# Single-site Rydberg addressing in 3D atomic arrays for quantum computing
with neutral atoms

Xiao-Feng Shi  School of Physics and Optoelectronic Engineering, Xidian
University, Xi’an 710071, China

###### Abstract

Neutral atom arrays are promising for large-scale quantum computing especially
because it is possible to prepare large-scale qubit arrays. An unsolved issue
is how to selectively excite one qubit deep in a 3D atomic array to Rydberg
states. In this work, we show two methods for this purpose. The first method
relies on a well-known result: in a dipole transition between two quantum
states driven by two off-resonant fields of equal strength but opposite
detunings $\pm\Delta$, the transition is characterized by two counter-rotating
Rabi frequencies $\Omega e^{\pm i\Delta t}$ [or $\pm\Omega e^{\pm i\Delta t}$
if the two fields have a $\pi$-phase difference]. This pair of detuned fields
lead to a time-dependent Rabi frequency $2\Omega\cos(\Delta t)$ [or
$2i\Omega\sin(\Delta t)$], so that a full transition between the two levels is
recovered. We show that when the two detuned fields are sent in different
directions, one atom in a 3D optical lattice can be selectively addressed for
Rydberg excitation, and when its state is restored, the state of any nontarget
atoms irradiated in the light path is also restored. Moreover, we find that
the Rydberg excitation by this method can significantly suppress the
fundamental blockade error of a Rydberg gate, paving the way for a high-
fidelity entangling gate with commonly used quasi-rectangular pulse that is
easily obtained by pulse pickers. Along the way, we find a second method for
single-site Rydberg addressing in 3D, where a selected target atom can be
excited to Rydberg state while preserving the state of any nontarget atom due
to a spin echo sequence. The capability to selectively address a target atom
in 3D atomic arrays for Rydberg excitation makes it possible to design large-
scale neutral-atom information processor based on Rydberg blockade.

##  I introduction

Neutral atom arrays have become a promising platform for large-scale quantum
computing Jaksch  _et al._ (2000); Saffman  _et al._ (2010, 2011); Saffman
(2016); Weiss and Saffman (2017); Adams  _et al._ (2019). The attraction lies
in that one can not only prepare large-scale qubit arrays and initialize the
state for the qubits with high fidelity, but can also prepare high-fidelity
single-qubit gate and easily read out the quantum information stored in the
qubits Aljunid  _et al._ (2009); Leung  _et al._ (2014); Nogrette  _et al._
(2014); Xia  _et al._ (2015); Zeiher  _et al._ (2015); Ebert  _et al._ (2015);
Wang  _et al._ (2016); Barredo  _et al._ (2016). The fidelity of two-qubit
entangling gates is growing steadily recently Wilk  _et al._ (2010); Isenhower
_et al._ (2010); Zhang  _et al._ (2010); Maller  _et al._ (2015); Zeng  _et
al._ (2017); Picken  _et al._ (2019); Levine  _et al._ (2018); Graham  _et
al._ (2019); Levine  _et al._ (2019), from the initial value of $73\%$
Isenhower  _et al._ (2010) to $97\%$ Levine  _et al._ (2019), which points to
the likeliness that high-fidelity universal gate set can be experimentally
realized in the near future for the required accuracy of measurement-free
fault-tolerant quantum computing Crow  _et al._ (2016). Nevertheless, there
are still many unsolved issues toward this goal as reviewed in Ref. Saffman
(2016).

One issue for large-scale quantum information processing in a 3-dimensional
(3D) qubit array is how to selectively excite one qubit deep in the lattice
between ground and Rydberg states. To date, there has been demonstration of
two-qubit gate in a two-dimensional (2D) array of neutral atoms Maller  _et
al._ (2015); Graham  _et al._ (2019), where the single-site addressing without
affecting nontarget atoms is guaranteed by sending laser fields perpendicular
to the 2D array, and thus is not applicable in a 3D lattice. There have been
several experiments on single-qubit gates in 3D Wang  _et al._ (2015, 2016) or
2D Xia  _et al._ (2015) atomic arrays by using a hybrid irradiation of laser
and microwave fields. These methods depend on shifting the atomic transition
frequency by sending laser fields upon target atoms, where the Stark shifts
(divided by the Planck constant) should be comparable to the inverse of the
gate durations of several hundred $\mu s$. To the best of our knowledge,
however, there is no method of single-site Rydberg addressing in a 3D lattice
where nontarget atoms in the light path are not influenced.

In this work, we present two methods for selectively exciting one qubit in any
site of a 3D qubit array to Rydberg state. The first method relies on a well-
known phenomenon: the action of two symmetrically detuned laser pulses is
equivalent to that of a monochromatic field whose amplitude is sinusoidally
modulated in time, as studied in Ref. Goreslavsky  _et al._ (1980). This means
that a full dipole transition can also occur by absorbing two fractional
photons, i.e., by absorbing half of a photon with energy $E_{ge}+\hbar\Delta$,
and half of another photon with energy $E_{ge}-\hbar\Delta$, where $\hbar$ and
$\Delta$ are the reduced Planck constant and frequency detuning, respectively.
This resonance can be called off-resonance-induced resonance (ORIR). By ORIR,
one laser light is sent along one direction, while the other sent along
another direction, and they together excite a target atom. The nontarget atom
can accumulate phase shift or even experience state evolution in general, but
we show that our ORIR-based theory can eliminate its effect when the state of
the target atom is restored. Besides the ORIR-based method, we show a second
method with a three-level ladder type system, where one target atom can be
excited to Rydberg state while preserving the state of any nontarget atom. In
the second method, the state of the target atom can pick a $\pi$ phase shift
upon the completion of the ground-Rydberg-ground state transfer; such a $\pi$
phase is crucial in the controlled-phase gate based on Rydberg blockade. These
methods make it possible to couple only one atom to the Rydberg state in 3D.

We further show that ORIR can lead to high fidelity in the Rydberg blockade
gate. In quantum computing with neutral atoms and Rydberg interactions Jaksch
_et al._ (2000); Saffman  _et al._ (2010, 2011); Saffman (2016); Weiss and
Saffman (2017); Adams  _et al._ (2019), it has been an outstanding challenge
to design a practical high-fidelity entangling gate Goerz  _et al._ (2014);
Theis  _et al._ (2016); Shi (2017); Petrosyan  _et al._ (2017); Shi (2018a);
Shen  _et al._ (2019); Shi (2019); Yu  _et al._ (2019); Levine  _et al._
(2019). A traditional method to achieve an entangling gate by Rydberg
interactions is via the blockade mechanism Jaksch  _et al._ (2000), in which
there is a fundamental blockade error. We show that when the Rabi frequency
$\Omega$ in the usual method is replaced by an ORIR-induced Rabi frequency
$i\Omega\sin(\Delta t)$, the blockade error can be suppressed by more than two
orders of magnitude. More important, this reduction of error by ORIR is robust
against the variation of the blockade interaction, which compares favorably to
other pulse-shaping-based methods for suppressing the blockade error. Thus,
ORIR can effectively remove the blockade error, making it possible to realize
a high-fidelity neutral-atom entangling gate with quasi-rectangular pulses
that are easily attainable by pulse pickers.

The remainder of this work is organized as follows. In Sec. II, we give
details about how ORIR appears. In Sec. III, we study two methods for single-
site Rydberg addressing in a 3D optical lattice; a comparison between the two
methods is given at the end of Sec. III. In Sec. IV, we show that ORIR can
suppress a fundamental rotation error in the Rydberg blockade gate. In Sec. V,
we discuss other applications of ORIR. Section VI gives a brief summary.

##  II Resonance from off resonance

Figure 1: (a) Schematic and population dynamics of Rabi oscillations between
$|g\rangle$ and $|e\rangle$ driven by an external field with a detuning of
$\Delta$ or $-\Delta$. The initial state is $|\psi(0)\rangle=|g\rangle$. (b)
When the transition $|g\rangle\leftrightarrow|e\rangle$ is driven by two
external coherent fields with opposite detunings $\pm\Delta$, an effective
Rabi frequency of $2\Omega\cos(\Delta t)$ appears. This leads to a resonant
transition between $|g\rangle$ and $|e\rangle$. The condition
$\Delta/\Omega=2/\pi$ is used in both (a) and (b).

The reason for ORIR to occur lies in that the dipole coupling between an atom
and two symmetrically detuned laser fields can be characterized by a
sinusoidal Rabi frequency $2i\Omega\sin(\Delta t)$ or $2\Omega\cos(\Delta t)$
Goreslavsky  _et al._ (1980). To put it in perspective we consider a pair of
external coherent electromagnetic fields of equal strength but with opposite
detunings $\pm\Delta$ applied for the dipole transition
$|g\rangle\leftrightarrow|e\rangle$. In the dipole approximation, these two
fields lead to a pair of counter-rotating Rabi frequencies
$\Omega_{\pm}=\Omega e^{\pm i\Delta t}$. When only one of these two detuned
fields is applied, the expectation value of the population in $|e\rangle$ has
an upper bound $\Omega^{2}/(\Omega^{2}+\Delta^{2})$ if the initial state is
$|g\rangle$ Shi (2017, 2018a), as shown in Fig. 1(a). But when the two
oppositely detuned fields are applied simultaneously and because
$\Omega_{+}+\Omega_{-}=2\Omega\cos(\Delta t)$, a full transition between
$|g\rangle$ and $|e\rangle$ becomes attainable within a time of less than
$|\pi/\Delta|$ as long as $|\Omega/\Delta|\geq\pi/2$. The population dynamics
in the two-level system in response to the detuned driving is shown in Fig. 1
when $\Omega/\Delta=\pi/2$. Figures 1(a) and 1(b) show that the dynamics in
the two-level system driven by the two oppositely detuned fields dramatically
differs from that when only one detuned driving is present. This phenomenon
results from the quantum interference between the two Rabi oscillations with
opposite detunings. Remarkably, the speed of this off-resonance-induced
resonant transition is comparable to its resonant counterpart: when
$|\Omega/\Delta|=\pi/2$, a full transition between the two states is achieved
within a time of $t_{\pi}\pi/2$ that is comparable to the duration,
$t_{\pi}=\pi/(2\Omega)$, of a $\pi$ pulse required for a full transition in
the resonant case with the Rabi frequency $2\Omega$. This type of multi-photon
ORIR can occur in various electric or magnetic dipole transitions of an atom
or molecule, either natural or artificial.

We give a detailed mathematical argument for the above discussion. For a two-
level system in Fig. 1, the eigenenergies of $|g\rangle$ and $|e\rangle$ are
zero and $E_{\text{ge}}$, respectively. The electric dipole transition between
$|g\rangle$ and $|e\rangle$ is driven by two laser fields, one with central
(angular) frequency $\omega+\Delta$, and the other with central frequency
$\omega-\Delta$, where $\omega=E_{\text{ge}}/\hbar$. The Hamiltonian (divided
by $\hbar$) of the matter-light coupling in dipole approximation is

$\displaystyle\hat{H}(t)$ $\displaystyle=$
$\displaystyle\omega|e\rangle\langle
e|+[\Omega_{1}(e^{it(\omega+\Delta)}+e^{-it(\omega+\Delta)})|e\rangle\langle
g|$ (1)
$\displaystyle+\Omega_{2}(e^{it(\omega-\Delta)}+e^{-it(\omega-\Delta)})|e\rangle\langle
g|+\text{H.c.}]/2,$

where we assume that $\Omega_{1}$ and $\Omega_{2}$ are real, and “H.c.”
denotes the Hermitian conjugate. In principle, there is a phase difference
$\varphi_{0}=(\mathbf{k}_{1}-\mathbf{k}_{2})\cdot\mathbf{r}$ between
$\Omega_{1}$ and $\Omega_{2}$, where the subscript $1(2)$ distinguishes the
two transitions with opposite detunings. If the Rydberg state $|e\rangle$ is
excited by couterpropagating fields along $\mathbf{z}$ via two-photon
excitation through an intermediate state with a GHz-scale detuning
$\delta_{1(2)}$, and if meanwhile the fields for $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}$ copropagate, $\varphi_{0}$ becomes
$2(\Delta+\delta_{2}-\delta_{1})z/c$, where $z$ is the z-component coordinate
of the atom, $c$ the speed of light. The difference between $\delta_{1}$ and
$\delta_{2}$ is much larger than $\Omega_{1(2)}$ so that a common intermediate
state is used for the two two-photon transitions; for instance, $\delta_{1}$
and $\delta_{2}$ can have different signs. Because $\Delta\sim\Omega_{1(2)}$
and the effective two-photon Rabi frequency $\Omega_{1(2)}$ is on the order of
MHz, $\varphi_{0}\approx 2(\delta_{2}-\delta_{1})z/c$. For atoms cooled to
temperatures around $1$ K or below, the change of $z$ within several $\mu$s is
on the order of $\mu$m, which means that $\varphi_{0}$ is almost constant;
furthermore, $|\delta_{2}-\delta_{1}|/c\ll k_{1},k_{2}$, and thus we can
assume that this phase has been compensated by adjustment of the overall
phases carried by the laser beams. This is technically possible since the
phase fluctuation of a laser field can be made negligible Saffman and Walker
(2005); Wineland  _et al._ (1998). But for single-site Rydberg addressing with
our methods (shown in Sec. III), the fields for $\Omega_{1}$ and those for
$\Omega_{2}$ propagate along different directions, e.g., one along
$\mathbf{z}$ and the other along $\mathbf{x}$, then
$\varphi_{0}\approx\mathcal{K}(z-x)$, where $\mathcal{K}$ is the difference of
the wavevectors of the upper and lower transitions. For configurations with
$\mathcal{K}$ of several $10^{6}m^{-1}$ Isenhower  _et al._ (2010), the
fluctuation of the qubit locations lead to large errors in $\varphi_{0}$, and
it is necessary to cool atoms to very low temperatures to establish ORIR.

Using the operator $\hat{\mathcal{R}}=\omega|e\rangle\langle e|$ for a
rotating-frame transform,
$e^{i\hat{\mathcal{R}}t}\hat{H}e^{-i\hat{\mathcal{R}}t}-\hat{\mathcal{R}}$,
the Hamiltonian becomes,

$\displaystyle\hat{H}(t)$ $\displaystyle=$
$\displaystyle\Omega_{1}(e^{it(2\omega+\Delta)}+e^{-it\Delta})|e\rangle\langle
g|/2$
$\displaystyle+\Omega_{2}(e^{it(2\omega-\Delta)}+e^{it\Delta})|e\rangle\langle
g|/2+\text{H.c.}.$

When $\Omega_{1}=\Omega_{2}=\Omega$, the above equation simplifies to,

$\displaystyle\hat{H}(t)$ $\displaystyle=$
$\displaystyle\Omega(e^{2it\omega}+1)\cos(t\Delta)|e\rangle\langle
g|+\text{H.c.}.$

We further assume $\Delta\sim|\Omega|\ll\omega$ so that the Bloch-Siegert
shift is negligible Cohen-Tannoudji  _et al._ (1998). In this case, $e^{\pm
2it\omega}$ is rapidly oscillating and can be discarded according to the
rotating wave approximation. This leads to the following Hamiltonian,

$\displaystyle\hat{H}(t)$ $\displaystyle=$
$\displaystyle\Omega\cos(t\Delta)(|e\rangle\langle g|+|g\rangle\langle e|).$

Starting from the initial state, $|\psi(0)\rangle=|g\rangle$, the system
wavefunction $|\psi(t)\rangle=C_{g}(t)|g\rangle+C_{e}(t)|e\rangle$ evolves
according to

$\displaystyle C_{g}(t)$ $\displaystyle=$
$\displaystyle\cos\left[\frac{\Omega}{\Delta}\sin(t\Delta)\right],$
$\displaystyle C_{e}(t)$ $\displaystyle=$
$\displaystyle-i\sin\left[\frac{\Omega}{\Delta}\sin(t\Delta)\right].$ (2)

When $\frac{|\Omega|}{\Delta}\geq\frac{\pi}{2}$ is satisfied, the transition
probability from the ground state to the excited state can reach 1. A critical
condition for a full transition is

$\displaystyle\frac{|\Omega|}{\Delta}=\frac{\pi}{2},~{}~{}t=\frac{\pi}{2\Delta}.$

Remarkably, the time for the transition from the ground state to the excited
state is $\frac{\pi}{2\Delta}$, which is only $\pi/2$ times longer than the
time $\frac{\pi}{2|\Omega|}$ of a $\pi$ pulse in a resonant transition with a
Rabi frequency $2\Omega$.

ORIR is a useful method in quantum control with atomic ions Sørensen and
Mølmer (1999). Below, we show its applicability in neutral atoms.

##  III Single-site Rydberg addressing in 3D arrays

ORIR can be used for Rydberg addressing of a single atom in a 3D optical
lattice. In general, the difficulty of Rydberg addressing in a 3D lattice lies
in two aspects. First, sending lasers to a target atom in a dense 3D lattice
can influence other atoms along the light path, and it is unlikely to leave
the state of a nontarget atoms intact after fully Rydberg exciting and
deexciting a target atom. Second, the atoms along the light path can exhibit
Rydberg blockade, bringing the problem a many-body complexity. One may image
that for a two-photon excitation of $s$\- or $d$-orbital Rydberg atoms, the
two laser beams can be sent along different directions; because both laser
fields are largely detuned, only the target atom at the intersection of the
two beams are Rydberg excited. However, phase twist can occur for the
nontarget atom even if it is pumped only by an off-resonant field [see, for
example, the dashed curve in Fig. 3(b) of Ref. Shi (2019)]. So, there can be a
phase shift to the atomic state of a nontarget qubit illuminated by the laser
fields addressing the lower transition in our problem. In fact, even if the
detuning of the laser field addressing the lower transition is 10 times larger
than its Rabi frequency $\Omega_{\text{g}}$, the phase shift to the ground
state can reach $\pi/2$ if the atom is irradiated by a time of
$20\pi/\Omega_{\text{g}}$. These issues can be tackled by using ORIR-based
optical spin echo in a very small lattice, and a microwave spin echo assisted
by ORIR in a relatively large one. Here, the optical spin echo is for
restoring the state of nontarget atoms if no Rydberg interaction exists
between the nontarget atoms, while the microwave spin echo is for removing
many-body effect if there is Rydberg blockade in the nontarget atoms.

We take a system shown in Fig. 2 for illustration. For a small array, the
laser spot is small enough so that only an atom at the beam axis can be
irradiated, and the laser intensity is of similar magnitude at the target atom
and any nontarget atom along the light path. So, the Rabi frequencies are of
similar magnitude for all the irradiated atoms in the problem. In Fig. 2, a
cubic lattice with $3\times 5\times 3$ sites is shown, the lattice constant is
$\mathcal{L}$, and the relevant transition is between a hyperfine state of
$F=2$ and a high-lying $s$-orbital Rydberg state of rubidium-87. The
quantization axis is along $[101]$ which can be specified by a magnetic field.
All laser fields are $\pi$-polarized along the quantization axis. We want to
excite the central atom (denoted by the red ball), which is located at the
origin of the Cartesian coordinate (the $\mathbf{x}-\mathbf{y}-\mathbf{z}$
arrows are only for clarifying the directions). In a rotating-frame as in Sec.
II, a two-photon transition $|1\rangle\leftrightarrow|r\rangle$ with Rabi
frequency $\Omega_{\perp}=\Omega e^{i(t\Delta_{\perp}+\varphi_{\perp})}$ is
created by sending focused Gaussian beams along
$\mathbf{l}_{\perp}=[\overline{1}21]$, with the foci at the center of the
target site. Here, $\varphi_{\perp}$ is determined by the phases of the laser
oscillators and the distance that the light travels from the laser sources to
the target atom; for brevity, we write
$\varphi_{\perp}=\mathbf{k}_{\perp}\cdot\mathbf{r}$, where
$\mathbf{k}_{\perp}$ is the wavevector of the fields. Furthermore,
$\Delta_{\perp}$ is the overall detuning of the two-photon transition between
the ground state $|1\rangle$ and the Rydberg state $|r\rangle$, which should
be much smaller than the detuning $\delta$ for the transition
$|1\rangle\rightarrow|e\rangle$. In our method, $\Delta$ is several MHz, thus
$\delta$ should be at least several hundred MHz. Meanwhile, another set of
lasers are sent along $\mathbf{l}_{\shortparallel}=[12\overline{1}]$ which
also focus at the same target atom. In method I, the lasers along
$\mathbf{l}_{\perp}$ and $\mathbf{l}_{\shortparallel}$ are of similar
wavelengths, but with some difference so that the two two-photon transitions
are built via very different one-photon detunings at the intermediate state.
In the same rotating frame, the lasers along $\mathbf{l}_{\shortparallel}$
drive the transition $|1\rangle\leftrightarrow|r\rangle$ by a Rabi frequency
$\Omega_{\shortparallel}=\Omega
e^{i(t\Delta_{\shortparallel}+\varphi_{\shortparallel})}$.

Our scheme requires the condition of
$e^{i\varphi_{\perp}},~{}e^{i\varphi_{\shortparallel}}=\pm 1$. Because both
$\varphi_{\perp}$ and $\varphi_{\shortparallel}$ are determined by the laser
sources and the length of the light path, it is possible to tune the laser
phases to the condition of
$e^{i\varphi_{\perp}},~{}e^{i\varphi_{\shortparallel}}=\pm 1$ if the position
fluctuation of qubits is negligible. For optically cooled qubits before each
experimental cycle, there is fluctuation of the qubit positions, leading to
fluctuation in $\varphi_{\perp}$ and $\varphi_{\shortparallel}$. For the
configuration in Fig. 2(b), we have
$|\mathbf{k}_{\perp}|=|\mathbf{k}_{\shortparallel}|\approx 5\times
10^{6}m^{-1}$ when the upper and lower fields propagate oppositely. This means
that the fluctuation of the qubit position along the light path should be much
smaller than $1~{}\mu$m to validate the method. In the experiment of Ref.
Graham  _et al._ (2019), the transverse (longitudinal) position fluctuation of
the qubit is $0.27~{}(1.47)~{}\mu$m, which will lead to large fluctuation of
$\varphi_{\perp}$ and $\varphi_{\shortparallel}$ if similar traps are used
here. Thus it is necessary to use sufficiently deep traps and efficient
cooling. This is why the methods shown below can not be implemented by sending
one laser field along $\mathbf{l}_{\perp}$ for a one-photon transition
$|1\rangle\rightarrow|e\rangle$, another field along
$\mathbf{l}_{\shortparallel}$ for the other one-photon transition
$|e\rangle\rightarrow|r\rangle$, where $|e\rangle$ is the intermediate state
shown in Fig. 2(b). This is because the wavevector for these two one-photon
transitions are much larger, which makes the condition of
$e^{i\varphi_{\perp}}=e^{i\varphi_{\shortparallel}}=\pm 1$ even more
challenging to be satisfied.

An alternative solution to the above issue is to use naturally existing
transitions with negligible wavevector. For instance, the Rydberg excitation
via two counterpropagating fields for $6^{1}S_{1}\rightarrow
6^{1}P_{1}\rightarrow n^{1}S_{0}$ (or $n^{1}D_{2}$) of ytterbium suffers from
a negligible Doppler dephasing because $k\lesssim 10^{5}m^{-1}$ (see, for
example, Fig.1 of Ref. Lehec  _et al._ (2018)). In this latter case, it is
necessary to store quantum information in the nuclear spin states of, e.g.,
171Yb (or 173Yb) for the purpose of quantum computing with Rydberg
interaction. For Rydberg excitation of one of the two nuclear spin qubit
states, one can first put the other nuclear spin qubit state to the metastable
excited state ${}^{3}P_{0}$ to avoid leakage. However, it is beyond the scope
of this work to give detail about this latter scheme of Rydberg gate based on
171Yb. We assume that the fluctuation of $\varphi_{\perp}$ and
$\varphi_{\shortparallel}$ has been suppressed.

Below, Secs. III.1 and III.2 show two optical spin-echo methods for the
excitation of target atoms, termed as method I and method II. The above
discussion is applicable to method I. In method II, the lasers along
$\mathbf{l}_{\perp}$ is for the transition
$|1\rangle\leftrightarrow|r\rangle$, but those along
$\mathbf{l}_{\shortparallel}$ is for the transition between $|r\rangle$ and
another Rydberg state $|R\rangle$ via a low-lying intermediate state. Because
$|r\rangle$ is near $|R\rangle$, the phase $\varphi_{\shortparallel}$ can be
easily set in method II. The discussion about the phase above is applicable
for both $\varphi_{\shortparallel}$ and $\varphi_{\perp}$ in method I and
$\varphi_{\perp}$ in method II.

It is useful to briefly show why two methods are introduced. For brevity, we
use “one pulse” when the laser fields along $\mathbf{l}_{\perp}$ and
$\mathbf{l}_{\shortparallel}$ irradiate the system simultaneously for a
certain duration. In method I, one pulse is used for the Rydberg excitation,
and the nontarget atoms can have some residual population in the Rydberg state
when the target atom is excited to the Rydberg state. After the second pulse,
both the target and nontarget atoms return to the ground state. No phase shift
occurs for the target atom because of the spin echo. In method II, two pulses
are used for Rydberg excitation. The two pulses form an optical spin echo
sequence for the nontarget atoms so that they have no population in the
Rydberg state when the target atom is in the Rydberg state. Similarly, two
similar pulses can pump the target atom back to its ground state. Because the
target atom experiences no spin echo, it can accumulate a $\pi$ phase shift
upon its state restoration. When there is Rydberg interaction between
nontarget atoms, a microwave spin echo is used to reverse the sign of the
Rydberg interaction, shown in Sec. III.3. In Sec. III.4, we study the issue of
divergence of the laser beam. In Sec. III.5, we study the application of the
methods in the Rydberg blockade gate. We give a detailed comparison between
the two methods in Sec. III.6.

Figure 2: (a) Diagram of targeting one atom by two detuned laser beams in a 3D
lattice of rubidium atoms. Origin of the coordinate system is at the center of
the target site. The quantization axis is along $[101]$; the two light paths
are along $\mathbf{l}_{\perp}=[\overline{1}21]$ and
$\mathbf{l}_{\shortparallel}=[12\overline{1}]$, respectively. The red atom
labeled as “C” is the target atom. Atom $E$ and atom $F$ (atom $A$ and atom
$G$) are nontarget atoms at the beam axis of fields along $\mathbf{l}_{\perp}$
($\mathbf{l}_{\shortparallel}$). (b) Atomic transition diagram. Both the light
fields propagating along $\mathbf{l}_{\perp}$ and
$\mathbf{l}_{\shortparallel}$ are polarized along the quantization axis,
inducing the same transition. The detuning for the transition
$|1\rangle\rightarrow|e\rangle$ is much larger than the two-photon detuning
$\Delta$. (c) Time-dependence of the Rabi frequencies for method I. The
effective Rabi frequency is $2i\Omega\sin(\Delta t)$ at the target atom.

###  III.1 Method I: one pulse for Rydberg excitation

We first show Rydberg excitation of the target atom by one pulse. There are
three stages to implement the optical spin echo, two optical pumping and one
wait time. The wait can have a duration of $t_{\mu}=2\pi/\Delta$, during which
a microwave field can be used to induce a transition between two different
Rydberg states as discussed later; for Rydberg blockade gate, another atom
near the target atom can be excited to the Rydberg state during the wait.

In the first stage, laser fields along $\mathbf{l}_{\perp}$ and along
$\mathbf{l}_{\shortparallel}$ induce the following transition

$\displaystyle\hat{H}_{\perp}^{(1)}$ $\displaystyle=$
$\displaystyle\Omega_{\perp}e^{it\Delta}|r\rangle_{\perp}\langle
1|/2+\text{H.c.},$ $\displaystyle\hat{H}_{\shortparallel}^{(1)}$
$\displaystyle=$
$\displaystyle-\Omega_{\shortparallel}e^{-it\Delta}|r\rangle_{\shortparallel}\langle
1|/2+\text{H.c.}$ (3)

during $t\in[0,~{}t_{0})$, where $t_{0}=\pi/\Delta$ and the subscript
$\alpha=\perp(\shortparallel)$ distinguishes notations for atoms in the two
different light paths. For Gaussian beams, the values of $\Omega_{\alpha}$
reach to their maximum at the target atom where $\Omega=\pi\Delta/4$. The
above Hamiltonian is in a rotating frame with
$\hat{\mathcal{R}}=\omega|r\rangle\langle r|$. To show the echo for the
nontarget atoms, we use the rotating frame with
$\hat{\mathcal{R}}_{\mp}\equiv\hat{\mathcal{R}}\mp\Delta|r\rangle\langle r|$
for the atoms along the two different light paths, and use $\psi~{}(\Psi)$ to
denote the wavefunction in the interaction (Schrödinger) picture. The
wavefunctions in the frame $\hat{\mathcal{R}}_{\mp}$ are

$\displaystyle|\psi(t)\rangle_{\perp}$ $\displaystyle=$ $\displaystyle
e^{it\hat{\mathcal{R}}_{-}}|\Psi(t)\rangle_{\perp},$
$\displaystyle|\psi(t)\rangle_{\shortparallel}$ $\displaystyle=$
$\displaystyle e^{it\hat{\mathcal{R}}_{+}}|\Psi(t)\rangle_{\shortparallel},$

and the Hamiltonians are

$\displaystyle\hat{\mathcal{H}}_{\perp}^{(1)}$ $\displaystyle=$
$\displaystyle\Delta|r\rangle_{\perp}\langle
r|+\Omega_{\perp}(|r\rangle_{\perp}\langle 1|+\text{H.c.})/2,$
$\displaystyle\hat{\mathcal{H}}_{\shortparallel}^{(1)}$ $\displaystyle=$
$\displaystyle-\Delta|r\rangle_{\shortparallel}\langle
r|-\Omega_{\shortparallel}(|r\rangle_{\shortparallel}\langle
1|+\text{H.c.})/2.$ (4)

The wavefunction evolves according to
$e^{-i\hat{\mathcal{H}}_{\alpha}t}|\psi(0)\rangle_{\alpha}$, where
$|\psi(0)\rangle_{\alpha}=|1\rangle_{\alpha}$ is the initial wavefunction. In
Eqs. (3) and (4), the Rabi frequency $\Omega$ for the target atom is equal to
the value of $\Omega_{\perp}$ (or $\Omega_{\shortparallel}$) at the foci of
the laser fields. Because of the Gaussian profile of the fields, $\Omega$ is
larger than $\Omega_{\perp}$ and $\Omega_{\shortparallel}$ of any nontarget
atom. Equation (3) means that the state of the target atom evolves according
to the Hamiltonian

$\displaystyle\hat{H}(t)$ $\displaystyle=$ $\displaystyle
i\Omega\sin(t\Delta)(|r\rangle\langle 1|-|1\rangle\langle r|)$ (5)

in the rotating frame of $\hat{\mathcal{R}}$, and one can show that when the
initial state of the target atom is $|1\rangle$, its state becomes $|r\rangle$
at the end of the first pulse.

Second, a wait time of duration $t_{\mu}$ is inserted when nothing is done. If
we would like to suppress the many-body effect as discussed later, a microwave
transition can be used to transfer the state $|r\rangle$ to another Rydberg
state $|r^{\prime}\rangle$. There can be a phase difference between $\langle
r|\psi(t_{0})\rangle$ and $\langle r^{\prime}|\psi(t_{0}+t_{\mu})\rangle$
determined by the microwave fields, and the subsequent laser fields in the
third stage should appropriately compensate this phase. But to show the
essence of the optical spin echo, we assume nothing is done during the wait.
The state of a nontarget atom evolves to

$\displaystyle|\Psi(t_{0}+t_{\mu})\rangle_{\perp}$ $\displaystyle=$
$\displaystyle
e^{-it_{\mu}\hat{\mathcal{R}}}e^{-it_{0}\hat{\mathcal{R}}_{-}}e^{-it_{0}\hat{\mathcal{H}}_{\perp}^{(1)}}|\psi(0)\rangle_{\perp},$
$\displaystyle|\Psi(t_{0}+t_{\mu})\rangle_{\shortparallel}$ $\displaystyle=$
$\displaystyle
e^{-it_{\mu}\hat{\mathcal{R}}}e^{-it_{0}\hat{\mathcal{R}}_{+}}e^{-it_{0}\hat{\mathcal{H}}_{\shortparallel}^{(1)}}|\psi(0)\rangle_{\shortparallel},$
(6)

at the end of the wait.

Third, during $t\in[t_{0}+t_{\mu},~{}2t_{0}+t_{\mu})$, laser fields along both
$\mathbf{l}_{\perp}$ and $\mathbf{l}_{\shortparallel}$ are sent for the
following transitions,

$\displaystyle\hat{H}_{\perp}^{(2)}$ $\displaystyle=$
$\displaystyle-\Omega_{\perp}e^{-it\Delta}|r\rangle_{\perp}\langle
1|/2+\text{H.c.},$ $\displaystyle\hat{H}_{\shortparallel}^{(2)}$
$\displaystyle=$
$\displaystyle\Omega_{\shortparallel}e^{it\Delta}|r\rangle_{\shortparallel}\langle
1|/2+\text{H.c.},$ (7)

so that the Hamiltonian for the target atom is still given by Eq. (5). When
$t_{\mu}=2\pi/\Delta$, one can show that the state of the target atom is
$|\Psi(t)\rangle=|1\rangle$ at the time $t=2t_{0}+t_{\mu}$. Now, we use the
rotating frame with $\hat{\mathcal{R}}_{\pm}$ for the atoms along the two
different light path so that the Hamiltonians become

$\displaystyle\hat{\mathcal{H}}_{\perp}^{(2)}$ $\displaystyle=$
$\displaystyle-\Delta|r\rangle_{\perp}\langle
r|-\Omega_{\perp}(|r\rangle_{\perp}\langle 1|+\text{H.c.}),$
$\displaystyle\hat{\mathcal{H}}_{\shortparallel}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta|r\rangle_{\shortparallel}\langle
r|+\Omega_{\shortparallel}(|r\rangle_{\shortparallel}\langle 1|+\text{H.c.}).$
(8)

The state of the nontarget atoms becomes,

$\displaystyle|\psi(2t_{0}+t_{\mu})\rangle_{\perp}$ $\displaystyle=$
$\displaystyle
e^{-it_{0}\hat{\mathcal{H}}_{\perp}^{(2)}}e^{i(t_{0}+t_{\mu})\hat{\mathcal{R}}_{+}}e^{-it_{\mu}\hat{\mathcal{R}}}e^{-it_{0}\hat{\mathcal{R}}_{-}}$
$\displaystyle\times
e^{-it_{0}\hat{\mathcal{H}}_{\perp}^{(1)}}|\psi(0)\rangle_{\perp},$
$\displaystyle|\psi(2t_{0}+t_{\mu})\rangle_{\shortparallel}$ $\displaystyle=$
$\displaystyle
e^{-it_{0}\hat{\mathcal{H}}_{\shortparallel}^{(2)}}e^{i(t_{0}+t_{\mu})\hat{\mathcal{R}}_{-}}e^{-it_{\mu}\hat{\mathcal{R}}}e^{-it_{0}\hat{\mathcal{R}}_{+}}$
(9) $\displaystyle\times
e^{-it_{0}\hat{\mathcal{H}}_{\shortparallel}^{(1)}}|\psi(0)\rangle_{\shortparallel},$

Because
$e^{i(t_{0}+t_{\mu})\hat{\mathcal{R}}_{+}}e^{-it_{\mu}\hat{\mathcal{R}}}e^{-it_{0}\hat{\mathcal{R}}_{-}}=e^{4i\pi|r\rangle\langle
r|}$, it is equal to the identity since it operates either on $|r\rangle$ or
$|1\rangle$; similarly,
$e^{i(t_{0}+t_{\mu})\hat{\mathcal{R}}_{-}}e^{-it_{\mu}\hat{\mathcal{R}}}e^{-it_{0}\hat{\mathcal{R}}_{+}}=\hat{1}$.
Because $\hat{\mathcal{H}}_{\alpha}^{(2)}=-\hat{\mathcal{H}}_{\alpha}^{(1)}$,
we have

$\displaystyle|\psi(2t_{0}+t_{\mu})\rangle_{\perp}$ $\displaystyle=$
$\displaystyle
e^{-it_{0}\hat{\mathcal{H}}_{\perp}^{(2)}}e^{-it_{0}\hat{\mathcal{H}}_{\perp}^{(1)}}|\psi(0)\rangle_{\perp}=|\psi(0)\rangle_{\perp},$
$\displaystyle|\psi(2t_{0}+t_{\mu})\rangle_{\shortparallel}$ $\displaystyle=$
$\displaystyle
e^{-it_{0}\hat{\mathcal{H}}_{\shortparallel}^{(2)}}e^{-it_{0}\hat{\mathcal{H}}_{\shortparallel}^{(1)}}|\psi(0)\rangle_{\shortparallel}=|\psi(0)\rangle_{\shortparallel},$

where $\alpha=\perp,\shortparallel$. So, the state of the nontarget atom is
restored whatever the magnitude of $\Omega_{\alpha}$ is. The value of
$t_{\mu}=2\pi/\Delta$ is chosen so that Eq. (9) can be simplified to Eq.
(LABEL:method01finalstate2). In fact, it can be an arbitrary value of
$t_{\mu}^{\prime}\in(0,~{}t_{\mu})$, but each set of the laser fields used
during the third stage should have an extra phase
$\varphi=t_{\mu}^{\prime}\Delta$ so that
$(\Omega_{\perp},~{}\Omega_{\shortparallel})$ becomes
$(\Omega_{\perp}e^{i\varphi},~{}\Omega_{\shortparallel}e^{-i\varphi})$.

Figure 3: Numerical result for the excitation and deexcitation of Rydberg
state $|r\rangle$ in Method I. (a) and (b) show the time evolution of the
population in $|1\rangle$ (solid curve) and $|r\rangle$ (dashed curve) for the
nontarget atom and target atom, respectively, with
$(\Delta,~{}\Omega)/2\pi=(4,~{}\pi)$ MHz, $t_{\mu}=2\pi/\Delta$, and the Rabi
frequency for the nontarget atom is $\Omega_{\perp}=0.73\Omega$. The same
state evolution occurs if $t_{\mu}=0$.

As shown above, the merit of the method is that it can avoid bringing other
atoms to Rydberg states upon the completion of the deexcitation of the target
atom. The method works well when the light intensity is different at atoms
along the light path. The focused Gaussian beam, even with a very small waist,
can still have a Rayleigh length as large as $X=26~{}\mu$m Wang  _et al._
(2016). At a spot away from the foci by $\mathfrak{l}_{0}$ for the beam
propagating along $\mathbf{l}_{\shortparallel}$, the laser intensity at the
beam axis is smaller than that at the foci by about
$X^{2}/(X^{2}+\mathfrak{l}_{0}^{2})$. This fact leads to different magnitudes
of Rabi frequency at different nontarget atoms, and its irrationality
indicates that it is impossible to simultaneously restore the states of all
the atoms along the light path unless spin echo is used.

We continue to analyze the condition when the blockade interaction is
included. If the lattice constant is $\mathcal{L}=10~{}\mu$m Isenhower  _et
al._ (2010), the values of $\Omega_{\alpha}=\Omega_{\shortparallel}$ (or
$\Omega_{\perp}$) for a nontarget atom and $\Omega$ of the target atom are
comparable because there are only two nontarget atoms in the light path along
$\mathbf{l}_{\shortparallel}$, denoted as atom $A$ and atom $G$ in Fig. 2(a).
Atom $A$ is away from the foci by $\mathfrak{l}_{0}=\pm\sqrt{6}\mathcal{L}$,
and thus its Rabi frequency $\Omega_{\alpha}$ is only about $0.73\Omega$ when
$X=26~{}\mu$m. We suppose that the van der Waals interaction between Rydberg
atoms is $V=C_{6}/\mathcal{L}^{6}$ and $V/\Omega\approx 12$ [see text
following Eq. (28) for the reason of this choice]. Then, the blockade
interaction between the target atom and the nontarget atom is only $V/216$. On
the other hand, the Rydberg population in the nontarget atom (if it is
initialized in the state $|1\rangle$) is tiny, shown above, and thus the
influence of the blockade interaction for the excitation of the target atom is
negligible. For the same reason, the Rydberg blockade does not perturb the
spin-echo time evolution in the nontarget atom since
$V/216\ll\Omega_{\alpha}$. So, the Rydberg addressing of one atom in a small
3D lattice with 45 sites in Fig. 2 is achievable.

Using Eqs. (3) and (7) in the interaction picture with $\hat{\mathcal{R}}$, we
numerically study the state evolution of the target atom and the nontarget
atom with parameters $(\Delta,~{}\Omega)/2\pi=(4,~{}\pi)$ MHz,
$t_{\mu}=2\pi/\Delta$ and, as an example,
$\Omega_{\perp}=\Omega_{\shortparallel}=0.73\Omega$. Because the population
evolution for the nontarget atom $\alpha=\perp$ is the same with that for
$\alpha=\shortparallel$, we take $\alpha=\perp$ as an example and ignore the
Rydberg blockade. The results are shown in Fig. 3, where the population in
$|r\rangle$ is 0.23 during $t\in[t_{0},~{}t_{0}+t_{\mu})$ for the nontarget
atom when the target atom is in the Rydberg state. As a result of the optical
spin echo, the state of the nontarget atom returns to the ground state at the
end of the pulse sequence.

One concern about the applicability of this method is that there is Rydberg-
state decay in any irradiated nontarget atom during the pulse sequence, which
can cause decoherence for many qubits. But the decay is negligible: if the
state is $|\psi(0)\rangle=|1\rangle$ for a nontarget atom, the probability of
decay for the atom is $T_{\text{de}}/\tau$, where $\tau$ is the lifetime of
the Rydberg state, and

$\displaystyle T_{\text{de}}$ $\displaystyle=$
$\displaystyle\int_{0}^{2t_{0}+t_{\mu}}|\langle r|\Psi(t)\rangle|^{2}dt,$ (11)

which is $0.093~{}\mu$s and $0.375~{}\mu$s for the processes in Figs. 3(a) and
3(b), respectively. For an $s$ or $d-$orbital state of rubidium with principal
value of $100$, $\tau$ is about $320~{}\mu$s in a temperature of $300$ K
Beterov  _et al._ (2009), which gives a decay error of about
$E_{\text{decay-n}}=2.9\times 10^{-4}$ for the nontarget atom, and $1.2\times
10^{-3}$ for the target atom; when the sequence in Fig. 3 is used for the
pumping of the control qubit in the Rydberg blockade gate Jaksch  _et al._
(2000), the decay of one nontarget atom contributes an error
$E_{\text{decay-n}}/2$ to the gate fidelity. Moreover, the state of a
nontarget qubit can also be $|0\rangle$ that doesn’t respond to the light
irradiation. The farther the nontarget atom is from the foci, the smaller the
value of $\Omega_{\perp}$ will be. With $\Omega_{\perp}=0.3\Omega$, we have
$T_{\text{de}}=0.02~{}\mu$s which gives a decay error of about five times
smaller than that in Fig. 3(a). These analyses show that the Rydberg-state
decay for the nontarget atoms is negligible.

###  III.2 Method II: two-pulse excitation of Rydberg states

For high fidelity quantum control, it is desirable to avoid Rydberg-state
decay whenever possible. The scheme in Sec. III.1 has a wait time during which
the nontarget atoms have some probability in the Rydberg state, shown in Fig.
3(a). This leads to extra Rydberg-state decay that hampers the protocol.
Besides, the target atom can not acquire any phase shift upon its state
restoration because of the spin echo. In this section, we show a method that
leaves no Rydberg population in the nontarget atom when the target atom is
excited to the Rydberg state. Moreover, the target atom can have a $\pi$ phase
shift when its state is restored. For brevity, we only show the Rydberg
excitation of the target atom since its deexcitation is achieved with a
similar process.

In the first pulse of duration $t_{0}=\pi/\Delta$, laser fields along
$\mathbf{l}_{\perp}$ similar to that in Eq. (3) are used for the following
transition,

$\displaystyle\hat{H}_{\perp}^{(1)}$ $\displaystyle=$
$\displaystyle\Omega_{\perp}e^{it\Delta}|r\rangle_{\perp}\langle
1|/2+\text{H.c.},$ (12)

and meanwhile laser fields along $\mathbf{l}_{\shortparallel}$ induce a two-
photon transition between $|r\rangle$ and $|R\rangle$ via a low-lying
intermediate state $|p\rangle$,

$\displaystyle\hat{H}_{\shortparallel}^{(1)}$ $\displaystyle=$
$\displaystyle\Omega_{\shortparallel}e^{-it\Delta}|R\rangle_{\shortparallel}\langle
r|/2+\text{H.c.},$ (13)

where $|p\rangle$ should be a state higher than $|e\rangle$, and the fields
used for $|r\rangle\rightarrow|p\rangle\rightarrow|R\rangle$ should be optical
or infrared.

After the first pulse, a wait duration $t_{\mu}$ elapses as in Sec. III.1,
where microwave pumping can be used if necessary Shi (2018a). As shown in Sec.
III.1, the duration of the wait can be adjusted without altering the optical
spin echo, and hence we can assume $t_{\mu}=0$ for brevity.

In the second pulse of duration $t_{0}$, laser fields along
$\mathbf{l}_{\perp}$ are sent for the following transition,

$\displaystyle\hat{H}_{\perp}^{(2)}$ $\displaystyle=$
$\displaystyle-\Omega_{\perp}e^{-it\Delta}|r\rangle_{\perp}\langle
1|/2+\text{H.c.},$ (14)

while the laser fields along $\mathbf{l}_{\shortparallel}$ are used for

$\displaystyle\hat{H}_{\shortparallel}^{(2)}$ $\displaystyle=$
$\displaystyle\Omega_{\shortparallel}e^{it\Delta}|R\rangle_{\shortparallel}\langle
r|/2+\text{H.c.}.$ (15)

Whatever the magnitude of $\Omega_{\perp}$ is, the state of a nontarget atom
irradiated by the laser fields along $\mathbf{l}_{\perp}$ can return to its
initial state, as shown in Sec. III.1. Because there is no population in
$|r\rangle$ or $|R\rangle$ for any nontarget atom irradiated by the fields
along $\mathbf{l}_{\shortparallel}$, nothing happens for them. Below, we study
the state evolution for the target atom.

Figure 4: Numerical result by Eqs. (12)-(15) for the excitation of the Rydberg
state $|R\rangle$ in Method II. (a) shows the time evolution of the population
in $|1\rangle$ (solid curve) and $|r\rangle$ (dotted curve) for the nontarget
atom $E$ (or atom $F$, see Fig. 2). The Rabi frequency for the nontarget atom
$E$ is $\Omega_{\perp}=0.73\Omega$ as estimated in Sec. III.1. (b) shows the
time evolution of the population in $|1\rangle$ (solid curve), $|R\rangle$
(dashed curve), and $|r\rangle$ (dotted curve) for the target atom (labeled as
atom $C$ in Fig. 2), with $\Delta/2\pi=4$ MHz, $\Omega=1.2247\Delta$, and
$t_{\mu}=0$. In (b), the initial state is $|1\rangle$, and the final state is
$i|R\rangle$ with an error smaller than $10^{-8}$. The same optical spin echo
for the nontarget atom and the Rydberg excitation of the target atom are
achieved with $t_{\mu}=2\pi/\Delta$.

Equations (12)-(15) are written in a rotating frame with

$\displaystyle\hat{\mathcal{R}}$ $\displaystyle=$
$\displaystyle\omega|r\rangle\langle r|+\omega_{R}|R\rangle\langle R|,$ (16)

where $\omega_{R}$ is the energy (divided by $\hbar$) of the state
$|R\rangle$. If we use a rotating frame with
$\hat{\mathcal{R}}_{\mp}=\hat{\mathcal{R}}\mp\Delta|r\rangle\langle r|$ for
the first (second) pulse, the Hamiltonians of the target atom become

$\displaystyle\hat{\mathcal{H}}^{(1)}$ $\displaystyle=$
$\displaystyle\Delta|r\rangle\langle
r|+(\Omega_{\shortparallel}|R\rangle\langle r|+\Omega_{\perp}|r\rangle\langle
1|+\text{H.c.})/2,$ $\displaystyle\hat{\mathcal{H}}^{(2)}$ $\displaystyle=$
$\displaystyle-\Delta|r\rangle\langle
r|+(\Omega_{\shortparallel}|R\rangle\langle r|-\Omega_{\perp}|r\rangle\langle
1|+\text{H.c.})/2.$

For $\Delta\gg\Omega_{\perp},\Omega_{\shortparallel}$, an effective pumping
emerges between $|1\rangle$ and $|R\rangle$ with a Rabi frequency of
$\Omega_{\perp}\Omega_{\shortparallel}/(2\Delta)$ for both
$\hat{\mathcal{H}}^{(1)}$ and $\hat{\mathcal{H}}^{(2)}$. But the time for a
complete excitation of $|R\rangle$ is
$2\pi\Delta/(\Omega_{\perp}\Omega_{\shortparallel})$. If $N$ cycles of optical
spin echo sequence are used, the condition of
$\Delta/\sqrt{\Omega_{\perp}\Omega_{\shortparallel}}=\sqrt{N}$ leads to
excitation of the target qubit. However, this is complicated. For an efficient
quantum control, the case of $N=1$ is best. We numerically found that with
$\Omega_{\perp}=\Omega_{\shortparallel}=1.2247\Delta$, a complete excitation
of the ground state $|1\rangle$ to the Rydberg state $i|R\rangle$ is achieved
with one spin-echo cycle, shown in Fig. 4; the residual population in
$|1\rangle$ and $|r\rangle$ is $1.9(5.6)\times 10^{-9}$ at the time of
$t=2t_{0}$ in Fig. 4(b). The time for the nontarget atom [atom $E$ of Fig.
2(a)] to be in the Rydberg state is $T_{\text{de}}=0.07~{}\mu$s in Fig. 4(a),
which means that the decay probability for the nontarget atom is
$T_{\text{de}}/\tau=2\times 10^{-4}$ if $\tau=320~{}\mu$s, i.e., negligible.

In Fig. 4(a), the population dynamics for atom $E$ of Fig. 2 was shown. As
estimated in the third paragraph counted backward from the end of Sec. III.1,
if the Rabi frequency in Eq. (12) for atom $C$ is $\Omega$, then atom $E$ is
pumped according to Eq. (12) with a Rabi frequency $0.73\Omega$, which was
used in Fig. 4(a). The reason to ignore the population of $|R\rangle$ in Fig.
4(a) lies in that the field amplitude for the transition
$|r\rangle\leftrightarrow|R\rangle$ at atom $E$ is negligible: the distance
from $E$ to the beam axis of the field propagating along
$\mathbf{l}_{\shortparallel}$ is $r_{\perp,E}\approx 2.3\mathcal{L}$, while
atom $B$ is only away from this beam axis by about $r_{\perp,B}\approx
0.46\mathcal{L}$. The field drops off by the law of $\propto
e^{-r_{\perp}^{2}/w^{2}}$ at a point away from the beam axis by $r_{\perp}$,
where $w$ is the beam radius. As detailed later below Eq. (22), we can suppose
that the field amplitude at atom $B$ is no more than $1/e$ of that at the beam
axis, i.e., $e^{-r_{\perp,B}^{2}/w^{2}}\leq 1/e$. Because
$r_{\perp,E}^{2}/r_{\perp,B}^{2}\approx 25$, the field amplitude for the
transition $|r\rangle\leftrightarrow|R\rangle$ at atom $E$ can be ignored.
Similarly, we can ignore any pumping of atom A (or atom G) by the field
propagating along $\mathbf{l}_{\perp}$. In Fig. 2(a), there are other atoms
very near to the beam axes, such as atom $D$. But it is away from the beam
axis of the fields along either $\mathbf{l}_{\perp}$ or
$\mathbf{l}_{\shortparallel}$ by about $0.9\mathcal{L}\approx 2r_{\perp,B}$,
which means that the field amplitude for either beam at atom $D$ is smaller
than $1/e^{4}$ of that at the beam axis if conditions like
$e^{-r_{\perp,B}^{2}/w^{2}}\leq 1/e$ apply to both beams. This means that only
the target atom, i.e., atom $C$ experiences pumping by the two laser beams.

In Fig. 4(b), the final phase $\pi/2$ for the state $|R\rangle$ arises in
response to the sequential pumping with
$(\hat{H}_{\perp}^{(1)},\hat{H}_{\shortparallel}^{(1)})$ and
$(\hat{H}_{\perp}^{(2)},\hat{H}_{\shortparallel}^{(2)})$ defined in Eqs.
(12)-(15); the same pulse sequence as in Fig. 4 can deexcite the target atom
from $i|R\rangle$ to $-|1\rangle$ which can lead to the conditional $\pi$
phase if method II is used for the Rydberg blockade gate Jaksch  _et al._
(2000). Note that if only
$(\hat{H}_{\perp}^{(1)},\hat{H}_{\shortparallel}^{(1)})$ is used for Rydberg
excitation with a time of $2t_{0}$, the final state of the target atom is
$-|R\rangle$, but a subsequent deexcitation by this sequence can not lead to a
conditional $\pi$ phase for entanglement generation.

Because the energy difference between $|r\rangle$ and $|R\rangle$ is smaller
than that between $|r(1)\rangle$ and $|p\rangle$ by several orders of
magnitude, the wavevector for $\Omega_{\shortparallel}$ is negligible when the
field for $|r\rangle\rightarrow|p^{\prime}\rangle$ and the field for
$|p^{\prime}\rangle\rightarrow|R\rangle$ copropagate, where
$|p(p^{\prime})\rangle$ is the intermediate state for
$|r\rangle\leftrightarrow|1(R)\rangle$. Thus, it is easy to set the phase for
$\Omega_{\shortparallel}$ although that for $\Omega_{\perp}$ is still an
issue. In this sense, the position distribution of the qubit will induce error
only through the phase fluctuation of $\Omega_{\perp}$, and the fidelity for
method II can be much larger than that of method I. For more detail about this
issue, see the third paragraph at the beginning of Sec. III.

###  III.3 Rydberg addressing in a relatively large 3D lattice

For a nontarget atom away from the foci of the Gaussian beam, the relevant
Rydberg blockade can be comparable to the Rabi frequency $\Omega_{\alpha}$ for
the nontarget atom in the light path along $\mathbf{l}_{\alpha}$, where
$\alpha=\perp,\shortparallel$. Then, we should include the blockade in the
analysis about the state evolution of the nontarget atom. First of all,
because any nontarget atom lying away from the beam axis is pumped by a very
small field, its population in Rydberg states is negligible and for a first
approximation it does not induce dynamical phase in the target atom. Second,
the blockade between two atoms along the beam axis is $V/216$ as shown in Sec.
III.1, so the blockade interaction between the target atom and a nontarget
atom can be neglected. Then, we can separate the atoms influenced by the
lasers into two groups: the target atom, and all the nontarget atoms. These
two groups do not disturb each other. But there is many-body physics in the
second group: for two nontarget atoms away from the foci of the laser beams by
$\mathfrak{l}_{0}\gg X$, $\Omega_{\alpha}$ can be comparable to the
interaction $V/216$ between them because $\Omega_{\alpha}/\Omega\propto
X/\sqrt{X^{2}+\mathfrak{l}_{0}^{2}}$. So, the single-particle optical spin
echo in Secs. III.1 and III.2 is not sufficient to restore the state of the
nontarget atoms.

####  III.3.1 Microwave spin echo for method I

A microwave spin-echo sequence can eliminate the many-body imprinted Rydberg
excitation in the nontarget atoms. As an example, we take method I shown in
Sec. III.1 to explain this. The required modification is that between the two
stages of time evolution shown in Hamiltonian (3) and (7), we should add a
two-photon microwave transition between $|r\rangle$ and another Rydberg state
$|r^{\prime}\rangle$

$\displaystyle\hat{H}_{\mu}$ $\displaystyle=$ $\displaystyle
i\Omega_{\mu}|r^{\prime}\rangle\langle r|/2+\text{H.c.},$ (18)

where the microwave field covers all atoms in the system, but will only
influence states that are initially in $|r\rangle$; one can assume that the
strength of the microwave field is the same for all atoms in the system. The
Rabi frequency of this microwave field should be much larger than the blockade
interactions between any two atoms in the problem, and the signs of the
blockade interactions of $|rr\rangle$ and $|r^{\prime}r^{\prime}\rangle$
should be opposite. This latter requirement is achievable if $|r\rangle$ and
$|r^{\prime}\rangle$ are $s$\- and $d$-orbital states of 87Rb, respectively
Walker and Saffman (2008). If the ratio between the $C_{6}$ coefficients of
$|r^{\prime}r^{\prime}\rangle$ and $|rr\rangle$ is $\varkappa$, the amplitude
and detuning of the laser fields used in the latter stage of Eq. (7) should be
$\varkappa\Omega$ and $\varkappa\Delta$, respectively. Then, the duration of
the second pulse in the microwave spin echo becomes
$t_{1}=\pi/|\varkappa\Delta|=t_{0}/|\varkappa|$. Because $\varkappa$ can
differ from $-1$, it is necessary to either add appropriate phases in the
laser fields, or add extra wait time. To explain this, we take a single
nontarget atom as an example. Suppose the energy of the state
$|r^{\prime}\rangle$ is $\hbar\omega^{\prime}$, then the rotating frame with
it is given by
$\hat{\mathcal{R}}^{\prime}=\hat{\mathcal{R}}+\omega^{\prime}|r^{\prime}\rangle\langle
r^{\prime}|$, and the operators $\hat{\mathcal{R}}_{\pm}$ in Secs. III.1 and
III.2 change to
$\hat{\mathscr{R}}_{\pm}=(\omega^{\prime}\pm|\varkappa|\Delta)|r^{\prime}\rangle\langle
r^{\prime}|$, which are applicable after the microwave pulse. If the microwave
field lasts for a duration of $t_{\mu}$, an extra wait time $t_{\text{w}}$
changes Eq. (9) to

$\displaystyle|\psi(t_{\text{f}})\rangle_{\perp}$ $\displaystyle=$
$\displaystyle
e^{-it_{1}\hat{\mathscr{H}}_{\perp}^{(2)}}e^{i(t_{\text{w}}+t_{\mu}+t_{0})\hat{\mathscr{R}}_{+}}e^{-i(t_{\text{w}}+t_{\mu}+t_{0})\hat{\mathcal{R}}^{\prime}}$
$\displaystyle\times
e^{-it_{\mu}\hat{H}_{\mu}}e^{it_{0}\hat{\mathcal{R}}^{\prime}}e^{-it_{0}\hat{\mathcal{R}}_{-}}e^{-it_{0}\hat{\mathcal{H}}_{\perp}^{(1)}}|\psi(0)\rangle_{\perp},$
$\displaystyle|\psi(t_{\text{f}})\rangle_{\shortparallel}$ $\displaystyle=$
$\displaystyle
e^{-it_{1}\hat{\mathscr{H}}_{\shortparallel}^{(2)}}e^{i(t_{\text{w}}+t_{\mu}+t_{0})\hat{\mathscr{R}}_{-}}e^{-i(t_{\text{w}}+t_{\mu}+t_{0})\hat{\mathcal{R}}^{\prime}}$
(19) $\displaystyle\times
e^{-it_{\mu}\hat{H}_{\mu}}e^{it_{0}\hat{\mathcal{R}}^{\prime}}e^{-it_{0}\hat{\mathcal{R}}_{+}}e^{-it_{0}\hat{\mathcal{H}}_{\shortparallel}^{(1)}}|\psi(0)\rangle_{\shortparallel},$

where $t_{\text{f}}=t_{1}+t_{\text{w}}+t_{\mu}+t_{0}$ is the time at the end
of the sequence, and

$\displaystyle\hat{\mathscr{H}}_{\perp}^{(2)}$ $\displaystyle=$
$\displaystyle-|\varkappa|\Omega_{\perp}e^{-it|\varkappa|\Delta}|r^{\prime}\rangle_{\perp}\langle
1|/2+\text{H.c.},$ $\displaystyle\hat{\mathscr{H}}_{\shortparallel}^{(2)}$
$\displaystyle=$
$\displaystyle|\varkappa|\Omega_{\shortparallel}e^{it|\varkappa|\Delta}|r^{\prime}\rangle_{\shortparallel}\langle
1|/2+\text{H.c.}.$ (20)

Compared to Eq. (9), the different magnitudes of the detuning of the laser
fields before and after the wait in Eq. (19) requires appropriate wait time to
validate the microwave spin echo. To show this, one can verify that with
$\Omega_{\mu}t_{\mu}=\pi$, the operator $e^{-it_{\mu}\hat{H}_{\mu}}$ is
equivalent to $(|r^{\prime}\rangle\langle r|-$H.c.) when acting on a state
$|r(r^{\prime})\rangle$, and is $\hat{1}$ otherwise. Thus the microwave spin
echo in the two equations in Eq. (19) requires
$|\varkappa|\Delta(t_{\text{w}}+t_{\mu}+t_{0})+t_{0}\Delta=2n\pi$, where $n$
is an integer, which gives

$\displaystyle t_{\text{w}}+t_{\mu}$ $\displaystyle=$
$\displaystyle\frac{2n\pi}{|\varkappa|\Delta}-t_{0}(1+1/|\varkappa|).$ (21)

For fast quantum control, the minimal positive integer $n$ leads to the
smallest $t_{\text{w}}+t_{\mu}$. When the above condition is satisfied, the
many-body Hamiltonian $\hat{H}^{\prime}$ during
$t\in[t_{\text{w}}+t_{\mu}+t_{0},~{}t_{\text{w}}+t_{\mu}+t_{0}+t_{1})$ is
exactly $\varkappa$ times the Hamiltonian during $t\in[0,~{}t_{0})$; the only
modification is that the involved Rydberg state is different. As a
consequence, the many-body system of the nontarget atoms will have their
states restored even though they experience many-body state evolution; in
fact, even if there is interaction between the target and nontarget qubit, the
microwave field also changes the state of the target atom, and hence the many-
body effect can still be removed. For a numerical simulation, see the appendix
of Ref. Shi (2018a) where the duration of $t_{\text{w}}+t_{\mu}$ is ignored
supposing the spin-echo condition is satisfied.

####  III.3.2 Microwave spin echo for method II

The process above is applicable to method II, too. Because there is no
nontarget atom in the light path along $\mathbf{l}_{\shortparallel}$, we focus
on the nontarget atoms in the light path along $\mathbf{l}_{\perp}$. As in
Sec. III.3.1, a microwave field pumps $|r\rangle$ to $|r^{\prime}\rangle$ so
as to reverse the sign of the Rydberg interaction. Because the microwave field
influences all atoms, the state component $|r\rangle$ of the target qubit also
changes likewise; but the target atom can have some population in $|R\rangle$,
too. To completely remove the many-body effect in the system, we should
reverse the sign of interaction in $|r^{\prime}R\rangle$, too. This can be
realized by using a superposition of two Rydberg states Shi  _et al._ (2016);
Shi and Kennedy (2017); Shi (2018b) as $|R\rangle$, so that its interaction
with a nontarget atom can also change by a ratio of $\varkappa$. For more
detail about this method, see Ref. Shi and Kennedy (2017).

In Fig. 5, we show the microwave spin echo with one atom by using
$\varkappa=-52.6/56.2$, which corresponds to the $100d(s)$ state of rubidium
Shi (2018a). Figure 5 shows that the nontarget atom irradiated by the light
along $\mathbf{l}_{\perp}$ indeed returns to the ground state, and the target
atom is excited to the Rydberg state $|R\rangle$. The final phase of the
Rydberg state is $\pi/2$; when the same spin-echo sequence in Fig. 5 is used
for deexcitation, the final state is $-|1\rangle$ for the target atom.

Figure 5: Numerical result for the excitation of the Rydberg state $|R\rangle$
in Method II with a microwave field $\pi$-pulse between the two optical
pulses. (a) shows the time evolution of the population in $|1\rangle$ (solid
curve), $|r\rangle$ (dotted curve), and $|r^{\prime}\rangle$ (dash-dotted
curve) for the nontarget atom $E$ (or atom $F$). (b) shows the time evolution
of the population in $|R\rangle$ (dashed curve), $|1\rangle$, $|r\rangle$, and
$|r^{\prime}\rangle$ for the target atom. Before the microwave pulse,
$\Delta/2\pi=4$ MHz and $\Omega=1.2247\Delta$; after the microwave pulse, the
detuning and Rabi frequency become $|\varkappa|\Delta$ and $|\varkappa|\Omega$
with $\varkappa=-0.936$ (see text), and the microwave field lasts for a
$\pi$-pulse duration of $t_{\mu}=(1/|\varkappa|-1)\pi/\Delta$ with
$\Omega_{\mu}=\pi/t_{\mu}$. The Rabi frequency for the nontarget atom is
$\Omega_{\perp}=0.73\Omega$ in (a).

###  III.4 Limitation from divergence of laser beam

The above method assumes that at most one nontarget atom is irradiated near
$\mathfrak{l}_{0}$ for each laser beam. If more than one nontarget atoms are
irradiated near $\mathfrak{l}_{0}$, the Rydberg interactions between the
nontarget atom at the axis of the beam and the atom off the axis can not have
a ratio of $\varkappa$ in the two pulses of the optical spin echo. This is
because the interaction between $s$-orbital Rydberg atoms is isotropic, but
that between $d$-orbital atoms is not Walker and Saffman (2008). Even the
Rydberg interaction along the the light path can change by a factor of
$\varkappa$ after the microwave irradiation, the interaction between two atoms
not along the light path can not.

We study condition that approximately validates the microwave spin echo in
Sec. III.3. We first consider two nontarget atoms shown in Fig. 2(a), labeled
as $A$ and $B$, where $B$ is not at the beam axis, but $A$ is, and thus atom
$A$ is irradiated. Because the field amplitude decays rapidly away from the
beam axis, atom $B$ is irradiated with a small field. For the configuration in
Fig. 2, the angle between $AC$ and $BC~{}(AB)$ is only $0.3~{}(0.4)$ radian,
so that the interaction between them is approximately isotropic Walker and
Saffman (2008), and if no atom that is even farther to the beam axis than B is
irradiated, the many-body spin echo is valid.

That no nontarget atoms farther than the atom B are irradiated is determined
by the lattice constant and the laser beam. To study the decay of the light
intensity away from the foci, we use $\mathfrak{l}_{0}$ as the longitudinal
coordinate, and $r_{\perp}$ as the radial coordinate. The foci is at the atom
$C$, which is the atom at the beam axis nearest to $A$. The coordinate of $B$
is $(\mathfrak{l}_{0},r_{\perp 0})$. From the geometry of the lattice, one can
locate another atom $B^{\prime}$ at $(\mathfrak{l}_{1},2r_{\perp 0})$, where
$\mathfrak{l}_{1}\approx\mathfrak{l}_{0}$. The beam radius of the laser field
near $(\mathfrak{l}_{1},0)$ is given by

$\displaystyle w(\mathfrak{l}_{1})$ $\displaystyle=$ $\displaystyle
w(0)\sqrt{X^{2}+\mathfrak{l}_{1}^{2}}/X$ (22) $\displaystyle=$
$\displaystyle\lambda\sqrt{\mathfrak{l}_{1}^{2}/w^{2}(0)+\pi^{2}w^{2}(0)/\lambda^{2}}/\pi$

where $w(0)$ is the beam radius at the beam waist and $\lambda$ is the
wavelength of the laser field. The electric field amplitude of the laser is
$\propto e^{-r_{\perp}^{2}/w^{2}(\mathfrak{l}_{1})}$, which means that the
electric field is $e^{-r_{\perp 0}^{2}/w^{2}(\mathfrak{l}_{1})}\mathcal{E}$
(or $e^{-4r_{\perp 0}^{2}/w^{2}(\mathfrak{l}_{1})}\mathcal{E}$) for the atom
$B$ (or $B^{\prime}$), where $\mathcal{E}$ is the amplitude at
$(\mathfrak{l}_{1},0)$. If it is reasonable to neglect atomic excitation when
the field amplitude of the laser light falls to $e^{-4}$ of that at the beam
axis, $r_{\perp 0}$ should be smaller than $w(0)$ so that $B^{\prime}$ is not
irradiated. From Eq. (22), one finds that
$w(\mathfrak{l}_{0})\geq\sqrt{2\lambda\mathfrak{l}_{0}/\pi}$. For Fig. 2(a),
one can show that $r_{\perp 0}=0.46\mathcal{L}$. So, it is necessary to have
$\mathfrak{l}_{0}<0.33\mathcal{L}^{2}/\lambda$ if we want the field at
$B^{\prime}$ to be smaller than $\mathcal{E}/e^{4}$. But for the sake of
convenience in experiments, it is useful to let the two laser fields propagate
only along the two directions, $[12\overline{1}]$ and $[\overline{1}21]$, so
that by only sweeping the laser fields parallel one can address any site of
the lattice. Then, we need to consider that the lasers can be focused at an
atom that is at any edge of the lattice so as to derive the permissible
largest lattice; one can find that the lattice size should be smaller than
$(1+\mathcal{N})\times(1+2\mathcal{N})\times(1+\mathcal{N})$, where
$\mathcal{N}$ is the largest integer that is smaller than
$0.13\mathcal{L}/\lambda$. This means that only a lattice with a large enough
lattice constant is applicable by our method. If small Rydberg blockade is
needed, relatively large lattice constants like $16.5~{}\mu$m are required Shi
(2019); in such a case and with $\lambda=0.78~{}\mu$m, a lattice of the size
$3\times 5\times 3$ in Fig. 2 can allow arbitrary single-site addressing by
our method. For $\lambda\leq 0.78~{}\mu$m in the case of rubidium,
$0.13\mathcal{L}/\lambda\approx 1$ if $\mathcal{L}=6~{}\mu$m, which means that
a $2\times 3\times 2$ lattice can be addressed only if the lattice constant is
as large as $6~{}\mu$m. This limitation from the beam divergence can be
tackled by sending laser fields along various directions other than the two
shown in Fig. 2 so that a larger lattice can be used for this purpose although
it involves more experimental complexity.

A possible solution to the problem of system size is to trap two types of
atoms in one lattice. In Fig. 9 of Ref. Beterov and Saffman (2015), a two-
dimensional lattice trapping both cesium and rubidium is studied. If such a
lattice is extended to 3D in a configuration so that the atoms nearest to a
rubidium are all cesium atoms, it should be possible to address the rubidium
atoms with laser fields that propagate through cesium atoms. This method may
allow more qubits to be trapped. However, it is an open question whether such
a 3D lattice can be prepared.

###  III.5 Application in Rydberg blockade gate

Till now, we have assumed that there is no Rydberg atom near the target atom
before exciting it to Rydberg states. In the context of Rydberg blockade gate
Jaksch  _et al._ (2000), if there is a Rydberg atom near the target qubit, it
is called the control qubit. Before proceeding, we note that the microwave
field used in the spin-echo sequence will influence both the control and
target qubits if the same Rydberg state is used for the two qubits, and thus
can transfer the Rydberg state of the control qubit. Then, the interaction
between the control qubit and any nontarget atoms reverses the sign,
validating the required spin-echo condition.

Compared to method I, method II is more useful when the control qubit is
already in the Rydberg state. We assume that the control qubit is away from
the target qubit by one lattice constant $\mathcal{L}$, and take the
configuration in Fig. 2(a) for analysis. For a concrete discussion, we assume
that the atom labeled by $D$ in Fig. 2(a) is the control qubit, which is
already in the Rydberg state $|R_{1}\rangle$, and we are to excite the target
atom $C$ to the Rydberg state $|R_{2}\rangle$ to prepare entanglement through
the blockade interaction $V$ of the state $|R_{1}R_{2}\rangle$. Now atom $D$
is very near to the laser beams along both $\mathbf{l}_{\shortparallel}$ and
$\mathbf{l}_{\perp}$, and one can choose different Rydberg states
$|R_{1}\rangle$ and $|R_{2}\rangle$ when exciting the control and target
qubit, respectively. When $|R_{1}\rangle$ and $|R_{2}\rangle$ have a GHz
energy separation, the light field used for the target will not alter the
state of the control qubit. If microwave spin echo is used in this case, three
sets of microwave fields should be employed, one to transfer $|r\rangle$ to
$|r^{\prime}\rangle$ (as in Fig. 5), and the other two for
$|R_{1(2)}\rangle\rightarrow|R_{1(2)}^{\prime}\rangle$. This can still realize
the microwave spin echo when the interactions of the states
$(|R_{1}r\rangle,|R_{1}R_{2}\rangle)$ and those of
$(|R_{1}^{\prime}r^{\prime}\rangle,|R_{1}^{\prime}R_{2}^{\prime}\rangle)$
change by a common ratio $\varkappa$; to realize this, superposition states
can be used as $|r^{\prime}\rangle$ and $|R_{1(2)}^{\prime}\rangle$ Shi  _et
al._ (2016); Shi and Kennedy (2017); Shi (2018b). But for method I, nontarget
atoms along the laser fields along both $\mathbf{l}_{\shortparallel}$ and
$\mathbf{l}_{\perp}$ can be excited to the Rydberg state. Then, because atom
$B$ and atom $D$ have a distance of $\sqrt{3}\mathcal{L}$, the interaction
between them is $V/3^{3}$, which can be comparable to the Rabi frequency and
hampers the gate operation. This means that it is necessary to avoid exciting
atom $B$ if method I is used, which puts a significant limit to the system
size. But if method II is used, the light field along
$\mathbf{l}_{\shortparallel}$ will not excite any Rydberg state, thus atom $B$
is not a problem. This means that both method I and method II can be used for
the excitation of the control qubit, but method II is more useful for the
$2\pi$ pulse for the target qubit in the Rydberg blockade gate Jaksch  _et
al._ (2000).

###  III.6 Comparison between method I and method II

The two methods shown in Secs. III.1 and III.2 have their own advantages and
shortcomings, and can be used for different purposes.

First, both methods will leave the state of any nontarget atoms intact when
the state of the target atom is restored to the ground state. However, method
I only needs one pulse for exciting the target atom to the Rydberg state,
while method II needs two.

Second, although the excitation and deexcitation of the target atom form an
optical spin-echo cycle in method I, the excitation pulse alone does not form
a spin echo. So, the nontarget atoms can have some residual population in
Rydberg state when the target atom is pumped to the Rydberg state. For method
II, Rydberg excitation of the target atom is by two sets of lasers: those
along $\mathbf{l}_{\shortparallel}$ form a spin-echo cycle, but those along
$\mathbf{l}_{\perp}$ do not. Although the field along $\mathbf{l}_{\perp}$ do
not form a spin echo sequence, it does not excite any nontarget atoms, shown
in Secs. III.2. So, no nontarget atom will have any population in the Rydberg
state when the target atom is completely excited to Rydberg states in method
II.

Third, the state of the target atom will not pick any phase upon the
completion of its state restoration in method I. For method II, the target
atom does not experience any spin echo, and a $\pi$ phase is imprinted when
its state is restored to the ground state. This means that for the Rydberg
gate in Ref. Jaksch  _et al._ (2000), method I can be used for the control
qubit, but can not be used for the target qubit, while method II can be used
for both the control and target qubit.

##  IV High-fidelity quantum gates

Another application of ORIR is in achieving high fidelity in a Rydberg quantum
gate Jaksch  _et al._ (2000); Saffman  _et al._ (2010); Saffman (2016); Weiss
and Saffman (2017). Although it is theoretically possible to achieve high
fidelity beyond $0.999$ in an entangling quantum gate by Rydberg interactions
Goerz  _et al._ (2014); Theis  _et al._ (2016); Shi (2017); Petrosyan  _et
al._ (2017); Shi (2018a, 2019), the experimental implementation is difficult
Wilk  _et al._ (2010); Isenhower  _et al._ (2010); Zhang  _et al._ (2010);
Maller  _et al._ (2015); Jau  _et al._ (2016); Zeng  _et al._ (2017); Picken
_et al._ (2019), with best fidelity below $0.98$ Graham  _et al._ (2019);
Levine  _et al._ (2019). This is partly due to a number of technical issues de
Léséleuc  _et al._ (2018). But even if these issues are removed with improved
technology Levine  _et al._ (2018); Graham  _et al._ (2019); Levine  _et al._
(2019), the fidelity of the best-known Rydberg quantum gate, i.e., the Rydberg
blockade gate Jaksch  _et al._ (2000), is limited by the blockade error Zhang
_et al._ (2012). Below, we show that ORIR can effectively suppress this
blockade error.

The Rydberg blockade gate maps the input states from
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ to
$\\{|00\rangle,-|01\rangle,-|10\rangle,-|11\rangle\\}$, and is realized in
three steps: apply a $\pi$ pulse to the control qubit; apply a $2\pi$ pulse to
the target; apply another $\pi$ pulse to the control qubit Jaksch  _et al._
(2000). These three steps induce the following state evolution for the input
state $|11\rangle$:

$\displaystyle|11\rangle\xrightarrow[\text{For
control}]{\pi}-i|r1\rangle\xrightarrow[\text{For
target}]{2\pi}-i|r1\rangle\xrightarrow[\text{For control}]{\pi}-|11\rangle,$

where $|r\rangle$ is a Rydberg state. The second step in the above equation is
perfect if the Rydberg blockade $V$ of the state $|rr\rangle$ is infinitely
large compared to the Rabi frequency $\Omega$ of the atom-field interaction;
for any finite $V$, it attains a blockade error on the order of
$\Omega^{2}/V^{2}$. There is also a phase error, but it can be removed
effectively by adding a phase difference in the laser field at the middle of
the second step of the gate. When a global phase shift to the target qubit is
inserted, the phase error is removed Zhang  _et al._ (2012), but the blockade
error sets a fundamental limit to the achievable fidelity of the gate. For
this reason, there are many theoretical proposals to suppress the blockade
error Goerz  _et al._ (2014); Theis  _et al._ (2016); Shi (2017); Petrosyan
_et al._ (2017); Shi (2018a, 2019); Shen  _et al._ (2019); Yu  _et al._
(2019); Levine  _et al._ (2019).

We show that the blockade error can be suppressed by replacing the usual Rabi
frequency $\Omega$ with a time-dependent Rabi frequency $i\Omega\sin(\Delta
t)$ formed by ORIR. In practice, this is done via replacing a resonant laser
of power $P$ by two laser beams of opposite phases, one with detuning $\Delta$
and power $P/4$, and the other with detuning $-\Delta$ and power $P/4$. This
results in an effective Rabi frequency $i\Omega\sin(\Delta t)$. In contrast to
the single-site Rydberg addressing studied in Sec. III, here the fields with
detuning $\Delta$ and those with detuning $-\Delta$ can propagate in one
direction, and hence it is easier to set the relative phase of the laser
fields to realize Eq. (1). In this case, the Hamiltonian for the input state
$|11\rangle$ during the second step of the traditional protocol

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}2V&\Omega\\\
\Omega&0\end{array}\right)$ (25)

is replaced by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}V&i\Omega\sin(\Delta t)/2\\\
-i\Omega\sin(\Delta t)/2&0\end{array}\right)$ (28)

in our scheme. The basis is $\\{|rr\rangle,~{}|r1\rangle\\}$ in the two
matrices of Eqs. (25) and (28). To induce the transform
$|01\rangle\rightarrow-|01\rangle$ for the input state $|01\rangle$, one can
choose $\Omega/\Delta=\pi$ and a duration $t=\pi/\Delta$ for the second step,
as can be derived in a similar method used for the derivation of Eq. (2).
Here, the condition of $\Omega/\Delta=\pi$ and $t=\pi/\Delta$ is not
compatible with an optical spin echo sequence because it does not allow the
transform from Eq. (9) to Eq. (LABEL:method01finalstate2), which is in
contrast to the case studied in Sec. III.1. This means that to use the method
in this section for addressing one atom in 3D lattice, only atoms near the
very edge of the atomic array can be excited so that no nontarget atoms will
be excited.

To analyze the blockade errors for the two schemes in Eqs. (25) and (28), we
take the setup in Levine  _et al._ (2018) to estimate the system parameters.
In Levine  _et al._ (2018), its Figure 1 shows that the laser fields for the
optical trap propagate (along $\mathbf{z}$) perpendicular to the quantization
axis (labeled as $\mathbf{x}$) of the atom. The supplemental material of
Levine  _et al._ (2018) shows that the r.m.s. fluctuation of the qubit spacing
along $\mathbf{x}$ is $\varsigma_{x}=0.2~{}\mu$m. If the position fluctuation
along $\mathbf{y}$ or $\mathbf{z}$ is about 10 times of that along
$\mathbf{x}$ as in Isenhower  _et al._ (2010), we can assume
$\varsigma_{y}=\varsigma_{z}=2~{}\mu$m. The two-qubit spacing is
$L=5.7~{}\mu$m in Levine  _et al._ (2018), which means that the actual
blockade $V$ can be very different from the chosen value $V_{0}$: fluctuation
along $\mathbf{x}$ can lead to $V/V_{0}=(1\pm
2\varsigma_{x}/L)^{-6}~{}(\approx 0.67,1.55)$, while that along $\mathbf{z}$
can lead to $V/V_{0}=(1+4\varsigma_{z}^{2}/L^{2})^{-3}~{}(\approx 0.3)$. For a
conserved estimation, we consider $(V-V_{0})/V_{0}\in[-0.25,~{}0.25]$. The
entanglement between the ground and Rydberg states in Levine  _et al._ (2018)
used $(\Omega,~{}V_{0})/2\pi=(2,~{}30)$ MHz, while the actual Rabi frequency
for the entanglement was $\sqrt{2}\Omega$ due to the many-body enhancement.
Then, we take $V_{0}/\Omega=12$ as an example to study the rotation error of a
Rydberg blockade process.

Figure 6: Population loss in the the state $-i|r1\rangle$ after the second
step of the Rydberg gate shown with the common logarithm and parameters
$V_{0}:\Omega:\Delta=12:1:\pi^{-1}$. $V_{0}$ is the blockade interaction of
$|rr\rangle$ when both qubits are located at the center of their traps; the
actual Rydberg blockade $V$ can differ from $V_{0}$ due to the thermal motion
of the qubits. The solid curve denotes results from propagating the
wavefunction by the Hamiltonian in Eq. (28) for a duration of $\pi/\Delta$.
The dashed curve shows the population error of the state $-i|r1\rangle$ after
an evolution time of $2\pi/\Omega$ under the Hamiltonian in Eq. (25). When
averaged over all the $V$ in the figure, the population errors are $9.5\times
10^{-6}$ and $3.7\times 10^{-3}$ for the ORIR-based method and the traditional
method, respectively. These two averages respectively become $1.7\times
10^{-5}$ and $4.6\times 10^{-3}$ if a larger interval of
$(V-V_{0})/V_{0}\in[-0.5,~{}0.5]$ is considered.

The population loss in the state $|r1\rangle$ after the second step of the
gate is shown in Fig. 6 for different $(V-V_{0})/V_{0}\in[-0.25,~{}0.25]$. The
dashed curve in Fig. 6 shows that the blockade error in the traditional method
is on the order of $10^{-3}$ for most values of $V$. Although at some special
values of $(V-V_{0})/V_{0}$ (for example, at around $-1/300$) a tiny
population leakage of less than $10^{-5}$ can appear with the traditional
method, the blockade error quickly rises to the level of $10^{-3}$ when $V$
deviates from that special value. Thus it is difficult to use the traditional
method to achieve a small blockade error since it is challenging to
sufficiently suppress the fluctuation of the qubit positions. On the other
hand, the solid curve in Fig. 6 shows that the population loss in $|r1\rangle$
is much smaller when the second step is characterized by the effective
Hamiltonian in Eq. (28). In fact, the blockade error in the ORIR-based
protocol of Fig. 6 is on the order of $10^{-5}$ for most values of $V$, which
is two orders of magnitude smaller compared with the errors of the traditional
method, effectively removing the blockade error.

The removal of blockade error by Eq. (28) is quite different from other
proposals of high-fidelity gates by time-dependent pumping. For example,
comparing to the method of analytical derivative removal by adiabatic gate in
Ref. Theis  _et al._ (2016), our method only needs rectangular pulse (or
quasi-rectangular pulse) that can be easily chopped from a continuous laser
field with a pulse picker. Moreover, the ORIR-based gate does not have a
strict dependence on specific blockade interaction for achieving the optimal
fidelity, which can be found by comparing the solid curve in Fig. 6 for our
method and Fig. 3 of Ref. Theis  _et al._ (2016) for the other method [when
comparing, we note that a fourth of $1-|\langle r1|\psi\rangle|^{2}$ in Fig. 6
contributes to the gate infidelity]. In fact, as an example, if the value of
$V_{0}/\Omega$ in Fig. 6 increases to $30~{}(50)$, the population errors
averaged over the interval $(V-V_{0})/V_{0}\in[-0.25,~{}0.25]$ become
$1.3\times 10^{-6}$ and $5.9\times 10^{-4}$ ($4.3\times 10^{-7}$ and
$2.1\times 10^{-4}$) in the ORIR-based method and the traditional method,
respectively. These data show that the ORIR-based protocol can still suppress
the blockade error by more than two orders of magnitude when $V/\Omega$ is
large.

Figure 7: Time evolution of $1-|\langle r1|\psi\rangle|^{2}$ during the second
step of the Rydberg gate shown by the common logarithm with the condition of
$V=V_{0}=12\Omega$. (a) shows the result from the ORIR-based method by using
the Hamiltonian in Eq. (28). (b) shows the results of the traditional method
by using the Hamiltonian in Eq. (25). The red circles denote the population
errors at the end of the second step of the gate, which are $4.3\times
10^{-6}$ and $1.2\times 10^{-4}$ in (a) and (b), respectively.

To understand the mechanism for the suppression of the blockade error by ORIR,
we show the time evolution of the population in $|rr\rangle$, which is equal
to $1-|\langle r1|\psi\rangle|^{2}$, during the second step for our method and
the traditional method in Fig. 7(a) and 7(b), respectively. Figure 7(a) shows
that the population leakage with ORIR increases with a minor oscillation
pattern during the first half of the pulse, but decreases with a similar
pattern during the latter half of the pulse. This removal of the blockade
error is similar to the suppression of the blockade error with the spin-echo
method studied in Ref. Shi (2018a), although the underlying physics is
different.

###  IV.1 Gate error from finite edges and timing errors of laser pulses

As shown in Appendix A, the finite rising and falling, and timing
synchronization of laser pulses are also factors that limit the accuracy of
the quantum gate. In most experiments on Rydberg quantum gates, an $s$\- or
$d$-orbital Rydberg state was used Wilk  _et al._ (2010); Isenhower  _et al._
(2010); Zhang  _et al._ (2010); Maller  _et al._ (2015); Zeng  _et al._
(2017); Picken  _et al._ (2019); Levine  _et al._ (2018); Graham  _et al._
(2019); Levine  _et al._ (2019). Their excitation is usually via a two-photon
excitation through a largely detuned intermediate state. In typical
experiments, the laser beams for the upper transition are left on for several
tens of nanoseconds longer than the lasers for the lower transitions Maller
_et al._ (2015). Then, the onset and cutoff of the optical pumping are
determined by the lasers of the lower transitions. So, we only need to
consider four timing errors
$\\{t_{\text{ge}+}^{\text{(s)}},t_{\text{ge}-}^{\text{(s)}},t_{\text{ge}+}^{\text{(e)}},t_{\text{ge}-}^{\text{(e)}}\\}$
for the two lower laser beams, where $t_{\text{ge}\pm}^{\text{(s)}}$ and
$t_{\text{ge}\pm}^{\text{(e)}}$ are the deviation of the actual start and
cutoff times from the desired start and cutoff times of the pulse (as
illustrated in Fig. 9); ideally, they should be zero. These errors will
influence both $|01\rangle$ and $|11\rangle$ among the four input states.
According to the discussion in the fourth paragraph of this section and Levine
_et al._ (2018), we choose $\Omega/2\pi=2$ MHz as the desired value of
$\Omega_{\text{ge}\pm}$. As for the Rydberg blockade, we assume
$V_{0}/\Omega=12$ and choose, as an example, $V=0.97V_{0}$, where the
population error in $|r1\rangle$ is about $10^{-5}$ ($\approx$ average error
for the solid curve in Fig. 6) in the ideal case. The optical excitation
occurs during $[0,~{}T]$ in the ideal case, where $T$ is the desired pulse
duration. We suppose that the rising or falling of the laser pulses need $20$
ns Urban  _et al._ (2009); Maller  _et al._ (2015); Levine  _et al._ (2018).
The optimal value of $T$ corresponds to the smallest population error in
$|01\rangle$. This is because the population leakage is less sensitive for the
input state $|11\rangle$ since the transition
$|r1\rangle\rightarrow|rr\rangle$ is blocked during the second step of the
gate. With a numerically found optimal $T$ of $795.3963$ ns, the population
leakage in $|01\rangle$ ($|r1\rangle$) is $4.4~{}(7.7)\times 10^{-6}$. We then
use this $T$ to investigate the population errors in $|01\rangle$ and
$|r1\rangle$ when the timing error is included.

Figure 8: Round (square) symbols show the average population loss in the state
$|01\rangle~{}(|r1\rangle)$ with a r.m.s. timing error $\sigma_{\text{t}}$ for
the start and end of the two laser pulses. Calculation was based on
propagating the state with the Hamiltonian in Eq. (LABEL:Hwithtimingerror)
during $[t_{\text{ge}\pm}^{\text{(s)}},~{}T+t_{\text{ge}\pm}^{\text{(e)}}]$.
Average was calculated with Eq. (34) by letting
$t_{\text{ge}\pm}^{\text{(s)}}$ and $t_{\text{ge}\pm}^{\text{(e)}}$ apply the
values
$\\{-5\sigma_{\text{t}},-4\sigma_{\text{t}},\cdots,~{}5\sigma_{\text{t}}\\}$.
Here $(\Omega,~{}V,~{}\Delta)/2\pi=(2,~{}23.28,~{}2/\pi)$ MHz.

To include possible timing error in the second step with optical pumping for
the target qubit, we cast Eq. (28) into

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}V&\frac{\Omega_{\text{ge}+}}{4}e^{i\Delta
t}-\frac{\Omega_{\text{ge}-}}{4}e^{-i\Delta t}\\\
\frac{\Omega_{\text{ge}+}}{4}e^{-i\Delta
t}-\frac{\Omega_{\text{ge}-}}{4}e^{i\Delta t}&0\end{array}\right),$ (31)

where $\Omega_{\text{ge}\pm}$ is nonzero during
$[t_{\text{ge}\pm}^{\text{(s)}},~{}T+t_{\text{ge}\pm}^{\text{(e)}}]$, and has
a rising and falling edge during the first and last $20$ ns of the pulse. Each
set of nonzero timing errors can result in gate error. Following Appendix A,
the average error is calculated with Eq. (34) where the summation runs through
the $11^{4}$ sets of
$\\{t_{\text{ge}+}^{\text{(s)}},t_{\text{ge}-}^{\text{(s)}},t_{\text{ge}+}^{\text{(e)}},t_{\text{ge}-}^{\text{(e)}}\\}$.
With the r.m.s. error $\sigma_{\text{t}}\in[0.2,~{}5]$ ns, the population
losses in $|01\rangle$ and $|r1\rangle$ are shown by the round and square
symbols in Fig. 8, respectively. One can find that even with a very large
fluctuation characterized by $\sigma_{\text{t}}=5$ ns, the lost population in
$|01\rangle$ ($|r1\rangle$) is only $3.0~{}(1.8)\times 10^{-4}$ after the
second step, which is much smaller than the blockade error in the traditional
method even without pulse defects (see the dashed curve in Fig. 6). This
suggests that ORIR is useful for high-fidelity quantum gate based on the
mechanism of Rydberg blockade.

##  V Discussions

The ORIR-based time-dependent Rabi frequencies can also be used in trapped
ions. Compared to neutral atoms, the entangling gates with trapped ions can
attain a much better accuracy Tan  _et al._ (2015); Ballance  _et al._ (2015,
2016); Gaebler  _et al._ (2016). But compared to ions, neutral atoms are easy
for building large-scale qubit arrays Wang  _et al._ (2015); Xia  _et al._
(2015); Barredo  _et al._ (2016); Wang  _et al._ (2016). Thus, it is useful to
combine these two merits together in one system; indeed, there is intense
interest to trap and manipulate Rydberg ions Higgins  _et al._ (2017a, b);
Engel  _et al._ (2018), as well as theoretical effort to design protocols for
quantum entanglement between Rydberg ions Müller  _et al._ (2008); Li  _et
al._ (2013); Li and Lesanovsky (2014). The methods in Müller  _et al._ (2008);
Li  _et al._ (2013); Li and Lesanovsky (2014) depend on time-dependent Rabi
frequencies and, in fact, the protocol in Li  _et al._ (2013) uses a Rabi
frequency $\propto\sin(\Delta t)$ that can be easily realized with quasi-
rectangular laser fields chopped from continuous lasers with pulse pickers.
Recently, adiabatic excitation of a single trapped Rydberg ion was
experimentally demonstrated by using two sinusoidal Rabi frequencies in a
ladder configuration Higgins  _et al._ (2017b). Reference Higgins  _et al._
(2017b) reported that the reason to have used sinusoidal Rabi frequencies is
that they result in a better fidelity than using Gaussian pulses. These
results show that ORIR, which can offer sinusoidal Rabi frequencies by only
using continuous laser and pulse pickers, is useful for Rydberg excitation in
trapped ions.

##  VI Conclusions

In conclusion, we have studied the application of an off-resonance-induced
resonance (ORIR) in single-site Rydberg addressing in a 3D lattice and high-
fidelity Rydberg gate. These applications benefit from a time-dependent Rabi
frequency $2\Omega\cos(\Delta t)$ [or $2i\Omega\sin(\Delta t)$] that naturally
arises from the action of two symmetrically detuned coherent fields of equal
[or opposite] phase. ORIR can be implemented with quasi-rectangular laser
pulse chopped from a continuous laser by an electro-optic modulator or an
acousto-optic modulator, a method commonly used in experiments with Rydberg
atoms.

In the study of quantum gates by Rydberg interactions of neutral atoms, ORIR
can enable single-site Rydberg addressing in a 3D optical lattice while
leaving irradiated nontarget atoms intact upon the completion of the Rydberg
deexcitation of the target atom; moreover, we show that ORIR can suppress a
fundamental rotation error in the Rydberg blockade gate, making it possible to
achieve high fidelity with only quasi-rectangular pulses. Along the way, we
find another method for single-site Rydberg addressing in a 3D lattice by
using a ladder-type system. In this latter method, spin echo is used for any
nontarget atom so that its state is restored when a single target atom is
pumped to the Rydberg state, and the target atom picks up a $\pi$ phase when
its state is restored to the ground state. These methods make it possible to
entangle two selected qubits deep in a 3D atomic array and paves the way to
large-scale quantum processors based on Rydberg atoms.

## ACKNOWLEDGMENTS

The author is grateful to Yan Lu and Tian Xia for fruitful discussions. This
work was supported by the National Natural Science Foundation of China under
Grant No. 11805146 and the 111 Project (No. B17035).

##  Appendix A Transfer error from failure of timing synchronization of laser
pulses

The ORIR-based application in the quantum gate and single-site addressing
depends on timing synchronization of laser pulses, which is discussed in this
appendix.

First, the timing synchronization of the laser pulses can be realized by using
acousto-optic multi-channel modulators made with tellurium dioxide crystals,
which can deflect multiple laser beams http://www.sintecoptronics.com/aom.asp
. To address arrays of rubidium atoms with two-photon excitation of Rydberg
states, a Doppler-free multiwavelength (780 and 480 nm) acousto-optic
deflector was experimentally demonstrated about a dozen years ago Kim  _et
al._ (2008). The deflector in Kim  _et al._ (2008) can simultaneously diffract
the two incident optical wavelengths with a common diffraction angle. So, it
is possible to realize a multi-channel modulator for deflecting several laser
lights. In this case, the durations of the two laser pulses in Fig. 9 will be
identical so that there will be no error due to failure of the timing
synchronization.

Second, the timing synchronization can be approximately achieved by using
commercially available high-precision digital delay generators. In Ref.
Saßmannshausen  _et al._ (2013), a single delay generator (Quantum Composer
9528) was used to control all timings that were relevant for Rydberg-state
excitation and detection. This type of delay generator has a resolution of
0.25 ns and r.m.s. jitter of 0.4 ns Qua , and is widely used in the
experimental study on quantum physics Matthews  _et al._ (2011); de Groote
_et al._ (2015), quantum chemistry Zhu  _et al._ (2019), combustion Eitel  _et
al._ (2017); Bonebrake  _et al._ (2019), accelerator Smith (2010), and
aerodynamics Watkins (2013). The precision of commercial digital delay
generators can also be surprisingly high. For example, the pulse generated by
a laser synchronization module in EKS  has a resolution of 25 ps, an accuracy
of 25 ps$+10^{-6}\times$delay, and jitter of less than 30 ps. However, ultra-
high precision is not required for most applications, so that such digital
delay generators are not well-known. To use these digital delay generators for
timing, the transmission time of the control signal should also be considered,
which is possible by adjusting the length of the cable.

Figure 9: Failure of timing synchronization of the laser pulses. The topmost
curve shows the shape of the two lower laser pulses when they have no timing
error. (a) and (b) show the pulse shapes with timing errors.

We suppose the pulse timing is controlled by a commercial digital delay
generator and assume that the r.m.s timing errors of the onset and cutoff of
the pulse are both $\sigma_{\text{t}}$. The beginning of the pulse is at a
time $t$ that obeys Gaussian distribution
$f(t|0,\sigma_{\text{t}})=e^{-t^{2}/(2\sigma_{\text{t}}^{2})}/\sqrt{2\pi\sigma_{\text{t}}^{2}}$,
while the end of the pulse is at a time $t$ that is distributed according to
$f(t|T,\sigma_{\text{t}})=e^{-(t-T)^{2}/(2\sigma_{\text{t}}^{2})}/\sqrt{2\pi\sigma_{\text{t}}^{2}}$,
where $T$ is the desired pulse duration. The timing diagram of the four laser
beams is shown in Fig. 9. Altogether, there are four timing errors, i.e., the
two timing errors of the pulse arrival, $t_{\text{ge}\pm}^{\text{(s)}}$, of
the two laser lights for the lower transition, and their cutoff,
$t_{\text{ge}\pm}^{\text{(e)}}$. The fidelity $\mathcal{F}$ for each
realization of the gate is a function of these four timing errors. Then, the
expected fidelity is given by

$\displaystyle\overline{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\int\mathcal{F}(t_{\text{ge}+}^{\text{(s)}},t_{\text{ge}-}^{\text{(s)}},t_{\text{ge}+}^{\text{(e)}},t_{\text{ge}-}^{\text{(e)}})$
(33)
$\displaystyle\cdot\mathcal{P}(t_{\text{ge}+}^{\text{(s)}},t_{\text{ge}-}^{\text{(s)}},t_{\text{ge}+}^{\text{(e)}},t_{\text{ge}-}^{\text{(e)}})\mathscr{D}_{4},$

where

$\displaystyle\mathcal{P}$ $\displaystyle=$ $\displaystyle
f(t_{\text{ge}+}^{\text{(s)}}|0,\sigma_{\text{t}})f(t_{\text{ge}-}^{\text{(s)}}|0,\sigma_{\text{t}})f(t_{\text{ge}+}^{\text{(e)}}|T,\sigma_{\text{t}})f(t_{\text{ge}-}^{\text{(e)}}|T,\sigma_{\text{t}}),$

and $\mathscr{D}_{4}$ indicates a four-dimensional integral given by
$dt_{\text{ge}+}^{\text{(s)}}dt_{\text{ge}-}^{\text{(s)}}dt_{\text{ge}+}^{\text{(e)}}dt_{\text{ge}-}^{\text{(e)}}$.
Each value of $\mathcal{F}$ in Eq. (33) is evaluated by integration of the
Schrödinger equation through the pumping process; this results in quite a long
time to evaluate the four-dimensional integration in Eq. (33). Since the
timing errors obey a normal distribution, the contribution from large timing
errors is negligible. For this reason, we approximate Eq. (33) by considering
eleven values $\in\\{\pm 5\sigma_{\text{t}},\pm 4\sigma_{\text{t}},\pm
3\sigma_{\text{t}},\pm 2\sigma_{\text{t}},\pm\sigma_{\text{t}},0\\}$ for the
integration over each of the four timing errors, so that

$\displaystyle\overline{\mathcal{F}}$ $\displaystyle\approx$
$\displaystyle\frac{\sum\mathcal{F}\mathcal{P}}{\sum\mathcal{P}},$ (34)

where ‘$\sum$’ sums over the $11^{4}$ sets of
$\\{t_{\text{ge}+}^{\text{(s)}},t_{\text{ge}-}^{\text{(s)}},t_{\text{ge}+}^{\text{(e)}},t_{\text{ge}-}^{\text{(e)}}\\}$.

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