# Controlled-NOT gate based on the Rydberg states of surface electrons

Jun Wang  Department of Physics, Applied Optics Beijing Area Major Laboratory,
Beijing Normal University, Beijing 100875, China  Wan-Ting He  Department of
Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal
University, Beijing 100875, China  Cong-Wei Lu  Department of Physics, Applied
Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing
100875, China  Yang-Yang Wang  Shaanxi Engineering Research Center of
Controllable Neutron Source, School of Electronic Information, Xijing
University, Xi’an 710123, China  Qing Ai  Department of Physics, Applied
Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing
100875, China  Hai-Bo Wang<EMAIL_ADDRESS>Department of Physics, Applied
Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing
100875, China

###### Abstract

Due to the long coherence time and efficient manipulation, the surface
electron (SE) provides a perfect two-dimensional platform for quantum
computation and quantum simulation. In this work, a theoretical scheme to
realize the controlled-NOT (CNOT) gate is proposed, where the two-qubit system
is encoded on the four-level Rydberg structure of SE. The state transfer is
achieved by a three-level structure with an intermediate level. By
simultaneously driving the SE with two external electromagnetic fields, the
dark state in the electromagnetically induced transparency (EIT) effect is
exploited to suppress the population of the most dissipative state and
increase the robustness against dissipation. The fidelity of the scheme is
0.9989 with experimentally achievable parameters.

surface electron, electromagnetically induced transparency, Rydberg state,
quantum gate

##  I Introduction

Universal quantum logic gates Barenco  _et al._ (1995); Sleator and Weinfurter
(1995) are the key elements of quantum information processing Chitambar and
Gour (2019) and quantum simulation Buluta and Nori (2009); Weimer  _et al._
(2010); Georgescu  _et al._ (2014); Mostame and Schützhold (2008). In recent
years, many schemes of quantum logic gates have been proposed in various
physical systems Buluta  _et al._ (2011); Xiang  _et al._ (2013); Zhang  _et
al._ (2017); Frisk Kockum  _et al._ (2019), such as superconducting qubits
Makhlin  _et al._ (2001); Yamamoto  _et al._ (2003); Liu  _et al._ (2005); You
and Nori (2005, 2011); Gu  _et al._ (2017), nuclear magnetic resonance (NMR)
systems Jones  _et al._ (1998); Feng  _et al._ (2013), cavity quantum
electrodynamics (QED) Turchette  _et al._ (1995); Rauschenbeutel  _et al._
(1999); Walther  _et al._ (2006); Burkard  _et al._ (2020), circuit QED
Chiorescu  _et al._ (2004); DiCarlo  _et al._ (2009); Scarlino  _et al._
(2019), ion traps Cirac and Zoller (1995); Poyatos  _et al._ (1998); Figgatt
_et al._ (2019), quantum dots Li  _et al._ (2003); Ciriano-Tejel  _et al._
(2021), and nitrogen-vacancy centers in diamond Jelezko  _et al._ (2004); Wei
and Deng (2013); Zhang  _et al._ (2020). Among the above proposals, the
controlled-NOT (CNOT) gate is one of the most attractive quantum gates,
because it can be used to realize universal quantum logic gates with the aid
of single-qubit gates Barenco  _et al._ (1995). A feasible quantum gate
requires the operation time to be shorter than the coherent time of the
system, thus the fast manipulation plays the significant role in quantum
gates. Some previous works have utilized the electromagnetically induced
transparency (EIT) effect Fleischhauer  _et al._ (2005); Liu  _et al._ (2014,
2016); Gu  _et al._ (2016); Wang  _et al._ (2018a) to reduce the influence of
dissipation and accelerate the manipulation Wang and Clerk (2012); McDonnell
_et al._ (2022). The EIT effect is due to the coherent interference among
different transition pathways, which is different from the Autler–Townes
splitting Peng  _et al._ (2014).

The surface electron (SE) on the surface of liquid helium provides a
controllable two-dimensional (2D) quantum system to study the behavior of
strongly-correlated electrons. The SE is attracted by the induced image charge
inside the liquid helium and concurrently repulsed by the helium atoms, and
therefore the motion perpendicular to the surface is confined and forms a
hydrogen-like spectrum Platzman and Dykman (1999). Meanwhile, the SE can move
freely parallel to the surface, forming a perfect 2D electron system free of
the defects and impurities present in semiconductor devices Kawakami  _et al._
(2021). The 2D electron system possesses the quantized orbital states when
electrons are trapped in an electrostatic potential Koolstra  _et al._ (2019).
Both the Rydberg and orbital states can be coupled to the spin states of
electrons Kawakami  _et al._ (2019); Schuster  _et al._ (2010); Kawakami  _et
al._ (2023), which have a much longer coherence time than other solid
materials Lyon (2006), making them an excellent resource for quantum
computing. The SE can be manipulated and detected by the circuit QED
architecture, which combines the superconducting coplanar-waveguide resonator
and the electron trap Koolstra  _et al._ (2019); Zhou  _et al._ (2022). In
addition, the SE can also be manipulated and transported through the
microchannel devices which are fabricated on the silicon substrate and filled
with the superfluid helium Glasson  _et al._ (2001); Ikegami  _et al._ (2009);
Rees  _et al._ (2011); Ikegami  _et al._ (2012); Rees  _et al._ (2016a, b);
Badrutdinov  _et al._ (2020); Zou and Konstantinov (2022). The unprecedented
transport efficiency of such microchannel devices Bradbury  _et al._ (2011)
manifests the applications of SE in the large-scale trapped-ion quantum
computing Kielpinski  _et al._ (2002).

The highly excited Rydberg state of neutral atoms is a promising candidate for
quantum information processing, benefiting from its long coherence time
Gallagher (1994) and strong long-range interactions Saffman  _et al._ (2010).
The SE system can be used to simulate Rydberg states Platzman and Dykman
(1999) because it has the same hydrogen-like energy spectrum as Rydberg atoms.
At low temperatures, the dissipation of SE is mainly due to the height
variations of the helium surface, which can be quantized as ripplons Kawakami
_et al._ (2021). The lifetime $T_{1}$ exceeds 10 $\mu$s at 10 mK, which is
sufficiently long compared to the Rabi frequency $\Omega$, i.e., $\Omega
T_{1}>10^{4}$ Platzman and Dykman (1999). A more realistic lifetime measured
by some recent works Monarkha and Sokolov (2006, 2007); Monarkha  _et al._
(2010); Kawakami  _et al._ (2021) is 1 $\mu$s, which is adopted by our work.

Here, we present a scheme to realize the CNOT gate in a single SE system. We
encode the two-qubit system in the four-level Rydberg structure of SE. The
first qubit represents whether the electron is close or far away from the
liquid surface, i.e., whether the electron is in the “lower” or “upper” mode.
Each mode comprises a two-level system, which is labeled by the second qubit.
This proposal is analogous to the hyperentanglement that combines several
degrees of freedom (DOFs) of a single particle, such as the spatial mode and
the polarization of a single photon Barreiro  _et al._ (2005); Yang  _et al._
(2005); Deng  _et al._ (2017). Although the multilevel structure is difficult
to be scaled up on a single electron, it can be scaled up with the assistance
of adjacent electrons or different DOFs. The dipole-dipole interaction of
neutral atoms has been used to realize the quantum gates Jaksch  _et al._
(2000); McDonnell  _et al._ (2022). Recent work based on the SE has proposed a
scheme to couple two adjacent electrons and scale up the system via the
dipole-dipole interaction of electron in different Rydberg states Kawakami
_et al._ (2023). Meanwhile, the Rydberg states and the spin states of the SE
can be coupled via an inhomogeneous magnetic field Tokura  _et al._ (2006);
Kawakami  _et al._ (2023). On the other hand, the SE is also a promising
platform for quantum simulation Buluta and Nori (2009); Weimer  _et al._
(2010); Georgescu  _et al._ (2014); Mostame and Schützhold (2008), and the
four-level structure in our scheme can be used to simulate the quantum
coherent effects of multilevel molecules such as the four-level
pigment–protein molecules in photosynthetic light harvesting Wang  _et al._
(2018b).

Since the highly excited Rydberg states are sensitive to the frequency
fluctuation of the driving fields Saffman  _et al._ (2010) and their level
spacing is narrow, the direct driving for state transfer could easily cause
the undesirable transitions to other neighboring states. Therefore, to
accurately achieve the transition between $|10\rangle$ and $|11\rangle$, we
use an intermediate level to avoid the undesired transitions to other highly
excited states. By applying two driving pulses simultaneously, we exploit the
dark state in the EIT effect Fleischhauer  _et al._ (2005); Wang  _et al._
(2018c) to reduce the population of the most dissipative intermediate level
and increase the robustness against dissipation.

This paper is organized as follows. In Sec. II, we describe the coherent-
driving scheme based on the Rydberg states of SE. In Sec. III, we compare our
scheme with other schemes and show how the CNOT gate was accelerated by the
two simultaneous driving fields at the same time. We also investigate the
effects of detuning and dissipation on the fidelity. Finally, we conclude the
work and give a prospect in Sec. IV. In Appendix A, we analyze the decay
mechanism of the excited states. In Appendix B, we provide the eigenvalues of
the non-Hermitian Hamiltonian by the perturbation theory.

##  II The model

Figure 1: Schematic diagram of the CNOT gate based on SE. (a) The SE on the
surface of liquid helium. (b) The two-qubit system is encoded in the four-
level SE Rydberg states, where two coherent driving fields with frequencies
$\omega_{j}$ and Rabi frequencies $\Omega_{j}$ ($j=1,2$) are applied
simultaneously. Here, $\Delta_{1}$ and $\Delta_{2}$ are the single-photon and
two-photon detunings, respectively.

As shown in Figure 1(a), along the direction $z$ perpendicular to the
interface, the SE is attracted by the induced image charge inside the liquid
helium and concurrently repulsed by the helium atoms, and therefore the motion
of the SE is confined by the hydrogen-like potential $V=-\Lambda e^{2}/z$ for
$z>0$, where $e$ is the charge of the electron and
$\Lambda=(\epsilon-1)/[4(\epsilon+1)]$ with the dielectric constant
$\epsilon\approx 1.057$. The quantized SE states possess the hydrogen-like
energy spectrum as Platzman and Dykman (1999); Monarkha and Kono (2004)

$\displaystyle\varepsilon_{n}^{(\perp)}=-\frac{m_{e}e^{4}\Lambda^{2}}{2\hbar^{2}n^{2}}=-\frac{R}{n^{2}},$
(1)

where $m_{e}$ is the mass of the electron, the positive integer $n$ labels the
SE state, and $R\approx 0.7$ meV Platzman and Dykman (1999) is the Rydberg
energy. The wave function of SE is Nieto (2000)

$\displaystyle\psi_{n}(z)=2n^{-5/2}r_{B}^{-3/2}z\exp\left(-\frac{z}{nr_{B}}\right)L_{n-1}^{(1)}\left(\frac{2z}{nr_{B}}\right),$
(2)

where $r_{B}=\hbar^{2}/(m_{e}e^{2}\Lambda)$ is the effective Bohr radius, and

$\displaystyle
L_{n}^{(\alpha)}(x)=\frac{e^{x}x^{-\alpha}}{n!}\frac{d^{n}}{dx^{n}}(e^{-x}x^{n+\alpha})$
(3)

is the Laguerre polynomial.

The expected positions $\langle
z\rangle_{n}=\langle\psi_{n}|z|\psi_{n}\rangle$ of the electrons in different
Rydberg states $|\psi_{n}\rangle$ are different. The electrons in higher
excited states are further away from the liquid surface. For example, $\langle
z\rangle_{2}/\langle z\rangle_{1}=4$, $\langle z\rangle_{3}/\langle
z\rangle_{1}=9$, $\langle z\rangle_{4}/\langle z\rangle_{1}=16$ and so on.
This phenomenon is remarkable for highly excited Rydberg states. The expected
position $\langle z\rangle$ has practical applications in physical systems.
For example, the electron with larger $\langle z\rangle$ has larger electric
dipole moment and induces larger dipole-dipole interaction with the adjacent
electrons, and thus can be used to couple adjacent electrons Jaksch  _et al._
(2000); McDonnell  _et al._ (2022). Meanwhile, in the gradient magnetic field,
the electrons with different $\langle z\rangle$ have different spin-level
shifts due to the Zeeman effect, and thus can be used to couple the Rydberg
states with the spin states Tokura  _et al._ (2006); Kawakami  _et al._
(2023).

Encoding a two-qubit system on a single electron is analogous to the
hyperentanglement which combines several DOFs of a single particle Barreiro
_et al._ (2005); Yang  _et al._ (2005); Deng  _et al._ (2017). We consider the
ground state and the first excited state of the SE as the “lower” mode and
consider the highly excited Rydberg states as the “upper” mode. The first bit
indicates which mode the SE is in. The “lower” mode is labeled as $|0\rangle$,
and the “upper” mode is labeled as $|1\rangle$. Each mode comprises a two-
level system, which is labeled by the second bit. Therefore, the ground state
with $n=1$ is labeled as $|00\rangle$, the first excited state with $n=2$ is
labeled as $|01\rangle$, the second excited state with $n=3$ is labeled as
$|10\rangle$, and the third excited state with $n=4$ is labeled as
$|11\rangle$, as shown in Figure 1(b). The CNOT gate in our scheme means that
the second bit flips only if the state is in the “upper” mode, while the
second bit doesn’t flip if the state is in the “lower” mode.

The level spacing of highly excited Rydberg states is narrow because the
energy space $\Delta E_{n}$ decreases with $n$. Moreover, since the dipole
moment matrix elements for the transitions between highly excited states are
large, the direct transition between these states is sensitive to the
frequency fluctuation of the driving fields Saffman  _et al._ (2010). This
sensitivity of the direct manipulation of highly excited states could easily
cause the undesirable transition to other neighboring states. Therefore, we
use the first excited state $|01\rangle$ as an intermediate state to realize
the state swap between $|10\rangle$ and $|11\rangle$.

Because the first excited state $|01\rangle$ is the most dissipative state
(see Appendix. A), we exploit the dark state in the EIT effect to reduce the
population on $|01\rangle$. Two driving fields with frequencies $\omega_{1}$
($\omega_{2}$) and Rabi frequencies $\Omega_{1}$ ($\Omega_{2}$) are used to
drive the transition $|01\rangle\rightleftharpoons|11\rangle$
($|01\rangle\rightleftharpoons|10\rangle$), as shown in Figure 1(b). The
single-photon detuning is $\Delta_{1}=\omega_{1}-(\omega_{11}-\omega_{01})$
and the two-photon detuning is
$\Delta_{2}=\omega_{1}-\omega_{2}-(\omega_{11}-\omega_{10})$, where
$\omega_{01}$, $\omega_{10}$ and $\omega_{11}$ are the energies of the states
$|01\rangle$, $|10\rangle$ and $|11\rangle$ respectively. As the driving
frequencies are far detuned from the transitions between $|00\rangle$ and
other states, we consider $|00\rangle$ to be a decoupled state. In the
subspace spanned by $\\{|01\rangle,|10\rangle,|11\rangle\\}$, the Hamiltonian
is

$\displaystyle H=~{}$ $\displaystyle\omega_{01}|01\rangle\langle
01|+\omega_{10}|10\rangle\langle 10|+\omega_{11}|11\rangle\langle 11|$
$\displaystyle-\Omega_{1}\cos\omega_{1}t(|01\rangle\langle
11|+|11\rangle\langle 01|)$
$\displaystyle-\Omega_{2}\cos\omega_{2}t(|01\rangle\langle
10|+|10\rangle\langle 01|),$ (4)

where $\hbar=1$. In the rotating frame with driving frequencies
$U=\exp[i\omega_{1}t|01\rangle\langle
01|+i(\omega_{1}-\omega_{2})t|10\rangle\langle 10|]$, under the rotating-wave
approximation Scully and Zubairy (1997); Ai  _et al._ (2010), and taking
$\omega_{11}=0$ as the zero point of energy, the matrix form of the
Hamiltonian reads

$\displaystyle H_{I}=~{}$ $\displaystyle
i\frac{dU^{{\dagger}}}{dt}U+U^{{\dagger}}HU$ $\displaystyle=~{}$
$\displaystyle-\frac{1}{2}\begin{pmatrix}-2\Delta_{1}&\Omega_{2}&\Omega_{1}\\\
\Omega_{2}&-2\Delta_{2}&0\\\ \Omega_{1}&0&0\end{pmatrix}.$ (5)

The evolution of the system can be described by the quantum master equation
Breuer and Petruccione (2002)

$\displaystyle\frac{\partial}{\partial
t}\rho=-i[H_{I},\rho]-\mathcal{L}(\rho),$ (6)

where the Lindblad operator is

$\displaystyle\mathcal{L}(\rho)=~{}$
$\displaystyle\kappa_{1}[|01\rangle\langle 01|\rho|01\rangle\langle
01|-\frac{1}{2}\\{|01\rangle\langle 01|,\rho\\}]$
$\displaystyle+\kappa_{2}[|10\rangle\langle 10|\rho|10\rangle\langle
10|-\frac{1}{2}\\{|10\rangle\langle 10|,\rho\\}]$
$\displaystyle+\kappa_{3}[|11\rangle\langle 11|\rho|11\rangle\langle
11|-\frac{1}{2}\\{|11\rangle\langle 11|,\rho\\}]$ (7)

with $\\{A,B\\}=AB+BA$ being the anti-commutator, and $\kappa_{1}$,
$\kappa_{2}$ and $\kappa_{3}$ being the decay rates of the states
$|01\rangle$, $|10\rangle$ and $|11\rangle$ respectively. In our scheme, the
electron is only confined by the image potential and no electric holding field
is applied. In this case, the decay rates of the Rydberg states of the SE
decrease with $n$, cf. Appendix A. Thus, in the analytic calculation, we
mainly consider the dissipations of the energy levels $|01\rangle$ and
$|10\rangle$ and neglect the dissipation of $|11\rangle$, while all of the
dissipations are considered in the numerical simulation. It is noteworthy that
in experiments there is usually an electric holding field $E_{z}$ applied
perpendicular to the liquid surface. For the experimental configuration such
as $E_{z}\approx 200$ V/cm Kawakami  _et al._ (2023) and $E_{z}\approx 1$
kV/cm Bradbury  _et al._ (2011); Zou and Konstantinov (2022), the decay rates
of the Rydberg states of the SE increase with $n$, cf. Appendix A. Recent
works have provided efficient quantum algorithms to simulate the quantum open
system both theoretically Zhang  _et al._ (2021) and experimentally Wang  _et
al._ (2018b), even for the non-Markovian process Chen  _et al._ (2022).

To solve the time evolution analytically, we neglect the quantum jump term and
describe the evolution by the Schrödinger equation with the following non-
Hermitian Hamiltonian

$\displaystyle H_{I}^{d}=~{}$ $\displaystyle
H_{I}-i\frac{\kappa_{1}}{2}|01\rangle\langle
01|-i\frac{\kappa_{2}}{2}|10\rangle\langle 10|$ $\displaystyle=~{}$
$\displaystyle-\frac{1}{2}\begin{pmatrix}-2\delta_{1}&\Omega_{2}&\Omega_{1}\\\
\Omega_{2}&-2\delta_{2}&0\\\ \Omega_{1}&0&0\end{pmatrix},$ (8)

where $\delta_{1}=\Delta_{1}-i\kappa_{1}/2$,
$\delta_{2}=\Delta_{2}-i\kappa_{2}/2$. When
$\delta_{2}\ll\Omega_{1},\Omega_{2}$, we take $\delta_{2}$ as the perturbation
term. The approximated eigenstates by taking $\delta_{2}=0$ are Fleischhauer
_et al._ (2005)

$\displaystyle\begin{split}&|a_{1}\rangle=-\sin\theta|10\rangle+\cos\theta|11\rangle,\\\
&|a_{2}\rangle=\cos\phi|01\rangle+\cos\theta\sin\phi|10\rangle+\sin\theta\sin\phi|11\rangle,\\\
&|a_{3}\rangle=-\sin\phi|01\rangle+\cos\theta\cos\phi|10\rangle+\sin\theta\cos\phi|11\rangle,\end{split}$
(9)

where the mixing angles are $\tan\theta=\Omega_{1}/\Omega_{2}$ and $\tan
2\phi=\Omega/\delta_{1}$, and $\Omega=\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}$.
It is noteworthy that $|a_{1}\rangle$ is a dark state since there is no
population on the state $|01\rangle$. As shown in Appendix B, the first-order
approximations of the eigenvalues are

$\displaystyle\begin{split}E_{1}\simeq&\frac{\Omega_{1}^{2}}{\Omega^{2}}\delta_{2},\\\
E_{2}\simeq&\frac{\Omega}{2}+\frac{\delta_{1}}{2}+\frac{\Omega_{2}^{2}}{2\Omega^{2}}\delta_{2},\\\
E_{3}\simeq&-\frac{\Omega}{2}+\frac{\delta_{1}}{2}+\frac{\Omega_{2}^{2}}{2\Omega^{2}}\delta_{2},\end{split}$
(10)

The time evolution of the initial state
$|\psi(0)\rangle=C_{1}|a_{1}\rangle+C_{2}|a_{2}\rangle+C_{3}|a_{3}\rangle$ is

$\displaystyle|\psi(t)\rangle=~{}$ $\displaystyle
C_{1}e^{-iE_{1}t}|a_{1}\rangle+C_{2}e^{-iE_{2}t}|a_{2}\rangle+C_{3}e^{-iE_{3}t}|a_{3}\rangle$
$\displaystyle=~{}$ $\displaystyle(C_{2}\cos\phi e^{-iE_{2}t}-C_{3}\sin\phi
e^{-iE_{3}t})|01\rangle$ $\displaystyle+(-C_{1}\sin\theta
e^{-iE_{1}t}+C_{2}\cos\theta\sin\phi e^{-iE_{2}t}$
$\displaystyle+C_{3}\cos\theta\cos\phi e^{-iE_{3}t})|10\rangle$
$\displaystyle+(C_{1}\cos\theta e^{-iE_{1}t}+C_{2}\sin\theta\sin\phi
e^{-iE_{2}t}$ $\displaystyle+C_{3}\sin\theta\cos\phi e^{-iE_{3}t})|11\rangle.$
(11)

The key point of the CNOT gate scheme is to swap the population of
$|10\rangle$ and $|11\rangle$, while maintaining the low population of
$|01\rangle$ in the final state. The state swapping is based on the population
oscillation. As $E_{1}\ll E_{2},~{}E_{3}$, the main oscillation factors are
$\exp(-iE_{2}t)$ and $\exp(-iE_{3}t)$. To achieve the maximum population
reversal of $|10\rangle$ and $|11\rangle$, these two oscillation terms
$\exp(-iE_{2}t)$ and $\exp(-iE_{3}t)$ should have the same period. The
synchronization of $\exp(-iE_{2}t)$ and $\exp(-iE_{3}t)$ requires
$E_{2}=-E_{3}$, so $\delta_{1}$ and $\delta_{2}$ must be much smaller than
$\Omega$, i.e.,

$\displaystyle\tan\phi=1.$ (12)

Meanwhile, the oscillation term in the coefficient of $|10\rangle$ and
$|11\rangle$ must be the same, which means that $\cos\theta=\sin\theta$, i.e.,

$\displaystyle\Omega_{1}=\Omega_{2}.$ (13)

Figure 2: The populations $P(t)$ of different states with the initial state
$|10\rangle$ and the dissipation rates $\kappa_{1}/\Omega=10^{-3}$,
$\kappa_{2}=0.3439\kappa_{1}$ and $\kappa_{3}=0.1520\kappa_{1}$. The solid
lines are the analytic solutions and the dots are the numerical solutions by
the quantum master equation Eq. (7).

Under these conditions, if the initial state is $|\psi(0)\rangle=|10\rangle$,
the final state is

$\displaystyle|\psi(t)\rangle=$
$\displaystyle\left(\frac{1}{2}\cos\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}+\frac{1}{2}e^{-i\frac{\delta_{2}}{2}t}\right)|10\rangle$
$\displaystyle-\frac{i}{\sqrt{2}}\sin\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}|01\rangle$
$\displaystyle+\left(\frac{1}{2}\cos\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}-\frac{1}{2}e^{-i\frac{\delta_{2}}{2}t}\right)|11\rangle.$
(14)

If the initial state $|\psi(0)\rangle=|01\rangle$, the final state is

$\displaystyle|\psi(t)\rangle=$
$\displaystyle-\frac{i}{\sqrt{2}}\sin\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}|10\rangle$
$\displaystyle+\cos\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}|01\rangle$
$\displaystyle-\frac{i}{\sqrt{2}}\sin\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}|11\rangle.$ (15)

If the initial state $|\psi(0)\rangle=|11\rangle$, the final state is

$\displaystyle|\psi(t)\rangle=$
$\displaystyle\left(\frac{1}{2}\cos\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}-\frac{1}{2}e^{-i\frac{\delta_{2}}{2}t}\right)|10\rangle$
$\displaystyle-\frac{i}{\sqrt{2}}\sin\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}|01\rangle$
$\displaystyle+\left(\frac{1}{2}\cos\frac{\Omega
t}{2}e^{-i\frac{2\delta_{1}+\delta_{2}}{4}t}+\frac{1}{2}e^{-i\frac{\delta_{2}}{2}t}\right)|11\rangle.$
(16)

The time evolution of different initial states is shown in Figure 2. The Rabi
frequency $\Omega=10^{9}~{}\textrm{s}^{-1}$ Platzman and Dykman (1999) and the
decay rate $\kappa_{1}=10^{6}~{}\textrm{s}^{-1}$ Monarkha and Sokolov (2007);
Kawakami  _et al._ (2021), thus $\kappa_{1}/\Omega=10^{-3}$. We take
$\kappa_{2}=0.3439\kappa_{1}$ and $\kappa_{3}=0.1520\kappa_{1}$, according to
Appendix A. The maximum population reversal between $|10\rangle$ and
$|11\rangle$ is reached when $\Omega t=2\pi$. The analytical solutions are in
good agreement with the numerical solutions calculated by Qutip Johansson  _et
al._ (2012, 2013).

##  III Fidelity analysis

Figure 3: (a) Schematic diagram of the coherent-driving scheme. (b) Schematic
diagram of the two-step driving scheme. (c) The fidelity $F$ of the state
transfer $|10\rangle\to|11\rangle$. The dissipation rates are the same as
Figure 2.

Since the driving frequencies are far detuned from the transition between
$|00\rangle$ and other states, we consider $|00\rangle$ to be a decoupled
state. From the time evolution in Eq. (14)-(16), the maximum population
reversal between $|10\rangle$ and $|11\rangle$ is achieved when $\Omega
t=2\pi$. The state fidelity $F$ between the final state $\rho(t)$ and the
ideal target state $\rho_{i}$ is defined as Uhlmann (1976); Jozsa (1994)

$\displaystyle
F=\left(\textrm{Tr}\sqrt{\sqrt{\rho_{i}}\rho(t)\sqrt{\rho_{i}}}\right)^{2}.$
(17)

In our coherent-driving scheme, two driving fields interact with the SE
simultaneously, with the dark and bright states being used equally for the
state transfer. The maximum fidelity is achieved when $t=2\pi/\Omega$, as
shown in Figure 3(c). For comparison, we show the result of the two-step
driving scheme in Figure 3(b). We derive the population reversal between
$|10\rangle$ and $|01\rangle$ by the first driving pulse $\Omega_{1}(t)$, and
then derive the population reversal between $|01\rangle$ and $|11\rangle$ by
the second driving pulse $\Omega_{2}(t)$. The maximum fidelity is achieved
when $t=2\sqrt{2}\pi/\Omega$, which is longer than the coherent-driving
scheme. In addition, the two-step driving scheme can only achieve the one-way
state transfer based on the driving pulse sequence, but a NOT-gate requires
the bidirectional transfer with the same driving pulse sequence.

In Sec. II, we have found that $\Delta_{1}$ need to be much smaller than
$\Omega$ in order to synchronize the oscillation terms $\exp(-iE_{2}t)$ and
$\exp(-iE_{3}t)$. When the single-photon resonance condition is invalid, the
fidelity of the state transfer decreases, as shown in Figure 4. It is
noteworthy that the oscillation period changes with $\Delta_{1}$. Thus, the
fidelity is calculated at the maximum in the first period. On the other hand,
when $\Delta_{2}$ becomes large, the perturbation method is invalid, but the
evolution of the states can still be derived from the master equation. As
shown in Figure 4, when the two-photon resonance condition is invalid, the
fidelity of the state transfer decreases because the EIT effect is suppressed.

Figure 4: State fidelity versus the single-photon detuning $\Delta_{1}$ and
the two-photon detuning $\Delta_{2}$. The dissipation rates are the same as
Figure 2. The solid (solid-dot) line shows the dependence on $\Delta_{1}$
($\Delta_{2}$) with $\Delta_{2}=0$ ($\Delta_{1}=0$).

Figure 3(c) only shows the fidelity of the state transfer with the input state
$|10\rangle$. In Table 1 we present the output state fidelities of the CNOT
gate under typical input states. When the control bit is $|0\rangle$, the
target bit remains in the initial state. When the control bit is $|1\rangle$,
the target bit flips. Meanwhile, when the input state is a superposition
state, the output also corresponds to the characteristic of the CNOT gate. The
output density matrices of $(|0\rangle+|1\rangle)\otimes|0\rangle/\sqrt{2}$
and $(|0\rangle+|1\rangle)\otimes|1\rangle/\sqrt{2}$ are shown in Figure 5.

Table 1: The output state fidelities of the CNOT gate under typical input states. The dissipation rates are the same as Figure 2.  Input state | Ideal output state | Fidelity
---|---|---
$|00\rangle$ | $|00\rangle$ | 1
$|01\rangle$ | $|01\rangle$ | 0.9987
$|10\rangle$ | $|11\rangle$ | 0.9987
$|11\rangle$ | $|10\rangle$ | 0.9987
$(|0\rangle+|1\rangle)\otimes|0\rangle/\sqrt{2}$ | $(|00\rangle+|11\rangle)/\sqrt{2}$ | 0.9990
$(|0\rangle+|1\rangle)\otimes|1\rangle/\sqrt{2}$ | $(|01\rangle+|10\rangle)/\sqrt{2}$ | 0.9980
Figure 5: The matrix elements of the output density operator when the initial
state is $(|0\rangle+|1\rangle)\otimes|0\rangle/\sqrt{2}$ for (a) and (b), and
the initial state is $(|0\rangle+|1\rangle)\otimes|1\rangle/\sqrt{2}$ for (c)
and (d). The dissipation rates are the same as Figure 2.

To demonstrate the characteristic of the entire gate, we calculate the gate
fidelity which is defined as Palao and Kosloff (2002); Wu  _et al._ (2017)

$\displaystyle F=\frac{1}{N}|\textrm{Tr}(e^{i\phi}U_{r}^{{\dagger}})U_{i}|,$
(18)

where $N$ is the dimension of the Hilbert space, $U_{i}$ is the ideal gate
operation, $U_{r}$ is the real operation in our scheme, and $\phi$ is a global
phase to maximize $F$. Since the transitions between $|00\rangle$ and other
states are negligible, we can analytically derive the operation matrix of the
CNOT gate in our scheme from the evolution of different initial states:

$\displaystyle U_{r}=$ $\displaystyle\begin{pmatrix}1&0&0&0\\\ 0&a&0&0\\\
0&0&b&c\\\ 0&0&c&b\end{pmatrix},$ (19)

where

$\displaystyle\begin{split}&a=e^{-\frac{(2\kappa_{1}+\kappa_{2})\pi}{4\Omega}},\\\
&b=\frac{1}{2}e^{-\frac{(2\kappa_{1}+\kappa_{2})\pi}{4\Omega}}-\frac{1}{2}e^{-\frac{\kappa_{2}\pi}{2\Omega}},\\\
&c=\frac{1}{2}e^{-\frac{(2\kappa_{1}+\kappa_{2})\pi}{4\Omega}}+\frac{1}{2}e^{-\frac{\kappa_{2}\pi}{2\Omega}}.\end{split}$
(20)

This formula is obtained by an additional phase operation on $|01\rangle$,
$|10\rangle$ and $|11\rangle$, which adds a $\pi$ phase to these three energy
levels. Compared to the ideal CNOT gate

$\displaystyle U_{i}=$ $\displaystyle\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 0&0&1&0\end{pmatrix},$ (21)

the fidelity of $U_{r}$ is

$\displaystyle
F=\frac{1}{4}|\textrm{Tr}(U_{r}^{T}U_{i})|=\frac{1+a}{4}+\frac{c}{2}.$ (22)

Experimental systems usually apply a vertical static electric field
$\vec{E}=E_{\perp}\vec{e}_{z}$ in order to tune the energy spacing between
Rydberg states. The lifetime of the excited state decreases with $E_{\perp}$,
and for typical $E_{\perp}=100$ V/cm it becomes 5 times shorter Monarkha and
Sokolov (2007). Thus, in Figure 6 we analyze the gate fidelity under different
dissipation rates. The fidelity $F>0.99$ in a wide range of $\kappa_{1}$, and
$F=0.9989$ with parameters $\kappa_{1}/\Omega=10^{-3}$ which is experimentally
achievable Platzman and Dykman (1999); Monarkha and Sokolov (2007); Kawakami
_et al._ (2021).

Figure 6: The gate fidelity between the ideal CNOT gate and the gate in our
scheme under different dissipation rates.

##  IV Conclusion and remarks

In this work, we present a scheme to realize the CNOT gate in the four-level
Rydberg structure of SE. We use a three-level structure to realize the state
transfer. By applying two driving pulses simultaneously, we exploit the dark
state in the EIT effect to suppress the population of the most-dissipative
state and increase the robustness against dissipation Huang  _et al._ (2022).
We obtain the time evolution of the system both analytically and numerically.
We optimize the Rabi frequencies and the detunings of the driving fields to
achieve the maximum population exchange of $|10\rangle$ and $|11\rangle$. The
optimal state transfer requires that both the single-photon and the two-photon
resonances are satisfied. We also calculate the fidelity of the state transfer
and of the entire gate. The fidelity is 0.9989 with experimentally achievable
parameters.

The CNOT operation involves a three-level structure whose configuration can be
the ladder type, $V$ type, or $\Lambda$-type Fleischhauer  _et al._ (2005).
The choice of configuration depends on the relative dissipation rates of the
three levels. In our scheme, we mainly consider the case without the electric
holding field. In this case the decay rates of the Rydberg states decrease
with $n$. Therefore, we choose the $V$-type configuration to consider the most
dissipative state as the intermediate state and use the EIT effect to suppress
the population on this state. For the case with a considerable electric
holding field, the decay rates are significantly different from those in the
zero-field case, and the $\Lambda$-type configuration of the three-level
structure should be chosen accordingly.

It’s noteworthy that both the iSWAP gate and CNOT gate can be used to
construct universal quantum computation. The efficiencies of these two gates
are determined by the character of the hardware. For example, in spin systems,
it’s easier to realize the XY interaction by spin-spin interaction. Thus, the
iSWAP gate is more efficient in this circumstance Lidar and Wu (2001); Schuch
and Siewert (2003); Tanamoto  _et al._ (2009). In our scheme we only use the
dipole transition of the surface electron, and thus iSWAP gates do not have
the advantage of CNOT gates. If the spin-spin interaction of surface electrons
is used for quantum computation, the iSWAP gate may be a more efficient
candidate.

In our scheme, the first logical bit of the two-bit system labels whether the
electron is close or far away from the surface because the expected positions
of different Rydberg states are different. The expected position $\langle
z\rangle$ has practical applications in physical systems. Recent works have
used the expected position of electrons to scale up the system. For example,
the electron with larger $\langle z\rangle$ has larger electric dipole moment
and induces larger dipole-dipole interaction with the adjacent electrons, and
thus can be used to couple adjacent electrons Jaksch  _et al._ (2000);
McDonnell  _et al._ (2022). Meanwhile, in the gradient magnetic field, the
electrons with different $\langle z\rangle$ have different spin-level shifts
due to the Zeeman effect, and thus can be used to couple the Rydberg states
with the spin states Tokura  _et al._ (2006); Kawakami  _et al._ (2023). Our
scheme demonstrates the advantages of precise manipulation on highly excited
Rydberg states with narrow energy space and therefore provides potential
applications in the schemes based on the dipole-dipole interaction of highly
excited Rydberg states and the coupling between the Rydberg states and the
spin states.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under Grant No. 61675028 and the Interdiscipline Research Funds of Beijing
Normal University. Q. Ai is supported by the Beijing Natural Science
Foundation under Grant No. 1202017 and the National Natural Science Foundation
of China under Grant Nos. 11674033, 11505007, and Beijing Normal University
under Grant No. 2022129. Y. Y. Wang is supported by the Natural Science Basic
Research Program of Shaanxi under Grant No. 2023-JC-QN-0092.

##  Appendix A Decay mechanism of the surface electron

The two-ripplon scattering in the short-wavelength range dominates the decay
mechanism at low temperature, and the decay rate of the first excited state
($n=2$) is expressed as Monarkha and Sokolov (2006, 2007); Monarkha  _et al._
(2010)

$\displaystyle\frac{1}{\tau_{2,k}}=\frac{m_{e}\kappa_{0}^{2}}{4\pi\hbar\alpha\rho}\left(\frac{\rho}{4\hbar^{2}\alpha}\right)^{1/3}\left(\frac{\partial
V}{\partial z}\right)_{11}\left(\frac{\partial V}{\partial
z}\right)_{22}\Delta_{21}^{2/3},$ (23)

where $V(z)$ is the electron potential, $\kappa_{0}$ is the penetration depth
of the electron wave function into liquid, $\alpha$ is the surface tension of
liquid helium, and $\rho$ is the liquid mass density. The decay rate is
determined by the diagonal matrix element of $\partial V/\partial z$ and the
energy difference $\Delta$ between the initial state and the lower-lying
Rydberg states that the initial state leaks to.

Since the spontaneous two-ripplon emission process decreases the energy of
surface electrons, we neglect the leakage to the higher-lying Rydberg states
when calculating the decay rate. The decay rate of the highly-excited Rydberg
state is the sum of the leakage to all lower-lying Rydberg states, i.e., the
decay rate $\kappa^{(n)}$ of the Rydberg state with the quantum number $n$ is

$\displaystyle\kappa^{(n)}=K\sum_{l=1}^{n-1}\left(\frac{\partial V}{\partial
z}\right)_{ll}\left(\frac{\partial V}{\partial
z}\right)_{nn}\Delta_{nl}^{2/3},$ (24)

where
$K=\frac{m_{e}\kappa_{0}^{2}}{4\pi\hbar\alpha\rho}(\frac{\rho}{4\hbar^{2}\alpha})^{1/3}$
is a constant.

Table 2: The decay rates of five lowest excited states under the electric holding field. Note that the dependence of $\kappa^{(n)}$ on $n$ is significantly different for the case with and without the applied electric field. $E_{z}$ (V/cm) | $\kappa^{(3)}/\kappa^{(2)}$ | $\kappa^{(4)}/\kappa^{(2)}$ | $\kappa^{(5)}/\kappa^{(2)}$ | $\kappa^{(6)}/\kappa^{(2)}$
---|---|---|---|---
0 | 0.3439 | 0.1520 | 0.0795 | 0.0465
100 | 0.9807 | 1.0050 | 1.0442 | 1.0890
200 | 1.1260 | 1.2418 | 1.3552 | 1.4663
500 | 1.3233 | 1.5950 | 1.8487 | 2.0927
1000 | 1.4701 | 1.8800 | 2.2689 | 2.6477

When there is no electric holding field, $V(z)=-\Lambda e^{2}/z$ is the image
potential. The decay rates can be calculated from the energy spectrum (1) and
the wave function (2). As shown in the first row of Table 2, the decay rate
decreases with increasing $n$. This is because highly-excited states are
further away from the surface. Since ripplons represent the height variations
of the helium surface, the couplings between the highly-excited electrons and
ripplons are weak.

In experiments, there is usually an electric holding field $E_{z}$ applied
perpendicular to the liquid surface. By considering $E_{z}$ as an uniform
field, the corresponding potential can be expressed as $eE_{z}z$. As shown in
the lower four rows of Table 2, the decay rates of the excited states under
different electric holding fields are calculated from the eigen values and
wave functions, which are solved numerically. The energy level spacings
between higher-laying Rydberg states enlarge with increasing $E_{z}$.
Meanwhile, the SE in the holding field is closer to the liquid surface than
that without the holding field, especially for the highly-excited states.
Thus, for the experimental configuration with a considerable electric holding
field, the decay rates are significantly different from the zero-field case
according to Eq. (24).

##  Appendix B Eigenvalues from the perturbation method

Assuming that $E_{n}=-x/2$, the secular equation of Eq. (8) is

$\displaystyle
x[x^{2}+2(\delta_{1}+\delta_{2})x+4\delta_{1}\delta_{2}-\Omega^{2}]-2\Omega_{1}^{2}\delta_{2}=0,$
(25)

where $\Omega=\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}$. To solve the secular
equation, we assume $\delta_{2}$ as a perturbation term and

$\displaystyle x_{j}\approx x_{0j}+A_{j}\delta_{2}.$ (26)

The zero-order terms satisfies

$\displaystyle
x_{0j}[x_{0j}^{2}+2(\delta_{1}+\delta_{2})x_{0j}+4\delta_{1}\delta_{2}-\Omega^{2}]=0,$
(27)

and the solutions are

$\displaystyle\begin{split}x_{01}&=0,\\\
x_{02}&=-\delta_{1}-\delta_{2}-\sqrt{(\delta_{1}-\delta_{2})^{2}+\Omega^{2}}\\\
&\simeq-\delta_{1}-\delta_{2}-\Omega-\frac{\delta_{1}-\delta_{2}}{2\Omega}(\delta_{1}-\delta_{2})\\\
&\simeq-\delta_{1}-\delta_{2}-\Omega,\\\
x_{03}&=-\delta_{1}-\delta_{2}+\sqrt{(\delta_{1}-\delta_{2})^{2}+\Omega^{2}}\\\
&\simeq-\delta_{1}-\delta_{2}+\Omega+\frac{\delta_{1}-\delta_{2}}{2\Omega}(\delta_{1}-\delta_{2})\\\
&\simeq-\delta_{1}-\delta_{2}+\Omega.\end{split}$ (28)

The approximation is valid when $\delta_{1},\delta_{2}\ll\Omega$. Thus, the
secular equation is transformed into

$\displaystyle(x-x_{01})(x-x_{02})(x-x_{03})-2\Omega_{1}^{2}\delta_{2}=0.$
(29)

Inserting $x_{j}\approx x_{0j}+A_{j}\delta_{2}$, we can obtain

$\displaystyle\begin{split}A_{1}&=\frac{2\Omega_{1}^{2}}{(x_{01}-x_{02})(x_{01}-x_{03})}\simeq-\frac{2\Omega_{1}^{2}}{\Omega^{2}},\\\
A_{2}&=\frac{2\Omega_{1}^{2}}{(x_{02}-x_{01})(x_{02}-x_{03})}\simeq\frac{\Omega_{1}^{2}}{\Omega^{2}},\\\
A_{3}&=\frac{2\Omega_{1}^{2}}{(x_{03}-x_{01})(x_{03}-x_{02})}\simeq\frac{\Omega_{1}^{2}}{\Omega^{2}},\end{split}$
(30)

and the eigenvalues to the first-order approximation as

$\displaystyle\begin{split}E_{1}&\simeq\frac{\Omega_{1}^{2}}{\Omega^{2}}\delta_{2},\\\
E_{2}&\simeq\frac{\Omega}{2}+\frac{\delta_{1}}{2}+\frac{\Omega_{2}^{2}}{2\Omega^{2}}\delta_{2},\\\
E_{3}&\simeq-\frac{\Omega}{2}+\frac{\delta_{1}}{2}+\frac{\Omega_{2}^{2}}{2\Omega^{2}}\delta_{2}.\end{split}$
(31)

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