# Cooling neutral atoms into maximal entanglement in the Rydberg blockade
regime

Wei-Lin Mu  Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun, 130024, China  Xiao-Xuan Li  Center for Quantum
Sciences and School of Physics, Northeast Normal University, Changchun,
130024, China<EMAIL_ADDRESS>Xiao-Qiang Shao  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
<EMAIL_ADDRESS>
###### Abstract

We propose a cooling scheme to prepare stationary entanglement of neutral
atoms in the Rydberg blockade regime by combination of periodically collective
laser pumping and dissipation. In each cycle, the controlled unitary dynamics
process can selectively pump atoms away from the non-target state while
maintaining the target state unchanged. The subsequent dissipative process
redistributes the populations of ground states through the engineered
spontaneous emission. After a number of cycles, the system will be eventually
stabilized into the desired steady state independent of the initial state.
This protocol does not rely on coherent addressing of individual neutral atoms
or fine control of Rydberg interaction intensity, which can in principle
greatly improve the feasibility of experiments in related fields.

††journal: ol

Quantum decoherence, resulting from the inevitable interaction between the
interested quantum system and its surroundings, has recently been reexamined
as a resource in quantum information processing. The earliest idea of using
dissipation in generation of entangled states dates back to the pioneering
work made by Plenio _et al._ , where they provided a probabilistic scheme for
generating an entangled state of two atoms by continuous monitoring of photons
leaking out of the cavity [1]. After that, the nontrivial dissipative
interactions combined with continuous and stationary control fields are
exploited to stabilize entangled states on various physical platforms in a
deterministic manner [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Neutral-atom systems
are considered to be powerful candidates among many physical systems for
quantum information processing because the highly excited Rydberg states
possess long lifetimes and extremely large dipole moments, especially the
latter feature can suppress the double excitation of Rydberg states from
ground-state atoms over a long distance relative to the atomic scale [12]. In
contrast with Rydberg blockade, the Rydberg antiblockade has more advantages
in dissipative preparation of entangled states due to the selective excitation
of ground states [13, 14, 15]. Nevertheless, such kind of facilitation
dynamics requires a delicate balance between the Rydberg interaction intensity
and the single-photon detuning parameter. Although the unconventional Rydberg
pumping mechanism proposed by our group can reduce the influence of
interatomic distance fluctuation to a certain extent, it is still not
applicable for preparation of multi-atom entanglement by dissipation [16, 17].
In this letter, we intend to incorporate the Rydberg blockade mechanism into
the dissipative protocols, such as multi-dimensional entanglement and
multipartite entanglement, which brings an advantage that the scheme becomes
first-order independent of the large blockade shift, so the defects in the
Rydberg antiblockade-based schemes can be avoided. Meanwhile, in order to
solve the problem of the stationarity of the target state under driving, we
introduce the pulsed laser instead of continuous laser, and only require the
target state to remain stable at the end of the pulse node, rather than the
entire driving process. The aforementioned unitary dynamics combined with the
atomic spontaneous emission constitutes a cycle of our scheme, and after many
cycles, the system will be stabilized into the maximally entangled state
irrespective to the initial state of system. This method can be described
mathematically as floquet-Lindblad dynamics and physically as modified laser
sideband cooling. It is noteworthy that the idea of pumping cycles has only
ever been used to improve continuous implementation in trapped-ion systems
[18, 19, 20, 21], while the alternate driving of various pulses is a
requirement for the current model due to the fact that continuous laser
driving is unable to provide selective excitation for the ground states in the
Rydberg blockade regime.

The relevant level structure of 87Rb atom is shown in Fig. 1. Three ground
states $|g\rangle=|F=1,m_{F}=1\rangle$, $|e\rangle=|F=2,m_{F}=1\rangle$, and
$|h\rangle=|F=2,m_{F}=2\rangle$ are the Zeeman sublevels of $5S_{1/2}$. The
intermediate states $|p_{1}\rangle=|F=2,m_{F}=2\rangle$ and
$|p_{2}\rangle=|F=2,m_{F}=0\rangle$ are temporary (short-lived) levels of
$5P_{3/2}$ and $6P_{3/2}$, respectively. $|r\rangle=|J=1/2,m_{J}=1/2\rangle$
is the Rydberg level of $100S_{1/2}$. $|g\rangle$ ($|e\rangle$) is driven to
$|p_{2}\rangle$ by a $\sigma^{+}$ polarized 420 nm laser of Rabi frequency
$\Omega_{a2}$, which is then pumped to the Rydberg state by a $\pi$ polarized
1011 nm laser of Rabi frequency $\Omega_{a1}$. The intermediate state
$|p_{2}\rangle$ can be eliminated adiabatically under large detuning
conditions, resulting in an effective pumping process driving the transition
$|g\rangle\leftrightarrow|r\rangle$ ($|e\rangle\leftrightarrow|r\rangle$) with
Rabi frequency $\Omega_{a}=\Omega_{a1}\Omega_{a2}/(2\Delta)$ [22, 23], where
$\Delta$ is the detuning parameter of the pumped lasers. In addition, we
introduce another $\sigma^{-}$ polarized 480 nm laser pulse of Rabi frequency
$\Omega_{b}$ to resonantly couple the Rydberg state $|r\rangle$ to the fast
decaying state $|p_{1}\rangle$ of lifetime $\gamma^{-1}=2.62\times 10^{-8}s$,
so as to modulate the spontaneous emission rate of the Rydberg state.
According to the selection rule for electric dipole transition, the branching
ratios of decay from $|p_{1}\rangle$ to $|g\rangle$, $|e\rangle$, and
$|h\rangle$ are 1/6, 1/2, and 1/3, respectively.

Figure 1: The internal states of atoms. This model incorporates three ground
states $|g\rangle$, $|e\rangle$, and $|h\rangle$ of the Zeeman sublevels of
the $5S_{1/2}$, temporary (short-lived) levels $|p_{1}\rangle$ of $5P_{3/2}$
and $|p_{2}\rangle$ of $6P_{3/2}$, and a Rydberg state $|r\rangle$ of
$100S_{1/2}$. Laser pulses of Rabi frequency $\Omega_{a}$ interacts resonantly
with the transition $|g\rangle\leftrightarrow|r\rangle$
($|e\rangle\leftrightarrow|r\rangle$) and laser pulses of Rabi frequency
$\Omega_{b}$ drive the transition $|r\rangle\leftrightarrow|p_{1}\rangle$. The
lifetime of the short-lived state $|p_{1}\rangle$ is $\gamma^{-1}=2.62\times
10^{-8}s$. Figure 2: (a) Population evolution of different ground states with
the number of cycles, where
$|\Phi^{-}\rangle=1/\sqrt{2}(|gg\rangle-|ee\rangle)$,
$|\Psi^{+}\rangle=1/\sqrt{2}(|eg\rangle+|ge\rangle)$, and
$|\Psi^{-}\rangle=1/\sqrt{2}(|eg\rangle-|ge\rangle)$. The population of the
target state $|\Phi^{+}\rangle$ under the effective master equation is shown
by the rhombus symbol, and the full dynamical evolution, including the natural
spontaneous emission of the Rydberg state, is represented by the cross symbol.
(b) Process of system evolution to the three-dimensional entangled state from
three different initial mixed states. (c) Population evolution of different
GHZ states with the number of cycles. (d) Population evolution of
$|\Phi^{+}\rangle$ under different Rydberg interaction intensities $U$. The
relevant parameters are $\Omega_{a}=2\pi\times 2$ MHz, $\Omega_{b}=2\pi\times
1.2$ MHz, $\Omega_{c}=2\pi\times 20$ kHz, and
$\delta=2\pi\times\sqrt{{2}/{3}}$ MHz.

In order to better understand the principle of the scheme, we first discuss
the dissipative generation of the Bell state of two atoms with form of
$|\Phi^{+}\rangle=1/\sqrt{2}(|gg\rangle+|ee\rangle)=1/\sqrt{2}(|++\rangle+|--\rangle)$,
where $|\pm\rangle={1}/{\sqrt{2}}(|g\rangle\pm|e\rangle)$. The whole process
consists of repeated cycles with each consisting of three steps. In step A, we
switch on the global laser driving the transition
$|g\rangle\leftrightarrow|r\rangle$. Under condition where the Rydberg
interaction $U$ is much stronger than the effective Rabi frequency
$\Omega_{a}$, the effective Hamiltonian in the interaction picture can be
described as $\hat{H}_{a}^{I}={\Omega_{a}}/{2}(|er\rangle\langle
eg|+|re\rangle\langle ge|)+\Omega_{a}/{2}(|rg\rangle+|gr\rangle)\langle
gg|+{\rm H.c.}.$ For atoms initialized in the ground states, the population of
$|gg\rangle$ is recovered after a driving period
$\tau_{a}={\sqrt{2}\pi}/{\Omega_{a}}$, while the populations of $|eg\rangle$
and $|ge\rangle$ cannot be completely restored due to the difference in the
Rabi frequency between the single and collective excitation processes. Then we
switch off the pump laser and start the dissipation process by switching on
the laser that drives the transition $|r\rangle\leftrightarrow|p_{1}\rangle$.
Under the premise of ignoring the natural Rydberg spontaneous emission, the
engineered spontaneous emission rate of $|r\rangle$ is derived as $\gamma_{\rm
eff}=\Omega_{b}^{2}/\gamma$ [17] when $\Omega_{b}\ll\gamma$. The corresponding
Markovian master equation reads
$\dot{\hat{\rho}}=\sum_{j=1}^{2}\\{{\gamma^{g}_{\rm{eff}}}/{2}\mathcal{\hat{L}}[|g\rangle_{jj}\langle
r|]+{\gamma^{e}_{\rm{eff}}}/{2}\mathcal{\hat{L}}[|e\rangle_{jj}\langle
r|]\\}\hat{\rho},$ where
$\mathcal{\hat{L}}[\hat{c}]\hat{\rho}=2\hat{c}\hat{\rho}\hat{c}^{+}-\\{\hat{c}^{+}\hat{c}\hat{\rho},\hat{\rho}\hat{c}^{+}\hat{c}\\}$.
This dissipation process can redistribute the population remaining on
$|r\rangle$ after executing the previous unitary dynamics to $|g\rangle$ and
$|e\rangle$ irreversibly. Note that the population of hyperfine ground state
$|h\rangle$ is ignored here because it can be recycled with a series of
$\pi$-polarized lights. The operations of step B and step C are similar to
that of step A, which both require controlled unitary evolution and
dissipation. The only difference is that in step B, we apply the drive from
$|r\rangle$ to $|e\rangle$, and in step C, we apply the drive from $|r\rangle$
to $|+\rangle$. The corresponding times should ensure that the population of
$|ee\rangle$ and the probability amplitude of $|++\rangle$ each undergo a
periodic oscillation, respectively. Therefore, the system will be stabilized
into $|\Phi^{+}\rangle$ after continuously cyclic evolution according to steps
A, B and C. Fig. 2(a) shows that after 20 cycles ($0.142$ms), the population
of the Bell state $|\Phi^{+}\rangle$ will be greater than 99$\%$ starting from
the fully mixed state $\rho_{0}=\sum_{i,j=g,e}|ij\rangle\langle ij|/4$, where
the parameters are $U=2\pi\times 400$ MHz, $\Delta=2\pi\times 1250$ MHz,
$\Omega_{a1}=2\pi\times 50$ MHz, $\Omega_{a2}=2\pi\times 50$ MHz,
$\Omega_{b}=2\pi\times 1.2$ MHz, $\gamma=2\pi\times 6$ MHz, and the relaxation
time in each step is chosen as $2\mu$s.

Based on the above scheme of preparing Bell state, the stationary three-
dimensional entanglement can be produced by introducing microwave fields
$\hat{H}_{mw}^{I}={\Omega_{c}}/{2}\sum_{j=1}^{2}(|e\rangle_{jj}\langle
g|+|h\rangle_{jj}\langle g|)+{\rm H.c.}$ to globally drive hyperfine level
transition [PhysRevA.89.052313]. Each cycle of this scenario only consists of
step A’ and step B’, which are similar to step A and step B in the preparation
of Bell state, but the driving time should ensure that the probability
amplitudes of $|gg\rangle$ ($|ee\rangle$) experiences a periodic evolution
rather than the population, and the above microwave fields need to be applied
after the engineered dissipation process characterized by
$\dot{\hat{\rho}}=\sum_{j=1}^{2}\\{\gamma^{g}_{\rm{eff}}/2\mathcal{\hat{L}}[|g\rangle_{jj}\langle
r|]+{\gamma^{e}_{\rm{eff}}}/{2}\mathcal{\hat{L}}[|e\rangle_{jj}\langle
r|]+{\gamma^{h}_{\rm{eff}}}/{2}\mathcal{\hat{L}}[|h\rangle_{jj}\langle
r|]\\}\hat{\rho}.$ There are three degenerate dark states for
$\hat{H}_{c}^{I}$, i.e.,
$|T_{1}\rangle={1}/{\sqrt{3}}(|ee\rangle-|gg\rangle+|hh\rangle)$,
$|T_{2}\rangle={1}/{\sqrt{3}}(|eh\rangle-|gg\rangle+|he\rangle)$, and
$|T_{3}\rangle={1}/{2}(|eg\rangle-|ge\rangle-|gh\rangle+|hg\rangle)$, of which
the pump lasers of step A’ and step B’ can destroy all states except
$|gg\rangle,|ee\rangle$, and $|hh\rangle$, so the system will be stable in
$|T_{1}\rangle$ after continuously cyclic evolution according to steps A’ and
B’. Fig. 2(b) depicts the process of system evolution to the target state with
three different initial mixed states $1/3(|gg\rangle\langle
gg|+|ee\rangle\langle ee|+|hh\rangle\langle hh|)$, $1/6(|gg\rangle\langle
gg|+|ee\rangle\langle ee|+|hh\rangle\langle hh|+|eg\rangle\langle
eg|+|ge\rangle\langle ge|+|eh\rangle\langle he|)$, and $1/2(|eg\rangle\langle
eg|+|ge\rangle\langle ge|)$. For all cases, after 60 cycles ($0.78$ ms), the
population of the target state can reach more than 99%, where we have chosen
$\Omega_{c}=2\pi\times 20$ kHz, $\tau_{c}={2\pi}/{(7\Omega_{c})}$, and other
parameters are the same as in the previous preparation of Bell state.

A prominent advantage of our current model over the antiblockade effect and
unconventional Rydberg pumping is that it can be directly extended to the
preparation of multi-atom GHZ states. We decompose each cycle of preparing an
$n$-atom GHZ state into step A”, step B”, and step C”. The steps A” and B” are
similar to steps A and B. The coupling strength between all atoms in
$|g\rangle$ $(|e\rangle)$ and the corresponding single excited collective
state is $\sqrt{n}\Omega_{a}/2$, thus the selection of time
$2\pi/(\sqrt{n}\Omega_{a})$ ensures that when $n$ atoms are initially in the
ground state $|g\rangle(|e\rangle)$, this probability will recover at the end
of the pulse. The duration of laser pulses applied in step C” must be chosen
in such a way that the probability amplitudes of states with even number of
$|+\rangle$ undergo a periodic evolution. This process can be easily realized
in the preparation of triatom GHZ state
${1}/{\sqrt{2}}(|ggg\rangle-|eee\rangle)={1}/2(|---\rangle+|++-\rangle+|+-+\rangle+|-++\rangle)$,
because the duration of laser pulses driving $|+\rangle$ to $|r\rangle$ is the
same as step C. But for preparing GHZ states of four and five atoms such as
$1/(2\sqrt{2})\sum_{k=0}^{2}|\mathcal{{X}}_{2K}^{4}\rangle$ and
$1/4\sum_{k=0}^{2}|\mathcal{{X}}_{2K}^{5}\rangle$ with
$|\mathcal{{X}}_{m}^{n}\rangle$ being the state incorporating $m$ atoms in
$|+\rangle$ and $(n-m)$ atoms in $|-\rangle$, the resonant lasers used in step
C” of preparing three-atom GHZ state cannot make the probability of amplitudes
of states $|\mathcal{{X}}_{2}^{n}\rangle$ and $|\mathcal{{X}}_{4}^{n}\rangle$
restored simultaneously, since the ratio of the Rabi frequencies coupled the
above states to the single excited Rydberg state is $1:\sqrt{2}$ rather than
an integer ratio. Fortunately, we can achieve our goal by introducing two-
photon-detuned pump lasers driving the transition
$|r\rangle\leftrightarrow|+\rangle$. The corresponding Hamiltonian reads
$\hat{H}_{c^{\prime\prime}}^{I}={{\sqrt{2}}/{2}\Omega_{a}e^{-i\delta
t}}\sum_{j=1}^{n}(|r\rangle_{jj}\langle+|+|+\rangle_{jj}\langle
r|)+\sum_{i<j}U_{i,j}|r_{i}r_{j}\rangle\langle r_{i}r_{j}|.$ If the system is
in the state of $|\mathcal{{X}}_{m}^{n}\rangle$ initially, the probability
amplitude that there still $m$ atoms in $|+\rangle$ at time $t$ is
${(P_{m}^{+}e^{P_{m}^{+}t}-P_{m}^{-}e^{P_{m}^{-}t})}/{iW_{m}}$, where
$W_{m}=\sqrt{\delta^{2}+2m\Omega^{2}_{a}}$ and $P_{m}^{\pm}={i}/{2}(-\delta\pm
W_{m})$. Therefore, in order to restore the probability amplitudes of
$|\mathcal{{X}}_{2}^{n}\rangle$ and $|\mathcal{{X}}_{4}^{n}\rangle$ at the
same time, the detuning $\delta$ and time $t$ should meet the following
conditions $\delta t=2k\pi,\sqrt{\delta^{2}+4\Omega_{a}^{2}}t=2l\pi$ and
$\sqrt{\delta^{2}+8\Omega_{a}^{2}}t=2j\pi$, where $k$, $l$, and $j$ are
integers. A set of solutions of the equations is
$\delta=\sqrt{{1}/{6}}\Omega_{a}$ and $t={\sqrt{6}\pi}/{\Omega_{a}}$. In Fig.
2(c), we simulate the generation of three-, four-, and five-atom GHZ states
with obtained parameters from fully mixed states respectively, and find the
populations of all target states will be over 98$\%$ after 80 cycles.

The Rydberg blockade regime requires the Rydberg interaction intensity to be
much greater than the effective Rabi frequency of laser driving the transition
between ground state and the excited Rydberg state. Fig. 2(d) illustrates the
evolution process of the desired Bell state under different Rydberg
interaction intensities. It can be seen that $U=20\Omega_{a}=2\pi\times
40\leavevmode\nobreak\ $MHz is large enough for blocking the bi-excitation
Rydberg state. For 87Rb$\leavevmode\nobreak\ {100}S_{1/2}$, we have
$C_{6}/2\pi$=$-$56.2 THz $\mu$m6 [24], thus the Rydberg interaction
$U$=$-C_{6}/d^{6}$ can achieve a strength of $2\pi\times
40\leavevmode\nobreak\ $MHz with interatomic distance
$10.6\leavevmode\nobreak\ \mu$m, and a bigger strength of $2\pi\times
4\leavevmode\nobreak\ $GHz can be obtained by setting the atomic spacing of
$4.9\leavevmode\nobreak\ \mu$m, both of which are easy to be realized
experimentally [25, 26, 27, 28].

Figure 3: (a) The specific process of Bell-state population changing with time
is given when the pump laser driving time increases by 5%. (b) The timing
error can be avoided after introducing Gaussian pulses. (c) Temperature
dependence of Bell state population at the end of cycles. (d) Effect of
dephasing on the population of Bell state.

As we mentioned in the introduction, our strategy requires the target state to
be stabilized at the end of the pulse node, not the entire drive process. As a
result, the system will be unable to perform precise Rabi oscillations if the
desired time for the regulated unitary dynamics is not met. Fig. 3(a) depicts
the population evolution of the whole process and assumes that the duration of
the laser pulse is increased by 5% compared with the ideal value. As the black
envelope shows, the population of the target state declines to 95% after 20
cycles. In fact, this timing error can be avoided by introducing time-
dependent Rabi frequencies such as Gaussian pulses. Now we replace the
original time-independent Rabi frequency $\Omega_{a}$ with
$\Omega_{a}e^{-{(t-\tau_{i})^{2}}/{(2\sigma_{i}^{2})}}$, where
$\tau_{i}=3\sigma_{i}$ ($i$ denotes the index of step in preparation of the
Bell state), and $\sigma_{A}=\sigma_{B}=1/(4\sqrt{\pi})$ and
$\sigma_{C}=1/(4\sqrt{2\pi})$. Fig. 3(b) shows the temporal evolution of
population for target state as the driving time increased by $2\tau_{i}\times
5\%$, starting from the same initial state as in Fig. 3(a). It is clear that
the Gaussian pulse has strong resistance to variations in the selected time,
and after 20 cycles, the fidelity is still better than 99%.

The variances of atomic position and momentum in the trap are shown as
$\langle x^{2}\rangle=\langle
y^{2}\rangle=\frac{\omega^{2}}{4}\frac{T}{U_{F}},\langle
z^{2}\rangle=\frac{\pi^{2}\omega^{4}}{2\lambda^{2}}\frac{T}{U_{F}}$ and
$\langle v_{x}^{2}\rangle=\langle v_{y}^{2}\rangle=\langle
v_{z}^{2}\rangle=\frac{T}{m}$ [29], where $U_{F}=\frac{\pi
c^{2}\Gamma}{2\omega_{0}^{2}}\frac{3}{\omega_{0}-\omega^{\prime}}I(r)$ is the
depth of the dipole potential, $\omega_{0}$ and $\omega^{\prime}$ are the
frequencies of $5P_{3/2}\leftrightarrow 5P_{1/2}$ transition and laser pluse
respectively, $\Gamma$ is the decay rate of $5P_{3/2}\rightarrow 5P_{1/2}$
transition and $I(r)=2P/(\pi\omega^{2})$ is the laser intensity determined by
the power $P$ and the optical tweezer parameter $\omega$. Therefore the
realistic Rabi frequency and Rydberg interaction felt by atoms are
characterized by $\Omega_{j}e^{ik_{j}(z+\Delta z)}$, and $U(r)=U\sqrt{(\Delta
x)^{2}+(\Delta y)^{2}+(z+\Delta z)^{2}},$ where $z$ is the interatomic
distance, and $\Delta\textbf{\emph{r}}_{i}$ denotes the change of atomic
position caused by both position fluctuation and velocity fluctuation
$\Delta\textbf{\emph{r}}_{i}=\delta\textbf{\emph{r}}_{i}+\int\delta\textbf{\emph{v}}_{i}(t)dt$.
In a recent experiment, the detected atomic temperature $T$ can be cooled to
$5.2\leavevmode\nobreak\ \mu$K with optical tweezer parameters
$\omega=1.2\leavevmode\nobreak\ \mu$m, laser wavelength
$\lambda=830\leavevmode\nobreak\ n$m, and $P=174\leavevmode\nobreak\ \mu$W
[28]. Referring to these experimental parameters, we plot the results of 100
stochastic simulations (light blue) for the variation of target-state
population with temperature $T$ in Fig. 3(c), where the spacing between two
atoms has been set as $z=6.3$ $\mu$m so that the relative phase caused by the
wave vectors of lasers can be ignored in the ideal case. The corresponding
average result is also shown as the dark blue line, from which we can see that
the population of the Bell state can be stabilized more than 99% within the
temperature of 0.13 mK.

Since the phase noise of the laser depends on the test results of specific
experiments, it is difficult to quantify directly in theory. However the
average result of noise will lead to dephasing of Rabi oscillations, and the
relevant Lindblad operators can be modeled as
$\hat{L}_{g}=\sqrt{\gamma^{g}/2}\sum_{j=1}^{2}(|p_{2}\rangle_{jj}\langle
p_{2}|-|g\rangle_{jj}\langle g|)$,
$\hat{L}_{e}=\sqrt{\gamma^{e}/2}\sum_{j=1}^{2}(|p_{2}\rangle_{jj}\langle
p_{2}|-|e\rangle_{jj}\langle e|)$, and
$\hat{L}_{p}=\sqrt{\gamma^{p}/2}\sum_{j=1}^{2}(|r\rangle_{jj}\langle
r|-|p_{2}\rangle_{jj}\langle p_{2}|)$, where we have considered the collective
dephasing channel since atoms are driven by lasers globally. Fig. 3(d) depicts
the relationship between the population of target state and three dephasing
rates by assuming $\gamma^{g}=\gamma^{e}$. It can be seen that the decoherence
rate $\gamma^{g(e)}$ plays a leading role in affecting the population of Bell
state. If the prepared target state population needs to be kept above 90%, it
is necessary to meet the condition
$\gamma^{g(e)}/(2\pi)<-5\gamma^{p}/(2\pi)+9$ kHz when $\gamma^{p}/(2\pi)$
varies from 0 to 20 kHz.

In summary, we have constructed an efficient scheme for dissipative
preparation of entangled states in the Rydberg blockade regime. The different
Rabi frequencies coupled to the transition between ground states and the
Rydberg state can be used as an effective means of selective excitation of
atoms. When combined with engineered spontaneous emissions and cyclic
evolution, Bell state, three-dimensional entangled states, triatomic,
quadatomic and pentaatomic GHZ states are all available from any initial
state, respectively. We briefly analyze the possible error sources in the
experiment and find that dephasing is still a dominant decoherence factor. In
the future, we will try to improve the fidelity of entanglement preparation
resorting to quantum error correction technology, and apply our scenario to
produce some interesting topological states in neutral-atom system.

Funding.  National Natural Science Foundation of China (NSFC) (11774047,
12174048).

Disclosures.  The authors declare no conflicts of interest.

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