# In-beam performance of a Resistive Plate Chamber operated with eco-friendly
gas mixtures

L. Quaglia on behalf of the ALICE and RPC ECOgas@GIF++ collaborations. The RPC
ECOgas@GIF++ collaboration: M. Abbrescia G. Aielli R. Aly M. C. Arena M.
Barroso L. Benussi S. Bianco D. Boscherini F. Bordon A. Bruni S. Buontempo M.
Busato P. Camarri R. Cardarelli L. Congedo D. De Jesus Damiao M. De Serio A.
Di Ciaccio L. Di Stante P. Dupieux J. Eysermans A. Ferretti G. Galati M.
Gagliardi R. Guida G. Iaselli B. Joly S.A. Juks K.S. Lee B. Liberti D. Lucero
Ramirez B. Mandelli S.P. Manen L. Massa A. Pastore E. Pastori D. Piccolo L.
Pizzimento A. Polini G. Proto G. Pugliese D. Ramos G. Rigoletti A. Rocchi M.
Romano A. Samalan P. Salvini R. Santonico G. Saviano M. Sessa S. Simone L.
Terlizzi M. Tytgat E. Vercellin M. Verzeroli N. Zaganidis

###### Abstract

ALICE (A Large Ion Collider Experiment) studies the Quark-Gluon Plasma (QGP):
a deconfined state of nuclear matter obtained in ultra-relativistic heavy-ion
collisions. One of the key probes for QGP characterization is the study of
quarkonia and open heavy flavour production, of which ALICE exploits the
muonic decay. In particular, a set of Resistive Plate Chambers (RPCs), placed
in the forward rapidity region of the ALICE detector, is used for muon
identification purposes.

The correct operation of these detectors is ensured by the choice of the
proper gas mixture. Currently they are operated with a mixture of C2H2F4,
i-C4H10 and SF6 but, starting from 2017, new EU regulations have enforced a
progressive phase-out of C2H2F4 because of its large Global Warming Potential
(GWP), which is making it difficult and costly to purchase. Moreover, CERN
asked LHC experiments to reduce greenhouse gases emissions, to which RPC
operation contributes significantly.

A possible candidate for C2H2F4 replacement is the C3H2F4 (diluted with other
gases, such as CO2), which has been extensively tested using cosmic muons.
Promising gas mixtures have been devised; the next crucial steps are the
detailed in-beam characterization of such mixtures as well as the study of
their performance under increasing irradiation levels.

This contribution will describe the methodology and results of beam tests
carried out at the CERN Gamma Irradiation Facility (equipped with a high
activity 137Cs source and muon beam) with an ALICE-like RPC prototype,
operated with several mixtures with varying proportions of CO2, C3H2F4,
i-C4H10 and SF6 . Absorbed currents, efficiencies, prompt charges, cluster
sizes, time resolutions and rate capabilities will be presented, both from
digitized (for detailed shape and charge analysis) and discriminated (using
the same front-end electronics as employed in ALICE) signals.

###### keywords:

Resistive Plate Chambers, Beam test, Eco-friendly gas mixtures

††journal: Nuclear Inst. and Methods in Physics Research, A

[1]organization=INFN Sezione di Torino, addressline=Via P. Giuria 1,
city=Torino, postcode=10125, state=Italy, country=

[2]organization=Politecnico di Bari, Dipartimento Interateneo di Fisica,
addressline=via Amendola 173, city=Bari, postcode=70125, state=Italy, country=

[3]organization=INFN Sezione di Bari, addressline=Via E. Orabona 4, city=Bari,
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[4]organization=INFN - Laboratori Nazionali di Frascati, addressline=Via
Enrico Fermi 54, city=Frascati (Roma), postcode=00044, state=Italy, country=

[5]organization=Sapienza Università di Roma, Dipartimento di Ingegneria
Chimica Materiali Ambiente, addressline=Piazzale Aldo Moro 5, city=Roma,
postcode=00185, state=Italy, country=

[6]organization=Università degli studi di Bari, Dipartimento Interateneo di
Fisica, addressline=Via Amendola 173, city=Bari, postcode=70125, state=Italy,
country=

[7]organization=INFN Sezione di Bologna, addressline=Via C. Berti Pichat 4/2,
city=Bologna, postcode=40127, state=Italy, country=

[8]organization=Helwan University, addressline=, city=Helwan, Cairo
Governorate, postcode=4037120, state=Egypt, country=

[9]organization=INFN Sezione di Pavia, addressline=Via A. Bassi 6, city=Pavia,
postcode=27100, state=Italy, country=

[10]organization=Ghent University, Dept. of Physics and Astronomy,
addressline=Proeftuinstraat 86, city=Ghent, postcode=B-9000, state=Belgium,
country=

[11]organization=Universidade do Estado do Rio de Janeiro, addressline=R. São
Francisco Xavier, 524, city=Maracanã, Rio de Janeiro - RJ, postcode=20550-013,
state=Brazil, country=

[12]organization=Università degli studi di Torino, Dipartimento di Fisica,
addressline=Via P. Giuria 1, city=Torino, postcode=10125, state=Italy,
country=

[13]organization=INFN Sezione di Roma Tor Vergata, addressline=Via della
Ricerca Scientifica 1, city=Roma, postcode=00133, state=Italy, country=

[14]organization=Clermont Université, Université Blaise Pascal, CNRS/IN2P3,
Laboratoire de Physique Corpusculaire, addressline=BP 10448, city=Clermont-
Ferrand, postcode=F-63000, state=France, country=

[15]organization=Universitè Claude Bernard Lyon I, addressline=43 Bd du 11
Novembre 1918, city=Villeurbanne, postcode=69100, state=France, country=

[16]organization=CERN, addressline=Espl. des Particules 1, city=Meyrin,
postcode=1211, state=Switzerland, country=

[17]organization=Università degli studi di Pavia, addressline=Corso Strada
Nuova 65, city=Pavia, postcode=27100, state=Italy, country=

[18]organization=INFN Sezione di Napoli, addressline=Complesso universitario
di Monte S. Angelo ed. 6 Via Cintia, city=Napoli, postcode=80126, state=Italy,
country=

[19]organization=Massachusetts Institute of Technology, addressline=77
Massachusetts Ave, city=Cambridge, MA, postcode=02139, state=USA, country=

[20]organization=Vrije Universiteit Brussel (VUB-ELEM), Dept. of Physics,
addressline=Pleinlaan 2, city=Brussels, postcode=1050, state=Belgium, country=

[21]organization=Università degli studi di Roma Tor Vergata, Dipartimento di
Fisica, addressline=Via della Ricerca Scientifica 1, city=Roma,
postcode=00133, state=Italy, country=

[22]organization=Universidad Iberoamericana, Dept. de Fisica y Matematicas,
addressline=, city=Mexico City, postcode=01210, state=Mexico, country=

[23]organization=Korea University, addressline=145 Anam-ro, city=Seongbuk-gu,
Seoul, postcode=, state=Korea, country=

[24]organization=Université Paris-Saclay, addressline=3 rue Joliot Curie,
Bâtiment Breguet, city=Gif-sur-Yvette, postcode=91190, state=France, country=

##  1 Introduction

A Large Ion Collider Experiment (ALICE) [1] is one of the four main LHC
experiments. Its main goal is the study of the Quark Gluon plasma (QGP) in
heavy-ion collisions. One of the exploited signatures for QGP studies in ALICE
is the suppression of quarkonia (bound states of heavy quarks and
corresponding anti-quark) production [2]. ALICE studies their dimuon decays
using its forward muon spectrometer [3] and the Resistive Plate Chambers
(RPCs), main subject of this contribution, are employed to identify the muons
among all the particles crossing the spectrometer since they are located
downstream of a hadron filter which absorbs any hadron traversing the muon
spectrometer (hence the name Muon IDentifier, MID)

RPCs are gaseous detectors with planar geometry and resistive electrodes. In
the case of the ALICE MID, they have a 2 mm single gas gap and 2 mm thick
bakelite electrodes ($\rho\approx$109-1010 $\Omega\cdot$cm); they are readout
by two perpendicular copper strips planes and they employ the FEERIC front-end
electronics [4]. The RPCs are operated in maxi-avalanche mode, with a mixture
of 89.7% C2H2F4, 10% i-C4H10 and 0.3% SF6 and, although this mixture satisfies
all the performance requirements, it contains a high fraction of fluorinated
greenhouse gases (F-gases/GHGs) (C2H2F4 and SF6).

Starting from 2017, new EU regulations [5] have imposed a progressive phase-
down in the production and usage of these gases, leading to an increase of
cost and reduction in availability. For this reason, CERN has adopted a policy
of F-gases reduction. Since RPCs represent a significant fraction of the total
GHG-gases emission of the LHC experiments, it is of the utmost importance to
search for more eco-friendly RPC gas mixtures. The first efforts are
concentrated on the replacement of C2H2F4 using a gas known as
tetrafluoropropene (C3H2F4) diluted with other gases to lower the detector
working voltage [6, 7, 8, 9]. Promising gas mixtures have been identified by
different research groups and their complete characterization in controlled
data-taking environments (such as beam tests), as well as the study of the
detectors long-term behavior when operated with eco-friendly gas mixtures is
now needed.

To this aim, the RPC ECOgas@GIF++ collaboration (between ALICE, ATLAS, CMS,
SHiP/LHCb and the CERN EP-DT group) was created to join forces among RPC
experts of the different LHC experiments to share knowledge and manpower. Each
group provided a detector prototype to be installed in a common support and
the results shown hereafter have been obtained with the detector provided by
the ALICE RPC group.

This contribution is divided as follows: Section 2 contains a description of
the experimental setup and the methodology used in the data-taking, Section 3
reports the main results obtained from a beam test campaign where mixtures
with different HFO/CO2 ratios have been beam-tested. Lastly, Section 4 is
dedicated to the conclusion and to possible outlooks for the future of this
work.

##  2 Experimental setup

The experimental setup is located at the CERN Gamma Irradiation Facility
(GIF++) [10], installed on the H4 secondary beam line of the CERN SPS. This
facility is equipped with a high activity ($\sim$12.5 TBq) 137Cs source and,
in dedicated beam time periods, it is exposed to a high energy ($\sim$150 GeV)
muon beam. The source can either be used to produce a high background
radiation on the detector, simulating long operation periods in shorter time-
spans (years in $\sim$months) or it can be combined with the muon beam to
study the detector performance under varying background levels.

Figure 1: Sketch of the experimental setup installed in GIF++ (a detailed
description of each component is reported in the text)

The $\gamma$ radiation can be modulated by means of lead filters and a total
of 27 combinations can be obtained. The attenuation is measured with a pure
number, which ranges from 1 to 46000 (maximum and minimum irradiation
respectively). Moreover, the source can be fully shielded, allowing one to
access the irradiation zone and to study the detector behavior without
background radiation.

The ALICE detector is 50x50 cm2 ALICE-like RPC prototype (made with the same
construction process as well as with the same features as a real-life MID
RPC). It is equipped with two perpendicular planes of 16 strips each (with a
pitch of $\approx$3 cm). Figure 1 reports a sketch of the experimental setup.
Its main components are: the gas mixing and distribution system (4 Mass Flow
Controllers to regulate the flow of each gas mixture component and a
humidifier to add water vapour to the mixture, up to 40% relative humidity),
the high voltage system (a CAEN SY1527
mainframe111https://www.caen.it/subfamilies/mainframes/ and
A1526P222https://www.caen.it/products/a1526/ high voltage board), the readout
system, which can either be VME TDCs333V1190 with 128 input channels and time
resolution of 100 ps coupled with the ALICE MID front-end discriminators
(FEERIC) [4] or a CAEN digitizer444DT5742, 16 input channels, 12 bit
resolution and 1, 2.5 and 5 Gs/s sampling rate directly connected to the
readout strips. Lastly, the DAQ machine supervising the whole data-taking
procedure (as well as the continuous logging of relevant environmental
parameters) is also shown in the figure.

A total of seven eco-friendly candidate mixtures has been tested, together
with the mixture currently employed by CMS for reference. Table 1 reports the
composition of the different beam-tested gas mixtures (MIX0-6 for the eco-
friendly and STD for the reference one).

Name | C2H2F4 % | HFO % | CO2 % | i-C4H10 % | SF6 %
---|---|---|---|---|---
STD | 95.2 | 0 | 0 | 4.5 | 0.3
MIX0 | 0 | 0 | 95 | 4 | 1
MIX1 | 0 | 10 | 85 | 4 | 1
MIX2 | 0 | 20 | 75 | 4 | 1
MIX3 | 0 | 25 | 69 | 5 | 1
MIX4 | 0 | 30 | 65 | 4 | 1
MIX5 | 0 | 35 | 60 | 4 | 1
MIX6 | 0 | 40 | 55 | 4 | 1
Table 1: Composition of the gas mixtures used in the tests described

##  3 Results

Section 3.1 describes the RPC response to the muon beam in the absence of
$\gamma$ background (source-off) while Section 3.2 reports the results for
varying levels of $\gamma$ background (source-on)

###  3.1 Source-off results

The first parameter that has been studied in the beam test is the RPC
efficiency without any gamma background from the 137Cs source. This value has
been computed using both the digitizer and the FEERIC front-end electronics.
Figure 2 shows the efficiency curves obtained when the digitizer was used. In
this case, the threshold was set on the signal amplitude. A time interval
where no muon signal is expected was identified as the ”noise window” and the
RMS of the signal amplitude was calculated in this interval. Then, if the
signal amplitude, in a time region where a muon-induced signal is expected, is
above five times the RMS calculated earlier(which usually translates to
$\approx$1.5 mV), the RPC is considered efficient. Figure 3 reports the
efficiency trend when the FEERIC front-end electronics are used in combination
with the VME TDCs (in this case only the STD and two of the seven mentioned
eco-friendly mixtures have been tested), the threshold was set to $\approx$150
fC before amplification. In general, the efficiency is calculated as the ratio
between the number of signals above the thresholds and the total number of
muon triggers.

Figure 2: Efficiency as a function of the effective high voltage without gamma
background. Results obtained when the readout strips are coupled directly to
the digitizer Figure 3: Efficiency as a function of the effective high voltage
without gamma background. Results obtained when the readout strips are
connected to the FEERIC discriminators

Note that the efficiency is shown as a function of the effective high voltage
(HVeff), i.e. the voltage corrected for temperature and pressure variations,
to keep the gain constant, according to the formula reported in [11].

In both cases, the efficiency curves have been interpolated using the logistic
function, reported in Equation 1

$\epsilon(HV)=\frac{\epsilon_{max}}{1+e^{-\lambda}(HV-HV_{50})}$ (1)

where $\epsilon_{max}$ represents the maximum asymptotic efficiency reached by
the detector, $\lambda$ is related to the steepness of the curve and HV50 is
the voltage where the efficiency reaches 50% of its maximum. These parameters
are used to extract the working point (WP, i.e. the voltage which grants full
efficiency) as WP=$\frac{log(19)}{\lambda}$+HV50 \+ 150 V, according to [11].

By looking at both Figures 2 and 3, one sees that if the HFO concentration
increases, so does the WP ($\approx$1 kV for every 10% HFO added to the
mixture). Moreover, the plateau efficiency reaches lower values if the HFO
concentration is below 20%. This could probably be explained by the lower
number of primary electron-ion pairs produced by a MIP crossing CO2 with
respect to crossing C2H2F4 [12].

Another useful quantity, which can be computed when the RPC analog signals are
directly sent to a digitizer, and the full waveform of each hit is saved, is
the signal prompt charge. This quantity is the time integral of the waveform
and it has been computed following the procedure described in [13]. Figure 4
shows the distribution of this quantity at the HVeff closest to the WP
estimated earlier.

Figure 4: Prompt charge distributions at the HVeff closest to the estimated WP
for selected gas mixtures

In general, two distinct populations (separated at the $\approx$20 pC) can be
observed; the one below 20 pC represents the avalanche signals (with smaller
charge content) while the one above refers to the, so-called, streamers
(larger signals due to avalanche degeneration). The STD gas mixture (in black)
shows a single, well-defined peak and almost no streamers. For what concerns
the eco-friendly mixtures, one sees that for lower HFO concentrations the
avalanche peak is shifted towards higher values and the amount of streamers is
higher (this translates to an increase of the absorbed current of
$\approx$1.6/1.7 times with respect to the STD gas mixture). If the HFO
concentration is increased (going from MIX0 to MIX6) the avalanche peak moves
towards smaller values and the number of streamers also decreases, this is due
to a quenching effect of more HFO in the mixture.

###  3.2 Source-on results

The study of the RPC response to the muon beam, when exposed to a high
intensity photon flux from the 137Cs source, is useful to simulate real-life
operating conditions at the LHC. The expected background rate for the MID
RPCs, during LHC Run 3 and 4, is $\approx$100 Hz/cm2 [14], for the most
exposed RPC during the Pb-Pb data-taking periods.

Figure 5 shows the efficiency (measured using the digitizer), as well as the
streamer contamination (calculated as the ratio between the number of signals
with a charge above 20 pC and the total number of signals), as a function of
HVeff for the MIX2 mixture, at increasing levels of $\gamma$ background.

Figure 5: Efficiency and large signal contamination as a function of the
effective high voltage for increasing $\gamma$ background rates for the MIX2
mixture

Three main effects can be pointed out: if the $\gamma$ background increases,
the efficiency curves tend to shift towards higher voltages, the plateau
efficiency decreases and the streamer contamination is reduced. A possible
explanation for this effect is that, when the $\gamma$ background increases,
so does the current absorbed by the RPC. This current flows through all the
detector elements, including the resistive electrodes. When it does so, it
leads to an Ohmic voltage drop across them, leading to a decrease of the high
voltage effectively applied to the gas, hence to a lower gain. All the three
effects highlighted earlier seem to point to a gain decrease.

Since the current-induced voltage drop is Ohmic, one can calculate it as the
product between the resistance of the electrodes and the flowing current. The
latter is measured during data-taking while the former can be estimated using
the Ar method [15]. Using those quantities one can calculate the actual HV
applied to the gas as: HVgas = HV - R$\times$I where HV is the high voltage
measured, R is the electrodes resistance and I the flowing current.

Figure 6: Efficiency and large signal contamination as a function of actual
high voltage applied to the gas, for increasing $\gamma$ background rates for
the MIX2 mixture

Figure 6 shows the efficiency and large signal (streamer) probability as a
function of of HVgas and one can see how the curves are now superimposed (the
same is true also for other mixtures, as reported in [13]).

##  4 Conclusions and outlook

Several eco-friendly mixtures, where C2H2F4 is replaced with different ratios
of HFO/CO2 have been beam-tested at the CERN GIF++. First of all, the working
point of the detectors increases by $\sim$1 kV for every 10% HFO added to the
mixture; secondly, the large signal contamination decreases if the HFO
concentration increases; mixtures with 30-40% HFO provide values of streamer
contamination similar to the ones given by the STD gas mixture, but streamers
grow faster for voltages above the WP, reducing the safe operating region.

When exposed to gamma background, the efficiency curves shift to higher
voltages and the maximum efficiency decreases (these observations can be
explained by the current-induced voltage drop on the bakelite electrodes). In
particular, for ALICE-like background rates ($\approx$100 Hz/cm2) the
efficiency drop is $\sim$1 percentage point (pp) for the standard gas mixture
and $\sim$3-4 pp for the eco-friendly alternatives.

The results obtained from the beam tests are promising and a long-term aging
test is now ongoing at GIF++, in order to study the stability of the detector
response when operated for a long time with eco-friendly alternatives
(preliminary results of these studies are reported in [16]).

## References

  * [1] ALICE collaboration, _ALICE: Technical proposal for a Large Ion collider Experiment at the CERN LHC_ , LHC technical proposal. ALICE, (1995)
  * [2] Matsui, T. and Satz, H., _$J/\psi$ Suppression by Quark-Gluon Plasma Formation_, Phys. Lett. B 178 416-422 (1986)
  * [3] ALICE collaboration, _ALICE Technical Design Report of the Dimuon Forward Spectrometer_ , CERN/LHCC 99-22 (1999)
  * [4] Manen, S. _et al._ , _FEERIC, a very-front-end ASIC for the ALICE muon trigger resistive plate chambers_ , 2013 IEEE Conference (2013)
  * [5] Council of European Union, _Council regulation (EU) no 517/2014_ (2014)
  * [6] Proto, G., _Study of the performance of the RPC detector with new eco-friendly gas mixtures_ , Il nuovo Cimento C (2021)
  * [7] Bianchi, A. _et al._ , _Studies on tetrafluoropropene-based gas mixtures with low environmental impact for Resistive Plate Chambers_ , JINST 15 C04039 (2020)
  * [8] Rigoletti, G. _et al._ , _Characterization of RPC detectors with LHC-like background radiation and new eco-friendly gas mixtures_ , JINST 15 C11003 (2020)
  * [9] Benussi, L. _et al._ , _A study of HFO-1234ze (1,3,3,3-Tetrafluoropropene) as an eco-friendly replacement in RPC detectors_ , INFN-REPORT-INFN-14-14-LNF (2015)
  * [10] Pfeiffer, D. _et al._ , _The radiation field in the Gamma Irradiation Facility GIF++ at CERN_ , NIM A, vol. 866 91-103 (2017)
  * [11] M. Shah et al., _The CMS RPC detector performance and stability during LHC RUN-2_ , JINST 14 C11012 (2019)
  * [12] F. Sauli, _Gaseous Radiation Detectors: Fundamentals and Applications_ , Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press (2014)
  * [13] L. Quaglia, _Development of eco-friendly gas mixtures for Resistive Plate Chambers_. PhD thesis the University of Turin Apr 26th, (2023)
  * [14] A. Ferretti, _The upgrade of the RPC-based ALICE Muon Trigger_ , JINST 14 C06011 (2019)
  * [15] G. Aielli _et al._ , _Fluoride production in RPCs operated with F-compound gases_ , Nuclear Physics B, 158 143–148 (2006)
  * [16] L. Quaglia _et al._ , _Preliminary results on the long term operation of RPCs with eco-friendly gas mixtures under irradiation at the CERN Gamma Irradiation Facility_ , submitted to EPJ Plus (2023)


# Ultralong-range Cs-RbCs Rydberg molecules: non-adiabaticity of dipole
moments

David Mellado-Alcedo  Departamento de Física, Universidad de Córdoba, 14071
Córdoba, Spain  Departamento de Física Atómica, Molecular y Nuclear,
Universidad de Granada, 18071 Granada, Spain  Alexander Guttridge  Department
of Physics, Durham University, South Road, Durham, DH1 3LE, United Kingdom
Simon L. Cornish  Department of Physics, Durham University, South Road,
Durham, DH1 3LE, United Kingdom  H. R. Sadeghpour  ITAMP, Center for
Astrophysics $|$ Harvard & Smithsonian , Cambridge, Massachusetts 02138, USA
Rosario González-Férez<EMAIL_ADDRESS>Departamento de Física Atómica,
Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain  Instituto
Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071
Granada, Spain

###### Abstract

We consider ultralong-range polyatomic Rydberg molecules formed by combining a
Rydberg cesium atom and a ground-state RbCs molecule. We explore the regime
where the charge-dipole interaction due to the Rydberg electron with the
diatomic polar molecule couples the quantum defect Rydberg states Cs($ns$) to
the nearest degenerate hydrogenic manifold. We consider polyatomic Rydberg
molecules in states which are amenable to production in optical tweezers and
study the influence of nonadiabatic coupling on the likelihood of their
formation. The decay rates of the vibrational states reflect the interference
signature of wave function spread in different coupled potential wells.

##  I Introduction

The exaggerated properties of Rydberg atoms, such as their large spatial
extents, high polarizabilities, and energies, make them an outstanding system
for precisely probing interactions with the surrounding environment [1]. In an
ultracold setting, a Rydberg atom, can couple to a ground state atom via the
zero-range interaction due to Rydberg electron low-energy scattering from the
ground state atom, or interact with another Rydberg atom via van der Waals or
dipole-dipole forces [2]. A Rydberg atom can also couple to the permanent
electric dipole moment of a polar molecule, via the electron-dipole
interaction [3, 4]. For dipole moments below the Fermi-Teller critical dipole
[5, 6], the electron-dipole interaction can produce a molecular bond capable
of supporting bound rovibrational states. Such states are predicted to exist
in a hybrid atom-molecule gas at ultracold temperatures. The prerequisite
charge-dipole energy shift was recently observed for the first time in an
ultracold hybrid system in an experiment with Rb Rydberg atoms and RbCs
molecules trapped in separate optical tweezers [7].

In Rydberg atoms, core-electron scattering and polarization effects are
significant for low angular momentum electronic states. Such states are
energetically shifted from the degenerate manifold by means of the so-called
quantum defects, which are the fingerprints that uniquely identify a Rydberg
atom. The $ns$-states of cesium atom possess a peculiar quantum defect, such
that Cs($ns$) states lie energetically near the corresponding hydrogenic
Cs$(n-4,l\geq 4)$ states, which are nearly degenerate. Based on this
proximity, trilobite Rydberg molecules Cs∗-Cs possessing sizable ($\sim 1$ kD)
permanent electric dipole moments, were observed for the spherically symmetric
non-degenerate states Cs($ns$) [8, 9, 10]. Similarly, when one considers the
interaction of a Cs Rydberg atom with a polar molecule, the Cs($ns$) quantum
defect facilitates hybridization of the Rydberg states Cs($ns$) and
Cs$(n-4,l\geq 4)$, which is mediated by the charge-dipole interaction [11]. As
a consequence, the optimal scheme for the realization of the ultralong-range
triatomic Rydberg molecules [11], based on this resonant coupling, also
benefits from the fact that the diatomic molecule is in the rotational ground
state.

Here, we provide a thorough theoretical study of the shift and formation of
the triatomic molecule (TriMol) Cs($ns$)-RbCs. We first review the basic
theory of the ultralong-range triatomic Rydberg molecules in Sec. II. In Sec.
III, we consider two experimental protocols, corresponding to holding the atom
and molecule in the same or separate optical tweezers, and present the results
for selected potential energy curves (PEC) of this TriMol and their
vibrational bound states. We analyze the impact on the energy shifts of the
scattering of the low-energy Rydberg electron from RbCs. Inspired by previous
studies on non-adiabatic effects on Rydberg systems [12, 13], we also explore
the role played by the avoided crossings, and provide the Franck-Condon
factors for excitation from the atomic ground state. The paper is concluded in
Sec. IV.

##  II The Born-Oppenheimer Hamiltonian

We consider a TriMol formed by a Rydberg atom and a heteronuclear diatomic
molecule in the electronic and rovibrational ground state. The ground state
diatomic molecule is described within the Born-Oppenheimer (BO) and rigid
rotor approximations, whereas the Rydberg atom is described as a single-
electron system. In this framework, the adiabatic Hamiltonian of the Rydberg
TriMol is

$H=H_{\text{A}}+H_{\text{mol}}+H_{\text{int}}\,,$ ((1))

where $H_{\text{A}}$ represents the single-electron Hamiltonian describing the
Rydberg atom

$H_{\text{A}}=-\frac{\hbar^{2}}{2m_{e}}\nabla^{2}_{r}+V_{l}(r),$ ((2))

where $V_{l}(r)$ is an $l$-dependent nonlocal model potential [14], with $l$
the angular momentum of the Rydberg electron. The rigid rotor Hamiltonian
describing the molecule is $H_{\text{mol}}=B\mathbf{N}^{2}$ with $B$ the
rotational constant, and $\mathbf{N}$ the rotational angular momentum.

In this ultracold hybrid system, where the interactions are long-range, the
Hamiltonian between the Rydberg atom and the molecule becomes

$H_{\text{int}}=-\mathbf{d}\cdot\mathbf{F}_{\text{ryd}}(\mathbf{R},\mathbf{r})+2\pi
a_{S}(k)\delta\left({\bf r}-{\bf R}\right),$ ((3))

where the first term represents the interaction of the Rydberg electron and
ionic core with the RbCs permanent electric dipole moment $\mathbf{d}$. The
electric field $\mathbf{F}_{\text{ryd}}(\mathbf{R},\mathbf{r})$ due to the
Rydberg electron and the ion at the position of the diatomic molecule,
$\mathbf{R}$, is

$\mathbf{F}_{\text{ryd}}(\mathbf{R},\mathbf{r})=\frac{e}{4\pi\epsilon_{0}}\left(\frac{\mathbf{R}}{R^{3}}+\frac{\mathbf{r}-\mathbf{R}}{|\mathbf{r}-\mathbf{R}|^{3}}\right),$
((4))

where $e$ is the electron charge and $\mathbf{r}$ is the position of the
Rydberg electron. Note that the center of the coordinate system is located at
the position of the ionic core, Cs+. In this work, we consider the RbCs
molecule with a rotational constant of $B=0.490$ GHz [15] in the electronic
and vibrational ground state with an electric dipole moment of ${d}=1.225$ D
[16], below the Fermi-Teller critical dipole, $d_{cr}=1.639$ D [5, 6]. The
subcritical dipole moment of RbCs ensures that the electron does not bind to
the molecule.

The second term in Eq. (3) represents the scattering of the low-energy,
nearly-free electron from the diatomic molecule. This is approximated by the
Fermi pseudopotential [17, 18], assuming that only the $L=0$ ($S$-wave)
scattering partial wave contributes to the collision. The $S$-wave scattering
length, $a_{S}(k)=-\tan(\delta_{S}(k))/{k}$, with $\delta_{S}(k)$ as the $L=0$
scattering phase shift, and $k$ the electron wave number, is approximated by a
constant value $a_{S}=1/\sqrt{2E_{A}}$ [19], where $E_{A}$ is the electron
affinity of RbCs.

For the numerical description of this system, we fix the center of the
laboratory fixed frame at the ionic core Cs+, and the diatomic molecule is
situated along the $Z$ axis. The Schrödinger equation associated with the
Hamiltonian (1) is solved for several values of the interspecies separation
$R$. Hence, the potential energy surfaces of the Rydberg molecule Cs-RbCs
reduce to curves, which only depend on $R$. To solve this Schrödinger
equation, we perform a basis set expansion in terms of the coupled molecular
basis [4]

$\Psi(\mathbf{r},\Omega;R)=\sum_{n,l,N,J}C_{n,l,N}^{J}(R)\Psi_{nl,N}^{JM_{J}}(\mathbf{r},\Omega),$
((5))

where $|l-N|\leq J\leq l+N$, and

$\Psi_{nl,N}^{JM_{J}}(\mathbf{r},\Omega)=\sum_{m_{l},M_{N}}\langle
lm_{l}NM_{N}|JM_{J}\rangle Y_{NM_{N}}(\Omega)\,\psi_{nlm_{l}}(\mathbf{r})$
((6))

with $\langle lm_{l}NM_{N}|JM_{J}\rangle$ the Clebsch-Gordan coefficients,
$J=|l-N|,\dots,l+N$, and $M_{J}=-J,\dots,J$. $\psi_{nlm_{l}}(\mathbf{r})$ is
the Rydberg electron wave function with $n$, $l$ and $m_{l}$ the principal,
orbital and magnetic quantum numbers, respectively, and $Y_{NM_{N}}(\Omega)$
the field-free rotational wave function of the diatomic molecule, whose
rotation is described with the Euler angles $\Omega=(\theta,\phi)$. The total
angular momentum of the Rydberg molecule, excluding an overall rotation, is
given by $\mathbf{J}=\mathbf{l}+\mathbf{N}$, where $\mathbf{l}$ is the orbital
angular momentum of the electron and $\mathbf{N}$ the rotational angular
momentum of the diatomic molecule. We include rotational excitations in
RbCs($N\leq 5$), the Cs($ns,np,nd,nf$) quantum defects, the degenerate
manifold, and investigate electronic states of the TriMol with $M_{J}=0$.

##  III Results

###  III.1 Zero-range electron-molecule scattering

Figure 1:  Adiabatic electronic potential curves of the Cs-RbCs Rydberg
molecule with $M_{J}=0$. (a) for Cs$(41p)$-RbCs($N=0$), (b) for
Cs$(40d)$-RbCs($N=0$). Blue solid lines include the electron scattering with
RbCs and orange dashed lines represent the curves with this term neglected. In
panels (a) and (b), the zero energies are set to the Rydberg energies
$E(\text{Cs}(41p))$ and $E(\text{Cs}(40d))$, respectively. Figure 2: For the
Cs-RbCs Rydberg molecule, adiabatic electronic potential energy curves for
$M_{J}=0$ evolving from (a) the Rydberg states Cs$(n=38,l\geq 4)$ and the RbCs
in the ground state, Cs$(42s)$-RbCs($N\leq 2$) and Cs$(38f)$-RbCs($N\leq 2$)
and (b) the Rydberg state Cs$(n=70,l\geq 4)$ and the RbCs in the ground state,
Cs$(70f)$-RbCs($N=0$), Cs$(74s)$-RbCs($N=0$) and Cs$(73p)$-RbCs($N=4$). The
labels $(n,l;N)$ are used to identify the Rydberg potentials
Cs$(nl)$-RbCs($N$). In panels (c) and (d), the left axis shows potential
curves (black solid lines) and the right axis shows the Landau-Zener coupling
$g=|\langle\Psi_{i}|\frac{\partial}{\partial R}|\Psi_{j}\rangle|$ (magenta
dashed lines) for the avoided crossings encircled in panels (a) and (b),
respectively.

We first quantify the effect of electron-molecule scattering on the adiabatic
electronic potentials. In the collision of an electron with
RbCs(X${}^{1}\Sigma$) negative ions could be formed, and the measured electron
affinity of RbCs is $E_{A}=0.478\pm 0.020$ eV [20]. Ab initio calculations of
the low-energy electron-molecule scattering length are notoriously difficult.
We therefore estimate the magnitude of the scattering length using
$a_{S}=(2E_{A})^{-1/2}\sim 5.34~{}a_{0}$. The existence of an $E_{A}$ for RbCs
implies the sign of the scattering length is positive. We use a
$k$-independent scattering length in the $S$-wave Fermi-pseudopotential Eq.
(3). The effect of this interaction is illustrated in Fig. 1 for the adiabatic
potentials of Cs($41p$)-RbCs($N=0$) and Cs($40d$)-RbCs($N=0$), which have been
computed with (blue solid lines) and without (orange dashed lines) the
pseudopotential in Eq. (3). Because $a_{S}>0$, the effect of the Fermi-
pseudopotential is to blue-shift the energies from those obtained with only
the charge-dipole interactions in Eq. (4). We find, as expected, that the
largest difference between these potentials is near the last maximum of the
Rydberg wave function where we calculate a difference of around $48.6$ MHz
between the two curves.

###  III.2 Cs∗-RbCs Rydberg polyatomic molecules

We consider two possible experimental protocols to create ultralong range
polyatomic Rydberg molecules. Firstly, the ground state Cs atom and the RbCs
molecule are prepared in the motional ground state of two separate optical
tweezers, as demonstrated for Rb in Ref. [7]. Then, the optical tweezers are
merged, ideally achieving a high occupation of the ground state of the
relative motion of Cs and RbCs. Finally, the Cs atom is excited to a Rydberg
state so that RbCs lies within the electron orbit. Within the same optical
tweezer, the spatial separation between the atom and molecule should be
smaller might be smaller than $200$ nm, and the condition of lying within the
orbit could be satisfied with a two-photon excitation to the Cs($42s$) Rydberg
state. An alternative route is to leave the atom and molecule in their
separate optical tweezers and to then excite the Cs atom. In this case, the
spatial separation between the centers of the tweezers of $\lambda/2\sim 500$
nm is achievable, and a higher principal quantum number for the Rydberg
excitation is required. In this work, we take Cs($74s$) as an example to
illustrate this second option.

The electronic structures of the TriMol Cs($42s$)-RbCs($N=0$) and
Cs($74s$)-RbCs($N=0$) are presented in Figs. 2 (a) and (b), respectively. Due
to the peculiar quantum defect of Cs($ns$), the adiabatic potential curve for
the TriMol Cs($ns$)-RbCs($N=0$) lies among those electronic states evolving
from the Rydberg degenerate manifold Cs($n-4$, $l\geq 4$) combined with
RbCs($N=0$). As a consequence, the electronic spectrum of these TriMol
Cs∗-RbCs is characterized by numerous avoided crossings, as observed in panels
(a) and (b) of Fig. 2. Furthermore, because of the small RbCs rotational
constant, the adiabatic potentials of the electronic states
Cs($42s$)-RbCs($N$) with $N\leq 2$, also become immersed among those evolving
from Cs($n=38$, $l\geq 4$)-RbCs($N=0$), creating a highly complex spectrum.

Figure 3: For the electronic states Cs($42s$)-RbCs($N=2$) and
Cs($38f$)-RbCs($N=1$) of the TriMol encircled in Fig. 2 (a), eigenenergies and
their vibrational wave functions obtained from (a) the coupled Schrödinger
equation ((8)) and (b) after stabilization technique is applied. Panels (c)
and (d) show similar results for the lowest Cs($n=70,l>3$)-RbCs($N=0$) and
Cs($70f$)-RbCs($N=0$) potentials encircled in Fig. 2 (b). The vibrational wave
functions are shifted from their corresponding energies and rescaled for
better visibility, and the components $\chi^{d}(R)$ and $\chi^{u}(R)$ are
plotted in blue and yellow, respectively.

The experimental observation of a TriMol requires the formation of a bound
vibrational state in one of these electronic potentials in Fig. 2, which would
be heralded by the appearance of a redshidted feature in the Cs Rydberg
spectrum. In the avoided crossings marked in Figs. 2 (a) and (b) the coupling
between neighboring electronic states is not negligible and provokes the
predissociation of the bound vibrational states. In these regions the Born-
Oppenheimer approximation breaks down, and the total wave function can be
expressed as

$\Phi({\bf r},\\!\Omega,\\!R)=\sum_{i=d,u}\Psi_{i}({\bf
r},\\!\Omega;\\!R)\frac{\chi^{i}(R)}{R},$ ((7))

where $\Psi_{i}({\bf r},\\!\Omega;\\!R)$ is the eigenfunction of the
Halmiltonian ((1)) and is given by Eq. (5), and the index $i=d,u$ indicates
the potentials down and up, respectively. The reduced vibrational wave
functions $\chi^{i}(R)$ satisfy the following coupled Schrödinger equation
[21]

$\begin{pmatrix}T+V_{d}+{\mathcal{A}}_{dd}&{\mathcal{A}}_{du}\\\
{\mathcal{A}}_{ud}&T+V_{u}+{\mathcal{A}}_{uu}\end{pmatrix}\begin{pmatrix}\chi^{d}\\\
\chi^{u}\end{pmatrix}=E\begin{pmatrix}\chi^{d}\\\ \chi^{u}\end{pmatrix},$
((8))

with $T=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dR^{2}}$ the reduced kinetic energy
and $m$ the reduced mass of the Rydberg TriMol. The coupling terms are
$\mathcal{A}_{ij}=\langle\Psi_{i}|T|\Psi_{j}\rangle-\frac{\hbar^{2}}{m}\langle\Psi_{i}|\frac{d}{dR}|\Psi_{j}\rangle\frac{d}{dR}$,
whereas the potentials $V_{i}=\langle\Psi_{i}|H|\Psi_{i}\rangle$ are the
eigenvalues of the adiabatic Hamiltonian ((1)), with $i,j=d,u$. Note that
equation ((8)) is only valid for bound states of the Rydberg molecule with
zero rotational angular momentum.

The states of the coupled potential energy curves Cs($42s$)-RbCs($N=2$) and
Cs($38f$)-RbCs($N=1$) computed within the non-adiabatic Schrödinger equation
((8)), are presented in Fig. 3 (a) together their wave functions. The
contributions to the wave functions from the potentials down and up, i. e.,
$\chi^{d}(R)$ and $\chi^{u}(R)$, respectively, are plotted in different
colors. To determine which states are truly bound, we solve the coupled
Schrödinger Eq. (8) varying the lower limit of the radial coordinate and use
the stabilization method [22]. The bound states are those levels whose energy
could be considered independent of $R_{min}$, and they are presented in panel
(b) of Fig. 3. For the sake of completeness, the corresponding stabilization
diagram of the energies in Fig. 3 (a) is shown in Fig. 8 of the appendix. A
similar study is done for the electronic potential evolving from
Cs($70f$)-RbCs($N=0$) taking into account non-adiabatic coupling to the upper
potential evolving from Cs($70,l\geq 4$)-RbCs($N=0$), all eigenstates for a
fixed radial range are presented in Fig. 3 (c). Whereas the truly bound ones,
obtained after the stabilization analysis is applied, are shown in Fig. 3 (d).

Figure 4:  For the vibrational states in the upper potentials in Figs. 3 (b)
and (d), panels (a) and (c) present weights, as defined in Eq. (13), of the
vibrational wave function in the lower and upper potentials i. e., ${\cal
W}_{i}^{d}$ (blue circles) and ${\cal W}_{i}^{u}$ (yellow triangles), as a
function of the vibrational quantum number in these upper potentials; (b) and
(d) present decay rates as defined in Eq. (15) as a function of the
vibrational quantum number in these upper potentials, respectively.

The broad avoided crossings hardly influence the deeply bound states in the
outer well potential, and their wave functions are dominated by the components
in the lower potential, i. e., ${\cal W}_{i}^{d}\sim 1$, and see Eq. (13) for
the weights, as the contribution from the upper potential is very small. In
contrast, the impact is significant for the states lying on the upper
potential. Their wave functions are dominated by the contribution from this
upper potential, but still possess a significant weight from lower one as
shown in Fig. 4 (a) and (c). The three lowest states within the right well of
Cs($38f$)-RbCs($N=1$) possess the largest mixing, together with the
$\nu_{u}=5$ state, which due to the spatial extension of its wave function
shows the highest coupling to the lower potential. Note that for the rightmost
well in the upper potential in Fig. 3 (c), which evolves from Cs($70,l\geq
4$)-RbCs($N=0$), we encounter three states associated with the first excited
vibrational state, i. e., the part of their wave functions on the upper
potential possess a node. For two of the states, the contributions from the
lower potential to their total wave functions, see Eq. (13), are larger than
$73\%$, whereas for third state, it is $47\%$. However, they are not bounded
because their energies are not stable as the lower limit of the radial
coordinate is changed when solving the coupled Schrödinger equation in the
stabilization diagram.

To illustrate the coupling induced at the avoided crossings, we have computed
the nonadiabatic decay rates Eq. (15) of the bound states in the upper
potential to the continuum states. The results for the vibrational states
within the Cs($38f$)-RbCs($N=1$) and the lowest electronic state from the
degenerate manifold Cs($70,l\geq 4$)-RbCs($N=0$) are shown in Figs. 4 (b) and
(d), respectively. The trend in the decay rate for the vibrational states in
Fig. 3 (b) is as expected; the higher vibrational states away from the avoided
crossing region decay more slowly than the states near the crossing. For those
states located in a potential well, the nodal structure of the vibrational
wave function gives rise to the oscillatory behaviour of the decay rate. For
the vibrational states in the upper potential in Fig. 3 (d), the oscillation
in the decay rate is a reflection of the interference of the wave function
spread in the three wells.

For the vibrational bound states within the lower and upper potentials in Fig.
3 (b) and (d), we have also analyzed the hybridization of the Rydberg partial
waves due to the charge-dipole interaction. The weights of the $s$ and $f$
partial waves, and the sum of the rest, see Eq. (14), to the wave functions of
the vibrational bound states of the Cs($42s$)-RbCs($N=2$) and
Cs($38f$)-RbCs($N=1$) potentials are shown in Fig. 5 (a) and (b),
respectively. The $s$-wave is dominant only for those bound states close to
the avoided crossing region, i. e., the high (low)-lying states in the lower
(upper) potential. This is due to the avoided crossing with the potentials
Cs($42s$)-RbCs($N=0$) and Cs($42s$)-RbCs($N=1$) at larger internuclear
separations.

Figure 5: Integrated weights of the Rydberg $l$-partial waves in the
electronic wave function, $W_{n,l=0}$ (red squares), $W_{n,l=3}$ (blue dots)
and $\sum_{l\neq 0,3}W_{n,l}$ (green triangles), see Eq. (10), for the bound
states within the lower and upper potential energy curves (a) and (b),
respectively, in Fig. 3 (b). Panels (c) and (d) show analogous results for the
bound states in the lower and upper potentials in Fig. 3 (d).

For all states, the Rydberg partial waves with $l\geq 3$ possess significant
weights, having the largest contribution for excited states on the upper
potentials. These results indicate a significant hybridization of the Rydberg
partial waves, giving rise to huge permanent electric dipole moments
($\sim$kD), as illustrated in Fig. 6 (a) and (b).

Figure 6: For the bound states within the lower and upper potentials in Fig. 3
(b), panels (a) and (b) show the permanent electric dipole moment defined in
Eq. (12), respectively. Panels (c) and (d) show analogous results for the
bound states in the lower and upper potentials in Fig. 3 (d). The sudden
decrease in the dipole moment for the vibrational quantum numbers $\nu_{d}=2$
and $4$ in panel (a) is because these states are localized in the inner well
of the down potential and, therefore, being less affected by the avoided
crossing in Fig. 3 (b).

Figures 5 (c) and (d) present the Rydberg partial waves contributions to the
vibrational states in the Cs($70f$)-RbCs($N=0$) and Cs($70,l\geq
4$)-RbCs($N=0$) potentials, respectively. We encounter a strong mixing,
indeed, they do not inherit a dominant $f$-wave character from the potential
correlating to the Cs($70f$)-RbCs($N=0$) dissociation limit. Again, the
contribution of the $s$-partial wave is dominant for those states close to the
avoided crossing region, i. e., the highest- and lowest-lying levels in the
lower and upper potentials in Fig. 3 (d), respectively. As a consequence, they
all possess massive electric dipole moments as presented in Fig. 6 (c) and
(d).

For the vibrational bound states within the lower potential energy curve, we
have computed the weighted Franck-Condon factors in Fig. 7, taking the Rabi
frequency for the two-photon Rydberg excitation to the $ns$ Rydberg states as
$\Omega_{ns}=1$. The Franck-Condon factors strongly depend on the nodal
structure of the vibrational wave function. For the Cs($42s$)-RbCs($N=2$)
potential, the lowest lying state of the left well possesses the largest
Franck-Condon factor, which is also due to its significant $s$-wave and $N=0$
contributions to the wave function. For the Cs($70f$)-RbCs($N=0$) potential,
we encounter that excited states possess larger Franck-Condon factors, for
instance, the lowest lying states within the leftmost potential well, and the
second excited state in the central well. This could be due to the larger
contribution of the Rydberg $s$ partial wave to their wave function. These
results indicate that in both cases, there exits several vibrarional bound
states that could be used for the photoassociation of these TriMols.

Figure 7: Weighted Franck-Condon factors with $l=0$ Rydberg partial wave and
$N=0$, assuming the Rabi frequency for the corresponding two-photon Rydberg
excitation $\Omega_{ns}=1$, see Eq. (17), to the bound states within the lower
potential energy curves Cs($42s$)-RbCs($N=2$) in Fig. 3 (b) (filled squares),
and Cs($70f$)-RbCs($N=0$) in Fig. 3 (d) (empty squares).

##  IV Concluding remarks and outlook

We have investigated triatomic ultralong-range Rydberg molecules formed by a
Rydberg cesium atom and a RbCs diatomic molecule. Our theoretical description
includes the charge-dipole interaction arising from the coupling of the
permanent dipole moment of RbCs with the electric field produced by the
Rydberg electron and core, and the $S$-wave scattering of the slow Rydberg
electron from the molecule. Due to the peculiar quantum defects of Cs, the
charge-dipole interaction induces a coupling between the $s$-wave quantum
defect Rydberg states and the nearest degenerate hydrogenic manifold, as both
states are combined with RbCs in its rotational ground state.

Assuming that the Rydberg atom and the diatomic molecule are confined in
optical tweezers [7], we focus on the Rydberg molecules evolving from
Cs($42s$)-RbCs($N=0$) and Cs($74s$)-RbCs($N=0$), whose interspecies
separations correspond to the two possible scenarios with Cs and RbCs being in
the same tweezer or in separate ones, respectively. For these electronic
states, we have explored the role played by the non-adiabatic coupling near
the avoided crossings, their main features and vibrational structure. In both
cases, we have identified vibrational bound states with large Franck-Condon
factors, that could be used to form the polyatomic Rydberg molecules.

Near resonant coupling between Rydberg states and molecular rotation induces
long-range dipole-dipole interactions. Charge-dipole coupling induces gigantic
shifts and permanent polyatomic molecular dipole moments. These state-specific
interactions can be utilized to perform conditional and nondestructive readout
of molecular state [23, 24, 25, 26, 27, 28]. In addition, quantum gates
between molecules could be implemented using the Rydberg states of Cs as
facilitators [27, 28]. A number of different ultracold bialkali molecules
containing Cs atoms have been successfully produced in a variety of settings.
For instance, production of molecules in the absolute ground state has been
achieved for LiCs using photoassociation [29], NaCs using magnetoassociation
in optical tweezers [30] and bulk gases [31], and RbCs molecules have been
produced in bulk gases [16, 32] and optical tweezers [7]. In addition, there
are good prospects for reaching the ground state of ultracold KCs [33]. For
all these polar alkali dimers, we present in Table 1 a selection of the pairs
of rotational levels and of Rydberg states in Cs, that could be used for this
near-resonant interaction, along with the energy difference between these
transitions. These energy splittings could be brought closer to resonance by
applying electric or magnetic fields to smoothly tune the Rydberg states.

The precision tunability of Rydberg atoms and the stability of molecular
dipoles combine to induce anisotropic long-range interactions with new scales
and functionality. This hybrid atom-molecule system holds potential for
investigation of an entire class of quantum processes, including quantum
magnetization and quantum simulations.

Table 1: Transitions of the Cs atom and diatomic molecules formed from Cs and energy differences of these transitions that could be used to create quantum gates. Cesium | Molecule and | Energy difference
---|---|---
Rydberg Transition | Rotational Transition | [MHz]
$95S_{1/2}\,\to\,92P_{3/2}$ | LiCs, $N=1\to 2$ | $2.3$
$47D_{5/2}\,\to\,48P_{3/2}$ | NaCs, $N=1\to 2$ | $4.3$
$118D_{3/2}\,\to\,115P_{1/2}$ | KCs, $N=0\to 1$ | $0.4$
$85D_{3/2}\,\to\,86P_{3/2}$ | RbCs, $N=0\to 1$ | $0.6$

###### Acknowledgements.

D.M.A. and R.G.F. gratefully acknowledge financial support by the Spanish
projects PID2020-113390GB-I00 (MICIN), PY20$\\_$00082 (Junta de Andalucía),
and A-FQM-52-UGR20 (ERDF-University of Granada) and the Andalusian research
group FQM-207. A.G and S.L.C. acknowledge support from the UK Engineering and
Physical Sciences Research Council (EPSRC) Grants EP/P01058X/1, EP/V047302/1,
and EP/W00299X/1, UK Research and Innovation (UKRI) Frontier Research Grant
EP/X023354/1, the Royal Society, and Durham University. H.R.S. acknowledges
support at ITAMP through a grant by the NSF.

##  Appendix A Expressions

In this appendix, we describe the quantities analyzed to get a better physical
insight of the features of the TriMol. To illustrate the induced coupling due
to the charge-dipole interaction, we compute the weight of a certain partial
wave of the Rydberg electron in a $k$-electronic state of the TriMol wave
function (5), which is given by

${\cal C}_{n,l}(R)=\sum_{N,J}|C_{n,l,N}^{J}(R)|^{2}\,,$ ((9))

where the sum in $J$ is for $|l-N|\leq J\leq l+N$, and satisfies
$\sum_{n,l}{\cal C}_{n,l}(R)=1$. For a vibrational bound state, we define the
weight of this Rydberg-electron partial wave as

$W_{n,l}^{\nu}=\int(\chi(R))^{*}{\cal C}_{n,l}(R)\chi(R)dR,\,\,$ ((10))

and $\sum_{n,l}W_{n,l}^{k}=1$. Note that $\chi(R)$ represents the reduced wave
function of a vibrational state, and satisfies the following Schrödinger
equation

$\Big{[}-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dR^{2}}+V(R)\Big{]}\chi(R)=E\chi(R)$
((11))

with $m$ being the reduced mass of the TriMol, and $V(R)$ the corresponding
adiabatic potential energy curve.

We also estimate the electric dipole moment of this vibrational state by

$\langle\mu\rangle=\int(\chi(R))^{*}D_{ryd}(R)\chi(R)dR\,,$ ((12))

with the $R$-dependent electric dipole moment given by

$D_{ryd}(R)=e\int\Psi^{*}(\mathbf{r},\Omega;R)r\cos\theta_{e}\Psi(\mathbf{r},\Omega;R)d^{3}rd\Omega\,,$

where $\theta_{e}$ is the polar angle of the Rydberg electron. This integral
is non-zero if there are partial waves with different parity, i. e., $\Delta
l=\pm 1$, contributing to the wave function (5).

For a bound state in the coupled electronic potentials, we define the weight
of the wave function in the potentials down and up as

${\cal W}_{i}^{k}=\int|\chi^{k}_{i}(R)|^{2}dR$ ((13))

where $\chi^{k}_{i}(R)$ represents the part of reduced wave function in the
$k$-potential with $k=d,u$, and $i$ identifies the vibrational states. For all
bound states, it holds ${\cal W}^{d}_{i}+{\cal W}^{u}_{i}=1$. For each of
these bound states, the weights of the Rydberg-electron wave function are
defined, respectively, as

$W_{n,l}=\sum_{k=d,u}W_{n,l}^{k}\,.\quad$ ((14))

The non-adiabatic decay rates due to coupling near the avoided crossings care
expressed as [13]

$\Gamma_{i}=\frac{2\pi}{\hbar}|\langle\chi_{i}^{d}|A_{du}|\chi_{j}^{u}\rangle|^{2}$
((15))

where
$\mathcal{A}_{du}=\langle\Psi_{d}|T|\Psi_{u}\rangle-\frac{\hbar^{2}}{m}\langle\Psi_{d}|\frac{d}{dR}|\Psi_{u}\rangle\frac{d}{dR}$,
with $\chi_{j}^{u}$ is the vibrational wave function of a bound state in the
upper potential, and $\chi_{i}^{d}$, is the the continuum wave function in the
lower potential at the same energy. Note that the scattering state
$\chi_{i}^{d}$ is energy normalized, whereas computationally it is $L^{2}$
normalized. A numerical way to obtain these energy normalized wave functions
is described in Ref. [34].

In the coupled picture, the electronic dipole moment of a bound state is

$\langle
i|\mu|i\rangle=\int(\chi_{i}^{k_{1}}(R))^{*}D_{ryd}^{k_{1},k_{1}}(R)\chi_{i}^{k_{1}}(R)dR\,,$
((16))

with the dipole moment of the electronic state $k_{1}$, with $k_{1}=d,u$,
given by

$D_{ryd}^{k_{1},k_{1}}(R)=e\int\Psi_{k_{1}}^{*}(\mathbf{r},\Omega;R)r\cos\theta_{e}\Psi_{k_{1}}(\mathbf{r},\Omega;R)d^{3}rd\Omega\,.$

To estimate the probability of forming these Rydberg molecules by a two photon
transition, we compute the Franck-Condon factors weighted with $l=0$ Rydberg
partial wave,

$F_{i,N}=\sum_{k=d,u}\int(\chi_{i}^{k}(R))^{*}\Omega_{ns}C^{J,k}_{n,l=0,N}(R)\psi_{scat}(R)RdR$
((17))

with $\psi_{scat}(R)$ being the scattering wave function of the initial open
channel and $\chi_{i}^{k}(R)$ the vibrational wave function of the final
potential energy curve of the TriMol, and $\Omega_{ns}$ the Rabi frequency for
a two-photon Rydberg excitation to the $ns$ state, which was taken equal to
$1$ in our calculations. We are assuming that the process is dominated by a
single $ns$ Rydberg state, i. e., there is no significant admixture of
different principle quantum numbers, then we can drop the sum over $n$.

In Fig. 8, we present the stabilization diagram of the electronic potentials
of Cs-RbCs shown in Fig. 3 (a). We have solved the Schrödinger equation ((8))
by using different discretization boxes for $R$ and modifying their lower
limit $R_{min}$ [34]. Those level whose energy could be considered independent
of $R_{min}$ in Fig. 8 are the states that are truly bound within these
potentials, and are shown in panel (b) of Fig. 3.

Figure 8: For the two bound states of the electronics states of Cs-RbCs
Rydberg molecule Cs($42s$)-RbCs($N=2$) and Cs($38f$)-RbCs($N=1$) presented in
Fig. 3 (a), stabilization diagram of the energies as a function of the lower
minimum value of the radial coordinate $R_{min}$ used in the numerical method
to solve the coupled Schrödinger equation ((8)). The vibrational quantum
number $\nu_{d}$ of the bound states is indicated in the right axis.

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