11institutetext: Dipartimento di Fisica E. Pancini, Università di Napoli
Federico II
Complesso Universitario di Monte Sant’Angelo, Via Cintia, I-80126 Napoli,
Italy 22institutetext: INFN, Sezione di Napoli, Complesso Universitario di
Monte Sant’Angelo, Via Cintia, I-80126 Napoli, Italy

#  Anomalies in the heavy sector

Giulia Ricciardi 1122<EMAIL_ADDRESS>
###### Abstract

Heavy flavour physics is well described by the Standard Model with the
exception of some measurements which could be pointing to new physics. We
briefly review status and prospects of $R(D^{(\ast)})$ and $R(K^{(\ast)})$
anomalies and also address the long standing tension in the $|V_{cb}|$ and
$|V_{ub}|$ exclusive/inclusive determinations.

##  1 Introduction

In a general scenario of optimal agreement within the Standard Model (SM), the
heavy flavour physics sector exhibits some measurements which present non-
significant but intriguing tensions with SM predictions. Here we briefly
address the discrepancy between measured ratios in semileptonic $B$ decays and
the corresponding SM predictions, which may hint at lepton flavour non-
universality or lepton flavour violation.

Semileptonic $B$ decays are also the processes of election when it comes to a
precise determination of the parameters $|V_{xb}|$ of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix. Their precise measurement allows a powerful check of the
unitarity of the CKM matrix. However, there is long-standing tension among the
$|V_{xb}|$ values, depending on whether they are extracted using exclusive or
inclusive semi-leptonic $B$ decays. We briefly review recent progress on
$|V_{xb}|$ determinations, prompted by the larger and larger samples of $B$
mesons collected at the $B$-factories and at LHCb, and by concomitant progress
in theoretical computations 111For brief overviews see for example
Ricciardi:2014aya ; Ricciardi:2016pmh ; Ricciardi:2017lne ; Koppenburg:2017mad
; Ricciardi:2019zph  and references therein..

##  2  $R(D)$ and $R(D^{\ast})$

Exclusive semi-tauonic $B$ decays were first observed by the Belle
Collaboration in 2007 Matyja:2007kt . Subsequent analyses by Babar and Belle
Aubert:2007dsa ; Bozek:2010xy ; Huschle:2015rga  measured branching fractions
above, although consistent with, the SM predictions. The ratio of branching
fractions (the denominator is the average for $l\in\\{e,\mu\\}$)

$R(D^{(\ast)})\equiv\frac{{\cal{B}}(B\to
D^{(\ast)}\tau\nu_{\tau})}{{\cal{B}}(B\to D^{(\ast)}l\nu_{l})}$ (1)

is typically used instead of the absolute branching fraction of $B\to
D^{(\ast)}\tau\nu_{\tau}$ decays to cancel uncertainties common to the
numerator and the denominator. These include the CKM matrix element $|V_{cb}|$
and several theoretical uncertainties on hadronic form factors and
experimental reconstruction effects. Since the strong force does not couple
directly to leptons, its effect on the semileptonic decays in different lepton
families is identical, aside from the difference in the phase space. The ratio
between the branching fractions of these decays is therefore predicted with
O(1%) precision. In 2012-2013 Babar has measured $R(D^{(\ast)})$ by using its
full data sample Lees:2012xj ; Lees:2013uzd , and reported a significant
excess over the SM expectation, confirmed in 2015 by LHCb Aaij:2015yra . In
2016 such excess has been confirmed also by the Belle collaboration, which has
performed the first measurement of $R(D^{\ast})$ using the semileptonic
tagging method, giving Belle:2016kgw

$R(D^{\ast})=0.302\pm 0.030\pm 0.011$ (2)

where the first error is statistic and the second one is systematic. By
selecting the previous measurements in the time-span 2012-2016 Lees:2012xj ;
Lees:2013uzd ; Aaij:2015yra ; Huschle:2015rga ; Belle:2016kgw , in 2016 the
HFLAV Collaboration found the average HFAG2016

$R(D)=0.397\pm 0.040\pm 0.028\qquad\qquad R(D^{\ast})=0.316\pm 0.016\pm 0.010$
(3)

where the first uncertainty is statistical and the second is systematic.
$R(D)$ and $R(D^{\ast})$ exceed the SM value $R(D)^{SM}=0.300\pm 0.008$, given
by the HPQCD collaboration Na:2015kha , and the SM value
$R(D^{\ast})^{SM}=0.252\pm 0.003$ Fajfer:2012vx  by 1.9$\sigma$ and
3.3$\sigma$, respectively. The combined analysis of $R(D)$ and $R(D^{\ast})$,
taking into account measurement correlations, finds that the deviation is
4$\sigma$ from the SM prediction.

Fast forward to 2022, the HFLAV analysis has added to the previous results
from Babar, Belle and LHCb Lees:2012xj ; Lees:2013uzd ; Huschle:2015rga ;
Aaij:2015yra  the most recent results from Belle Belle:2016dyj ; Belle:2017ilt
; Belle:2019rba  and LHCb LHCb:2017smo ; LHCb:2017rln ; LHCbpreli22 . The
HFLAV determination finds lower average values and smaller errors, namely
Amhis:2022mac

$R(D)=0.358\pm 0.025\pm 0.012\qquad\qquad R(D^{\ast})=0.285\pm 0.010\pm 0.008$
(4)

New SM calculations have become available since the 2016. The results are
compatible with the previous ones, although generally more accurate or robust.
The HFLAV has performed their arithmetic average resulting in Amhis:2022mac
$R(D)^{SM}=0.298\pm 0.004$ and $R(D^{\ast})^{SM}=0.254\pm 0.005$. The
experimental averages (4) exceed these SM values by 2.16$\sigma$ and
2.26$\sigma$, respectively. The combined analysis of $R(D)$ and $R(D^{\ast})$
finds that the deviation is 3.2$\sigma$ from the SM prediction.

The actual recorded luminosity at Belle II is 428 fb-1, to be compared with
Belle (988 fb-1) and BaBar (513 fb-1); the ultimate goal is to reach 50 ab-1.
Investigating the anomaly through precision measurements of $R(D^{(\ast)})$ is
a chief goal of Belle II. At Belle II a better understanding of backgrounds
tails under the signal and a reduction of the uncertainty to 3% for
$R(D^{\ast})$ and 5% for $R(D)$ is expected at 5 ab-1Belle-II:2022cgf .

Belle II has the unique potential to measure the ratio $R(X)$ of the inclusive
rate $B\to X\tau\nu$ to the lower-mass lepton counterparts. This ratio would
probe both electron and muon modes with a precise consistency check whose
phenomenological interpretation is independent from LQCD uncertainties that
affect the other observables. However, this is a challenging measurement
attempted and never completed at previous B-factory experiments. The main
experimental challenge is to control the significant systematic uncertainties
associated with background composition Belle-II:2022cgf . Belle II is
preparing and this year it has measured the ratio $R(X_{e/\mu})$ of the
inclusive rate $B\to X_{e}\nu$ to $B\to X_{\mu}\nu$, which is in good
agreement with the SM Junkerkalefeld:3125 .

The $B$-factories have performed the most precise measurements of $R(D)$ and
$R(D^{\ast})$ to date thanks to their ability to significantly constrain the
kinematics of these neutrinos by leveraging their knowledge of the
$e^{+}e^{-}$ collision energy. LHCb, however, is expected to surpass the
B-factories precision and reach uncertainties down to the percent level from
the analysis of the enormous data samples expected from the operation of the
Upgrades I and II detectors, provided that the relevant systematic
uncertainties can be properly controlled LHCb:2022ine .

Other ratios are currently under investigation. LHCb has found the value
LHCb:2017vlu  $R(J/\Psi)=0.71\pm 0.17\pm 0.18$, which is 2.1$\sigma$ from the
LQCD value Harrison:2020nrv  $R(J/\Psi)^{\mathrm{HPQCD}}=0.2582\pm 0.0038$.
Another LHCb measurement gives LHCb:2022piu

$R(\Lambda_{c})\equiv\frac{{\cal{B}}(\Lambda_{b}\to\Lambda_{c}\tau\nu_{\tau})}{{\cal{B}}(\Lambda_{c}\to\Lambda_{c}\mu\nu_{\mu})}=0.242\pm
0.026\pm 0.040\pm 0.059$ (5)

where the external branching fraction uncertainty from the channel
$\Lambda^{0}_{b}\to\Lambda^{+}_{c}\mu^{-}\bar{\nu}_{\mu}$ contributes to the
last term. The last ratio seems can be related to the values of $R(D)$ and
$R(D^{*})$ Blanke:2018yud .

##  3 $R(K)$ and $R(K^{*})$

While $R(B)$ is defined as the ratio of branching fractions of decays that
occur at tree level in the SM at the lowest perturbative order, the observable
$R(K)$ and $R(K^{*})$ are defined as the ratios of branching fractions of rare
decays. In general, for semileptonic decays of $B$ hadrons, where
$B=B^{+},B^{0},B^{0}_{s}$ or $\Lambda^{0}_{b}$, the ratio $R(H)$ is defined in
the dilepton mass-squared range
$q^{2}_{\mathrm{min}}<q^{2}<q^{2}_{\mathrm{max}}$ as

$R(H)=\frac{\int_{q^{2}_{\mathrm{min}}}^{q^{2}_{\mathrm{max}}}\frac{d{\cal{B}}(B\to
H\mu^{+}\mu^{-})}{dq^{2}}{\tiny
dq^{2}}}{\int_{q^{2}_{\mathrm{min}}}^{q^{2}_{\mathrm{max}}}\frac{d{\cal{B}}(B\to
He^{+}e^{-})}{dq^{2}}dq^{2}}$ (6)

where $H$ can be an hadron as $K$ or a combination of particles such as a
proton and charged kaon, $pK^{-}$.

LHCb has provided the most precise measurements of the branching fractions of
the $B^{+}\to K^{(\ast)+}\mu^{+}\mu^{-}$ and $B^{0}\to
K^{(\ast)0}\mu^{+}\mu^{-}$ decays LHCb:2014cxe ; LHCb:2012juf ; LHCb:2013zuf .
All these measurements are below the SM prediction. At LHCb, $R(K)$ has been
measured to be Aaij:2014ora

$R(K^{+})=0.745^{+0.090}_{-0.074}\pm 0.036$ (7)

where the first error is statistical and the second one is systematic. The
measurement was performed across the dilepton invariant-mass-squared range
[1,6] GeV2. This result is 2.6$\sigma$ deviations away from the SM prediction
$R(K)^{SM}=1.0003\pm 0.0001$ Bobeth:2007dw . The impact of radiative
corrections has been estimated not to exceed a few % Bordone:2016gaq . The
result in Eq. (7) has been superseded in 2021 by an LHCB analysis which uses
essentially identical techniques but an additional 4 fb-1 of data collected in
2017 and 2018 LHCb:2021trn . Hence, the entire amount of data was recorded
during the years 2011, 2012 and 2015â2018, in which the centre-of-mass energy
of the collisions was 7, 8 and 13 TeV, and correspond to an integrated
luminosity of 9 fb-1. The analysis LHCb:2021trn  gives, in the dilepton
invariant-mass-squared range [1,6] GeV2, the value

$R(K^{+})=0.846^{+0.042\,+0.013}_{-0.039\,-0.012}$ (8)

where the first uncertainty is statistical and the second systematic.
Combining the uncertainties they obtain $R(K^{+})=0.846^{+0.044}_{-0.041}$.
There are previous measurements by Babar BaBar:2012mrf  and Belle
BELLE:2019xld , but the LHCb measurement is the most precise to date and is
consistent with the SM expectation at the level of 3.1 standard deviations.

The other ratio measurements from LHCb include $R(K^{\ast 0})$ with the Run 1
data LHCb:2017avl  in two regions of the dilepton invariant mass squared; the
value found for the interval [0.045,1.1] GeV2 is

$R(K^{\ast 0})=0.66^{+0.11}_{-0.07}\pm 0.03$ (9)

and the value found for the interval [1.1,6.0] GeV2 is

$R(K^{\ast 0})=0.69^{+0.11}_{-0.07}\pm 0.05$ (10)

These results, which represent the most precise measurements of $R(K^{\ast
0})$ to date, are compatible with the Standard Model expectations at the level
of 2.1-2.3 and 2.4-2.5 standard deviations (depending on the theoretical
predictions) in the two $q^{2}$ regions. Previous results from Babar
BaBar:2012mrf  and Belle Belle:2009zue  are also available. LHCb current
results are reported in figure 1, where also the relevant dilepton invariant
mass squared interval is indicated.

Figure 1: Current LHCb values for semileptonic ratios $R(H)$, where the
decaying particle is also indicated. The first two results are taken from
LHCb:2017avl , the baryon decay has been analyzed in LHCb:2019efc , $R(K^{+})$
in LHCb:2021trn , and the last two measurements in LHCb:2021lvy .

##  4 Data interpretation

The alleged breaking of lepton-flavour universality suggested by most of the
data is quite large, and several theoretical models have been tested against
the experimental results. A model independent approach, where the SM is
considered as an effective low-energy theory, is mostly used. The SM Effective
Theory (SMEFT) is the effective theory above the electroweak (EW) scale,
incorporating the full SM content, with non renormalizable operators which are
gauge-invariant under the SM gauge group. The main assumption inherent to the
SMEFT framework is that new physics interactions have been integrated out at
some high scale. The SMEFT aims at characterizing experimental deviations from
the predictions of the Standard Model and pointing towards the structures of
its possible extensions above the EW scale. The Weak Effective Theory (WET) is
the effective theory valid below the EW scale which includes all SM particles
except the the gauge bosons, the Higgs and the $t$ quark. In contrast to the
SMEFT case, the dimension-six operators in WET are not invariant under the
full SM gauge group, but only under $SU(3)_{c}\times U(1)_{em}$, as this EFT
is defined below the EW scale where $SU(2)_{L}\times U(1)_{Y}$ is already
broken. For a given set of SMEFT dimension-six operators with the
corresponding Wilson coefficients specified at the higher scale, one has to
perform the renormalization group evolution of the SMEFT Wilson coefficients
down to the EW scale, match the given set of SMEFT operators onto the WET ones
at the EW scale and finally perform the RGE of the WET Wilson coefficients
from the EW down to the dynamical scale of the $B$ meson. This way, one can
then bridge the gap between the SMEFT Lagrangian and the low-energy
measurements in $B$ physics.

New physics effects in semiletonic decays into light quarks are stringently
constrained. For instance the ratio Belle:2018ezy

$\frac{{\cal{B}}(B^{0}\to D^{\ast-}e^{+}\nu_{e})}{{\cal{B}}(B^{0}\to
D^{\ast-}\mu^{+}\nu_{\mu})}=1.01\pm 0.01\pm 0.03$ (11)

is in agreement with the SM. Hence new physics effects are generally assumed
in semitauonic $B$ decays. The relevant terms of the WET Lagrangian for the
underlying $b\to c\tau\nu$ transition are Aebischer:2015fzz ;
Aebischer:2017gaw ; Tanaka:2012nw

$\mathcal{L}_{\text{eff}}=-\frac{4G_{F}}{\sqrt{2}}V_{cb}\Big{[}(1+C_{V}^{L})O_{V}^{L}+C_{S}^{R}O_{S}^{R}+C_{S}^{L}O_{S}^{L}+C_{T}O_{T}\Big{]}+\text{h.c.}$
(12)

where $G_{F}$ is the Fermi constant, $V_{cb}$ is an element of the CKM matrix
and the dimension-six four-fermion operators are

$\displaystyle
O_{V}^{L}=\left(\bar{c}\gamma^{\mu}P_{L}b\right)\left(\bar{\tau}\gamma_{\mu}P_{L}\nu_{\tau}\right)\,,$
$\displaystyle
O_{S}^{R}=\left(\bar{c}P_{R}b\right)\left(\bar{\tau}P_{L}\nu_{\tau}\right)\,,$
$\displaystyle
O_{T}=\left(\bar{c}\sigma^{\mu\nu}P_{L}b\right)\left(\bar{\tau}\sigma_{\mu\nu}P_{L}\nu_{\tau}\right)\,,$
$\displaystyle
O_{S}^{L}=\left(\bar{c}P_{L}b\right)\left(\bar{\tau}P_{L}\nu_{\tau}\right)\,.$

Non-zero Wilson coefficients relate to the possible tree-level NP mediators:
(i) exchange of an heavy $W^{\prime}$ boson with left-handed couplings.
Simplified models (spin-1 colorless weak triplet, two Higgs doublet models, a
spin-0 or spin-1 leptoquark) are in tension with existing $\tau^{+}\tau^{-}$
LHC results Faroughy:2016osc . These constraints do no apply to the low mass
region in the $W^{\prime}$ and vector leptoquark models; (ii) exchange of a
charged Higgs boson. This scenario is in tension with the LHC mono-$\tau$ data
and it can be related to (large) branching ratios for $B_{c}\to\tau\nu$
decays. (iii) leptoquark scenarios. They generally evade the LHC mono-$\tau$
tests, but they are subject to LHC constraints from their pair-production and
t-channel mediated di-lepton processes. Other constraints may come from
characteristics of the models, as for instance induced CP violations or the
presence of colour-octet resonances, often introduced together in UV-complete
models. In general, and in all the above models, more severe constraints can
appear in concrete UV completions.

The $b\to s\ell^{+}\ell^{-}$ transitions are described by WET Lagrangian

$\mathcal{L}_{\text{eff}}=\frac{4G_{F}}{\sqrt{2}}V_{tb}^{*}V_{ts}\frac{e^{2}}{16\pi^{2}}\sum_{i}(C_{i}O_{i}+C^{\prime}_{i}{O}^{\prime}_{i})+\text{h.c.}\,,$
(13)

where the operators most sensitive to new physics are:

$O_{7}^{(\prime)}=\frac{m_{b}}{e}(\bar{s}\sigma_{\mu\nu}P_{R(L)}b)F^{\mu\nu}\quad
O_{9,10}^{(\prime)}=(\bar{s}b)_{V\mp A}(\bar{\ell}\ell)_{V,A}$ (14)

and their corresponding Wilson coefficients have contributions from the SM
processes as well as from new physics ones. Several global fits have been
performed in the literature (see for instance Capdevila:2017bsm ;
Celis:2017doq ; Alok:2017sui ; Camargo-Molina:2018cwu ; Datta:2019zca ;
Aebischer:2019mlg ; Aoude:2020dwv ). They can be fully data driven, which
means that no assumption is done about charming penguins, or partly or fully
model dependent, assuming assume LCSR results for charming penguins. There is
a general agreement among the fits, which show that even the simple one-
dimensional new physics scenarios, such as
$C_{9}^{\mathrm{NP}\,\mu}\simeq-0.73$ or
$C_{9}^{\mathrm{NP}\,\mu}=-C_{10}^{\mathrm{NP}\,\mu}\simeq-0.39$, can lead to
a significant improvement of the quality of the fit, compared to the SM-only
hypothesis. Mainstream models are loop-induced new physics, as well as tree-
level new physics contributions, mediated by $Z^{\prime}$ gauge bosons or
leptoquarks. At a variance with the $b\to c$ anomalies, the new physics scale
is found of order 40 TeV, so one cannot check direct production of new
particles at the (HL-)LHC. There is a limited reach for searches for
deviations from the SM in high $p_{T}$ di-muon tails. The couplings are
stringently constrained by $B^{0}_{s}$ meson mixing data. One has also to
remark that vector leptoquarks can provide a unique solution for both $b\to c$
and $b\to$ s anomalies. Last but not least, the discrepancies with SM
predictions have also generated alternative data analyses, as for example the
Dispersive Matrix method, claiming the values $R(D)=0.296\pm 0.008$ and
$R(D^{\ast})=0.261\pm 0.020$, which differ by about 1.4$\sigma$ from the
latest experimental determinations PhysRevD.105.034503 .

##  5 $|V_{xb}|$ determination

Improving our knowledge of $|V_{cb}|$ and $|V_{ub}|$ and reducing their
uncertainty are crucial to the so-called $|V_{cb}|$ and $|V_{ub}|$ puzzles,
the longstanding tension between the values obtained from exclusive and
inclusive determinations.. The latest global analysis of the inclusive $B\to
X_{c}\ell\nu$ by HFLAV HFLAV:2019otj  gives
$|V_{cb}^{\mathrm{incl}}|=(42.19\pm 0.78)\times 10^{-3}$. It has been done in
the framework of kinetic scheme, where $|V_{cb}|$ is extracted together with
the $b$ and $c$ quark masses and 4 non-perturbative parameters (namely
$\mu^{2}_{\pi}$, $\mu^{2}_{G}$, $\rho^{3}_{D}$ and $\rho^{3}_{LS}$). Details
on the extraction can be found for instance in Ref. Ricciardi:2019zph . More
recent results are generally consistent with the HFLAV analysis, as
Bordone:2021oof  $|V_{cb}^{\mathrm{incl}}|=(42.16\pm 0.50)\times 10^{-3}$, or
the first determination using moments of the dilepton invariant mass
Bernlochner:2022ucr  $|V_{cb}^{\mathrm{incl}}|=(41.69\pm 0.63)\times 10^{-3}$.
One can compare these results with the latest FLAG determination for exclusive
decays FlavourLatticeAveragingGroupFLAG:2021npn
$|V_{cb}^{\mathrm{excl}}|=(39.36\pm 0.68)\times 10^{-3}$. In the case of Ref.
Bordone:2021oof , for instance, one finds a 2.6$\sigma$ difference. At a
variance with previous determinations, in the dispersion matrix approach
Martinelli:2022skp  the $|V_{cb}|$ inclusive/exclusive values are compatible
within the 1$\sigma$ level. Novel approaches have been recently put forward,
that allow one to address inclusive decays in LQCD, see for instance Ref.
Gambino:2022dvu  and references therein.

In order to extract $|V_{ub}|$ from semileptonic $B\to X_{u}\ell\nu$ decays
one has to reduce the $b\to c$ semileptonic background through experimental
cuts. Such cuts enhance the relevance of the so-called threshold region in the
phase space, jeopardizing the use of the heavy quark expansion. In order to
face this problem, that is absent in the inclusive determination of
$|V_{cb}|$, different theoretical schemes have been devised, which are
tailored to analyze data in the threshold region, but differ in their
treatment of perturbative corrections and the parametrization of non-
perturbative effects. The value of $|V_{ub}|$ has been extracted by BaBar
Lees:2011fv ; Beleno:2013jla , Belle Urquijo:2009tp ; Belle:2021ymg ;
Cao:2021uwy  and HFLAV Amhis:2016xyh  collaborations. It is based on the
measured partial branching fractions value with the state-of-the-art theory
predictions on decay rate in four different theoretical different approaches:
ADFR Aglietti:2004fz ; Aglietti:2006yb ; Aglietti:2007ik , BLNP Lange:2005yw ;
Bosch:2004th ; Bosch:2004cb , DGE Andersen:2005mj  and GGOU Gambino:2007rp .
The arithmetic average of the most precise determinations for the phase-space
region $E_{\ell}^{B}>1$ gives (Belle 2022 Cao:2021qpa )
$|V_{ub}^{\mathrm{incl}}|=(4.10\pm 0.09\pm 0.22\pm 0.15)\times 10^{-3}$, where
the uncertainties are statistical, systematic and from the theory calculation,
respectively. This can be compared with the world average of exclusive results
$|V_{ub}^{\mathrm{excl}}|=(3.67\pm 0.09\pm 0.12)\times 10^{-3}$. The 2022
Belle inclusive value is smaller than the previous inclusive ones, reducing
the discrepancy with the exclusive measurement of about 2-3 to 1.3 standard
deviations. It is also compatible with the value expected from CKM unitarity
from a global fit Charles:2004jd  $|V_{ub}|=(3.62^{+0.11}_{-0.08})\times
10^{-3}$ within 1.6 standard deviations.

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# High-fidelity remote entanglement of trapped atoms mediated by time-bin
photons

Sagnik Saha  Corresponding author<EMAIL_ADDRESS>Duke Quantum Center,
Departments of Electrical and Computer Engineering and Physics, Duke
University, Durham, NC 27708  Mikhail Shalaev  Duke Quantum Center,
Departments of Electrical and Computer Engineering and Physics, Duke
University, Durham, NC 27708  Jameson O’Reilly  Present Address: Department of
Physics, University of Oregon, Eugene, OR 97331 Duke Quantum Center,
Departments of Electrical and Computer Engineering and Physics, Duke
University, Durham, NC 27708  Isabella Goetting  Duke Quantum Center,
Departments of Electrical and Computer Engineering and Physics, Duke
University, Durham, NC 27708  George Toh  Duke Quantum Center, Departments of
Electrical and Computer Engineering and Physics, Duke University, Durham, NC
27708  Ashish Kalakuntla  Duke Quantum Center, Departments of Electrical and
Computer Engineering and Physics, Duke University, Durham, NC 27708  Yichao Yu
Duke Quantum Center, Departments of Electrical and Computer Engineering and
Physics, Duke University, Durham, NC 27708  Christopher Monroe  Duke Quantum
Center, Departments of Electrical and Computer Engineering and Physics, Duke
University, Durham, NC 27708

(August 28, 2024)

###### Abstract

Photonic interconnects between quantum processing nodes are likely the only
way to achieve large-scale quantum computers and networks. The bottleneck in
such an architecture is the interface between well-isolated quantum memories
and flying photons. We establish high-fidelity entanglement between remotely
separated trapped atomic qubit memories, mediated by photonic qubits stored in
the timing of their pulses. Such time-bin encoding removes sensitivity to
polarization errors, enables long-distance quantum communication, and is
extensible to quantum memories with more than two states. Using a measurement-
based error detection process and suppressing a fundamental source of error
due to atomic recoil, we achieve an entanglement fidelity of 97% and show that
fidelities beyond 99.9% are feasible.

High-quality and fast photonic links between good quantum memories are crucial
for large-scale modular quantum computers and networks Awschalom  _et al._
(2021). Trapped atomic ions set the standard for quantum memory, with
indefinite idle quantum coherence times Wang  _et al._ (2021), near-unit
efficiency state-preparation-and-measurement (SPAM) Ransford  _et al._ (2021),
and the highest-fidelity local quantum gates Harty  _et al._ (2014); Ballance
_et al._ (2016); Srinivas  _et al._ (2021); Clark  _et al._ (2021). Shuttling
between multiple zones on a single trap may allow scaling to hundreds or
thousands of qubits Kielpinski  _et al._ (2002); Moses  _et al._ (2023), but
scaling beyond this in a modular fashion will likely require optical photonic
interconnects regardless of the qubit platform Awschalom  _et al._ (2021).

Qubits stored in trapped atomic ion arrays are natural single photon emitters
and have led the way in remote qubit entanglement protocols Moehring  _et al._
(2007a); Hucul  _et al._ (2015); Stephenson  _et al._ (2020); O’Reilly  _et
al._ (2024). In these demonstrations, photonic qubits emitted from and
entangled with their parent trapped ions are interfered and detected in an
entanglement-swapping procedure Żukowski  _et al._ (1993) that leaves the two
trapped ion memories entangled Barrett and Kok (2005); Moehring  _et al._
(2007b); Duan and Monroe (2010). Similar protocols have been demonstrated with
neutral atoms Ritter  _et al._ (2012); van Leent  _et al._ (2022) and solid-
state emitters Bernien  _et al._ (2013); Knaut  _et al._ (2024); Ruskuc  _et
al._ (2024).

Photonic qubits are commonly encoded in the polarization degree of freedom,
allowing qubit manipulation and diagnosis with simple polarization optics.
Remote trapped ions have been entangled through photonic polarization qubits
with a fidelity of $F=0.960(1)$ Nadlinger  _et al._ (2022), an entanglement
rate of 250 s-1 O’Reilly  _et al._ (2024), and over a distance of 230 m
Krutyanskiy  _et al._ (2023). Remote neutral atoms have been similarly
entangled through polarization photonic qubits with a post-selected fidelity
of 0.987(22) Ritter  _et al._ (2012) and a rate of 0.004 s-1. However,
polarization qubits are susceptible to uncontrolled birefringence in optical
elements, windows, and optical fibers, limiting performance in these and other
experiments. As system complexity grows by extending to nearby local qubit
memories, additional optical elements, more connected nodes, longer distances
between nodes, or higher-dimensional quantum memories, polarization qubits
become even less tenable Bersin  _et al._ (2024); Knaut  _et al._ (2024).
Alternative photonic degrees of freedom such as frequency Maunz  _et al._
(2009) or time-bin Bernien  _et al._ (2013) are therefore preferable for a
scalable architecture.

Here, we report for the first time the entanglement of two remote individual
atomic qubits via time-bin photons, with a measured Bell state fidelity of
${F=0.970(4)}$. Visible time-bin encoded photons emitted from remote trapped
ions are collected into single-mode optical fibers, and their interference and
coincident detection heralds the entanglement of the ion memories Barrett and
Kok (2005). We identify and suppress fidelity limits from the timing of the
atomic excitations as well as the random times of photon detection. We also
flag erasure errors from the atomic qubits to purify the resulting Bell state
with little overhead Kang  _et al._ (2023). This demonstration shows that the
fidelity limits for remote entanglement based on photons can be better than
$0.999$, allowing modular scaling of quantum computers based on atomic qubits,
long-distance quantum communication between quantum nodes, and the
entanglement of high-dimensional quantum memories Wang  _et al._ (2020).

Figure 1: (a) Schematic of the apparatus, including two ion trap chambers
Alice (A) and Bob (B). Here e(l) denotes a photon in the early(late) time-bin.
Photons collected with high numerical aperture (NA) objectives are interfered
on a beamsplitter (BS) and detected with avalanche photodiodes (APD). For
subsequent atomic qubit state readout, ion fluorescence is detected on
photomultiplier tubes (PMT). (b) Relevant energy levels of the 138Ba+system
(nuclear spin $I=0$). The atomic qubit is stored in the
${\mid\downarrow\rangle_{q}=|{{}^{2}S_{1/2},m_{J}=-\frac{1}{2}}\rangle}$ and
${\mid\uparrow\rangle_{q}=|{{}^{2}D_{5/2},m_{J}=\frac{1}{2}}\rangle}$ states
and coherently driven by narrowband laser radiation at 1762 nm. State
preparation and measurement (SPAM) is provided through optical pumping beams
at 493 nm, 650 nm, and 614 nm and Doppler laser-cooling is performed on the
493 nm and 650 nm transitions. Single photons are generated with ultrafast
laser pulses at 493 nm (see Supp. Info. .1, .4). (c) Procedure applied to each
138Ba+ ion at A and B for heralded remote entanglement using time-bin encoded
photons.

## Time-Bin Entanglement Protocol

We entangle two ${}^{138}\textrm{Ba}^{+}$ ions, each trapped in separate
vacuum chambers a few meters apart, henceforth referred to as Alice (A) and
Bob (B). The experimental schematic is shown in Figure 1a. Each chamber
contains a four-rod radiofrequency Paul trap loaded with a single 138Ba+ ion
with energy levels indicated in Fig. 1b. The atomic qubit $q\in\\{A,B\\}$ is
encoded in the states
${\mid\downarrow\rangle_{q}=|{{}^{2}S_{1/2},m_{J}=-\frac{1}{2}}\rangle}$ and
${\mid\uparrow\rangle_{q}=|{{}^{2}D_{5/2},m_{J}=-\frac{1}{2}}\rangle}$.

The remote entanglement protocol proceeds following Fig. 1c. Each ion is first
laser-cooled and initialized in the $|{\downarrow}\rangle_{q}$ state through
optical pumping, then prepared in the superposition state
$\mid\downarrow\rangle_{q}+\mid\uparrow\rangle_{q}$ by driving a $\pi/2$-pulse
between the qubit states with 1762 nm radiation Dietrich  _et al._ (2010) (see
Supp. Info. .1). At time $t_{e}$, population in the
$\mid\downarrow\rangle_{q}$ state is then driven with a probability
$P_{\mathrm{exc}}>0.80$ to the excited state
${|{e}\rangle_{q}=|{{}^{2}P_{1/2},m_{J}=+\frac{1}{2}}\rangle}$ using a
circularly-polarized 493 nm ultrafast laser pulse (see Supp. Info. .4). With
probability $73\%$, the spontaneous emission from $|{e}\rangle_{q}$ produces a
single 493 nm photon wavepacket distributed over time given an exponential
radiative lifetime of $\tau_{R}=7.85$ ns Arnold  _et al._ (2019), and 2/3 of
those decays return the population to $\mid\downarrow\rangle_{q}$ for a net
branching ratio of $\beta=49\%$. Decay to the other ground state
($|{{}^{2}S_{1/2},m_{J}=+\frac{1}{2}}\rangle$) via a $\pi$-polarized photon is
rejected with a polarization filter. Decay to the ${}^{2}D_{3/2}$ state
($27\%$ branching ratio) by emission of a 650 nm photon is rejected through
spatial and spectral filtering.

The ideal unnormalized state of each ion $q$ and its collected photon mode is
now

$\sqrt{1-p_{q}}\mid\downarrow\rangle_{q}|{0_{e}}\rangle_{q}+\sqrt{p_{q}}e^{i\phi_{qe}}\mid\downarrow\rangle_{q}|{1_{e}}\rangle_{q}+\mid\uparrow\rangle_{q}|{0_{e}}\rangle_{q}$
(1)

where $|{1_{e}}\rangle$ ($|{0_{e}}\rangle$) denotes the presence (absence) of
a photon in the first (early) time-bin and $p_{q}$ is the probability a single
photon has been collected. The phase $\phi_{qe}=\mathbf{\Delta
k}\cdot\mathbf{r}_{q}(t_{e})+\phi^{*}_{qe}$ includes the position
$\mathbf{r}_{q}(t_{e})$ of ion $q$ at time $t_{e}$ of the early time-bin, and
${\mathbf{\Delta k}=\mathbf{k}^{\prime}-\mathbf{k}}$ is the difference between
the excitation pulse wavevector $\mathbf{k}^{\prime}$ and that of the emitted
photon $\mathbf{k}$ (both of magnitude $k$). The small random phase
$\phi^{*}_{qe}$ accounts for the narrow distribution of emission times and is
discussed below.

To generate the second (late) time-bin photon, the populations
$\mid\downarrow\rangle_{q}$ and $\mid\uparrow\rangle_{q}$ are swapped with a
1762 nm $\pi$-pulse (see Supp. Info. .1), then at time $t_{l}$ the
$\mid\downarrow\rangle_{q}$ state is again excited to the $|{e}\rangle_{q}$
state. With probability $p_{q}$, there is now a single-photon time-bin qubit
entangled with its parent ion qubit, ideally in the state

$e^{i\phi_{qe}}\mid\uparrow\rangle_{q}|{1_{e}0_{l}}\rangle_{q}+e^{i\phi_{ql}}\mid\downarrow\rangle_{q}|{0_{e}1_{l}}\rangle_{q},$
(2)

where $|{n_{e}n_{l}}\rangle_{q}$ denotes a state of $n_{e}$ ($n_{l}$) photons
in the early (late) time-bin from ion $q$.

The photons are then directed to a non-polarizing 50:50 fiber beamsplitter
(BS), which erases their “which-path” information through Hong-Ou-Mandel
interference Hong  _et al._ (1987). Subsequent detection of early and late
photons ideally projects the ions into a Bell state Moehring  _et al._ (2007a)

$\Psi^{\pm}=\mid\downarrow\rangle_{A}\mid\uparrow\rangle_{B}\pm
e^{i\phi}\mid\uparrow\rangle_{A}\mid\downarrow\rangle_{B}$ (3)

where the phase is ${\phi=(\phi_{Ae}-\phi_{Be})-(\phi_{Al}-\phi_{Bl})}$. The
$\Psi^{+}$ $(\Psi^{-})$ state is heralded by early and late detections on the
same (opposite) BS output channels.

Bell state | $P_{\uparrow_{A}\downarrow_{B}}+P_{\downarrow_{A}\uparrow_{B}}$ | $C$ | $F$ | $\delta t$ | $Y$
---|---|---|---|---|---
$\Psi^{+}$ | 0.990(4) | 0.927(6) | 0.959(4) | 50 ns | 0.998
$\Psi^{-}$ | 0.996(3) | 0.931(6) | 0.963(3) | 50 ns | 0.998
$\Psi^{+}$ | 0.990(4) | 0.948(6) | 0.968(4) | 10 ns | 0.714
$\Psi^{-}$ | 0.996(3) | 0.949(6) | 0.972(3) | 10 ns | 0.714

Figure 2: The table shows measured populations, parity fringe contrast, and
fidelities for the Bell states $\Psi^{\pm}$ for two cases of detection window
$\pm\delta t$ and the corresponding yield $Y$. The plot shows the measured
state parity fringe for $\Psi^{-}$ (red) and $\Psi^{+}$ (blue) following
$\pi/2$ rotations of each qubit while scanning their relative phase ($\delta
t=10$ ns).

## Entanglement Rate and Fidelity

The maximum success probability of ion-ion entanglement is
$P_{E}=\frac{1}{2}p_{A}p_{B}=2.3\times 10^{-5}$. The factor of 1/2 accounts
for the rejection of events where both photons are collected in the same time-
bin (and on the same detector). We select coincident events occurring within a
detection time window $\pm\delta t$ of the mean photon arrival time of each
time-bin, resulting in a small reduction in $P_{E}$ by the yield factor
$Y=1-e^{-\delta t/\tau_{R}}$, given the atomic lifetime $\tau_{R}$. With a
detection time window of $\delta t=10$ ns, we observe a mean entanglement rate
of ${0.35\,s^{-1}}$. A full description of photon collection parameters is
provided in Supp. Info. .3.

We characterize the entangled state by measuring qubit correlations in
different bases. The fidelity with respect to the nominal Bell state in Eq. 3
is given by $F=(P_{\mathrm{odd}}+C)/2$, where $P_{\mathrm{odd}}$ is the
population of the odd parity states
$\mid\downarrow\rangle_{A}\mid\uparrow\rangle_{B}$ or
$\mid\uparrow\rangle_{A}\mid\downarrow\rangle_{B}$ and $C$ is the contrast of
parity oscillations of the two qubit states as the relative phase of analysis
$\pi/2$ rotations on each qubit is scanned Sackett  _et al._ (2000). Fig. 2
shows measurements for both states $\Psi^{\pm}$. We detect the ion qubit
states by shining 493 nm and 650 nm light on the ions, which causes the ions
in the $|{\downarrow}\rangle_{q}$ state to fluoresce, while ions in the
$|{\uparrow}\rangle_{q}$ state remain dark. This provides deterministic qubit
state detection with a fidelity exceeding 99.5$\%$ (see Supp. Info. .1). As
shown in Fig. 2, the measured fidelities of the entangled states (uncorrected
for SPAM) are $F=0.968(4)$ for the state $\Psi^{+}$ and $F=0.972(3)$ for the
state $\Psi^{-}$.

We attribute most of the observed fidelity imperfection to slow drifts in the
intensity of the 1762 nm qubit laser. We observe $\sim 1\%$ fluctuations over
the few-hour time period of an experimental run, while slow drifts are
eliminated between runs through periodic calibration. These fluctuations
degrade SPAM and are expected to contribute to a fidelity error of $1\%$.
Other sources of error include leakage via the $\pi$ decay channel and
residual entanglement with motion (both discussed separately below);
micromotion Doppler shifts; and optical system imperfections. The overall
error budget is discussed in Supp. Info. .7.

## Correction of Erasure Errors

During the photon emission process, there is a $\sim 24\%$ probability that
each ion decays to the wrong ground state
$|{X}\rangle=|{{}^{2}S_{1/2},m_{J}=+\frac{1}{2}}\rangle$. Although the
corresponding $\pi$-polarized photons are blocked by a polarizer with $>98\%$
efficiency, polarization mixing from imperfect alignment Kim  _et al._ (2011)
or drifts of the fiber birefringence make it difficult to passively eliminate
these false positives. However, we can flag this qubit erasure error by
shelving the state $|{X}\rangle$ to
$|{X^{\prime}}\rangle=|{{}^{2}D_{5/2},m_{J}=+\frac{1}{2}}\rangle$ before state
detection. After state detection, we de-shelve $|{X^{\prime}}\rangle$ back to
$|{X}\rangle$ and perform another round of state detection to check for the
error (see Fig. 1c). This allows for the suppression of erasure errors to
below 0.1% Wu  _et al._ (2022) with very little loss in success rate. This
erasure-veto technique will play an increasingly important role in suppressing
errors when single mode fibers susceptible to polarization drifts are used for
long-distance quantum communication Bersin  _et al._ (2024).

Table 1: Measured harmonic motional frequencies $\omega_{qi}$ for the two atomic ions $q=A,B$ along direction $i$ and their commensurability with the photonic excitation rate $\tau^{-1}=165.35$ kHz ($\tau=6048$ ns). The six mode frequencies are set to be nearly integer multiples of the excitation rate, suppressing errors from residual entanglement with motion. Also shown are the Lamb-Dicke parameters $\eta_{qi}$ and $\zeta_{qi}$ with respect to the excitation/emission wavevector difference and the wavevector of emission, respectively. $q$ (ion) |  $i$ (mode) |  $\frac{\omega_{qi}}{2\pi}$ (kHz) | $\frac{\omega_{qi}\tau}{2\pi}$ | $\eta_{qi}$ |  $\zeta_{qi}$00
---|---|---|---|---|---
A | Axial | 991.5 | 5.996 | 0.055 | 0 00
A | Radial 1 | 1157.5 | 7.000 | 0.086 | 0.051
A | Radial 2 | 1488.0 | 8.999 | 0.013 | 0.045
B | Axial | 330.3 | 1.997 | 0.095 | 0 00
B | Radial 1 | 826.7 | 4.999 | 0.066 | 0.0067
B | Radial 2 | 992.0 | 5.999 | 0.073 | 0.077

## Timing and Atomic Recoil Errors

The fidelity of photonically-heralded atom-atom entanglement is sensitive to
time differences in photon detection, owing to a betrayal of “which-path”
information from differences in atomic recoil. After tracing over the motion
of both ions, the average parity fringe contrast degrades to (see Supp. Info.
.5 and Refs. Johnson  _et al._ (2015),Yu  _et al._ (2024))

$\displaystyle C$
$\displaystyle=\prod_{qi}e^{-(2\bar{n}_{qi}+1)\left[\eta_{qi}^{2}(1-\cos\omega_{qi}\tau)+\zeta_{qi}^{2}W\omega_{qi}^{2}\tau_{R}^{2}\right]}$
(4)

Here, $\omega_{qi}$ and $\bar{n}_{qi}$ are the frequency and thermal phonon
occupation number for mode $i$ of ion $q$.

The first decoherence term in Eq. 4 stems from an entanglement between the
qubits and the atomic motion from the separated time of excitation
$\tau=t_{l}-t_{e}$ (seen in the phase of Eq. 3) and is specific to time-bin
encoding schemes. The Lamb-Dicke recoil parameter is $\eta_{qi}=\Delta
k_{i}\sqrt{\hbar/2m\omega_{qi}}$ for ion $q$ with mass $m$ with respect to the
wavevector difference between excitation and emission along $i$. We see that
when $\omega_{qi}\tau$ is an integer multiple of $2\pi$ for all modes, each
ion is excited from the same position in each time bin, and this source of
decoherence vanishes. In the experiment, we ensure this condition by tuning
the mode frequencies to be commensurate and setting the excitation rate
$1/\tau$ to be their greatest common divisor ($\tau=6048$ ns), as summarized
in Table 1. We characterize this effect by scanning the difference in
excitation times $\tau$ about the nominal value, as shown in Fig. 3. We
estimate that the residual fidelity error from the drift in mode frequencies
is less than $0.1\%$.

The second decoherence term in Eq. 4 is more fundamental and stems from
fluctuations in the random detection times of the photons in each time bin
through the random phase $\phi^{*}_{qe}$ given by the finite lifetime of the
emitting atoms. This generates residual entanglement between the qubits and
their motion as above. But in this case, the Lamb-Dicke recoil parameter
${\zeta_{qi}=k_{i}\sqrt{\hbar/2m\omega_{qi}}}$ is with respect to the emission
wavevector only and not $\Delta k_{i}$. This decoherence can be controlled by
narrowing the detection window $\delta t$ characterized by the scaled variance
$0<W<1$ (see Supp. Info. .5), but this also degrades the yield $Y$ and hence
the rate of entanglement. Figure 4 shows the observed fidelity and yield as we
vary $\delta t$ from $2$ ns ($W\approx 0.01$, $Y=0.22$) to $50$ ns ($W\approx
0.95$, $Y=0.998$). The measurements are consistent with the model of Eq. 4,
assuming thermal states of motion near the Doppler laser-cooling limit for all
modes. We observe a $\sim 1\%$ improvement in the fidelity by decreasing the
window from 50 ns to 10 ns ($Y=0.71$), with a residual fidelity error of $\sim
0.2\%$. This decoherence from random photon arrival times is universal to all
photonic encoding schemes for recoiling emitters but has not been previously
observed.

Figure 3: Entanglement fidelity degradation as time-bin period $\tau$ is tuned
away from being commensurate with trapped ion motion (in all three directions
of motion for both ions). As predicted in Eq. 4, the loss of fidelity due to
asynchronous timing depends on the thermal occupation number of each of the
six modes of motion, with the value $\tau=6048$ ns nearly eliminating this
error (see Table 1). Points are measured fidelity values, re-scaled to a
maximum of 1, and the curves are theoretical fidelity limits from the first
term of Eq. 4 at various levels of cooling. The red line corresponds to
Doppler cooling in the experiment (Doppexp) given the geometry of the cooling
beams with respect to the principal axes in the experiment (see Supp. Fig. 7).
The green line is for Doppler cooling with an optimal beam geometry (Doppopt)
and the blue line is for zero-point cooling. Figure 4: Observed fidelity as a
function of the detection time window $\delta t$ between the excitation and
emission of photons from each ion. The shaded region indicates the expected
fidelity including motional recoil from Eq. 4, assuming Doppler cooling for
all modes given the beam geometry (see Supp. Info. Fig. 7 and Table 2). This
band was normalized to match the experimental value at $\delta t=0$ with a
width reflecting the uncertainty in the angles of the ion trap principal axes.
The green line (right axis) shows the measured (points) and theoretical (line)
yield for each value of $\delta t$, with no free fit parameters.

We note that for fixed emitters such as color-centers in solid-state hosts
Bernien  _et al._ (2013), the emitter mass becomes so large that
$\eta_{qi},\zeta_{qi}\rightarrow 0$ and the above recoil-induced decoherence
is negligible. For very weakly-bound emitters such as neutral atoms Ritter
_et al._ (2012); Covey  _et al._ (2023), where $\omega_{qi}\tau,\zeta_{qi}\ll
1$, these effects can be prominent, depending on the level of cooling (see
Supp. Info. .5).

## Summary

Remote entanglement of trapped ions using time-bin encoded photons is the
preferred method for high fidelity or long-distance photonic interconnects
between quantum memory nodes. By stabilizing the 1762 nm laser power and using
hyperfine clock qubits with indefinitely long idle coherence times, stabilized
mode frequencies on the order of MHz, low $\bar{n}<1$ via sub-Doppler cooling,
and improved SPAM, it should be possible to reach remote entanglement
fidelities in excess of 0.999. Furthermore, by increasing and stabilizing the
intensity of the 1762 nm laser to achieve stable Rabi frequencies exceeding a
MHz should enable the operation of this protocol at rates approaching $10^{3}$
per second. Such high-fidelity and high-rate entanglement will be critical for
scaling of a photonically networked quantum computer, as well as quantum
repeaters and other long-distance quantum communication protocols Ramette  _et
al._ (2023). Moreover, by extending the above protocol to any number of time-
bins, this type of photonic interconnect can easily interface with high-
dimensional qubit registers Wang  _et al._ (2021); Jiang Low  _et al._ (2023)
to generate particular entangled qudit states for applications in networking
Islam  _et al._ (2017) and quantum computation Sun  _et al._ (2023).

This work is supported by the DOE Quantum Systems Accelerator (DE-FOA-0002253)
and the NSF STAQ Program (PHY-1818914). J.O. is supported by the NSF Graduate
Research Fellowship (DGE 2139754) and A.K. by the AFOSR National Defense
Science and Engineering Graduate (NDSEG) Fellowship.

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## Supplementary Materials

###  .1 Optical qubit SPAM and coherent rotations

The atomic qubit spans
${\mid\downarrow\rangle=|{{}^{2}S_{1/2},m_{J}=-\frac{1}{2}}\rangle}$ and
${\mid\uparrow\rangle=|{{}^{2}D_{5/2},m_{J}=-\frac{1}{2}}\rangle}$ with an
optical interval at a wavelength of 1762 nm. The linear Zeeman splitting
between the two levels is $0.56$ MHz/G Kurz  _et al._ (2010). A static
magnetic field of approximately 4 G is applied to each chamber and balanced to
under 1 mG to bring the optical qubit frequencies within 200 Hz of each other.

Coherent individual qubit rotations are performed using a 1762 nm Tm-doped,
distributed-feedback fiber laser from NKT Photonics. The laser light is
directed through a fiber amplifier for a total output of 450 mW. By locking to
a high-finesse cavity, the laser frequency is stabilized to a linewidth of
less than 200 Hz. The free-space 1762 nm light is split into two paths and
coupled into individual fiber AOMs for Alice and Bob. The rf waveforms that
drive the two AOMs are generated by a multi-channel arbitrary waveform
generator to ensure a stable relative phase between the pulses. In each
system, we direct about 20 mW of laser power focused down to a 20 $\mu$m waist
and drive a $\pi$ transition from
$\mid\downarrow\rangle=|{{}^{2}S_{1/2},m_{J}=-\frac{1}{2}}\rangle$ to
$\mid\uparrow\rangle=|{{}^{2}D_{5/2},m_{J}=-\frac{1}{2}}\rangle$ in $\sim 4.2$
$\mu$s (Rabi frequency of $\sim 120$ kHz).

State preparation involves optical pumping into $\mid\downarrow\rangle$ with
$\sigma^{-}$-polarized 493 nm light, and 650 nm and 614 nm re-pump beams to
clear the D manifolds (see Fig. 1b). Measurement is performed by ion-
fluorescence-based detection. Since ${}^{2}D_{5/2}$ has a lifetime of
$\tau\approx 30s$, the $\mid\uparrow\rangle$ state is dark when the ion is
exposed to 493 nm and 650 nm light, while the $\mid\downarrow\rangle$ state
fluoresces, emitting photons that are counted by a photomultiplier tube (PMT)
detector. We perform fluorescence detection for a period of 1 ms, with
histograms of fluorescence counts shown in Fig. 5 for Alice. To differentiate
between bright and dark states, we use a threshold of 2.5 counts and 10.5
counts for Alice and Bob, respectively. Employing this method, we achieve a
state preparation and measurement (SPAM) fidelity exceeding $99.5\%$ for both
Alice and Bob with the residual SPAM error dominated by fluctuations in the
1762 nm operations. None of the data presented here is corrected for SPAM
errors.

Figure 5: Histogram of ion-fluorescence counts in Alice for qubit state
prepared in $\mid\downarrow\rangle$ (dark) and $\mid\uparrow\rangle$ (bright).
The fluorescence integration time is 1 ms per measurement, averaged over
20,000 repetitions. Setting a threshold of 2.5 counts allows the two qubit
states to be distinguished in a single shot with measurement fidelity
$>99.5\%$.

###  .2 Measurement of qubit coherence time ($T_{2}^{*}$)

The coherence of the desired entangled state of Eq. 3 is insensitive to common
mode qubit decoherence. In order to determine the effect of differential qubit
decoherence on the observed state fidelity (presumably from differential
magnetic field noise), we perform a Ramsey experiment on qubit A as observed
from the frame of qubit B Chwalla  _et al._ (2007); Olmschenk  _et al._
(2007). After preparing both ions in the $\mid\downarrow\rangle$ state, we use
the 1762 nm laser to apply a $\pi/2$ pulse to both qubits, wait for some delay
time $\Delta t$, and then apply another $\pi/2$ pulse with a scrambled phase.
By varying the relative phase difference of the second pulse between the two
ions, we measure and fit the parity of the two qubit states to extract the
contrast as shown in Fig. 6a. Since the two qubits are not initially
entangled, the maximum parity expected without decoherence is 0.5. After
repeating the measurement for different $\Delta t$, we fit the parity
amplitudes to measure the differential decoherence time $T_{2}^{*}$. From the
data shown in Fig. 6b, we fit to $\exp[-(t/T_{2}^{*})^{2}]$ and obtain
$T_{2}^{*}=2.10(4)$ ms. This is consistent with an rms differential magnetic
field noise of $\sim 1$ mG over the measurement bandwidth.

Figure 6: Measurement of differential qubit coherence time between the qubits
in Alice and Bob. a Measured oscillations of the two qubit parity with
different relative phase for delay times of 0.5 ms and 1.6 ms. (b) Parity
contrast as we vary the delay time between two $\pi/2$ pulses. Because the two
ions are not entangled, a maximum parity contrast of 0.5 is expected. The fit
to a Gaussian envelope gives a $1/e$ coherence time of $T_{2}^{*}=2.10(4)$ ms.

###  .3 Entanglement rate calculation

The maximum success probability of ion-ion entanglement is
$P_{E}=\frac{1}{2}p_{A}p_{B}=2.3\times 10^{-5}$. The individual collection and
detection probabilities of each node are
$p_{q}=p_{\mathrm{exc}}\beta\epsilon_{F}T\epsilon_{D}(d\Omega_{q}/4\pi)$. Here
$p_{exc}=0.8$ is the probability of excitation, $\beta=0.49$ is the effective
branching ratio into the correct subspace, $\epsilon_{F}\approx 19\%$ is the
fiber coupling efficiency (including polarization rejection by the fiber Kim
_et al._ (2011)), $T\approx 90\%$ is the transmission through optical
elements, $\epsilon_{D}=0.71$ is the detector efficiency of the avalanche
photodiodes (APD), and $d\Omega_{q}$ is the solid angle of light collection
from chamber $q$. The objective lenses have numerical apertures (NA) of 0.6 in
Alice and 0.8 in Bob Carter  _et al._ (2024)), so $d\Omega_{A}/4\pi=10\%$ and
$d\Omega_{B}/4\pi=20\%$. For a detection time window of $\delta t=10$ ns
(yield $Y=0.71$), the mean entanglement rate is ${P_{E}YRf=0.35\,s^{-1}}$ at a
repetition rate of $R\approx 70$ kHz and a duty cycle of $f=30\%$ due to
laser-cooling interruptions.

###  .4 Photon generation with ultrafast 493 nm laser

Single photon generation relies on excitation pulses of duration
$t_{p}\ll\tau_{R}$, where $\tau_{R}=7.855$ ns is the radiative lifetime of the
excited $|{e}\rangle$ state. To generate fast, solitary pulses at 493 nm, we
use a mode-locked Ti:Sapphire laser (Coherent Mira 900P) at 986 nm, producing
$t_{p}\approx 3$ ps pulses at a repetition rate of $f_{\mathrm{rep}}=76.226$
MHz O’Reilly  _et al._ (2024). The pulses are sent through an electro-optic
pulse picker that selectively transmits a single pulse when triggered. These
pulses are then frequency-doubled with a MgO-doped, periodically-poled lithium
niobate crystal to 493 nm, quadratically increasing the extinction ratio of
neighboring pulses to a level below $10^{-4}$. After passing through an AOM
that further extinguishes subsequent pulses, the single pulse is split in two
paths and fiber-coupled into polarization-maintaining optical fibers directed
to Alice and Bob.

The probability of two photons being emitted from either atom after a single
pulse is estimated to be
$P_{\mathrm{exc}}^{2}\beta^{2}(t_{p}/8\tau_{R})<10^{-5}$. Here
$P_{\mathrm{exc}}\sim 0.8$ is the excitation probability over the duration of
the pulse and $\beta=0.49$ is the successful branching ratio back to the
$\mid\downarrow\rangle$ state.

We connect the clock output of the Ti:Sapphire laser to the control system to
synchronize the start of the experimental procedure with the laser repetition
rate. This synchronization leads to a more precise time-stamp of the arrival
photons, removing a potential unsynchronized $1/f_{\mathrm{rep}}\approx 13$ ns
timing jitter.

###  .5 Fidelity limits from atomic recoil over time

Multiple excitation times in the time-bin protocol or even the distribution of
emission times within a single time-bin can lead to entanglement between the
photon qubit and motion of the each atom from atomic recoil. We model the
resulting decoherence on the atomic qubits by calculating the reduced density
matrix of each ion and its emitted photon and tracing over the motion Yu  _et
al._ (2024). We first consider the temporal separation of the time bins and
neglect the random distribution of emission times by taking $\phi^{*}_{q}=0$.

The initial state of ion $q\in(A,B)$, including its time-bin photon emission
modes and motion is

$\rho_{q}=\frac{(\mid\uparrow\rangle_{q}+\mid\downarrow\rangle_{q})(\langle{\uparrow}|_{q}+\langle{\downarrow}|_{q})}{2}\otimes|{0_{e}0_{l}}\rangle_{q}\langle{0_{e}0_{l}}|_{q}\otimes
M_{q},$ (5)

where $|N_{e}N_{l}\rangle_{q}$ denotes $N_{e}$ ($N_{l}$) photons emitted in
the early (late) time bin. The initial motional density matrix $M_{q}$ is
expressed as a thermal state in the basis of coherent states Glauber (1963):

$M_{q}=\prod_{i}\frac{1}{\pi\bar{n}_{qi}}\int\mathrm{d}^{2}\alpha_{i}|\alpha_{i}\rangle\langle\alpha_{i}|e^{-|\alpha_{i}|^{2}/\bar{n}_{qi}},$
(6)

where $\bar{n}_{i}$ is the average thermal motional quantum number in
direction $i$.

The final state after the early ($e$) excitation, qubit swap, and free
evolution in the ion trap for time $\tau$ between photon emissions is

$\rho_{q}^{\prime}=(L_{q}e^{-iH_{q}\tau}X_{q}E_{q})\rho_{q}(L_{q}e^{-iH_{q}\tau}X_{q}E_{q})^{\dagger}.$
(7)

Here, the Hamiltonian $H_{q}=\sum_{i}\omega_{qi}(n_{qi}+\frac{1}{2})$
describes the free evolution of the atomic motion in the trap with harmonic
frequencies $\omega_{qi}$ and phonon occupation numbers $n_{qi}$ in all three
dimensions. The Pauli spin-flip operator $X_{q}$ describes the qubit swap
between emission attempts. The production of a photon from ion $q$ in the
early or late time-bin is given by the evolution operators

$\displaystyle E_{q}$
$\displaystyle=\mid\downarrow\rangle_{q}\langle{\downarrow}|_{q}\left(\sqrt{p_{q}}e^{i\mathbf{\Delta
k}\cdot\mathbf{r}_{q}}a^{\dagger}_{q}+\sqrt{1-p_{q}}\right)+\mid\uparrow\rangle_{q}\langle{\uparrow}|_{q}$
(8) $\displaystyle L_{q}$
$\displaystyle=\mid\downarrow\rangle_{q}\langle{\downarrow}|_{q}\left(\sqrt{p_{q}}e^{i\mathbf{\Delta
k}\cdot\mathbf{r}_{q}}b^{\dagger}_{q}+\sqrt{1-p_{q}}\right)+\mid\uparrow\rangle_{q}\langle{\uparrow}|_{q},$
(9)

where $p_{q}$ is the success probability of photon collection, $\mathbf{\Delta
k}$ is the difference in the wavevector between the excitation and emitted
photons, $\mathbf{r}_{q}$ is the (time-independent) atomic position operator
of ion $q$, and $a^{\dagger}_{q}$ ($b^{\dagger}_{q}$) is the creation operator
for ion $q$ producing a photon in the early (late) time-bin mode. The phase
factor in Eqs. 8-9 is recognized as a momentum kick $e^{i\mathbf{\Delta
k}\cdot\mathbf{r}_{q}}=\prod_{i}\mathcal{D}_{i}(i\eta_{qi})$, where
$\mathcal{D}_{i}$ is the coherent displacement operator in the $i$th dimension
of phase space Glauber (1963) and ${\eta_{qi}=\Delta
k_{i}\sqrt{\hbar/2m\omega_{qi}}}$ is the Lamb-Dicke parameter of ion $q$
associated with $\Delta k_{i}$.

By tracing over the motion we find that with probability $p_{q}$, the reduced
density matrix for qubit $q$ and its photonic channel becomes the mixed state

$\displaystyle\mathrm{tr}_{M_{q}}({\rho^{\prime}_{q}})=\left[\begin{array}[]{cc}1/2&e^{-i\phi_{q0}}C^{\prime}_{q}/2\\\
e^{i\phi_{q0}}C^{\prime}_{q}/2&1/2\end{array}\right]$ (12)

written in the basis of the two states
$\mid\downarrow\rangle_{q}|{0_{e}1_{l}}\rangle_{q}$ and
$\mid\uparrow\rangle_{q}|{1_{e}0_{l}}\rangle_{q}$, where the coherence
amplitude is

$\displaystyle C^{\prime}$
$\displaystyle=\prod_{q}C^{\prime}_{q}=\prod_{qi}e^{-\eta_{qi}^{2}(2\bar{n}_{qi}+1)(1-\cos\omega_{qi}\tau)}$
(13)

and its zero-point phase offset is
$\phi_{q0}=\eta_{qi}^{2}\sin\omega_{qi}\tau$, which is very small for
$\eta_{qi}\ll 1$.

In addition to the reduction in coherence from the time-bin separation $\tau$
of excitation pulses, there is another reduction from the random times of
photon detection. This stems from the finite lifetime $\tau_{R}$ of each
atomic emitter resulting in the random phase $\phi^{*}_{qe}$ that appears in
Eq. 3. Similar to the above treatment, we find that the coherence amplitude is
reduced further by the factor

$\displaystyle
C^{\prime\prime}_{q}=\prod_{i}e^{-\zeta_{qi}^{2}(2\bar{n}_{qi}+1)[1-\cos\omega_{qi}(\tau^{*}-\tau)]},$
(14)

with an additional negligible phase offset Yu  _et al._ (2024). Here,
$\tau^{*}$ is the measured difference in detection time of the two photons in
the early and late time bins for each event. (This assumes balanced optical
path lengths after the BS, but any imbalance can be factored into the data
analysis with no degradation.) This is similar to the form of Eq. 13, except
the relevant Lamb-Dicke parameter
${\zeta_{qi}=k_{i}\sqrt{\hbar/2m\omega_{qi}}}$ is associated with only the
recoil from emission. The random variable $\tau^{*}$ follows a double-sided
exponential (Laplace) distribution with mean $0$ and variance $2\tau_{R}^{2}$.
We can reduce the impact of this variance by symmetrically truncating this
distribution by post-selecting events with photon detection times within
$\pm\delta t$ of the nominal value, or $|\tau^{*}-\tau|<\delta t$. This
results in $\tau^{*}-\tau$ following a truncated Laplace distribution with
mean $0$ and variance $2\tau_{R}^{2}W$, where the variance parameter

$\displaystyle W=\frac{1-(1+w+w^{2}/2)e^{-w}}{1-e^{-w}}$ (15)

smoothly increases from 0 to 1 as the relative window size $w\equiv\delta
t/\tau_{R}$ increases from 0 to $\infty$. The yield of accepted events is
$Y=1-e^{-w}$. For $\omega_{qi}\tau_{R}\ll 1$, the above average results in

$\displaystyle C^{\prime\prime}$
$\displaystyle\approx\prod_{qi}e^{-\zeta_{qi}^{2}(2\bar{n}_{qi}+1)W\omega_{qi}^{2}\tau_{R}^{2}}.$
(16)

Furthermore, if also $\zeta_{qi}^{2}(2\bar{n}_{qi}+1)\omega_{qi}\tau_{R}\ll
1$, then

$\displaystyle C^{\prime\prime}$ $\displaystyle\approx
1-\sum_{qi}\zeta_{qi}^{2}(2\bar{n}_{qi}+1)W\omega_{qi}^{2}\tau_{R}^{2}.$ (17)

The net entanglement contrast is $C=C^{\prime}C^{\prime\prime}$, as written in
Eq. 4 in the main body.

###  .6 Beam geometry and Lamb-Dicke Parameters for optimal cooling

The geometry of the input beams and emitted photons for the traps is shown in
the schematic, Fig. 7. The excitation and cooling beams at 493 nm are
delivered colinearly at an angle of $\beta_{q}=45^{\circ}$ degrees to the
axial ($z$) axis of both traps. The emission direction is perpendicular to the
$z$-axis and at an angle $\alpha_{q}$ to each trap’s principal $x$-axis.

Figure 7: The orientation of trap principal axes $x$, $y$ and $z$ with respect
to pulsed excitation, Doppler cooling and single photon emission directions
(see Table 2). The excitation and Doppler cooling wavevector $k^{\prime}$ is
perpendicular to the emission wavevector $k$.

We are interested in the angles $\theta_{qi}$ between the excitation/cooling
wavevectors and the principal axes $i$ as well as the angles $\psi_{qi}$
between the emission wavevector and the principal axes:

$\displaystyle\cos\theta_{qx}$
$\displaystyle=-\sin{\beta_{q}}\sin{\alpha_{q}}$ $\displaystyle\cos\psi_{qx}$
$\displaystyle=\cos\alpha_{q}$ (18) $\displaystyle\cos\theta_{qy}$
$\displaystyle=\sin{\beta_{q}}\cos{\alpha_{q}}$ $\displaystyle\cos\psi_{qy}$
$\displaystyle=\sin\alpha_{q}$ (19) $\displaystyle\cos\theta_{qz}$
$\displaystyle=-\cos\beta_{q}$ $\displaystyle\cos\psi_{qz}$ $\displaystyle=0.$
(20)

Hence, the Lamb-Dicke recoil parameters are given by

$\displaystyle\eta_{qi}$ $\displaystyle=\sqrt{\frac{\hbar
k^{2}}{2m\omega_{qi}}}|\cos\psi_{qi}-\cos\theta_{qi}|$ (21)
$\displaystyle\zeta_{qi}$ $\displaystyle=\sqrt{\frac{\hbar
k^{2}}{2m\omega_{qi}}}|\cos\psi_{qi}|$ (22)

and the expected Doppler cooling limit isWineland and Itano (1979)

$\displaystyle\bar{n}^{\mathrm{D}}_{qi}=\frac{\gamma}{4\omega_{qi}}\left[\frac{\Delta}{\gamma}+\frac{\gamma(1+s)}{4\Delta}\right]\left(1+\frac{1}{3\cos^{2}\theta_{qi}}\right).$
(23)

Here $\Delta\sim\gamma/2$ is the red detuning of the Doppler cooling beam from
resonance and $s=I/I_{\mathrm{sat}}\sim 2$ is the saturation parameter with
$I$ the laser intensity and $I_{\mathrm{sat}}$ the saturation intensity. We
note that one particular principal axis in Bob (B) is nearly orthogonal to the
cooling beam ($\theta_{By}\approx 87^{\circ}$). This single direction has a
poor Doppler cooling limit and hence is the dominant source of error in Eq. 4.
We estimate an uncertainty in all geometrical angles to be $<3^{\circ}$.

Table 2: Summary of beam angles, Lamb-Dicke recoil parameters, and Doppler cooling limits $\bar{n}^{\mathrm{D}}_{qi}$ for each motional normal mode $i$ of ion $q$. $q$ | $i$ |  $\frac{\omega_{qi}}{2\pi}$ (kHz) | $\alpha_{q}$ | $\beta_{q}$ | $\theta_{qi}$ | $\psi_{qi}$ | $\eta_{qi}$ | $\zeta_{qi}$ | $\bar{n}^{\mathrm{D}}_{qi}$
---|---|---|---|---|---|---|---|---|---
A | $z$ | 991.5 | - | 45∘ | 135∘ | 90∘ | 0.055 | 000 | 13
A | $x$ | 1157.5 | 45∘ | 45∘ | 120∘ | 45∘ | 0.086 | 0.051 | 15
A | $y$ | 1488.0 | 45∘ | 45∘ | 60∘ | 45∘ | 0.013 | 0.045 | 12
B | $z$ | 330.3 | - | 45∘ | 135∘ | 90∘ | 0.095 | 000 | 38
B | $x$ | 826.7 | 85.5∘ | 45∘ | 134.8∘ | 85.5∘ | 0.066 | 0.0067 | 15
B | $y$ | 992.0 | 85.5∘ | 45∘ | 86.8∘ | 4.5∘ | 0.073 | 0.077 | 826

###  .7 Error budget

The largest source of error in the entangled state fidelity is intensity
fluctuations in the 1762 nm laser that drives coherent qubit rotations,
contributing to SPAM and the swap of qubit states in the protocol. This and
several additional sources of errors and noise are summarized in Table 3. The
temporal wavefunctions of the photons are matched to within 30 ps at the BS by
equalising the path length between the excitation laser and the BS, leading to
a small error of $0.004$. We expect a fidelity error of $0.2\%$ from residual
entanglement with ion motion due to the recoil from the spread of photon
detection times within each time-bin, at a detection window of $\delta t=10$
ns (see Fig. 4). Dark counts on the photon detectors and background scattered
light from the excitation pulse are expected to contribute $<$0.2%. Imbalance
in the fiber BS (measured to be less than 2$\%$ from the nominal 50:50) and
imperfect BS mode matching are expected to limit fidelity errors to below
$0.1\%$. We observe a differential qubit coherence time of $2.1$ ms, likely
due to differential magnetic field noise between the two qubits. This is
expected to reduce the fidelity by $<10^{-4}$ during the $\sim 8$ $\mu$s dwell
time between the early photon detection and the analysis $\pi/2$-pulse (see
Supp. Info. .2). Residual rf micromotion of trapped ions Berkeland  _et al._
(1998) can result in a fluctuating frequency of the emitted photons, causing a
phase error in the final entangled state and a reduction in the fidelity. We
measure a micromotion-induced Doppler shift of under 200 kHz through a photon
autocorrelation procedure Berkeland  _et al._ (1998), and expect this to
contribute to a fidelity error of less than $10^{-4}$.

Table 3: Sources of error affecting fidelity and their magnitudes. | Fidelity
---|---
Source of error | Error
SPAM / 1762 intensity fluctuations | 0.0100
Photon wavepacket overlap | 0.004
Atom recoil, $\delta t=10$ ns | 0.002
Background counts | $<0.002$
Atom recoil, $\omega_{qi}$ fluctuation | $<0.001$
Beamsplitter imperfection | $<0.001$
Residual erasure errors | $<0.001$
Micromotion | $<0.0001$
Coherence time | $<0.0001$
TOTAL |  0.0200