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Nuclear electric resonance for spatially-resolved spin control
via pulsed optical excitation in the UV-visible spectrum∗
Johannes K. Krondorfer1and Andreas W. Hauser1, †
1Institute of Experimental Physics, Graz University of Technology, Petersgasse 16, 8010 Graz
(Dated: January 30, 2025)
Nuclear electric resonance (NER) spectroscopy is currently experiencing a revival as a tool for nu-
clear spin-based quantum computing. Compared to magnetic or electric fields, local electron density
fluctuations caused by changes in the atomic environment provide a much higher spatial resolution
for the addressing of nuclear spins in qubit registers or within a single molecule. In this article, we
investigate the possibility of coherent spin control in atoms or molecules via nuclear quadrupole res-
onance from first principles. An abstract, time-dependent description is provided which entails and
reflects on commonly applied approximations. This formalism is then used to propose a new method
we refer to as ‘optical’ nuclear electric resonance (ONER). It employs pulsed optical excitations in
the UV-visible light spectrum to modulate the electric field gradient at the position of a specific
nucleus of interest by periodic changes of the surrounding electron density. Possible realizations and
limitations of ONER for atomically resolved spin manipulation are discussed and tested on 9Be as
an atomic benchmark system via electronic structure theory.
Keywords: nuclear quadrupole resonance, quantum computing, nuclear spin, electric field gradient, spin
manipulation
I. INTRODUCTION
Quantum technologies are attracting increasing atten-
tion in recent years, above all the field of quantum com-
puting. This interest stems from the potential of quan-
tum systems to outperform classical computers in tasks
such as the simulation of complex quantum systems,
optimization problems, or cryptography [1–5]. Current
paradigms of quantum computers are superconducting
circuits [6,7], trapped ions [8,9] or atoms [10,11], solid-
state systems such as semiconductors [12,13] or topo-
logical qubits [14,15] and nuclear spin-based quantum
systems [16–18].
A critical factor for the realization of quantum comput-
ers is the coherence time of a single qubit. Among the
many systems being studied, nuclear spins are of par-
ticular interest due to their comparably large coherence
time. In large ensembles, their control and detection via
magnetic resonance is widely exploited. Early propos-
als for solid-state quantum computers utilized nuclear
magnetic resonance to realize quantum search and fac-
toring algorithms [19–21]. However, despite the success,
possible applications are intrinsically limited by the fact
that oscillating magnetic fields cannot be easily confined
or screened at the nanoscale. As a consequence, iden-
tical nuclear spins within a large region respond to the
same signal and cannot be addressed individually. This
presents a challenge for the up-scaling and the integra-
tion of nuclear systems into multi-spin devices based on
magnetic control only.
∗©2023 American Physical Society. The published version
is available at Phys. Rev. A 108, 053110 (2023), DOI:
10.1103/PhysRevA.108.053110.
†andreas.w.hauser@gmail.com
Control via electric fields, on the other hand, would re-
solve this problem, since electric fields can be efficiently
routed and confined via industrial standard procedures.
Recently, there has been significant progress in using
the electron-nuclear hyperfine interaction to transduce
electric signals into magnetic fields for nuclear spin con-
trol [11,22], and first universal gates have been realized
in this way for trapped ytterbium atoms [11]. Yet, al-
though working in principle, this type of coupling also
opens a channel for nuclear spin decoherence. To main-
tain maximum coherence and allow for individual con-
trol of the nuclear spins, a direct, exclusively electrical
control over spin states might turn out as a viable solu-
tion. Here, the use of radio frequency electric fields is
believed to be suitable for an up-scaling of nuclear spin-
based quantum devices, taking advantage of the nuclear
quadrupole interaction (NQI), a well-known effect lead-
ing to line shifts of nuclear magnetic resonance (NMR)
signals. Despite much earlier suggestions of coherent
quadrupole coupling [23], a first experimental demon-
stration of a controlled spin manipulation succeeded only
very recently: Avoiding the mediation via a magnetic
field, a coherence time of 0.1 s could be achieved for
a high-spin nucleus in silicon [18]. Motivated by these
findings, we provide a comprehensive theoretical analysis
and discuss commonly applied phenomenological approx-
imations that have been used to either describe nuclear
electric resonance (NER) or nuclear acoustic resonance
(NAR) [18,24,25]. Based on this theoretical framework,
we then propose a new protocol of nuclear spin control
using electric fields in the visible regime, a technique we
refer to as optical nuclear electric resonance (ONER).
Bringing optical excitation into play, the entire field of
optoelectronics and nanophotonics might enter the quest
for viable quantum computing technologies based on nu-
clear spin processes.
arXiv:2501.17575v1 [quant-ph] 29 Jan 2025
2
Our article is structured as follows. In a first step,
we introduce a nuclear quadrupole Hamiltonian for a
molecular system and describe the interaction of the nu-
clear quadrupole moment with the electric field gradient
(EFG). Relevant properties, such as energy correction
terms and transition elements of the quadrupole Hamilto-
nian, are investigated for an open two-level system. Sec-
ond, we derive an abstract description of time-dependent
nuclear quadrupole interaction for NER and NAR from
quantum mechanical principles. In a third step, we use
our formalism to propose a protocol for ONER as a new
paradigm and apply it to a single beryllium atom in a
benchmark study.
II. METHODS
A. Nuclear quadrupole Hamiltonian
Atomic nuclei consist of protons and neutrons and
therefore exhibit a charge distribution. The latter has
a vanishing dipole moment with respect to the center of
charge of the nucleus, but might feature a non-vanishing
quadrupole moment, which is related to the nuclear spin
Iand interacts with the electric field gradient (EFG), the
second derivative of the electric potential, at the position
of the nucleus. The corresponding Hamiltonian can be
written as
HQ=IµQµν Iν,(1)
with Qµν =q
2I(2I−1) Φµν , where Φµν is the EFG tensor
and qis the scalar quadrupole moment of the nucleus.
Note that we implicitly sum over double appearing Greek
indices in this expression, a convention we keep through-
out the manuscript. A detailed derivation of this Hamil-
tonian can be found in Appendix B. The numerical values
of the scalar quadrupole moments of different nuclei are
tabulated in Refs. [26,27]. Note that only nuclei with
nuclear spin I > 1
2can have a non-vanishing quadrupole
tensor, as indicated by the proportionality constant. We
will refer to the tensor Qas NQI tensor in the further
discussion.
1. Quadrupole energy splitting
Before continuing with the derivation of a suitable
model, it is convenient to investigate a few properties of
the spin quadrupole Hamiltonian. Although quadrupole
splitting also occurs if no external magnetic field is
present, it is reasonable to apply an external magnetic
field to the nucleus of interest in order to obtain a suf-
ficiently large splitting of the nuclear spin states. In an
external magnetic field B=B0ez, the spin Hamiltonian
for a quadrupolar nucleus reads
H=HB+HQ=−γnB0Iz+IµQµν Iν,(2)
with γndenoting the gyromagnetic moment of the nu-
cleus. If the energy splitting due to the magnetic field
is large compared to the quadrupole line splitting, the
energy correction due to NQI can be treated perturba-
tively. In good approximation, the eigenstates of the
total Hamiltonian can be described by the eigenstates
of the Zeeman-Hamiltonian, which are just the orienta-
tional nuclear spin states |mI⟩for a fixed spin quantum
number I. In first order, the corrected energies are given
by
E(1)
mI=⟨mI|H|mI⟩
=−γnB0mI+3m2
I
2−I(I+ 1)
2Qzz .(3)
This leads to corrected transition energies of
∆E(mI−1→mI) = −γnB0+3
2(2mI−1)Qzz
∆E(mI−2→mI) = −2γnB0+3
2(4mI−4)Qzz .
(4)
Note that this correction opens the possibility to ad-
dress specific transitions individually, which is not pos-
sible in the case of equidistant Zeeman-splitting alone.
These corrected transition energies become relevant when
choosing a suitable driving frequency for the nuclear spin
system.
2. Transition elements and time-dependent couplings
In order to drive transitions between different spin
states |mI⟩via direct quadrupole coupling a time-
dependent variation of the EFG tensor is necessary. In
the following, we calculate the transition elements for a
quadrupolar nucleus in an external, constant magnetic
field B=B0ez, similar to the situation discussed in the
supplementary material of Ref. [18]. The spin Hamilto-
nian for a nucleus of spin I, subject to a time-dependent
NQI tensor Qµν (t), is given by (2). The time-dependence
is inherited from the EFG tensor, which can be manipu-
lated. The Rabi frequency for transitions from some mI
to m′
Iis mainly determined by the transition amplitudes,
denoted as gmI→m′
Ibelow.
Any quadratic combination of x, y, z components of the
spin operators can appear in the interaction Hamilto-
nian, but it is immediately clear that only transitions
mI→mI±1 and mI→mI±2 are possible since the in-
teraction is quadratic in the spin operators. Transitions
with ∆mI=±1 are driven by the terms IxIz,IyIzand
their corresponding adjoints. Since the quadrupole inter-
action is symmetric, we can combine the terms to derive
the proportionality factor of Qxz(t). For mI→mI−1
transitions one obtains the transition amplitude
gmI→mI−1(t) = αmI−1↔mI(Qxz(t)+iQyz (t)) ;
αmI−1↔mI=1
2|2mI−1|pI(I+ 1) −mI(mI−1).
(5)
3
Transitions with ∆mI=±2 are driven by the terms
quadratic in the xand yspin operators, i.e. I2
x,I2
y,IxIy
and IyIx. Calculating the transition amplitude for the
corresponding mI→mI−2 transitions yields
gmI→mI−2(t) =βmI−2↔mI(Qxx(t)−Qyy(t) + 2iQyx(t))
βmI−2↔mI=1
4p(I(I+ 1) −(mI−1)(mI−2))
×p(I(I+ 1) −mI(mI−1)).
(6)
Note that the modulus of these transition amplitudes will
determine the Rabi frequency of the respective spin level
transitions in the dynamics simulations later.
B. The open two-level system
In order to derive a protocol for ONER we consider
pulsed excitations of a two-level system to drive the nu-
clear transitions. The energy of the ground state |g⟩is
set to zero, the energy of the excited state |e⟩is denoted
as ω0. We model the interaction in the presence of an
external electric field E(t) = E0cos(ωt)ˆεof amplitude
E0and linear polarization ˆε. The detuning of the elec-
tric field is denoted as ∆ = ω−ω0. Within the dipole
approximation, the dynamics is described by the Hamil-
tonian
H2L=ω0σ†σ− P · E(t),(7)
with P=µσ +µ∗σ†as dipole transition operator, with
σ=|g⟩⟨e|as the lowering operator and µas the corre-
sponding dipole transition matrix element. It is standard
practice to perform the rotating wave approximation, ne-
glecting fast oscillating terms in the Hamiltonian, and to
transform the system in the rotating frame by a uni-
tary transformation [28]. This yields a time-independent
Hamiltonian
H′=−∆σ†σ+Ω
2σ+Ω∗
2σ†,(8)
with Ω = −⟨g|P · ˆε|e⟩E0denoting the Rabi frequency,
which depends on the relative orientation of the dipole
moment and the polarization of the electric field.
If interactions with an environment are taken into ac-
count, decay and dephasing terms may enter through a
Linblad Master equation (a brief introduction to open
quantum systems and density operators can be found in
Appendix C). Switching to the density operator formal-
ism, the general form reads
iℏ∂tρ′= [H′, ρ′]+iℏΓL[σ]ρ′+ iℏγc
2L[σz]ρ′,(9)
with σinducing decays with rate Γ from the excited state
to the ground state and σz=|e⟩⟨e| − |g⟩⟨g|inducing co-
herence loss, quantified by the decay rate γc. The super-
operator Lis defined as
L[c]ρS=cρSc†−1
2c†cρS+ρSc†c,(10)
for some collapse operator c.
The representation as Master equation is convenient
for a general numerical implementation and generaliza-
tions to more complicated systems. Possible solutions
can be obtained numerically, e.g. via the Python library
QuTiP [29,30].
Whereas the isolated system does not have any steady-
state solutions, the open system tends toward an equilib-
rium for t→ ∞. Demanding ∂tρ= 0 and solving the
remaining homogeneous linear equation, one obtains the
steady-state solutions for the density operator matrix el-
ements,
ρee(t→ ∞) = Ω2
2γ⊥Γ
1
1 + ∆2
γ2
⊥
+Ω2
γ⊥Γ
ρ′
eg(t→ ∞) = iΩ
2γ⊥
1 + i∆
γ⊥
1 + ∆2
γ2
⊥
+Ω2
γ⊥Γ
,
(11)
with the definition γ⊥=Γ
2+γc. These solutions, and
thus the dynamics of the system, depend on the ratio
of Rabi frequency and the decay rate. The higher the
decay rate, the faster the convergence to the steady-state
solution.
III. RESULTS AND DISCUSSION
A. Nuclear electric resonance
Having established the interaction principles of a
quadrupolar nucleus and the EFG, we are now interested
in the possibility of controlling the nuclear spin coher-
ently. This will be achieved through a modulation of the
EFG at the position of the nucleus we aim to address.
We start by a general formulation of the quantum me-
chanical equations and derive an evolution equation for
the spin system. For the sake of a reduced formalism,
only a single nucleus will be assumed; a generalization
of this interaction to systems containing several nuclei is
straight-forward.
1. General description
We consider a generic molecular system exposed either
to external strain in case of NAR or to external fields in
the case of NER. All non-spin related contributions to
the Hamiltonian are collected in a ‘molecular’ part, i.e.
kinetic energies of all particles and Coulomb interactions
plus external strain or external electric field effects, and a
‘spin’ part including spin interactions with the external
magnetic field and with the electric field gradient. In
compact form, the total Hamiltonian reads
H(t) = HM(t)⊗1+1⊗HB
+Qµν ⊗IµIν,(12)
4
with HM(t) denoting the molecular Hamiltonian, HBde-
noting the Zeeman interaction Hamiltonian of the nuclear
spin, and Qµν as the NQI tensor derived in Section II A.
The molecular part might also contain coupling terms
to the environment in form of Lindblad superoperators
that only act on the molecular system. This accounts
for the fact that decay rates of electronic states are typ-
ically very fast compared to time scales of the nuclear
spin system. Note that the two separate Hilbert spaces
are coupled only through the nuclear quadrupole inter-
action. For the purpose of this work, we assume that ad-
ditional spin interactions, such as hyperfine coupling or
spin-spin coupling with neighboring atoms, are negligi-
ble, and will choose our benchmark system accordingly.
A similar analysis might be possible considering addi-
tional interactions; however, this is much more involved,
and might even hinder coherent control via quadrupole
interaction, as was suggested by Ref. [18]. Since the cou-
pling between molecular system and spin system is small,
we will further neglect any reverse impact of the spin sys-
tem onto the molecular system in the time evolution of
the molecular system. This step greatly simplifies the
computational treatment, since all relevant properties of
the isolated molecular system become accessible via stan-
dard computational chemistry packages. In this work,
we will use the Molpro suite of programs [31–33] for the
computation of EFG fluctuations caused by electronic
excitation.
Since the coupled system obeys the von Neumann
equation, we have to take partial traces to obtain dy-
namical equations for the molecular system and the spin
system, respectively. For details, we refer to Appendix C.
Supposing a weak coupling only due to quadrupole inter-
action, i.e. ∥Q∥:= maxµν |Qµν |, we may assume that the
total density operator remains decomposable over time,
i.e. that the Born approximation
ρ(t) = ρM(t)⊗ρS(t) (13)
holds throughout time evolution, where ρMand ρSare
the partial density operators of the molecular system and
the spin system, respectively. This approximation is rea-
sonable since the coupling strength of the spin system
is negligibly small compared to the energy scale of the
molecular Hamiltonian. With these approximations in
place, we obtain an evolution equation for the molecular
part of the form
i∂tρM= i∂ttrS{ρ}= trS{[H, ρ]}
≈[HM(t), ρM]
=1
z }| {
trS{ρS}+ρM
=0
z }| {
trS{[HB, ρS]}
+ [Qµν , ρM] trS{IµIνρS}
= [HM(t), ρM] + O(∥Q∥).
(14)
Note that the error of the molecular density operator of
the isolated system with respect to the coupled system
is of the order of O(∥Q∥). Furthermore, we obtain an
evolution equation for the spin part by taking the partial
trace over the molecular part, trM, which yields
i∂tρS= i∂ttrM{ρ}= trM{[H, ρ]}
=ρS
=0
z }| {
trM{[HM(t), ρM]}+ [HB, ρS]
=1
z }| {
trM{ρM}
+ [IµIν, ρS] trM{Qµν ρM}
|{z }
=⟨Qµν ⟩(t)
= [HB+⟨Qµν ⟩(t)IµIν, ρS].
(15)
Hence, we are left with the two dynamical equations
i∂tρM= [HM(t), ρM] + O(∥Q∥)
i∂tρS= [HB+⟨Qµν ⟩(t)IµIν, ρS],(16)
with
⟨Qµν ⟩(t) = trM{ρM(t)Qµν }+O∥Q∥2.(17)
Both can be solved via pure states, i.e. via a Schr¨odinger
equation instead of the von Neumann ansatz for density
operators.
2. Solution for the molecular system
According to (16) and (17), the molecular equation
needs to be solved first, since the spin part can only be
solved once the time-dependence of the NQI tensor is
known. Depending on the actual problem setting the
solution of the molecular system necessitates further ap-
proximations. Since the time-dependence of the external
field in the molecular Hamiltonian in NER as well as
NAR is slow compared to the characteristic time of elec-
tron movement, an adiabatic behavior may be assumed
to obtain the time-dependence of the molecular system.
A discussion of the units and orders of magnitude of NQI
parameters is given in Appendix D. Employing the adi-
abatic theorem [34], the wavefunction of the molecular
system can be written as
|ψM(t)⟩=e−iγ(t)e−iRt
0E(τ) dτ|ϕM(t)⟩,(18)
with γ(t) as a time-dependent phase and |ϕM(t)⟩,E(t)
as eigenfunction and eigenvalue, respectively, of the adi-
abatic Hamiltonian equation
HM(t)|ϕM(t)⟩=E(t)|ϕM(t)⟩.(19)
Choosing the adiabatic ground state due to environmen-
tal coupling and short relaxation times in comparison to
the long timescale of the nuclear quadrupole interaction,
the time-dependence of the NQI tensor may be written
as an expectation value of the corresponding electronic
wavefunction,
⟨Qµν ⟩(t) = ⟨ψM(t)|Qµν |ψM(t)⟩,(20)
which can be easily calculated via common electronic
structure methods, which provide the EFG tensor com-
ponents Φµν (see equation (1)) at the position of the
nucleus.
5
B. Optical nuclear electric resonance
With all prerequisites in place, we are now in the posi-
tion to propose our protocol for an optical stimulation of
nuclear spin processes via pulsed light. As a special case
of the above, we consider an electronic two-level system
coupled to a nuclear spin system via quadrupole interac-
tion. The environment needs to be taken into account,
since typical lifetimes of electronically excited states of
atoms are in the range of 10−9seconds, i.e. are much
smaller than the typical timescales for nuclear spin con-
trol via quadrupole interaction (10−3to 10−6s).
1. General derivation of ONER
The Hamiltonian of a two-level system coupled to a
nuclear spin in an external constant magnetic field B=
B0ezand time-dependent electric field E(t) is given by
H= (H2L+HE(t)) ⊗1+1⊗HB+HQ
= (ω0|e⟩⟨e| − P · E(t)) ⊗1
−1⊗γnB0Iz+Qµν ⊗IµIν,
(21)
with all constants defined as above. Note that the electric
field gradient, and thus the quadrupole coupling tensor
Q, is an operator in the Hilbert space of the two-level
system, i.e. a 2 ×2 matrix in the basis {|g⟩,|e⟩} contain-
ing blocks of EFG tensors for the spin system. Only the
last term in (21) couples the two-level system and the
nuclear spin system. The dynamics of the two-level sys-
tem is mainly influenced by the time-dependent (dipole)
Hamiltonian, inducing transitions of the two-level sys-
tem. It will be treated via a Born-Markov Master equa-
tion with a decay constant Γ ≫ |γnB0|,∥Q∥; see Sec-
tion II B for details. This is reasonable since the decay
frequency lies in the range of GHz for non-metastable
excited states of isolated or weakly interacting atoms,
whereas the Zeeman splitting and the quadrupole split-
ting are in the MHz and kHz regime, respectively. The
Rabi frequency of the two-level system is chosen to be of
the order of GHz, which can be adjusted by the intensity
of the laser field. The energy range of the electronic exci-
tation lies in the order of PHz. Thus, we have established
the necessary conditions for our protocol,
ω≈ω0≫Γ∼Ω≫ |γnB0|,∥Q∥.(22)
The effect of the spin system on the two-level system is
negligible, as the dominant decay channel Γ of the two-
level system is governed by the interaction with the en-
vironment and the dynamics of the two-level system is
much faster than the dynamics of the spin system. This
also justifies the Born-like assumption that the total den-
sity matrix of the two-level system and the spin system
remains decomposable throughout time-evolution. Thus,
the same derivation as above is applicable, and by tak-
ing the respective partial traces we obtain the dynamical
equations
i∂tρ2L≈[H2L+HE(t), ρ2L] + iΓL[σ]ρ2L+O(∥Q∥)
⟨Qµν ⟩(t) = tr2L{ρ2L(t)Qµν }+O∥Q∥2
i∂tρS= [HB+⟨Qµν ⟩(t)IµIν, ρS],
(23)
where we are denoting the density operator of the elec-
tronic two-level system and of the spin system as ρ2L
and ρS, respectively. The interaction of the open two-
level system with an external electric field will lead to
damped oscillations (see Section II B and panels (a) and
(b) of Figure 2), asymptotically approaching the steady-
state solution (11) with a decay rate of Γ. If the external
electric field is turned off, the two-level system will decay
to its ground state. The key feature of our proposed pro-
tocol is now the following: Enforcing a periodic repetition
of this process via a pulsed excitation, the nuclear spin
system can be controlled by its quadrupole-mediated in-
teraction with the electronic two-level system, which fea-
tures different EFG tensor values in different electronic
states. A graphical illustration of the proposed scheme
is given in Figure 1, where the pulse duration τ, the fre-
quency of the external undetuned field ω≈ω0, the decay
rate Γ, as well as the EFG tensors of ground and excited
state are illustrated. This is the basic principle of ONER.
Formally, we introduce a pulsed external electric field,
corresponding to a cosine modulated with a square pulse,
i.e. E(t) = E0Θ(tmod τ∈(0, τ /2)) cos(ωt), with
tmod τdenoting the fraction of tthat is in an inter-
val (nτ, (n+ 1)τ) for some n∈N. This way, a pulsed
modulation of the population of the excited state can be
achieved, as illustrated in the panels (a) and (b) of Fig-
ure 2. It shows one period of the square pulse envelop-
ing function of E(t) (red dashed line) and the resulting
population of the excited state of the two-level system
(blue solid line). The repetition rate τ−1of the square
pulse is chosen such that it matches the transition energy
of the spin system for a specific transition. Note that
1
τ≪Ω = O(GHz), since the Zeeman energy splitting of
the spin system is of the order of MHz. As long as (22)
is satisfied, the general analysis does not change since
exp (−Γτ/2) ≪1. This means that the excited state
fully decays into the ground state after half a period of
the pulse sequence. This way, a periodically pulsed mod-
ulation of the excited state population is achieved, which
translates into a corresponding modulation of the EFG
tensor and therefore also the NQI tensor quantities. For
a sufficiently large decay rate, the steady-state solution is
reached quickly, and the population of the excited state
resembles a pulsed square function in good approxima-
tion. For lower decay rates, more Rabi oscillations ap-
pear before a steady-state is reached and the edge decays
more slowly after the signal has been turned off, as it is
illustrated in Figure 2.
In any case, the density matrix of the two-level system
is periodically modulated with period τ. For convenience,
6
FIG. 1: Schematic illustration of a pulsed excitation of a two-level system {|g⟩,|e⟩} with frequency ω0, decay rate Γ,
external pulse duration τand external field frequency ω. For visibility, the frequency of the driving field is scaled.
The EFG tensors of ground and excited state are illustrated at the respective state level. More information on the
visualization of EFG tensors can be found in Appendix E.
the Fourier coefficients of the excited state population are
compared in panels (c) and (d) of Figure 2. The zeroth
and first Fourier component are well approximated by
the Fourier components of a square pulse, i.e. the steady-
state solution.
From equation (23) it follows that the time depen-
dence of the quadrupole interaction is directly inherited
from the time dependence of the two-level density ma-
trix. This can be used to determine the effective en-
ergy splitting of the spin system and the corresponding
Rabi frequencies for specific transitions. In order to ob-
tain analytical results of these quantities we investigate
the effective quadrupole coupling tensor ⟨Q⟩. Since the
quadrupole interaction tensor is real and symmetric, and
the two-level density operator is hermitian with trace
one, we get
⟨Q⟩(t) = tr2L{ρ2L(t)Q}
=ρ2L,ee(t)⟨e|Q|e⟩+ (1 −ρ2L,ee (t)) ⟨g|Q|g⟩
+ 2Re {ρ2L,eg (t)⟨e|Q|g⟩}.
(24)
Note that the off-diagonal elements of the quadrupole
coupling tensor, i.e. ⟨e|Q|g⟩and ⟨g|Q|e⟩, can be cho-
sen to be real and can therefore be pulled out of the
real part in the last term. Furthermore, as can be
seen from (11), the off-diagonal steady-state components
of the two-level density operator in the rotating frame,
ρ′
2L,eg and ρ′
2L,ge, are (mainly) imaginary and thus can-
cel effectively when taking the real part and time average
over a relevant timescale of the spin system. The time
average vanishes since the off-diagonal elements of the
density matrix in the non-rotating frame can be written
as ρ2L,eg(t) = ρ′
2L,ege−iω t, with ω≈ω0the frequency
of the driving field. Thus, on the timescale of the spin
system, we find
Re{ρ2L,eg(t)}= Re(1
TZt+T
t
ρ2L,eg(t′) dt′)≈0,(25)
where 2π
ω≪T≪τis some averaging time. This is
even more the case if dephasing terms are added for the
two-level system, since the off-diagonal density matrix
elements will decay even faster. This implies that only
the diagonal NQI tensor components, i.e. the quadrupole
coupling tensor of the ground and excited state, are rele-
vant for the time-dependence of the effective quadrupole
coupling in the spin system. Both can be expressed via
the excited state population of the two-level system, as is
immediately clear from (24) by neglecting the last term.
In order to extract the relevant time dependency of
the excited state population we investigate its Fourier
expansion, which is also illustrated in panels (c) and (d)
of Figure 2. Since the excited state population is real, we
may also use an expansion in a real Fourier series, which
yields
ρ2L,ee(t) = a(0)
2+X
nb(n)sin(ωnt) + a(n)cos(ωnt)
(26)
with ωn= 2πn/τ and a(n)=2
τRτ
0ρ2L,ee cos(ωnt) dtand
b(n)=2
τRτ
0ρ2L,ee sin(ωnt) dt. Due to the time symme-
try of the square pulse the sine coefficients are clearly
dominating, and the cosine part can be neglected except
for a constant contribution. If the conditions (22) hold,
7
(a) (b)
(c) (d)
FIG. 2: Population of the excited state of a two-level system under pulsed excitation with decay with
Rabi-frequency Ω ≫1
τ. The dashed red line in panels (a) and (b) indicates the enveloping function of the laser
pulse. The x-axes shows one pulse duration τ. Panel (a) shows the case of a large decay rate, Γ ≫1
τ, whereas a
moderate decay rate is shown in panel (b). In both cases the steady state solution is reached. In panel (b) the
excited state population oscillates in the beginning. After turning off the external pulse, the system relaxes into the
ground state, according to their respective decay rate. The Fourier analysis of the respective populations and a
comparison with the steady-state solution, i.e. a perfect square signal, is shown in panels (c) and (d). The x-axis
refers to the Fourier component n, with corresponding frequency ωn=2πn
τ. It can be seen that the steady-state
solution is a good approximation for the dominant Fourier modes in both cases. This becomes relevant in the
theoretical analysis of the ONER protocol.
the steady-state approximation can be applied (see also
panels (c) and (d) of Figure 2) and the coefficients are
well approximated by the coefficients of the step function
with height ρ∞
2L,ee, which are given by
a(0) =ρ∞
2L,ee
a(n)= 0,for n > 0
b(n)=(2ρ∞
2L,ee / πn, n odd,
0, n even. .
(27)
Since the repetition frequency τ−1of the pulse is ad-
justed to the transition energy of the spin system, the
8
zeroth and first order contribution are most relevant.
Higher frequencies of the Fourier decomposition of the
excited state population have negligible influence on spin
level transitions due to large detuning. Therefore, we
may write
⟨Q⟩(t)≈ρ2L,ee(t)⟨e|Q|e⟩+ (1 −ρ2L,ee (t))⟨g|Q|g⟩
≈ ⟨g|Q|g⟩+ (⟨e|Q|e⟩−⟨g|Q|g⟩)ρ2L,ee(t)
≈ ⟨g|Q|g⟩+ ∆Q(e, g)a(0)
2+b(1) sin 2πt
τ,
(28)
with ∆Q(e, g) := ⟨e|Q|e⟩−⟨g|Q|g⟩, by applying the
steady state approximation and neglecting high fre-
quency contributions. For given NQI tensors in the
ground state and the excited state of the two-level sys-
tem, respectively, it is then easy to obtain the actual
dynamics of the spin system by solving the respective
evolution equation (23). Within the steady-state approx-
imation, the spin Hamiltonian in an external magnetic
field B=B0ezcan then be written as
H≈ −γnB0Iz+IµIνQ(0)
µν (e, g)
+IµIνQ(1)
µν (e, g) sin 2πt
τ,(29)
where the constant quadrupole interaction term
Q(0)(e, g) = ⟨g|Q|g⟩+ρ∞
2L,ee
2∆Q(e, g),(30)
determines the quadrupole energy splitting (see Sec-
tion II A1 (4)) and the harmonically modulated
quadrupole tensor
Q(1)(e, g) = 2ρ∞
2L,ee
π∆Q(e, g),(31)
determines the magnitude of the Rabi frequency of the
spin system (see Section II A2 (5) and (6)). The EFG
tensor of the ground state and the excited state of the
two-level system can be obtained via ab initio methods.
The repetition rate (i.e. the energy splitting between spin
states of interest) and the resulting Rabi frequency of the
spin level transitions can then be calculated analogously
to Section II A1 and Section II A2, respectively.
Note that it is important to include the quadrupole
splitting which stems from Q(0) into the calculation of the
repetition rate τ−1in order to avoid detuning. Further-
more, the quadrupole splitting and the spin-level Rabi
frequency only depend on Ω/Γ via the steady-state pop-
ulation of the excited state ρ∞
2L,ee, if the conditions (22)
are satisfied. Thus, the intensity of the laser field should
be chosen such that the Rabi frequency of the two-level
system lies in the range of GHz. For a dipole moment
of approximately 1 Debye, this corresponds to an electric
field amplitude in the order of 105V
m.
2. Numerical simulation for 9Be
We pick a single 9Be atom for a first numerical simula-
tion of ONER, using the Molpro program package [31–33]
to calculate the relevant parameters, i.e. the 1Po←1S
transition energy as well as the electric field gradient Φµν
of the both states as a function of an applied external,
electric field. The choice of beryllium is motivated by the
fact that it features optical excitations in the UV/visi-
ble regime and a singlet electronic ground state; the to-
tal electron spin is zero and hyperfine coupling effects,
most noteworthy the substantial contributions stemming
from the Fermi contact term for s orbitals, are not rel-
evant. Employing the aug-cc-pVTZ basis set [35], we
combine a multiconfigurational self-consistent field cal-
culation (MCSCF [36,37]) with a follow-up multirefer-
ence configuration interaction approach (MRCI [38,39])
to account for dynamic correlation. For computational
efficiency, the atom is treated within the C2vmolecular
symmetry group. A symmetrically balanced active space
involving the orbitals 6/3/3/0 has been chosen with re-
spect to the internal ordering A1/B1/B2/A2. With this
setup, a perfect degeneracy of the three sublevels of the
P state is preserved at zero electric field, and an excellent
excitation energy of 5.30 eV is obtained for the 1Po←1S
transition, which deviates from the experimental value of
5.28 eV tabulated at NIST by less than 0.5 percent [40].
An evaluation of the electric field gradient of Be in its
well-studied 3Polowest triplet state, calculated with the
same settings, produces a value of -0.1199 a.u., which
agrees well (about 4% deviation) with earlier calculated
values from literature [41]. Sternheimer shielding effects
lie within this uncertainty [42]. Note that a single value
is sufficient to characterize the EFG tensor in this case
due to spherical symmetry [43].
9Be has a non-zero scalar quadrupole moment of
0.0529(4) barn [26,27]. Its gyromagnetic moment is
given by γBe9
n=−1.17749(2) µN≈8.9755 MHz
Tas given
in Ref. [26], where µN= 7.622593285(47) MHz
Tis the nu-
clear magneton. 9Be has a total nuclear spin quantum
number of I= 3/2, leading to an energy scale of several
MHz for the Zeeman Hamiltonian of the spin system. We
assume that the 9Be atom is subject to a constant mag-
netic field B. The direction of the latter will be used
to define a reference axis of the spin system, as the Zee-
man splitting is dominant for the nuclear spin. Also, a
constant electric field Eis applied to set a reference axis
for the two-level system and to tune the Rabi frequency
of the spin transitions. An illustration of the coordinate
frames is given in Figure 3. Quantities which are frame-
dependent will be marked with a superscript Eor Bfor
the coordinate frame being either aligned with the elec-
tric or the magnetic field, respectively. Both frames have
the same x-axis, which we choose to be the direction of
propagation of the laser field. The angle between the
zB-axis and the zE-axis is denoted as θ.
In Figure 4, the non-zero components of the NQI ten-
sor are shown in the E-frame. The energy splitting of
9
FIG. 3: Orientations of the external magnetic field B,
the electric field E, the propagation direction of the
pulsed signal kx, and the three sublevels (pE
x,pE
y,pE
z) of
the 1Poelectronically excited state of the 9Be atom.
The angle between the zB-axis and the zE-axis is
denoted as θ.
the NQI lies in the range of several kHz, which justifies
the treatment of the energy correction via perturbation
theory as discussed in Section II A1.
In more detail, Figure 4 compares the electric field-
dependent NQI tensors for sE-ground state ⟨sE|QE|sE⟩,
the pE
y-excited state ⟨pE
y|QE|pE
y⟩, and the pE
z-excited
state ⟨pE
z|QE|pE
z⟩. Note that the NQI for the pE
x-excited
state is identical with ⟨pE
y|QE|pE
y⟩but with xx and yy
components swapped. The shapes of the respective NQI
tensors of ground state and excited state, shown in Fig-
ure 4, are crucial for the calculation of the repetition rate
and the obtained Rabi frequency of the spin level transi-
tions. Due to spherical symmetry of the sE-ground state
it has a vanishing NQI at zero field. Also, for a finite
field strength, the respective NQI lies in the range of a
few kHz and is thus negligibly small compared to the
NQI of the excited states. The pE-excited states have a
non-vanishing quadrupole interaction of the same magni-
tude at zero field but with interchanged axis. This is the
expected behavior, since p-orbitals have a non-vanishing
EFG tensor at the origin. The different behavior of pE
z-
excited state and pE
y-excited state is caused by the con-
stant electric field in zE-direction.
The corresponding electronic excitation energies of the
1Po←1Stransition are illustrated in Figure 5. As
expected for a Stark splitting in the external field, the
pE
x- and pE
y-transitions remain degenerate, while the pE
z-
transition occurs at a different energy. The excitation
energy for this two-level system lies around 1.28 PHz,
which corresponds to 234 nm or 5.30 eV. We assume no
detuning, i.e. ω≈ω0, and choose the amplitude of the
electric field such that a Rabi frequency in the range of
GHz is obtained for the two-level system. For a dipole
moment of approximately 1 Debye this corresponds to an
FIG. 4: Non-zero components (xx,yy,zz) of the
nuclear quadrupole interaction (NQI) tensor for 9Be are
shown for a laboratory frame aligned with the electric
field. The sE-ground state NQI ⟨sE|QE|sE⟩is shown
with dotted lines, the pE
z-excited state NQI ⟨pE
z|QE|pE
z⟩
is shown with a solid line and the pE
y-excited state NQI
components ⟨pE
y|QE|pE
y⟩are shown with dashed lines.
For the sE-ground NQI and the pE
z-excited state NQI
the xx and the yy component overlap, due to symmetry.
At zero field the sE-ground state is spherical symmetric
and thus has vanishing NQI. Also for finite field
strength the NQI of the sE-ground state is negligibly
small. Both pE
z-excited state and pE
y-excited state have
similar NQI for zero field, but with swapped axis. For
non-zero field the behavior differs, since the field is
aligned with the zE-axis.
electric field amplitude of the order of 105V
m. The Rabi
frequency of the two-level system should be chosen such
that the damping Γ is approximately of the same order;
for the further discussion we choose 0.4 Ω. This ensures
that the steady-state approximation is valid and leads to
a steady-state population of
ρ∞
2L,ee =25
54,(32)
as obtained via (11) with γ⊥= Γ/2. With the param-
eter values set as motivated above, the steady-state ap-
proximation holds and we can apply the analysis of the
previous section.
From the NQI tensor components shown in Figure 4
we can calculate the constant NQI tensor Q(0),E(pE
i, sE)
via (30) and the harmonically modulated NQI tensor
Q(1),E (pE
i, sE) from (31) in the E-frame for i∈ {x, y, z}.
Since the laboratory frame of the spin system is the B-
frame we have to apply a rotation to the respective NQI
tensors. The corresponding matrix for a rotation around
10
FIG. 5: Stark splitting of the 1Po←2Stransition of
9Be in an external electric field. pE
x- and pE
y-transitions
remain degenerate, while the pE
z-transition occurs at a
lower energy.
the common x-axis by an angle θis given by
R(θ) =
1 0 0
0 cos(θ)−sin(θ)
0 sin(θ) cos(θ)
.(33)
The respective NQI tensors in the B-frame are then given
by
Q(j),B(pE
i, sE) = R(θ)Q(j),E (pE
i, sE)R(θ)⊤,(34)
for j∈ {0,1}and i∈ {x, y, z}. Note that the E-frame
still remains the reference for electronic states. The NQI
tensor values in the B-frame can now be used to calculate
the quadrupole energy correction and thus the correct
repetition rate of the laser pulse, the ∆mI=±1 and
∆mI=±2 transition elements, and the corresponding
Rabi frequency of the spin level transitions. In Figure 6,
the transition energy corrections for the different spin
level transitions are given in units of kHz for different
electric field strengths Eand angles θbetween zEand zB.
Only the quadrupole corrections are shown, i.e. equa-
tion (4) without the Zeeman splitting. The magnitude
of the Zeeman splitting depends on the magnitude of the
external magnetic field. However, we assume that the ex-
ternal field is in the order of Tesla, leading to a MHz Zee-
man splitting. We denote the quadrupole energy correc-
tion for the spin level transition from state |mI⟩to |m′
I⟩
for given electronic excitation as ∆EB(mI→m′
I|pE
i, sE)
with i∈ {x, y, z}. Figure 6 shows the transition energy
corrections for the 3/2↔1/2 transition, which are iden-
tical to those for the 3/2↔ −1/2 transition. The cor-
rection energies for the −1/2↔ −3/2 transition and the
1/2↔ −3/2 only differ in sign. The 1/2↔ −1/2 tran-
sition is not shown since it has a vanishing transition
element, as can be seen from (5). The quadrupole correc-
tion energy for the pE
z-excited state NQI shows stronger
changes with increasing electric field strength of the con-
stant field E, while the quadrupole correction for the pE
y-
excited state NQI differs much less. This is reasonable,
since the electric field is aligned with the zEaxis.
FIG. 6: Quadrupole energy corrections in kHz for
different electric field strengths Eand different angles θ
between zEand zBfor electronic excitations into the
pE
y-state (solid line) and pE
z-state (dashed line),
respectively. The corrections are shown for the
3/2↔1/2 transition, but are identical to those for the
3/2↔ −1/2 transition. The corrections for
−1/2↔ −3/2 and the 1/2↔ −3/2 only differ in sign.
All quadrupole energy corrections are in the order of
100 kHz. The calculations were performed for a
two-level steady-state population of ρ∞
2L,ee =25
54 , which
corresponds to Γ = 0.4Ω.
In total, this leads to a repetition rate τ−1of the pulse
sequence in the range of several MHz. The quadrupole
corrections are in the order of 100 kHz, and the repeti-
tion rate has to be chosen such that it matches the cor-
rected transition energy of a spin level transition. The
quadrupole correction also enables individual address-
ability of the spin transitions, since equidistant Zeeman
transition energies are shifted individually by NQI. Given
a suitable repetition rate, we observe Rabi oscillations of
the spin level transitions. The Rabi frequency of the
spin level transitions is determined by the modulus of
the respective transition element given in (5) and (6) for
the corresponding harmonically modulated NQI tensor
Q(1),B(pE
i, sE). We denote the respective transition ele-
ments from state |mI⟩to state |m′
I⟩for given electronic
excitation by gB(mI→m′
I|pE
i, sE), with i∈ {x, y, z}.
For a nuclear spin quantum number of I= 3/2 the
prefactors αand βof the ∆m=±1 and ∆m=±2
11
transitions in (5) and (6)), respectively, are given by
β3/2↔−1/2=β1/2↔−3/2=√3/2
α3/2↔1/2=α−1/2↔−3/2=√3
α1/2↔−1/2= 0.
(35)
The 1/2↔ −1/2 transition is forbidden, whereas the
other transitions might have a non-vanishing transition
amplitude, depending on the respective NQI tensor.
Figure 7 illustrates the dependence of the resulting
Rabi frequencies on the relative angle θbetween E-
frame and B-frame for different values of the electric
field strength E. Panel (a) shows the Rabi frequency
for the 3/2↔1/2 transition, which is the same for the
−1/2↔ −3/2 transition, and panel(b) shows the Rabi
frequency for the 3/2↔ −1/2 transition, which is also
the same for the 1/2↔ −3/2 transition.
In general, the Rabi frequency for the NQI tensor of
the pE
z-excited state can be more easily controlled with
the external constant electric field. The Rabi frequency
for the NQI tensor of the pE
y-excited state varies much
less; for the ∆mI=±1 transitions in particular, almost
no dependence can be observed. Note that, for zero field,
the transition elements of the NQI tensor for both excited
states coincide.
The general shape of the θ-dependence of the Rabi
frequency can be explained as follows. For an angle of
θ= 0 the NQI of both excited states is diagonal and
therefore no ∆mI=±1 transitions can be driven as the
transition element vanishes. This is immediate from (5).
The same happens for θ=π/2. For θ=π/4, the off-
diagonal elements are maximized, as can be easily derived
from the shape of the rotation matrix.
For the pE
z-excited state also ∆mI=±2 transitions
cannot be driven at θ= 0, since xx and yy components
coincide (see (6)). For the pE
y-excited state, the xx and yy
components of the corresponding NQI tensor are maxi-
mally different in this case, leading to a maximum of the
transition element at θ= 0. For the NQI of the pE
z-
excited state, the xx and yy components are maximally
different for an angle of π/2, since the corresponding ro-
tation into the B-frame swaps the yy and zz component.
Depending on the transitions one is interested in, the an-
gle has to be chosen accordingly. A reasonable choice
would be θ=π/4, since all transitions can then be ad-
dressed. The resulting Rabi frequencies lie in the range
of 100 kHz, depending on the orientation and the mag-
nitude of the constant electric field. For comparison, the
Rabi frequency in the NER studies of Ref. [18] was found
at 68 kHz.
Finally, in Figure 8, we show the resulting Rabi oscil-
lations from a numerical simulation of the coupled sys-
tem, i.e. a numerical simulation of the open two-level
system coupled to a spin system via nuclear quadrupole
interaction. The total Hamiltonian from (II A) with de-
cay operators for the two-level system is simulated for
an external pulsed excitation with repetition rate τ−1
with the standard Master equation solver of the Python
library QuTiP [29,30]. The simulation parameters are
chosen as Γ = 0.4Ω and ∆ = 0, with Ω = 1 GHz, simi-
lar to the above analytical investigation of the system.
The angle between electric field and magnetic field is
chosen as θ=π/4, the constant electric field strength
was chosen to be E= 0.01 a.u. and the magnetic field
strength was set to 1 Tesla. An electronic excitation into
the pE
z-excited state is assumed. The repetition rate is
chosen such that it matches the desired spin level tran-
sition energy including the quadrupole correction term.
Note that the analytical treatment is necessary for this
step, as it delivers the corrected repetition rate of the
laser pulses. Figure 8 shows the Rabi oscillations of the
numerical simulation of the spin system for different pulse
frequencies matching the transition energies of the spin
system for a 3/2↔1/2 and a 3/2↔ −1/2 transition,
as derived from the analytical considerations. The time
axis is normalized to the analytically obtained Rabi fre-
quency of the respective transition. It can be seen that
there is a slight deviation of the numerically obtained
Rabi frequency and the analytical one, since the maxima
and minima are not perfectly at integer and half-integer
values. However, they are in good agreement.
To summarize, this shows that a nuclear spin system
can be controlled by pulsed electronic excitation of a cou-
pled two-level system. Since the transitions appear at
different energies, they can be addressed individually by
selecting an appropriate repetition rate for the external
pulses. An additional decay channel of the two-level sys-
tem, stemming from the interaction with the spin system,
is not problematic since the decay of the two-level system
is mainly governed by external couplings. In fact, the
two-level system can be considered effectively indepen-
dent of the spin system, which enables a simplified anal-
ysis of the combined system and the imposed quadrupole
interaction between electronic excitation and nuclear spin
manipulation. The simplified analytic discussion is cor-
roborated by the numerical results obtained for the total
coupled system.
IV. CONCLUSION
Fundamental principles and practical implications of
nuclear quadrupole resonance as a tool for quantum com-
puting were investigated. A detailed theoretical descrip-
tion of nuclear quadrupole coupling was developed from
first principles. The quadrupole interaction Hamiltonian
was derived from the molecular Hamiltonian in a con-
sistent manner by employing Taylor expansions for non-
pointlike nuclei. Important time-independent and time-
dependent properties of the nuclear quadrupole Hamil-
tonian were discussed, and the coupling of electric field
gradient and nuclear spin was investigated in detail.
Within the adiabatic approximation, and assuming a
quasi-independence of the electronic system from the nu-
clear spin system, a consistent, general description for
future discussions of NER and NAR experiments was ob-
12
(a) (b)
FIG. 7: Rabi frequencies in kHz for different electric field strengths Eand different angles θbetween zEand zBfor
pE
y-excited state (solid line) and for pE
z-excited state (dashed line) NQI. Panel (a) shows the Rabi frequencies for the
∆mI=±1 transitions, whereas panel (b) is showing the Rabi frequencies for the ∆mI=±2 transitions. In both
cases the Rabi frequencies are in the order of up to 100 kHz. The calculations were performed for a two-level
steady-state population of ρ∞
2L,ee =25
54 , which corresponds to Γ = 0.4Ω.
(a) (b)
FIG. 8: Rabi oscillations of the numerical simulation of a quadrupolar nucleus with nuclear spin I= 3/2 via pulsed
excitation of a coupled two-level system subject to an external constant magnetic field of magnitude 1 Tesla. The
simulation parameters are chosen for an electric field strength of E= 0.01 a.u. with an angle of θ=π/4 to the
magnetic field axis. The two-level system parameters were chosen as Γ = 0.4Ω and ∆ = 0 with Ω = 1 GHz.
Electronic excitation was performed into the pE
z-excited state. The y-axis shows the population of the respective
spin states mI. Panel (a) shows the 3
2↔ −1
2transition, panel (b) the 3
2↔1
2transition. The x-axes are normalized
to the analytically calculated Rabi frequency. It can be seen that in both cases the analytical frequency and the
numerical are in good agreement. Slight deviations can be seen, since maxima and minima are not located at integer
and half-integer values of the normalized time.
tained (a summary of the applied approximations can be
found in Appendix G). Our formalism exceeds simpler
phenomenological models and lays a consistent founda-
tion for commonly applied approximations, which can be
obtained as special cases of our more general description
(a typical phenomenological model is discussed in Ap-
pendix F).
Putting this general description to practice, we further
propose a new scheme for nuclear electric resonance us-
ing pulsed electronic excitation. Polarized laser pulses
in the UV/visible spectrum can be used to drive spin
transitions via a coupling of the EFG tensors in different
electronic states of an atomic or molecular system to the
nuclear quadrupole moment. We refer to this technique
13
as ‘optical nuclear electric resonance’ (ONER). It exploits
changes in the electric field gradient at the position of the
nucleus, which are induced by electronic excitation, e.g.
from the electronic ground state into a suitable excited
state. In a first numerical test on atomic 9Be as a bench-
mark, Rabi oscillations for nuclear spin transitions in the
order of several kHz are predicted.
Based on these first results, we believe that ONER
has the potential to link coherent nuclear spin manipu-
lation with well-established concepts of optoelectronics
and nanophotonics. As a third paradigm aside NER and
NAR, this new protocol offers the advantage of an in-
creased flexibility with respect to the addressing of molec-
ular spin systems: It combines a selectivity which is in-
trinsic to electronic excitations and their specific effect
on the electric field gradient at the various nuclear po-
sitions of a molecular system. Furthermore, it has the
ability to select specific spin transitions of these nuclei
via pulsed laser light, by tuning the repetition rate to a
certain transition of interest.
Future research will be devoted to the simulation of
larger, more complicated but experimentally accessible
systems with the possibility to address and couple specific
nuclear spins, e.g. within the same molecule or a given
molecular qubit register, through different electronically
excited states. Potential candidate systems are metal
complexes with reduced magnetic noise and minimal vi-
brational coupling [44]. Alternatively, regarding the in-
dividual addressing of spatially separated quantum sys-
tems such as cold atoms, ions, or solid-state qubits e.g.
via light-shift gradients [45,46] or non-linear response in
two-level systems [47], new techniques of quantum opti-
cal control may emerge from a combination of methods.
As a final comment, we would like to emphasize that
the current formalism does not involve any coupling to
the electron spin, which is assumed zero. In singlet sys-
tems, the only spin-spin coupling takes place between nu-
clei, which might be a technical advantage, and should
be the concern of future publications on the subject.
Appendix A: Overview
Appendix B contains a detailed derivation of the nu-
clear quadrupole Hamiltonian. Appendix C provides a
brief introduction to density operators and open quan-
tum systems. Appendix D gives an overview of common
units and the order of energy splittings due to quadrupole
effects. Appendix E contains a description of graphi-
cal illustrations of EFG tensors for the sake of an im-
proved, visual understanding of the tensor components
occurring in the main text. In Appendix F we provide a
phenomenological description of the nuclear quadrupole
coupling that is commonly used to describe nuclear elec-
tric resonance or nuclear acoustic resonance. Appendix G
summarizes the approximations made in the theoretical
description of NER, NAR, and ONER.
Appendix B: Derivation of the nuclear quadrupole
Hamiltonian
We start from the usual many-particle Hamiltonian of
molecular physics in the absence of external fields. In
natural units (ℏ=1
4πϵ0=me=a0= 1) it reads
H(0) =
TE
z }| {
X
i−1
2∇2
i+
T(0)
N
z }| {
X
A−1
2MA∇2
A+
VEE
z }| {
X
i<j
1
r(i)−r(j)
+X
A<B
ZAZB
R(A)−R(B)
| {z }
V(0)
NN
+X
i,A −ZA
r(i)−R(A)
| {z }
V(0)
EN
,
(B1)
with lower- and upper-case indices for electron and nu-
clear coordinates, respectively. The position of the i-th
electron is denoted as r(i), and ∇iis the derivative with
respect to this position. The position of the A-th nucleus
is denoted as R(A)and ∇Ais the derivative with respect
to this position. Charge and mass of the A-th nucleus
are denoted as ZAand MA, respectively. Thus, TErep-
resents the electron kinetic energy, T(0)
Nthe nuclei kinetic
energy, VEE the electron-electron interaction, V(0)
NN the
nucleus-nucleus interaction and V(0)
EN the electron-nuclei
interaction. The superscript (0) will turn out convenient
in the further discussion.
In this simplified Hamiltonian, the nuclei are thought
of as point-like particles of mass MAand charge ZA; their
actual, non-trivial charge distribution is neglected. How-
ever, since the deviation of proton positions from the
center-of-charge of the respective nucleus can be consid-
ered small, we can apply a Taylor series expansion up to
second order to obtain correction terms for a non-point-
like charge distribution of the respective nuclei. Introduc-
ing center-of-charge coordinates R(A)=1
ZAPpAR(A,pA)
and denoting the deviation of each proton from this cen-
ter as δR(A,pA)=R(A,pA)−R(A), we obtain the corrected
total Hamiltonian
H=H(0) +1
6X
A
Φ(A)
µν Q(A)
µν ,(B2)
with Φ(A)
µν as the total electric field gradient tensor at the
position of the A-th nucleus,
Φ(A)
µν =X
B=A
ZB
R(A,B)53R(A,B)
µR(A,B)
ν−δµν R(A,B)
2
−X
i
1
r(i,A)53r(i,A)
µr(i,A)
ν−δµν r(i,A)
2,
(B3)
with definitions r(i,A):= r(i)−R(A)and R(A,B):= R(A)−
R(B)for electron-nucleus and nucleus-nucleus difference
14
vectors, respectively. Q(A)
nm denotes the quadrupole mo-
ment of the respective nucleus, which can be expressed
as
Q(A)
µν =X
pA3δR(A,pA)
µδR(A,pA)
ν−δµν δR(A,pA)
2.
(B4)
The nuclear quadrupole moment of each nucleus can
be related to the total nuclear spin by employing the
Wigner-Eckart theorem. The nuclear quadrupole tensor
is a traceless symmetric second rank tensor. It is pro-
portional to the symmetric traceless second rank total
angular momentum tensor 3
2(IµIν+IνIµ)−δµν I2in a
subspace with constant nuclear spin quantum number
I. This is satisfied, in good approximation, for the case
of nuclear electric resonance, since the orbital angular
momentum of the nucleus can be considered constant.
Thus, it is sufficient to calculate the matrix elements in
the basis |I, mI⟩of the magnetic quantum numbers of the
nuclear spin, with spin quantum number Iand magnetic
quantum number mI. Calculating the proportionality
constant in the |I, I ⟩state yields
Qµν =q
I(2I−1) 3
2(IµIν+IνIµ)−δµν I2,(B5)
with q:= ⟨II|Q33 |II⟩as the scalar quadrupole mo-
ment of the nucleus. The numerical values of the scalar
quadrupole moments of different nuclei are tabulated in
Refs. [26,27]. Note that only nuclei with nuclear spin
I > 1
2can have a non-vanishing quadrupole tensor, as
indicated by the proportionality constant.
Exploiting that the electric field gradient tensor Φ is
symmetric and traceless, the nuclear quadrupole Hamil-
tonian for a single nucleus can be rewritten as
HQ=1
6Φµν Qµν =IµQµν Iν,(B6)
with Qµν =q
2I(2I−1) Φµν .
Appendix C: Open quantum systems
There are several different timescales involved and in-
teractions with the environment of the molecular system
need to be taken into account. This is achieved by intro-
ducing the density operator ρ, which has the properties
ρ=ρ†≥0,tr{ρ}= 1,(C1)
and, in the case of an isolated system, obeys the von
Neumann equation
iℏ∂tρ= [H, ρ].(C2)
As a hermitian operator the density operator can be ex-
pressed in its eigenbasis via ρ=Pαpα|ψα⟩⟨ψα|. The
expectation value of an observable Ois calculated by the
trace tr{ρO}=Pαpα⟨ψα|O|ψα⟩, highlighting the sta-
tistical nature of the density operator.
In open systems, the system of interest interacts with
an environment. The total system, SE, consisting of the
system of interest Sand the environment E, is treated
via a joint density operator ρSE whose dynamics is given
by the von Neumann equation. Since we are only inter-
ested in the dynamics of the system S, we want to reduce
this density matrix to a density matrix of that particular
system only, such that expectation values of observables
and matrix elements of the system remain the same as for
the total system SE. This is reasonable because the dy-
namics of the environment is unknown in most cases and
observables are only accessible for the system S. For that
purpose, the reduced density operator ρSis introduced
as the partial trace of the total density operator,
ρS= trE{ρSE }=X
mE⟨mE|ρSE |mE⟩,(C3)
for some basis |mE⟩of the environment Hilbert space.
Due to the linearity of the trace operator, it is immedi-
ately clear that expectation values of observables of the
system OScan be calculated via
⟨OS⟩= tr {ρSE OS}= trS{OStrEρS E }
= trS{OSρS}.(C4)
Additionally, it is easy to check that the reduced den-
sity operator fulfills the conditions for a density operator
stated in (C1) in the Hilbert space of the system S. In
general, the Hamiltonian of the total system may be writ-
ten as
H=HS⊗IE+IS⊗HE+HSE ,(C5)
with HSand HEas operators acting exclusively on the
system and the environment, respectively, and HSE as
a coupling term. IEand ISare denoting unit opera-
tors in their corresponding Hilbert space. If the interac-
tion between system and environment is negligible, i.e.
HSE ≈0, then the dynamical equation for the reduced
density operator is given by
iℏ∂tρS= [HS, ρS] (C6)
as can be checked easily. In this case, the system can be
treated as an isolated system. However, if the interac-
tion between system and environment is not negligible,
the treatment of the dynamics of the system becomes
much more involved. A common ansatz to handle such
cases is the Born-Markov approximation, which leads to
a non-unitary evolution equation for the reduced density
operator, often referred to as Linblad Master equation.
A full derivation can be found in Ref. [28]. One starts
from the evolution equation for the total system
iℏ∂tρSE = [H, ρS E ],(C7)
and the assumptions of initial separability, i.e. ρSE (0) =
ρS(0)⊗ρE(0), separability during time evolution and con-
stant environment (Born approximation), i.e. ρSE (t) =
15
ρS(t)⊗ρE, a short memory environment (Markov approx-
imation) and a coarse grained system dynamics (secular
approximation).
This leads to the Born-Markov Master equation of the
reduced system
iℏ∂tρS= [HS, ρS]+iℏX
α
kαL[cα]ρS,(C8)
with decay strengths kαand the Linblad superoperator
Ldefined by
L[c]ρS=cρSc†−1
2c†cρS+ρSc†c(C9)
for a so-called collapse operator c. The prefactors kαof
the non-unitary evolution terms can be interpreted as the
rate of the process described by the coupling operators,
for example the strength of dissipation due to coupling
to the environment. Note that a Master equation of this
type is trace-preserving and ensures that the density op-
erator is hermitian, which is important to ensure that
the solution to the Lindblad Master equation is indeed a
density operator.
Appendix D: Units and order estimates
In this section we provide numerical estimates of typ-
ical energy scales within the context of quadrupole in-
teraction. So far, atomic units have been used, i.e.
ℏ=1
4πϵ0=me=a0= 1. Within the SI, the EFG tensor
is given in units of V
m2. The corresponding atomic units
are 1 au = EH
ea2
0, with a0denoting the Bohr length, ethe
elementary charge and EHthe energy in Hartree. The
conversion factor to SI units is 1 au = 9.717 ×1021 V
m2.
Typical values for the scalar quadrupole moment of
a nucleus are of the order of 10−3−1 barn, where
1 barn = 10−28 m2, as can be seen from the table of
scalar quadrupole moments given in Refs. [26,27]. Since
qis very small, the EFG necessary to generate a sig-
nificant quadrupole interaction needs to be very large.
Typically, only a microscopic mechanism in a crystal lat-
tice or molecule, such as the distortion of covalent bonds
in the vicinity of the nucleus, creates a significant EFG;
values of the latter range from 1016 V
m2to 1021 V
m2. For
scalar quadrupole moments in the range of 10−3−1 barn
this yields an interaction strength of the order of kHz to
MHz.
Often, a magnetic field is applied as well to generate a
Zeeman-splitting of the nuclear energy levels. The mag-
nitude of the gyromagnetic moment is of the order of
1−50 MHz
Tas shown in Refs. [26,27]. Assuming a mag-
netic field of approximately 1 T, this results in a level
splitting in the MHz regime. In comparison to that, the
gyromagnetic moment of the electron is given by approx-
imately 28 GHz
T. Hence, besides direct stimulation via
radio frequency adsorption, only an acoustic phonon ex-
citation in solids offers a possible coupling within this
energy regime. These options give rise to the possibility
of nuclear electric resonance (NER) or nuclear acoustic
resonance (NAR) as experimental techniques that have
been studied so far in recent years [18,24,25].
Appendix E: Graphical illustration of EFG tensors
In order to improve our understanding of the structure
and the origin of EFG tensor components we want to dis-
play them graphically and investigate different examples
of combinations of atomic orbitals, since, in the general
case, linear combinations of atomic orbitals have to be
considered. As defined in Ref. [48], we will consider the
function
f(r) = sX
ij
ri⟨Φij ⟩rj=s∥r∥2g(ϕ, θ) (E1)
with ⟨Φij ⟩being the expectation value of the EFG tensor
in the state to investigate and sa scaling parameter, to
illustrate the EFG tensor as a three-dimensional surface
plot. We set ∥r∥=|g(ϕ, θ)|and plot the resulting surface
in blue if g(ϕ, θ)>0 and orange if g(ϕ, θ)<0. This is a
general scheme that can be used to illustrate symmetric
second rank tensors graphically. An example is given
in Figure 9, where the isotropic case and three feasible
EFG tensors for different asymmetry parameters ηare
presented. The asymmetry parameter is defined as
η=|Φy′y′−Φx′x′|
|Φz′z′|,(E2)
where the x′, y′, z′are the eigenvectors of the EFG ten-
sor, with z′corresponding to the largest eigenvalue. If
the EFG tensor is not diagonal in the chosen laboratory
frame, it will appear as a rotation of a diagonal tensor
in the principle axis system to the laboratory frame. In
molecular systems, the origin of the tensor is shifted to
the origin of the nucleus of interest.
Appendix F: Phenomenological description of
nuclear quadrupole coupling
The theoretical treatment in Section III A1 entails
and justifies a commonly applied model for nuclear
quadrupole coupling, which is briefly discussed here. We
assume an EFG tensor created by a specific charge dis-
tribution n. The EFG tensor at the position of a nucleus
Aof a molecular system (with fixed nuclei), denoted as
Φ(A)
µν , is given in (B3).
Since only one-electron operators appear in this ex-
pression, we can relate the expectation value ⟨ψ|Φ(A)
µν |ψ⟩,
for a given state |ψ⟩of the system, to an integral over
16
(a) isotropic, diag(1,1,1) (b) axial, η= 1, diag(1,−1,0)
(c) axial, η= 0, diag(1,1,−2) (d) intermediate, η= 1/3, diag(−3,2,1)
FIG. 9: Graphical representation of a second rank tensor with varying asymmetry parameters ηin the principle axis
system. (a) shows an isotropic second rank tensor, (b), (c) and (d) traceless second rank tensors with asymmetry
parameters η= 1, η= 0 and η= 1/3, respectively.
the total charge density
n(r(1)) = −NZψ(r(1) , . . . , r(N))
2d3r(2) . . . d3r(N)
+X
B
ZBδ(r(1) −R(B)),
(F1)
with Nas the total number of electrons in the system,
and obtain
Φ(A)
µν =Z1
r(A)53r(A)
µr(A)
ν−δµν r(A)
2n(r) d3r,
(F2)
again with r(A)=r−R(A), as defined in Appendix B.
Note that this description incorporates also movements
of the nuclei, yet in a classical manner, within the Born-
Oppenheimer approximation. Assuming that the posi-
tion of the nucleus Ais fixed, which can always be done
by coordinate transformation, the first term in equa-
tion (F2) does not change under application of an electric
field Eor mechanical strain εto the system. Thus, a vari-
ation of the EFG tensor is mediated through the change
of the total charge density.
In a linear expansion around the zero-point position,
i.e. zero electric field and strain, we can write
Φ(A)
µν ≈Zd3r3r(A)
µr(A)
ν−δµν r(A)2
r(A)5
× n(r)E=0
ε=0
+δn(r)
δεαβ E=0
ε=0
εαβ +δn(r)
δEγE=0
ε=0
Eγ!.
(F3)
This expression suggests the functional form
Φ(A)
µν ≈Φ(A),(0)
µν +Sµναβ ϵαβ +Rµν γ Eγ(F4)
for the electric field gradient tensor at the position of
nucleus A, with Φ(A),(0)
µν denoting the EFG tensor of the
unperturbed system, Sµν αβ as a fourth rank coupling ten-
sor of strain and the EFG tensor, and Rµνγ as a third
rank coupling tensor of electric field and the EFG tensor.
17
Note that we sum over Greek indices appearing twice in
one term. The tensors Sµν αβ and Rµνγ are determined
by the change of the total charge density due to the in-
teraction of the system with strain and electric field. In
a simplified picture, we can deduce that the distortion of
the electron hull or the atomistic environment leads to
an electric field gradient at the position of the nucleus.
Due to the ∼1
r3characteristics of the EFG tensor, we
would expect that only effects in the close neighborhood
of the nucleus of interest are relevant. The coupling of
an external static electric field to the EFG tensor is a
well-known phenomenon, which is also known as linear
quadrupole Stark effect [49,50].
The derived relation between EFG tensor components
and external quantities can be generalized to the time-
dependent case, if we assume that the density relaxes in-
stantaneously to the ground state, i.e. that the timescale
of the external field variations is much larger than the
timescale of the intrinsic dynamics of the electronic sys-
tem. As can be seen from equations (F3) and (F4), a
possible time-dependence of the strain εαβ or the electric
field Eγis not entering the derivation. Therefore, the
obtained results are only valid in the adiabatic approx-
imation for slowly varying strain or electric field, i.e. if
the assumption, that the electronic system remains in its
adiabatic ground state during evolution, is justified.
Given the functional dependence of the EFG tensor in
a linearized adiabatic approximation on the electric field
or strain, we immediately see that the time dependence of
the EFG tensor is inherited from the time dependence of
the external quantity. Thus, an application of an electric
field E(t) = E0cos(ωt) with a frequency in the radio-
frequency regime, matching the transition frequency of
nuclear spin states, may be used to locally modulate the
EFG and thereby control nuclear spin state occupations
coherently. This scheme has already been verified exper-
imentally for a 123Sb (spin 7/2) nucleus in silicon [18].
The interaction strength is determined by the intensity
of the electric field and the quadrupole coupling tensor.
Note that this scheme enables a pure electric control of
the nuclear spin system without the need for oscillating
magnetic fields in the radiofrequency regime.
Appendix G: List of approximations in the model
For convenience, all approximations applied in the
main part of this article are summarized and listed below:
•Born approximation for total density matrix, i.e.
decomposable density matrix throughout time-
evolution
•negligible effect of spin system on electronic system
due to relatively small coupling
•negligible effect of hyperfine coupling between elec-
tron spin and nuclear spin
•adiabatic approximation for the electronic system
All of these approximations are reasonable if the inter-
action of spin system and electronic system can be con-
sidered small and the time-dependence of the external
field is quasi-static in comparison to typical time-scales
of the electronic system. Both conditions are well sat-
isfied in the case of the nuclear quadrupole interaction
for NER and NAR. However, these conditions are also
satisfied for excitations of the electronic system, as it is
shown for the ONER protocol in Section III B.
ACKNOWLEDGMENTS
Financial support by the Austrian Science Fund
(FWF) under Grant P-36903N is gratefully acknowl-
edged. We further thank the IT Services (ZID) of the
Graz University of Technology for providing high perfor-
mance computing resources and technical support.
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