{cav}\approx\frac{3}{4}\mathrm{NA}_{lens}$ reduces the finesse to
54.

With the choice $\mathrm{NA}_{cav}\approx\frac{1}{2}\mathrm{NA}_{lens}$ out of
the way, we now assume we actually end up with a substantially _lower_ finesse
limited by material constraints (glass absorption, imperfect AR coatings,
etc…); in the absence of cavity outcoupling, the maximum achievable finesse is
assumed to be $F_{max}$; choosing to an outcoupler with transmission $\beta$
times the internal cavity losses reduces the finesse to
$\frac{F_{max}}{1+\beta}$, but outcouples a fraction $\frac{\beta}{1+\beta}$
of the light within the cavity.

As such, if, in the absence of the cavity, an atom scatters into $4\pi$ at a
rate $\Gamma_{4\pi}$, the cavity will collect light and outcouple it at a rate
$\Gamma_{cav}=\Gamma_{4\pi}\times\frac{6F_{max}}{\pi(1+\beta)}\times\mathrm{NA}_{cav}^{2}\times\frac{\beta}{1+\beta}$.
If we optimize this rate over $\beta$, we find that we should choose
$\beta=1$, halving the cavity finesse, resulting in
$\Gamma_{cav}=\Gamma_{4\pi}\times\frac{3F_{max}}{2\pi}\times\mathrm{NA}_{cav}^{2}$.
Noting that $\mathrm{NA}_{cav}\approx\frac{1}{2}\mathrm{NA}_{lens}$ we have:
$\Gamma_{cav}=\Gamma_{4\pi}\times\frac{3F_{max}}{8\pi}\times\mathrm{NA}_{lens}^{2}$.

We should now compare this to the single-lens collection efficiency. A lens
with $\mathrm{NA}_{lens}\ll 1$ subtends a solid angle
$\Omega_{lens}\approx\pi\mathrm{NA}_{lens}^{2}$. This means that the lens
collects light at a rate
$\Gamma_{lens}=\Gamma_{4\pi}\times\frac{\Omega_{lens}}{4\pi}=\Gamma_{4\pi}\times\frac{\mathrm{NA}_{lens}^{2}}{4}$.

We can now compute the cavity collection efficiency enhancement
$R=\frac{\Gamma_{cav}}{\Gamma_{lens}}=\frac{3F_{max}}{2\pi}\approx\frac{F_{max}}{2}$.
That is, the total improvement in collection that the cavity provides is given
by half of its finesse.

_Note: We have neglected polarizations, assumed that cavity scattering does
not induce extra heating, and worked for the lens in the low NA limit. These
assumptions can be made most accurate with an optically pumped atom at a
cavity anti-node, detected in cavity transmission._

##  Supplement N Cavity Structure Mechanical Design

Supplemental Information Fig. S7: Molasses mounts. (a) The vertical flexure
allows for control of the y-position of the curved mirror. Piezo-electric
stacks slide into both holders depicted here. (b) 2D flexure allows for
control of the x and z positions of the aspheric lens. Together with the
vertical flexure, all degrees of freedom in the relative alignment between the
asphere and curved mirror are accounted for. Piezo stacks slide into only half
off the holders here. (b) Molasses mounts, with 3D angles set by proper
alignment of beams to the small waist.  Supplemental Information Fig. S8:  In
vacuum installation of the completed flexure assembly, with aspheric lens and
curved mirror glued into mounts.

As covered in Supplement H, the difficulty in operating cavities at the small
waist points of the stability diagram arises from the increased sensitivity of
the cavity axis to transverse misalignments. This sensitivity is much improved
for the small waist cavity geometry outlined in this paper. Nonetheless,
precise in-vacuum positioning is still necessary to account for drifts in
cavity alignment during the bake and to tune the relative position of optics
to the stable point. Our custom designed/machined flexure mounts allow us to
accomplish this task without the use of commercial integrated three-axis piezo
stages, which can have long lead times, less throw, and high cost.

The cavity mount uses two waterjet-cut flexures for 3D control of the relative
transverse alignment between the C140 asphere and the ROC = 1 cm curved
mirror:

  1. 1.

A single-axis vertical flexure mount for the curved mirror which provides
control over the $y$-axis control

  2. 2.

A two-axis horizontal flexure that attaches to the asphere mount and provides
$x$/$z$ motion

Both flexures employ a similar lever arm design to amplify the base
$45\leavevmode\nobreak\ \mu$m throw of the NAC2003-H32 Noliac piezo stacks.
Initial designs used a straight lever arm, for which the theoretical
amplification is given simply by the constrained ratio of displacements in
piezo, set by the angle of the arm. Space constraints motivated switching to
the curved metal flexing elements shown in Figure S7. To simulate the
predicted amplification for this design, we used Fusion’s stress analysis
software with factors included to account for the stress/strain parameters
(Young’s modulus, etc.) of stainless steel. For the vertical mount, two piezos
were used (one for each side to preserve symmetry), and we use a Mach-Zehnder
interferometer to measure an amplified throw of $\sim 96\leavevmode\nobreak\
\mu$m. The horizontal flexure operates in 2D, with mobile central platform
supported by four lever arm slots. Only two of these four slots are mounted
with piezo stacks – one for control in $x$ and the other for $z$. We measure
and amplified throw of $80\leavevmode\nobreak\ \mu$m in both directions.

The completed structure is secured using vented silver screws, with the
vertical flexure sitting upright on the horizontal flexure. The entire
assembly sits on top of a base plate to properly align the component relative
to the loadlock translator arm. We use silver prism mirrors for the 3D
molasses beams. These slide into custom machined molasses mounts (see Figure
S7) with angles set to achieve cooling at the small waist.

##  Supplement O Model-Free Estimation of the Atom Detection Efficiency

The figures of merit for assessing the performance of the cavity-enhanced
readout demonstrated in this paper are (1) the atom detection fidelity for a
measurement of time $t$ and (2) the survival rate of the atom during that
measurement. To estimate these quantities, it is customary to tailor
individual fitting routines to the separate sections of the single-atom
histogram (background peak, atom peak, signal tails, etc). However, for
fidelities above $99\%$, the extrapolated fidelity becomes highly dependent on
the functional form of the histogram in its low probability wings
madjarov2021entangling. In a cavity the dependence of $g$ on the axial
position of the atom further complicate this choice of functional forms by
introducing averaging of the dipole signal over the nodes and anti-nodes of
the readout mode, or by creating an axial variation in coupling strength
beyond the Raleigh range, relevant for lattice signals where the signal atom
can in principle be localized one of many sites.

To circumvent this issue we employ a model-free estimation scheme for
calculating the atom detection efficiencies, as detailed in Section 2.6.7 of
reference madjarov2021entangling. The method relies on extrapolating the
survival rate S and fidelity F from the set of probabilities $p_{x_{1}x_{2}}$,
where $x_{1}$ is 0 if the the signal is measured above the threshold in the
first of two consecutive shots, and $x_{2}$ defined similarly for the the
second shot in the measurement. The probabilities $p_{x_{1}x_{2}}$ can than be
enumerated individually and related to F and S in a system of equations:

$p_{11}=fF_{1}^{2}S+(1-f)(1-F_{0})^{2}+fF_{1}(1-S)(1-F_{0})$ (O.1)

$p_{10}=fF_{1}S(1-F_{1})+fF_{1}(1-S)F_{0}+(1-f)(1-F_{0})F_{0}$ (O.2)

$p_{01}=f(1-F_{1})SF_{1}+f(1-F_{1})(1-S)(1-F_{0})+(1-f)F_{0}(1-F_{0})$ (O.3)

where $f$ is the filling fraction, and the overall fidelity
$F=fF_{1}+(1-f)F_{0}$. From the two shot data, it is easy to extrapolate $f$,
$p_{11}$, $p_{10}$, and $p_{01}$ at a given threshold and then to extrapolate
F and S from the equation above. Generally, there will be two solutions, one
of which can be discarded since as it will predict F < 0.5. For fidelities up
to $99.9\%$, the final F is robust to fluctuations in the loading fraction
$f$, though the component fidelities $F_{0}$ and $F_{1}$ can be highly
sensitive to it. There is only a narrow range of the loading fraction for
which all solved parameters are physical, providing a check on the
reasonableness of the extrapolated $f$.

##  Supplement P Comparing Cooperativity Expressions

Here we begin with the most commonly used expression for cooperativity,
$C\equiv\frac{4g^{2}}{\kappa\Gamma}$, and show that, up to numerical factors,
it is equivalent to (a) the closed-transition cooperativity
$C_{closed}=\frac{24F}{\pi}\frac{1}{(kw_{0})^{2}}$, and (b) the Purcell
factor, $F_{P}=\frac{3}{4\pi^{2}}\lambda^{3}\frac{Q}{V}$.

To show that $C=C_{closed}$, we note that for a closed transition, the atomic
linewidth can be written using the Weisskopf-Wigner formula:
$\Gamma\equiv\frac{\omega^{3}d^{2}}{3\pi\epsilon_{0}\hbar c^{3}}$, where $d$
is the dipole moment of the transition. The cavity linewidth can be written in
terms of the cavity length $L$ and finesse $F$ according to
$\kappa=2\pi\times\frac{c}{2L}\times{1}{F}$. Finally, the vacuum Rabi coupling
$g$ can be written as $g=dE_{1}/\hbar$, where $E_{1}$ is the electric field of
a single photon in the cavity at the location of the atom. By conservation of
energy
$\frac{1}{2}\epsilon_{0}E_{1}^{2}\times\frac{\pi}{2}w_{0}^{2}L=\frac{1}{2}\hbar\omega$,
so $g^{2}=\frac{2d^{2}\omega}{\hbar\epsilon_{0}\pi w_{0}^{2}L}$. Combining all
of these results yields:
$C\equiv\frac{4g^{2}}{\kappa\Gamma}=\frac{24F}{\pi}\times\frac{1}{(kw_{0})^{2}}=C_{closed}$,
as anticipated.

To show the equivalence of the Purcell factor $F_{P}$, we just note that for a
standing wave resonator of length $L$ and finesse $F$, the quality factor is
given by $Q=\frac{F(2L)}{\lambda}$, and the mode volume by
$V=\frac{1}{2}\frac{\pi}{2}w_{0}^{2}L$. Combining these expressions yields
$F_{p}=\frac{12F}{2\pi^{3}}\frac{\lambda^{2}}{w_{0}^{2}}=\frac{24F}{\pi}\frac{1}{(kw_{0})^{2}}$,
equal to $C$,$C_{closed}$.

In total, we can understand this equivalence to mean that, at least for atoms
with closed transitions, the probability that the atom emits into a cavity is
entirely independent of both transition’s dipole moment, and length of the
cavity. Atoms with narrow transitions (small dipole moments) emit slowly into
cavities, but they emit into freespace _even more slowly_. Similarly, a long
cavity has a small $g$, and hence a small light-matter coupling strength, but
it also has a narrow linewidth, so the weak coupling has more time to
coherently reinforce itself.

The only caveat to this story is that weak transitions and long cavities
result in slower atom emission into the resonator, and hence _slower_ readout;
if readout speed is truly the currency, then choosing shorter cavities and
stronger transitions is sensible.

##  Supplement Q Photon Loss Budget

The speed with which high-fidelity single atom detection can be performed is
set by the photon collection rate at the photodetector, accounting for all
loss channels in the system. We enumerate these for our system at the high
outcoupling (20$\%$) point:

  1. 1.

Cavity Collection: 15$\%$. At optimal Pellicle outcoupling, the cavity
collects 15$\%$ of photons (see SI A). This is double the free space
collection efficiency 7.8 $\%$ of the ideal 0.56 NA of the C140. Our
expression accounts for the birefringence of the cavity due to the EOM and
Pellicle beamsplitter and for the delocalization due to the finite temperature
of the atom. Without the birefringence the predicted collection efficiency
increases to 25 $\%$, and for an atom perfectly localized to an anti-node this
further rises to 35 $\%$.

  2. 2.

Fiber Coupling: 50 $\%$. Light that leaks out of the cavity on both sides of
the Pellicle beamsplitter propagates past an 808 nm line filter, where locking
and signal paths are divided, and through to the fiber incoupler. The 780 nm
reflection of the LL01-808 Max Line filter from AVR optics can be optimized to
be effectively loss (R > 99.9 $\%$). We measure the fiber coupling efficiency
of both paths to be $\approx{50}\%$.

  3. 3.

Quantum Efficiency: 64 $\%$. We use single photon counter module SPCM-AQR-14,
with a quantum efficiency of 64 $\%$ at 780 nm.

Overall, we calculate a total collection efficiency of 4.8 $\%$. These
parameters are far from optimized. With a more sophisticated locking and
outcoupling scheme to obviate the need for birefringent intracavity optics
together with the use of an EMCCD which does not require fiber-coupling and
has a higher overall quantum efficiency (80 $\%$ at 780 nm), a fourfold
increase in total collection efficiency to 20 $\%$ should be achievable.
Assuming that detection time goes linearly with improvements in efficiency,
this will enable readout with 99.5 $\%$ fidelity in 27 $\mu$s. Beyond these
improvements, the use of lenses with better anti-reflection coatings should
enable further gains in the photon collection rate of the cavity by increasing
the finesse and therefore the Purcell factor. At present, we extrapolate an
internal loss of the cavity of $\approx{12\%}$, based on our measured finesse
at 4$\%$ outcoupling. This loss includes the relatively poor anti-reflection
coating on the C140-TMD-B asphere with round trip loss of 1.2 $\%$ in addition
to loss due to clipping. Both issues can be circumvented with better polished
aspheres. This will enable high-fidelity detection in 10 $\mu$s.

## References