# Optimized nonadiabatic holonomic quantum computation based on Förster
resonance in Rydberg atoms

Shuai Liu1,2, Jun-Hui<EMAIL_ADDRESS>Ri-Hua Zheng1,2, Yi-Hao
Kang1,2, Zhi-Cheng Shi1,2, Jie Song4, and Yan<EMAIL_ADDRESS>1Fujian Key Laboratory of Quantum Information and Quantum Optics (Fuzhou
University), Fuzhou 350116, China
2Department of Physics, Fuzhou University, Fuzhou 350116, China
3School of Rail Transportation, Fujian Chuanzheng Communications College,
Fuzhou 350007, China
4Department of Physics, Harbin Institute of Technology, Harbin 150001, China

###### Abstract

In this paper, we propose a scheme for implementing the nonadiabatic holonomic
quantum computation (NHQC+) of two Rydberg atoms by using invariant-based
reverse engineering (IBRE). The scheme is based on Förster resonance induced
by strong dipole-dipole interaction between two Rydberg atoms, which provides
a selective coupling mechanism to simply the dynamics of system. Moreover, for
improving the fidelity of the scheme, the optimal control method is introduced
to enhance the gate robustness against systematic errors. Numerical
simulations show the scheme is robust against the random noise in control
fields, the deviation of dipole-dipole interaction, the Förster defect, and
the spontaneous emission of atoms. Therefore, the scheme may provide some
useful perspectives for the realization of quantum computation with Rydberg
atoms.

##  I Introduction

The quantum computation, due to the characteristic of coherent superposition
of quantum states, has shown many advantages in speeding up the processing of
certain complex problems, such as factoring large integers and searching
unsorted databases Grover (1997); Long (2001); Paredes  _et al._ (2005); Fan
_et al._ (2018); Fan and Zhu (2019). In quantum computing, the realization of
high-fidelity gates is necessary for fault-tolerant quantum computing.
However, the gate fidelity is generally limited by control errors in gate
parameters and decoherence induced by dissipation. To diminish the effects of
control errors and decoherence, various proposals of quantum computation have
been put forward. One promising proposal is nonadiabtic holonomic quantum
computation (NHQC) based on a non-Abelian geometric phases Sjöqvist  _et al._
(2012); Xu  _et al._ (2012); Zhao  _et al._ (2019, 2020); Xing  _et al._
(2020); Zhao  _et al._ (2018); Xu  _et al._ (2017); Chen  _et al._ (2018a); Xu
_et al._ (2020a); Li  _et al._ (2021). On one hand, the NHQC is based on
geometric phases determined by global properties of evolution paths.
Therefore, it is insensitive to classical parameter fluctuation. On the other
hand, the NHQC is free from the limitation of the adiabatic condition, thus
the evolution process can be accelerated. Accordingly, this can reduce the
influence of decoherence on the system and improve the fidelity of the gate.
However, the NHQC requires strict conditions to be satisfied for the
Hamiltonian at all times, which undoubtedly restricts the choice of parameters
Sjöqvist  _et al._ (2012); Xu  _et al._ (2012).

To overcome the difficulty, a flexible scheme Liu  _et al._ (2019) called as
the NHQC+ has recently been proposed, which relaxes the requirement of
parameters by selecting auxiliary basis. With the relaxation of the
conditions, the NHQC+ approach can be compatible with various optimal control
methods including counteradiabatic driving (CD) Berry (2009); Chen  _et al._
(2010); del Campo (2013); Qi and Jing (2020), dynamical decoupling (DD)
Khodjasteh and Lidar (2005); Souza  _et al._ (2011); Genov  _et al._ (2017),
single-shot-shaped pulse (SSSP) Daems  _et al._ (2013); Yu  _et al._ (2018),
invariant based reverse engineering (IBRE) Lewis and Riesenfeld (1969); Chen
_et al._ (2011), etc., which further improves the robustness against
systematic errors. To date, many NHQC+ schemes have been put forward in
different physical systems, for example, superconducting circuit Li  _et al._
(2020); Yan  _et al._ (2019); Xu  _et al._ (2020b), spin qubits Kang  _et al._
(2020a); Zhang  _et al._ (2020), and Rydberg atoms Guo  _et al._ (2020); Kang
_et al._ (2020b); Liu  _et al._ (2020); Chen  _et al._ (2020).

The Rydberg atom system is one promising candidate platform for physical
implementation of quantum computing due to its long coherence time and strong
interatomic interaction Saffman  _et al._ (2010); Urban  _et al._ (2009);
Gaëtan  _et al._ (2009); Barredo  _et al._ (2015); Su  _et al._ (2017a); Kang
_et al._ (2018); Su  _et al._ (2018); Li and Shao (2019); Chen  _et al._
(2018b); Zheng  _et al._ (2020a, b); Su  _et al._ (2020); Shi (2021); Zheng
_et al._ (2021); Yin  _et al._ (2020). In Rydberg atom systems, the most
representative phenomenon is Rydberg blockade Jaksch  _et al._ (2000); Lukin
_et al._ (2001). That is, when more than one atom are excited in Rydberg
states, strong dipole-dipole or van der Waals interactions between Rydberg
atoms exist, which prevents neighboring Rydberg atoms simultaneously being
excited to Rydberg states. In contrast to the Rydberg blockade, the Rydberg
antiblockade Ates  _et al._ (2007); Amthor  _et al._ (2010); Su  _et al._
(2017b) is more selective for computational basis states. For a two-qubit
system, the Rydberg antiblockade allows only one computational basis state to
be excited to the doubly excited Rydberg state, while the remaining three
computational states keep unchanged. Apart from the Rydberg blockade and
Rydberg antiblockade, the interaction among neutral atoms can exhibit many
peculiar phenomena, such as, Rydberg dressing Müller  _et al._ (2014); Keating
_et al._ (2015); Mitra  _et al._ (2020) and Förster resonance Huang  _et al._
(2018); Beterov  _et al._ (2018a).

Among these different phenomena, the Starked Förster resonance Förster (1948)
is an useful way to control the interaction between atoms, where two pairs of
Rydberg states allow dipole transitions in between can be shifted into
resonance by dc or microwave electric fields Ryabtsev  _et al._ (2010);
Gorniaczyk  _et al._ (2016). Moreover, the coherent coupling at Förster
resonance has been also experimentally demonstrated Ravets  _et al._ (2014)
and exploited to realize quantum computing Wu  _et al._ (2010); Shao (2020).
For example, Beterov _et al._ proposed a scheme for two-qubit gates using
double adiabatic passage of the Stark-tuned Förster resonances of Rydberg
states Beterov  _et al._ (2016). Huang _et al_. proposed a scheme to implement
the two-qubit controlled-Z gate via the Stark-tuned Förster interaction of
Rydberg atom Beterov  _et al._ (2018b). In view of that we are led to ask if
it is possible to implement NHQC+ using the Förster resonance. The answer is
positive.

In this paper, we propose a scheme to realize NHQC+ based on Förster resonance
in Rydberg atoms. The scheme has several advantages as follows: Firstly,
inspired by Refs. Li and Shao (2018); Shao (2020), we realize a selective
coupling mechanism for specific initial state in the regime of Rydberg Förster
resonance, which is utilized to simplify the dynamics of system. Therefore,
the mechanical effect Li  _et al._ (2005) and the possible ionization Li  _et
al._ (2005); Amthor  _et al._ (2007a, b) can be effectively suppressed because
there have no the simultaneous excitation of Rydberg atoms. Secondly, the
evolution is studied by IBRE, where the paths for NHQC+ are naturally
constructed by eigenvectors of an dynamic invariant and the corresponding
control fields are also reversely designed. Moreover, the scheme can be
compatible with the zero-systematic-error-sensitivity optimal control method
Ruschhaupt  _et al._ (2012), so that the robustness of the control fields
against systematic errors is further enhanced. The performance of the scheme
is estimated with the numerical simulations and the results indicate that the
scheme is insensitive to the random noise in control fields, the deviation of
dipole-dipole interaction and the Förster defect. Thirdly, the decoherence is
also taken into account. Because the populations of partial Rydberg states are
decoupled with the effective Hamiltonian and quantum information is coded on
the atoms’ ground states, the scheme is robust against the spontaneous
emission of atoms. Therefore, the scheme may provide useful perspectives in
the realization of high-fidelity quantum computation.

The outline of the paper is as follows. In Sec. II, we review the general
theories for realizing NHQC+ with IBRE. In Sec. III, we implement nonadiabatic
holonomic quantum gates in the regime of Förster resonance by using IBRE. In
Sec. IV, we analyze the influence of errors and decoherence on the fidelity of
the quantum gate based on the NHQC+ (we call it as NHQC+ gate throughout the
paper) via numerical simulations. Finally, conclusions are presented in Sec.
V.

##  II theoretical preparation

###  II.1 Lewis-Riesenfeld invariant theory

For the sake of elaborating the scheme more clearly, let us first review the
Lewis-Riesenfeld theory in a nutshell Lewis and Riesenfeld (1969). We now
consider a system with Hamiltonian $H(t)$. By introducing a Hermitian
invariant operator $I(t)$ obeying the equation ($\hbar=1$)

$\displaystyle i\frac{\partial}{\partial t}I(t)-[H(t),I(t)]=0,$ (1)

an arbitrary solution of the time-dependent Schrödinger equation
$i\frac{\partial}{\partial t}|\Psi(t)\rangle=H(t)|\Psi(t)\rangle$ can be
expressed by eigenvectors $\\{|\vartheta_{k}\rangle\\}$ of $I(t)$ as

$\displaystyle|\Psi(t)\rangle$ $\displaystyle=$
$\displaystyle\sum_{k}c_{k}|\psi_{k}(t)\rangle,$ (2)
$\displaystyle|\psi_{k}(t)\rangle$ $\displaystyle=$
$\displaystyle\exp[i\alpha_{k}(t)]|\vartheta_{k}(t)\rangle,$ (3)

where $k=0,1,...$, $c_{k}=\langle\vartheta_{k}(0)|\Psi(0)\rangle$ are
corresponding coefficients and the Lewis-Riesenfeld phases $\alpha_{k}(t)$ are
defined as

$\displaystyle\alpha_{k}(t)=\int_{0}^{t}\langle\vartheta_{k}(t^{\prime})|i\frac{\partial}{\partial
t^{\prime}}-H(t^{\prime})|\vartheta_{k}(t^{\prime})\rangle dt^{\prime}.$ (4)

With the help of the dynamical invariant $I(t)$, the Hamiltonian $H(t)$ can be
inversely derived by Eq. (1). Therefore, dynamical invariant $I(t)$ is helpful
for analyzing the evolution of the system.

###  II.2 Requirements of the NHQC+ approach

In this section, we briefly review the requirement of the NHQC+ approach Liu
_et al._ (2019). Compared to the previous NHQC schemes, the difference is that
the Hamiltonian in the NHQC+ approach Liu  _et al._ (2019) has fewer
restrictions Sjöqvist  _et al._ (2012); Xu  _et al._ (2012). To facilitate the
description of the difference more clearly, we consider an $M$-dimensional
quantum system with Hamiltonian $H(t)$. Then, a complete set of basis can be
represented as $\\{|\phi_{m}(t)\rangle,m=1,2,...,M\\}$, where all the basis
$|\phi_{m}\rangle$ follow the Schrödinger equation. Generally speaking, the
NHQC is realized by using a $L$-dimensional subspace satisfying the cyclic
evolution and parallel transport conditions Sjöqvist  _et al._ (2012); Xu  _et
al._ (2012), i.e.,

$\displaystyle(i)$
$\displaystyle\sum_{m=1}^{L}|\phi_{m}(T)\rangle\langle\phi_{m}(T)|=\sum_{m=1}^{L}|\phi_{m}(0)\rangle\langle\phi_{m}(0)|,$
(5) $\displaystyle(ii)$
$\displaystyle\langle\phi_{m}(t)|H(t)|\phi_{m^{\prime}}(t)\rangle=0,(m,m^{\prime}=1,...,L).$
(6)

The above two conditions (i) and (ii) ensure a cyclic evolution with only pure
geometric phases being accumulated during the time interval $[0,T]$. In the
NHQC case, the condition (ii) shown in Eq. (5) requires restrictions for all
possible $m,m^{\prime}$ and at every moment, which hinders the combination of
NHQC and some optimal control methods Liu  _et al._ (2019).

To remove these constraints, the NHQC+ approach suggests to find a set of
auxiliary basis $\\{|\tilde{\vartheta}_{m}(t)\rangle,m=1,2,...,L\\}$, which
satisfies cyclic condition, i.e.,
$|\tilde{\vartheta}_{m}(T)\rangle=|\tilde{\vartheta}_{m}(0)\rangle=|\phi_{m}(0)\rangle$.
According to the results of Ref. Liu  _et al._ (2019), the defined projector
$\Pi_{m}(t)=|\tilde{\vartheta}_{m}(t)\rangle\langle\tilde{\vartheta}_{m}(t)|$
should satisfy von Neumann equation

$\displaystyle\frac{d}{dt}\Pi_{m}(t)=-i[H(t),\Pi_{m}(t)].$ (7)

Besides, for each $|\widetilde{\vartheta}_{m}(t)\rangle$, the dynamic phase
acquired in the whole evolution process should vanish, i.e.,

$\displaystyle\theta_{m}(T)=-\int^{T}_{0}\langle\tilde{\vartheta}_{m}(t)|H(t)|\tilde{\vartheta}_{m}(t)\rangle
dt=0,$ (8)

such that the evolution becomes purely geometric as

$\displaystyle
U(T,0)=\sum_{m}e^{i\Theta_{m}(T)}|\tilde{\vartheta}_{m}(T)\rangle\langle\tilde{\vartheta}_{m}(T)|,$
(9)

with the geometric phase

$\displaystyle\Theta_{m}(T)=i\int_{0}^{T}\langle\tilde{\vartheta}_{m}(t)|\dot{\tilde{\vartheta}}_{m}(t)\rangle
dt.$ (10)

Compared with the condition (ii) in Eq. (5), the condition in Eq. (8) shows
that the Hamiltonian in the NHQC+ approach removes the constraints for $m\neq
m^{\prime}$ and the dynamical phase is only required to vanish for the
integral in the time interval $[0,T]$. Thus, the NHQC+ approach may be
compatible with different optimal control methods.

##  III Implementation of nonadiabatic holonomic quantum gates

In this section, we introduce the physical model, the design methods of
control fields (IBRE), the optimal control method (zero-systematic-error-
sensitivity optimal control) to implement the NHQC+ gate. In Sec. III.1, we
give the physical model and derive the effective Hamiltonian in the Förster
resonance regime. Based on the effective Hamiltonian in Sec. III.1, we further
combine IBRE with the optimal control method to realize the optimized NHQC+
gate in Sec. III.2. Specially, a clear schematic diagram for the scheme is
given in Fig. 1.

Figure 1: Schematic diagram for the implementation of optimized NHQC+ gates.

###  III.1 Physical model and effective Hamiltonian

Figure 2: Level configurations and coherent couplings of atom $a$ and atom $b$
with a dipole-dipole interaction strength $V$.

The model for implementing the nondiabatic holonomic quantum gates is shown in
Fig. 2. We consider two identical Rubidium atoms individually trapped in
optical tweezers. Each atom includes two ground states
$|0\rangle=|5S_{1/2},F=1,m_{F}=0\rangle$ and
$|1\rangle=|5S_{1/2},F=2,m_{F}=0\rangle$, and three Rydberg states
$|r\rangle=|59D_{3/2},m_{j}=3/2\rangle$,
$|r_{+}\rangle=|61P_{1/2},m_{j}=1/2\rangle$, and
$|r_{-}\rangle=|57F_{5/2},m_{j}=5/2\rangle$ Walker and Saffman (2005). The
transition $|0\rangle_{1}(|1\rangle_{1})\leftrightarrow|r\rangle_{1}$ for atom
1 is resonantly driven by control field with Rabi frequency $\Omega_{1}(t)$
($\Omega_{2}(t)$). For the atom 2, the off-resonant transition
$|0\rangle_{2}\leftrightarrow|r\rangle_{2}(|r_{+}\rangle_{2})$ is driven by
control field of Rabi frequency $\Omega_{3}$ with red detuning $\Delta$ (blue
detuning $\Delta$). Besides, the off-resonant transition
$|1\rangle_{2}\leftrightarrow|r\rangle_{2}(|r_{+}\rangle_{2})$ is driven by
control field of $\Omega_{4}$ with red detuning $\Delta$ (blue detuning
$\Delta$). When both atoms are excited to Rydberg states, the states
$|rr\rangle$ and $(|r_{+}r_{-}\rangle+|r_{-}r_{+}\rangle)/\sqrt{2}$ are
coupled by the dipolar interaction based on the Förster process with a small
Förster defect, which the pair states $|rr\rangle$, $|r_{+}r_{-}\rangle$, and
$|r_{-}r_{+}\rangle$ are almost degenerate Walker and Saffman (2005, 2008).
Noticing that the Förster defect can be eliminated by an external field Ravets
_et al._ (2014). Therefore, the resonant Förster interaction between two
Rydberg atoms is described as

$\displaystyle H_{F}=V|rr\rangle(\langle r_{+}r_{-}|+\langle
r_{-}r_{+}|)/\sqrt{2}+\mathrm{H.c.}$ (11)

The Hamiltonian $H_{F}$ in Eq. (11) can be further diagonalized as
$V(|\varpi_{+}\rangle\langle\varpi_{+}|+|\varpi_{-}\rangle\langle\varpi_{-}|)$
with $|\varpi_{\pm}\rangle=(|rr\rangle\pm|R\rangle)/\sqrt{2}$, where we define
$|R\rangle=(|r_{+}r_{-}\rangle+|r_{-}r_{+}\rangle)/\sqrt{2}$. To parameterize
the Rabi frequencies of control fields $\Omega_{1}(t)$, $\Omega_{2}(t)$,
$\Omega_{3}$ and $\Omega_{4}$, we assume $\Omega_{1}(t)=\Omega_{a}(t)\cos
v_{a}e^{i\varphi_{a}(t)}$, $\Omega_{2}(t)=\Omega_{a}(t)\sin
v_{a}e^{i\varphi_{a}(t)}$, $\Omega_{3}=\Omega_{b}\cos v_{b}$, and
$\Omega_{4}=\Omega_{b}\sin v_{b}$. In the interaction picture, the Hamiltonian
$H_{I}$ of the system can be written as

$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle H_{1}+H_{2}+H_{F},$ (12)
$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\Omega_{a}(t)e^{i\varphi_{a}(t)}|\xi_{+}\rangle_{a}\langle
r|+\mathrm{H.c.},$ (14) $\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\Omega_{b}e^{-i\Delta
t}|r_{+}\rangle_{b}\langle\xi_{+}|+\Omega_{b}e^{i\Delta
t}|r\rangle_{b}\langle\xi_{+}|+\mathrm{H.c.},$ (16)

where $|\xi_{+}\rangle_{l}=\cos v_{l}|0\rangle_{l}+\sin v_{l}|1\rangle_{l}$
($l=a,b$).

Figure 3: Setup for effective couplings of the two-atom collective ground
states and the two-atom collective Rydberg state. The blue solid lines
represent the resonance interaction between these two states. The red solid
lines and green solid lines represent the interaction with red detuning
$\Delta$ and blue detuning $\Delta$, respectively. $E$ denotes the energy of
the single atom in the Rydberg state $|r\rangle$.

Besides, we also define $|\xi_{-}\rangle_{l}=\cos v_{l}|1\rangle-\sin
v_{l}|0\rangle_{l}$, such that $\\{|\xi_{-}\rangle_{l},|\xi_{+}\rangle_{l}\\}$
can form a complete orthogonal basis for atom $l$ instead of
$\\{|0\rangle_{l},|1\rangle_{l}\\}$.

After considering the condition $\Delta=V$, the Hamiltonian $H_{I}$ in Eq.
(12) is reformulated in a rotating frame with respect to
$R=e^{iVt(|\varpi_{+}\rangle\langle\varpi_{+}|-|\varpi_{-}\rangle\langle\varpi_{-}|)}$,

$\displaystyle H^{r}_{1}=RHR^{\dagger}+i\dot{R}R^{\dagger}=H_{r}+H_{h},$ (18)

where the Hamiltonians $H_{r}$ and $H_{h}$ are respectively given by

$\displaystyle H_{r}$ $\displaystyle=$
$\displaystyle\Omega_{a}(t)e^{i\varphi_{a}(t)}(|\xi_{+}\xi_{-}\rangle\langle
r\xi_{-}|+|\xi_{+}\xi_{+}\rangle\langle r\xi_{+}|$ (20)
$\displaystyle+|\xi_{+}r_{+}\rangle\langle
rr_{+}|)+\frac{\Omega_{b}}{\sqrt{2}}|r\xi_{+}\rangle\langle\varpi_{-}|+\mathrm{H.c.},$
$\displaystyle H_{h}$ $\displaystyle=$ $\displaystyle\Omega_{b}e^{-i\Delta
t}(|\xi_{-}\xi_{+}\rangle\langle\xi_{-}r|+|\xi_{-}r_{+}\rangle\langle\xi_{-}\xi_{+}|$
(26)
$\displaystyle+|\xi_{+}\xi_{+}\rangle\langle\xi_{+}r|+|\xi_{+}r_{+}\rangle\langle\xi_{+}\xi_{+}|+|rr_{+}\rangle\langle
r\xi_{+}|)$
$\displaystyle+\frac{\Omega_{a}(t)e^{i\varphi_{a}(t)}}{\sqrt{2}}e^{-i\Delta
t}(|\varpi_{-}\rangle\langle\xi_{+}r|+|\xi_{+}r\rangle\langle\varpi_{+}|)$
$\displaystyle+\frac{\Omega_{b}}{\sqrt{2}}e^{-2i\Delta
t}|r\xi_{+}\rangle\langle\varpi_{+}|+\mathrm{H.c.}$

In the regime of the large detuning limit $\Delta\gg\Omega_{b}$, the
Hamiltonian $H_{h}$ in Eq. (20) can be reduced as James and Jerke (2007)

$\displaystyle H^{\prime}_{h1}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{b}^{2}}{\Delta}(|\xi_{-}r\rangle\langle\xi_{-}r|+|\xi_{+}r\rangle\langle\xi_{+}r|-|\xi_{-}r_{+}\rangle\langle\xi_{-}r_{+}|$
(33)
$\displaystyle+|\xi_{+}r\rangle\langle\xi_{+}r|-|\xi_{+}r_{+}\rangle\langle\xi_{+}r_{+}|+|r\xi_{+}\rangle\langle
r\xi_{+}|$ $\displaystyle-|rr_{+}\rangle\langle
rr_{+}|)+\frac{\Omega_{b}^{2}}{4\Delta}(|\varpi_{+}\rangle\langle\varpi_{+}|-|r\xi_{+}\rangle\langle
r\xi_{+}|)$
$\displaystyle+\frac{\Omega_{a}^{2}}{2\Delta}(|\varpi_{+}\rangle\langle\varpi_{+}|-|\varpi_{-}\rangle\langle\varpi_{-}|),$
$\displaystyle H^{\prime}_{h2}$ $\displaystyle=$
$\displaystyle-\frac{\Omega_{b}\Omega_{a}(t)e^{i\varphi_{a}(t)}}{\sqrt{2}\Delta}|\varpi_{-}\rangle\langle\xi_{+}\xi_{+}|+\mathrm{H.c.},$
(34)

where, the term $H^{\prime}_{h1}$ can be identified with Stark shifts
associated with the control fields and the term $H^{\prime}_{h2}$ represents
the Raman transition
$|\xi_{+}\xi_{+}\rangle\leftrightarrow|\varpi_{-}\rangle$. If the system is
initial in subspace
$\mathcal{S}=\\{|\xi_{-}\xi_{-}\rangle,|\xi_{-}\xi_{+}\rangle,|\xi_{+}\xi_{-}\rangle,|\xi_{+}\xi_{+}\rangle\\}$,
by neglecting decoupled terms, the full Hamiltonian $H_{1}^{r}$ in Eq. (18)
can be simplified as

$\displaystyle H^{r}_{2}$ $\displaystyle=$
$\displaystyle\frac{3\Omega^{2}_{b}}{4\Delta}|r\xi_{+}\rangle\langle
r\xi_{+}|+\Omega_{a}(t)e^{i\varphi_{a}(t)}(|\xi_{+}\xi_{-}\rangle\langle
r\xi_{-}|$ (37) $\displaystyle+|\xi_{+}\xi_{+}\rangle\langle
r\xi_{+}|)+\frac{\Omega_{b}}{\sqrt{2}}|r\xi_{+}\rangle\langle\varpi_{-}|+\mathrm{H.c.}$

As shown in Fig. 3, only the transitions
$|\xi_{+}\xi_{-}\rangle\leftrightarrow|r\xi_{-}\rangle$,
$|\xi_{+}\xi_{+}\rangle\leftrightarrow|r\xi_{+}\rangle$, and
$|r\xi_{+}\rangle\leftrightarrow|\varpi_{-}\rangle$ are allowed in the regime
of the large detuning limit $\Delta\gg\Omega_{b}$.

In order to achieve a selective coupling interaction, we further suppress the
transition $|\xi_{+}\xi_{+}\rangle\leftrightarrow|r\xi_{+}\rangle$ by
diagonalizing the Hamiltonian
$\frac{\Omega_{b}}{\sqrt{2}}|r\xi_{+}\rangle\langle\varpi_{-}|+\mathrm{H.c.}$
in Eq. (37). That is, the term
$\frac{\Omega_{b}}{\sqrt{2}}|r\xi_{+}\rangle\langle\varpi_{-}|+\mathrm{H.c.}$
in Eq. (37) is transformed to the form
$\frac{\Omega_{b}}{\sqrt{2}}(|E_{+}\rangle\langle E_{+}|-|E_{-}\rangle\langle
E_{-}|)$ with
$|E_{\pm}\rangle=(|r\xi_{+}\rangle\pm|\varpi_{-}\rangle)/\sqrt{2}$. In such a
situation, the Hamiltonian $H_{2}^{r}$ in Eq. (37) yields

$\displaystyle H^{r}_{3}$ $\displaystyle=$
$\displaystyle\frac{3\Omega_{b}^{2}}{8\Delta}(|E_{+}\rangle+|E_{-}\rangle)(\langle
E_{+}|+\langle E_{-}|)$ (40)
$\displaystyle+\Omega_{a}(t)e^{i\varphi_{a}(t)}\big{[}|\xi_{+}\xi_{-}\rangle\langle
r\xi_{-}|+\frac{1}{\sqrt{2}}|\xi_{+}\xi_{+}\rangle(\langle E_{+}|$
$\displaystyle+\langle
E_{-}|)+\mathrm{H.c.}\big{]}+\frac{\Omega_{b}}{\sqrt{2}}(|E_{+}\rangle\langle
E_{+}|-|E_{-}\rangle\langle E_{-}|).$

By considering the limit of $\Omega_{b}\gg\Omega_{a}(t)$, the collective state
$|\xi_{+}\xi_{+}\rangle$ is decoupled with $|E_{+}\rangle$ and $|E_{-}\rangle$
in the regime of large detuning. It is worth noting that the term
$H^{\prime}_{h2}$ in Eq. (33) is very small compared with the Hamiltonian
$H_{r}$ in Eq. (20) under the limit condition
$\Delta\gg\Omega_{b}\gg\Omega_{a}(t)$, so that we have neglected it in the
above derivations. Therefore, the Hamiltonian $H_{3}^{r}$ in Eq. (40) can be
further simplified as

$\displaystyle H_{\rm
eff}=\Omega_{a}(t)e^{i\varphi_{a}(t)}|\xi_{+}\xi_{-}\rangle\langle
r\xi_{-}|+\mathrm{H.c.},$ (42)

which signifies that the Hamiltonian provides a selective coupling mechanism
that only the initial state $|\xi_{+}\xi_{-}\rangle$ can evolve in the
subspace $\mathcal{S}$.

Figure 4: Sketch map of deriving the effective Hamiltonian.

In a word, such a course of deriving the effective Hamiltonian is shown in
Fig. 4.

###  III.2 The realization of NHQC+ gate by using IBRE

In order to conveniently design the $H_{\rm eff}$ in Eq. (42) with IBRE, we
denote $\Omega_{x}(t)=2\Omega_{a}(t)\cos\varphi_{a}(t)$,
$\Omega_{y}(t)=2\Omega_{a}(t)\varphi_{a}(t)$, and $\Omega_{z}=0$. Therefore,
the effective Hamiltonian $H_{\rm eff}$ in Eq. (42) can be represented as

$\displaystyle H_{\rm
eff}=\frac{\Omega_{x}(t)}{2}\sigma_{x}+\frac{\Omega_{y}(t)}{2}\sigma_{y}+0\times\sigma_{z},$
(43)

with $\sigma_{x}=|r\xi_{-}\rangle\langle\xi_{+}\xi_{-}|+\mathrm{H.c.}$,
$\sigma_{y}=-i|r\xi_{-}\rangle\langle\xi_{+}\xi_{-}|+\mathrm{H.c.}$, and
$\sigma_{z}=|r\xi_{-}\rangle\langle
r\xi_{-}|-|\xi_{+}\xi_{-}\rangle\langle\xi_{+}\xi_{-}|$. According to Sec. II,
we can make use of IBRE to design the control fields. For a two-level quantum
system, the invariant $I(t)$ can be constructed by the superposition of three
group generators of SU(2) algebra with time-dependent parameters $\mu_{1}$ and
$\mu_{2}$ Kang  _et al._ (2020b),

$\displaystyle
I(t)=\frac{u}{2}(\cos\mu_{1}\sigma_{z}+\sin\mu_{1}\sin\mu_{2}\sigma_{x}+\sin\mu_{1}\cos\mu_{2}\sigma_{y}),$
(44)

where $u$ is an arbitrary constant with units of frequency to keep $I(t)$ with
dimensions of energy Chen  _et al._ (2011). Substituting Eq. (44) into Eq.
(1), $\mu_{1}(t)$ and $\mu_{2}(t)$ are satisfied the differential equations as

$\displaystyle\dot{\mu}_{1}(t)$ $\displaystyle=$
$\displaystyle\Omega_{y}(t)\sin\mu_{2}(t)-\Omega_{x}(t)\cos\mu_{2}(t),$ (46)
$\displaystyle\dot{\mu}_{2}(t)$ $\displaystyle=$
$\displaystyle\cot\mu_{1}(t)(\sin\mu_{2}(t)\Omega_{x}(t)+\cos\mu_{2}(t)\Omega_{y}(t)).$
(48)

By using Eq. (46), we can reversely solve the control fields as

$\displaystyle\Omega_{x}(t)$ $\displaystyle=$
$\displaystyle\sin\mu_{2}\tan\mu_{1}\dot{\mu}_{2}-\cos\mu_{2}\dot{\mu}_{1},$
(50) $\displaystyle\Omega_{y}(t)$ $\displaystyle=$
$\displaystyle\cos\mu_{2}\tan\mu_{1}\dot{\mu}_{2}+\sin\mu_{2}\dot{\mu}_{1}.$
(52)

In addition, the orthogonal eigenvector of the invariant $I(t)$ with the
eigenvalues $\pm\mu/2$ can also be obtained as

$\displaystyle|\vartheta_{1}(t)\rangle$ $\displaystyle=$
$\displaystyle\cos\frac{\mu_{1}}{2}|r\xi_{-}\rangle+ie^{-i\mu_{2}}\sin\frac{\mu_{1}}{2}|\xi_{+}\xi_{-}\rangle,$
(53) $\displaystyle|\vartheta_{2}(t)\rangle$ $\displaystyle=$ $\displaystyle
ie^{i\mu_{2}}\sin\frac{\mu_{1}}{2}|r\xi_{-}\rangle+\cos\frac{\mu_{1}}{2}|\xi_{+}\xi_{-}\rangle.$
(55)

According to the results in Eqs. (7) and (8), the time derivatives of dynamic
phase and geometric phase acquired for the basis states
$|\vartheta_{1}(t)\rangle$ and $|\vartheta_{2}(t)\rangle$ can be calculated as

$\displaystyle\dot{\theta}_{1}(t)$ $\displaystyle=$
$\displaystyle-\frac{\dot{\mu}_{2}\sin^{2}\mu_{1}}{2\cos\mu_{1}},\dot{\Theta}_{1}(t)=\dot{\mu}_{2}\sin^{2}\frac{\mu_{1}}{2},$
(57) $\displaystyle\dot{\theta}_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{\dot{\mu}_{2}\sin^{2}\mu_{1}}{2\cos\mu_{1}},\dot{\Theta}_{2}(t)=-\dot{\mu}_{2}\sin^{2}\frac{\mu_{1}}{2},$
(58)

respectively.

To construct NHQC+ dynamics, we choose the auxiliary basis
$|\tilde{\vartheta}_{1}(t)\rangle=|\xi_{-}\xi_{-}\rangle$ and
$|\tilde{\vartheta}_{2}(t)\rangle=|\vartheta_{2}(t)\rangle$. Obviously, the
state $|\xi_{-}\xi_{-}\rangle$ is satisfied the conditions of NHQC+. Next, we
only need to make sure that whether or not the basis
$|\vartheta_{2}(t)\rangle$ satisfies the conditions of NHQC+. First, to
satisfy the cyclic condition
$|\vartheta_{2}(0)\rangle=|\vartheta_{2}(T)\rangle=|\xi_{+}\xi_{-}\rangle$, we
consider the boundary conditions satisfy $\mu_{1}(0)=0$, $\mu_{1}(T)=0$. In
the case, the selection of parameter $\mu_{2}(t)$ is irrelevant to the cyclic
condition according to Eq. (53). Second,
$|\tilde{\vartheta}_{2}(t)\rangle\langle\tilde{\vartheta}_{2}(t)|$ satisfies
the von Neumann equation, which is easily proved in Appendix A. In order to
make the dynamic phase in the evolution process vanish and obtain a pure
geometric phase $\Theta_{g}$, we design the parameters $\mu_{1}(t)$ and
$\mu_{2}(t)$ by dividing the time interval $[0,T]$ into two parts including
$[0,T/2]$ and $[T/2,T]$. The parameters $\mu_{1}(t)$ and $\mu_{2}(t)$ are set
as $\mu_{1}(t)=\mu_{1}(T-t)$ with $\mu_{1}(T/2)=\pi$ in the time interval
$[0,T]$ and $\mu_{2}(t)=-\Theta_{g}+\mu_{2}(T-t)$ in the time interval
$[T/2,T]$, respectively Kang  _et al._ (2020b). With the assumptions of
parameters $\mu_{1}(t)$ and $\mu_{2}(t)$, the dynamic phase acquired in the
evolution time $[0,T]$ can be demonstrated to be equal to zero (see Appendix B
for details). Thus, the NHQC+ dynamics can be constructed and the evolution
operator can be described as

$\displaystyle
U_{0}(T,0)=|\xi_{-}\xi_{-}\rangle\langle\xi_{-}\xi_{-}|+e^{i\Theta_{g}}|\xi_{+}\xi_{-}\rangle\langle\xi_{+}\xi_{-}|,$
(59)

i.e., we can achieve the gate operation at $t=T$ when the geometric phase
$\Theta_{g}=\pi$,

$\displaystyle U_{0}(T,0)=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\\\ \end{bmatrix},$ (60)

in the subspace $\mathcal{S}$. Furthermore, back the computational basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, the nontrivial two-qubit
NHQC+ gate is represented as

$\displaystyle
U_{0}(v_{a},v_{b})=\begin{bmatrix}\sin^{2}v_{a}+\cos^{2}v_{a}\cos
2v_{b}&\cos^{2}v_{a}\sin 2v_{b}&-\sin 2v_{a}\sin^{2}v_{b}&\frac{1}{2}\sin
2v_{a}\sin 2v_{b}\\\ \\\ \cos^{2}v_{a}\sin
2v_{b}&\sin^{2}v_{a}-\cos^{2}v_{a}\cos 2v_{b}&\frac{1}{2}\sin 2v_{a}\sin
2v_{b}&-\sin 2v_{a}\cos^{2}v_{b}\\\ \\\ -\sin
2v_{a}\sin^{2}v_{b}&\frac{1}{2}\sin 2v_{a}\sin
2v_{b}&\cos^{2}v_{a}+\sin^{2}v_{a}\cos 2v_{b}&\sin^{2}v_{a}\sin 2v_{b}\\\ \\\
\frac{1}{2}\sin 2v_{a}\sin 2v_{b}&-\sin 2v_{a}\cos^{2}v_{b}&\sin^{2}v_{a}\sin
2v_{b}&\cos^{2}v_{a}-\sin^{2}v_{a}\cos 2v_{b}\\\ \\\ \end{bmatrix}.$ (61)

By choosing different values of $v_{a}$ and $v_{b}$, different two-qubit NHQC+
gates $U_{0}(v_{a},v_{b})$ in Eq. (61) are realized. For $v_{a}=\pi/2$,
$v_{b}=\pi$, $U_{0}(\pi/2,\pi)$ is

$\displaystyle U_{0}(\pi/2,\pi)=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&-1\\\ \end{bmatrix},$ (63)

which is a holonomic controlled phase. While for $v_{a}=\pi/2$, $v_{b}=\pi/4$,
a holonomic controlled-NOT (CNOT) gate $U_{0}(\pi/2,\pi/4)$ is

$\displaystyle U_{0}(\pi/2,\pi/4)=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 0&0&1&0\\\ \end{bmatrix}.$ (64)

Until now, we have constructed the NHQC+ dynamics and implemented nontrivial
two-qubit NHQC+ gate. However, the systematic errors of control fields always
exist in the experiment, which will inevitably reduce the fidelity of gate.
The previous schemes of the NHQC are sensitive to the systematic errors and
difficult to incorporate optimal control technique without additional
adjustable parameters. Here, for improving the fidelity of the NHQC+ gate, we
introduce zero-systematic-error-sensitivity optimal control method to further
enhance the robustness of the systematic error in the implementation of
quantum gate Ruschhaupt  _et al._ (2012). In the presence of the systematic
error $\varepsilon$ of the control fields, the effective Hamiltonian in Eq.
(43) can be written as

$\displaystyle H_{\rm
eff}=(1+\varepsilon)\frac{(\Omega_{x}+i\Omega_{y})}{2}|\xi_{+}\xi_{-}\rangle\langle
r\xi_{-}|+\mathrm{H.c.}$ (65)

By using perturbation theory up to $O(\varepsilon^{2})$, we can obtain
$|\psi^{\varepsilon}_{2}(T)\rangle=|\psi_{2}(T)\rangle-i\varepsilon\int_{0}^{T}dtU_{0}(T,t)H_{\rm
eff}(t)|\psi_{2}(t)\rangle-\varepsilon^{2}\int_{0}^{T}dt\int_{0}^{t}dt^{\prime}U_{0}(T,t)H_{\rm
eff}(t)U_{0}(t,t^{\prime})H_{\rm
eff}(t^{\prime})|\psi_{2}(t^{\prime})\rangle+O(\varepsilon^{3})$ Sjöqvist  _et
al._ (2012), where $|\psi_{2}(t)\rangle$ ($|\psi^{\varepsilon}_{2}(t)\rangle$)
are the state of the system in the absence (presence) of systematic errors. As
the evolution of system can be described by
$|\psi_{2}(t)\rangle=e^{i\alpha_{2}(t)}|\vartheta_{2}(t)\rangle$, the fidelity
of evolution is estimated as

$\displaystyle P_{\varepsilon}$ $\displaystyle=$
$\displaystyle\left|\langle\psi_{2}(T)|\psi_{2}^{\varepsilon}(T)\rangle\right|^{2}$
(66) $\displaystyle=$ $\displaystyle
1-\varepsilon^{2}\left|\int_{0}^{T}dte^{2i\alpha_{2}(t)}\langle\vartheta_{1}(t)|H_{\rm
eff}(t)|\vartheta_{2}(t)\rangle\right|^{2},$ (67)

with $\alpha_{1}(t)=-\alpha_{2}(t)$ being considered.

To evaluate the influence of the static systematic error, the systematic error
sensitivity Ruschhaupt  _et al._ (2012) defined as $q_{s}=-\frac{\partial
P_{\varepsilon}}{2\partial\varepsilon^{2}}$ is introduced. The systematic
error sensitivity can be calculated as

$\displaystyle q_{s}=-\frac{\partial
P_{\varepsilon}}{2\partial\varepsilon^{2}}=|\int_{0}^{T}dt\exp(i\chi)\dot{\mu}_{1}\sin^{2}\mu_{1}|^{2},$
(69)

with $\chi(t)=\mu_{2}(t)+2\alpha_{2}(t)$. Here, we set
$\chi(\mu_{1})=\eta[2\mu_{1}-\sin(2\mu_{1})]$ to minimize the $q_{s}$ with
$\eta$ being a time-independent optimized coefficient. In order to make the
improvement of the gate performance attributed to the optimal control, the
maximum value of the optimized control fields is bounded by the maximal
amplitude $\Omega_{max}$ of control fields $\Omega_{x}(t)$ and
$\Omega_{y}(t)$. Under the restriction of $\Omega_{max}$, the evolution time
$T$ increases with the value of $\eta$, which aggravates the influence of
decoherence on the fidelity of the quantum gate. For this choice of $\chi$, we
can obtain $q_{s}=\frac{\sin^{2}(\eta\pi)}{\eta^{2}}$. Clearly, when $\eta$=1,
2, 3, …, the value of $q_{s}$ equal to zero, which means that the minimum of
the systematic error sensitivity is achieved. To trade off between the
robustness against systematic errors and decoherence, we choose the optimized
coefficient $\eta=1$. Accordingly, we have $\mu_{2}(t)=4\sin^{3}\mu_{1}(t)/3$
in the time interval $[0,T/2]$ and $\mu_{2}(t)=4\sin^{3}\mu_{1}(t)/3-\pi$ in
the time interval $[T/2,T]$. To make the control fields continuous and vanish
at the boundary, we choose $\mu_{1}(t)=\pi\sin^{2}(\pi t/T)$ in the time
interval $[0,T]$.

Figure 5: (a) Dependences on $t$ of control fields $\Omega_{x}(t)$,
$\Omega_{y}(t)$ with $\eta=1$. (b) The performance of the average fidelity of
the holonomic CNOT gate with systematic error $\varepsilon$ for different
optimized coefficients $\eta$.

With the choice of the parameters designed above, we plot the control fields
$\Omega_{x,y}(t)$ versus $t$ in Fig. 5(a). From Fig. 5(a), we can further
obtain the maximal value of control field $\Omega_{max}=36.05/T$. In order to
better satisfy the condition $V\gg\Omega_{b}$, the Rydberg interaction
strength is set as $V=18000/T$ and the control field $\Omega_{b}=600/T$ to
build up the effective Hamiltonian. In this case, the total operation time is
$T=18000/V$. For the dipole-dipole interaction $V$, its strength
$\sqrt{2}C_{3}/R^{3}$ can be continuously varied in the range of
$2\pi\times[3.38,216.32]$ MHz Ravets  _et al._ (2014, 2015), where
$C_{3}=2\pi\times 2.39$ GHz $\mu{\rm m}^{3}$ and $R$ represents the
interatomic distance. Besides, the Rabi coupling strength of the ground state
and the Rydberg state can be up to $2\pi\times 5$ MHz realized by a two-photon
process Ravets  _et al._ (2015). Here, we consider the experimentally feasible
parameter $V=2\pi\times 133.04$ MHz at the distance of $R\approx 2.94$ $\mu$m,
leading to short evolution time $T=21.5$ $\mu$s. Correspondingly, the control
field $\Omega_{b}=2\pi\times 4.43$ MHz and the maximal value of control field
$\Omega^{max}_{0}=2\pi\times 0.27$ MHz, which is available in the experiment.
Under the current parameter condition, it is important to verify the validity
of the effective Hamiltonian $H_{\rm eff}$ in Eq. (43). In Figs. 6(a)-6(d), we
plot the time evolution of populations for the states
$|\xi_{-}\xi_{-}\rangle$, $|\xi_{-}\xi_{+}\rangle$, $|\xi_{+}\xi_{-}\rangle$
and $|\xi_{+}\xi_{+}\rangle$ in the subspace $\mathcal{S}$ governed by the
full Hamiltonian in Eq. (12), respectively. It can be seen that this result is
in good agreement with the expectation. As shown in Fig. 6, the states
$|\xi_{-}\xi_{-}\rangle$, $|\xi_{-}\xi_{+}\rangle$, and
$|\xi_{+}\xi_{+}\rangle$ always keep in their initial states with higher
fidelities and the state $|\xi_{+}\xi_{+}\rangle$ can realize the coherent
population transfer. Therefore, the validity of effective Hamiltonian $H_{\rm
eff}$ in Eq. (43) is verified.

Figure 6: The time evolution of populations for the states
$|\xi_{-}\xi_{-}\rangle$, $|\xi_{-}\xi_{+}\rangle$, $|\xi_{+}\xi_{-}\rangle$
and $|\xi_{+}\xi_{+}\rangle$ in the subspace $\mathcal{S}$ in (a)-(d),
respectively. The time evolution of population is defined as
$Q^{\prime}=|\langle\Phi_{0}^{\prime}(T)|\Phi_{0}^{\prime}(t)\rangle|^{2}$ for
the states in the subspace $\mathcal{S}$, where $|\Phi_{0}^{\prime}(T)\rangle$
is the target state and $|\Phi_{0}^{\prime}(t)\rangle$ is the state of system
at time $t$. The parameters are set $\eta=1$, $V=2\pi\times 133.04$ MHz,
$\Omega_{b}=2\pi\times 4.43$ MHz, and $\Omega^{max}_{0}=2\pi\times 0.27$ MHz.
Figure 7: (a) Average fidelity $F_{\rm CN}$ of the implementation of the
holonomic CNOT gate versus $t$ with the full Hamiltonian in Eq. (12), where we
choose $\eta=1$. (b) The time evolution of population for the computational
basis states $|00\rangle$, $|01\rangle$, $|10\rangle$ and $|11\rangle$. The
time evolution of population is defined as
$Q=|\langle\Phi_{0}(T)|\Phi_{0}(t)\rangle|^{2}$ for the computational basis
states, where $|\Phi_{0}(T)\rangle$ is the target state and
$|\Phi_{0}(t)\rangle$ is the state of system at time $t$. Corresponding
parameters are the same as that in Fig. 6.

As an example, we check the robustness of the implementation of the holonomic
CNOT gate against systematic errors of control fields and show the results of
optimization by numerical simulation. The final average fidelities $F_{\rm
CN}(T)$ versus error $\varepsilon$ with different optimized coefficient $\eta$
are plotted in Fig. 5(b), where $F_{\rm CN}(t)$ is defined as Zanardi and
Lidar (2004); Pedersen  _et al._ (2007)

$\displaystyle F_{\rm
CN}(t)=\frac{1}{N(N+1)}\\{Tr[M(t)M(t)^{\dagger}]+|Tr[M(t)]|^{2}\\},$ (70)

with $M(t)=P_{c}U_{\rm CN}^{\dagger}U(t)P_{c}$, $U_{\rm CN}=|00\rangle\langle
00|+|01\rangle\langle 01|+|10\rangle\langle 11|+|11\rangle\langle 10|$,
$P_{c}=|00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle
10|+|11\rangle\langle 11|$ being the projection operator onto the
computational subspace, and $N=4$ representing the dimension of the
computational subspace. Seen from the blue solid line in Fig. 5(b), $F_{\rm
CN}(T)$ with the optimized coefficient $\eta=1$ is always higher than 99.8$\%$
when $\varepsilon\in[-0.1,0.1]$, which shows that the implementation of the
holonomic CNOT gate is quite insensitive to systematic errors. We also see
that the average fidelity falls to 0.9747 when $\varepsilon=\pm 0.1$ with
$\eta=0$. Noticing that when $\eta=0$ the current scheme reduces to previous
NHQC cases Xue  _et al._ (2015, 2016, 2017); Zhao  _et al._ (2017); Xu  _et
al._ (2018). Thus, the scheme can enhance the robustness against systematic
errors.

Figure 8: The geometric phase $\Theta_{2}(t)$ and the dynamic phase
$\theta_{2}(t)$ acquired in the implementation of the holonomic CNOT gate
versus $t$ with the effective Hamiltonian $H_{\rm eff}$ in Eq. (43).
Corresponding parameters are the same as that in Fig. 6.

In addition, we also simulate the average fidelity $F_{\rm CN}(t)$ in Fig.
7(a). As seen from Fig. 7(a), we obtain the average fidelity of the holonomic
CNOT gate as $F_{\rm CN}(T)=99.89\%$ at $t=T$, which is in accordance with the
expectation. Specially, we also choose a special initial state
$|\Phi_{0}(0)\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ to verify the
validity of the CNOT gate. Correspondingly, the target state is
$|\Phi_{0}(T)\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. As shown in
Fig. 7(b), the computational basis states $|00\rangle$ and $|01\rangle$ keep
in the initial states and the states $|10\rangle$ and $|11\rangle$ accomplish
the coherent population transfer. Moreover, the acquired geometric phase
$\Theta_{2}(t)$ and dynamic phase $\theta_{2}(t)$ are plotted in Fig. 8. We
can see from Fig. 8 that the dynamic phase $\theta_{2}(T)$ finally vanishes at
$t=T$, while the geometric phase reaches predetermined value
$\Theta_{2}(T)=\Theta_{g}=\pi$, which means that a pure geometric phase is
acquired in the evolution process. Therefore, the holonomic CNOT gate can be
successfully realized with high fidelity.

##  IV Analysis of errors and decoherence on the fidelity of the NHQC+ gate

In addition to the systematic errors $\varepsilon$ of the control fields
discussed in Sec. III.2, there may exists other errors and decoherence
affecting the performance of the NHQC+ gate. In a practical experiment
environment, the analysis of the robustness against these disturbing factors
is helpful for estimating the performance of the scheme. As an example, we
analyze the robustness for implementation of the holonomic CNOT gate in the
following discussion.

Figure 9: $1-F_{\rm CN}$ versus simulation count with signal-to-noise ratios
(a) $SNR=10$ and (b) $SNR=2$, respectively. Corresponding parameters are the
same as that in Fig. 6.

###  IV.1 Random noise in control fields

In the experiment, the random noise is an error source reducing the average
fidelity of gate in experiment. Because the fluctuation of Rabi frequencies of
the control fields is always random, thus it is worthwhile to study the
implementation of gate in a noisy environment. As the additive white Gaussian
noise (AWGN) is a nice model to simulate the process of many random noises,
here we study the performance of gate by adding AWGN to the original control
fields. The control fields under the influence of the AWGN can be described as

$\displaystyle\Omega^{\prime}_{x}(t)$ $\displaystyle=$
$\displaystyle\Omega_{x}(t)+{\rm AWGN}(\Omega_{x}(t),SNR),$ (72)
$\displaystyle\Omega^{\prime}_{y}(t)$ $\displaystyle=$
$\displaystyle\Omega_{y}(t)+{\rm AWGN}(\Omega_{y}(t),SNR),$ (74)

where ${\rm AWGN}(\Omega_{x}(t),SNR)$ (${\rm AWGN}(\Omega_{y}(t),SNR))$
denotes a function generating AWGN with the signal-to-noise ratio $R$ for
control field $\Omega_{x}(t)$ ($\Omega_{y}(t)$). Because the AWGN is random in
every single simulation, we perform 50 numerical simulations for the
implementation of the CNOT gate with $SNR=10$ and $SNR=2$, respectively. The
infidelities $1-F_{\rm CN}$ for corresponding signal-to-noise in each
simulation are shown in Figs. 9(a) and 9(b), respectively. As seen from Fig.
9, the infidelities $1-F_{\rm CN}$ fluctuate around 0.00205 and 0.00209 with
$SNR=10$ and $SNR=2$, respectively. Compared with the case without AWGN, the
noise only reduces the fidelity of the gate by $10^{-3}$. Therefore, the
results show that the scheme is robust against AWGN. The physical mechanism
behind this phenomenon is that the random values of AWGN added to the original
control fields are zero average, thus the total effect of AWGN on the
evolution of system can be nearly neglected. As the accumulations of geometric
phases are mainly related to the global property of the evolution path, the
local fluctuations caused by AWGN can not cause more errors Kang  _et al._
(2020a).

Figure 10: (a) The effect of deviation $\delta^{\prime}$ on the average
fidelity of the holonomic CNOT gate. (b) The effect of the Förster defect
$\delta$ on the average fidelity of the holonomic CNOT gate. Corresponding
parameters are the same as that in Fig. 6.

###  IV.2 Deviation of dipole-dipole interaction

In the process of obtaining the effective Hamiltonian $H_{\rm eff}$ in Eq.
(42), we have assumed the condition $\Delta=V$. Nevertheless, it is difficult
to strictly control the interaction between Rydberg states to satisfy the
condition in the experimental operation. In order to evaluate the performance
of deviation of dipole-dipole interaction on the average fidelity of the
holonomic CNOT gate, we plot the average fidelity of the gate versus the
deviation $\delta^{\prime}$ in Fig. 10(a), where we suppose
$V=2\pi\times(133.04+\delta^{\prime})$ MHz. As can be seen from Fig. 10(a),
the current scheme is insensitive to the movement of the distance between
atoms, and the average fidelity of the gate is higher than 94.5$\%$. This
infidelity is mainly due to the term
$\frac{\Omega_{b}}{\sqrt{2}}|r\xi_{+}\rangle\langle\varpi_{-}|+\mathrm{H.c.}$
of $H_{r}$ in Eq. (20). When the deviation of dipole-dipole interaction
exists, the transition between $|r\xi_{+}\rangle$ and $|\varpi_{-}\rangle$
becomes not resonant. Under the condition $\Omega_{b}\gg\Omega_{a}(t)$, the
state $|\xi_{+}\xi_{+}\rangle$ can not be completely suppressed in the
evolution process.

###  IV.3 Förster defect

In the presence of Förster defect, the interaction Hamiltonian between atoms
$H_{F}$ in Eq. (11) is modified in the basis $\\{|rr\rangle,|R\rangle\\}$ by

$\displaystyle H^{\prime}_{F}=\begin{pmatrix}0&V\\\ V&\delta\\\
\end{pmatrix},$ (75)

where $\delta$ is the Förster defect between states $|rr\rangle$ and
$|R\rangle$. The Förster defect changes the eigenvalue with a small deviation
$|V-\frac{1}{2}(\delta+\sqrt{4V^{2}+\delta^{2}})|$, which is approximately
equal to $\delta/2+\delta^{2}/8V$ for a small ratio $\delta/V$. In Fig. 10(b),
we numerically simulate the effect of the Förster defect on the average
fidelity of the holonomic CNOT gate. As shown in Fig. 10(b), the average
fidelity still reaches about 97$\%$ when $\delta/V=\pm 0.1$. Considering the
experimental parameter of defects $\delta=2\pi\times 8.5$ MHz in the absence
of an electric field Ravets  _et al._ (2014), the average fidelity of the gate
is 98.64$\%$, which shows that the scheme is still effective. On the whole,
the scheme is robust against the deviation of Förster defect.

###  IV.4 Decoherence induced by dissipation

Figure 11: (a) Fidelities $F_{\rm CN}$ of the implementation of the holonomic
CNOT gate versus $t$ with decay $\gamma=1$ kHz and without decay by choosing
initial state $(|00\rangle+|10\rangle)/\sqrt{2}$. (b) Final fidelities $F_{\rm
CN}(T)$ of the holonomic CNOT gate versus decay rate $\gamma$ with initial
state $(|00\rangle+|10\rangle)/\sqrt{2}$. Corresponding parameters are the
same as that in Fig. 6.

So far we do not consider the influence of decoherence on the scheme. In the
practical implementation, the system is always inevitably affected by the
environment. Therefore, we check the performance of decoherence on the
implementation of the holonomic CNOT gate with the initial state
$(|00\rangle+|10\rangle)/\sqrt{2}$ in the end of this section. The dissipation
dynamics process of the system is govern by the master equation,

$\displaystyle\dot{\rho}(t)$ $\displaystyle=$ $\displaystyle
i\big{[}H_{I},\rho(t)\big{]}+\sum_{p=1}^{2}\sum_{p^{\prime}=r,r_{+},r_{-}}\big{[}L_{pp^{\prime}}\rho(t)L_{pp^{\prime}}^{\dagger}-$
(77)
$\displaystyle-\frac{1}{2}L^{\dagger}_{pp^{\prime}}L_{pp^{\prime}}\rho(t)-\frac{1}{2}\rho(t)L^{\dagger}_{pp^{\prime}}L_{pp^{\prime}}\big{]},$

where $\rho(t)$ being the density operator of the system, $p$ being the atomic
number, and $p^{\prime}$ being the Rydberg state. Besides, the corresponding
decay from excited states $|r\rangle$, $|r_{+}\rangle$, $|r_{-}\rangle$ into
ground states $|0\rangle$ and $|1\rangle$ can be expressed as the operator
$L_{pp^{\prime}}=\sqrt{\gamma_{pp^{\prime}}}|j\rangle_{p}\langle p^{\prime}|$
$(j=0,1$). For simplicity, we assume the atomic spontaneous emission rates
$\gamma_{pp^{\prime}}=\gamma$.

Figure 12: Truth table for the scheme of the holonomic CNOT gate in the
presence of decay $\gamma=1$ kHz. Corresponding parameters are the same as
that in Fig. 6.

We plot the fidelities of the evolution of initial state versus $t$ without
decay and with decay in Fig. 11(a). As seen from the green solid line in Fig.
11(a), the fidelity of the holonomic CNOT gate is $F_{\rm CN}(T)=99.89\%$ at
$t=T$ without decay. Moreover, the blue solid line in Fig. 11(a) represents
the fidelity versus $t$ in the presence of decay with $\gamma=1$ kHz, which
indicates $F_{\rm CN}(T)=99.42\%$. We also plot the final fidelity $F_{\rm
CN}(T)$ of the holonomic CNOT gate with the initial state
$(|00\rangle+|10\rangle)/\sqrt{2}$ versus decay $\gamma$ in Fig. 11(b). From
Fig. 11(b), we can find the final fidelity $F_{\rm CN}(T)$ is very high (above
$98\%$). Thus, the scheme is robust against the atomic spontaneous emission.
This is because the populations of partial Rydberg states are decoupled with
the effective Hamiltonian in the limits of large detuning
$\Delta\gg\Omega_{b}$ and $\Omega_{b}\gg\Omega_{a}(t)$. In order to better
characterize the performance of the holonomic CNOT gate in the presence of
decay, we also numerically calculate the truth table Reed  _et al._ (2012) in
Fig. 12. Specifically, we need to perform the gate operation on each of the
computational basis states (Input state $|\Phi_{In}\rangle$). By calculating
the populations defined as $|\langle\Phi_{Out}|U_{\rm
CN}(T)|\Phi_{In}\rangle|^{2}$, we can obtain the corresponding population of
output state ($|\Phi_{Out}\rangle$) for each input state, where $U_{\rm
CN}(T)$ is the evolution operator for the gate at $t=T$. From the numbers on
the bars shown in Fig. 12, the minimum values of population of each input
state exceeds 98.7$\%$. Thus, the scheme is helpful to realize a perfect two-
qubit gate operation in Rydberg atom systems.

##  V Conclusion

In conclusion, we have proposed a scheme to realize the NHQC+ in the regime of
Rydberg Förster resonance. In this scheme, assisted by the strong dipole-
dipole interaction between Rydberg atoms, a selective coupling mechanism for
the subspace $\mathcal{S}$ is proposed to simply the dynamics of system. In
the whole evolution process, the simultaneous excitation of Rydberg atoms is
avoided. Based on the derived effective Hamiltonian, the NHQC+ dynamics can be
naturally constructed and the corresponding control fields are designed to
implement the quantum gate with the help of IBRE. The advantages of IBRE in
NHQC+ have been shown in the paper. On one hand, the eigenvectors of the
dynamical invariant provide natural evolution paths to realize NHQC+. On the
other hand, IBRE can be compatible with the systematic-error-sensitivity
optimal control method, which further enhances the robustness against
systematic errors of control fields. By minimizing the systematic error
sensitivity $q_{s}$, the optimal Rabi frequencies of control fields can be
designed. Moreover, the control fields do not involve sudden changes in the
whole process by setting proper boundary conditions for the time derivatives
of control parameters, which means that the control fields are continuous. In
addition, the performance of the NHQC+ gate is estimated in the presence of
the random noise in control fields, the deviation of dipole-dipole
interaction, the Förster defect, and decoherence based on the master equation.
Both the theoretical and numerical results show that the scheme can realize
high-fidelity quantum gate.

Compared with previous schemes Qi and Jing (2020); Xue  _et al._ (2016, 2017);
Zhao  _et al._ (2017); Xu  _et al._ (2018), the current scheme has flexible
parameter selections so that it can be compatible with many control and
optimal methods. For example, in contrast to Ref. Qi and Jing (2020), the
design of Hamiltonian via IBRE method in the current scheme is different from
that via transitionless quantum driving (TQD) method. The design of
Hamiltonian by the TQD method is to steer the dynamics along the instantaneous
eigenstates of the original Hamiltonian without transitions among them. The
additional Hamiltonians need to be added to compensate the nonadiabatic errors
according to the original Hamiltonian. Thus, the original Hamiltonian needs to
be designed first. However, the design of Hamiltonian by the IBRE method is to
reversely derive the Hamiltonian by constructing a dynamic invariant. In this
case, the eigenvectors of the dynamic invariant as the exact evolution paths
can be obtained by designing the boundary condition of parameters. Thus, the
IBRE method has more freedom to design the parameters. Accordingly, the
convenience of parameter selections make the scheme well incorporate the
systematic-error-sensitivity optimal control method. This enables the scheme
to maintain high fidelity in the presence of errors. As the scheme is fully
compatible with the advantages of geometric phases, reverse engineering, and
the systematic-error-sensitivity optimal control method, it can be helpful for
precise quantum computation in Rydberg atom systems.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grants No. 11575045, No. 11874114, and No. 11674060, the Natural Science
Funds for Distinguished Young Scholar of Fujian Province under Grant
2020J06011 and Project from Fuzhou University under Grant JG202001-2.

##  Appendix A Demonstration of
$|\tilde{\vartheta}_{2}(t)\rangle\langle\tilde{\vartheta}_{2}(t)|$ meeting the
von Neumann equation

From Eq. (2),
$|\psi_{2}(t)\rangle=\exp[i\alpha_{2}(t)]|\vartheta_{2}(t)\rangle=\exp[i\alpha_{2}(t)]|\tilde{\vartheta}_{2}(t)\rangle$
is satisfied the Schrödinger equation

$\displaystyle i\frac{d|\psi_{2}(t)\rangle}{dt}=H_{\rm
eff}|\psi_{2}(t)\rangle.$ (78)

Define
$\Pi_{2}(t)=|\tilde{\vartheta}_{2}(t)\rangle\langle\tilde{\vartheta}_{2}(t)|=|\psi_{2}(t)\rangle\langle\psi_{2}(t)|$.
We can demonstrate the $\Pi_{2}(t)$ meets the von Neumann equation by
$|\psi_{2}(t)\rangle\langle\psi_{2}(t)|$ . Then,

$\displaystyle\frac{d\Pi_{2}(t)}{dt}$ $\displaystyle=$
$\displaystyle\frac{d|\psi_{2}(t)\rangle}{dt}\langle\psi_{2}(t)|+|\psi_{2}(t)\rangle\frac{d\langle\psi_{2}(t)|}{dt}$
(79) $\displaystyle=$ $\displaystyle-iH_{\rm
eff}|\psi_{2}(t)\rangle\langle\psi_{2}(t)|+i|\psi_{2}(t)\rangle\langle\psi_{2}(t)|H_{\rm
eff}$ (81) $\displaystyle=$ $\displaystyle-i[H_{\rm
eff},|\psi_{2}(t)\rangle\langle\psi_{2}(t)|].$ (83)

Therefore, the $|\tilde{\vartheta}_{2}(t)\rangle$ satisfies the von Neumann
equation

$\displaystyle\frac{d\Pi_{2}(t)}{dt}=-i[H_{\rm eff},\Pi_{2}(t)].$ (84)

##  Appendix B Demonstration of dynamical phase in the evolution process equal
to zero

In order to eliminate the dynamic phase, the evolution of the system is
divided into two time intervals [0,$T/2$] and [$T/2$,$T$]. First, we calculate
the time derivative of $\mu_{2}(t)$ by setting $\tilde{t}=T-t$ in the time
interval [$T/2$,$T$] as

$\displaystyle\frac{d\mu_{2}(t)}{dt}=\frac{d\tilde{t}}{dt}\frac{d}{d\tilde{t}}[-\Theta_{g}+\mu_{2}(\tilde{t})]=-\frac{d}{d\tilde{t}}\mu_{2}(\tilde{t}).$
(85)

According to Eqs. (8) and (10) , we have

$\displaystyle\theta_{2}(T)-\theta_{2}(T/2)$ $\displaystyle=$
$\displaystyle\int_{T/2}^{T}\frac{\dot{\mu}_{2}\sin^{2}\mu_{1}}{2\cos\mu_{1}}dt$
(86) $\displaystyle=$
$\displaystyle\int_{T/2}^{0}\frac{\dot{\mu}_{2}(\tilde{t})\sin^{2}\mu_{1}(\tilde{t})}{2\cos\mu_{1}(\tilde{t})}d\tilde{t}$
(88) $\displaystyle=$ $\displaystyle-\theta_{2}(T/2),$ (90)
$\displaystyle\Theta_{2}(T)-\Theta_{2}(T/2)$ $\displaystyle=$
$\displaystyle\int_{T/2}^{T}\dot{\mu}_{2}(t)\sin^{2}\frac{\mu_{1}(t)}{2}dt$
(92) $\displaystyle=$
$\displaystyle\int_{T/2}^{0}\dot{\mu}_{2}(\tilde{t})\sin^{2}\frac{\mu_{1}(\tilde{t})}{2}d\tilde{t}$
(94) $\displaystyle=$ $\displaystyle-\Theta_{2}(T/2).$ (96)

It is proved that the dynamic phase and geometric phase acquired in the time
interval [0,$T/2$] is nullified by that acquired in the time interval
[$T/2$,$T$]. The geometric phase is only attributed by making the value of
$\mu_{2}$ increase $-\Theta_{g}$ at the moment of $t=T/2$. This can be
interpreted as

$\displaystyle\Theta_{2}(T/2+\Delta t)-\Theta_{2}(T/2)$ $\displaystyle=$
$\displaystyle-\int_{T/2}^{T/2+\Delta
t}\dot{\mu}_{2}(t)\sin^{2}\frac{\mu_{1}(t)}{2}dt$ (97) $\displaystyle=$
$\displaystyle\mu_{2}(T/2)-\mu_{2}(T/2+\Delta t)$ (99) $\displaystyle=$
$\displaystyle\Theta_{g},$ (101)

when $\Delta t\rightarrow 0$. In the time interval [$T/2$, $T/2+\Delta t$],
the $\mu_{1}(t)=\pi$. According to Eq. (57), the dynamic phase acquired is
zero. Thus the total dynamic phase acquired in the time interval [0, $T$] is
zero.

## References

  * Grover (1997) L. K. Grover, Phys. Rev. Lett. 79, 325 (1997).
  * Long (2001) G. L. Long, Phys. Rev. A 64, 022307 (2001).
  * Paredes  _et al._ (2005) B. Paredes, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 95, 140501 (2005).
  * Fan  _et al._ (2018) Y.-J. Fan, Z.-F. Zheng, Y. Zhang, D.-M. Lu, and C.-P. Yang, Front. Phys. 14, 21602 (2018).
  * Fan and Zhu (2019) H. Fan and X. Zhu, Front. Phys. 14, 61201 (2019).
  * Sjöqvist  _et al._ (2012) E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, New J. Phys. 14, 103035 (2012).
  * Xu  _et al._ (2012) G. F. Xu, J. Zhang, D. M. Tong, E. Sjöqvist, and L. C. Kwek, Phys. Rev. Lett. 109, 170501 (2012).
  * Zhao  _et al._ (2019) P. Z. Zhao, G. F. Xu, and D. M. Tong, Phys. Rev. A 99, 052309 (2019).
  * Zhao  _et al._ (2020) P. Z. Zhao, K. Z. Li, G. F. Xu, and D. M. Tong, Phys. Rev. A 101, 062306 (2020).
  * Xing  _et al._ (2020) T. H. Xing, X. Wu, and G. F. Xu, Phys. Rev. A 101, 012306 (2020).
  * Zhao  _et al._ (2018) P. Z. Zhao, X. Wu, T. H. Xing, G. F. Xu, and D. M. Tong, Phys. Rev. A 98, 032313 (2018).
  * Xu  _et al._ (2017) G. F. Xu, P. Z. Zhao, T. H. Xing, E. Sjöqvist, and D. M. Tong, Phys. Rev. A 95, 032311 (2017).
  * Chen  _et al._ (2018a) T. Chen, J. Zhang, and Z.-Y. Xue, Phys. Rev. A 98, 052314 (2018a).
  * Xu  _et al._ (2020a) Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, Phys. Rev. Lett. 124, 230503 (2020a).
  * Li  _et al._ (2021) S. Li, P. Shen, T. Chen, and Z.-Y. Xue, Front. Phys. 16, 51502 (2021).
  * Liu  _et al._ (2019) B.-J. Liu, X.-K. Song, Z.-Y. Xue, X. Wang, and M.-H. Yung, Phys. Rev. Lett. 123, 100501 (2019).
  * Berry (2009) M. V. Berry, J. Phys. A 42, 365303 (2009).
  * Chen  _et al._ (2010) X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010).
  * del Campo (2013) A. del Campo, Phys. Rev. Lett. 111, 100502 (2013).
  * Qi and Jing (2020) S.-f. Qi and J. Jing, J. Opt. Soc. Am. B 37, 682 (2020).
  * Khodjasteh and Lidar (2005) K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95, 180501 (2005).
  * Souza  _et al._ (2011) A. M. Souza, G. A. Álvarez, and D. Suter, Phys. Rev. Lett. 106, 240501 (2011).
  * Genov  _et al._ (2017) G. T. Genov, D. Schraft, N. V. Vitanov, and T. Halfmann, Phys. Rev. Lett. 118, 133202 (2017).
  * Daems  _et al._ (2013) D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, Phys. Rev. Lett. 111, 050404 (2013).
  * Yu  _et al._ (2018) X.-T. Yu, Q. Zhang, Y. Ban, and X. Chen, Phys. Rev. A 97, 062317 (2018).
  * Lewis and Riesenfeld (1969) H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
  * Chen  _et al._ (2011) X. Chen, E. Torrontegui, and J. G. Muga, Phys. Rev. A 83, 062116 (2011).
  * Li  _et al._ (2020) S. Li, T. Chen, and Z.-Y. Xue, Adv. Quantum Technol. 3, 2000001 (2020).
  * Yan  _et al._ (2019) T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, Phys. Rev. Lett. 122, 080501 (2019).
  * Xu  _et al._ (2020b) J. Xu, S. Li, T. Chen, and Z.-Y. Xue, Front. Phys 15, 41503 (2020b).
  * Kang  _et al._ (2020a) Y.-H. Kang, Z.-C. Shi, B.-H. Huang, J. Song, and Y. Xia, Phys. Rev. A 101, 032322 (2020a).
  * Zhang  _et al._ (2020) C. Zhang, T. Chen, S. Li, X. Wang, and Z.-Y. Xue, Phys. Rev. A 101, 052302 (2020).
  * Guo  _et al._ (2020) C.-Y. Guo, L.-L. Yan, S. Zhang, S.-L. Su, and W. Li, Phys. Rev. A 102, 042607 (2020).
  * Kang  _et al._ (2020b) Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, Phys. Rev. A 102, 022617 (2020b).
  * Liu  _et al._ (2020) B.-J. Liu, S.-L. Su, and M.-H. Yung, Phys. Rev. Research 2, 043130 (2020).
  * Chen  _et al._ (2020) T. Chen, P. Shen, and Z.-Y. Xue, Phys. Rev. Appl. 14, 034038 (2020).
  * Saffman  _et al._ (2010) M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010).
  * Urban  _et al._ (2009) E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, Nat. Phys. 5, 110 (2009).
  * Gaëtan  _et al._ (2009) A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, Nat. Phys. 5, 115 (2009).
  * Barredo  _et al._ (2015) D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, Phys. Rev. Lett. 114, 113002 (2015).
  * Su  _et al._ (2017a) S.-L. Su, Y. Gao, E. Liang, and S. Zhang, Phys. Rev. A 95, 022319 (2017a).
  * Kang  _et al._ (2018) Y.-H. Kang, Y.-H. Chen, Z.-C. Shi, B.-H. Huang, J. Song, and Y. Xia, Phys. Rev. A 97, 042336 (2018).
  * Su  _et al._ (2018) S. L. Su, H. Z. Shen, E. Liang, and S. Zhang, Phys. Rev. A 98, 032306 (2018).
  * Li and Shao (2019) D. X. Li and X. Q. Shao, Phys. Rev. A 99, 032348 (2019).
  * Chen  _et al._ (2018b) Y.-H. Chen, Z.-C. Shi, J. Song, Y. Xia, and S.-B. Zheng, Phys. Rev. A 97, 032328 (2018b).
  * Zheng  _et al._ (2020a) R.-H. Zheng, Y.-H. Kang, D. Ran, Z.-C. Shi, and Y. Xia, Phys. Rev. A 101, 012345 (2020a).
  * Zheng  _et al._ (2020b) R.-H. Zheng, Y.-H. Kang, S.-L. Su, J. Song, and Y. Xia, Phys. Rev. A 102, 012609 (2020b).
  * Su  _et al._ (2020) S.-L. Su, F.-Q. Guo, L. Tian, X.-Y. Zhu, L.-L. Yan, E.-J. Liang, and M. Feng, Phys. Rev. A 101, 012347 (2020).
  * Shi (2021) X.-F. Shi, Front. Phys 16, 52501 (2021).
  * Zheng  _et al._ (2021) R.-H. Zheng, Y. Xiao, S.-L. Su, Y.-H. Chen, Z.-C. Shi, J. Song, Y. Xia, and S.-B. Zheng, Phys. Rev. A 103, 052402 (2021).
  * Yin  _et al._ (2020) H.-D. Yin, X.-X. Li, G.-C. Wang, and X.-Q. Shao, Opt. Express 28, 35576 (2020).
  * Jaksch  _et al._ (2000) D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).
  * Lukin  _et al._ (2001) M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 037901 (2001).
  * Ates  _et al._ (2007) C. Ates, T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 98, 023002 (2007).
  * Amthor  _et al._ (2010) T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, Phys. Rev. Lett. 104, 013001 (2010).
  * Su  _et al._ (2017b) S.-L. Su, Y. Tian, H. Z. Shen, H. Zang, E. Liang, and S. Zhang, Phys. Rev. A 96, 042335 (2017b).
  * Müller  _et al._ (2014) M. M. Müller, M. Murphy, S. Montangero, T. Calarco, P. Grangier, and A. Browaeys, Phys. Rev. A 89, 032334 (2014).
  * Keating  _et al._ (2015) T. Keating, R. L. Cook, A. M. Hankin, Y.-Y. Jau, G. W. Biedermann, and I. H. Deutsch, Phys. Rev. A 91, 012337 (2015).
  * Mitra  _et al._ (2020) A. Mitra, M. J. Martin, G. W. Biedermann, A. M. Marino, P. M. Poggi, and I. H. Deutsch, Phys. Rev. A 101, 030301 (2020).
  * Huang  _et al._ (2018) X.-R. Huang, Z.-X. Ding, C.-S. Hu, L.-T. Shen, W. Li, H. Wu, and S.-B. Zheng, Phys. Rev. A 98, 052324 (2018).
  * Beterov  _et al._ (2018a) I. I. Beterov, G. N. Hamzina, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, and I. I. Ryabtsev, Phys. Rev. A 97, 032701 (2018a).
  * Förster (1948) T. Förster, Ann. Phys 437, 55 (1948).
  * Ryabtsev  _et al._ (2010) I. I. Ryabtsev, D. B. Tretyakov, I. I. Beterov, and V. M. Entin, Phys. Rev. Lett. 104, 073003 (2010).
  * Gorniaczyk  _et al._ (2016) H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsky, and S. Hofferberth, Nat. Commun. 7 (2016).
  * Ravets  _et al._ (2014) S. Ravets, H. Labuhn, D. Barredo, L. Béguin, T. Lahaye, and A. Browaeys, Nat. Phys. 10, 914 (2014).
  * Wu  _et al._ (2010) H.-Z. Wu, Z.-B. Yang, and S.-B. Zheng, Phys. Rev. A 82, 034307 (2010).
  * Shao (2020) X.-Q. Shao, Phys. Rev. A 102, 053118 (2020).
  * Beterov  _et al._ (2016) I. I. Beterov, M. Saffman, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, S. Bergamini, E. A. Kuznetsova, and I. I. Ryabtsev, Phys. Rev. A 94, 062307 (2016).
  * Beterov  _et al._ (2018b) I. I. Beterov, I. N. Ashkarin, E. A. Yakshina, D. B. Tretyakov, V. M. Entin, I. I. Ryabtsev, P. Cheinet, P. Pillet, and M. Saffman, Phys. Rev. A 98, 042704 (2018b).
  * Li and Shao (2018) D. X. Li and X. Q. Shao, Phys. Rev. A 98, 062338 (2018).
  * Li  _et al._ (2005) W. Li, P. J. Tanner, and T. F. Gallagher, Phys. Rev. Lett. 94, 173001 (2005).
  * Amthor  _et al._ (2007a) T. Amthor, M. Reetz-Lamour, S. Westermann, J. Denskat, and M. Weidemüller, Phys. Rev. Lett. 98, 023004 (2007a).
  * Amthor  _et al._ (2007b) T. Amthor, M. Reetz-Lamour, C. Giese, and M. Weidemüller, Phys. Rev. A 76, 054702 (2007b).
  * Ruschhaupt  _et al._ (2012) A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, New J. Phys. 14, 093040 (2012).
  * Walker and Saffman (2005) T. G. Walker and M. Saffman, J. Phys. B 38, S309 (2005).
  * Walker and Saffman (2008) T. G. Walker and M. Saffman, Phys. Rev. A 77, 032723 (2008).
  * James and Jerke (2007) D. F. James and J. Jerke, Can. J. Phys. 85, 625 (2007).
  * Ravets  _et al._ (2015) S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, Phys. Rev. A 92, 020701 (2015).
  * Zanardi and Lidar (2004) P. Zanardi and D. A. Lidar, Phys. Rev. A 70, 012315 (2004).
  * Pedersen  _et al._ (2007) L. H. Pedersen, N. M. Møller, and K. Mølmer, Phys. Lett. A 367, 47 (2007).
  * Xue  _et al._ (2015) Z.-Y. Xue, J. Zhou, and Z. D. Wang, Phys. Rev. A 92, 022320 (2015).
  * Xue  _et al._ (2016) Z.-Y. Xue, J. Zhou, Y.-M. Chu, and Y. Hu, Phys. Rev. A 94, 022331 (2016).
  * Xue  _et al._ (2017) Z.-Y. Xue, F.-L. Gu, Z.-P. Hong, Z.-H. Yang, D.-W. Zhang, Y. Hu, and J. Q. You, Phys. Rev. Appl. 7, 054022 (2017).
  * Zhao  _et al._ (2017) P. Z. Zhao, X.-D. Cui, G. F. Xu, E. Sjöqvist, and D. M. Tong, Phys. Rev. A 96, 052316 (2017).
  * Xu  _et al._ (2018) Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y. P. Song, Z.-Y. Xue, Z.-q. Yin, and L. Sun, Phys. Rev. Lett. 121, 110501 (2018).
  * Reed  _et al._ (2012) M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature 482, 382 (2012).