# Reevaluation of Stark-induced transition polarizabilities in cesium

H. B. Tran Tan  D. Xiao  A. Derevianko<EMAIL_ADDRESS>Department of Physics,
University of Nevada, Reno, 89557, USA

###### Abstract

Extracting electroweak observables from experiments on atomic parity violation
(APV) using the Stark interference technique requires accurate knowledge of
transition polarizabilities. In cesium, the focus of our paper, the
$6S_{1/2}\rightarrow{7S_{1/2}}$ APV amplitude is deduced from the measured
ratio of the APV amplitude to the vector transition polarizability, $\beta$.
This ratio was measured with a $0.35\%$ uncertainty by the Boulder group
[Science 275, 1759 (1997)]. Currently, there is a sizable discrepancy in
different determinations of $\beta$ critically limiting the interpretation of
the APV measurement. The most recent value [Phys. Rev. Lett. 123, 073002
(2019)] of $\beta=27.139(42)\,\mathrm{a.u.}$ was deduced from a semi-empirical
sum-over-state determination of the scalar transition polarizability $\alpha$
and the measured $\alpha/\beta$ ratio [Phys. Rev. A 55, 2 (1997)]. This value
of $\beta$, however, differs by $\sim 0.7\%$ or $2.8\sigma$ from the previous
determination of $\beta=26.957(51)$ by [Phys. Rev. A 62, 052101 (2000)] based
on the measured ratio $M1/\beta$ of the magnetic-dipole
$6S_{1/2}\rightarrow{7S_{1/2}}$ matrix element to $\beta$. Here, we revise the
determination of $\beta$ by [Phys. Rev. Lett. 123, 073002 (2019)], using a
more consistent and more theoretically complete treatment of contributions
from the excited intermediate states in the sum-over-state $\alpha/\beta$
method. Our result of $\beta=26.887(38)\,\mathrm{a.u.}$ resolves the tension
between the $\alpha/\beta$ and $M1/\beta$ approaches. We recommend the value
of $\beta=26.912(30)$ obtained by averaging our result and that of [Phys. Rev.
A 62, 052101 (2000)].

##  I Introduction

Atomic parity violation (APV) plays an important role in probing the
electroweak sector of the standard model (SM) of elementary particles at low
energy. The information derived from table-top APV experiments is both
complementary to and in competition with that from large-scale particle
colliders (see, e.g., the review [1] and references therein). To date, the
1997 Boulder experiment [2] searching for APV in ${}^{133}\mathrm{Cs}$ remains
the most accurate. A substantial body of work has been devoted to the
interpretation of and the extraction of electroweak observables from the
Boulder results.

In its setup, the Boulder experiment [2] employed the 133Cs
$6S_{1/2}\rightarrow{7S_{1/2}}$ transition, whose $E1$ amplitude nominally
vanishes due to the parity selection rule. However, parity non-conserving
(PNC) weak interactions between the atomic nucleus and electrons admix small
components of $P_{1/2}$-states into the nominal $S_{1/2}$-states, thus opening
the $E1$ channel. Using the parity-mixed multi-electron states
$|6S^{\prime}_{1/2}\rangle$ and $|7S^{\prime}_{1/2}\rangle$ and the hyperfine
basis (see Eq. (4) below), the APV transition amplitude may be written as

$\displaystyle A^{\mathrm{PNC}}_{fi}$ $\displaystyle=\langle
7S_{1/2}^{\prime},\,F_{f}M_{f}|-\mathbfcal{E}_{L}\cdot{\boldsymbol{D}}|6S_{1/2}^{\prime},\,F_{i}M_{i}\rangle$
$\displaystyle=i\mathrm{Im}(E1_{\mathrm{PNC}})\mathbfcal{E}_{L}\cdot\langle
F_{f}M_{F_{f}}|\boldsymbol{\sigma}|F_{i}M_{F_{i}}\rangle\,,$ (1)

where $\mathbfcal{E}_{L}$ is the laser electric field driving the $E1$
transition, $\boldsymbol{D}$ is the electric dipole operator, and
$\boldsymbol{\sigma}$ is the Pauli matrix.

Due to the smallness of $E1_{\rm{PNC}}$, which is on the order of $\sim
10^{-11}$ in atomic units, measuring the PNC transition amplitude
$A^{\mathrm{PNC}}_{fi}$ directly is a formidable challenge. To overcome this
difficulty, it was suggested that one uses the Stark-interference technique
[3, 4], which relies on the mixing of states of opposite parities due to an
externally applied electric field. The transition rate $R$ between the parity-
mixed states then includes contributions from the Stark-induced $E1$,
magnetic-dipole $M1$, and the PNC-induced amplitudes [5]

$R\propto|A^{\mathrm{Stark}}_{fi}+A^{M1}_{fi}+A^{\mathrm{PNC}}_{fi}|^{2}\,.$
(2)

Upon expansion, the right-hand side of Eq. (2) yields the Stark-PNC
interference term, $2\mathrm{Re}[A^{\rm{Stark}}_{fi}(A^{\rm{PNC}}_{fi})^{*}]$,
whose sign is subject to the handedness of the experiment measuring $R$. Thus,
the PNC amplitude $A^{\mathrm{PNC}}_{fi}$ can be extracted from the Stark-PNC
interference term by measuring the changes in $R$ under parity reversals.
Based on the Stark-interference technique, the Boulder group [5] reported the
following values

$\frac{\mathrm{Im}(E1_{\mathrm{PNC}})}{\beta}=\begin{cases}-1.6349(80)\,\mathrm{mV/cm}\\\
\,\,\,\,{\rm
for}\,\,\,6S_{1/2},\,{F_{i}=4}\rightarrow{7S_{1/2},\,{F_{f}=3}}\,,\\\
-1.5576(77)\,\mathrm{mV/cm}\\\ \,\,\,\,{\rm
for}\,\,\,6S_{1/2},\,{F_{i}=3}\rightarrow{7S_{1/2},\,{F_{f}=4}}\,,\end{cases}$
(3)

where $\beta$ is the atomic vector polarizability.

A weighted average of the two values in Eq. (3) yields the nuclear-spin-
independent observable, i.e., the nuclear weak charge, while their difference
determines nuclear-spin-dependent effects, e.g., the nuclear anapole moment.
For the extraction of these quantities, knowledge of the vector transition
polarizability $\beta$ is essential and substantial attention [6, 7, 8, 9, 10,
11, 12] has been paid over the years to determining its value. Since 2000, the
most accurate value of $\beta$ has been determined based upon combining a
semi-empirical calculation of the hyperfine-induced magnetic-dipole
$6S_{1/2}\rightarrow 7S_{1/2}$ transition amplitude $M1$ [12] with a
measurement of the ratio $M1/\beta$ [13]. Another approach to estimating
$\beta$ combines a calculation of the scalar polarizability $\alpha$ with the
measurement of the ratio $\alpha/\beta$ [14]. The latest most accurate
determination of $\beta$ was published by the Purdue group [10] who adopted
the most accurate value of $\alpha/\beta=9.905(11)$ [14] and used the sum-
over-state (SoS) method to calculate $\alpha$. Their calculation of $\alpha$
were carried out using experimentally and theoretically determined matrix
elements and energies. Although the uncertainties of the $\alpha/\beta$ [14,
10] and $M1/\beta$ [13, 12] approaches are comparable, both approximately at
the level of $0.2\%$, their central values differ by $\sim 0.7\%$ or
2.7$\sigma$. This difference critically undermines the accuracy of extracting
electroweak observables from the Boulder APV measurement.

Recently, our theory group performed the most sophisticated to date ab initio
calculations of the $E1$ transition matrix elements in Cs [15]. Here, we use
these newly determined $E1$ matrix elements and a SoS approach to reevaluate
the scalar and vector polarizabilities $\alpha$ and $\beta$. We show that the
updated value of $\beta$ agrees well with that obtained from the $M1/\beta$
method of Refs. [13, 12], thus reconciling the two alternative approaches (see
Fig. 1).

Figure 1: Comparison of our value for the vector transition polarizability
$\beta$ with previous results [6, 7, 8, 9, 10, 11, 12]. The previous
determinations of $\beta$ are identified by the initial three letters of the
first author’s last name and the abbreviated publication year. The left panel
presents results from the sum-over-state approach, the middle panel those from
the $M1/\beta$ determination, and the right panel shows our recommended value
for $\beta$ obtained by taking a weighted average of our result and that of
Ref. [12].

The paper is organized as follows. In Sec. II, we provide a review and
derivation for the second-order transition polarizabilities $\alpha$ and
$\beta$. In Sec. III, we detail the numerical methods employed for the
computation of these quantities. In Sec. IV, we present our numerical values
and error estimates and provide a comparison with previous results. Unless
stated otherwise, atomic units are used throughout.

##  II Stark-induced $E1$ transitions and transition polarizabilities

In the presence of a DC electric field, the initial and final $S$ states of Cs
admix states of opposite parities, thus enabling the otherwise forbidden $E1$
transition between the $6S_{1/2}$ and $7S_{1/2}$ states. In this section, we
rederive the conventional results for the Stark-induced $E1$ transition
amplitudes. The reader may refer to the original paper by Bouchiat and
Bouchiat [4] for an alternative derivation.

We start by introducing the hyperfine basis

$|n\,({IJ})FM_{F}\rangle=\sum\limits_{M_{J}M_{I}}C^{FM_{F}}_{JM_{J}IM_{I}}|n\,JM_{J}\rangle|IM_{I}\rangle\,,$
(4)

whose members are formed by coupling electronic states $|n\,JM_{J}\rangle$ of
angular momentum $\boldsymbol{J}$ and nuclear states $|IM_{I}\rangle$ of spin
$\boldsymbol{I}$ to form states of definite total angular momentum
$\boldsymbol{F}=\boldsymbol{I}+\boldsymbol{J}$. Here, $M_{F}$, $M_{J}$, and
$M_{I}$ are the magnetic quantum numbers, $n$ stands for the remaining quantum
numbers, such as the principal quantum number of the electronic state, and
$C^{FM_{F}}_{JM_{J}IM_{I}}$ is the conventional Clebsch-Gordan coefficients.

In the hyperfine basis, Eq. (4), the initial and final states involved in the
$i\rightarrow f$ transition are

$\displaystyle|i\rangle$
$\displaystyle\equiv|n_{i}(IJ_{i})F_{i}M_{i}\rangle\,,$ (5a)
$\displaystyle|f\rangle$
$\displaystyle\equiv|n_{f}(IJ_{f})F_{f}M_{f}\rangle\,.$ (5b)

In the presence of an externally applied static electric field
$\mathbfcal{E}_{S}$, the initial and final states acquire the admixtures

$\displaystyle|\delta i\rangle$ $\displaystyle=-\sum_{a\neq
i}|a\rangle\frac{{\mathbfcal{E}}_{S}\cdot\boldsymbol{D}_{ai}}{\Delta{E_{ia}}}\,,$
(6a) $\displaystyle|\delta f\rangle$ $\displaystyle=-\sum_{a\neq
f}|a\rangle\frac{{\mathbfcal{E}}_{S}\cdot\boldsymbol{D}_{af}}{\Delta{E_{fa}}}\,,$
(6b)

where $\Delta{E_{ab}}\equiv{E_{a}-E_{b}}$ and
$\boldsymbol{D}_{ab}\equiv\langle a|\boldsymbol{D}|b\rangle$ is the electric
dipole matrix element.

If a laser is now applied, it can drive the transition $i\rightarrow{f}$,
whose Stark-induced $E1$ transition amplitude is given by

$\displaystyle A_{fi}$ $\displaystyle=-\langle\delta
f|\boldsymbol{{\mathbfcal{E}}}_{L}\cdot{\boldsymbol{D}}|i\rangle-\langle
f|\boldsymbol{{\mathbfcal{E}}}_{L}\cdot{\boldsymbol{D}}|\delta
i\rangle={{\mathcal{E}}}_{L}{{\mathcal{E}}}_{S}a_{fi}\,,$ (7)

where $\mathbfcal{E}_{L}$ is the laser electric field. In the last step of Eq.
(7), we have factored out the amplitudes of the electric fields and defined
the Stark-induced transition polarizability

$\displaystyle a_{fi}$ $\displaystyle\equiv\sum_{a\neq
f}\frac{(\boldsymbol{\hat{\varepsilon}}\cdot{\boldsymbol{D}}_{fa})(\boldsymbol{\hat{e}}\cdot{\boldsymbol{D}}_{ai})}{\Delta{E_{fa}}}$
$\displaystyle+\sum_{a\neq
i}\frac{(\boldsymbol{\hat{e}}\cdot{\boldsymbol{D}}_{fa})(\boldsymbol{\hat{\varepsilon}}\cdot{\boldsymbol{D}}_{ai})}{\Delta{E_{ia}}}\,.$
(8)

Note that $a_{fi}$ still depends on the polarization vectors
$\hat{\boldsymbol{e}}\equiv\boldsymbol{\mathcal{E}}_{S}/\mathcal{E}_{S}$ and
$\hat{\boldsymbol{\varepsilon}}\equiv\boldsymbol{\mathcal{E}}_{L}/\mathcal{E}_{L}$
of the DC and laser fields.

The expression for $a_{fi}$ may be cast into a form more convenient for
angular reduction. To this end, one uses the recoupling identity [16]

$\displaystyle(R^{(k_{1})}\cdot S^{(k_{1})})(U^{(k_{2})}\cdot{V^{(k_{2})}})$
$\displaystyle=\sum\limits_{Q}(-1)^{Q-k_{1}-k_{2}}$
$\displaystyle\times\\{R^{(k_{1})}\otimes U^{(k_{2})}\\}^{(Q)}$
$\displaystyle\cdot{\\{S^{(k_{1})}\otimes V^{(k_{2})}}\\}^{(Q)}\,,$ (9)

where the operators ${P^{(k_{1})}}$, ${Q}^{(k_{1})}$, ${R}^{(k_{2})}$, and
${S}^{(k_{2})}$ are irreducible tensor operators (ITOs) of ranks $k_{1}$ and
$k_{2}$. In Eq. (II), a scalar product of two rank-$k$ ITOs is understood as
the following sum over their spherical components

${P^{(k)}}\cdot{{Q}^{(k)}}=\sum_{q=-k}^{k}(-1)^{q}P^{(k)}_{q}{Q}^{(k)}_{-q}\,,$
(10)

and a compound ITO of rank $Q$ is defined as

$\\{P^{(k_{1})}\otimes{R^{(k_{2})}}\\}_{q}^{(Q)}=\sum_{q_{1}q_{2}}C^{Qq}_{k_{1}q_{1}k_{2}q_{2}}P^{(k_{1})}_{q_{1}}R^{(k_{2})}_{q_{2}}\,,$
(11)

where $q_{1}$ and $q_{2}$ label the spherical basis components of the ITOs.
The possible values of $Q$ are limited by the triangular selection rule, i.e.,
${|k_{1}-k_{2}|}\leq{Q}\leq{k_{1}+k_{2}}$.

In our case of the electric dipole couplings, the polarization and dipole
operators in Eq. (II) are ITOs of rank $1$. As a result, one has

$\displaystyle a_{fi}$
$\displaystyle=\sum\limits_{Q=0}^{2}(-1)^{Q}\\{\hat{\boldsymbol{\varepsilon}}\otimes\hat{\boldsymbol{e}}\\}^{(Q)}\cdot\left(\sum_{a\neq
f}\frac{\\{\boldsymbol{D}_{fa}\otimes{{\boldsymbol{D}_{ai}}}\\}^{(Q)}}{\Delta{E_{fa}}}\right.$
$\displaystyle\left.+(-1)^{Q}\cdot\sum_{a\neq
i}\frac{\\{\boldsymbol{D}_{fa}\otimes{\boldsymbol{D}_{ai}}\\}^{(Q)}}{\Delta{E_{ia}}}\right)\,,$
(12)

where we have used
$\\{\hat{\boldsymbol{\varepsilon}}\otimes\hat{\boldsymbol{e}}\\}_{q}^{(Q)}=(-1)^{Q}\\{\hat{\boldsymbol{e}}\otimes\hat{\boldsymbol{\varepsilon}}\\}_{q}^{(Q)}$.
The term in Eq. (II) with $Q=0$ corresponds to the scalar, that with $Q=1$ —
to the vector, and the one with $Q=2$ — to the tensor (quadrupole)
contributions to the transition polarizability.

To simplify Eq. (II) further, one may introduce the effective ITOs,

$a[k]_{q}^{(Q)}\equiv\\{\boldsymbol{D}\otimes{R_{k}}\boldsymbol{D}\\}^{(Q)}_{q}\,,$
(13)

with the resolvent operator $R_{k}\equiv{(E_{k}-H_{0})^{-1}}$, where $H_{0}$
stands for the unperturbed atomic Hamiltonian. Since $R_{k}$ has the spectral
resolution

$\displaystyle
R_{k}=(E_{k}-H_{0})^{-1}=\sum\limits_{a\neq{k}}\Delta{E_{ka}^{-1}}|a\rangle\langle
a|\,,$ (14)

and is a scalar (so that the combination $R_{k}\boldsymbol{D}$ remains a
rank-1 ITO), Eq. (II) may be written as

$\displaystyle a_{fi}$
$\displaystyle=\sum\limits_{Q=0}^{2}(-1)^{Q}\\{\hat{\boldsymbol{\varepsilon}}\otimes\hat{\boldsymbol{e}}\\}^{(Q)}$
$\displaystyle\cdot\left(\langle f|a[f]^{(Q)}|i\rangle+(-1)^{Q}\langle
f|a[i]^{(Q)}|i\rangle\right)\,.$ (15)

which, upon applying the Wigner-Eckart theorem, further simplifies to

$\displaystyle a_{fi}$
$\displaystyle=\sum\limits_{Q=0}^{2}w_{Q}(\hat{\boldsymbol{\varepsilon}},\hat{\boldsymbol{e}})$
$\displaystyle\times\left[\langle f||a[f]^{(Q)}||i\rangle+(-1)^{Q}\langle
f||a[i]^{(Q)}||i\rangle\right]\,,$ (16)

where the multipolar polarization weights
$w_{Q}(\hat{\boldsymbol{\varepsilon}},\hat{\boldsymbol{e}})$ are defined as

$\displaystyle w_{Q}(\hat{\boldsymbol{\varepsilon}},\hat{\boldsymbol{e}})$
$\displaystyle=(-1)^{Q}\sum_{q}(-1)^{q+F_{f}-M_{f}}$
$\displaystyle\times\begin{pmatrix}F_{f}&Q&F_{i}\\\
-M_{f}&-q&M_{i}\end{pmatrix}(\hat{\boldsymbol{\varepsilon}}\otimes{\hat{\boldsymbol{e}}})_{q}^{(Q)}\,,$
(17)

where $\begin{pmatrix}F_{f}&Q&F_{i}\\\ -M_{f}&-q&M_{i}\end{pmatrix}$ is the
$3j$ symbol.

Finally, by summing over magnetic quantum numbers, we obtain

$\displaystyle\langle f||a[f]^{(Q)}||i\rangle$
$\displaystyle=(-1)^{F_{i}+I-J_{i}}[F_{f},Q,F_{i}]^{1/2}$
$\displaystyle\times\begin{Bmatrix}Q&J_{f}&J_{i}\\\
I&F_{i}&F_{f}\end{Bmatrix}\sum\limits_{n_{a}J_{a}}\begin{Bmatrix}Q&J_{i}&J_{f}\\\
J_{a}&1&1\end{Bmatrix}$ $\displaystyle\times\frac{\langle
n_{f}J_{f}||D||n_{a}J_{a}\rangle\langle
n_{a}J_{a}||D||n_{i}J_{i}\rangle}{E_{n_{f}J_{f}}-E_{n_{a}J_{a}}}\,,$ (18)

where $[J_{1},J_{2},\ldots,J_{n}]\equiv(2J_{1}+1)(2J_{2}+1)\ldots(2J_{n}+1)$.
The reduced matrix elements $\langle f||a[i]^{(Q)}||i\rangle$ are given by the
same formula, but with $E_{n_{i}J_{i}}$ replacing $E_{n_{f}J_{f}}$ in the
energy denominator. We point out that due to the $6j$ symbols in Eq. (II), the
term with $Q=2$ vanishes for $J_{f}=J_{i}=1/2$, in particular for the
transition $6S_{1/2}\rightarrow{7S}_{1/2}$ of interest, as expected.

Conventionally, the Stark-induced transition polarizability $a_{fi}$ is
expressed as a linear combination of the second-order scalar and vector
polarizabilities [4], $\alpha$ and $\beta$

$\displaystyle a_{fi}$
$\displaystyle=\alpha\,(\hat{\boldsymbol{e}}\cdot\hat{\boldsymbol{\varepsilon}})\delta_{F_{f}F_{i}}\delta_{M_{f}M_{i}}$
$\displaystyle+i\beta\,(\hat{\boldsymbol{e}}\times\hat{\boldsymbol{\varepsilon}})\cdot{\langle
F_{f}M_{f}|\boldsymbol{\sigma}|F_{i}M_{i}\rangle}\,.$ (19)

These two terms map into the $Q=0$ and $Q=1$ contributions in Eq. (II),
respectively. In other words,

$\displaystyle a_{fi}$
$\displaystyle=-\sqrt{3[F_{f}]}w_{0}(\hat{\boldsymbol{\varepsilon}},\hat{\boldsymbol{e}})\alpha$
$\displaystyle-\sqrt{2}\langle F_{f}||{\sigma}||F_{i}\rangle
w_{1}(\hat{\boldsymbol{\varepsilon}},\hat{\boldsymbol{e}})\beta\,,$ (20)

where, for the $S_{1/2}$ states, the reduced matrix element $\langle
F_{f}||{\sigma}||F_{i}\rangle$ in the hyperfine basis (4) is given by

$\displaystyle\langle F_{f}||{\sigma}||F_{i}\rangle$
$\displaystyle=\sqrt{6}(-1)^{I+F_{i}-1/2}$
$\displaystyle\times\sqrt{[F_{f},F_{i}]}\begin{Bmatrix}1/2&F_{f}&I\\\
F_{i}&1/2&1\end{Bmatrix}\,,$ (21)

where we have used $\langle S=1/2||{\sigma}||S=1/2\rangle={\sqrt{6}}$.

In accordance with Eq. (II), one may then write

$\displaystyle\alpha=$ $\displaystyle-\frac{\langle
f||a[f]^{(0)}||i\rangle+\langle
f||a[i]^{(0)}||i\rangle}{\sqrt{3(2F_{f}+1)}}\,,$ (22a) $\displaystyle\beta=$
$\displaystyle-\frac{\langle f||a[f]^{(1)}||i\rangle-\langle
f||a[i]^{(1)}||i\rangle}{\sqrt{2}\langle F_{f}||{\sigma}||F_{i}\rangle}\,,$
(22b)

or, as explicit sums over intermediate states,

$\displaystyle\alpha$
$\displaystyle=\delta_{J_{f}J_{i}}\sqrt{\frac{1}{6}}\sum\limits_{n_{a}J_{a}}\frac{(-1)^{J_{a}-J_{i}}}{\sqrt{3(2J_{i}+1)}}$
$\displaystyle\times\langle n_{f}J_{f}||D||n_{a}{J_{a}}\rangle\langle
n_{a}{J_{a}}||D||n_{i}J_{i}\rangle$
$\displaystyle\times\left(\frac{1}{E_{n_{f}J_{f}}-E_{{n_{a}J_{a}}}}+\frac{1}{E_{n_{i}J_{i}}-E_{{n_{a}J_{a}}}}\right)\,,$
(23a) $\displaystyle\beta$
$\displaystyle=-\frac{1}{2}\sum\limits_{n_{a}J_{a}}\begin{Bmatrix}1&J_{i}&J_{f}\\\
J_{a}&1&1\end{Bmatrix}$ $\displaystyle\times\langle
n_{f}J_{f}||D||n_{a}J_{a}\rangle\langle n_{a}J_{a}||D||n_{i}J_{i}\rangle$
$\displaystyle\times\left(\frac{1}{E_{n_{f}J_{f}}-E_{n_{a}J_{a}}}-\frac{1}{E_{n_{i}J_{i}}-E_{n_{a}J_{a}}}\right)\,.$
(23b)

These equations recover the conventional expressions in the literature, see,
e.g., formulae in Ref. [17] specialized for the initial and final states of
the $S_{1/2}$ character. In this case, the $E1$ selection rules fix the
intermediate states to the $P_{1/2}$ and $P_{3/2}$ angular characters.

##  III Evaluation of the transition polarizabilities

In the last section, we derived the second order transition polarizabilities
$\alpha$ and $\beta$. In this section, we present the numerical methods with
which these polarizabilities are calculated. Our approach is a blend of
relativistic many-body methods of atomic structure and high-precision
experimental values for atomic level energies.

Since the Cs atom has $55$ electrons, its electronic structure is relatively
simple: it has a single valence electron outside the [Xe]-like closed-shell
core. This simplicity greatly facilitates accounting for many-body effects due
to the residual electron-electron interaction (correlation). In what follows,
we describe several approximations of increasing complexity through which the
correlation contributions to $\alpha$ and $\beta$ are computed.

The lowest-order approximation in the electron-electron interaction is the
mean-field Dirac-Hartree-Fock (DHF) method, wherein each electron experiences
an “averaged” influence from all other electrons (and of course the Coulomb
interaction with the nucleus). Within the DHF approach, we use the “frozen-
core” approximation, where atomic orbitals in the [Xe]-like closed-shell core
are computed self-consistently, and the valence orbitals are determined
afterward in the resulting $V^{N-1}$ DHF potential of the core.

We point out that even at this lowest-order DHF level, the intermediate states
involved in calculating $\alpha$ and $\beta$, Eqs. (23), span a countable yet
infinite set of bound states and an innumerable set of states in the
continuum. Since this Hilbert space is infinitely large, direct numerical
summations, while possible, require different numerical implementations for
various many-body methods. An elegant way to handle this issue is the
$B$-spline approach popularized by the Notre Dame group [18, 19, 20]. This
approach generates a finite and numerically complete basis set that has been
proven useful in evaluating otherwise infinite sums. In this approach, the set
of eigenfunctions is a linear combination of $B$-spline functions covering a
radial grid extending from the origin to $R_{\mathrm{max}}$, the radius of an
artificially imposed spherical cavity. The Notre Dame approach is further
refined by employing the dual-kinetic balance $B$-spline basis set [21] which
helps mitigate the issue of spurious states and improve the numerical quality
of orbitals both near and far away from the nucleus.

The low-$n$ orbitals from a $B$-spline finite basis set closely resemble those
obtained with the conventional finite-difference techniques with a
sufficiently large radial grid extent. We refer to these low-$n$ orbitals as
“physical” states. As $n$ increases, this mapping deteriorates, so higher-$n$
basis orbitals often differ substantially from their finite-difference
counterparts; we refer to such states as “non-physical” states.

The value of $n$ separating the physical and non-physical parts of the pseudo-
spectrum primarily depends on $R_{\rm max}$ and to some extent on the number
of basis functions. The dependence on the cavity’s radius is easily understood
by recalling that low-$n$ orbitals decays exponentially with increasing
distance from the nucleus (origin) so they cannot “know about” the existence
of a cavity of sufficiently large radius. In contrast, high-$n$ orbitals have
their maxima at larger distances and therefore are much more susceptible to
the cavity’s presence. Our $B$-spline basis set contains $N=60$ basis
functions of order $k=9$ per partial wave generated in a cavity of radius
$R_{\mathrm{max}}=250\,\mathrm{a.u.}$. These parameters are chosen so that the
fractional differences in the DHF eigenenergies between the basis set and the
finite-difference approach for physical states ($n^{\prime}\leq 12$) are
within $0.015\%$. Similarly, the basis-set values of the $E1$ matrix elements
involving physical states differ from their finite-difference counterparts by
less than $0.1\%$. A detailed discussion of the proper mapping of the finite
basis set orbitals to the physical states may be found in Ref. [15].

With the finite basis set, one may further facilitate the numerical
evaluations of $\alpha$ and $\beta$ by splitting the summations in Eqs. (23a)
and (23b) into the “core-valence” (“cv”), “main”, and “tail” contributions

$\displaystyle\alpha$
$\displaystyle=\alpha_{\rm{cv}}+\alpha_{\rm{main}}+\alpha_{\rm{tail}}\,,$
(24a) $\displaystyle\beta$
$\displaystyle=\beta_{\rm{cv}}+\beta_{\rm{main}}+\beta_{\rm{tail}}\,,$ (24b)

where the cv terms correspond to summations over $2\leq{n_{a}}\leq{5}$, the
main terms to summations over $6\leq{n_{a}}\leq{12}$, and the tail terms to
summations over $13\leq{n_{a}}\leq{\infty}$, respectively. The cv term comes
from the core particle-hole intermediate states with excitations to the
valence orbital blocked by the Pauli exclusion principle [22]. The infinity in
$13\leq{n_{a}}\leq{\infty}$ corresponds to the maximum number of basis set
orbitals of a given angular character. We disregard the summation over Dirac
negative-energy states, as their contribution in the length-gauge for dipole
operators is suppressed by $\alpha_{\rm fs}^{4}$, where $\alpha_{\rm
fs}\approx 1/137$ is the fine-structure constant. We have chosen the boundary
$n_{a}=12$ between the main and tail terms with the convention of the earlier
work, Ref. [10], in mind. Since we have carefully chosen our finite basis set
so that that $B$-spline single-electron orbitals with $n_{a}\leq 12$ coincide
with their finite-difference counterparts, the intermediate many-body states
$|n_{a}J_{a}\rangle$ in $\alpha_{\rm{main}}$ and $\beta_{\rm{main}}$ map into
physical states.

The next-level approximation is the Brueckner orbitals (BO) method which
incorporates certain many-body effects beyond the DHF treatment. BOs
qualitatively describe the phenomenon where the valence electron charge causes
the atomic core to become polarized, thus inducing a dipole and higher-rank
multipolar moments within the core. Consequently, the redistributed charges
within the core attract the valence electron. Compared to the DHF
approximation, the BO method improves the theory-experiment agreement for
valence electron removal energies. In our work, the BO basis set is obtained
by rotating the DHF set using the second-order self-energy operator, see Ref.
[15] for further details.

A further improvement upon the DHF and BO methods is the random phase
approximation (RPA), which is a linear response theory implemented within the
mean-field framework [23, 24]. The primary function of RPA is to account for
the screening of externally applied fields by the core electrons. The main
advantages of the RPA formalism are that RPA is an all-order method and the
RPA transition amplitudes are gauge-independent. For more details about our
finite-basis-set implementation of RPA, the reader is referred to Ref. [15].
The RPA(BO) approach incorporates both the core polarization and the core
screening effects. The quality of the RPA(DHF) and RPA(BO) dipole matrix
elements is substantially improved over the DHF or BO methods, see, again Ref.
[15].

To proceed beyond RPA(DHF) and RPA(BO), we employ the all-order relativistic
many-body coupled-cluster (CC) approach, which systematically accounts for
correlation contributions at each level of approximation. In our recent work
[15], $E1$ matrix elements between the $6,7S_{1/2}$ and $nP_{1/2,3/2}$ states
for $6\leq n\leq 12$ were computed using the CCSDpTvT method. This method
incorporates single (S), double (D), and triple (T) excitation from the
reference DHF state [15] in the CC formalism. The “pTvT” qualifier in CCSDpTvT
refers to a perturbative treatment of core triples and a full treatment of
valence triples. In addition to an accurate treatment of the many-body
effects, the CCSDpTvT $E1$ matrix elements values include scaling, dressing,
Breit, and QED corrections [15]. These CCSDpTvT values are the most complete
theoretical determinations of the $E1$ matrix elements in Cs to date. The
results are complete through the fifth order of many-body perturbation theory
and include some chains of topologically-similar diagrams to all orders. As
such, the CCSDpTvT method is the most theoretically complete applied to
correlation effects in Cs so far. Since the finite basis set used in Ref. [15]
is identical to that employed in this work, we identify the CCSDpTvT many-body
states with $6\leq n\leq{12}$ with the physical states and use the CCSDpTvT
matrix elements to compute the main contribution to transition
polarizabilities.

##  IV Numerical results and discussions

We have provided an overview of the numerical approaches employed in our
calculations of the transition polarizabilities $\alpha$ and $\beta$. In what
follows, we present our numerical results and estimates of uncertainties.

In Table 1, we compile our numerical results for $\alpha$ and $\beta$. In
addition to the DHF, BO, RPA(DHF), RPA(BO), and CCSDpTvT results, we list our
values obtained from CC calculations of varying complexity. In particular, in
the SD approximation, only linear singles and doubles are included, the CCSD
approximation additionally incorporates nonlinear effects, the CCSDvT
approximation includes full valence triples on top of CCSD, and finally
CCSDpTvT(scaled) indicates a CCSDpTvT value rescaled using experimental values
for the removal energies. See Ref. [15] for further details. The final values
for $\alpha$ and $\beta$ are obtained by adding to the scaled CCSDpTvT values
the Breit, QED, and basis extrapolation contributions to the $E1$ matrix
elements, as mentioned in Sec. III. Note that the different CC approximations
only apply to the $E1$ matrix elements in Eqs. (23). For the energy
denominators, we have used the DHF, BO, RPA(DHF), RPA(BO) values in the
corresponding approximations and experimental values for all CC
approximations. Note also that the CC approximations were only used to compute
the main terms, as mentioned in Sec. III. The cv and tail terms are only
calculated up to RPA(BO), since (i) their contributions are much smaller than
those of the main terms, (ii) full CCSDpTvT calculations are expensive, and
(iii) the disparity between the high-$n$ states in the tail terms and their
physical counterparts is significant. The semi-empirical result for $\beta$ is
obtained by dividing our theoretically determined result for $\alpha$ by the
experimentally measured ratio $\alpha/\beta=9.905(11)$ [14] (see below for
further details on this point).

| $\alpha$ | $\beta$
---|---|---
This work
DHF | $-348.50$ | $29.278$
BO | $-339.53$ | $26.483$
RPA(DHF) | $-276.17$ | $29.318$
RPA(BO) | $-273.68$ | $26.364$
SD | $-272.73$ | $26.934$
CCSD | $-279.55$ | $27.214$
CCSDvT | $-266.04$ | $27.370$
CCSDpTvT | $-265.85$ | $27.324$
CCSDpTvT(scaled) | $-266.23$ | $27.227$
Final | $-266.31(23)$ | $27.023(114)$
Semi-empirical $\boldsymbol{(\alpha/\beta)}$ |  | $\textbf{26}.\textbf{887(38)}$
Other works
Toh19 [10] (Sum over states $\alpha$) | $-268.81(30)$ | $27.139(42)$
Dzu02 [9] (Sum over states $\alpha$) |  | $27.15(11)$
Vas02 [8] (Sum over states $\alpha$) | $-269.7(1.1)$ | $27.22(11)$
Dzu00 [12] ($M1$ calculation) |  | $26.957(51)$
Ben99 [25] ($M1/\beta$ experiment) |  | $27.024(80)$
Saf99 [7] (Sum over states $\alpha$) | $-268.6(2.2)$ | $27.11(22)$
Saf99 [7] (Sum over states $\beta$) |  | $27.16$
Dzu97 [6] (Sum over states $\alpha$) | $-269.0(1.3)$ | $27.15(13)$
Blu92 [17] (Sum over states $\beta$) |  | $27.0(2)$
Table 1: Numerical results for the scalar and vector
$6S_{1/2}\rightarrow{7S_{1/2}}$ transition polarizabilities in 133Cs in the
Dirac-Hartree-Fock (DHF) approximation, the Brueckner orbitals (BO)
approximation, the random-phase approximation (RPA) implemented on a DHF basis
set, RPA implemented on a BO basis set, the coupled-cluster (CC) approximation
with only linear singles and doubles (SD), the CC approximation with nonlinear
treatment singles and doubles (CCSD), the CC approximation with linear and
nonlinear singles and doubles and valence triples (CCSDvT), the CC
approximation with linear and nonlinear singles and doubles, perturbative core
triples, and valence triples (CCSDpTvT). The CCSDpTvT(scaled) result is
obtained by using experimental results for removal energies to rescale single
and double amplitudes. The final result is obtained by adding to
CCSDpTvT(scaled) the Breit, QED, and basis extrapolation contributions. The
semi-empirical result for $\beta$ (in bold) is obtained by combining the
theoretically determined result for $\alpha$ with the experimentally measured
ratio $\alpha/\beta=9.905(11)$ [14]. All values are given in atomic units.

As shown in Table 1, the BO correction has a larger impact on $\beta$ than on
$\alpha$, differing by 9.6% from the DHF value of $\beta$, while being only
2.6% away from the DHF value for $\alpha$. In contrast, the RPA contribution
appears to be much more important for $\alpha$ than $\beta$, with the RPA(DHF)
and RPA(BO) values for $\alpha$ differing from the DHF and BO values by around
20%, whereas the RPA(DHF) and RPA(BO) values for $\beta$ are only 0.1–0.5%
away from the corresponding DHF and BO values. SD shifts $\alpha$ by 0.35%
away from its RPA(BO) value while the SD change for $\beta$ is at 2.2%. CCSD
moves the SD value for $\alpha$ by 2.5% and that for $\beta$ by 1%. Adding
valence triples amounts to a 4.9% shifts for $\alpha$ and a 0.6% shift for
$\beta$ while perturbative core triples give rise to a 0.07% shift for
$\alpha$ and a 0.2% shift for $\beta$. Semi-empirical scaling of removal
energies changes $\alpha$ by 0.2% and $\beta$ by 0.4%, while Breit, QED, and
basis extrapolation corrections are at the level of 0.03% for $\alpha$ and
0.8% for $\beta$.

In Table 2, the cv, main, and tail contributions to $\alpha$ and $\beta$ in
different approximations are presented explicitly. This allows us to determine
the central values for our computations and estimate our uncertainties. The
uncertainties in $\alpha_{\rm main}$ and $\beta_{\rm main}$ may be estimated
by considering the convergence patterns of these terms across various
approximations. Indeed, Fig. 2 shows the diminishing of contributions from
terms of higher and higher order in many-body perturbation theory: the RPA
contributions are large, the additional effects of nonlinear core singles and
doubles and valence triples, although significantly smaller, are still
substantial, whereas additional core triples and scaling effects are generally
small. As a result of this observation, we estimate our uncertainty
$\sigma_{CC}$ as half the difference between the CCSDpTvT and scaled CCSDpTvT
values. This uncertainty represents missing contributions from higher-order CC
diagrams. The uncertainty $\sigma_{\rm Breit+QED+basis}$ from the Breit, QED,
and basis extrapolation contributions are assumed, conservatively, to be half
the difference between the final and scaled CCSDpTvT values. The total
uncertainties $\sigma_{\rm main}$ in $\alpha_{\rm main}$ and $\beta_{\rm
main}$ are obtained by adding $\sigma_{CC}$ and $\sigma_{\rm Breit+QED+basis}$
in quadrature.

The contributions and uncertainties of the tail terms may be estimated by
considering how much the DHF, BO, RPA(DHF), and RPA(BO) values for
$\alpha_{\rm main}$ and $\beta_{\rm main}$ differ from the final CCSDpTvT
results of these main terms. We observe that the RPA(DHF) and RPA(BO)
approximations generally give better agreement with the final values, as to be
expected since RPA is known to be responsible for a large portion of the
electron correlation effects. As a result, we assume that the contributions
from the tail terms are the average of the corresponding RPA(DHF) and RPA(BO)
values, and that the uncertainties $\sigma_{\rm tail}$ are half of the
corresponding RPA(DHF) and RPA(BO) differences.

Finally, since the cv terms are the same in the RPA(DHF) and RPA(BO)
approaches, we take these values as our estimates for $\alpha_{\rm cv}$ and
$\beta_{\rm cv}$. The uncertainties $\sigma_{\rm cv}$ in these contributions
are assumed to be half the corresponding BO and RPA(BO) differences. The total
uncertainties in our evaluations of $\alpha$ and $\beta$ are obtained by
adding $\sigma_{\rm cv}$, $\sigma_{\rm main}$, and $\sigma_{\rm tail}$ in
quadrature.

| $\alpha_{\rm main}$ | $\alpha_{\rm cv}$ | $\alpha_{\rm tail}$ | $\beta_{\rm main}$ | $\beta_{\rm cv}$ | $\beta_{\rm tail}$
---|---|---|---|---|---|---
DHF | $-348.24$ | $0.20$ | $-0.46$ | $29.221$ | $0.001$ | $0.056$
BO | $-338.80$ | $0.21$ | $-0.94$ | $26.379$ | $0.002$ | $0.103$
RPA(DHF) | $-276.33$ | $0.40$ | $-0.24$ | $29.309$ | $0.003$ | $0.006$
RPA(BO) | $-273.69$ | $0.40$ | $-0.39$ | $26.319$ | $0.003$ | $0.042$
SD | $-272.81$ |  |  | $26.907$ |  |
CCSD | $-279.63$ |  |  | $27.187$ |  |
CCSDvT | $-266.12$ |  |  | $27.343$ |  |
CCSDpTvT | $-265.93$ |  |  | $27.297$ |  |
CCSDpTvT(scaled) | $-266.31$ |  |  | $27.200$ |  |
Final | $-266.39$ | $0.40$ | $-0.32$ | $26.996$ | $0.003$ | $0.024$
Uncertainty | $0.20$ | $0.10$ | $0.09$ | $0.113$ | $0.001$ | $0.018$
Table 2: The behaviors of the core-valence, main, and tail contributions to $\alpha$ and $\beta$ across several approximations employed in this work. See the caption of Table 1 for an explanation of the notation. All values are given in atomic units. Figure 2: Convergence patterns for the main contributions $\alpha_{\rm main}$ and $\beta_{\rm main}$ to the second order scalar atomic polarizabilities with increasing complexity of the approximations for electron correlation effects. | $n_{a}$ | Toh19 [10] | SD | Final | Difference
---|---|---|---|---|---
$n_{a}P_{1/2}$ | 6 | $-32.54$ |  | $-32.44$ | $9.94[-2]$
7 | $-36.84$ |  | $-36.84$ | $5.04[-2]$
8 | $-5.463[-2]$ | $-5.500[-2]$ | $-4.551[-1]$ | $8.92[-2]$
9 | $-7.988[-2]$ | $-7.824[-2]$ | $-5.611[-2]$ | $2.41[-2]$
10 | $-2.301[-2]$ | $-2.239[-2]$ | $-1.374[-2]$ | $9.48[-3]$
11 | $-9.313[-3]$ | $-9.036[-3]$ | $-4.839[-3]$ | $4.38[-3]$
12 | $-4.614[-3]$ | $-4.472[-3]$ | $-2.007[-3]$ | $2.81[-3]$
$n_{a}P_{3/2}$ | 6 | $-92.93$ |  | $-92.68$ | $2.55[-1]$
7 | $-102.1$ |  | $-101.1$ | $1.00$
8 | $-2.428$ | $-2.461$ | $-2.215$ | $2.13[-1]$
9 | $-4.692[-1]$ | $-4.685[-1]$ | $-4.042[-1]$ | $6.51[-2]$
10 | $-1.653[-1]$ | $-1.650[-1]$ | $-1.372[-1]$ | $2.81[-2]$
11 | $-7.788[-2]$ | $-7.774[-2]$ | $-6.375[-2]$ | $1.41[-2]$
12 | $-4.342[-2]$ | $-4.329[-2]$ | $-3.489[-2]$ | $8.53[-3]$
Main (6, 7) |  | $-264.86$ |  | $-263.00$ | $1.86$
Main (8–12) |  | $-3.847$ |  | $-3.387$ | $0.46$
Core-valence |  | $0.2$ |  | $0.40$ | $0.20$
Tail |  | $-0.30$ |  | $-0.32$ | $-0.02$
Total |  | $-268.81$ |  | $-266.31$ | $2.50$
Table 3: Comparison of individual contributions to the scalar transition
polarizability $\alpha$ from intermediate states $n_{a}P_{J}$ with
$n_{a}=6,\ldots,12$, as well as core-valence and tail terms, as computed by
Ref. [10] and by us. The notation $x[y]$ stands for $x\times 10^{y}$. See the
caption of Table 1 for an explanation of other notations. All values are given
in atomic units.

In Table 3, the main term of $\alpha$ is further broken down into
contributions from intermediate states $n_{a}P_{J}$ with different $n_{a}$.
This facilitates a detailed comparison between the result of this work and
that of Ref. [10]. We first remind the reader that Ref. [10] estimated the
contributions from $6,7P_{J}$ by using experimental values for the $E1$ matrix
elements between $6,7S_{1/2}$ and these $P$ states. In contrast, we estimate
the contributions from $6,7P_{J}$ by using theoretical CCSDpTvT values for the
matrix elements from Ref. [15]. For $6,7P_{1/2}$, the two approaches agree
quite well, reflecting the fact that the theoretical CCSDpTvT matrix elements
$\langle 6,7S_{1/2}||D||6,7P_{1/2}\rangle$ from Ref. [15] are in good
agreement with experiments. On the other hand, our estimates for the
contributions from $6,7P_{3/2}$ disagree quite substantially with those from
Ref. [10], due to tensions between theoretical values for $\langle
6,7S_{1/2}||D||6,7P_{3/2}\rangle$ from Ref. [15] and experimental results.

Another noticeable feature of Table 3 is the significant difference between
our values for the contributions from $n_{a}P_{J}$ with $n_{a}=8,\ldots,12$
and those of Ref. [10]. This discrepancy is due to the fact that Ref. [10]
used for matrix elements $\langle 6,7S_{1/2}||D||n_{a}P_{J}\rangle$
($n_{a}=8,\ldots,12$) theoretical values from Ref. [26], which computed them
in the SDpT approximation. In Table 3, we present our SD values for the
contributions with $n_{a}=8,\ldots,12$. One observes that our SD values
generally agree with those used by Ref. [10], with the small deviations coming
from the pT contributions and the fact that Ref. [26] used a different basis
from ours, with less accurate mapping to the physical states.

We next point out that Ref. [10] estimated the tail contribution by first
computing $\alpha_{\rm tail}$ in the DHF approximation, then rescaling this
DHF result based on the fact that the DHF values for contributions from
$n_{a}=8,\ldots,12$ are $\sim$ 30% higher than the more accurate SDpT values.
In this work, we adopt a slightly different method mentioned earlier, where we
estimate $\alpha_{\rm tail}$ by averaging the corresponding RPA(DHF) and
RPA(BO) values. This approach stems from the observation that for individual
contributions to $\alpha_{\rm main}$ from $n_{a}=8,\ldots,12$, the average of
our RPA(DHF) and RPA(BO) values agree well with the final CCSDpTvT results.
Reassuringly, our final value for $\alpha_{\rm tail}$ is in good agreement
with that of Ref. [10]. We note also that while our DHF value for $\alpha_{\rm
cv}$ agrees with that of Ref. [10], we choose to estimate this term using the
RPA(DHF) and RPA(BO) methods, since these are more complete theoretical
treatments.

From Table 3, the origin of the difference between our estimate for $\alpha$
and that of Ref. [10] is also clear. Out of the total disagreement of 2.50
a.u., 1.86 (74%) comes from the disagreement between experimental and
theoretical values for $\langle 6,7S_{1/2}|D||6,7P_{J}\rangle$, 0.46 (18%)
originates from our use of the CCSDpTvT instead of SDpT values for the main
contributions with $n_{a}=8,\ldots,12$, and the remaining 0.20 (8%) comes from
the cv contribution.

We close by noting that, as may be observed from Fig. 2, the computation of
$\beta_{\rm main}$ does not converge as well with increasingly complex
approximations as that for $\alpha_{\rm main}$. Indeed, whereas the final
uncertainty in $\alpha_{\rm main}$ is at 0.075%, the final uncertainty in
$\beta_{\rm main}$ is about six times worse, at 0.42%. This may be understood
by noting that in $\beta$, contributions from the $nP_{3/2}$ intermediate
states add with an opposite sign to those from $nP_{1/2}$, whereas in
$\alpha$, contributions from $nP_{1/2}$ and $nP_{3/2}$ add with the same sign,
due to the prefactor $(-1)^{J_{a}-J_{i}}$ in Eq. (23a). Since the $nP_{1/2}$
and $nP_{3/2}$ states are degenerate in the nonrelativistic limit, $\beta$ is
nonzero solely due to relativistic effects and is thus suppressed compared to
$\alpha$. This may also be understood from the observation that the matrix
element of a rank-1 tensor (the vector polarizability) between the $L=0$
states (the $S$ states in the nonrelativistic limit) vanishes due to the
angular selection rules, while the same matrix element between the $S_{1/2}$
($J=1/2$) states does not. The cancellation of terms and the resulting
suppression of $\beta$ render SoS computations of the vector polarizability
less reliable than those for $\alpha$. An improved evaluation of $\beta$
involves, as in previous works, combining the theoretically determined value
of $\alpha=-266.30(21)$ with the experimentally measured ratio [14]
$\alpha/\beta=9.905(11)$ to obtain $\beta=26.887(38)$. This semi-empirical
value (in bold) for $\beta$ is also presented in Table 1. It differs from the
value of $\beta=27.139(42)$ of Ref. [10] by 0.94% or 4.4$\sigma$ while is only
0.26% or 1.1$\sigma$ away from the $M1/\beta$ value of $\beta=26.957(51)$ of
Ref. [12]. A comparison between our new value for $\beta$ with previous
results is presented in Fig. 1. We conclude that our determination of $\beta$
brings the two alternative approaches ($\alpha/\beta$ and $M1/\beta$ into an
essential agreement. A weighted average of our value for $\beta$ and that of
Ref. [12] results in

$\beta=26.912(30),$

the most accurate determination of the vector polarizability in Cs to date.
Since these two results were obtained using different methods, potential
cross-correlation effects are anticipated to be suppressed when taking the
weighted average. Note that taking weighted average over all the values in
Fig. 1 is strictly speaking incorrect, as all the values on the left panel are
statistically correlated.

## Acknowledgements

This work was supported in part by the U.S. National Science Foundation grants
PHY-1912465 and PHY-2207546, by the Sara Louise Hartman endowed professorship
in Physics, and by the Center for Fundamental Physics at Northwestern
University.

## References

  * Safronova  _et al._ [2018] M. S. Safronova, D. Budker, D. DeMille, D. F. Jackson-Kimball, A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90, 25008 (2018).
  * Wood  _et al._ [1997] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner, and C. E. Wieman, Science 275, 1759 (1997).
  * Bouchiat and Bouchiat [1974] M. A. Bouchiat and C. Bouchiat, J. Phys. 35, 899 (1974).
  * Bouchiat and Bouchiat [1975] M. A. Bouchiat and C. Bouchiat, J. Phys. 36, 493 (1975).
  * Wood  _et al._ [1999] C. S. Wood, S. C. Bennett, J. L. Roberts, D. Cho, and C. E. Wieman, Can. J. Phys. 77, 7 (1999).
  * Dzuba  _et al._ [1997] V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, Phys. Rev. A 56, R4357 (1997).
  * Safronova  _et al._ [1999] M. S. Safronova, W. R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999).
  * Vasilyev  _et al._ [2002] A. A. Vasilyev, I. M. Savukov, M. S. Safronova, and H. G. Berry, Phys. Rev. A 66, 020101(R) (2002).
  * Dzuba  _et al._ [2002] V. A. Dzuba, V. V. Flambaum, and J. S. M. Ginges, Phys. Rev. D 66, 076013 (2002).
  * Toh  _et al._ [2019] G. Toh, A. Damitz, C. E. Tanner, W. R. Johnson, and D. S. Elliott, Phys. Rev. Lett. 123, 073002 (2019).
  * Bennett and Wieman [1999a] S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2484 (1999a).
  * Dzuba and Flambaum [2000] V. A. Dzuba and V. V. Flambaum, Phys. Rev. A 62, 052101 (2000).
  * Bennett and Wieman [1999b] S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2484 (1999b).
  * Cho  _et al._ [1997] D. Cho, C. S. Wood, S. C. Bennett, J. L. Roberts, and C. E. Wieman, Phys. Rev. A 55, 1007 (1997).
  * Tran Tan and Derevianko [2023] H. B. Tran Tan and A. Derevianko, Phys. Rev. A 107, 042809 (2023).
  * Varshalovich  _et al._ [1988] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii,  _Quantum Theory of Angular Momentum_ (World Scientific, Singapore, 1988).
  * Blundell  _et al._ [1992] S. A. Blundell, J. Sapirstein, and W. R. Johnson, Phys. Rev. D 45, 1602 (1992).
  * Johnson  _et al._ [1988] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 307 (1988).
  * Sapirstein and Johnson [1996] J. Sapirstein and W. R. Johnson, J. Phys. B 29, 5213 (1996).
  * Johnson [2007] W. R. Johnson,  _Atomic Structure Theory: Lectures on Atomic Physics_ (Springer, New York, NY, 2007).
  * Beloy and Derevianko [2008] K. Beloy and A. Derevianko, Comp. Phys. Comm. 179, 310 (2008).
  * Derevianko  _et al._ [1999] A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb, Phys. Rev. Lett. 82, 3589 (1999).
  * Amusia [1990] M. Amusia,  _Atomic Photoeffect_ (Springer US, 1990).
  * Johnson [1989] W. R. Johnson,  _Relativistic Random-Phase Approximation_, edited by D. Bates and B. Bederson, Advances in Atomic and Molecular Physics, Vol. 25 (Academic Press, 1989) pp. 375–391.
  * Bennett and Wieman [1999c] S. Bennett and C. Wieman, Phys. Rev. Lett. 83, 889(E) (1999c).
  * Safronova  _et al._ [2016] M. S. Safronova, U. I. Safronova, and C. W. Clark, Phys. Rev. A 94, 012505 (2016).


# A Review of Quantum communication using high-dimensional Hilbert spaces

Yuval Idan and Avihai Didi

###### Abstract

In this project we examine several different quantum key distribution
protocols which we divide into ones utilizing qubits whose Hilbert spaces are
two dimensional and ones whose Hilbert space dimension is greater than two,
these units of data in quantum computers are known as qudits and in the papers
we’ll examine are implemented using the orbital angular momentum of twisted
photons. In sections 3 and 4 the specific procedures of each protocol are
briefly described and followed by an examination of the theoretical and
experimental merits of each protocol. These merits are measured in the bit
error rate tolerance $e_{b}$, which quantifies the maximum channel noise and
the key rate $R$ which quantifies the rate at which data is transferred in the
protocol. In section 7 we present a unified view of all the relevant data for
the different protocols, and argue for the benefits and drawbacks of the
different protocols for different applications.

##  1 Introduction

In classical computers the basic unit of information is a bit. Each bit can
equal $0$ or $1$, and using a combination of multiple bits we are able to
encode any data we should wish using $\log_{2}(|\Sigma|)$ bits where $\Sigma$
is the alphabet describing the data in question and $|\Sigma|$ is the
cardinality (size) of the alphabet. Different types of data are represented by
different alphabets, such as unicode for text, RGB values for images, and so
on. In principle, the use of a binary unit of information is arbitrary, and
units of information that can take on more than two values could equivalently
be used, in which case the number of such units needed to represent
information in some alphabet $\Sigma$ will be $\log_{d}(|\Sigma|)$ where $d$
is the number of different values the basic unit of information can take on.
However, in practice testing has definitively shown that using a binary unit
of data is much simpler and cheaper to implement, and as such it is the
dominant form of data storage in classical computers

In quantum computing, the basic unit of information is the qubit. Qubits can
take on either of the values a classical bit may take on, but may also be in a
state of super-position of the two, in practice these qubits are usually
implemented using two level systems such as the spin of a spin-half particle,
a super conduction circuit, or the polarization of a photon. Said super-
positions may be manipulated using methods studied within the field of quantum
mechanics, and this manipulation forms the basis of the majority of quantum
computing. Qubit are commonly represented using the Bloch sphere, which, for a
qubit in a normalized superposition of the states $\ket{0}$ and $\ket{1}$ with
a phase difference of $\phi$ is written as

$\ket{\psi}=\cos{(\theta/2)}\ket{0}+e^{i\phi}\sin{(\theta/2)}\ket{1}$ (1)

This representation gives an intuitive visualization method for the effects
that various quantum operators have when acting on a qubit.

Analogously to classical computing, information in quantum computers may also
be represented by units whose Hilbert spaces have a dimension greater than
two, and these units are called qudits. Just like qubits, there are many
competing ways to implement qudits, although no specific implementation has
been definitively shown to be the best means of implementation yet (in the way
that the voltage drop on a circuit has for classical bits) photon based
solutions seem promising due to their long decoherence time and comparatively
weak interactions with other particles. In addition to polarization which is
commonly used as a binary unit of data in qubit based systems, a photon may
also carry orbital angular momentum (OAM) [1], corresponding to helical
wavefronts. Whereas polarization is naturally bi-dimensional, being typically
represented using the basis $\\{\\{\ket{L}\text{, }\ket{R}\\}\text{,
}\\{\ket{\leftrightarrow}\text{, }\ket{\updownarrow}\\}\\}\\}$ (or any other
complete $2D$ basis), the dimensionality of OAM by comparison is infinite
($\aleph_{0}$ to be exact), as it can take on any value of $\hbar l$ where $l$
is an integer [7, 9]. These light waves which are in an arbitrary coherent
superposition of different polarization of and spatial modes are referred to
as structured photons, a visual example of some of the states that the photons
can take on is given in figure 1 ,and in the examples we’ll examine later in
this project the OAM of these light waves is used to experimentally implement
qudits.

Figure 1: (Image taken from [9]). A pair of mutually unbiased bases for $l=2$,
encoding information in OAM (a) and polarization (b) where the intensity is
used to denote the amplitude, the white lines denote the transverse
polarization, and the color is used to denote the phase of the wave function.
Each basis is orthonomalized and mutually unbiased with respect to the other
basis such that the absolute value squared of the projection of any state from
one basis onto the other equals $1/4$. These MUBs posses identical intensity
profiles and are shape-invariant upon free space propagation, making them
suitable for long range communication.

While the practicality of using units of information which can take on more
than two states has been thoroughly ruled out by now in the field of classical
computation, the same can not be said for the emerging field of quantum
computation, where the applicability of such units of data to real world use
cases remains up for debate. This possibility is the primary focus of our work
here, we will study it by comparing the application of qubit and qudit based
protocols for quantum key distribution (henceforth QKD), to see the advantages
and disadvantaged of qudits compared to qubits. It should also be noted that
unlike in classical computers where the only noteworthy application of units
of information that can take on more than two states is the possibility of
representing data using less than $\log_{2}(|\Sigma|)$ units of information,
as we’ll see in the protocols we’ll examine the higher dimension can instead
be used to improve the error resilience of the protocol or decrease the
probability that an eavesdropper could successfully intercept vital
information while still only using the qudit to represent one classical bit of
data.

Our primary goal in this project is to form a comprehensive overview of the
current state of research, theoretical and experimental alike, into the
applicability of qudit based protocols for quantum key distribution and to
contrast those with analogous qubit based protocol so as to ascertain the
relative benefits and drawbacks of each. It should be noted that the
applicability of qubit based solutions to problems besides QKD lies outside
the scope of this project and should be considered separately.

##  2 Quantum Key Distribution (QKD)

In order for two computers to safely communicate without fear that their data
is being read by a third party it is common practice for them two to encrypt
the information so that even if a third party does intercept the data being
sent they would have no use for it without the decryption key. The issue then
becomes how the two could share an encryption key between them over a
compromised channel, since if they simply send it as is the key itself could
also be intercepted. This issue was solved using the RSA algorithm, first
published in 1977. The algorithm relies of the fact that the problem of
integer factorization is an NP problem, meaning that unless $P=NP$ is proven
(which most mathematicians think is highly unlikely) RSA based encryption
remains impractically difficult to solve on a classical computer, making any
data encrypted with it, while technically possible to decrypt, completely safe
from the eyes of third parties, as the average time it would take to decrypt
any data encrypted using RSA is many orders of magnitude greater than the age
of the universe.

While RSA encryption is practically impregnable by classical computers, in
1994 Peter Shor published an algorithm designed for quantum computers which
can solve integer factorization problems in polynomial time, which in theory
allows it to crack RSA encryption with relative ease. The development of
Shor’s algorithm greatly endangered the safety of classical key distribution
and necessitated a transition to quantum based key distribution methods.
Thankfully by this point researches have already began developing such
methods, which, once implemented, allow for the sharing of encryption keys
without risk of Shor’s algorithm. At time of writing, current quantum
computers are not capable of running Shor’s algorithm for numbers large enough
for them to pose an actual threat to RSA encryption, but the study of QKD
protocols seeks to find a solution to this problem before it manifests.

##  3 Qubit based protocols

###  3.1 BB84 protocol

The BB84 protocol operates using qubits which are represented using the
orthonormalized wave functions $\ket{0}$ and $\ket{1}$ .

####  3.1.1 Description of the protocol

In the first step of the protocol, Alice, who wishes to send an encryption key
to Bob, generates an $n$ bit long key which we’ll label $\vec{a}$ and a second
$n$ bit long vector which we’ll call $\vec{b}$, both should be chosen
randomly. She then constructs the wave function that she’ll send to Bob such
that each of the $n$ qubits is determined by both $\vec{a}$ and $\vec{b}$ as
follows:

$\begin{split}\ket{\psi_{00}}&=\ket{0}\\\ \ket{\psi_{10}}&=\ket{1}\\\
\ket{\psi_{01}}&=\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})\\\
\ket{\psi_{11}}&=\ket{-}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})\end{split}$ (2)

Figure 2: A schematic representation of the four states used in the BB84
protocol. The squared projection of any state with another of it’s basis is
zero whereas the squared projected amplitude onto a vector from the other
basis is $1/2$.

Next, Alice sends this wave function over to Bob over a public channel, it is
assumed that Eve has access to this channel and can freely measure the wave
function Alice intended to send to Bob and then resend the resulting wave
function to him. Once Bob receives the wave function, he generates his own $n$
bit long vector which we’ll call $\vec{c}$, this vector determines the basis
which he’ll use when measuring each of the qubits in the wave function he
received, where for $0$ he’ll measure in the $\\{\ket{0}\text{, }\ket{1}\\}$
basis and for $1$ he’ll measure in the $\\{\ket{+}\text{, }\ket{-}\\}$ basis.
Bob will save the results of this measurement as the ”Raw Key”. When measuring
in the same basis that Alice happened to choose, which on average will happen
for $n/2$ if the qubits, Bob will receive the exact bit that Alice intended to
encode, whereas for the cases where he chooses the wrong basis the result
he’ll get is random. After this, Alice publicly announces the basis she
encoded her key in $\vec{b}$ over a classical communication channel. Hearing
this, Bob will keep only the parts of the raw key where his basis of
measurement $\vec{c}$ happened to match Alice’s basis. Assuming that no noise
or third party was involved, the results of his measurements there will allow
him to recover the a key which will on average be $n/2$ bits long. If Alice or
Bob suspect that there may have been eavesdropping they can publicly announce
the results of a subset of the key bits so as to compare what they got, if no
eavesdropping attempt was made the two of them should obtain the same result
but if Eve interfered with the measurement the two subsets will be different
and Alice and Bob will know that they should discard the key and try again.

####  3.1.2 Key rate and error resistance

The key rate for the BB84 protocol in 2 dimensions is:

$R=1-2h(e_{b})$ (3)

where $h(x)=-x\log_{2}(x)-(1-x)\log(1-x)$ is the Shannon entropy $e_{b}$ is
the QBER. The key rate is a measure of the average number of bits of
information that Bob will be able to recover from each unit of information
Alice sends to him which has passed trough the sifting process, those units
being single qubits for qubit based protocols and single qudits for qudit
based protocols. It should be noted that this is not necessarily the same as
the average number of bits Bob is able to recover from each unit of
information Alice sends, that would be given by the product of the key rate
and the sifting rate which is always less than one. We address this
distinction and its implications in greater detail in section 7. The QBER is
defined as the probability that a generic bit in Bob’s sifted string is
different from the corresponding bit in Alice’s sifted string [8]. In order to
compute the QBER, Alice and Bob perform a session of parameter estimation,
where they agree to disclose a random subset of their data. Comparing these
bits (later discarded), they can quantify the QBER and check if this is lower
or higher than a certain security threshold of the protocol.

The quantum bit error rate (henceforth QBER) threshold for the BB84 protocol
is $e_{b}=11\%$, meaning that the protocol is only reliable so long as the
channel the information is being transmitted on only flips bits less than
$11\%$ of the time. One way to increase the QBER threshold is the six state
protocol. The six state protocol is the brother of BB84, but with all the
three MUB in two dimension. Thus, $\ket{x},\ket{y},\ket{z}$, because there are
more options to Eve she produce more QBER then the normal BB84, In addition
the six state protocol have more noise tolerance then the BB84. The
unconditional key rate against coherent attacks has the following expression
[8]:

$R_{\text{6state}}=\frac{1}{3}(1+\frac{3e_{b}}{2})\log_{2}(\frac{e_{b}}{2})+(1-\frac{3e_{b}}{2})\log_{2}(1-\frac{3e_{b}}{2}))$
(4)

The security threshold value is about $12.6\%$, slightly improving of the BB84
protocol and have more immunity against eavesdrop. The security threshold is a
measure of how much noise Eve is able to make trough her eavesdropping
attempts before she is detected relative to the allowed QBER. This is the MUB
version of the 2d BB84 protocol. We discuss higher dimensional MUB protocols
in section 4.3.

####  3.1.3 Eavesdropping immunity

In quantum information the medium we use for communication is noisy, and this
noise is ascribed to Eve the Eavesdropper in the worst-case scenario. Eve will
attempt to intercept the communications send to Bob, measure them herself, and
then resend the post-measurement result to Bob. The issue Eve faces is that
she has a $50\%$ chance of picking the wrong basis for her measurement, for
each qubit. In the cases where she picks wrong, even if Bob happened to pick
the same basis as Alice the results he’ll obtain will be random due to the
collapse of the wave function which occurred when Eve measured it. The
probability that they all happen to pick the same basis and Eve’s
eavesdropping goes unnoticed is $1/4$ per qubit The total probability that
Eve’s eavesdropping won’t be detected is $(1/4)^{n}$ which exponentially
converges to zero, meaning that the protocol is safe from eavesdropping. This
safety from eavesdropping is guaranteed by the fact that any attempt at
intercepting the message Alice sent to Bob by Eve will create a lot of noise
in the system which will bring the QBER up to $e_{b}=25\%$ [8] significantly
higher than the allowed rate of $11\%$. In practice Eve can also use the
faults of the system (false positives/negatives in the detector, imperfections
in the pulses generated by the laser, etc…) in order to gain information about
the key which is independent of the protocol. In section 5 we discuss one
example of this which is the use of high dimensional quantum cloning machines,
which can produce imperfect copies of the data Alice sent, to intercept the
information.

###  3.2 Chau02 protocol

This protocol was introduced by H. F. Chau in this 2002 paper [4]. We’ve taken
to calling it Chau02 after we were unable to find a specific name for it in
the literature. It is the qubit based QKD protocol with the highest bit error
rate tolerance according to [5, 4] with a tolerance of up to $27.6\%$. In the
introduction of [5] H. F. Chau directly contrasts the bit error rate tolerance
of this protocol with that of the qudit based version, we will address this
comparison in greater detail in section 7 after we introduce the higher
dimension version of this protocol in section 4.2.

####  3.2.1 Description of the protocol

Alice generates a set of $n$ qubits each of which is randomly set to one of
the six eigenstates of the Pauli matrices $\\{\sigma_{x}\text{,
}\sigma_{y}\text{, }\sigma_{z}\\}$. The total wave function of the $n$ qubits
$\ket{\psi}$ is then sent to Bob over an insecure channel. Bob acknowledges
the reception of the qubits and then randomly chooses one of the three Pauli
matrices for each qubit, he then measures each qubit in the eigenbasis of the
Pauli matrix he chose for it. Once this step is complete, Alice and Bob both
publicly announce the basis they used over a classical channel and keep only
the ones they both happened to prepare in the same basis. The next (several)
steps of this protocol consist of the application of several privacy
amplification procedures which lie outside the scope of our word, suffice it
to say that this process greatly improves the maximum bit error rate the
protocol can handle.

####  3.2.2 Key rate and error resistance

The theoretical bit error rate for this protocol is
$e_{b}^{\text{max}}=1/2-0.1\sqrt{5}\approx 27.6\%$, this gives us a
theoretical key rate of $R(0)=1$. We were unfortunately unable to find any
papers detailing an experimental implementation of this protocol, so the
theoretical maximum values are all we have to work with.

##  4 Qudit based protocol(s).

###  4.1 BB84

Following up on our previous discussion of BB84 in section 3.1 where we
presented the 2 dimensional version of the protocol, we’ll now show the $d$
dimensional version. The primary advantage of increasing the dimension of the
data carrying unit we use by switching from qubits to qudits is that the
number of mutually unbiased basis (MUBs) with which Alice may encode
information increases. In $d$ dimensions we have $d+1$ MUBs where $d$ is a
power of a prime number, for instance, for $d=2$ we can use the eigenbasis of
the three Pauli matrices as a set of 3 MUBs. In the paper we use as a source
for the discussion of this protocol [2] high dimensional BB84 is implemented
in $d=2,4,8$ dimensions. As the first basis we use pure orbital angular
momentum (OAM), i.e $\ket{\phi}\in\\{-d/2,...,d/2\\}$. For the second basis we
use the Fourier basis which is given by the discrete quantum Fourier transform
of the position space basis
$\ket{\phi_{i}}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}w_{d}^{ij}\ket{\psi_{i}}$
with $w_{d}=e^{2\pi i/d}$. The explicit form of this basis is given in [6].

####  4.1.1 Key rate and error resistance

The experimental quantum bit error rate results obtained in [2] for the three
dimensions tested are $e^{d=2}=0.628\%$, $e^{d=4}=3.51\%$, and
$e^{d=8}=10.9\%$. The experimental quantum key rate results obtained are
$R^{d=2}=0.8901$, $R^{d=4}=1.4500$, and $R^{d=8}=1.3942$. Although the maximum
key rate found in this paper is at $d=4$, the higher bit error tolerance of
$d=8$ might make it preferable for real life applications for the particular
implementation used here (Orbital angular momentum of twisted photons).

###  4.2 Chau15 protocol

The Chau15 protocol, described in [5] uses $2^{N}$ dimensional qudits, where
$N$ is an integer greater than one, which were experimentally implemented as
the orbital angular momentum of photons in [2]. The basis vectors of the
Hilbert space are $\cup_{i=1}^{2^{N}}\\{\ket{i}\\}$.

####  4.2.1 Description of the protocol

First, Alice generates a string of $n$ bits which we’ll denote as $\vec{a}$.
She then generates a wave-function consisting of $n$ qudits as follows. For
each bit $a_{k}$ in $\vec{a}$, Alice chooses a random pair of numbers
$\\{i\text{, }j\\}$ where $i\neq j$ and encodes the bit as
$(\ket{i}+(-1)^{a_{k}}\ket{j})/\sqrt{2}$. Once this is done for every bit in
$\vec{a}$ Alice transfers the completed wave function to Bob, the medium used
to transfer the data is presumed to be unsafe. Once Bob has received the wave
function, for each qudit he chooses a random pair of numbers
$\\{i^{\prime}\text{, }j^{\prime}\\}$ where $i^{\prime}\neq j^{\prime}$ and
measures the qudit in the subspace
$\\{(\ket{i^{\prime}}+\ket{j^{\prime}})/\sqrt{2}\text{,
}(\ket{i^{\prime}}-\ket{j^{\prime}})/\sqrt{2}\\}$. He keeps all of the
$i^{\prime}\text{, }j^{\prime}$ pairs he kept and all of the results of the
measurements. If a measurement of a subspace showed that the qudit is
$(\ket{i^{\prime}}+\ket{j^{\prime}})/\sqrt{2}$ he saves it as zero, and
otherwise he saves it as one. At this point, Alice publishes the list of
$i\text{, }j$ pairs that she picked on a classical public channel, Bob in turn
publishes his list of $i^{\prime}\text{, }j^{\prime}$ pairs. The two of them
then keep only the elements of the key where $i=i^{\prime}$ and
$j=j^{\prime}$. By checking if the results he obtained by measuring projecting
these qudits onto $(\bra{i^{\prime}}+\bra{j^{\prime}})/\sqrt{2}$ is able to
fully determine the elements of $\vec{a}$ which correspond to these qudits.
Alice and Bob can now use those elements of $\vec{a}$ as a key.

####  4.2.2 Key rate and error resistance

For $4\leq N$ ($16\leq d$) the theoretical bit error rate tolerance of the
Chau15 protocol is $50\%$ [5]. In [2] the bit and dit error rates, as well as
the key rates were tested for $N=2$ and $N=3$ ($d=4$ and $d=8$ respectively).
For $N=2$ ($d=4$) they obtained an average bit error rate of
$e_{b}^{d=4}=0.778\%$, an average dit error rate of $e_{d}^{d=4}=3.79\%$ ,and
an average key rate of $R^{(d=4)}=0.8170$ bits per sifted photon. For $N=3$
($d=8$) they obtained an average bit error rate of $e_{b}^{d=8}=3.11\%$, an
average dit error rate of $e_{d}^{d=8}=0.82\%$ ,and an average key rate of
$R^{(d=8)}=0.8172$ bits per sifted photon. Theoretically, given a small enough
dit error rate, bit error rates of up to $e_{b}^{max}=50\%$ may be tolerated.
For orbital angular momentum of twisted photons which were the qudits used in
this experiment this isn’t a clear advantage since bit and dit error rates
will usually correlate since they are the result of similar sources (e.g
apparatus misalignment, turbulence, or optical aberration). However, for other
kinds of high dimensional quantum information carriers where the distinction
between bit and dit error rates is less ambiguous the Chau15 protocol may be
preferable [2].

Figure 3: (Image taken from [2]). Theoretical predictions and experimental
results for the probability of detection matrices for the Chau15 protocol in
dimensions $d=4$ and $d=8$. The strong agreement between the theory and
experiment is clearly visible. The rows correspond to the states sent by Alice
whereas the columns correspond to Bob’s projections of those states made by
Bob during his measurements. The sifted data corresponds to the on-diagonal 2
by 2 blocks.

###  4.3 MUB-based protocol

####  4.3.1 Description of the protocol

A MUB protocol is one wherein all $d+1$ MUBs that are available in dimension
$d$ are used. By using all of the MUBs we can perform QKD protocols (such as
BB84) at a higher effective dimension than the one actually possessed by the
qudits we use. We can also only use some of the MUBs, choosing any amount
between $2$ and $n+1$. In some cases it might be beneficial to use less MUBs
than we can since they have a scaling efficiency of $1/(1+d)$. One example of
an MUB protocol is the six state protocol in dimension two that we mentioned
in section 3.1, which uses all six eigenfunctions of the Pauli matrices, as
opposed to just the eigenfunctions of $\sigma_{x}$ and $\sigma_{z}$. The
protocol may be generalized to high dimensions, and in [2] the two and four
dimensional versions were considered.

####  4.3.2 Key rate and error resistance

For $d=2$, [2] obtained a QBER of $e_{b}^{d=2}=0.923\%$ and a key rate of
$R^{d=2,m=3}=0.8727$. For $d=4$, they obtained a QBER $e_{b}^{d=4}=3.87\%$ and
a key rate of $R^{d=4,m=5}=1.5316$.

##  5 High-dimensional quantum cloning

A high dimensional cloning machine was proposed by [3], although prefect
cloning of an unknown quantum wave function is forbidden by the no cloning
theorem, it nevertheless remains possible to create an imperfect clone of a
wave function. This was the subject of the paper at hand where they considered
the potential venerability of high dimensional quantum states, such as qudits,
to attempts to intercept and eavesdrop on communications without detection. In
the paper this issue was considered by first defining a variable called the
cloning fidelity which was denoted as $F$. This variable is the absolute value
squared overlap of the imperfect clone onto the original wave function
$F=|\braket{\psi_{\text{clone}}}{\psi_{\text{original}}}|^{2}$, and by the no
cloning theorem it is always smaller than one. The optimal cloned state
allowed by the rules of quantum mechanics has a bounded cloning fidelity of
$F_{est}=\frac{2}{1+d}$, where d is the dimension of the quantum state. Here,
we concentrate on the $1\xrightarrow{}2$ universal optimal quantum cloning
machine, for which the optimal fidelity of the two cloned copies is given by
$F_{clo}=\frac{1}{2}+\frac{1}{1+d}$.

###  5.1 Cloning attack

As mentioned, the primary focus of the paper [3] was the potential application
of such a cloning machine to eavesdrop on a quantum system, such as the
transmission of an encryption key over a vulnerable network. In the paper this
was experimentally tested by using a cloning machine to attempt to attack a
high dimensional version of the BB84 protocol (the one we described in section
4.1), the researches were able to demonstrate that the high dimension of the
protocol made Eve’s attempt’s at interception clearly visible to Alice and Bob
even without the utilization of any error correction or privacy amplification
procedures, as Eve’s attempted cloning attack brought the experimentally
measured bit error rate up from $16\%$ to a clearly noticeable $57\%$, more
than $3.5$ times greater. In the $d=7$ case the error bound on the BB84
protocol is $D_{coh}=23.72\%$, meaning that by using error correction and
privacy amplification procedures Alice and Bob may obtain a completely secure
and error free shared encryption key. However, Eve’s cloning attack attempt
will increase the error rate far above the bound that the BB84 protocol can
handle, meaning that in those cases Alice and Bob can easily detect her and
simply discard the compromised key.

In order to compare this result with the two dimensional case the researchers
attempted a similar attack on a two dimensional version of the BB84 protocol
(the one we described in section 3.1) and obtained quantum bit error rates of
$19\%$ and $0.7\%$ with and without the attempted cloning attack respectively,
well above and below the security bound of the 2 dimensional BB84 protocol of
$D_{coh}=11.00\%$. This however, is a smaller bit error rate than the one
incurred for a standard BB84 hack, hence, it is clear that high-dimensional
quantum cryptography leads to higher signal disturbance in the presence of an
optimal cloning attack, resulting in a larger tolerance to noise in the
quantum channel.

Figure 4: (Image taken from [3]). This figure is about the implementation of
BB84 with and without Eavesdrop in $d=7$. In figure A the results without
eavesdropping are shown and in sub figure B the results with eavesdropping are
shown. On the left sides of both figures are the experimentally measured
probability matrices and on the right side are Alice’s messages before
encryption and after Bob received and decrypted them using the shared key.

In addition to the aforementioned measurements, the researchers also tested
the mutual information rate between Alice and Bob, which may be calculated
form:

$I^{d}_{AB}=\log_{2}(d)+(1-e^{N}_{B})\log_{2}(1-e^{N}_{B})+e^{N}_{B}\log_{2}(\frac{e^{N}_{B}}{(d-1)})$
(5)

where $e^{N}_{B}$ is Bob’s error rate. Experimental values of 0.36 and 1.73
bits per photon were obtained for Alice and Bob’s mutual information with and
without Eve’s attempted cloning attack respectively.

The results presented in this paper illustrated the feasibility of high
dimensional optimal quantum cloning of orbital angular momentum states of
single photons. This capacity was then put to use to test the venerability of
the BB84 protocol to cloning based attacks, which demonstrated the robustness
of high dimensional quantum cryptography upon quantum hacking.

##  6 Practical implementation and resolution

In this section we describe in detail one particular experimental
implementation of qudit based communications as an example, in particular we
examine the one described in the paper [9], where the researches used the BB84
protocol is 4 dimensions. The researches who wrote the paper in question built
a free-space link between the rooftops of two building $0.3$ km apart and $40$
meters above the ground, in the University of Ottawa campus, a schematic
description of the system overlaid over an image of the two buildings is given
in figure 5.

Figure 5: (Image taken from [9]). Schematic representation of the experimental
setup, detailed descriptions of Alice and Bob’s setups are given in sections
6.1.1 and 6.1.2 respectively.

###  6.1 The experiment setup

####  6.1.1 Alice setup

Alice’s system consists of a heralded single-photon source as well as a ppKTP
crystal, two dichronic mirrors (DM), a polarizing beam splitter (PBS), two
wave plates (WP), a q-plate (QP), and a telescope build from a pair of lenses
whose focal lengths are $f_{1}=75$mm and $f_{2}=400$mm (items are listed in
order of application as seen in figure 5, the specific number and order of WP
and QP depends on the MUB being encoded). The photons emitted by the laser
become pairs of photons due to the spontaneous parametric down conversion
process in the ppKTP crystal which is pumped by a laser diode. The idler
photon and the signal photon are each given a different wavelength and the two
wavelengths are chosen to be non-degenerate in order to better separate the
two; only the signal photon is encoded with information. It should be noted
that because of this, the scheme is still immune to number-splitting attacks,
since the idler photon does not contain any of the polarization or OAM
information of the signal photon. The signal and idler photons are each
coupled to a separate single-mode fiber (SMF) to spatially filter the photons
into the fundamental mode. Band-pass filters are placed in front of the fiber
couplers in order to select for the correct photon pairs. Using a combination
of wave plates and q-plates Alice then prepares the signal photon into on of
the different MUBs, the specific combination and order of the plates depends
on the chosen MUB Alice wishes to prepare the photon in. The idler and signal
photon are then recombined using another dichroic mirror and passed trough the
telescope in order to minimize divergence upon propagation on their way before
they are sent to Bob.

####  6.1.2 Bob setup

Bob’s system consists of a telescope made from a pair of lenses whose focal
lengths are $f_{3}=400$mm and $f_{4}=50$mm, a DM, two WP, a QP, a PBS, and a
pair of single photon detectors (items are listed in order of application as
seen in figure 5). In order to measure to received photons Bob first
demagnifies their structure using the telescope and then he uses a DM to
separate the idler photon and the signal photon. The idler photon is the
coupled into a SMF to act as a herald for the signal photon. Meanwhile, the
signal photon goes trough a series of WP and QP abd PBS which are a mirrored
copy of the set Alice made. Once this step is done Bob can Bob can make a
measurement on the signal photon by projecting it onto one of the states from
one of the MUBs. By mirroring the system Alice prepared to set up her signal
photon Bob has a spatial mode filter which will make it so that if he projects
it onto the same state that Alice prepared it in the signal photon will be
phase-flattened and optimally detected. By using APDs and a coincidence logic
box (marked in the diagram by $\otimes$), the received idler photon acts as a
trigger for the arrival of the signal photon and the coincidence rates are
recorded.

###  6.2 Results

As previously discussed, the key rate is defined as:

$R(e_{b})=log_{2}(d)-2h(e_{b})$ (6)

where $d$ is the dimension of Hilbert space, $h(e_{b})$ is the Shannon entropy
in dimension $d$, and $e_{b}$ is the bit error rate. The theoretical maximum
bit error rates for the BB84 protocol are $e_{b}^{d=2\text{ (max)}}=11\%$ and
$e_{b}^{d=4\text{ (max)}}=18.9\%$ in two and four dimensions respectively, and
the theoretical maximum key rates are $R(0)=1$ and $R(0)=2$ for two and four
dimensions respectively. In this experiment the values obtained in 4d were
$e_{b}^{4=2}=14\%$ and $R^{4d}=0.39$, these values are sufficient for a secure
key to be transmitted even without the use of any corrections. For comparison
they also performed the experiment with qubits and obtained $e_{b}^{d=2}=5\%$
and $R^{2d}=0.43$, the difference between the performance of the protocol in
these two different dimensions is illustrated in figure 6

Figure 6: (Image taken from [9]). A visual illustration of the difference in
key rates for $d=4$ and $d=2$. We can see that in the higher dimension the
higher key rate allows more information to be transferred.

It should be noted that unlike in the experiments detailed in [2], the
experiments detailed in this paper [9] were performed with an open air channel
over a large distance, which greatly affected the resulting bit error rate.

##  7 Comparison between qubit and qudit based protocols

When comparing the performance of different protocols there are five main
subjects we should consider:

  * •

The bit error rate tolerance, which is the ability of the protocol to continue
working as intended after a percentage of the bits were unintentionally
flipped. This is the primary indicator of the protocol’s viability for use in
noisy quantum channels. The maximum theoretical bit rate can be computed in
theory, and also measured in practice in experimental systems.

  * •

The key rate, defined as the number of bits of secret key established divided
by the number of sifted photons.

  * •

The sifting rate, this indicates what portion of the bits that were
successfully transferred from Alice to Bob will be used in the actual
encryption key that the two of them will use. This fraction is determined by
the design of the protocol.

  * •

The effective key rate, which is given by the product of the key rate and the
sifting rate, this number indicates what portion of each qudit of information
Alice sends to Bob will become a part of the final encryption key.

  * •

The difficulty and cost of implementing the hardware and software required for
the protocol. Since at this moment all present implementations of these
protocols are experimental models made in laboratory settings there is not
clear way to ascertain the relative commercial viability of large scale
implementations of these protocols, meaning that there is not much for us to
go on in writing such a comparison at the moment. That said, this point
nevertheless will have a large impact on what communication protocols will
actually see large scale use in the future.

Protocol | $d$ | $e_{b}^{\text{max}}$ | $e_{b}^{\text{exp}}$ | $R(0)$ | $R^{\text{exp}}$ | Sifting |  $R^{\text{exp}}$ x Sifting
---|---|---|---|---|---|---|---
Chau02 [4] | $2$ | $27.6\%$ | - | $1$ | - | $1/2$ |  -
Chau15 [2] | $4$ | $50\%$ | $0.778\%$ | $1$ | $0.8170$ | $1/6$ |  $0.1362$
| $8$ | $50\%$ | $3.11\%$ | $1$ | $0.8172$ | $1/28$ |  $0.0292$
BB84 [2] | $2$ | $11.00\%$ | $0.628\%$ | $1$ | $0.8901$ | $1/2-1*$ |  $0.4451-0.8901$
| $4$ | $18.93\%$ | $3.51\%$ | $2$ | $1.4500$ | $1/2-1*$ |  $0.7250-1.4500$
| $8$ | $24.70\%$ | $10.9\%$ | $3$ | $1.3942$ | $1/2-1*$ |  $0.6971-1.3942$
BB84 [9] | $2$ | $11.00\%$ | $5.0\%$ | $1$ | $0.43$ | $1/2-1*$ |  $0.215-0.43$
| $4$ | $18.93\%$ | $14\%$ | $2$ | $0.39$ | $1/2-1*$ |  $0.195-0.39$
MUB [2] | $2$ | $12.62\%$ | $0.923\%$ | $1$ | $0.8727$ | $1/3-1*$ |  $0.2909-0.8727$
| $4$ | $23.17\%$ | $3.87\%$ | $2$ | $1.5316$ | $1/5-1*$ |  $0.3063-1.5316$
Table 1:  The data for this table is taken from [2, 4, 5, 9]. The theoretical
maximum tolerance for the QBER $e_{b}^{\text{max}}$ and the theoretical
maximum key rate $R(0)$ are presented alongside their experimentally measured
values $e_{b}^{\text{exp}}$ and $R^{\text{exp}}$. *In the BB84 and MUB
protocols depending on the size of the encryption key Alice wishes to send to
Bob the choice of basis can be biased to get a a larger sifting efficiency
than $1/2$ and $1/(d+1)$ for BB84 and MUB, respectively [10]. In the infinite
key limit, the sifting efficiency can be made to approach 1 for these
protocols.

We find that the use of qudits in place of qubits increases error resilience
and should expect it to also increase the key rate, however the exact effects
differ by protocol.
For the Chau15 protocol, we obtain a significant increase in the error
tolerance of the protocol ($27.6\%$ to $50\%$). This great error resistance
would make it suitable to noisy channels, but would make it unwise to
implement in comparatively low noise channels since in such cases all that
increasing the error tolerance would do is give Eve a better window to attempt
to intercept Alice’s messages without being noticed, although this is somewhat
counteracted by the high sifting rate. While increasing the dimension of the
qudit used in Chau15 has no effect on the key rate, it does have a significant
effect on the sifting rate which equals $2/(d^{2}-d)$, meaning that for high
dimensions the effective key rate can become so small so as to make usage of
it impractically slow, such as the effective key rate of $0.0292$ found for
Chau15 in 8 dimensions (assuming that the experimental key rate for $d=16$
would also be $R^{\text{exp}}=0.817$, we would get an effective key rate of
$0.0068$, requiring an average of $147$ qudits per encryption key bit).
Overall, the Chau15 protocol seems suitable for use in high noise environments
and at a relatively low dimension, unless eavesdropping is a major concern in
which case the high sifting rate might be preferable.
The BB84 protocol appears to benefit to most from the use of qudits out of the
protocols we’ve examined, since the error resilience and key rate both
increase as we increase the qudit dimension without affecting the sifting
rate. We also found in [9] that the protocol remains reliable even in noisy
environments (open space propagation across a relatively large distance).
Overall, BB84 seems fairly versatile in applicability for different
environments, but best suited for relatively low noise environments (compared
to Chau15). In such environments we would be able to use a high dimensional
qudit to fully benefit from the increased key rate, however this would also
create a greater risk of eavesdropping, so the dimension of the qudit should
be chosen carefully the balance these conflicting concerns.
MUB-based protocols offer a middle ground between the two. Their error
resilience benefits more from the increased dimension than BB84 but not as
much as Chau15. Their (Theoretical) key rate increase as $\log_{2}(d)$ but the
sifting rate increases as $1/(d+1)$, so while the effective key rate doesn’t
decay as quickly as in Chau15, it does decay, unlike in BB84, though this can
be counteracted by biasing the choice of basis [10].
Overall, we think that in theory, difficulty of implementation not
withstanding, qudits are preferable over qubits for use in QKD. However, due
to the difficulty of implantation in practice, qubits may be more reliable for
real world applications.

##  8 Conclusion

In our work we have examined several different QKD protocol examining both
qubit and qudit based implementations and in particular we were intereseted in
comparing the benefits and drawbacks of each of them. We found across the
different protocols that we’ve examined that qudit based protocols seem to be
preferable, given that a protocol which is appropriate to the operational
environment is chosen. However, this apparent benefit does not account for the
increased difficulty of implementation, which may make qubit based protocols
preferable in spite of their lesser performance in these regards.

## References

  * [1] L. Allen, M.. Beijersbergen, R… Spreeuw and J.. Woerdman  “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes”  In _Phys. Rev. A_ 45 American Physical Society, 1992, pp. 8185–8189  DOI: 10.1103/PhysRevA.45.8185
  * [2] Frédéric Bouchard et al.  “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons”  In _Quantum_ 2 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2018, pp. 111  DOI: 10.22331/q-2018-12-04-111
  * [3] Frédéric Bouchard, Robert Fickler, Robert W. Boyd and Ebrahim Karimi  “High-dimensional quantum cloning and applications to quantum hacking”  In _Science Advances_ 3.2 American Association for the Advancement of Science, 2017  DOI: 10.1126/sciadv.1601915
  * [4] H.. Chau  “Practical scheme to share a secret key through a quantum channel with a 27.6 percent bit error rate”  In _Phys. Rev. A_ 66 American Physical Society, 2002, pp. 060302  DOI: 10.1103/PhysRevA.66.060302
  * [5] H.. Chau  “Quantum key distribution using qudits that each encode one bit of raw key”  In _Phys. Rev. A_ 92 American Physical Society, 2015, pp. 062324  DOI: 10.1103/PhysRevA.92.062324
  * [6] THOMAS DURT, BERTHOLD-GEORG ENGLERT, INGEMAR BENGTSSON and KAROL ŻYCZKOWSKI  “ON MUTUALLY UNBIASED BASES”  In _International Journal of Quantum Information_ 08.04, 2010, pp. 535–640  DOI: 10.1142/S0219749910006502
  * [7] Alois Mair, Alipasha Vaziri, Gregor Weihs and Anton Zeilinger  “Entanglement of the orbital angular momentum states of photons”  In _Nature_ 412.6844, 2001, pp. 313–316  DOI: 10.1038/35085529
  * [8] S. Pirandola et al.  “Advances in quantum cryptography”  In _Adv. Opt. Photon._ 12.4 OSA, 2020, pp. 1012–1236  DOI: 10.1364/AOP.361502
  * [9] Alicia Sit et al.  “High-dimensional intracity quantum cryptography with structured photons”  In _Optica_ 4.9 OSA, 2017, pp. 1006–1010  DOI: 10.1364/OPTICA.4.001006
  * [10] Gisin Nicolas Tomamichel Marco and Renner Renato  “Tight finite-key analysis for quantum cryptography”  In _Nature Communications_ , 2012  DOI: 10.1038/ncomms1631