# Implementation of electromagnetic analogy to gravity mediated entanglement

Ji Bian  School of Physics and Astronomy, Sun Yat-sen University, Zhuhai
519082, China  Center of Quantum Information Technology, Shenzhen Research
Institute of Sun Yat-Sen University, Shenzhen 518087, China  Teng Liu  School
of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China
Pengfei Lu  School of Physics and Astronomy, Sun Yat-sen University, Zhuhai
519082, China  Center of Quantum Information Technology, Shenzhen Research
Institute of Sun Yat-Sen University, Shenzhen 518087, China  Qifeng Lao
School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China
Xinxin Rao  School of Physics and Astronomy, Sun Yat-sen University, Zhuhai
519082, China  Feng Zhu  School of Physics and Astronomy, Sun Yat-sen
University, Zhuhai 519082, China  Center of Quantum Information Technology,
Shenzhen Research Institute of Sun Yat-Sen University, Shenzhen 518087, China
Yang Liu  School of Physics and Astronomy, Sun Yat-sen University, Zhuhai
519082, China  Center of Quantum Information Technology, Shenzhen Research
Institute of Sun Yat-Sen University, Shenzhen 518087, China  Le Luo
<EMAIL_ADDRESS>School of Physics and Astronomy, Sun Yat-sen
University, Zhuhai 519082, China  Center of Quantum Information Technology,
Shenzhen Research Institute of Sun Yat-Sen University, Shenzhen 518087, China
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University, Guangzhou 510275, China  International Quantum Academy, Shenzhen,
518048, China

###### Abstract

Recently, experiments aimed at measuring gravity mediated entanglement (GME)
using quantum information techniques have been proposed, based on the
assumption that if two systems get entangled through local interactions with
gravitational field, then this field must be quantum. While there is a debate
about what could be drawn from GME, quantum simulation might provide some
clarification. Here, we present electromagnetic analogy of GME using magnetic-
field mediated interaction between the electron and nucleus in a single atom.
Our work successfully implements the general procedures of GME experiments and
confirms that the mediating field does not support the mean-field description.
It also clarifies that, without considering the light-crossing time, the GME
experiment would not distinguish a quantum-field-theory description from a
quantum-controlled classical field one. Furthermore, this work provides a
novel method to construct two-qubit systems in a single atom, and providing
the first quantum simulation of GME using material qubits. It helps to
conceive the future GME experiments on the scale of light-crossing time.

##  I Introduction

Understanding gravity within the framework of quantum mechanics is one of the
great challenges of modern physics. Due to the lack of empirical evidence,
there is a debate on whether gravity is a quantum entity [1]. Recently,
experiments aimed at measuring quantum gravitational effects, for example, the
gravity mediated entanglement (GME) [2, 3] and the non-Gaussianity [4], in
table-top experiments using quantum information techniques have been proposed,
which might resolve this debate in the end. The GME proposals are based on a
subtle logic that, if two quantum systems (e.g., two masses that could be
prepared in spatial superposition) get entangled through local interactions
with a third system (i.e., a gravitational field), then this third system
itself must be quantum [2, 3]. Developments in quantum control of larger
masses and measurement of gravitational fields of smaller masses may soon
bring this important experimental test into practice [5, 6, 7]. At that time,
both a positive (the masses are entangled) and a negative result will convey
valuable information about quantum gravity [8]. However, there is also an on-
going debate about the precise conclusions that could be drawn from the
detection of GME [8, 9, 10, 11, 12, 13]. As suggested in Ref. [9], GME
supports the view that gravitational fields are sourced coherently by
superposition of masses, instead of a mean field description; However, it
could not tell whether gravity admits a quantum field description. Instead,
measuring on the light-crossing time between the two masses would truly reveal
quantum features of the gravitational interaction, as shown in Ref. [11].

Based on the detailed correspondence between electromagnetism and general
relativity [14, 15], conducting a GME-like experiment by analogies in
electromagnetism could bring additional insights to the above debates. In
addition, with the rapid development of quantum information science, quantum
simulation of the fundamental properties of spacetime has achieved fruitful
results [16, 17, 18]. Thus, studies of the GME experiment using gravito-
electromagnetic analogies [14, 19, 20] are highly attracting. In this article,
we propose and perform a quantum simulation experiment of the GME effect,
using magnetic-field mediated interaction between the electron and nucleus in
a single atom. While the magnetic field is the analog of gravitational field,
the spins of electron and nucleus play two roles from the GME experiment. The
first role is the position of each mass: different spin states result in
different magnetic fields, in analog to the case that different positional
combinations of the two masses create different gravitational fields. The
second role is the spins possessed by the masses which are used to detect
entanglement. We take quantum field theory (QFT) of electromagnetic field as a
premise, and provide a preview of the general experimental procedure of the
GME experiment. Observation of entanglement in the final spin state rules out
classical mean-field description of the mediating field. Moreover, our
experimental result confirms that the QFT description and the quantum-
controlled classical field one could not be distinguished by original GME
proposals, which stresses the necessity of measuring on the order of light-
crossing time [11]: demonstrating that entanglement due to interaction with
gravitational field is established between the two masses at a time smaller
than the light-crossing time (entanglement harvesting [21]) will help to
confirm the quantum nature of the mediating field (with an exception of the
yet to be developed gravitational absorber theory [9]). We also point out that
existing experiments on entanglement between remote trapped-ion qubits [22,
23, 24] could be modified to a simulation of such measurements in
electromagnetism.

The experiment utilizes the $d(>2)$-level system (qudit) in a single ion [27,
25, 26]. Moreover, since the four level structure is created by the hyperfine
interaction between the nucleus (spin-$1/2$) and the valence electron
(spin-$1/2$), the $4$-dimensional Hilbert space is naturally partitioned into
two $2$-dimensional subspace. This allows us speak about entanglement of two
spins even in a single ion, and makes the experiment easier, as otherwise two
trapped ions have to be used to simulate the GME experiment [28, 29]. This
work thus demonstrates the advantage that multilevel structure of a quantum
unit offers, i.e, simplification of the experimental setup [30, 31, 32], and
may shed new light on quantum simulation using trapped ions [33].

##  II Realization of the single-atom GME quantum-simulator

The GME proposal [3, 2] is illustrated in Fig.1(a) (see Appendix VI.1).
Consider two objects $a$ and $b$ with gravitational interaction, each has mass
$m$ and posses spin-$1/2$ , as shown in Fig.1(a). Denote the angular momentum
operators

$I_{x}=\frac{\hbar}{2}\begin{pmatrix}0&1\cr 1&0\end{pmatrix},\
I_{y}=\frac{\hbar}{2}\begin{pmatrix}0&-i\cr i&0\end{pmatrix},\
I_{z}=\frac{\hbar}{2}\begin{pmatrix}1&0\cr 0&-1\end{pmatrix},$

and eigenstates of $I_{z}$ to be $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$, with corresponding eigenvalues $\hbar/2$ and
$-\hbar/2$ (we take $\hbar=1$). Each mass could be prepared in spatial quantum
superposition of two places: up ($u$) and down ($d$), correlated with their
spin (e.g., via some Stern-Gerlach scheme). In this paper we presume the
interaction is not instantaneous and is mediated by some field. The derivation
of this scenario using a QFT-like framework of gravitational field is already
given in, e.g., Ref. [10, 9, 11]. Here as the light crossing time
$t_{c}\approx d_{0}/c$ is negligible, a quantum-controlled local classical
field derivation would give essentially the same result. In the following we
give a Hilbert space to the gravitational field merely to make easy comparison
with the electromagnetic simulation. Denote the state of the whole system as
$|\psi_{a}\rangle\otimes|\psi_{b}\rangle\otimes|g\rangle:=|\psi_{a}\rangle|\psi_{b}\rangle|g\rangle$,
where $|\psi_{a,b}\rangle$ is the spin state of $a,b$ and $|g\rangle$
represents the gravitational field. The initial state reads
$\frac{1}{2}(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)|g_{0}\rangle.$
After the gravitational interaction for time $\tau\gg t_{c}$, the state
becomes

$\displaystyle\frac{1}{2}(e^{-i\phi^{\prime}_{\uparrow\uparrow}}\left|\uparrow\right\rangle\left|\uparrow\right\rangle|g_{uu}\rangle+e^{-i\phi^{\prime}_{\uparrow\downarrow}}\left|\uparrow\right\rangle\left|\downarrow\right\rangle|g_{ud}\rangle$
(1)
$\displaystyle+e^{-i\phi^{\prime}_{\downarrow\uparrow}}\left|\downarrow\right\rangle\left|\uparrow\right\rangle|g_{du}\rangle+e^{-i\phi^{\prime}_{\downarrow\downarrow}}\left|\downarrow\right\rangle\left|\downarrow\right\rangle|g_{dd}\rangle),$

where
$\phi^{\prime}_{\uparrow\uparrow}=\Phi_{uu}\tau/\hbar,\phi^{\prime}_{\uparrow\downarrow}=\Phi_{ud}\tau/\hbar,\phi^{\prime}_{\downarrow\uparrow}=\Phi_{du}\tau/\hbar,\phi^{\prime}_{{\downarrow\downarrow}}=\Phi_{dd}\tau/\hbar,$
and $\Phi_{nj}=-Gm^{2}/d_{nj},$ $(n,j=u,d)$ is the gravitational potential
energy when $a$ is in position $n$ and $b$ is in position $j$,
$|g_{nj}\rangle$ is the corresponding gravitational field state. Finally, one
preserves the spin while disentangle the gravitational field, this could be
done by coherently bringing $a$ and $b$ back to the middle position in all
branches and factor out $|g_{0}\rangle$. Neglecting the time it takes for this
procedure and omitting a global phase, one arrives at the final state (See
Appendix VI.1)

$\frac{1}{2}(e^{-i\phi_{\uparrow\uparrow}}\left|\uparrow\right\rangle\left|\uparrow\right\rangle+\left|\uparrow\right\rangle\left|\downarrow\right\rangle+\left|\downarrow\right\rangle\left|\uparrow\right\rangle+e^{-i\phi_{\uparrow\uparrow}}\left|\downarrow\right\rangle\left|\downarrow\right\rangle)|g_{0}\rangle,$
(2)

where $\phi_{\uparrow\uparrow}=(\Phi_{uu}-\Phi_{ud})\tau/\hbar$. As long as
$e^{-2i\phi_{\uparrow\uparrow}}\neq 1$, $a$ and $b$ are entangled [34]. To
detect entanglement, one could utilize Bell inequalities, entanglement
witness, or quantum state tomography as explained later. The observation of
entanglement will support the view that gravitational fields are sourced
coherently by the superposition of sources, and rule out classical mean field
descriptions. In the following, we present two intuitive analogies of GME in
electromagnetism, one involves two electric charges with electric interaction,
the other involves two spins with magnetic interaction. While the first case
could be realized by two trapped-ion qubits, the latter is suitable for the
single-atom simulation implemented in this work.

First, we describe the case of two electric charges. Consider the same
situation as in Fig.1(a), and suppose the gravitational interaction and the
spin-spin interaction between $a$ and $b$ are too small and could be
neglected. But now they both posses electric charges $q$ and interact through
electric field. The derivation of this scenario using a QFT description of
electromagnetic field is already given in, e.g., Ref. [10, 9]. Here as the
light crossing time is negligible, a quantum-controlled classical field
derivation [11] would give a result with negligible difference. Due to the
same reason, a non-relativistic (instantaneous interaction) calculation is
further applied merely as a convenient approximation. The state of the whole
system before final measurement is then

$\frac{1}{2}(e^{-i\phi_{\uparrow\uparrow}}\left|\uparrow\right\rangle\left|\uparrow\right\rangle+\left|\uparrow\right\rangle\left|\downarrow\right\rangle+\left|\downarrow\right\rangle\left|\uparrow\right\rangle+e^{-i\phi_{\uparrow\uparrow}}\left|\downarrow\right\rangle\left|\downarrow\right\rangle)|E_{0}\rangle,$
(3)

where $|E_{0}\rangle$ represents the electric field when $a$ and $b$ are both
in the middle position. $\phi_{\uparrow\uparrow}$ has the same form as (2),
and $\Phi_{nj}(n,j=u,d)$ now represents electric potential energy. In this
example, the quantum nature (ability to stay in quantum superposition
correlated with the source) of electric field is necessary to generate the
final entanglement between $a$ and $b$. It is worth mentioning that two-qubit
entangling gates in trapped-ion quantum information processors essentially
utilize similar mechanism as given above [35, 36, 37, 38, 39, 40, 41, 42]. In
Sec.IV.2, we provide a reinterpretation of existing experiments [39] as a
simulation of GME.

Figure 1: (a) Configuration of the GME experiment considered in this work. The
left-right configuration in the original GME proposal [2] is modified to be
up-down here to simplify the simulation experiment, they are essentially the
same. Objects $a$ and $b$ could be put into spatial superposition of $u$ and
$d$ correlated with their spin states, and there is gravitational interaction
between them. (b) The s-electron (e) in the outermost shell and the nucleus
(n) in a Hydrogen-like atom, or in an alkaline-like ion. The electron spin and
the nuclear spin interacts through magnetic field and results in the
interaction Hamiltonian $H_{f}$. In the 171Yb+ ion used here, there is a
magnetic field $B_{0}$ in $z$ providing the quantization axis. The
corresponding four-level subspace is controlled by microwave fields. The total
population on $|1\rangle$, $|2\rangle$, and $|4\rangle$ is measured by
applying the $369.5$ nm detection laser beam and measuring fluorescence. (c)
The experimental pulse sequence in $f_{O}$ (see Sec.VI.3). After initializing
to $|3\rangle$ by optical pumping, we prepare $|\Psi_{0}\rangle$ by
sequentially applying microwave pulses. We then evolve the system by (9) and
arrive at $|\Psi_{f}\rangle$. Finally we apply tomographic pulses and readout
the population on $|3\rangle$ by fluorescence measurement. Here as an example
the tomographic pulses correspond to measuring Re($\langle
1|\rho_{f}|4\rangle$). The labels on pulses are explained in the main text.

Secondly, consider the scenario of two spins $a$ and $b$ in free space, they
interact with each other through the magnetic field they produce, i.e., the
magnetic dipole-dipole interaction [43, 44]. The derivation of this
interaction in QFT framework is given in Sec.VI.4 [45, 46]. When the light
crossing time is negligible, the interaction Hamiltonian is well approximated
by
$H_{dd}=\lambda[3(\mathbf{I}_{a}\cdot\mathbf{r})(\mathbf{I}_{b}\cdot\mathbf{r})-\mathbf{I}_{a}\cdot\mathbf{I}_{b}],$
where $\mathbf{I}_{a(b)}$ is angular momentum operator with components
$I_{a(b),k},(k=x,y,z)$, $\mathbf{r}$ is the unit vector in the direction of
the line joining the two spins, and $\lambda$ is a constant depending on the
distance between the spins and their gyromagnetic ratios (the tensor products
‘$\otimes$’ between operators are omitted) [43]. When there is a strong
magnetic field present in $z$, $H_{dd}$ could be further simplified to

$H_{zz}=\lambda I_{a,z}I_{b,z},$ (4)

which is called the secular approximation [43]. This form of interaction will
result in a final state similar to (2), and could be alternatively realized by
choosing some proper rotating reference frame, as explained later. Denote the
eigenvalues of $H_{zz}$ to be $S_{nj}$ ($n,j$ could be $\uparrow,\downarrow$),
with corresponding eigenstates $|n\rangle|j\rangle$. $S_{nj}$ equals the
magnetic potential energy when $a$ is in $|n\rangle$ and $b$ is in
$|j\rangle$. Simulating the GME experiment using magnetic field under $H_{zz}$
is slightly different from the electric field. This is because spin states
also determine magnetic field, unlike in GME or the electric field case, where
the gravitational field or the electric field directly depends on the position
of $a$ and $b$ only. Different spin states $|\psi_{a}\rangle|\psi_{b}\rangle$
could give same gravitational field (as long as the position configurations
are the same), while this is generally not true for the magnetic field case.
Consequently, the protocol using magnetic field is as follows.

Assume at the beginning, $a$ and $b$ are far apart (or magnetically shielded)
from each other such that the interaction between them could be ignored, and
their spin states are separable. First prepare each spin to be
$\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)$.
To see the time-evolved state, first note that starting from
$|n\rangle|j\rangle|M_{0}\rangle$, where $n,j=$ $\uparrow,\downarrow$ and
$|M_{0}\rangle$ is the initial magnetic field state, the total state of spin
and field at time $t$ under the QFT interaction Hamiltonian $H_{\textrm{F}}$
(containing the spin$+$field degrees of freedom, see Sec.VI.4) could be
written as
$\alpha(t)|n\rangle|j\rangle|M_{nj}(t)\rangle+\sum_{n^{\prime}j^{\prime}}\beta_{n^{\prime}j^{\prime}}(t)|n^{\prime}\rangle|j^{\prime}\rangle|M_{n^{\prime}j^{\prime}}(t)\rangle$,
where $n^{\prime}j^{\prime}$ are combinations different from $nj$,
$|\alpha(t)|^{2}+\sum_{n^{\prime}j^{\prime}}|\beta_{n^{\prime}j^{\prime}}|^{2}=1$
and $|M_{nj(n^{\prime}j^{\prime})}(t)\rangle$ is the corresponding magnetic
field state. As $H_{\textrm{F}}$ could be simplified to $H_{zz}$, we have
$|\alpha(t)|\approx 1$ and $|M_{nj}(t)\rangle\approx|M_{0}\rangle$, that is,
the field state remains unchanged approximately and the initial state is an
approximate eigenstate of $H_{\textrm{F}}$ hence only generates a phase during
the interaction period. On the other hand, the difference of
$|M_{nj}(t)\rangle$ for different $nj$, although tiny, is crucial in
generating the final entanglement of spins, as explained in Sec.VI.4. So we
write $|M_{nj}(t)\rangle=|M_{nj}\rangle$ as a constant state and set
$|\alpha(t)|=1$. Thus after letting the two spins interact through magnetic
field for time $\tau$, the final state is

$\displaystyle|\Psi_{f}\rangle=$
$\displaystyle\frac{1}{2}(e^{-i\phi_{\uparrow\uparrow}}\left|\uparrow\right\rangle\left|\uparrow\right\rangle|M_{\uparrow\uparrow}\rangle+\left|\uparrow\right\rangle\left|\downarrow\right\rangle|M_{\uparrow\downarrow}\rangle$
(5)
$\displaystyle+\left|\downarrow\right\rangle\left|\uparrow\right\rangle|M_{\downarrow\uparrow}\rangle+e^{-i\phi_{\uparrow\uparrow}}\left|\downarrow\right\rangle\left|\downarrow\right\rangle|M_{\downarrow\downarrow}\rangle),$

omitting the irrelevant global phase, where
$\phi_{\uparrow\uparrow}=(S_{\uparrow\uparrow}-S_{\uparrow\downarrow})\tau=\lambda\tau/2$.

Finally, we are going to measure and detect entanglement. We choose to perform
quantum state tomography, as it has the ability to conclude a definitive
negative experimental outcome (i.e., no entanglement), and is also more
straightforward to implement on our setup. Unlike the gravitational field
case, we are not able to disentangle the magnetic field state. However, as
explained in Sec.VI.2 and Sec.VI.4, the fact that $H_{F}$ could be simplified
to $H_{zz}$ indicates $|M_{nj}\rangle\approx|M_{n^{\prime}j^{\prime}}\rangle$
in (5). Denote $\rho_{f}$ the density matrix after tracing out the field
degrees of freedom in (5), we could thus apply conventional two-qubit state
tomography technique and calculate the entanglement of formation $W(\rho_{f})$
[47] to detect entanglement. This also suggests that if the field states in
different branches are made close to each other in future GME experiments,
then it might not be necessary to coherently bring back the masses (as this
might introduce additional experimental error). One could trace out the field
and detect entanglement between the spins.

We are going to simulate the GME experiment using an electron spin and a
nuclear spin in a single atom, as shown in Fig.1(b). The idea is essentially
the same as explained above. Consider the s-electron in the outermost shell
and the nucleus in a Hydrogen-like atom, or in an alkaline‐like ion, e.g., the
171Yb+ ion used here. The s-electron forms a spherically symmetric charge
distribution surrounding the nucleus. The nuclear spin feels the magnetic
field produced by the electron and vice versa. The system could be treated as
a uniformly magnetised sphere interacts with a spin at its center [48]. The
magnetic dipole-dipole interaction between two dipoles is already derived in
sec.VI.4, under QFT framework, and it could be well approximated by
$H_{\textrm{dd}}$ when the two dipoles are close to each other. So if we
neglect the light crossing time (thus a QFT or a quantum-controlled classical
field derivation could be conveniently approximated by a non-relativistic
quantum mechanics one), to find the approximation of the QFT description, we
could directly starting from $H_{\textrm{dd}}$. Then following e.g., Ref. [48]
we obtain the hyperfine interaction
$H_{h}=A(I_{a,x}I_{b,x}+I_{a,y}I_{b,y}+I_{a,z}I_{b,z}),$ where $A$ is the
hyperfine interaction constant, $a$ represents nucleus and $b$ electron. The
influence of the remaining symmetrical core of paired electrons could be
approximated as a correction term that contributes to the interaction constant
[49]. The ion is typically in an external magnetic field $B_{0}$ providing the
quantization axis (denote it $z$ axis), so the free evolution Hamiltonian is

$\displaystyle H_{f}$
$\displaystyle=A(I_{a,x}I_{b,x}+I_{a,y}I_{b,y}+I_{a,z}I_{b,z})$ (6)
$\displaystyle-B_{0}(\gamma_{a}I_{a,z}I+\gamma_{b}II_{b,z})$
$\displaystyle=E_{1}|1\rangle\langle 1|+E_{2}|2\rangle\langle
2|+E_{3}|3\rangle\langle 3|+E_{4}|4\rangle\langle 4|,$

where $\gamma_{a,b}$ are the gyromagnetic ratios, $I$ is the $2\times 2$
identity matrix. $|1\rangle$, $|2\rangle$, $|3\rangle$, $|4\rangle$ and
$\left|\uparrow\right\rangle\left|\uparrow\right\rangle$,
$\left|\uparrow\right\rangle\left|\downarrow\right\rangle$,
$\left|\downarrow\right\rangle\left|\uparrow\right\rangle$,
$\left|\downarrow\right\rangle\left|\downarrow\right\rangle$ are related by a
mapping operator $R$ as explained in sec.VI.3:

$\displaystyle|1\rangle=\left|\uparrow\right\rangle\left|\uparrow\right\rangle,\quad|2\rangle=\cos\frac{\theta}{2}\left|\uparrow\right\rangle\left|\downarrow\right\rangle-\sin\frac{\theta}{2}\left|\downarrow\right\rangle\left|\uparrow\right\rangle$
(7)
$\displaystyle|3\rangle=\sin\frac{\theta}{2}\left|\uparrow\right\rangle\left|\downarrow\right\rangle+\cos\frac{\theta}{2}\left|\downarrow\right\rangle\left|\uparrow\right\rangle,\quad|4\rangle=\left|\downarrow\right\rangle\left|\downarrow\right\rangle,$

where $\theta$ depends on $B_{0}$ ($\theta\approx-\frac{\pi}{2}$ here). In
171Yb+, (6) corresponds to the ${}^{2}S_{1/2}$ four-level subspace, as shown
in Fig.1(b). Although (6) is different from (4) in the lab frame, (6) seen
from a rotating reference frame $f_{O}$ will take the same form as (4): Define
a rotating frame $f_{O}$ with rotation operator $O$,
$|\psi\rangle_{R}=O|\psi\rangle_{L}$, where $|\psi\rangle_{L}$ is the state in
the lab frame and $|\psi\rangle_{R}$ in the rotating frame. The evolution
operators and Hamiltonian are also changed accordingly. The rotation operator
is

$\displaystyle O=$
$\displaystyle\textrm{exp}[i\int_{0}^{t^{\prime}}(\delta_{1}(t)|1\rangle\langle
1|+\delta_{2}(t)|2\rangle\langle 2|$ (8)
$\displaystyle+\delta_{3}(t)|3\rangle\langle 3|+\delta_{4}(t)|4\rangle\langle
4|)dt].$

Working in this frame makes control-sequence-designing easier and it does not
affect the population readout. In the interaction period $\tau$, we set
$\delta_{1,4}=E_{1,4}-\Delta$ and $\delta_{2,3}=E_{2,3}$. The evolution
operator in this period is then

$\displaystyle U(\Delta,\tau)$
$\displaystyle=\textrm{exp}[-i(OH_{f}O^{\dagger}+i\dot{O}O^{\dagger})\tau]$
(9) $\displaystyle=\textrm{exp}[-i(\Delta|1\rangle\langle
1|+\Delta|4\rangle\langle 4|)\tau]$
$\displaystyle=\textrm{exp}[-i(\frac{\Delta}{2}|1\rangle\langle
1|-\frac{\Delta}{2}|2\rangle\langle 2|$
$\displaystyle-\frac{\Delta}{2}|3\rangle\langle
3|+\frac{\Delta}{2}|4\rangle\langle
4|)\tau]\textrm{exp}[-i\frac{\Delta}{2}\tau]$
$\displaystyle=\textrm{exp}(-i2\Delta
I_{a,z}I_{b,z}\tau)\textrm{exp}(-i\frac{\Delta}{2}\tau).$

This is the desired evolution under the Hamiltonian of the form $H_{zz}$
(ignoring the irrelevant global phase $\textrm{exp}(-i\frac{\Delta}{2}\tau)$).
The fact that the last equality holds is crucial as this allows a
straightforward implementation of the experiment. We also set
$\delta_{1,2,3,4}=E_{1,2,3,4}$ during the initial state preparation and final
tomography steps. With appropriate choice of the control-pulse phases, in this
$f_{O}$, the interaction between spins is “tailored” to the form of $H_{zz}$
during the interaction period. The fact that (8) is itself an entangling
operation does not affect the validity of the simulator [See sec.VI.2 for
explanation].

To sum up, the protocol using a single ion is

1) First prepare the state

$\displaystyle|\Psi_{0}\rangle=$
$\displaystyle\frac{1}{2}(\left|\uparrow\right\rangle\left|\uparrow\right\rangle|M_{\uparrow\uparrow}\rangle+\left|\uparrow\right\rangle\left|\downarrow\right\rangle|M_{\uparrow\downarrow}\rangle$
(10)
$\displaystyle+\left|\downarrow\right\rangle\left|\uparrow\right\rangle|M_{\downarrow\uparrow}\rangle+\left|\downarrow\right\rangle\left|\downarrow\right\rangle|M_{\downarrow\downarrow}\rangle)$

in $f_{O}$.

2) Then let the system evolves for time $\tau$. In $f_{O}$, the evolution is
under the Hamiltonian $H=2\Delta I_{z}I_{z},$ where $\Delta$ could be varied
by adjusting $\delta_{1,2,3,4}$ in (8), and the final state

$\displaystyle|\Psi_{f}\rangle=$
$\displaystyle\frac{1}{2}(e^{-i\phi_{\uparrow\uparrow}}\left|\uparrow\right\rangle\left|\uparrow\right\rangle|M_{\uparrow\uparrow}\rangle+\left|\uparrow\right\rangle\left|\downarrow\right\rangle|M_{\uparrow\downarrow}\rangle$
(11)
$\displaystyle+\left|\downarrow\right\rangle\left|\uparrow\right\rangle|M_{\downarrow\uparrow}\rangle+e^{-i\phi_{\uparrow\uparrow}}\left|\downarrow\right\rangle\left|\downarrow\right\rangle|M_{\downarrow\downarrow}\rangle),$

where $\phi_{\uparrow\uparrow}=\Delta\tau$ is achieved.

3) Detect entanglement of the two spins in (11) after tracing out the field.

##  III Observation of the field mediated entanglement

The experiment is performed on the trapped 171Yb+ ion quantum information
processor, as shown in Fig.1(b). A magnetic field with strength $B_{0}$ along
$z$ provides the quantization axis. The 2S1/2 hyperfine energy levels result
from the hyperfine interaction between the ${}^{171}\textrm{Yb}$ nuclear spin
and the s-electron spin in the valence shell. A microwave field with tunable
frequency, amplitude and phase is used to control the spin states. The
population on $|3\rangle$, $P_{3}$ could be readout by first measuring the
total population $P_{1}+P_{2}+P_{4}$ on $|1\rangle$, $|2\rangle$, and
$|4\rangle$ through applying the $369.5$ nm detection beam and measuring the
fluorescence, then $P_{3}=1-(P_{1}+P_{2}+P_{4})$. The initial state of each
experiment is $|3\rangle$, which is dissipatively prepared by optical pumping
(note interestingly that $|3\rangle$ is actually already an entangled state).
Details of the experimental apparatus and parameters of the four-level
subspace are given in Sec.VI.3.

Figure 2: (a) Real parts of the quantum state tomography of
$\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=0,\tau=0)$ (lower panel) and the
corresponding $\rho_{\textrm{th}}$ (upper panel). The imaginary parts of
$\rho_{\textrm{th}}$, $\textrm{Im}(\rho_{\textrm{th}})$ are all zero,
$|\textrm{Im}(\rho_{\textrm{exp}})|$ are all smaller than $0.06$ and are not
shown. The fidelity $F\approx 0.97\pm 0.01$. $|i\rangle|j\rangle$ are shown as
$|ij\rangle$, ($i,j=$ $\uparrow$ or $\downarrow$) for simplicity. (b) Real and
imaginary parts of the quantum state tomography of
$\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=\pi/2,\tau=2$ $\mu$s), $F\approx
0.98\pm 0.01$

The experimental sequence is illustrated in Fig.1(c), in the following, the
states and operators are all in $f_{O}$:

1) After initializing the system to $|3\rangle$ through optical pumping,
prepare the state $|\Psi_{0}\rangle$ by sequentially applying microwave pulses
resonant with transitions $|3\rangle\leftrightarrow|4\rangle$,
$|1\rangle\leftrightarrow|3\rangle$, $|2\rangle\leftrightarrow|3\rangle$. Here
pulses in transitions $|n\rangle\leftrightarrow|m\rangle$ ($n<m$) with labels
$\pm y,\theta$ represent evolution operators $U_{p}=\textrm{exp}[\mp
i(-i|n\rangle\langle m|+i|m\rangle\langle n|)\theta]$. The desired evolution
could be realized by adjusting the pulse width, microwave field strength and
phase, as explained in Sec.VI.3, $\theta_{13,23}$ are also given there.

2) The interaction period. Evolve the system by $U(\Delta,\tau)$ (eq.(9)) for
$\tau$. To achieve this, recall that we are in $f_{O}$, we just need to let
the system evolve freely for $\tau$. For this to work, we have to apply an
extra $\phi=\Delta\tau$ phase shift for the following readout microwave
pulses. This is a common technique in quantum information processing
experiments to achieve operations under $H_{zz}$, e.g., when implementing
controlled-phase (CZ) gates [50, 51].

3) Finally, we apply quantum state tomography to obtain the full density
matrix of the two-spin system, and detect entanglement. As the direct
observable is $|3\rangle\langle 3|$, we convert the populations and coherence
to the $\\{|2\rangle,|3\rangle\\}$, $\\{|1\rangle,|3\rangle\\}$ , or
$\\{|3\rangle,|4\rangle\\}$ subspace by $\pi$ pulses, and then read them out
by standard qubit-tomography techniques. For example, in Fig.1(c) we show the
readout pulses obtaining Re($\langle 1|\rho_{f}|4\rangle$). Assume after this
sequence the population on $|3\rangle$ is measured to be $P^{\prime}_{3}$, and
we have already measured $P_{1}$ and $P_{4}$ (by $\pi$ pulses on
$|1,4\rangle\leftrightarrow|3\rangle$ and readout), then $\textrm{Re}(\langle
1|\rho_{f}|4\rangle)=(P_{1}+P_{4})/2-P^{\prime}_{3}$. After obtaining all
$\langle u|\rho_{f}|v\rangle$, ($u,v=$ $1$, $2$, $3$, $4$), we easily recover
$\langle nj|\rho_{f}|kl\rangle$, $(n,j,k,l=$ $\uparrow$ or $\downarrow$) by
applying the mapping operator $R$.

When $\tau=0$, the system is in the initial separable state. The theoretical
and experimental density matrix $\rho_{\textrm{th}}$ and $\rho_{\textrm{exp}}$
are shown in Fig.2(a). The state fidelity
$F=\left|\mathrm{Tr}(\rho_{\textrm{th}}\rho_{\textrm{exp}})\right|/\sqrt{\mathrm{Tr}(\rho_{\textrm{th}}\rho_{\textrm{th}})\mathrm{Tr}(\rho_{\textrm{exp}}\rho_{\textrm{exp}})}\approx
0.97\pm 0.01$. To further obtain the degree of entanglement, we calculate the
entanglement of formation $W(\rho)$ of $\rho$.
$W[\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=0)]=0$, confirming the
separability.

After an interaction period $\tau$, the two spins are generally entangled. For
example, we choose $\tau=2$ $\mu$s, $\Delta=\pi/(2\tau)=0.785$ MHz, and the
final state will be maximally entangled ($\phi_{\uparrow\uparrow}=\pi/2$).
This could serve to simulate a GME experiment with same
$\phi_{\uparrow\uparrow}$, e.g, $m=10^{-14}$ kg, $\tau=2.5$ s, $d_{uu}=200$
$\mu$m, $d_{ud}=280$ $\mu$m, which could be realized by future interferometers
using masses with embedded spins [2]. The experimental results are shown in
Fig.2(b), where $F\approx 0.98\pm 0.01$.
$W[\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=\pi/2)]=0.66\pm 0.06>0$, which
means the state is entangled. Deviation from the ideal value $1$ is mainly due
to pulse errors and decoherence in steps 1) and 3). We also vary $\Delta$ with
fixed $\tau=2$ $\mu$s, the results are, e.g.,
$W[\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=\pi/4)]=0.40\pm 0.02$ with
$F=0.98\pm 0.01$ ($W[\rho_{\textrm{th}}(\phi_{\uparrow\uparrow}=\pi/4)]=0.5$),
and $W[\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=\pi)]=0.02^{+0.03}_{-0.02}$
with $F=0.98\pm 0.01$
($W[\rho_{\textrm{th}}(\phi_{\uparrow\uparrow}=\pi)]=0$).

If a negative outcome is encountered in future GME experiments, it could
results from the following three possibilities: 1. Gravity does not have
standard quantum mechanical properties; 2. There are experimental noises such
as magnetic noise and gravitational noise that causes decoherence; 3. There
are other new mechanisms present, such as the spontaneous collapse mechanism
[52] that leads to a strong loss of coherence on the time scale of the
experiment. We observe the decoherence effect in the simulation experiment.
While in our case such an effect is due to the magnetic noise described in
possibility $2$, it also provides a mechanism to simulate possibility $3$. In
the $zz$ basis (defined in Sec.VI.3), denote the density matrix corresponds to
the spin part of (2)
$\rho_{f}(\tau)=1/4\sum_{n,j,k,l}\alpha_{njkl}|nj\rangle\langle kl|$, ignoring
the gravitational part as it is already factored out. Assume for simplicity
that the decoherence introduces a decay factor $e^{-\tau/\beta_{njkl}}$ to
each nondiagonal element, then
$\alpha_{njkl}=e^{-\tau/\beta_{njkl}}e^{i\varphi_{njkl}}$, where
$e^{i\varphi_{njkl}}$ are phases without decoherence. Denote the final state
of the simulator
$\rho^{s}_{f}(\tau)=1/4\sum_{n,j,k,l}\alpha^{s}_{njkl}|nj\rangle\langle kl|$,
also ignoring the magnetic field states. The simulator is also undergoing
decoherence process, as shown by the Ramsey $T^{*}_{2}$ measurement among
different transitions. Through the mapping operator $R$, the exponential decay
of the coherence $\langle b|\rho^{s}_{f}(\tau)|a\rangle$, ($a,b=$ $1$, $2$,
$3$, $4$) manifest themselves as the decay factors
$e^{-\tau/\beta^{s}_{njkl}}$ in $\alpha^{s}_{njkl}$. Thus
$\alpha^{s}_{njkl}=e^{-\tau/\beta^{s}_{njkl}}e^{i\varphi^{s}_{njkl}}$, where
$e^{i\varphi^{s}_{njkl}}$ are phases without decoherence. Clearly the
decoherence in GME could be simulated by the decoherence process in the
simulator. We fix $\phi_{\uparrow\uparrow}=\pi/2$, and vary $\tau$ from $0$ to
$400$ $\mu$s ($\Delta$ is varied accordingly). The resulting
$W[\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=\Delta\tau=\pi/2)]$ is
illustrated in Fig.3. When $\tau$ is small, $W$ is relatively large,
indicating an entangled state with high-degree of entanglement. The deviation
of $W$ from ideal value $1$ is due to the error and decoherence in the
preparation and tomography steps. As $\tau$ is increased, $W$ decreases, and
eventually approaches $0$. This demonstrates the importance of minimizing
decoherence due to technical noise in future GME experiments. It would also be
crucial to have a good characterization of the magnetic noise if one wishes to
further tell apart decoherence due to spontaneous collapse from decoherence
due to magnetic noise.

Figure 3: Decoherence of spins due to magnentic noise. The entanglement of
formation $W[\rho_{\textrm{exp}}(\phi_{\uparrow\uparrow}=\Delta\tau=\pi/2)]$
is plotted with varying $\tau$ and $\Delta$. The light red curve is added to
guide the eye.

##  IV Discussion

###  IV.1 Indistinguishability of the quantum-controlled classical field and
the QFT description

The result of entanglement detection in this work “rules out” mean-field
source for the electromagnetic potential [9]. As described in Sec.VI.4, the
result is in consistent with both the QFT description of electromagnetic field
and the quantum-controlled classical field description. So, the
distinguishability of the two scenarios is lacking in the current GME
experiments. To further tell these two descriptions apart, measuring on the
order of the light-crossing time in future GME experiments is required, that
is, demonstrating the two masses begin to get entangled due to gravitational
interaction within the light crossing time, or measuring tinier quantitative
effects around the light-crossing time [11]. It is interesting to point out
that existing experiments on entanglement between remote trapped-ion qubits
[22, 23, 24] could be modified to a demonstration of this effect in
electromagnetism. For example, in Ref.[22], starting from time $t=0$, two
171Yb+ atomic qubits ($|2,3\rangle$ form the qubit levels) remotely located
with light crossing time $t_{c}$ get entangled mediated by electromagnetic
field, i.e., the photons they produce. To achieve this, each atom first emit a
single photon to create an entangled state between the atomic qubit and the
photonic qubit (frequency qubit). The photons are then directed to a beam
splitter to perform two-photon-interference. Coincident photon detection at
time $t_{d}$ in the two output ports of the beam splitter projects the atomic
qubits to an entangled state. The desired observable of atomic qubits is then
measured by internal-state control and readout with a total duration $t_{r}$.
If the experimental setup could be modified such that $t_{d}+t_{r}<t_{c}$, one
would be able to show that the entanglement is established within the light
crossing time between the two atomic qubits, thus falsify the quantum-
controlled classical field description. The implication of this for GME is
worth further study.

###  IV.2 Two-ion entanglling gate as a simulation of GME

Electromagnetic field’s ability to stay in quantum superposition correlated
with the source is necessary for two-qubit entanglling gates based on Coulomb
interaction in trapped ions, e.g., the Cirac-Zoller gate [35], Mølmer-Sørensen
gate [36, 37], light-shift gate [38], and fast gates that are not confined
within the Lamb-Dicke regime [39]. Starting from
$1/2(\left|\uparrow\uparrow\right\rangle+\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle+\left|\downarrow\downarrow\right\rangle)$,
during these gates, entangled states between internal degrees of freedom
(spin) and the collective motional degrees of freedom (phonon) are generated.
After motional states in all branches (correspond to different collective spin
states) become the same and factored out, an entangled state of two spins is
generated. Relative phases among different branches are coherently accumulated
under the joint action of spin energy, motional energy, interaction energy
with the control field, and importantly, the electromagnetic field energy
caused by the electromagnetic interaction between two ions. If there is no
such interaction, the center of mass mode and relative motion mode will have
same mode frequency, then
$\phi_{\uparrow\uparrow}+\phi_{\downarrow\downarrow}=\phi_{\uparrow\downarrow}+\phi_{\downarrow\uparrow}$,
hence no entanglement between the two spins will be generated. If the
electromagnetic field is not quantum, instead it admits a mean-field
description and takes the same state in all branches, then the situation is
the same as if there is no electromagnetic interaction between the two ions
(plus a global electric field which only creates a global phase), hence no
entanglement is generated either.

Take the experiment in Ref.[39] as an example, two trapped 171Yb+ ions
($|2,3\rangle$ form the qubit levels) get entangled through mutual electrical
interaction and interaction with laser control field. One basic operation of
the experiment is the “spin-dependent-kick” (SDK) that creates entangled
states of spin and motion of each ion, which is similar to Stern-Gerlach
operation. The SDK evolution operator for the two-qubit system in the rotating
frame is
$U_{\textrm{SDK}}=e^{2i\phi(t)}\sigma_{1+}\sigma_{2+}D_{\textrm{C}}(i\eta_{\textrm{C}})+e^{-2i\phi(t)}\sigma_{1-}\sigma_{2-}D_{\textrm{C}}(-i\eta_{\textrm{C}})+\sigma_{1+}\sigma_{2-}D_{\textrm{R}}(i\eta_{\textrm{R}})+\sigma_{1-}\sigma_{2+}D_{\textrm{R}}(-i\eta_{\textrm{R}})$,
where
$\sigma_{1,2+(-)}=\left|\downarrow\right\rangle\left\langle\uparrow\right|(\left|\uparrow\right\rangle\left\langle\downarrow\right|)$
for spin $1$ and $2$, $\phi(t)$ and $\Delta k$ is related to the laser control
field, $x_{1,2}$ are position operators for ion $1$ and $2$,
$D_{\textrm{C},\textrm{R}}(i\eta_{\textrm{C},\textrm{R}})$ are coherent state
displacement operators for the center of mass mode and relative motion mode,
and $\eta_{C,R}$ are the corresponding Lamb-Dicke parameters. Starting from
$1/2(\left|\uparrow\uparrow\right\rangle+\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle+\left|\downarrow\downarrow\right\rangle)$,
combination of several $U_{\textrm{SDK}}$ and free evolution disentangles the
motional state and generates phase difference among different branches. The
final entangled state is
$|\Psi_{e}\rangle=e^{i\Phi_{g}}/2(e^{i\gamma}\left|\downarrow\right\rangle\left|\downarrow\right\rangle+e^{-i\gamma}\left|\uparrow\right\rangle\left|\uparrow\right\rangle)-1/2(\left|\uparrow\right\rangle\left|\downarrow\right\rangle+\left|\downarrow\right\rangle\left|\uparrow\right\rangle)$,
where $\Phi_{g}=\pi/2.06$ and $\gamma$ is a constant in each experiment. The
difference of electromagnetic field in different branches is crucial to
generate $\Phi_{g}\neq n\pi$ ($n$ is an integer), i.e., entanglement. As a
simulation of GME, the electromagnetic field plays the role of gravitational
field. Different motional states of the two ions play the role of different
positional combination of the masses. The combination of $U_{\textrm{SDK}}$
and free evolution is the anolog of the interaction period (for $\tau$) plus
“bringing back to the middle position” in GME.

##  V Conclusion

To conclude, by taking the QFT of electromagnetic field as a premise, this
work propose and perform a quantum simulation experiment of the GME effect. It
could provide a preview of the general experimental procedure of the future
GME experiment. Observing that the experimental result could not distinguish
between a QFT description from a quantum-controlled classical field one, this
work helps to stress the benefits of measuring on the order of the light-
crossing time. It also points out that existing experiments on entanglement
between remote trapped-ion qubits [22, 23, 24] could be modified to
demonstrate such measurements in electromagnetism.

The experimental system consists of two spins and the magnetic field they
create which mediates their interaction, in a single atom. The use of the
naturally-existing multilevel hyperfine structure in a single ion reduces the
experimental cost of simulating the GME experiment. The quantum control
techniques presented here could be further developed to achieve experimental
complete-control of qudit systems in 171Yb+. This work is also among the first
to abstract a naturally-existing two-spin system from a single
${}^{171}\textrm{Yb}^{+}$ ion [31] and perform two-qubit simulation
experiments upon it, which may shed new light on quantum simulation using
trapped ions [33].

##  VI Appendix

###  VI.1 Brief description of GME.

Consider two objects $a$ and $b$, each has mass $m$ and posses spin-$1/2$ , as
shown in Fig.1(a). The radius of the masses are neglected,
$d_{uu}=d_{dd}=d_{0}$, $d_{ud}=d_{du}>d_{0}$ and the spin-spin interaction
between $a$ and $b$ is so weak that could be neglected. Initially one prepares
the spin state of each mass to be
$\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)$
and hold the position of $a$ and $b$ to be in the middle of $u$ and $d$.
Denote the state of the gravitational field corresponds to this configuration
$\left|g_{0}\right\rangle$, then the total system becomes
$\frac{1}{2}(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)|g_{0}\rangle.$
Next, one releases $a$ and $b$. If there is only one mass, the mass would be
in spatial superposition of $u$ and $d$ (neglecting the time it takes to
develop this superposition). As explained in Sec.II, after the gravitational
interaction for time $\tau$ the state becomes (1). Each component of (1) is
approximately an eigenstate of the total Hamiltonian hence only generates a
phase during the interaction period [8]. The phase acquired by the internal
(spin) energy $\hbar\omega_{nj}$ $(n,j=$ $\uparrow\textrm{or}\downarrow)$ is
ignored, as one could compensate this by going to the rotating frame. As
$\Phi_{uu}=\Phi_{dd}$, $\Phi_{du}=\Phi_{ud}$, we could subtract an irrelevant
global phase factor and define the new relative phases before each branch as
$\phi_{\uparrow\uparrow}=(\Phi_{uu}-\Phi_{ud})\tau/\hbar,\phi_{\uparrow\downarrow}=0,\phi_{\downarrow\uparrow}=0,\phi_{\downarrow\downarrow}=(\Phi_{uu}-\Phi_{ud})\tau/\hbar.$
Thus one obtains the final state (2). In the full language of general
relativity, there are higher-order corrections to the phases. For example,
consider the leading correction term to $\phi^{\prime}_{\uparrow\uparrow}$,
$\tau/\hbar
E_{\textrm{m}}=\tau/\hbar[\Phi_{uu}\hbar(\omega_{\uparrow}+\omega_{\uparrow})/(mc^{2})]$,
where $E_{\textrm{m}}$ could be intuitively think of as originated from the
internal energy contribution to the weight of the system and thus also couples
to gravity [53]. These terms are neglected in the original GME proposals, as
$\hbar(\omega_{n}+\omega_{j})/(mc^{2})\ll 1$ for typical experimental
parameters. Interestingly, it is argued that measuring these terms would allow
one take a step forward from concluding the superposition of gravitational
fields to concluding the superposition of spacetime [13, 8], as these terms
correspond to the gravitational time dilation (red shift) and could be
measured by quantum clocks [54, 55].

###  VI.2 Details of the single-atom GME simulator

We first explain entanglement detection in (5). Unlike the gravitational field
case, one is not able to disentangle the magnetic field state. Nevertheless,
as is derived in Sec.VI.4, the QFT Hamiltonian $H_{F}$ is well approximated by
$H_{dd}$, this means the electromagnetic field state is almost unchanged, as
$H_{dd}$ only affects the spins, not the electromagnetic field. Starting from
$H_{dd}$, one obtains $H_{h}$, and hence $H_{zz}$ in the simulator. This
results in
$|M_{\uparrow\uparrow}\rangle\approx|M_{\uparrow\downarrow}\rangle\approx|M_{\downarrow\uparrow}\rangle\approx|M_{\downarrow\downarrow}\rangle\approx|M_{0}\rangle$.
So tracing out the field state in (11), one obtains a density matrix that is
negligibly different from the density matrix after disentangling the field.
Note however that the tiny difference between field states in different
branches is required to generate different phases, hence the final
entanglement, as explained in Sec.VI.4. To further obtain the degree of
entanglement, for a two-qubit state $\rho$ (pure or mixed), one could
calculate its entanglement of formation $W(\rho)$ [47]. A larger $W$ indicates
a higher degree of entanglement: $W=0$ indicates a separable state while $W=1$
a maximally entangled state.

We then explain the rotating operator (8). Observing that (8) is an entangling
operation might raises concern: whether the entanglement of the spins is due
to the quantum properties of the magnetic field, or is merely an artifact due
to the rotating frame transformation. This is not a problem, as already
mentioned that the logic of this work is to take the quantum nature of
electromagnetic fields as premise, and conduct a simulation experiment that
might bring insights to the GME experiment. Transforming to the rotating frame
merely makes the simulation more close to the original GME experiment, i.e.,
their final entangled states have the same form. We could have stayed in the
lab frame and utilize the non-separable evolution under $H_{f}$, to achieve an
entangled spin state, but the form of the entangled state will not be the same
as (5). The introduction of $f_{O}$ merely “tailored” the naturally-existing
entangling-interaction between spins to the form of $H_{zz}$ during the
interaction period.

###  VI.3 Experimental setup

The magnetic field $B_{0}$ defining the quantization axis ($z$ axis) is
created by the electric current in a Helmholtz coil. The electric current is
produced by a dc power supply, and is stabilized at $1.20152\pm 0.00002$ A by
a proportional-integral (PI) controller. This allows us determine the
resonance frequencies of all three transitions (by Ramsey curves) within $3$
kHz during data collection. Hence we obtain the frequencies of the on-
resonance microwaves to be applied (Table.1), and $B_{0}$ (used to design the
pulse sequence and calculate $R$) is calculated to be $5.615$ Gs by the
transition $|3\rangle\leftrightarrow|1\rangle$. From simulation results, the
final state fidelity corresponds to, e.g.,
$\rho(\phi_{\uparrow\uparrow}=\pi/2,\tau=2$ $\mu$s) is still above $0.97$ when
there is $3$ kHz mismatch in all three resonance frequencies
($|3\rangle\leftrightarrow|4\rangle$, $|3\rangle\leftrightarrow|1\rangle$,
$|3\rangle\leftrightarrow|2\rangle$), plus $2$ mGs miscalibration (corresponds
to the $3$ kHz mismatch) in the magnetic field. This allows us achieve high-
fidelity experimental results.

Figure 4: (a) Rabi curve of $|3\rangle\leftrightarrow|4\rangle$.
$\Omega/(2\pi)$ is fitted to be $55.2$ kHz. (b) Ramsey measurement of
$T^{*}_{2}$ of $|2\rangle\leftrightarrow|4\rangle$. $T^{*}_{2}$ is fitted to
be $440.7$ $\mu$s.

The frequencies of the applied on-resonance microwaves
$\omega=\omega_{0}+\omega^{\prime}$ ($\omega_{0}=2\pi\times 12.61157173$ GHz),
amplitude of Rabi frequencies $|\Omega|$ and coherence times $T^{*}_{2}$ of
the relevant transitions within the four-level subspace (Fig.1(b)) are given
in Table.1. As an example, the Rabi curve of
$|3\rangle\leftrightarrow|4\rangle$ is shown in Fig.4(a), and the Ramsey
measurement of $T^{*}_{2}$ of $|2\rangle\leftrightarrow|4\rangle$ is shown in
Fig.4(b). The $T^{*}_{2}$ between different Zeeman sublevels could be easily
upgraded to above $1$ ms using triggering to ac-line techniques [56].

Details of the quantum-control technique for the four-level/two-qubit system
is given below. We first introduce a mapping operator that simplifies the
discussion. Define the “zz basis” where
$\left|\uparrow\right\rangle\left|\uparrow\right\rangle_{\textrm{zz}}=[1,0,0,0]^{\textrm{T}}$,
$\left|\uparrow\right\rangle\left|\downarrow\right\rangle_{\textrm{zz}}=[0,1,0,0]^{\textrm{T}}$,
$\left|\downarrow\right\rangle\left|\uparrow\right\rangle_{\textrm{zz}}=[0,0,1,0]^{\textrm{T}}$,
$\left|\downarrow\right\rangle\left|\downarrow\right\rangle_{\textrm{zz}}=[0,0,0,1]^{\textrm{T}}$,
the subscript indicates zz basis. Also define the “number basis” where
$|1\rangle_{\textrm{n}}=[1,0,0,0]^{\textrm{T}}$,
$|2\rangle_{\textrm{n}}=[0,1,0,0]^{\textrm{T}}$,
$|3\rangle_{\textrm{n}}=[0,0,1,0]^{\textrm{T}}$,
$|4\rangle_{\textrm{n}}=[0,0,0,1]^{\textrm{T}}$, the subscript “n” indicates
number basis. Define a mapping operator $R$ (depends on $B_{0}$) satisfying

$\displaystyle
R|k\rangle_{\textrm{n}}=|k\rangle_{\textrm{zz}}\quad(k=1,2,3,4),$ (12)

i.e., it maps the matrix form of an operator in number basis to that in zz
basis, and vice versa. The explicit form of $R$ reads

$R=\begin{pmatrix}1&0&0&0\cr 0&\cos(\theta/2)&\sin(\theta/2)&0\cr
0&-\sin(\theta/2)&\cos(\theta/2)&0\cr 0&0&0&1\end{pmatrix},$ (13)

where $\theta=2\textrm{tan}^{-1}(\lambda)$ and

$\lambda=\frac{-B_{0}\gamma_{a}+B_{0}\gamma_{b}-\sqrt{A^{2}+B^{2}_{0}\gamma^{2}_{a}+B^{2}_{0}\gamma^{2}_{b}-2B^{2}_{0}\gamma_{a}\gamma_{b}}}{A}.$
(14)

When $A\gg B_{0}\gamma_{a,b}$, as in our case, $\theta\approx-\pi/2$.
Nevertheless we still use the explicit form of (13). During step 1) in the
experiment, the Hamiltonian in $f_{O}$ after rotating wave approximation (RWA)
reads

$H_{c}=c_{13}(t)|1\rangle\langle 3|+c_{23}(t)|2\rangle\langle
3|+c_{34}(t)|3\rangle\langle 4|+\textrm{H.c.},$ (15)

where $c_{13,23,34}$ are complex numbers determined by the strength and phase
of the on-resonance microwave control fields. We tune the microwave field
parameters to realize $U_{p}$ in step $1$ of the experiment.
$\theta_{13}=2\sin^{-1}(1/\sqrt{3})$, and

$\displaystyle\theta_{23}=$ $\displaystyle
2\cos^{-1}\\{\frac{-\sqrt{2}/2[1-\tan^{-1}(\theta/2)]}{\cos(\theta/2)\tan^{-1}(\theta/2)+\sin(\theta/2)}\\}.$
(16)

Step 2) is realized by free evolution plus corresponding phase correction in
the following microwave readout pulses, as explained in Sec.III.

With $H_{f}$ and the available control fields, the four-level system is
completely controllable, as could be proved following Ref.[57]. This means
arbitrary $\textrm{SU}(4)$ quantum operations could be achieved in this
subspace. In this work, we only pick out several special pieces from the
complete control-toolbox, namely, a state-to-state operation achieving (10),
$\pi$ and $\pi/2$ rotations among the available transitions, and the
entangling operation (9). It is promising to demonstrate complete
controllability in this system by realizing a universal gate set, i.e.,
arbitrary single-spin rotations and CNOT gates via, e.g., numerical optimal
control [58]. Or in the language of qudit, realizing arbitrary single-qudit
rotation gates [27, 25]. The fidelities of the controls and the coherence time
could be further upgraded by magnetic shielding [56].

Table 1: Parameters of the four-level subspace. |  $\omega^{\prime}/(2\pi)$ $(\textrm{MHz})\;$ |  $|\Omega|/(2\pi)$ $(\textrm{kHz})\;$ |  $T^{*}_{2}$ $(\mu s)$ |
---|---|---|---|---
$|1\rangle\leftrightarrow|3\rangle$ | $39.1096$ | $49.6$ | $404.3$ |
$|2\rangle\leftrightarrow|3\rangle$ | $31.2500$ | $107.8$ | $>5\times 10^{5}$ |
$|3\rangle\leftrightarrow|4\rangle$ | $23.3814$ | $55.2$ | $403.9$ |
$|1\rangle\leftrightarrow|2\rangle$ | / | / | $410.9$ |
$|2\rangle\leftrightarrow|4\rangle$ | / | / | $440.7$ |
$|1\rangle\leftrightarrow|4\rangle$ | / | / | $138.5$ |

###  VI.4 Magnetic dipole-dipole interaction mediated by electromagnetic field

Following Ref. [45, 46], when the two magnetic dipoles (spin-$1/2$)
$\hat{m}_{i}=\sum_{x,y}\vec{m}^{xy}_{i}\hat{\tau}^{xy}_{i}$
($x,y=\uparrow\textrm{or}\downarrow$, $i=1,2$), with the dipole moment
amplitude $\vec{m}^{xy}_{i}:=\langle x|\hat{m}_{i}|y\rangle_{i}$, and
$\hat{\tau}^{xy}_{i}:=|x\rangle_{i}\langle y|$, are placed in the EM field,
they change the original field dynamics. This influence generates the
interaction between the two dipoles. In the Heisenberg picture, the quantized
magnetic field in some volume $V$ is

$\hat{\mathbf{B}}(\mathbf{x},t)=\sum_{\mathbf{k}\sigma}i\hat{e}_{\mathbf{k}\tilde{\sigma}}Z_{\mathbf{k}}[\hat{a}_{\mathbf{k}\sigma}(t)e^{i\mathbf{k}\cdot\mathbf{x}}-\textrm{H.c.}],$
(17)

where $Z_{k}=\sqrt{\mu_{0}\omega_{\mathbf{k}}/(2V)}$,
$\hat{e}_{\mathbf{k}\tilde{\sigma}}:=\hat{e}_{\mathbf{k}}\times\hat{e}_{\mathbf{k}\sigma}$
are polarization directions. It interacts with the dipole through the
interaction Hamiltonian $-\hat{m}_{i}\cdot\hat{\mathbf{B}}(\mathbf{x},t)$. The
part of the field resulting from the presence of dipole $1$ reads

$\displaystyle\hat{\mathbf{B}}_{D1}=$
$\displaystyle\sum_{\mathbf{k}\sigma,xy}\hat{e}_{\mathbf{k}\tilde{\sigma}}Z_{\mathbf{k}}(g^{xy}_{1,\mathbf{k}\sigma})^{*}$
(18) $\displaystyle\times
e^{i\mathbf{k}\cdot\mathbf{x}}\int_{0}^{t}dse^{-i\omega_{\mathbf{k}}s}\hat{\tau}^{yx}_{1}(t-s)+\textrm{H.c.},$

where
$g^{xy}_{i,\mathbf{k}\sigma}=-i(\vec{m}^{xy}_{i}\cdot\hat{e}_{\mathbf{k}\tilde{\sigma}})Z_{k}e^{i\mathbf{k}\cdot\mathbf{x}_{i}}$.
Note $\hat{\tau}^{yx}_{1}(t-s)$ are operators in Heisenberg picture, they
evolve in time (through the Heisenberg equation) and in general act non-
trivially on both spin and field states. Its interaction with dipole $2$
results in

$\displaystyle H_{1\rightarrow 2}$
$\displaystyle=-\hat{m}_{2}(t)\cdot\hat{\mathbf{B}}_{D1}(\mathbf{x}_{2},t)$
(19) $\displaystyle=\sum_{uv,xy}\int_{0}^{t}dsD^{yx,uv}_{1\rightarrow
2}(s)\hat{\tau}^{yx}_{1}(t-s)\cdot\hat{\tau}^{uv}_{2}(t)+\textrm{H.c.},$

where

$\displaystyle D^{yx,uv}_{1\rightarrow
2}(s):=\sum_{\mathbf{k}\sigma}-i(g^{xy}_{1,\mathbf{k}\sigma})^{*}g^{uv}_{2,\mathbf{k}\sigma}e^{-i\omega_{\mathbf{k}}s}$
(20)
$\displaystyle:=-i\int_{0}^{\infty}\frac{d\omega}{2\pi}J^{yx,uv}_{1\rightarrow
2}(\omega)e^{-i\omega s},$

and

$\displaystyle J^{yx,uv}_{1\rightarrow
2}(\omega):=2\pi\sum_{\mathbf{k}\sigma}(g^{xy}_{1,\mathbf{k}\sigma})^{*}g^{uv}_{2,\mathbf{k}\sigma}\delta(\omega-\omega_{\mathbf{k}}).$
(21)

For free-space EM field,

$\displaystyle J^{yx,uv}_{1\rightarrow 2}(\omega)=$
$\displaystyle\frac{\mu_{0}}{2\pi
r^{3}}\\{\vec{m}^{yx}_{1}\cdot\vec{m}^{uv}_{2}[\eta^{2}\sin\eta+\eta\cos\eta-\sin\eta]$
(22)
$\displaystyle-3(\vec{m}^{yx}_{1}\cdot\hat{e}_{\mathbf{r}})(\vec{m}^{uv}_{2}\cdot\hat{e}_{\mathbf{r}})$
$\displaystyle\times[\frac{1}{3}\eta^{2}\sin{\eta}+\eta\cos{\eta}-\sin{\eta}]\\},$

where $\eta:=kr=\omega r/c$, and $r=|\mathbf{x}_{1}-\mathbf{x}_{2}|$ is the
distance between the two dipoles. The interaction (19) has a retarded form,
and $D^{yx,uv}_{1\rightarrow 2}(s)$ characterizes the propagation of the
interaction. One could similarly obtain $H_{2\rightarrow 1}$, hence the QFT-
interaction Hamiltonian $H_{\textrm{F}}=1/2(H_{1\rightarrow 2}+H_{2\rightarrow
1})$. From the above derivation, we see that $H_{\textrm{F}}$ in general
entangles the spins with field. The field is perturbed and deviates from the
initial state during the interaction, as could be seen from (18).

Because $D^{yx,uv}_{1\rightarrow 2}(s)$ approaches zero rapidly with
increasing $s$ [46], the interaction is approximately local in time. As
$D^{yx,uv}_{1\rightarrow 2}(s)$ is of the second order of
$g^{xy}_{i,\mathbf{k}\sigma}$, we could make the approximation

$\hat{\tau}^{yx}_{1}(t-s)\approx\hat{\tau}^{yx}_{1}(t)\cdot
e^{i\Omega^{yx}_{1}s},$ (23)

where $\Omega^{yx}_{1}:=E^{(y)}_{1}-E^{(x)}_{1}$ is the energy difference
between the two levels $|y,x\rangle_{1}$, hence the interaction becomes time-
local. This approximation means we could neglect the higher-order interaction
between the dipole and field. As $r/c$ is small, the time to build up the
dipole-dipole interaction is small compared to the interaction period in the
experiment. Thus we could extend the time integral to infinity (Markovian
approximation), the interaction reduces to a time-independent form

$H_{1\rightarrow 2}=\sum_{xy,uv}[K^{yx,uv}_{1\rightarrow
2}(\Omega^{yx}_{1})-\frac{i}{2}J^{yx,uv}_{1\rightarrow
2}(\Omega^{yx}_{1})]\hat{\tau}^{yx}_{1}\hat{\tau}^{uv}_{2},$ (24)

where

$\displaystyle K^{yx,uv}_{1\rightarrow 2}(\Omega)=\frac{\mu_{0}}{4\pi
r^{3}}\\{\vec{m}^{yx}_{1}\cdot\vec{m}^{uv}_{2}$ (25)
$\displaystyle[-\eta^{\prime
2}\cos\eta^{\prime}+\eta^{\prime}\sin\eta^{\prime}+\cos\eta^{\prime}]-3(\vec{m}^{yx}_{1}\cdot\hat{e}_{\mathbf{r}})(\vec{m}^{uv}_{2}\cdot\hat{e}_{\mathbf{r}})$
$\displaystyle\times[-\frac{1}{3}\eta^{2}\cos{\eta^{\prime}}+\eta^{\prime}\sin{\eta^{\prime}}+\cos{\eta^{\prime}}]\\},$

and $\eta^{\prime}=\Omega r/c$. $H_{2\rightarrow 1}$ could be similarly
simplified. The total interaction is then

$H^{\prime}_{F}=\sum_{xy,uv}(G^{(P)}_{yx,uv}+G^{(D)}_{yx,uv})\hat{\tau}^{yx}_{1}\hat{\tau}^{uv}_{2},$
(26)

where

$G^{(P)}_{yx,uv}=\frac{1}{2}[K^{yx,uv}_{1\rightarrow
2}(\Omega^{yx}_{1})+K^{uv,yx}_{2\rightarrow 1}(\Omega^{uv}_{2})],$ (27)

and

$G^{(D)}_{yx,uv}=\frac{1}{4i}[J^{yx,uv}_{1\rightarrow
2}(\Omega^{yx}_{1})+J^{uv,yx}_{2\rightarrow 1}(\Omega^{uv}_{2})].$ (28)

From the simplified (26), we see that the field state is almost unperturbed.
In our experiment, $\Omega^{yx}_{1,2}r/c\rightarrow 0$, then all
$G^{(D)}_{xy,uv}$ approach zero, and all $G^{(P)}_{yx,uv}$ approach
$K^{yx,uv}_{1\rightarrow 2}(0)$. Thus

$\displaystyle\hat{H}_{1\leftrightarrow 2}$ $\displaystyle=\frac{\mu_{0}}{4\pi
r^{3}}[\hat{m}_{1}\cdot\hat{m}_{2}-3(\hat{m}_{1}\cdot\hat{e}_{\mathbf{r}})(\hat{m}_{2}\cdot\hat{e}_{\mathbf{r}})],$
(29)

which could be written in the form
$\lambda[3(\mathbf{I}_{1}\cdot\mathbf{r})(\mathbf{I}_{2}\cdot\mathbf{r})-\mathbf{I}_{1}\cdot\mathbf{I}_{2}]$,
this is just the approximation $H_{dd}$ used in the main text. From $H_{dd}$
one further obtains $H_{zz}$. The initial state
$1/2(\left|\uparrow\uparrow\right\rangle+\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle+\left|\downarrow\downarrow\right\rangle)$
evolves to
$1/2(e^{-i\phi_{\uparrow\uparrow}}\left|\uparrow\uparrow\right\rangle+\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle+e^{-i\phi_{\downarrow\downarrow}}\left|\downarrow\downarrow\right\rangle$
under $H_{zz}$. This could be described as resulting from the phase
accumulation in different branches by a quantum-controlled classical field,
without giving a Hilbert space to the field. The accumulated phases in each
branch could be calculated by classical magnetic dipole interaction. Thus we
see that due to $r/c$ being small, the QFT description and the quantum-
controlled classical field one is indistinguishable.

## Acknowledgement

This work is supported by the Key-Area Research and Development Program of
Guang Dong Province under Grant No.2019B030330001, the National Natural
Science Foundation of China under Grant No.11774436, No.11974434 and No.
12074439, Natural Science Foundation of Guangdong Province under Grant
2020A1515011159, Science and Technology Program of Guangzhou, China
202102080380, the Central -leading-local Scientific and Technological
Development Foundation 2021Szvup172. Ji Bian receives support from China
Postdoctoral Science Foundation under Grant No.2021M703768. Le Luo
acknowledges the support from Guangdong Province Youth Talent Program under
Grant No.2017GC010656. The authors are thankful to Dr. Rongxin Miao and Dr.
Longlong Feng for their discussion.

## References

  * [1] Daniele Oriti.  Approaches to quantum gravity: Toward a new understanding of space, time and matter.  Cambridge University Press, 2009.
  * [2] Sougato Bose, Anupam Mazumdar, Gavin W Morley, Hendrik Ulbricht, Marko Toroš, Mauro Paternostro, Andrew A Geraci, Peter F Barker, MS Kim, and Gerard Milburn.  Spin entanglement witness for quantum gravity.  Physical review letters, 119(24):240401, 2017.
  * [3] Chiara Marletto and Vlatko Vedral.  Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity.  Physical review letters, 119(24):240402, 2017.
  * [4] Richard Howl, Vlatko Vedral, Devang Naik, Marios Christodoulou, Carlo Rovelli, and Aditya Iyer.  Non-gaussianity as a signature of a quantum theory of gravity.  PRX Quantum, 2(1):010325, 2021.
  * [5] Lorenzo Magrini, Philipp Rosenzweig, Constanze Bach, Andreas Deutschmann-Olek, Sebastian G Hofer, Sungkun Hong, Nikolai Kiesel, Andreas Kugi, and Markus Aspelmeyer.  Real-time optimal quantum control of mechanical motion at room temperature.  Nature, 595(7867):373–377, 2021.
  * [6] Tobias Westphal, Hans Hepach, Jeremias Pfaff, and Markus Aspelmeyer.  Measurement of gravitational coupling between millimetre-sized masses.  Nature, 591(7849):225–228, 2021.
  * [7] Daniel Carney, Philip CE Stamp, and Jacob M Taylor.  Tabletop experiments for quantum gravity: a user’s manual.  Classical and Quantum Gravity, 36(3):034001, 2019.
  * [8] Marios Christodoulou and Carlo Rovelli.  On the possibility of laboratory evidence for quantum superposition of geometries.  Physics Letters B, 792:64–68, 2019.
  * [9] Vasileios Fragkos, Michael Kopp, and Igor Pikovski.  On inference of quantization from gravitationally induced entanglement.  AVS Quantum Science, 4(4):045601, 2022.
  * [10] Marios Christodoulou, Andrea Di Biagio, Markus Aspelmeyer, Časlav Brukner, Carlo Rovelli, and Richard Howl.  Locally mediated entanglement in linearized quantum gravity.  Physical Review Letters, 130(10):100202, 2023.
  * [11] Eduardo Martín-Martínez and T Rick Perche.  What gravity mediated entanglement can really tell us about quantum gravity.  arXiv preprint arXiv:2208.09489, 2022.
  * [12] Viqar Husain, Irfan Javed, and Suprit Singh.  Dynamics and entanglement in quantum and quantum-classical systems: lessons for gravity.  Physical Review Letters, 129(11):111302, 2022.
  * [13] QISS Conference 2022.  https://www.youtube.com/watch?v=r_PeKe4toZ8.
  * [14] Roy Maartens and Bruce A Bassett.  Gravito-electromagnetism.  Classical and Quantum Gravity, 15(3):705, 1998.
  * [15] Yu-Qing Cui, Tian-Ming Zhao, Rong-Xin Miao, Jin-Dong Wang, and Huanyang Chen.  Schrodinger’s cat state of optical parallel universes.  arXiv preprint arXiv:2110.12438, 2021.
  * [16] Iulia M Georgescu, Sahel Ashhab, and Franco Nori.  Quantum simulation.  Reviews of Modern Physics, 86(1):153, 2014.
  * [17] Daniel Jafferis, Alexander Zlokapa, Joseph D Lykken, David K Kolchmeyer, Samantha I Davis, Nikolai Lauk, Hartmut Neven, and Maria Spiropulu.  Traversable wormhole dynamics on a quantum processor.  Nature, 612(7938):51–55, 2022.
  * [18] Run-Qiu Yang, Hui Liu, Shining Zhu, Le Luo, and Rong-Gen Cai.  Simulating quantum field theory in curved spacetime with quantum many-body systems.  Physical Review Research, 2(2):023107, 2020.
  * [19] L Filipe O Costa and José Natário.  Gravito-electromagnetic analogies.  General Relativity and Gravitation, 46(10):1–57, 2014.
  * [20] Emanuele Polino, Beatrice Polacchi, Davide Poderini, Iris Agresti, Gonzalo Carvacho, Fabio Sciarrino, Andrea Di Biagio, Carlo Rovelli, and Marios Christodoulou.  Photonic implementation of quantum gravity simulator.  arXiv preprint arXiv:2207.01680, 2022.
  * [21] Alejandro Pozas-Kerstjens and Eduardo Martín-Martínez.  Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms.  Physical Review D, 94(6):064074, 2016.
  * [22] David L Moehring, Peter Maunz, Steve Olmschenk, Kelly C Younge, Dzmitry N Matsukevich, L-M Duan, and Christopher Monroe.  Entanglement of single-atom quantum bits at a distance.  Nature, 449(7158):68–71, 2007.
  * [23] L Luo, D Hayes, TA Manning, DN Matsukevich, P Maunz, S Olmschenk, JD Sterk, and C Monroe.  Protocols and techniques for a scalable atom–photon quantum network.  Fortschritte der Physik, 57(11-12):1133–1152, 2009.
  * [24] John Hannegan, James D Siverns, and Qudsia Quraishi.  Entanglement between a trapped-ion qubit and a 780-nm photon via quantum frequency conversion.  Physical Review A, 106(4):042441, 2022.
  * [25] Martin Ringbauer, Michael Meth, Lukas Postler, Roman Stricker, Rainer Blatt, Philipp Schindler, and Thomas Monz.  A universal qudit quantum processor with trapped ions.  Nature Physics, pages 1–5, 2022.
  * [26] Xinxing Yuan, Yue Li, Mengxiang Zhang, Chang Liu, Mingdong Zhu, Xi Qin, Nikolay V Vitanov, Yiheng Lin, and Jiangfeng Du.  Preserving multi-level quantum coherence by dynamical decoupling.  Physical Review A, 106(2):022412, 2022.
  * [27] Yuchen Wang, Zixuan Hu, Barry C Sanders, and Sabre Kais.  Qudits and high-dimensional quantum computing.  Frontiers in Physics, 8:589504, 2020.
  * [28] KL. Wang, M. Johanning, M. Feng, F. Mintert, C. Wunderlich  Quantum gates using electronic and nuclear spins of Yb+ in a magnetic field gradient  The European Physical Journal D, 63, 157-164, 2011.
  * [29] M. Feng, YY. Xu, F. Zhou, D. Suter  Nuclear spin qubits in a trapped-ion quantum computer  Physical Review A, 79(5):052331, 2009.
  * [30] Wesley C Campbell and Eric R Hudson.  Polyqubit quantum processing.  arXiv preprint arXiv:2210.15484, 2022.
  * [31] H-X Yang, J-Y Ma, Y-K Wu, Ye Wang, M-M Cao, W-X Guo, Y-Y Huang, Lu Feng, Z-C Zhou, and L-M Duan.  Realizing coherently convertible dual-type qubits with the same ion species.  Nature Physics, pages 1–4, 2022.
  * [32] HCJ Gan, Gleb Maslennikov, Ko-Wei Tseng, Chihuan Nguyen, and Dzmitry Matsukevich.  Hybrid quantum computing with conditional beam splitter gate in trapped ion system.  Physical review letters, 124(17):170502, 2020.
  * [33] Christopher Monroe, Wes C Campbell, L-M Duan, Z-X Gong, Alexey V Gorshkov, PW Hess, R Islam, K Kim, Norbert M Linke, Guido Pagano, et al.  Programmable quantum simulations of spin systems with trapped ions.  Reviews of Modern Physics, 93(2):025001, 2021.
  * [34] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki.  Quantum entanglement.  Reviews of modern physics, 81(2):865, 2009.
  * [35] Juan I Cirac and Peter Zoller.  Quantum computations with cold trapped ions.  Physical review letters, 74(20):4091, 1995.
  * [36] Anders Sørensen and Klaus Mølmer.  Quantum computation with ions in thermal motion.  Physical review letters, 82(9):1971, 1999.
  * [37] G Kirchmair, J Benhelm, F Zähringer, R Gerritsma, Christian F Roos, and Rainer Blatt.  Deterministic entanglement of ions in thermal states of motion.  New Journal of Physics, 11(2):023002, 2009.
  * [38] Dietrich Leibfried, Brian DeMarco, Volker Meyer, David Lucas, Murray Barrett, Joe Britton, Wayne M Itano, B Jelenković, Chris Langer, Till Rosenband, et al.  Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate.  Nature, 422(6930):412–415, 2003.
  * [39] JD Wong-Campos, SA Moses, KG Johnson, and C Monroe.  Demonstration of two-atom entanglement with ultrafast optical pulses.  Physical Review Letters, 119(23):230501, 2017.
  * [40] GJ Milburn, S Schneider, and DFV James.  Ion trap quantum computing with warm ions.  Fortschritte der Physik: Progress of Physics, 48(9-11):801–810, 2000.
  * [41] Rainer Blatt and David Wineland.  Entangled states of trapped atomic ions.  Nature, 453(7198):1008–1015, 2008.
  * [42] Colin D Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M Sage.  Trapped-ion quantum computing: Progress and challenges.  Applied Physics Reviews, 6(2):021314, 2019.
  * [43] Malcolm H Levitt.  Spin dynamics: basics of nuclear magnetic resonance.  John Wiley & Sons, 2013.
  * [44] Michael Johanning, Andrés F Varón, and Christof Wunderlich.  Quantum simulations with cold trapped ions.  Journal of Physics B: Atomic, Molecular and Optical Physics, 42(15):154009, 2009.
  * [45] Jiaxuan Wang, Hui Dong, and Sheng-Wen Li.  Magnetic dipole-dipole interaction induced by the electromagnetic field.  Physical Review A, 97(1):013819, 2018.
  * [46] Fang-qi Hu, Qing Zhao, and Sheng-Wen Li.  The field-induced interaction between non-resonant magnetic dipoles.  Journal of Physics B: Atomic, Molecular and Optical Physics, 53(6):065003, 2020.
  * [47] William K Wootters.  Entanglement of formation and concurrence.  Quantum Inf. Comput., 1(1):27–44, 2001.
  * [48] Christopher J Foot.  Atomic physics, volume 7.  OUP Oxford, 2004.
  * [49] S Ahmad, J Andriessen, K Raghunathan, and TP Das.  Theory of hyperfine interactions in alkaline-earth ions isoelectronic with alkali-metal atoms—ba+ 137.  Physical Review A, 25(6):2923, 1982.
  * [50] Lieven MK Vandersypen and Isaac L Chuang.  Nmr techniques for quantum control and computation.  Reviews of modern physics, 76(4):1037, 2005.
  * [51] Hartmut Häffner, Christian F Roos, and Rainer Blatt.  Quantum computing with trapped ions.  Physics reports, 469(4):155–203, 2008.
  * [52] Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P Singh, and Hendrik Ulbricht.  Models of wave-function collapse, underlying theories, and experimental tests.  Reviews of Modern Physics, 85(2):471, 2013.
  * [53] Igor Pikovski, Magdalena Zych, Fabio Costa, and Časlav Brukner.  Universal decoherence due to gravitational time dilation.  Nature Physics, 11(8):668–672, 2015.
  * [54] Chin-Wen Chou, David B Hume, Till Rosenband, and David J Wineland.  Optical clocks and relativity.  Science, 329(5999):1630–1633, 2010.
  * [55] Tobias Bothwell, Colin J Kennedy, Alexander Aeppli, Dhruv Kedar, John M Robinson, Eric Oelker, Alexander Staron, and Jun Ye.  Resolving the gravitational redshift across a millimetre-scale atomic sample.  Nature, 602(7897):420–424, 2022.
  * [56] Ruster, Thomas and Schmiegelow, Christian T and Kaufmann, Henning and Warschburger, Claudia and Schmidt-Kaler, Ferdinand and Poschinger, Ulrich G.  A long-lived Zeeman trapped-ion qubit.  Applied Physics B, 122(10):1–7, 2016.
  * [57] Francesca Albertini and Domenico D’Alessandro.  The lie algebra structure and controllability of spin systems.  Linear algebra and its applications, 350(1-3):213–235, 2002.
  * [58] Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J Glaser.  Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms.  Journal of magnetic resonance, 172(2):296–305, 2005.