# Dissipative stabilization of high-dimensional GHZ states for neutral atoms

Yue Zhao These authors contributed equally to this work.  Center for Quantum
Sciences and School of Physics, Northeast Normal University, Changchun 130024,
China  Yu-Qing Yang These authors contributed equally to this work.  Center
for Quantum Sciences and School of Physics, Northeast Normal University,
Changchun 130024, China  Weibin Li  Authors to whom correspondence should be
addressed<EMAIL_ADDRESS>and<EMAIL_ADDRESS>School of
Physics and Astronomy, and Centre for the Mathematics and Theoretical Physics
of Quantum Non-equilibrium Systems, The University of Nottingham, Nottingham
NG7 2RD, United Kingdom  Xiao-Qiang Shao  Authors to whom correspondence
should be addressed<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China  Center for Advanced Optoelectronic
Functional Materials Research, and Key Laboratory for UV Light-Emitting
Materials and Technology of Ministry of Education, Northeast Normal
University, Changchun 130024, China

###### Abstract

High-dimensional quantum entanglement characterizes the entanglement of
quantum systems within a larger Hilbert space, introducing more intricate and
complex correlations among the entangled particles’ states. The high-
dimensional Greenberger-Horne-Zeilinger (GHZ) state, symbolic of this type of
entanglement, is of significant importance in various quantum information
processing applications. This study proposes integrating a neutral atom
platform with quantum reservoir engineering to generate a high-dimensional GHZ
state deterministically. Leveraging the advantages of neutral atoms in a
modified unconventional Rydberg pumping mechanism, combined with controlled
dissipation, we achieve a three-dimensional GHZ state with a fidelity
surpassing 99% through multiple pump and dissipation cycles. This innovative
approach paves the way for experimentally feasible, deterministic preparation
of high-dimensional GHZ states in Rydberg atom systems, thereby advancing the
capabilities of quantum information processing.

††preprint: AIP/123-QED

Entanglement within higher-dimensional quantum systems introduces more
intricate and complex correlations among the states of the entangled
particles.Vaziri, Weihs, and Zeilinger (2002); O’Sullivan-Hale  _et al._
(2005); Huber and de Vicente (2013); Kues  _et al._ (2017); Wang  _et al._
(2018); Cozzolino  _et al._ (2019); Luo  _et al._ (2019); Hu  _et al._
(2020a); Erhard, Krenn, and Zeilinger (2020) They can prove beneficial for
quantum computation and simulation,Lanyon  _et al._ (2009); Neeley  _et al._
(2009); Kaltenbaek  _et al._ (2010); Bocharov, Roetteler, and Svore (2017)
enhance security in quantum key distribution protocols,Cerf  _et al._ (2002);
Durt  _et al._ (2004); Huber and Pawłowski (2013) provide increased capacity
and noise resistance in quantum communications,Bechmann-Pasquinucci and Tittel
(2000); Wang  _et al._ (2005); Walborn  _et al._ (2006); Ali-Khan, Broadbent,
and Howell (2007); Zhang  _et al._ (2013); Islam  _et al._ (2017) and produce
stronger violations of Bell-type inequalities. Kaszlikowski  _et al._ (2000);
Durt, Kaszlikowski, and Żukowski (2001) These remarkable advantages of high-
dimensional quantum entanglement underscore its significance as a valuable
resource for quantum information processing, which has led researchers to
actively explore methods for preparing high-dimensional entangled states in
various physical systems.Shao  _et al._ (2014a, b); Malik  _et al._ (2016);
Hebbache (2016); Zhang  _et al._ (2017); Wang  _et al._ (2018); Guo  _et al._
(2018); Reimer  _et al._ (2019); Hu  _et al._ (2020b); Ringbauer  _et al._
(2022); Hrmo  _et al._ (2023)

The high-dimensional GHZ state is defined by the expression
$|\textup{GHZ}(N,d)\rangle=\sum_{k=0}^{d-1}|k\rangle^{\bigotimes N}/\sqrt{d}$,
involving $N$ particles, with each particle encoding a qudit of dimension $d$.
This state is of notable significance in various applications, such as quantum
key distributionPivoluska, Huber, and Malik (2018); Zhang  _et al._ (2020) and
programmable qudit-based quantum processors.Chi  _et al._ (2022) In the realm
of linear optics, only a limited number of theoretical and experimental
strategies have been proposed to create high-dimensional GHZ states. These
strategies take advantage of various degrees of freedom of photons as primary
carriers of information.Krenn  _et al._ (2017); Krenn, Gu, and Zeilinger
(2017); Erhard  _et al._ (2018); Gu  _et al._ (2019); Paesani  _et al._
(2021); Bell  _et al._ (2022); Bao  _et al._ (2023); Xing  _et al._ (2023) One
such scheme suggests superposition of photon pairs with different crystals and
alignment of photon paths to engineer multiphoton GHZ states in higher
dimensions Krenn  _et al._ (2017) with an efficiency of $E=d/[(N\cdot
d)/2]^{N/2}$. Subsequently, the same group, drawing inspiration from the
computer algorithm MELVIN,Krenn  _et al._ (2016) develops a novel multiport
design to generate a genuine three-dimensional GHZ state. This approach
requires two pairs of three-dimensionally entangled photon pairs as the
entanglement resource.Erhard  _et al._ (2018) Another theoretical approach is
proposed, employing a Fourier transform matrix within a linear optical circuit
in a photonic system to generate high-dimensional states.Paesani  _et al._
(2021) While this successfully addresses the scalability issue present in
postselected schemes,Krenn  _et al._ (2017); Erhard  _et al._ (2018) the
success probability diminishes exponentially with an increase in dimension and
photon number. For example, the success probability for generating the
$|\textup{GHZ}(3,3)\rangle$ state is merely $10^{-10}$. Even with optimization
efforts, it can only be enhanced to $10^{-4}$.

(a)

(b)

Figure 1: Schematic for a potential experimental setup. (a) The atoms are
confined using optical tweezers and arranged in an equilateral triangle within
the $x$–$y$ plane. The use of counterpropagating circularly polarized light
beams with various wavelengths significantly reduces the influence of the
Doppler effect during atomic excitation from the ground state to the Rydberg
state. Simultaneously, co-propagating circularly polarized light beams with
the same wavelength prevents the creation of unpredictable relative phases
between ground states. (b) Depiction of pertinent levels in 87Rb and the
laser-atom interaction model illustrate the implementation of the modified
unconventional Rydberg pumping.

The characteristics of the linear optical system determine that all the
schemes based on it are nondeterministic. Therefore, to prepare a
deterministic high-dimensional GHZ state suitable for quantum information
processing, it is necessary to leverage the coupling of light and matterBell
_et al._ (2022) or explore the development of alternative matter qubits. An
experimental demonstration and certification of a three-dimensional GHZ state
with superconducting transmon qutrits was recently reported.Cervera-Lierta
_et al._ (2022) This scheme is also grounded in the quantum circuit employed
in MELVIN, Krenn  _et al._ (2016) and the experiment is executed using IBMQ
quantum processors. Nevertheless, due to the inevitable interaction between
the superconducting quantum processor and its surroundings, various
decoherence channels emerge, particularly qubit crosstalk, a common occurrence
in superconducting transmon devices. This results in a fidelity of
approximately 76% for the final prepared quantum entangled state.

In this Letter, our objective is to deterministically prepare a high-
dimensional GHZ state by integrating quantum reservoir engineering into a
neutral-atom platform which provides a controlled manner for trapping,
cooling, and manipulating atoms. Glaetzle  _et al._ (2015); Prajapati  _et
al._ (2021); Hu  _et al._ (2022); Fancher  _et al._ (2023) This approach
converts the decoherence factors, such as atomic spontaneous emission, of the
quantum system into valuable resources and operates independently of the
preparation of the initial state, which represents a fundamentally distinct
scenario from previous preparations of high-dimensional GHZ states in linear
optical systems and superconducting systems.

The neutral atom exhibits a plethora of stable hyperfine ground states, with
examples including 8 for Rubidium and 16 for Cesium. Leveraging these states
proves effective in expanding the dimensionality of a qudit.Saffman, Walker,
and Mølmer (2010) By engineering pairwise additive interactions between
excited Rydberg states, entangling operations can be facilitated, potentially
simplifying the scaling of quantum systems.Jaksch  _et al._ (2000); Lukin  _et
al._ (2001); Tong  _et al._ (2004); Ates  _et al._ (2007); Urban  _et al._
(2009); Gaëtan  _et al._ (2009); Amthor  _et al._ (2010); Pupillo  _et al._
(2010); Johnson and Rolston (2010); Honer  _et al._ (2010); Omran  _et al._
(2019); Browaeys and Lahaye (2020); Bluvstein  _et al._ (2021); Scholl  _et
al._ (2021); Graham  _et al._ (2022); Bluvstein  _et al._ (2022); Srakaew  _et
al._ (2023); Evered  _et al._ (2023) By incorporating the inherent advantages
of neutral atoms into the modified unconventional Rydberg pumping, as outlined
in the supplementary material, we can effectively stabilize the system within
a high-dimensional GHZ state throughout multiple pump and dissipation cycles.
To elucidate the mechanism of the current scheme more comprehensively, this
work predominantly focuses on a detailed discussion of the preparation of
three-dimensional GHZ states. Concerning GHZ states with higher dimensions,
their extension can be readily achieved, in principle.

The requisite experimental configuration for our proposed scheme is
illustrated in Fig. 1. We envision three 87Rb atoms confined using optical
tweezers and arranged in an equilateral triangle within the $x$–$y$ plane,
where the interatomic distance ensures effective operation of the van der
Waals (vdW) regime. Each qutrit is encoded into three hyperfine ground states:
$|0\rangle=|5S_{1/2},F=2,m_{F}=-1\rangle$,
$|1\rangle=|5S_{1/2},F=1,m_{F}=0\rangle$, and
$|2\rangle=|5S_{1/2},F=2,m_{F}=1\rangle$. Simultaneously, we introduce the
degenerate subspace $\\{|r^{\pm}\rangle=|80S_{1/2},m_{J}=\pm 1/2\rangle\\}$ of
the highly excited Rydberg state to modulate the interaction between the
atoms. The incorporation of an additional $s$-orbital Rydberg sublevel alters
the original Hamiltonian of the system, rendering it time-independent.Li and
Shao (2018) This adjustment facilitates the implementation of rigorous and
efficient numerical simulations. In the two-photon process mediated by the
$5P_{1/2}$ states, the Rabi frequencies and the red (blue) detuning for the
transition from the ground states $|0\rangle$, $|1\rangle$, and $|2\rangle$ to
$|r^{+}\rangle$ are indicated by $\pm\Omega_{b}$ and $\Delta_{1}$,
respectively. Similarly, the Rabi frequencies and red (blue) detuning for the
transition from ground states to $|r^{-}\rangle$ are denoted as
$\pm\Omega_{a}$ and $\Delta_{2}$ ($\Delta_{2}+\Delta$). Furthermore, coupling
between the $5P_{1/2}$ state and the two Rydberg states is achieved through
distinct polarized light sources, thus minimizing the influence of light with
varying frequencies and intensities. For simplification purposes, we
temporarily overlook the influence of atomic position on the relative phase
between individual atoms, reserving this discussion for the supplementary
material.

Aligned with the laser cooling approach, our methodology comprises repetitive
cycles of pumping and dissipation, as shown in Fig. 2. Within each cycle, we
systematically perform six distinct coherent operations, each succeeded by a
consistent dissipative operation. Significantly, at each stage, the utilized
laser serves the purpose of collectively exciting three atoms, obviating the
need for individual addressing.

Figure 2:  The flowchart illustrates the step-by-step process for the
preparation of the target state $|\textup{GHZ(3,3)}\rangle$. The light green
shading in B or C indicates the excitation of some of these states. In the
middle section, lasers are applied to different ground states at each step.
The lower part of this section details the timing of the driving in each step,
denoted as $\pi/\Omega_{2}$ or $\pi/(\sqrt{2}\Omega_{2})$. The light pink
segment portrays the dissipation occurring in each step, and its lower portion
specifies the dissipation time $\tau$.

In the interaction picture, the corresponding Hamiltonian of the coherent
operation at step $i$ can be described by ($\hbar$=1)

$\displaystyle{\hat{H}}^{(i)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{3}\frac{\alpha^{i}_{j}}{2}(\Omega_{1}|r^{-}_{j}\rangle\langle
a_{j}^{i}|+\Omega_{2}|r^{+}_{j}\rangle\langle a_{j}^{i}|)+{\rm
H.c.}-\Delta|r^{-}_{j}\rangle\langle r^{-}_{j}|$ (1)
$\displaystyle-\sum_{k\neq j}[\frac{C_{6}}{2R^{6}}(|r^{+}_{j}\rangle\langle
r^{+}_{j}|\otimes|r^{+}_{k}\rangle\langle r^{+}_{k}|+|r^{-}_{j}\rangle\langle
r^{-}_{j}|\otimes|r^{-}_{k}\rangle\langle r^{-}_{k}|)$
$\displaystyle+\frac{C\textquoteright_{6}}{R^{6}}|r^{+}_{j}\rangle\langle
r^{+}_{j}|\otimes|r^{-}_{k}\rangle\langle r^{-}_{k}|],$

where we have obtained the effective single-photon transition with Rabi
frequencies
$\Omega_{1}={\Omega_{a}^{2}(2\Delta_{2}+\Delta)}/[{4\Delta_{2}(\Delta_{2}+\Delta)}]$
and $\Omega_{2}=\Omega_{b}^{2}/({2\Delta_{1}})$ between ground states and
Rydberg states after adiabatically eliminating the intermediate state
$|5P_{1/2}\rangle$ in Fig. 1(b). The state $|a_{j}^{i}\rangle$ represents the
laser-driven normalized ground state of the $j$th atom, with $\alpha_{j}^{i}$
serving as the associated collective enhancement factor. It should be noted
that in the first three steps, the parameter $\alpha_{j}^{i}$ is set to 1,
while for subsequent steps it is adjusted to $\sqrt{2}$. Additionally, the
dispersion coefficients $C_{6}$ and $C^{\prime}_{6}$ for the high-lying $s$
state of Rydberg atoms demonstrate isotropy with respect to the polar angle,
representing the orientation of the interatomic connection to the quantization
axis. Specifically, for $n=80$, their values are $C_{6}/(2\pi)=-4160$ GHz
$(\mu\mathrm{m})^{6}$ and $C^{\prime}_{6}/(2\pi)=-4213$ GHz
$(\mu\mathrm{m})^{6}$. Consequently, the interaction strength between Rydberg
atoms can be tuned by altering the atomic spacing $R$. In the scenario of
$C^{\prime}_{6}/R^{6}=-\Delta$, and under the limit of
$\Delta\gg\Omega_{1}\gg\Omega_{2}$, an effective Hamiltonian can be derived as
(see the supplementary material for details)

$\displaystyle\hat{H}_{\textup{eff}}^{(i)}=$
$\displaystyle\frac{\alpha^{i}\Omega_{2}}{2}\sum_{m,n}(|r^{+}_{1}\rangle\langle
a_{1}^{i}|P_{2}^{m}P_{3}^{n}+P_{1}^{m}|r^{+}_{2}\rangle\langle
a_{2}^{i}|P_{3}^{n}$ (2)
$\displaystyle+P_{1}^{m}P_{2}^{n}|r^{+}_{3}\rangle\langle
a_{3}^{i}|)+\mathrm{H.c.},$

where $P^{m}$ and $P^{n}$ represent the projection operators onto two distinct
bases that are orthogonal to the state $|a^{i}\rangle$ within the atomic
ground state. Following Eq. (2), the system allows only a single excitation
process associated with the ground state, allowing the ground state
$|a^{i}\rangle$ to undergo excitation to the Rydberg state $|r^{+}\rangle$ at
intervals of $\pi/\Omega_{2}$ during the initial three steps, and with a
period of $\pi/(\sqrt{2}\Omega_{2})$ during the subsequent steps. The jump
operators that characterize each atomic spontaneous emission, taking into
account the decay rate, are defined as follows:

$\hat{L}^{l}=\sqrt{\frac{\Gamma_{1}}{3}}|l\rangle\langle r^{+}|,\ \
(l=0,1,2).$ (3)

Here, the effective decay rate $\Gamma_{1}$ can be significantly increased
beyond the natural linewidth of $|80S_{1/2}\rangle$ for 87Rb atoms using
controlled dissipation techniques. Simonelli  _et al._ (2017); Shao  _et al._
(2023); Bégoc  _et al._ (2023) This enhancement facilitates the acceleration
of the atomic relaxation process to the desired ground state.

Figure 3:  The temporal evolutions of fidelity and purity of the target state
$|{\textup{GHZ(3,3)}}\rangle$ are illustrated against time (number of cycles)
starting from an arbitrary initial state. The selected parameters are
$\Omega_{1}=2\pi\times 4$ MHz, $\Omega_{2}=2\pi\times 0.04$ MHz,
$\Delta=2\pi\times 200$ MHz, $\Gamma_{1}=2\pi\times 0.23$ MHz, and
$\tau=3.19~{}\mu$s. The time required for 30 cycles is approximately 2.49 ms.
The inset provides a detailed view of the fidelity in the first circle for
each step.

To elaborate on the details of the scheme, we partition the ground state space
of three atoms into three distinct subspaces. Subspace A is represented by
$\\{(|000\rangle+|111\rangle+|222\rangle)/\sqrt{3}\\}$, which corresponds to
the target state $|{\rm GHZ(3,3)}\rangle$. The subspace B
$=\\{(2|222\rangle-|000\rangle-|111\rangle)/\sqrt{6},(|000\rangle-|111\rangle)/\sqrt{2}\\}$
consists of states that are orthogonal to the target state. Together, subspace
A and subspace B form a complete set of subspace
$\\{|000\rangle,|111\rangle,|222\rangle\\}$. Lastly, subspace C encompasses
the remaining 24 orthogonal basis states. Under the action of Eq. (2), the
initial three steps illustrated in Fig. 2 allow one of the three atoms in the
ground state to be excited to the Rydberg state $|r^{+}\rangle$ if and only if
the pumped atom is initialized in the $|a^{i}\rangle$. After controlled
dissipation, the distribution of each component of the atomic ground state is
adjusted. We find that under the assumption of considering the first three
steps in general, the stability of subspace C is destroyed and the system will
be stabilized to a mixed state composed of states in subspaces A and B. The
operational steps in IV, V, and VI are analogous to the initial three steps.
The impact of these lasers becomes more apparent when we express the bare
ground states of all atoms in terms of the dressed states, as deduced in the
supplementary material. Let us consider step IV as an example. The state
$|a^{IV}\rangle=(|0\rangle-|1\rangle)/\sqrt{2}$ can be pumped to
$|r^{+}\rangle$ after a period $\pi/(\sqrt{2}\Omega_{2})$, and then
$|r^{+}\rangle$ decays towards $|0\rangle$, $|1\rangle$, and $|2\rangle$
independently. This process decreases the population of $|a^{IV}\rangle$ and
improves the population of
$|a^{IV}_{\perp}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$. Consequently, within
the subspaces of A and B, all mixed states with components like
$(|000\rangle-|111\rangle)/\sqrt{2}=(|a^{IV}a^{IV}a^{IV}\rangle+|a_{\perp}^{IV}a_{\perp}^{IV}a^{IV}\rangle+|a_{\perp}^{IV}a^{IV}a_{\perp}^{IV}\rangle+|a^{IV}a_{\perp}^{IV}a_{\perp}^{IV}\rangle)/2$
are selectively pumped out. In contrast, components containing
$(|000\rangle+|111\rangle)/\sqrt{2}=(|a_{\perp}^{IV}a_{\perp}^{IV}a_{\perp}^{IV}\rangle+|a^{IV}a^{IV}a_{\perp}^{IV}\rangle+|a^{IV}a_{\perp}^{IV}a^{IV}\rangle+|a_{\perp}^{IV}a^{IV}a^{IV}\rangle)/2$
in the mixed states persistently increase.

Similarly, in steps V and VI, the components of
$(|000\rangle+|222\rangle)/\sqrt{2}$ and $(|111\rangle+|222\rangle)/\sqrt{2}$
also experience an increase. Hence, it becomes evident that after an
evolutionary period, only the population of the target state $|{\rm
GHZ(3,3)}\rangle$ remains unaffected by laser pumping, gradually increasing
through controlled spontaneous emission. This is illustrated by the light blue
shading in the inset of Fig. 3. After multiple cycles, the initial population
in any state will be transferred to the target state. To underscore the
robustness of the scheme against variations in the initial state, we
intentionally choose a completely mixed state,
$\rho(0)=\sum_{i,j,k=0,1,2}|ijk\rangle\langle ijk|/27$, as the initial
condition. The fidelity, defined by the population ${\mathcal{F}}=\langle{\rm
GHZ(3,3)}|\rho(t)|{\rm GHZ(3,3)}\rangle$, of the target state $|{\rm
GHZ(3,3)}\rangle$, and the purity ${\mathcal{P}}={\rm Tr}[\rho(t)^{2}]$ are
then plotted against time (number of cycles) in Fig. 3, providing insight into
the performance of the scheme. The corresponding parameters,
$\Omega_{1}=2\pi\times 4$ MHz, $\Delta=50\Omega_{1}$ and
$\Omega_{2}=0.01\Omega_{1}$, are chosen to satisfy the conditions of
unconventional Rydberg pumping. After 30 cycles, the fidelity of
$|{\textup{GHZ(3,3)}}\rangle$ reaches 99.1%, and the entire process is
completed in approximately 2.49 ms, showcasing the efficacy of the chosen
parameters within the specified temporal constraints.

In the context of Rydberg atom platform experiments, various imperfections are
inherent, encompassing factors such as the random thermal motion of atoms
within the optical tweezers, spontaneous emission from the intermediate state,
laser phase noise, and the Doppler effect.Saffman and Walker (2005); de
Léséleuc  _et al._ (2018); Lee  _et al._ (2019) These imperfections
significantly impact the coherence between the ground state and the excited
state of Rydberg atoms. Addressing these factors is essential for overcoming
challenges and realizing efficient quantum information processing in the
Rydberg atomic system. In our proposed scheme, the spontaneous emission of
atoms has been transformed into a valuable resource, and the remaining factors
can be attributed to the influence of finite temperature. Specifically, the
dephasing effect induced by laser phase noise can be equivalently regarded as
the Doppler effect.

Given the atomic arrangement illustrated in Fig.1(a), for atoms at a
temperature $T_{a}$, the time-averaged variances of position and momentum are
approximated as $\langle x^{2}\rangle=\langle
y^{2}\rangle=k_{B}T_{a}/(\omega^{2}m)$ and $\langle v_{x}^{2}\rangle=\langle
v_{y}^{2}\rangle=k_{B}T_{a}/m$. Here, $k_{B}$ represents the Boltzmann
constant, $\omega=\omega_{x(y)}$ is the oscillation frequency of the trap, and
$m$ is the atomic mass. Longitudinal position fluctuations combine in
quadrature, making a lesser contribution to distance fluctuations.Bernien  _et
al._ (2017) Therefore, the deviation of the interatomic distance is expressed
as $\sigma=\sqrt{2k_{B}T_{a}/(\omega^{2}m)}$. For a typical trap frequency of
$\omega=2\pi\times 90~{}$kHz,Béguin  _et al._ (2013) an atomic temperature of
$10~{}\mu$K results in $\sigma=77.37~{}$nm. The lower atomic temperatures
experimentally obtained, $T_{a}=5.2~{}\mu$K and $T_{a}=1~{}\mu$K, lead to
reduced position fluctuations of $\sigma=55.79$ nm and $\sigma=24.47~{}$nm,
respectively.Fu  _et al._ (2022); Zhao  _et al._ (2023) In a recent
experiment, a higher trap frequency with
$(\omega_{x},\omega_{y})=2\pi\times(147,117)~{}$kHz Chew  _et al._ (2022) can
be used to further suppress fluctuation of the atomic spacing.

Figure 4:  The influence of distance and timing errors is showcased in two
distinct scenarios. In (a), a rectangular pulse with a Rabi frequency of
$\Omega_{2}=2\pi\times 0.04$ MHz is considered, whereas in (b), the analysis
entails a rectangular pulse with a reduced Rabi frequency of
$\Omega_{2}=2\pi\times 0.02$ MHz, achieving a more robust effect.

To assess the resilience of our approach to variations in atomic spacing
$\overline{\delta R}$ around the target value of $R_{0}=5.255~{}\mu$m and to
account for timing errors $\delta t$ during coherent operations, we illustrate
a contour plot of the target state population after 30 cycles as a function of
$\overline{\delta R}$ and $\delta t$ in Fig. 4(a). The target state population
remains consistently above 90% within the $\overline{\delta R}$ range of
$[-50,50]~{}$nm. The impact of timing errors ${\delta t}/t_{i}\in[-0.2,0.2]$
with $t_{i}$ corresponding to the timing for each step in Fig. 2, denoted as
$\pi/\Omega_{2}$ or $\pi/(\sqrt{2}\Omega_{2})$ in the population is
nonmonotonic, manifesting itself as oscillations. In the supplementary
material, we present evidence demonstrating that selecting a smaller Rabi
frequency for the (effective single-photon) resonant laser enhances the
resilience of the target state population against variations in atomic
spacing, as illustrated in Fig. 4(b). While this improvement is attained by
extending the duration of the coherent operations and may be accompanied by
potential influences of various unknown decoherence factors, it nonetheless
stands as a viable option to effectively tackle challenges arising from
fluctuations in atomic spacing.

In our setup, where linearly polarized light and circularly polarized light
are employed, the effective wave vector for the two-photon excitation from the
ground state to the excited state of the Rydberg atom is conservatively
estimated to be $k_{\rm eff}\approx 1.54\times 10^{7}~{}{\rm m^{-1}}$ at
maximum (orthogonal beams). As a result, the atom is subjected to an extra
detuning of the excitation laser, which is represented as a random variable
with a Gaussian probability distribution centered around zero and a standard
deviation of $k_{\rm eff}\sqrt{k_{B}T_{a}/m}\approx
2\pi\times(76,55,24)~{}$kHz. This effect, however, has no effect on the two-
photon off-resonance pathway from the ground state to the Rydberg state
$|r^{-}\rangle$. The higher Rabi frequency $\Omega_{1}$ associated with this
process is attributed to this,de Léséleuc  _et al._ (2018) and the frequency
detuning induced by the Doppler effect is negligible compared to the detuning
of $\Delta$ in our scheme. On the other hand, for the resonant Rydberg atomic
transition, the counterpropagating 474-nm and 795-nm wavelengths yield a small
effective wave vector of $k_{\rm eff}\approx 5.35\times 10^{6}~{}{\rm
m^{-1}}$. This implies a corresponding $k_{\rm eff}\sqrt{k_{B}T_{a}/m}\approx
2\pi\times(26,19,8.3)~{}$kHz. Applying this factor to the resonant Rydberg
atomic transition using standard random sampling and average processing is
unquestionably time-consuming. Within this particular framework, we simplify
the procedure by replacing the detuning frequency with the standard deviation
previously mentioned and qualitatively determine the influence of the Doppler
effect on our scheme. In Fig. 5, we track the progression of target state
fidelity as influenced by these three parameters over multiple cycles,
respectively. We permit some populations to remain in the ground state under
each coherent control due to the periodic pumping and dissipation of the
scheme itself, these residual populations can be progressively transferred to
the target entangled state of the atom in the subsequent cyclic operation.
Consequently, we observe that it is still possible to attain a greater
population of the target state by increasing the number of cycles.

Figure 5:  The influence of Doppler effect. The driving field with Rabi
frequency $\Omega_{2}$ have an extra detuning of $2\pi\times(8.3,19,26)$ kHz.

In conclusion, we have presented a scheme for the deterministic preparation of
high-dimensional GHZ states, leveraging periodic pumping and dissipation in a
neutral atom system. This method remains independent of the initial state
preparation and exhibits resilience to fluctuations in atomic spacing and
timing errors. Moreover, it proves advantageous in mitigating the impact of
the Doppler effect, as a higher fidelity target state can still be achieved by
increasing the number of cycles. While extending our approach to a high-
dimensional GHZ state is straightforward, it necessitates additional coherent
manipulations and evolutionary cycles. However, the spatial arrangement
structure limitations between Rydberg atoms pose a challenge, rendering the
current experimental parameters insufficient to guarantee the robust creation
of a high-dimensional GHZ state with more than three particles. We anticipate
that ongoing developments in atomic manipulation technology and advancements
in atomic cooling technology will pave the way for the realistic preparation
of a high-dimensional multi-particle GHZ state in the future.

See the supplementary material for a detailed description of the effective
two-level system, the effective Hamiltonian, the controlled spontaneous
emission of Rydberg states, the relative phases of atoms induced by wave
vectors, and the extension of three-dimensional GHZ states into higher
dimensions.

This work is supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 12174048. W.L. acknowledges support from the EPSRC
through Grant No. EP/W015641/1, and the Going Global Partnerships Programme of
the British Council (Contract No. IND/CONT/G/22-23/26).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Yue Zhao: Numerical simulation (equal); Writing-original draft (equal). Yu-
Qing Yang: Numerical simulation (equal); Writing-original draft (equal).
Weibin Li: Writing-review & editing (equal); Supervision (equal). Xiao-Qiang
Shao: Conceptualization; Funding acquisition; Writing-review & editing
(equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

## REFERENCES

## References

  * Vaziri, Weihs, and Zeilinger (2002) A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
  * O’Sullivan-Hale  _et al._ (2005) M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled $d=3$ and $d=6$ qudits,” Phys. Rev. Lett. 94, 220501 (2005).
  * Huber and de Vicente (2013) M. Huber and J. I. de Vicente, “Structure of multidimensional entanglement in multipartite systems,” Phys. Rev. Lett. 110, 030501 (2013).
  * Kues  _et al._ (2017) M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
  * Wang  _et al._ (2018) J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
  * Cozzolino  _et al._ (2019) D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
  * Luo  _et al._ (2019) Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, A. Zeilinger, and J.-W. Pan, “Quantum teleportation in high dimensions,” Phys. Rev. Lett. 123, 070505 (2019).
  * Hu  _et al._ (2020a) X.-M. Hu, C. Zhang, B.-H. Liu, Y. Cai, X.-J. Ye, Y. Guo, W.-B. Xing, C.-X. Huang, Y.-F. Huang, C.-F. Li, and G.-C. Guo, “Experimental high-dimensional quantum teleportation,” Phys. Rev. Lett. 125, 230501 (2020a).
  * Erhard, Krenn, and Zeilinger (2020) M. Erhard, M. Krenn, and A. Zeilinger, “Advances in high-dimensional quantum entanglement,” Nature Reviews Physics 2, 365–381 (2020).
  * Lanyon  _et al._ (2009) B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nature Physics 5, 134–140 (2009).
  * Neeley  _et al._ (2009) M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, A. N. Cleland, M. R. Geller, and J. M. Martinis, “Emulation of a quantum spin with a superconducting phase qudit,” Science 325, 722–725 (2009).
  * Kaltenbaek  _et al._ (2010) R. Kaltenbaek, J. Lavoie, B. Zeng, S. D. Bartlett, and K. J. Resch, “Optical one-way quantum computing with a simulated valence-bond solid,” Nature Physics 6, 850–854 (2010).
  * Bocharov, Roetteler, and Svore (2017) A. Bocharov, M. Roetteler, and K. M. Svore, “Factoring with qutrits: Shor’s algorithm on ternary and metaplectic quantum architectures,” Phys. Rev. A 96, 012306 (2017).
  * Cerf  _et al._ (2002) N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using $\mathit{d}$-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
  * Durt  _et al._ (2004) T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
  * Huber and Pawłowski (2013) M. Huber and M. Pawłowski, “Weak randomness in device-independent quantum key distribution and the advantage of using high-dimensional entanglement,” Phys. Rev. A 88, 032309 (2013).
  * Bechmann-Pasquinucci and Tittel (2000) H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
  * Wang  _et al._ (2005) C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
  * Walborn  _et al._ (2006) S. P. Walborn, D. S. Lemelle, M. P. Almeida, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
  * Ali-Khan, Broadbent, and Howell (2007) I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
  * Zhang  _et al._ (2013) Z. Zhang, M. Tengner, T. Zhong, F. N. C. Wong, and J. H. Shapiro, “Entanglement’s benefit survives an entanglement-breaking channel,” Phys. Rev. Lett. 111, 010501 (2013).
  * Islam  _et al._ (2017) N. T. Islam, C. C. W. Lim, C. Cahall, J. Kim, and D. J. Gauthier, “Provably secure and high-rate quantum key distribution with time-bin qudits,” Science Advances 3, e1701491 (2017).
  * Kaszlikowski  _et al._ (2000) D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled $\mathit{N}$-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418–4421 (2000).
  * Durt, Kaszlikowski, and Żukowski (2001) T. Durt, D. Kaszlikowski, and M. Żukowski, “Violations of local realism with quantum systems described by n-dimensional hilbert spaces up to $n=16$,” Phys. Rev. A 64, 024101 (2001).
  * Shao  _et al._ (2014a) X.-Q. Shao, T.-Y. Zheng, C. H. Oh, and S. Zhang, “Dissipative creation of three-dimensional entangled state in optical cavity via spontaneous emission,” Phys. Rev. A 89, 012319 (2014a).
  * Shao  _et al._ (2014b) X.-Q. Shao, J.-B. You, T.-Y. Zheng, C. H. Oh, and S. Zhang, “Stationary three-dimensional entanglement via dissipative rydberg pumping,” Phys. Rev. A 89, 052313 (2014b).
  * Malik  _et al._ (2016) M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and Z. Anton, “Multi-photon entanglement in high dimensions,” Nature Photonics 10, 248–252 (2016).
  * Hebbache (2016) M. Hebbache, “Generation of a three-qudit ghz state with diamond defect spins,” Europhysics Letters 115, 28003 (2016).
  * Zhang  _et al._ (2017) D. Zhang, Y. Zhang, X. Li, D. Zhang, L. Cheng, C. Li, and Y. Zhang, “Generation of high-dimensional energy-time-entangled photon pairs,” Phys. Rev. A 96, 053849 (2017).
  * Guo  _et al._ (2018) Y. Guo, X.-M. Hu, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C. Guo, “Experimental realization of path-polarization hybrid high-dimensional pure state,” Opt. Express 26, 28918–28926 (2018).
  * Reimer  _et al._ (2019) C. Reimer, S. Sciara, P. Roztocki, M. Islam, L. Romero Cortés, Y. Zhang, B. Fischer, S. Loranger, R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, W. J. Munro, J. Azaña, M. Kues, and R. Morandotti, “High-dimensional one-way quantum processing implemented on d-level cluster states,” Nature Physics 15, 148–153 (2019).
  * Hu  _et al._ (2020b) X.-M. Hu, W.-B. Xing, C. Zhang, B.-H. Liu, M. Pivoluska, M. Huber, Y.-F. Huang, C.-F. Li, and G.-C. Guo, “Experimental creation of multi-photon high-dimensional layered quantum states,” npj Quantum Information 6, 88 (2020b).
  * Ringbauer  _et al._ (2022) M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, “A universal qudit quantum processor with trapped ions,” Nature Physics 18, 1053–1057 (2022).
  * Hrmo  _et al._ (2023) P. Hrmo, B. Wilhelm, L. Gerster, M. W. van Mourik, M. Huber, R. Blatt, P. Schindler, T. Monz, and M. Ringbauer, “Native qudit entanglement in a trapped ion quantum processor,” Nature Communications 14, 2242 (2023).
  * Pivoluska, Huber, and Malik (2018) M. Pivoluska, M. Huber, and M. Malik, “Layered quantum key distribution,” Phys. Rev. A 97, 032312 (2018).
  * Zhang  _et al._ (2020) X.-H. Zhang, X.-Y. Yan, Y.-Q. Wang, and L.-H. Gong, “Tripartite layered quantum key distribution scheme with a symmetrical key structure,” Phys. Rev. A 59, 562–573 (2020).
  * Chi  _et al._ (2022) Y. Chi, J. Huang, Z. Zhang, J. Mao, Z. Zhou, X. Chen, C. Zhai, J. Bao, T. Dai, H. Yuan, M. Zhang, D. Dai, B. Tang, Y. Yang, Z. Li, Y. Ding, L. K. Oxenløwe, M. G. Thompson, J. L. O’Brien, Y. Li, Q. Gong, and J. Wang, “A programmable qudit-based quantum processor,” Nature Communications 13, 1166 (2022).
  * Krenn  _et al._ (2017) M. Krenn, A. Hochrainer, M. Lahiri, and A. Zeilinger, “Entanglement by path identity,” Phys. Rev. Lett. 118, 080401 (2017).
  * Krenn, Gu, and Zeilinger (2017) M. Krenn, X. Gu, and A. Zeilinger, “Quantum experiments and graphs: Multiparty states as coherent superpositions of perfect matchings,” Phys. Rev. Lett. 119, 240403 (2017).
  * Erhard  _et al._ (2018) M. Erhard, M. Malik, M. Krenn, and Z. Anton, “Experimental greenberger–horne–zeilinger entanglement beyond qubits,” Nature Photonics 12, 759–764 (2018).
  * Gu  _et al._ (2019) X. Gu, M. Erhard, A. Zeilinger, and M. Krenn, “Quantum experiments and graphs ii: Quantum interference, computation, and state generation,” Proceedings of the National Academy of Sciences 116, 4147–4155 (2019).
  * Paesani  _et al._ (2021) S. Paesani, J. F. F. Bulmer, A. E. Jones, R. Santagati, and A. Laing, “Scheme for universal high-dimensional quantum computation with linear optics,” Phys. Rev. Lett. 126, 230504 (2021).
  * Bell  _et al._ (2022) T. J. Bell, J. F. F. Bulmer, A. E. Jones, S. Paesani, D. P. S. McCutcheon, and A. Laing, “Protocol for generation of high-dimensional entanglement from an array of non-interacting photon emitters,” New Journal of Physics 24, 013032 (2022).
  * Bao  _et al._ (2023) J. Bao, Z. Fu, T. Pramanik, J. Mao, Y. Chi, Y. Cao, C. Zhai, Y. Mao, T. Dai, X. Chen, J. Bao, Z. Fu, T. Pramanik, J. Mao, Y. Chi, Y. Cao, C. Zhai, Y. Mao, T. Dai, X. Chen, X. Jia, L. Zhao, Y. Zheng, B. Tang, Z. Li, J. Luo, W. Wang, Y. Yang, Y. Peng, D. Liu, D. Dai, Q. He, A. L. Muthali, L. K. Oxenløwe, C. Vigliar, S. Paesani, H. Hou, R. Santagati, J. W. Silverstone, A. Laing, M. G. Thompson, J. L. O’Brien, Y. Ding, Q. Gong, and J. Wang, “Very-large-scale integrated quantum graph photonics,” Nature Photonics 17, 573–581 (2023).
  * Xing  _et al._ (2023) W.-B. Xing, X.-M. Hu, Y. Guo, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Preparation of multiphoton high-dimensional ghz states,” Opt. Express 31, 24887–24896 (2023).
  * Krenn  _et al._ (2016) M. Krenn, M. Malik, R. Fickler, R. Lapkiewicz, and A. Zeilinger, “Automated search for new quantum experiments,” Phys. Rev. Lett. 116, 090405 (2016).
  * Cervera-Lierta  _et al._ (2022) A. Cervera-Lierta, M. Krenn, A. Aspuru-Guzik, and A. Galda, “Experimental high-dimensional greenberger-horne-zeilinger entanglement with superconducting transmon qutrits,” Phys. Rev. Appl. 17, 024062 (2022).
  * Glaetzle  _et al._ (2015) A. W. Glaetzle, M. Dalmonte, R. Nath, C. Gross, I. Bloch, and P. Zoller, “Designing frustrated quantum magnets with laser-dressed rydberg atoms,” Phys. Rev. Lett. 114, 173002 (2015).
  * Prajapati  _et al._ (2021) N. Prajapati, A. K. Robinson, S. Berweger, M. T. Simons, A. B. Artusio-Glimpse, and C. L. Holloway, “Enhancement of electromagnetically induced transparency based Rydberg-atom electrometry through population repumping,” Applied Physics Letters 119, 214001 (2021).
  * Hu  _et al._ (2022) J. Hu, H. Li, R. Song, J. Bai, Y. Jiao, J. Zhao, and S. Jia, “Continuously tunable radio frequency electrometry with Rydberg atoms,” Applied Physics Letters 121, 014002 (2022).
  * Fancher  _et al._ (2023) C. T. Fancher, K. L. Nicolich, K. M. Backes, N. Malvania, K. Cox, D. H. Meyer, P. D. Kunz, J. C. Hill, W. Holland, and B. L. Schmittberger Marlow, “A self-locking Rydberg atom electric field sensor,” Applied Physics Letters 122, 094001 (2023).
  * Saffman, Walker, and Mølmer (2010) M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
  * Jaksch  _et al._ (2000) D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85, 2208–2211 (2000).
  * Lukin  _et al._ (2001) M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
  * Tong  _et al._ (2004) D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93, 063001 (2004).
  * Ates  _et al._ (2007) C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in rydberg excitation of an ultracold lattice gas,” Phys. Rev. Lett. 98, 023002 (2007).
  * Urban  _et al._ (2009) E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. Yavuz, T. Walker, and M. Saffman, “Observation of rydberg blockade between two atoms,” Nature Physics 5, 110–114 (2009).
  * Gaëtan  _et al._ (2009) A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the rydberg blockade regime,” Nature Physics 5, 115–118 (2009).
  * Amthor  _et al._ (2010) T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of antiblockade in an ultracold rydberg gas,” Phys. Rev. Lett. 104, 013001 (2010).
  * Pupillo  _et al._ (2010) G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly correlated gases of rydberg-dressed atoms: Quantum and classical dynamics,” Phys. Rev. Lett. 104, 223002 (2010).
  * Johnson and Rolston (2010) J. E. Johnson and S. L. Rolston, “Interactions between rydberg-dressed atoms,” Phys. Rev. A 82, 033412 (2010).
  * Honer  _et al._ (2010) J. Honer, H. Weimer, T. Pfau, and H. P. Büchler, “Collective many-body interaction in rydberg dressed atoms,” Phys. Rev. Lett. 105, 160404 (2010).
  * Omran  _et al._ (2019) A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Generation and manipulation of schrödinger cat states in rydberg atom arrays,” Science 365, 570–574 (2019).
  * Browaeys and Lahaye (2020) A. Browaeys and T. Lahaye, “Many-body physics with individually controlled rydberg atoms,” Nature Physics 16, 132–142 (2020).
  * Bluvstein  _et al._ (2021) D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuletić, and M. D. Lukin, “Controlling quantum many-body dynamics in driven rydberg atom arrays,” Science 371, 1355–1359 (2021).
  * Scholl  _et al._ (2021) P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, “Quantum simulation of 2d antiferromagnets with hundreds of rydberg atoms,” Nature 595, 233–238 (2021).
  * Graham  _et al._ (2022) T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, X. Jiang, A. Marra, B. Grinkemeyer, M. Kwon, M. Ebert, J. Cherek, M. T. Lichtman, M. Gillette, J. Gilbert, D. Bowman, T. Ballance, C. Campbell, E. D. Dahl, O. Crawford, N. S. Blunt, B. Rogers, T. Noel, and M. Saffman, “Multi-qubit entanglement and algorithms on a neutral-atom quantum computer,” Nature 604, 457–462 (2022).
  * Bluvstein  _et al._ (2022) D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, _et al._ , “A quantum processor based on coherent transport of entangled atom arrays,” Nature 604, 451–456 (2022).
  * Srakaew  _et al._ (2023) K. Srakaew, P. Weckesser, S. Hollerith, D. Wei, D. Adler, I. Bloch, and J. Zeiher, “A subwavelength atomic array switched by a single rydberg atom,” Nature Physics 19, 714–719 (2023).
  * Evered  _et al._ (2023) S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletić, and M. D. Lukin, “High-fidelity parallel entangling gates on a neutral-atom quantum computer,” Nature 622, 268–272 (2023).
  * Li and Shao (2018) D. X. Li and X. Q. Shao, “Unconventional rydberg pumping and applications in quantum information processing,” Phys. Rev. A 98, 062338 (2018).
  * Simonelli  _et al._ (2017) C. Simonelli, M. Archimi, L. Asteria, D. Capecchi, G. Masella, E. Arimondo, D. Ciampini, and O. Morsch, “Deexcitation spectroscopy of strongly interacting rydberg gases,” Phys. Rev. A 96, 043411 (2017).
  * Shao  _et al._ (2023) X. Q. Shao, F. Liu, X. W. Xue, W. L. Mu, and W. Li, “High-fidelity interconversion between greenberger-horne-zeilinger and $w$ states through floquet-lindblad engineering in rydberg atom arrays,” Phys. Rev. Appl. 20, 014014 (2023).
  * Bégoc  _et al._ (2023) B. Bégoc, G. Cichelli, S. P. Singh, F. Perciavalle, D. Rossini, L. Amico, and O. Morsch, “Controlled dissipation for rydberg atom experiments,” arXiv preprint arXiv:2310.20687 (2023).
  * Saffman and Walker (2005) M. Saffman and T. G. Walker, “Analysis of a quantum logic device based on dipole-dipole interactions of optically trapped rydberg atoms,” Phys. Rev. A 72, 022347 (2005).
  * de Léséleuc  _et al._ (2018) S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, “Analysis of imperfections in the coherent optical excitation of single atoms to rydberg states,” Phys. Rev. A 97, 053803 (2018).
  * Lee  _et al._ (2019) W. Lee, M. Kim, H. Jo, Y. Song, and J. Ahn, “Coherent and dissipative dynamics of entangled few-body systems of rydberg atoms,” Phys. Rev. A 99, 043404 (2019).
  * Bernien  _et al._ (2017) H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
  * Béguin  _et al._ (2013) L. Béguin, A. Vernier, R. Chicireanu, T. Lahaye, and A. Browaeys, “Direct measurement of the van der waals interaction between two rydberg atoms,” Phys. Rev. Lett. 110, 263201 (2013).
  * Fu  _et al._ (2022) Z. Fu, P. Xu, Y. Sun, Y.-Y. Liu, X.-D. He, X. Li, M. Liu, R.-B. Li, J. Wang, L. Liu, and M.-S. Zhan, “High-fidelity entanglement of neutral atoms via a rydberg-mediated single-modulated-pulse controlled-phase gate,” Phys. Rev. A 105, 042430 (2022).
  * Zhao  _et al._ (2023) L. Zhao, M. D. K. Lee, M. M. Aliyu, and H. Loh, “Floquet-tailored rydberg interactions,” Nature Communications 14, 7128 (2023).
  * Chew  _et al._ (2022) Y. Chew, T. Tomita, T. P. Mahesh, S. Sugawa, S. de Léséleuc, and K. Ohmori, “Ultrafast energy exchange between two single rydberg atoms on a nanosecond timescale,” Nature Photonics 16, 724–729 (2022).