# Interspecies Förster resonances of Rb-Cs Rydberg $d$-states for enhanced
multi-qubit gate fidelities

P. M. Ireland  D. M. Walker  J. D. Pritchard  0000-0003-2172-7340
<EMAIL_ADDRESS>Department of Physics, SUPA, Strathclyde
University, Glasgow, G4 0NG, UK

###### Abstract

We present an analysis of interspecies interactions between Rydberg $d$-states
of rubidium and cesium. We identify the Förster resonance channels offering
the strongest interspecies couplings, demonstrating the viability for
performing high-fidelity two- and multi-qubit $C_{k}Z$ gates up to $k=4$,
including accounting for blockade errors evaluated via numerical
diagonalization of the pair-potentials. Our results show $d$-state orbitals
offer enhanced suppression of intraspecies couplings compared to $s$-states,
making them well suited for use in large-scale neutral atom quantum
processors.

##  I Introduction

Neutral atom arrays provide a versatile platform for performing both
programmable quantum simulation and quantum computation [1, 2, 3, 4]. Arrays
of optical tweezers offer a scalable route to creating deterministically
loaded, defect-free qubit registers in up to three dimensions [5, 6, 7, 8].
Interactions between qubits can be engineered using highly-excited Rydberg
states to perform high-fidelity two- and multi-qubit gate operations [9, 10,
11, 12, 13, 14, 15, 16] for implementing digital quantum algorithms [17, 18],
or for encoding solutions to classical optimization problems using coherent
quantum annealing [19, 20, 21]. Using this architecture for quantum simulation
[22, 23] has enabled novel topological phases [24] and quantum spin liquids
[25] to be observed. Combining these approaches with dynamically
reconfigurable tweezers [18] provides a route to observation of fast-
scrambling dynamics [26] or efficient implementation of low-density parity
check codes for quantum error correction [27], with recent demonstrations of
transversal gates between logical qubits making a first step towards fault-
tolerant operation [28].

Whilst there has been significant progress to advance two-qubit gate
fidelities $\mathcal{F}>99.5\%$ [29, 30] along with recent demonstrations of
scalable non-destructive readout [31] and mid-circuit measurement [32, 33,
34], most experiments have used only a single atomic species. This introduces
limitations in localized qubit readout due to cross-talk between atoms, and in
multi-qubit gate operations where fidelities are limited by parasitic
interactions between Rydberg states[13, 20, 16].

Figure 1: Dual-species interactions. (a) Schematic of a dual species array of
Cs and Rb atoms using state-selective optical tweezers. For performing
interspecies $CZ$ or $C_{4}Z$ gates (indicated by dashed outlines), the
orientation of the internuclear axis relative to the quantization axis
($\hat{z}$) is important, with quantization along the tweezer axis in (b)
leading to interactions at a range of angles, whilst quantization
perpendicular to the array plane (c) ensures all interactions occur for
$\theta=90^{\circ}$. (d) Interaction angle $\theta$ relative to the
quantization axis, with an example Förster-resonant state occuring when the
initial pair state ${|a\rangle}_{A}{|b\rangle}_{B}$ is dipole-coupled to
${|\alpha\rangle}_{A}{|\beta\rangle}_{B}$ with a small energy defect $\Delta$.

One approach to overcome this limitation is to use dual-species arrays, which
naturally provides a separation in readout wavelengths to suppress cross-talk,
whilst allowing engineering of different inter and intra-species couplings.
Additionally, this offers a route to universal quantum computation using
globally driven pulses [35]. Early work demonstrated two-qubit gates and
arrays between different isotopes of rubidium (Rb) [11, 36], with recent
developments showing continuous loading and measurement-feedback onto a cesium
(Cs)-Rb array [37, 38]. This same approach of using heterogeneous tweezers has
enabled assembly of polar molecules [39, 40, 41, 42, 43], leading to first
demonstrations of hybrid systems [44, 45] based on coupling a Rydberg atom to
a polar molecule [46].

Previous theoretical work exploring dual-species Rydberg interactions has
focused on studying Rydberg interactions in $s$-state orbitals for Rb-Cs [47]
and $s$ and $d$ states for Rb-K atoms [48, 49], where suitable Förster
resonances have been identified to obtain strong intraspecies couplings with
dispersion coefficients calculated in the asymptotic limit. This forms the
basis of proposals for quantum non-demolition readout [47], quantum error
correction using dual-species surface codes [50], and high-fidelity
heteronuclear C2NOT2 gates [51] which suppress the unwanted interspecies
target-target and control-control interactions.

In this paper we extend the dual-species analysis to study interspecies
interactions between Cs and Rb Rydberg atoms in the $d$-state orbitals,
utilising open source libraries [52, 53] for calculating Rydberg-atom
interaction potentials via direct diagonalization of the dipole-dipole
Hamiltonian to go beyond evaluating the asymptotic dispersion coefficients. We
identify suitable Förster resonances for maximising interspecies coupling
strengths, carefully considering the angular dependence to address
applications of performing two- and multi-qubit gates in neutral atom arrays
for different quantization axis choices as illustrated in Fig. 1. We quantify
blockade leakage errors for different Förster-pair states, and further
evaluate the intrinsic gate errors based on the canonical three-pulse
controlled phase gate protocol [54] for $C_{k}Z$ gates up to $k=4$,
demonstrating enhanced fidelity for multi-qubit operations compared to single-
species approaches. We show the advantages of using $d$-state rather than
$s$-state resonances identified in previous studies due to the inherent
reduction in intraspecies couplings. These results are relevant to teams
working in both quantum computation and simulation, and provide a guide to
appropriate choices of state and polarization when designing dual-species
experiments.

The paper is organized as follows. In Sec. II we provide a brief introduction
to Rydberg atom interactions, then in Sec. III we identify the dominant
Förster resonances for the Rb-Cs $d$-states, studying the angular dependence
and comparing to previously identified $s$-state resonances. In Sec. IV we
evaluate the blockade and leakage errors associated with Förster-resonant
pairs, considering the performance for achieving both strong blockaded
interactions for Rb-Cs pairs and weak intraspecies Rb-Rb and Cs-Cs couplings.
From this analysis the Rb $59d_{5/2}$ \- Cs $6d_{3/2}$ resonance offers the
best performance for extension to $C_{k}Z$ gates and in Sec. V we provide
estimates of gate fidelity, before summarizing our findings in Sec. VI.

##  II Rydberg Atom Interactions

To begin we briefly summarize the methods used to calculate Rydberg atom
interactions. For a pair of atoms separated by distance $R$ at angle $\theta$
with respect to the quantization axis, as illustrated in Fig. 1(b), the
dipole-dipole interaction is given by [55]

$V(R)=\frac{\bm{\mu}_{\text{A}}\cdot\bm{\mu}_{\text{B}}-3(\hat{\bm{n}}\cdot\bm{\mu}_{\text{A}})(\hat{\bm{n}}\cdot\bm{\mu}_{\text{B}})}{R^{3}},$
(1)

where $\bm{\mu}_{A}$ and $\bm{\mu}_{B}$ are the electric dipole moments for
atom A and atom B respectively for coupling target pair states
$|a\rangle_{A}|b\rangle_{B}$ to pair states
$|\alpha\rangle_{A}|\beta\rangle_{B}$, and $\hat{\bm{n}}$ is the unit vector
along the atom separation axis $R$.

The sign and strength of the resulting interaction is determined by the dipole
coupled pair state
$|{\alpha^{A}\beta^{B}}\rangle\equiv|\alpha\rangle_{A}|\beta\rangle_{B}$ with
the smallest energy defect $\Delta=(E_{\alpha}-E_{a})+(E_{\beta}-E_{b})$ with
respect to the target pair state ${|ab\rangle}$. For $|\Delta|\gg V(R)$ the
resulting interaction is in the van der Waals regime with energy shifts
scaling as $C_{6}/R^{6}$, whilst for $|\Delta|<V(R)$ the system experiences a
strong first-order energy shift into branches with energies of $\pm
C_{3}/R^{3}$. Alternatively, dc electric fields can be used to tune a near-
resonant pair to achieve $\Delta=0$ [56, 57, 58].

To estimate the sign and magnitude of the coupling strengths for a given pair
state, it is instructive to calculate the $C_{3,k}$ channel coefficient
introduced in Ref. [47] defined as

$C_{3,k}=\frac{e^{2}}{4\pi\epsilon_{0}}\frac{\langle\alpha_{k}||r_{A}||a\rangle\langle\beta_{k}||r_{B}||b\rangle}{\sqrt{(2j_{\alpha_{k}}+1)(2j_{\beta_{k}}+1)}},$
(2)

where ${|\alpha^{A}_{k}\beta^{B}_{k}\rangle}$ is a Rydberg pair state which is
dipole coupled to the target pair state ${|a^{A}b^{B}\rangle}$, and
$\langle\phi_{k}||r_{X}||\psi\rangle$ is the reduced matrix element for
transitions between states $\psi$ and $\phi_{k}$ in the fine-structure basis
for atom $X$ associated with interaction channel $k$. $j_{\psi}$ is the total
angular momentum quantum number associated to the state ${|\psi\rangle}$.

Asymptotic dispersion coefficients $C_{6}$ and $C_{3}$ can be calculated by
summing over the relevant dipole-coupled pair states [59, 47], however for the
typical atomic separations used in current tweezer array experiments
$3<R<15~{}\mu$m accurate calculation of the resulting interatomic interactions
requires exact diagonalization of the resulting pair-state Hamiltonian
including higher-order dipole-quadrupole and quadrupole-quadrupole terms.
Below we use the _pair-interaction_ library [52] to evaluate Rydberg state
couplings, which exploits measured quantum defects and numerical integration
of the atomic wavefunctions based on model potentials to calculate Rydberg
state energies and matrix elements respectively.

To engineer strong interspecies couplings it is necessary to identify Förster
resonant pair states that have small energy defects $\Delta$ and large
coupling strenthgs $C_{3,k}$ [47]. For two independent species or isotopes,
this is achieved by choosing relevant quantum numbers including principal
quantum number and orbital angular momentum, resulting in a large degree of
tunabiltiy and control in choosing Forster resonant pairs. For the
intraspecies couplings however, the energy defects of a given pair is
determined entirely by the inherent quantum defects for each species, meaning
there is typically only a single value of $n$ which is near-resonant and that
predominantly the intraspecies interactions are in the weaker van der Waals
regime which drops off rapidly with $R$.

##  III Rubidium-Cesium $d$-state Förster Resonances

We identify suitable Förster-resonant pairs of Rb and Cs in $d$ orbital
angular momentum states by calculating the energy defect $\Delta$ and channel
coefficient $C_{3,k}$ for all dipole-coupled pair states from atoms initially
in $d_{3/2}$ or $d_{5/2}$. Due to the large number of possible Förster
resonant pair states, we follow the approach of Ref. [47] to consider only
resonances with $50\leq n\leq 90$ that have $|C_{3,k}|>1$ GHz$\,\mu$m3 and an
energy defect $|\Delta|<0.005$ of the level spacing. Complete tables of
resonances are given in the Appendix, but we focus here on identifying the
dominant channels for different combinations of values of the total angular
momentum quantum number $j$.

###  III.1 Rb $d_{5/2}$ \- Cs $d_{5/2}$

The $n_{\mathrm{Rb}}d_{5/2}n_{\mathrm{Cs}}d_{5/2}$ pair-states are dipole
coupled through a total of 9 possible channels associated with angular quantum
numbers of $n^{\prime}_{\mathrm{Rb}}p_{3/2}n^{\prime}_{\mathrm{Cs}}p_{3/2}$,
$n^{\prime}_{\mathrm{Rb}}fn^{\prime}_{\mathrm{Cs}}p_{3/2}$,
$n^{\prime}_{\mathrm{Rb}}p_{3/2}n^{\prime}_{\mathrm{Cs}}f$ and
$n^{\prime}_{\mathrm{Rb}}fn^{\prime}_{\mathrm{Cs}}f$ where the $f$-states can
be either the $j=5/2$ or $j=7/2$ magnetic sublevels. From the resonances
listed in Tab. 2 there is only a single resonance via the
$n^{\prime}_{\mathrm{Rb}}p_{3/2}n^{\prime}_{\mathrm{Cs}}p_{3/2}$ channel,
whilst the strongest coupling coefficient’s $C_{3,k}$ occur for the
$(n_{\mathrm{Rb}}+1)p_{3/2}(n_{\mathrm{Cs}}-2)f_{7/2}$ channel. Additional
dominant channels are $(n_{\mathrm{Rb}}-2)f(n_{\mathrm{Cs}}-2)f$ and
$(n_{\mathrm{Rb}}-2)f(n_{\mathrm{Cs}}+2)p_{3/2}$.

Figure 2: Example Rb-Cs $n_{\mathrm{Rb}}d_{5/2}n_{\mathrm{Cs}}d_{5/2}$
resonances. (a) Rb $72d_{5/2}$-Cs $63d_{5/2}$, resonant with Rb $70f$-Cs $61f$
(b) Rb $74d_{5/2}$-Cs $82d_{5/2}$, resonant with Rb $72f$-Cs $84p_{3/2}$ (c)
Rb $70d_{5/2}$-Cs $80d_{5/2}$, resonant with Rb $71p_{3/2}$-Cs $78f$.
Calculated for $\theta=0^{\circ}$ and
$m_{j}^{\mathrm{Rb}}=m_{j}^{\mathrm{Cs}}=+5/2$.

Using these states we now examine the characteristic pair-potentials for each
of these channels via direct matrix diagonalization, calculating potentials
for $\theta=0^{\circ}$ and considering atoms in magnetic sublevels
$m_{j}^{\mathrm{Rb}}=m_{j}^{\mathrm{Cs}}=+5/2$. For the
$(n_{\mathrm{Rb}}-2)f(n_{\mathrm{Cs}}-2)f$ channel, Fig. 2(a) shows the
coupling for Rb $72d_{5/2}$-Cs $63d_{5/2}$ which is resonant with Rb $70f$-Cs
$61f$ states with $\Delta/2\pi=19.1$ MHz, with datapoint intensity coloured to
reflect the overlap of the target state with the calculated eigenstate
${|\phi_{i}\rangle}$ as $|{\langle rr|}\phi_{i}\rangle|^{2}$. Whilst this
resonance possesses a strong channel coefficient of $|C_{3,k}|=7.64~{}$GHz
µm3, it is evident that rather than a strong resonant interaction at short
range ($R<10\,$µm), the off-resonant coupling to other pair-states suppresses
the shift of the upper excitation branch, leading to a weak interaction that
is poorly suited to realising strongly blockaded excitation.

In Fig. 2(b) we show the Rb $74d_{5/2}$-Cs $82d_{5/2}$ state, which is near
resonant with Rb $72f$-Cs $84p_{3/2}$ states. Around $R=4-6~{}\mu$m we observe
the $f$ states split due to coupling with other branches, resulting in a
strongly-shifted upper excitation branch and mixing in a third eigenstate that
is only weakly shifted and preventing use of the
$(n_{\mathrm{Rb}}-2)f(n_{\mathrm{Cs}}+2)p_{3/2}$ resonances for high-fidelity
gates.

In Fig. 2(c) we show the $72d_{5/2}$-Cs $80d_{5/2}$, which is strongly coupled
to the resonant Rb $72f$-Cs $84p_{3/2}$ states with
$|C_{3,k}|=23~{}\mathrm{GHz}\,\mu\mathrm{m}^{3}$. This reveals the expected
symmetric splitting and $1/R^{3}$ scaling, making the
$(n_{\mathrm{Rb}}+1)p_{3/2}(n_{\mathrm{Cs}}-2)f_{7/2}$ resonances ideal for
engineering strong interspecies couplings at $\theta=0^{\circ}$. These results
highlight the importance not only of evaluating the pair defects and channel
couplings, but also examining the real pair curves over the desired separation
ranges.

Figure 3: Example Rb $d_{3/2}$-Cs $d_{3/2}$ Förster resonances. (a) Rb
$76d_{3/2}$-Cs $67d_{3/2}$, resonant with Rb $74f_{5/2}$-Cs $65f_{5/2}$ (b) Rb
$70d_{3/2}$-Cs $77d_{3/2}$, resonant with Rb $68f_{5/2}$-Cs $79p_{1/2}$ (c) Rb
$73d_{3/2}$-Cs $83d_{3/2}$, resonant with Rb $74p_{1/2}$-Cs $81f_{5/2}$.
Calculated for $\theta=0^{\circ}$ and
$m_{j}^{\mathrm{Rb}}=m_{j}^{\mathrm{Cs}}=+3/2$.

###  III.2 Rb $d_{3/2}$ \- Cs $d_{3/2}$

For the Rb $d_{3/2}$-Cs $d_{3/2}$ states there are again 9 possible channels
associated with orbital angular momentum quantum numbers of
$n^{\prime}_{\mathrm{Rb}}pn^{\prime}_{\mathrm{Cs}}p$,
$n^{\prime}_{\mathrm{Rb}}pn^{\prime}_{\mathrm{Cs}}f_{5/2}$,
$n^{\prime}_{\mathrm{Rb}}f_{5/2}n^{\prime}_{\mathrm{Cs}}p$ and
$n^{\prime}_{\mathrm{Rb}}f_{5/2}n^{\prime}_{\mathrm{Cs}}f_{5/2}$ where $p$ can
be either $j=1/2,3/2$. Table 3 shows the filtered resonances, with the
strongest $|C_{3,k}|$ channel coupling via the
$(n_{\mathrm{Rb}}+1)p_{1/2}(n_{\mathrm{Cs}}-2)f_{5/2}$ resonances.

In Fig. 3 example resonances for the different channels are plotted, with
similar behavior observed as for the $d_{5/2}-d_{5/2}$ resonances where the
channels with pair-states resonant with Rb $f_{5/2}$ states (shown in (a) and
(b)) provide unsuitable resonances at short range. The strongest resonance is
for Rb $73d_{3/2}$-Cs $83d_{3/2}$ which is coupled to Rb $74p_{1/2}$-Cs
$81f_{5/2}$ with $\Delta/2\pi=-0.84$ MHz and $C_{3,k}=-27.3~{}$GHz$\,\mu$m3
plotted in Fig. 3(c) showing a strong Förster resonant coupling at short
range, with similar behavior observed for resonances for pairs involving Rb
$p_{3/2}$ state but with a weaker coupling due to the larger defects. At short
range ($R<5\mu$m), comparing the result here to the approximately equivalent
resonance shown in Fig. 2(c) at comparable quantum numbers, the
$d_{3/2}-d_{3/2}$ resonance mixes with more pair states than the
$d_{5/2}-d_{5/2}$ case, leading to the two characteristic branches splitting
with the target pair state now appearing across a larger number of pair
eigenstates.

###  III.3 Rb $d_{3/2}$ \- Cs $d_{5/2}$ and Rb $d_{5/2}$ \- Cs $d_{3/2}$

Förster resonances for the Rb $d_{3/2}$-Cs $d_{5/2}$ and Rb $d_{5/2}$-Cs
$d_{3/2}$ states are presented in Tab. 5 and 4 respectively. These mixed $j$
pairs follow a similar trend to those above, with the strongest resonances
observed for
$(n_{\mathrm{Rb}}+1)p_{j_{\mathrm{Rb}}-1}(n_{\mathrm{Cs}}-2)f_{j_{\mathrm{Cs}}+1}$
couplings, and resonances with Rb $p$ and Cs $f$ states producing the desired
$1/R^{3}$ resonant splitting at $\theta=0^{\circ}$. In comparison to the Rb
$d_{3/2}$-Cs $d_{5/2}$ states, Rb $d_{5/2}$-Cs $d_{3/2}$ pairs typically have
larger $\Delta$ values, however there are more states with strong $C_{3,k}$ at
lower values of $n_{\mathrm{Rb}}$.

As will be shown below in Sec. V, the resonances via Rb $d_{3/2}$ states are
also less favorable for scaling to multi-qubit interactions due to the
presence of a near Förster resonance for Rb $58d_{3/2}$-Rb $58d_{3/2}$,
leading to anomalously large intraspecies $C_{6}$ coefficients for nearby
$n_{\mathrm{Rb}}d_{3/2}$ states.

###  III.4 Angular Dependence

In the analysis above we consider atoms at $\theta=0^{\circ}$, which results
in the total magnetic quantum number $M=m_{j}^{A}+m_{j}^{B}$ being preserved
by the dipole-dipole interaction with $\Delta M=0$ selection rules. For
experiments in 2D tweezer arrays, this condition is only met for the case of
atoms oriented with the internuclear axis along the quantization axis. For
performing multi-qubit gate operations, or exploiting long-range couplings for
quantum simulation, placing a quantization axis parallel to one of the axis
coordinates results in atoms interacting across a range of angles from
$\theta=0^{\circ}$–$90^{\circ}$ as shown in Fig. 1(b). Alternatively, the
quantization axis can be aligned normal to the array plane, meaning all atoms
interact at $90^{\circ}$ as shown in Fig. 1(c), providing an isotropic
coupling of an atom to its neighbors, where at this angle $V(R)$ couples pair
states with $\Delta M=0,\pm 2$. In this section we explore the angular
dependence of the $d$-state Förster resonances, considering the impact on
choice of magnetic sublevel.

Figure 4 shows angular potentials for the Förster resonance between Rb
$55d_{5/2}$ \- Cs $63d_{5/2}$ for different combinations of
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(5/2,\pm 5/2)$ and for angles
$\theta=0^{\circ}$ and $90^{\circ}$. Considering first the $(5/2,5/2)$
combination, Fig. 4(a) shows at $0^{\circ}$ we obtain a strong Förster
resonance splitting, however at $90^{\circ}$ the additional coupling terms
driving $\Delta M=\pm 2$ cause the target Rydberg states to be mixed with a
large number of other pair eigenstates, several of which have weak or
vanishing interaction shifts. Conversely, for the $(5/2,-5/2)$ combination
with $M=0$ we see a strong suppression of the interaction at $0^{\circ}$ in
Fig. 4(c) leading to a flat potential curve for $R>5~{}\mu$m due to the
presence of a Förster-zero in the interaction channel [60], whilst for
$90^{\circ}$ we recover the desired Förster splitting as shown in Fig. 4(d).
Comparing the results in (a) and (d) shows that the $\theta=90^{\circ}$ offers
improved symmetry between upper and lower excitation branches of the pair
potentials with respect to the unshifted pair state, extending the useful
blockade range which is limited by the smallest shift of either branch. This
shows that operation at $90^{\circ}$ with a $(5/2,-5/2)$ combination is
preferable not only for achieving isotropic couplings across the array, but
also for improved long-range interactions.

Alongside the Rb-Cs pair potentials, the figures also show intraspecies
interaction curves for Rb $55d_{5/2}$ \- $55d_{5/2}$ in blue and Cs
$63d_{5/2}$-$63d_{5/2}$ in red. For both Fig. 4(a) and (d) with strong
interspecies Förster resonance, we see the intraspecies interaction curves
remain flat until $R\lesssim 5~{}\mu$m before showing a van der Waals like
$C_{6}/R^{6}$ shift. Despite having a lower quantum number of
$n_{\mathrm{Rb}}=55$, the Rb-Rb $C_{6}$ coefficient is comparable to that of
the $n_{\mathrm{Cs}}=63$ due to the Rb quantum defects yielding smaller pair-
defects for the intraspecies $d$-states than that of Cs. This also shows the
advantage of using dual-species interactions, with the Rb-Cs coupling offering
a strong blockaded interaction at $R=6~{}\mu$m where the intraspecies coupling
is negligible.

Figure 4: Angular dependence of the Rb $55d_{5/2}$ \- Cs $63d_{5/2}$ Förster
resonance for different $(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})$. (a)
$\theta=0^{\circ}$, $(5/2,3/2)$ (b) $\theta=90^{\circ}$, $(5/2,5/2)$ (c)
$\theta=0^{\circ}$, $(5/2,-5/2)$ (d) $\theta=90^{\circ}$, $(5/2,-5/2)$.
Interspecies pair-potentials are plotted black with opacity proportional to
state overlap. Also shown are intraspecies interaction curves are shown for
Rb-Rb in $55d_{5/2}$ (red) and Cs-Cs $63d_{5/2}$ (blue). (e,f) The variation
of energy eigenstates with $\theta$ for $(5/2,\pm 5/2)$ calculated at
$R=6~{}\mu$m revealing the Förster resonance can only be used at $0^{\circ}$
and $90^{\circ}$.

For intermediate angles, Fig. 4(e) and (f) show the pair-state eigenenergies
as a function of $\theta$ calculated for a separation of $R=6~{}\mu$m. Unlike
the characteristic $C_{3}\propto(1-3\cos^{2}(\theta))$ dependence obtained for
single-species couplings between atoms in identical states [61], the angular
profiles show that operating more than around 5 degrees away from the optimal
values of $\theta=0^{\circ}$ and $\pi/2$ the pair potentials show a complex
energy landscape with a large number of additional eigenstates appearing that
have weak or vanishing energy shifts, preventing realisation of strong
blockaded interactions at these intermediate angles.

For the other interaction channels, a similar behavior is observed with stable
resonances observed for
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(+j^{\mathrm{Rb}},+j^{\mathrm{Cs}})$
at $\theta=0^{\circ}$ and $(+j^{\mathrm{Rb}},-j^{\mathrm{Cs}})$ at
$90^{\circ}$. An example resonance for the Rb $59d_{5/2}$ \- Cs $68d_{3/2}$
resonance at $\theta=90^{\circ}$ and $(5/2,-3/2)$ is shown in Fig. 5(a), which
provides one of the best states for achieving low blockade and leakage errors
as detailed later.

###  III.5 Comparison to $s$-state Förster resonances

To illustrate the advantage of the $d$-state resonances over $s$-states, Fig.
5(b) shows an example Förster resonance for the Rb $72s_{1/2}$ \- Cs
$70s_{1/2}$ state previously identified in Ref. [47] for $\theta=90^{\circ}$
and $(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(1/2,-1/2)$. This reveals a
strong interspecies resonance with comparable magnitude to that of Rb
$59d_{5/2}$ \- Cs $68d_{3/2}$ shown in Fig. 5(a), but with the downside being
that the intraspecies Rb-Rb and Cs-Cs interactions have a similar magnitude,
negating the advantage of being able to exploit strong Rb-Cs interaction
whilst suppressing the intraspecies couplings.

A further benefit is that when performing two-photon excitation to the Rydberg
states, the dipole-matrix elements to the intermediate state scale as
$n^{-3/2}$. As well as obtaining comparable interspecies interactions at lower
principal quantum numbers, the $d$-states have a pre-factor in this scaling
approximately twice as strong as the $s$-states yielding enhanced Rabi
frequencies at the same laser power.

Figure 5: Comparison of $d$ and $s$ state Förster resonances with
$\theta=90^{\circ}$ with magnetic sublevels denoted
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})$. (a) Rb $59d_{5/2}$ \- Cs
$68d_{3/2}$ Förster resonance for $(5/2,-3/2)$, and (b) Rb $72s_{1/2}$ \- Cs
$75s_{1/2}$ Förster resonance for $(1/2,-1/2)$. Interspecies pair-potentials
are plotted black with opacity proportional to state overlap. Intraspecies
interaction curves are shown for Rb (red) and Cs (blue).

Rb - Cs Förster Resonances $\Delta/2\pi$ $C_{3,k}$ Rb-Cs at $R=6~{}\mu$m Rb-Rb
at $R=8.5~{}\mu$m Cs-Cs at $R=8.5~{}\mu$m Target Pair Resonant Pair (MHz)
(GHz$\,\mu$m3) $P_{1r}\left(t_{\pi}\right)$ $\tilde{C}_{3}$ (GHz$\,\mu$m3)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6) *$55d_{5/2}$ \-
$63d_{5/2}$ $56p_{3/2}$ \- $61f_{5/2}$ 12.66 -2.24 0.9969 -11.30 0.9989 21.5
0.9840 -98.8 $62d_{5/2}$ \- $71d_{5/2}$ $63p_{3/2}$ \- $69f_{5/2}$ -9.53 -3.66
0.9997 15.79 0.9855 64.1 0.7747 -416.8 $69d_{5/2}$ \- $79d_{5/2}$ $70p_{3/2}$
\- $77f_{7/2}$ -19.14 -21.97 0.9998 26.34 0.7855 192.8 0.0013 -1448.9
$70d_{5/2}$ \- $80d_{5/2}$ $71p_{3/2}$ \- $78f_{7/2}$ 13.69 -23.21 0.9990
-29.03 0.7442 223.9 0.0238 -1812.1 $76d_{5/2}$ \- $87d_{5/2}$ $77p_{3/2}$ \-
$85f_{5/2}$ -19.30 -8.41 0.9997 40.58 0.1156 450.0 0.0388 -4482.5 *$52d_{5/2}$
\- $60d_{3/2}$ $53p_{3/2}$ \- $58f_{5/2}$ 15.19 -6.70 0.9964 -8.37 0.9997 12.2
0.9958 -45.0 $59d_{5/2}$ \- $68d_{3/2}$ $60p_{3/2}$ \- $66f_{5/2}$ 13.40
-11.23 0.9992 -14.36 0.9951 41.6 0.9410 -206.1 $66d_{5/2}$ \- $76d_{3/2}$
$67p_{3/2}$ \- $74f_{5/2}$ 11.12 -17.74 0.9997 -22.84 0.9176 114.4 0.4467
-692.9 $79d_{5/2}$ \- $91d_{3/2}$ $80p_{3/2}$ \- $89f_{5/2}$ -12.53 -36.95
0.9999 49.07 0.0070 619.2 0.0052 -3596.5 $80d_{5/2}$ \- $92d_{3/2}$
$81p_{3/2}$ \- $90f_{5/2}$ 7.42 -38.76 0.9999 -51.12 0.0233 705.2 0.0144
-5756.9 *$53d_{3/2}$ \- $60d_{5/2}$ $54p_{1/2}$ \- $58f_{5/2}$ -4.41 -1.97
0.9968 7.12 0.9989 19.7 0.9938 -57.00 $76d_{3/2}$ \- $87d_{5/2}$ $77p_{3/2}$
\- $85f_{5/2}$ 6.49 2.81 0.9986 -52.91 0.0129 723.9 0.0186 -5038.3 $65d_{3/2}$
\- $74d_{3/2}$ $66p_{1/2}$ \- $72f_{5/2}$ -13.07 -17.02 0.9974 16.67 0.7416
234.6 0.8297 -375.1 $73d_{3/2}$ \- $83d_{3/2}$ $74p_{1/2}$ \- $81f_{5/2}$
-0.70 -27.26 0.9995 28.54 0.1360 452.0 0.1082 -2156.1 $80d_{3/2}$ \-
$91d_{3/2}$ $81p_{1/2}$ \- $89f_{5/2}$ -13.66 -39.62 0.9998 48.51 0.0440
1263.6 0.0234 -3178.2 $81d_{3/2}$ \- $92d_{3/2}$ $82p_{1/2}$ \- $90f_{5/2}$
4.39 -41.55 0.9997 -47.30 0.0590 1411.2 0.0153 -3836.9 $72s_{1/2}$ \-
$70s_{1/2}$ $71p_{3/2}$ \- $70p_{1/2}$ -7.90 8.51 0.9992 9.82 0.0040 -1106.6
0.4005 -605.3

Table 1: Summary of the strongest Rb-Cs $d$-state Förster resonances offering
$P_{1r}(t_{\pi})\sim 0.999$ for a pair of Rb-Cs atoms separated by
$R=6~{}\mu$m calculated at $90^{\circ}$ for
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(+j^{\mathrm{Rb}},-j^{\mathrm{Cs}})$assuming
a single-photon Rabi frequency of $\Omega/2\pi=2$ MHz compared against the
blockade errors expected for multi-qubit gates due to intraspecies coupling
for Rb-Rb and Cs-Cs atoms at $8.5~{}\mu$m. Along with the asymptotic pair
energy defect $\Delta$ and channel coefficient $C_{3,k}$, we give the
effective $\tilde{C}_{3}$ and $\tilde{C}_{6}$ coefficients obtained by fitting
the pair-potentials in the region $4<R<15~{}\mu$m. States labelled with an
asterisk $(*)$ are suitable for multi-qubit gate with weak intraspecies
couplings. Also shown is the only $s$-state resonance acheiving
$P_{1r}(t_{\pi})>0.999$.

##  IV Blockade Leakage Errors

In order to exploit Rydberg atom interactions for performing high-fidelity
controlled gate operations, we use the blockade regime where the interaction
strength $V$ greatly exceeds the excitation linewidth determined by the Rabi
frequency $\Omega$ describing the rate of coupling from the computational
ground-state to the target Rydberg state, preventing more than a single
Rydberg excitation to be created.

For the Förster resonances identified above, the exact pair potentials include
coupling not just to a single level, but to many pair-eigenstates
${|\phi_{i}\rangle}$ each with energy $\epsilon_{i}$, and it is necessary to
consider not only the blockade effect arising from the strongly shifted
eigenstates but also blockade leakage due to coupling to weakly shifted
eigenstates. To determine the blockade leakage error for a given target pair
state ${|r_{t}^{A},r_{t}^{B}\rangle}$, we calculate the probability of
exciting this double Rydberg state following a resonant $\pi$-pulse applied to
atom $A$ for atoms initialized at time $t=0$ in the state
${|\Psi(0)\rangle}={|1^{A},r_{t}^{B}\rangle}$, where ${|1^{A}\rangle}$ is the
computational state of atom $A$, taking into account coupling to all possible
eigenstates ${|\phi_{i}\rangle}$ found through diagonalizing the pair state
interactions.

The Hamiltonian for the coupled system is given by

$\hat{H}=\hbar\sum_{i=1}\left(\left[\frac{\eta_{i}\Omega}{2}{|\phi_{i}\rangle}{\langle
0^{A},r_{t}^{B}|}+\text{h.c.}\right]-\epsilon_{i}{|\phi_{i}\rangle}{\langle\phi_{i}|}\right),$
(3)

where $\Omega$ is the Rabi frequency for coupling
${|1^{A}\rangle}\rightarrow{|r_{t}^{A}\rangle}$ and $\eta_{i}={\langle
r_{t}^{A},r_{t}^{B}|}\phi_{i}\rangle$ denotes the relative overlap of the
target pair state with the energy eigenstate, and the sum is taken for all
eigenstates with $|\eta_{i}|>10^{-3}$. The resulting wavefunction at time $t$
is given by ${|\Psi(t)\rangle}=e^{-i\hat{H}t/\hbar}{|\Psi(0)\rangle}$, from
which the probability of a double Rydberg excitation is given by

$P_{rr}(t)=\sum_{i=1}\left|\langle\phi_{i}|\Psi(t)\rangle\right|^{2},$ (4)

and the probability of finding the atom in the initial ground state is
$P_{1r}(t)=\left|\langle 1^{A},r_{t}^{B}|\Psi(t)\rangle\right|^{2}$.

For each of the possible Förster-resonant pairs, we evaluate the blockade
leakage error calculated for an Rb-Cs pair at a spacing of $R=6~{}\mu$m for
$\theta=90^{\circ}$ and using $(+m_{j}^{\mathrm{Rb}},-m_{j}^{\mathrm{Cs}})$
following application of a resonant $\pi$-pulse for assuming a value of
$\Omega/2\pi=2$ MHz. Results are presented in Tab. 1 which is filtered to
select states with $P_{1r}(t_{\pi})>0.999$ consistent with strong blockade and
leakage errors below $10^{-3}$. For the Rb $d_{3/2}$-Cs $d_{5/2}$ interactions
no pair states reach this limit, with two example resonances included for
completeness, whilst for the other interaction channels several candidate pair
states are identified with leakage errors as low as $10^{-4}$ for Rb
$79d_{5/2}$-Cs $91d_{3/2}$ and Rb $80d_{5/2}$-Cs $92d_{3/2}$.

Since one of the motivations for using dual species interactions is not only
to obtain strong Rb-Cs couplings, but also to provide a route to realising
multi-qubit gates suppressing target-target and control-control couplings, we
additionally consider the impact of intraspecies interactions. We assume atoms
are arranged on a square lattice as shown in Fig. 1, and calculate the
probability of creating doubly excited Rb-Rb and Cs-Cs pair states at a
minimum distance of $\sqrt{2}R=8.5~{}\mu$m. For these states we evaluate both
the probability of creating a double excitation after a $\pi$-pulse, and of
returning back to the initial state following a $2\pi$ pulse which in the
ideal case of zero intraspecies interactions would return unity in both cases.
We also tabulate the effective $\tilde{C}_{3}$ and $\tilde{C}_{6}$
coefficients obtained by fitting the dominant interaction curves in the region
$4<R<15~{}\mu$m for the Rb-Cs, Rb-Rb and Cs-Cs interaction curves to enable
extrapolation of the interaction strength within these distances typically
used for tweezer experiments.

From Table 1 it can be seen that whilst many suitable Rb-Cs Förster resonances
can be found, those offering the highest suppression of blockade leakage
typically correspond to states with higher principal quantum number ($n\gtrsim
65$) where the residual intraspecies interactions at 8.5 $\mu$m are
sufficiently strong that there is a signficant blockade effect visible for the
Rb-Rb and Cs-Cs curves, making these poorly suited to extension to multi-qubit
gates. Whilst in practise these errors can be suppressed to some extent by
using increased single-photon Rabi frequencies, this comes at the cost of
increased leakage errors for the Rb-Cs interactions. This also highlights the
limitations of using $n_{\mathrm{Rb}}d_{3/2}$ states above, as from the table
it can be seen that for $n_{\mathrm{Rb}}\geq 58$ these have much a much
stronger blockade error for Rb-Rb than the comparable $n_{\mathrm{Rb}}d_{5/2}$
states. For completeness we note only a single $s$-state resonance achieves
$P_{1r}(t_{\pi})>0.999$ for which the resulting Rb-Rb and Cs-Cs interactions
are too strongly blockaded for use for multi-qubit gates in agreement with the
pair-potential shown above in Fig. 5(b).

Imposing the requirement to not only obtain low leakage errors for Rb-Cs but
also reduced errors for Rb-Rb and Cs-Cs operations, we identify a number of
possible states denoted by an asterisk in the table that balance the different
error channels. From these candidates $55d_{5/2}63d_{5/2}$,
$52d_{5/2}60d_{3/2}$, and $53d_{3/2}60d_{5/2}$ offer comparable performance of
$P>0.99$ for all metrics, and $59d_{5/2}68d_{3/2}$ achieves the lowest
intraspecies error with $P_{rr}(t_{\pi})=0.995$ for Rb-Rb whilst achieving
$P_{1r}(t_{\pi})>0.999$ for Rb-Cs. In the next section we use this state to
estimate realistic multi-qubit gate performance.

##  V Dual-species multi-qubit gate operations

In the analysis above the only error considered in performing gate operations
has been that of blockade errors due to errors associated with leakage of
population into doubly-occupied states for the Rb-Cs interactions, and the
suppression of doubly occupied states for the Rb-Rb and Cs-Cs case due to
residual intraspecies couplings. To accurately estimate gate fidelity, it is
necessary to also include errors associated with spontaneous emission from the
Rydberg state as well as scattering from the intermediate excited state during
the two-photon excitation of atoms to the Rydberg state.

To efficiently simulate these errors, we adopt the effective model introduced
in Ref. [16] to model each atom using three computational levels with an
effective single-atom Hamiltonian describing atom-light interactions of the
form

$\displaystyle\hat{H}_{\mathrm{eff}}=\frac{\hbar}{2}\Bigl{[}$ $\displaystyle-
iP_{e}^{1}\Gamma_{e}{|1\rangle}{\langle 1|}+\Omega_{R}({|r\rangle}{\langle
1|}+{|1\rangle}{\langle r|})$ (5)
$\displaystyle-(\delta+i\Gamma_{r}+P_{e}^{r}\Gamma_{e}){|r\rangle}{\langle
r|}\Bigr{]},$

where $\Omega_{R}$ is the effective two-photon Rabi frequency, $\delta$ is the
two-photon detuning, $\Gamma_{e,r}$ are the excited and Rydberg state
linewidths and $P_{e}^{1}$ and $P_{e}^{r}$ are excited state probabilities
determined from performing adiabatic elimination of the intermediate excited
states. For gates performed using the Rb $59d_{5/2}$-Cs $68d_{3/2}$ Förster
resonance, we first determine suitable parameters for the two-photon Rabi
frequency to ensure atoms obtain a $\pi$-phase shift when excited through
$2\pi$ by tuning the ratio between the two-photon Rabi frequencies [10]. For
both species, we assume a single-photon detuning of 10 GHz from the excited
state transition and determine parameters giving $\Omega_{R}/2\pi=5$ MHz. For
Rb, excitations are modelled via the $6p_{3/2}$ intermediate state with
$\Gamma_{e}/2\pi=1.4$ MHz and for Cs via $7p_{1/2}$ with $\Gamma_{e}/2\pi=1.0$
MHz. Full details of the excitation parameters are given in Appendix B.

Using the effective Hamiltonian, we model the canonical phase-gate protocol
based on a $\pi$-pulse on the control qubit, followed by a $2\pi$-pulse on the
target and a third $\pi$-pulse on the control qubit for an initial symmetric
superposition of all possible input states. We note that whilst this is not an
optimal pulse sequence, it provides a simple approach to extending to $C^{k}Z$
gates yielding the correct output states up to single qubit rotations [13].
For these diagonal controlled phase-gates, this also enables efficient
estimation of the resulting gate fidelity by calculating
$\mathcal{F}=|{\langle\Psi_{I}|}\Psi\rangle|^{2}$ where ${|\Psi\rangle}$ is
the final wavefunction and ${|\Psi_{I}\rangle}$ is the output of an ideal
gate.

We first consider a two-qubit $CZ$ gate for Rb-Cs atoms separated by
$R=6~{}\mu$m, which experience an interaction shift of $V_{\mathrm{Rb-
Cs}}/2\pi=65.6$ MHz. Using Rb as the control qubit give
$\mathcal{F}_{CZ}=0.9953$, whilst for Cs as the control qubit
$\mathcal{F}_{CZ}=0.9955$, competitive against recent experimental
demonstrations [29, 30]. Scaling to multi-qubit gates, we recall that this
pair-state was chosen due to the suppressed Rb-Rb interactions and proceed to
use a single Cs target atom and increase the number of neighboring Rb control
atoms.

For the $CCZ$ gate, placing atoms on a line with the Rb control atoms either
side of a central Cs target, the Rb-Rb interaction at 16 $\mu$m corresponds to
$V_{\mathrm{Rb-Rb}}/2\pi=0.11$ MHz, resulting in $\mathcal{F}_{CCZ}=0.994$,
whilst moving the second Rb control atom to the site below the Cs (at a
reduced Rb-Rb distance of $8.5~{}\mu$m) increases the control-control coupling
to $V_{\mathrm{Rb-Rb}}/2\pi=0.14$ MHz and results in
$\mathcal{F}_{CCZ}=0.983$. Extending to a $C^{3}Z$ gate with three Rb atoms in
an equilateral triangle configuration with the Cs atom at the centre gives
$\mathcal{F}_{C^{3}Z}=0.988$, and for the square geometry shown in Fig. 1(c)
we find a $C^{4}Z$ gate gives $\mathcal{F}_{C^{4}Z}=0.913$.

Whilst even with the engineered interspecies interactions we still observe a
reduction in fidelity due to residual intraspecies coupling, decomposition of
the $CCZ$ gate requires 6 $CZ$ operations [62], which would require
$\mathcal{F}_{CZ}>0.999$ and $0.997$ respectively for the line and square
configurations, demonstrating the benefit of using dual-species couplings for
native multi-qubit gate implementations. Additionally, further optimizaton of
gate pulses taking account of additional technical noise sources could yield
higher gate performance, but lies beyond the scope of this current work.

##  VI Conclusion

In this paper we have presented a detailed analysis of the $d$-state Förster
resonances for engineering strong Rb-Cs interactions to enable high-fidelity
gate operations between heterogeneous atom pairs. We have identified dominant
interaction channels yielding strong $C_{3,k}$ coefficients and small pair
defects, and demonstrated the importance of performing diagonalization of the
pair-state potentials to identify suitable resonances for achieving strong
short-range interactions.

From studying the angular dependence of the pair potentials we have shown that
for the $d$-states the Förster resonances are only robust at either
t$\theta=0^{\circ}$ or $90^{\circ}$ dependent upon choice of magnetic
sublevel, with
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(+j^{\mathrm{Rb}},-j^{\mathrm{Cs}})$
providing the best performance at $\theta=90^{\circ}$, but with additional
benefits compared to the $s$-state resonances through much greater suppression
of the intra-species interactions and providing stronger resonant interactions
at lower principal quantum number.

By calculating blockade leakage and excitation errors using the full pair-
state eigenvalues, we have identified states offering the best performance for
realising robust blockade for controlled gate operations, and demonstrated the
ability to use these strong interspecies couplings to realize high-fidelity
$C^{k}Z$ gates due to suppression of the residual intraspecies couplings
between control qubits compared to single-species gate implementations,
providing a route to native multi-qubit gates with performance exceeding that
expected for the equivalent two-qubit gate decompositions using current best-
known protocols [29, 16].

These results highlight the potential advantages of developing scalable
neutral atom processors based on dual atomic species, and will guide future
demonstrations of gate operations on these emerging Rb-Cs platforms [37, 46]
by providing improved choice over the best Förster-resonant pair states to
target.

###### Acknowledgements.

This work was funded by the Royal Academy of Engineering Senior Research
Fellowship with M Squared Lasers and the Engineering and Physical Sciences
Research Council (Grant No. EP/X025055/1). The data presented in the paper are
available here [63].

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##  Appendix A Tables of Rb-Cs $d$-state Förster Resonances

Below in Tables 2 to 5 we present
$n_{\mathrm{Rb}}d_{j}$-$n_{\mathrm{Cs}}d_{j}$ Förster resonant pair states
with $|\Delta|/2\pi\lesssim 20$ MHz and $|C_{3,k}|>1~{}$GHz$\,\mu$m3 for the
different combination of $j=3/2,5/2$. For each state we additionally show
$P_{1r}(t_{\pi})$ for Rb-Cs at $R=6~{}\mu$m to quantify blockade leakage
errors, and $P_{rr}(t_{\pi})$ for Rb-Rb and Cs-Cs interactions at
$R=8.5~{}\mu$m representative of expected errors for dual-species multi-qubit
gates on a square grid. Finally, we include the dispersion coefficients
$\tilde{C}_{3}$ and $\tilde{C}_{6}$ for inter- and intra-species couplings
respectively obtained by fitting the pair-eigenstate with the largest target
state probability in the region $5\leq R\leq 15~{}\mu$m. For comparison Tab. 6
includes the parameters obtained for $s$-state Förster-resonant pairs
previously identified in Ref. [47], which typically have higher leakage errors
than the $d$-states and much stronger blockade error for intraspecies
couplings with similar magnitude and range as the Rb-Cs coupling.

##  Appendix B Two-Photon Excitation Parameters

In this paper we consider interactions between $d$-orbital Rydberg states of
Cs and Rb which are created using a two-photon excitation process via an
intermediate excited state. To include the effects of excitation and
spontaneous decay from the intermediate states, we use the simplified
effective Hamiltonian introduced in Eq. 5 which is obtained by performing
adiabatic elimination of excited state hyperfine levels $f_{e}$ requiring
calculation of the following terms [16]

$\displaystyle\Omega_{R}=$
$\displaystyle\sum_{f_{e}}\frac{\Omega_{1}^{f_{e}}\Omega_{f_{e}}^{r}}{2\Delta_{f_{e}}},$
(6) $\displaystyle\delta_{\mathrm{AC}}=$
$\displaystyle\sum_{f_{e}}\frac{|\Omega_{1}^{f_{e}}|^{2}-|\Omega_{f_{e}}^{r}|^{2}}{4\Delta_{f_{e}}},$
(7) $\displaystyle P_{e}^{1}=$
$\displaystyle\sum_{f_{e}}\frac{|\Omega_{1}^{f_{e}}|^{2}}{4\Delta^{2}_{f_{e}}},$
(8) $\displaystyle P_{e}^{r}=$
$\displaystyle\sum_{f_{e}}\frac{|\Omega^{r}_{f_{e}}|^{2}}{4\Delta^{2}_{f_{e}}},$
(9)

where $\Omega_{R}$ is the effective two-photon Rabi frequency,
$\Delta_{f_{e}}$ is the single-photon detuning with respect to the
intermediate hyperfine level, and $P_{e}^{1,r}$ is the residual excited state
population due to coupling between the ground and Rydberg state transitions
respectively.

Using this effective model to represent the transition from ${|1\rangle}$ and
${|r\rangle}$ to appear as a spin-$1/2$ system enforces the the desired result
that after a $2\pi$ rotation the wavefunction acquires a $\pi$-phase shift.
However for a real two-photon excitation with multiple intermediate excited
states, the exact phase $\phi_{2\pi}$ accumulated after a $2\pi$-pulse can
take any value between $0$–$2\pi$ dependent upon the relative power of the
lasers and intermediate state detuning [10]. To ensure accurate parameters are
chosen for estimating the gate fidelity with the effective model, for each of
the transitions in Cs and Rb we first generate an exact single atom model
including all excited states without adiabatic ellimination, and adjust the
power-ratios in the two lasers to tune the system to give an error in the
phase-shift of $|\phi_{2\pi}-\pi|<10^{-5}$.

For both transitions we calculate single-photon Rabi frequencies
$\Omega_{1}^{f_{e}}$ and $\Omega_{f_{e}}^{r}$ assuming a centre of mass
detuning from the intermediate state of $10$ GHz, targetting a two-photon Rabi
frequency of $\Omega_{R}/2\pi=5$ MHz typical of recent experimental
demonstrations, and quote powers for a beam waist of 10 $\mu$m to allow easy
scaling to other beam parameters.

Excitation in Rb from ${|1\rangle}={|2,0\rangle}$ to
${|r\rangle}={|59d_{5/2},m_{j}=5/2\rangle}$ with $\Gamma_{r}/2\pi=0.76$ kHz is
modelled using transitions via the $6p_{3/2}$ intermediate state with
$\Gamma_{e}/2\pi=1.4$ MHz using a pair of $\sigma^{+}$-polarized photons.
Choosing powers of 20 mW and 150 mW for the lower and upper transitions
results in a $\pi$-phase shift with $P_{e}^{1,r}=(5.0,1.7)\times 10^{-4}$
respectively. Excitation in Cs from ${|1\rangle}={|4,0\rangle}$ to
${|r\rangle}={|68d_{3/2},m_{j}=-3/2\rangle}$ with $\Gamma_{r}/2\pi=0.86$ kHz
is modelled using transitions via the $7p_{1/2}$ intermediate state with
$\Gamma_{e}/2\pi=1.0$ MHz using a pair of $\sigma^{-}$-polarized photons.
Choosing powers of 5.2 mW and 240 mW for the lower and upper transitions
results in a $\pi$-phase shift with $P_{e}^{1,r}=(5.0,1.2)\times 10^{-4}$
respectively.

Rb - Cs Förster Resonances $\Delta/2\pi$ $C_{3,k}$ Rb-Cs at $R=6~{}\mu$m Rb-Rb
at $R=8.5~{}\mu$m Cs-Cs at $R=8.5~{}\mu$m Target Pair Resonant Pair (MHz)
(GHz$\,\mu$m3) $P_{1r}\left(t_{\pi}\right)$ $\tilde{C}_{3}$ (GHz$\,\mu$m3)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6) $55d_{5/2}$ \-
$53d_{5/2}$ $57p_{3/2}$ \- $50f_{7/2}$ 18.71 -2.73 0.9860 -3.13 0.9989 21.4
0.9999 -7.4 $55d_{5/2}$ \- $63d_{5/2}$ $56p_{3/2}$ \- $61f_{5/2}$ 12.66 -2.24
0.9969 -11.30 0.9989 21.5 0.9840 -98.8 $55d_{5/2}$ \- $61d_{5/2}$ $53f_{7/2}$
\- $63p_{3/2}$ -12.19 -1.78 0.9747 2.08 0.9989 21.4 0.9923 -62.1 $62d_{5/2}$
\- $71d_{5/2}$ $63p_{3/2}$ \- $69f_{5/2}$ -9.53 -3.66 0.9997 15.79 0.9855 64.1
0.7747 -416.8 $69d_{5/2}$ \- $79d_{5/2}$ $70p_{3/2}$ \- $77f_{7/2}$ -19.14
-21.97 0.9998 26.34 0.7855 192.8 0.0013 -1448.9 $69d_{5/2}$ \- $66d_{5/2}$
$71p_{3/2}$ \- $63f_{5/2}$ -4.29 -1.79 0.9933 7.49 0.7855 192.8 0.9538 -164.5
$77d_{5/2}$ \- $88d_{5/2}$ $78p_{3/2}$ \- $86f_{7/2}$ 1.91 -34.24 0.9998
-47.42 0.2966 393.5 0.0282 -4303.1 $74d_{5/2}$ \- $82d_{5/2}$ $72f_{7/2}$ \-
$84p_{3/2}$ 5.63 -6.15 0.9967 -16.85 0.3269 367.5 0.1115 -2258.4 $70d_{5/2}$
\- $80d_{5/2}$ $71p_{3/2}$ \- $78f_{7/2}$ 13.69 -23.21 0.9990 -29.03 0.7442
223.9 0.0238 -1812.1 $76d_{5/2}$ \- $87d_{5/2}$ $77p_{3/2}$ \- $85f_{5/2}$
-19.30 -8.41 0.9997 40.58 0.1156 450.0 0.0388 -4482.5 $84d_{5/2}$ \-
$80d_{5/2}$ $86p_{3/2}$ \- $77f_{7/2}$ 11.74 -15.58 0.9996 -24.33 0.0514
1318.7 0.0148 -1692.0 $80d_{5/2}$ \- $70d_{5/2}$ $78f_{7/2}$ \- $68f_{5/2}$
-18.70 3.05 0.9932 2.38 0.0229 698.0 0.8301 -356.6 $83d_{5/2}$ \- $79d_{5/2}$
$85p_{3/2}$ \- $76f_{7/2}$ -10.78 -14.80 0.9975 12.46 0.0957 1108.5 0.0013
-1448.9

Table 2: Rb-Cs Förster resonances from Rb $d_{5/2}$ \- Cs $d_{5/2}$ states
with $|\Delta|/2\pi<20~{}$MHz and $|C_{3,k}|>1$ GHz$\,\mu$m3.
$P_{1r}(t_{\pi})$ and $P_{rr}(t_{\pi})$ along with fitted $\tilde{C}_{3}$ and
$\tilde{C}_{6}$ coefficients are calculated for $\theta=90^{\circ}$ and
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(5/2,-5/2)$ assuming
$\Omega/2\pi=2$ MHz.

Rb - Cs Förster Resonances $\Delta/2\pi$ $C_{3,k}$ Rb-Cs at $R=6~{}\mu$m Rb-Rb
at $R=8.5~{}\mu$m Cs-Cs at $R=8.5~{}\mu$m Target Pair Resonant Pair (MHz)
(GHz$\,\mu$m3) $P_{1r}\left(t_{\pi}\right)$ $\tilde{C}_{3}$ (GHz$\,\mu$m3)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6) $51d_{3/2}$ \-
$59d_{3/2}$ $52p_{3/2}$ \- $57f_{5/2}$ -18.12 2.08 0.9912 0.83 0.9995 13.8
0.9995 -15.8 $50d_{3/2}$ \- $57d_{3/2}$ $51p_{1/2}$ \- $55f_{5/2}$ 16.13 -5.82
0.9958 -6.33 0.9997 11.4 0.9998 -10.9 $51d_{3/2}$ \- $62d_{3/2}$ $50f_{5/2}$
\- $59f_{5/2}$ -15.98 4.52 0.9814 3.15 0.9995 13.8 0.9995 -13.2 $65d_{3/2}$ \-
$75d_{3/2}$ $66p_{3/2}$ \- $73f_{5/2}$ 8.32 5.58 0.9955 -36.55 0.7416 234.6
0.7704 -440.2 $69d_{3/2}$ \- $66d_{3/2}$ $71p_{1/2}$ \- $63f_{5/2}$ -6.16
-7.35 0.9979 6.81 0.4719 224.4 0.9706 -141.0 $64d_{3/2}$ \- $61d_{3/2}$
$66p_{3/2}$ \- $58f_{5/2}$ 3.96 1.64 0.8913 -0.35 0.7767 211.9 0.9939 -63.6
$65d_{3/2}$ \- $74d_{3/2}$ $66p_{1/2}$ \- $72f_{5/2}$ -13.07 -17.02 0.9974
16.67 0.7416 234.6 0.8297 -375.1 $76d_{3/2}$ \- $67d_{3/2}$ $74f_{5/2}$ \-
$65f_{5/2}$ 2.04 8.99 0.9868 -12.99 0.0129 723.9 0.9587 -163.3 $77d_{3/2}$ \-
$73d_{3/2}$ $79p_{3/2}$ \- $70f_{5/2}$ -6.12 3.50 0.9789 3.82 0.0692 820.6
0.8729 -308.5 $71d_{3/2}$ \- $79d_{3/2}$ $69f_{5/2}$ \- $81p_{3/2}$ 9.09 1.63
0.9944 -11.82 0.2858 329.2 0.3238 -545.4 $70d_{3/2}$ \- $77d_{3/2}$
$68f_{5/2}$ \- $79p_{1/2}$ 7.97 -5.30 0.9961 -10.84 0.3789 287.4 0.6138 -629.1
$72d_{3/2}$ \- $83d_{3/2}$ $73p_{3/2}$ \- $81f_{5/2}$ 10.33 8.46 0.9985 -42.76
0.2049 390.5 0.1082 -2156.1 $79d_{3/2}$ \- $91d_{3/2}$ $80p_{3/2}$ \-
$89f_{5/2}$ 10.38 12.33 0.9998 -51.08 0.1280 1112.0 0.0234 -3178.2 $78d_{3/2}$
\- $90d_{3/2}$ $79p_{3/2}$ \- $88f_{5/2}$ -10.66 11.75 0.9998 21.39 0.1607
957.1 0.0262 -2568.1 $73d_{3/2}$ \- $83d_{3/2}$ $74p_{1/2}$ \- $81f_{5/2}$
-0.70 -27.26 0.9995 28.54 0.1360 452.0 0.1082 -2156.1 $81d_{3/2}$ \-
$92d_{3/2}$ $82p_{1/2}$ \- $90f_{5/2}$ 4.39 -41.55 0.9997 -47.30 0.0590 1411.2
0.0153 -3836.9 $84d_{3/2}$ \- $74d_{3/2}$ $82f_{5/2}$ \- $72f_{5/2}$ -8.89
13.56 0.9906 3.10 0.0368 1937.6 0.6502 -484.5 $84d_{3/2}$ \- $80d_{3/2}$
$86p_{1/2}$ \- $77f_{5/2}$ 11.47 -16.47 0.9953 -24.02 0.0368 1937.6 0.7637
-388.1 $80d_{3/2}$ \- $89d_{3/2}$ $78f_{5/2}$ \- $91p_{3/2}$ 4.98 2.68 0.9966
-21.96 0.0440 1263.6 0.0135 -2640.7 $80d_{3/2}$ \- $88d_{3/2}$ $78f_{5/2}$ \-
$90p_{1/2}$ -3.06 -9.23 0.9357 9.12 0.0440 1263.6 0.0007 -2226.4 $80d_{3/2}$
\- $91d_{3/2}$ $81p_{1/2}$ \- $89f_{5/2}$ -13.66 -39.62 0.9998 48.51 0.0440
1263.6 0.0234 -3178.2 $83d_{3/2}$ \- $79d_{3/2}$ $85p_{1/2}$ \- $76f_{5/2}$
-10.65 -15.65 0.9983 10.75 0.0603 1610.2 0.0904 -1035.9

Table 3: Rb-Cs Förster resonances from Rb $d_{3/2}$ \- Cs $d_{3/2}$ states
with $|\Delta|/2\pi<20~{}$MHz and $|C_{3,k}|>1$ GHz$\,\mu$m3.
$P_{1r}(t_{\pi})$ and $P_{rr}(t_{\pi})$ along with fitted $\tilde{C}_{3}$ and
$\tilde{C}_{6}$ coefficients are calculated for $\theta=90^{\circ}$ and
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(3/2,-3/2)$ assuming
$\Omega/2\pi=2$ MHz.

Rb - Cs Förster Resonances $\Delta/2\pi$ $C_{3,k}$ Rb-Cs at $R=6~{}\mu$m Rb-Rb
at $R=8.5~{}\mu$m Cs-Cs at $R=8.5~{}\mu$m Target Pair Resonant Pair (MHz)
(GHz$\,\mu$m3) $P_{1r}\left(t_{\pi}\right)$ $\tilde{C}_{3}$ (GHz$\,\mu$m3)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6) $59d_{5/2}$ \-
$68d_{3/2}$ $60p_{3/2}$ \- $66f_{5/2}$ 13.40 -11.23 0.9992 -14.36 0.9951 41.6
0.9410 -206.1 $50d_{5/2}$ \- $37d_{3/2}$ $49f_{7/2}$ \- $38p_{1/2}$ -5.91
-2.93 0.9864 3.87 0.9999 8.2 1.0000 -0.2 $52d_{5/2}$ \- $60d_{3/2}$
$53p_{3/2}$ \- $58f_{5/2}$ 15.19 -6.70 0.9964 -8.37 0.9997 12.2 0.9958 -45.0
$66d_{5/2}$ \- $76d_{3/2}$ $67p_{3/2}$ \- $74f_{5/2}$ 11.12 -17.74 0.9997
-22.84 0.9176 114.4 0.4467 -692.9 $68d_{5/2}$ \- $60d_{3/2}$ $66f_{5/2}$ \-
$58f_{5/2}$ -8.50 1.52 0.9272 3.71 0.8373 170.4 0.9993 -18.9 $60d_{5/2}$ \-
$66d_{3/2}$ $58f_{7/2}$ \- $68p_{1/2}$ -14.00 -2.87 0.9906 2.65 0.9929 48.3
0.9698 -144.1 $67d_{5/2}$ \- $49d_{3/2}$ $66f_{7/2}$ \- $50p_{1/2}$ -5.80
-9.64 0.9932 11.22 0.8889 142.5 1.0000 -2.7 $78d_{5/2}$ \- $74d_{3/2}$
$80p_{3/2}$ \- $71f_{5/2}$ 2.73 -11.14 0.9977 -16.52 0.0056 557.1 0.8297
-375.1 $79d_{5/2}$ \- $91d_{3/2}$ $80p_{3/2}$ \- $89f_{5/2}$ -12.53 -36.95
0.9999 49.07 0.0070 619.2 0.0052 -3596.5 $73d_{5/2}$ \- $84d_{3/2}$
$74p_{3/2}$ \- $82f_{5/2}$ 9.09 -26.72 0.9949 -36.07 0.4852 314.6 0.1085
-2230.7 $77d_{5/2}$ \- $56d_{3/2}$ $76f_{7/2}$ \- $57p_{1/2}$ -20.99 -16.90
0.9912 17.86 0.2966 393.5 0.9996 -14.4 $81d_{5/2}$ \- $90d_{3/2}$ $79f_{7/2}$
\- $92p_{3/2}$ 16.57 2.90 0.9953 -19.25 0.0649 765.4 0.0024 -3282.6
$83d_{5/2}$ \- $73d_{3/2}$ $81f_{5/2}$ \- $71f_{5/2}$ 15.70 3.43 0.9939 -12.68
0.0957 1108.5 0.8729 -308.5 $81d_{5/2}$ \- $89d_{3/2}$ $79f_{5/2}$ \-
$91p_{1/2}$ 5.56 -2.58 0.9802 -23.45 0.0858 839.1 0.0261 -2866.1 $84d_{5/2}$
\- $61d_{3/2}$ $86f_{7/2}$ \- $61p_{3/2}$ -6.70 -0.00 0.9991 26.62 0.0774
1227.1 0.9939 -63.6 $80d_{5/2}$ \- $92d_{3/2}$ $81p_{3/2}$ \- $90f_{5/2}$ 7.42
-38.76 0.9999 -51.12 0.0233 705.2 0.0144 -5756.9

Table 4: Rb-Cs Förster resonances from Rb $d_{5/2}$ \- Cs $d_{3/2}$ states
with $|\Delta|/2\pi\lesssim 20~{}$MHz and $|C_{3,k}|>1$ GHz$\,\mu$m3.
$P_{1r}(t_{\pi})$ and $P_{rr}(t_{\pi})$ along with fitted $\tilde{C}_{3}$ and
$\tilde{C}_{6}$ coefficients are calculated for $\theta=90^{\circ}$ and
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(5/2,-3/2)$ assuming
$\Omega/2\pi=2$ MHz.

Rb - Cs Förster Resonances $\Delta/2\pi$ $C_{3,k}$ Rb-Cs at $R=6~{}\mu$m Rb-Rb
at $R=8.5~{}\mu$m Cs-Cs at $R=8.5~{}\mu$m Target Pair Resonant Pair (MHz)
(GHz$\,\mu$m3) $P_{1r}\left(t_{\pi}\right)$ $\tilde{C}_{3}$ (GHz$\,\mu$m3)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6) $59d_{3/2}$ \-
$57d_{5/2}$ $61p_{1/2}$ \- $54f_{5/2}$ 2.93 -1.03 0.9823 -5.56 0.6608 187.8
0.9981 -32.2 $53d_{3/2}$ \- $60d_{5/2}$ $54p_{1/2}$ \- $58f_{5/2}$ -4.41 -1.97
0.9968 7.12 0.9989 19.7 0.9938 -57.0 $64d_{3/2}$ \- $71d_{5/2}$ $62f_{5/2}$ \-
$73p_{3/2}$ 4.08 -3.26 0.9909 -7.31 0.7767 211.9 0.7383 -435.9 $78d_{3/2}$ \-
$88d_{5/2}$ $79p_{1/2}$ \- $86f_{5/2}$ 6.51 -9.49 0.9915 -39.56 0.1607 957.1
0.0085 -5492.7 $76d_{3/2}$ \- $87d_{5/2}$ $77p_{3/2}$ \- $85f_{5/2}$ 6.49 2.81
0.9986 -52.91 0.0129 723.9 0.0186 -5038.3 $80d_{3/2}$ \- $70d_{5/2}$
$78f_{5/2}$ \- $68f_{5/2}$ 3.69 2.95 0.9920 -11.62 0.0440 1263.6 0.8048 -377.9
$83d_{3/2}$ \- $79d_{5/2}$ $85p_{3/2}$ \- $76f_{7/2}$ 8.93 4.92 0.9918 -32.33
0.0522 1813.4 0.0000 -1466.9 $90d_{3/2}$ \- $86d_{5/2}$ $92p_{1/2}$ \-
$83f_{5/2}$ 2.55 -5.81 0.9961 -40.04 0.0066 3762.9 0.0201 -4212.1

Table 5: Rb-Cs Förster resonances from Rb $d_{3/2}$ \- Cs $d_{5/2}$ states
with $|\Delta|/2\pi\lesssim 20~{}$MHz and $|C_{3,k}|>1$ GHz$\,\mu$m3.
$P_{1r}(t_{\pi})$ and $P_{rr}(t_{\pi})$ along with fitted $\tilde{C}_{3}$ and
$\tilde{C}_{6}$ coefficients are calculated for $\theta=90^{\circ}$ and
$(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(3/2,-5/2)$ assuming
$\Omega/2\pi=2$ MHz.

Rb - Cs Förster Resonances $\Delta/2\pi$ $C_{3,k}$ Rb-Cs at $R=6~{}\mu$m Rb-Rb
at $R=8.5~{}\mu$m Cs-Cs at $R=8.5~{}\mu$m Target Pair Resonant Pair (MHz)
(GHz$\,\mu$m3) $P_{1r}\left(t_{\pi}\right)$ $\tilde{C}_{3}$ (GHz$\,\mu$m3)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6)
$P_{rr}\left(t_{\pi}\right)$ $\tilde{C}_{6}$ (GHz$\,\mu$m6) $48s_{1/2}$ \-
$51s_{1/2}$ $48p_{3/2}$ \- $50p_{1/2}$ -5.53 1.69 0.6281 3.00 0.9998 -9.5
0.9996 -14.5 $59s_{1/2}$ \- $57s_{1/2}$ $58p_{1/2}$ \- $57p_{1/2}$ -16.45 3.54
0.3752 1.35 0.9767 -114.2 0.9943 -56.6 $69s_{1/2}$ \- $68s_{1/2}$ $68p_{1/2}$
\- $68p_{3/2}$ -9.89 6.92 0.9959 12.69 0.3230 -701.2 0.6503 -430.2 $68s_{1/2}$
\- $67s_{1/2}$ $67p_{1/2}$ \- $67p_{3/2}$ 2.63 6.50 0.9727 -9.12 0.4660 -594.3
0.7432 -367.5 $61s_{1/2}$ \- $65s_{1/2}$ $61p_{1/2}$ \- $64p_{1/2}$ 2.77 4.80
0.9735 -2.62 0.9487 -168.9 0.8689 -263.5 $72s_{1/2}$ \- $75s_{1/2}$
$72p_{1/2}$ \- $74p_{3/2}$ 2.70 9.65 0.9835 -12.40 0.0040 -1106.6 0.0359
-1187.3 $77s_{1/2}$ \- $81s_{1/2}$ $77p_{3/2}$ \- $80p_{1/2}$ -2.10 12.28
0.9969 16.42 0.0100 -2103.4 0.0344 -2151.8 $72s_{1/2}$ \- $70s_{1/2}$
$71p_{3/2}$ \- $70p_{1/2}$ -7.90 8.51 0.9992 9.82 0.0040 -1106.6 0.4005 -605.3
$71s_{1/2}$ \- $69s_{1/2}$ $70p_{3/2}$ \- $69p_{1/2}$ 9.45 8.01 0.9987 -10.66
0.0643 -955.7 0.5348 -518.5

Table 6: Rb-Cs Förster resonances from Rb $s_{1/2}$ \- Cs $s_{1/2}$ states
previously identified [47]. $P_{1r}(t_{\pi})$ and $P_{rr}(t_{\pi})$ along with
fitted $\tilde{C}_{3}$ and $\tilde{C}_{6}$ coefficients are calculated for
$\theta=90^{\circ}$ and $(m_{j}^{\mathrm{Rb}},m_{j}^{\mathrm{Cs}})=(1/2,-1/2)$
assuming $\Omega/2\pi=2$ MHz.