# Polarizabilities as Probes for P, T, and PT Violation

Sebastian Lahs, Daniel Comparat

###### Abstract

Searches for violations of the fundamental symmetries of parity $P$ and time
reversal $T$ in atomic and molecular systems provide a powerful tool for
precise measurements of the physics of and beyond the standard model. In this
work, we investigate how these symmetry violations affect the response of
atoms and molecules to applied electric and magnetic fields. We recover well-
known observables like the $P$-odd, $T$-odd spin-electric field coupling that
is used for searches of the electron electric dipole moment (eEDM) or the
effect of $P$-odd, $T$-even optical rotation in atomic gases. Besides these,
we obtain several other possible observables. This includes, in particular,
effects that can only be seen when using oscillating or inhomogeneous fields.

##  1 Introduction

At the present day, precision measurements in atomic and molecular systems are
among the most powerful tools to test the standard model of particle physics
(SM) at low energy scales [1, 2]. This further enables searches for new
particles or interactions that are proposed to answer some of the open
questions in the foundations of physics. The interaction of atoms and
molecules with these new particles would lead to small shifts in their
spectra. The magnitude of almost all of these small corrections is far below
the uncertainty at which absolute energy levels can presently be calculated.
To avoid this problem, it makes sense to search for observables that are
forbidden by the symmetries of atomic physics. If we detect non-zero values of
those, their size would directly correspond to some new or exotic interaction.
Two such symmetries are the ones of Parity $P$ and time reversal $T$.
$P$-violating processes were previously used to proof the validity of the
Glashow-Weinberg-Salam theory at atomic energy scales [2]. Further, numerous
searches for simultaneous violations of $P$ and $T$ were performed in the
context of electric dipole moment (EDM) measurements [3, 4]. Only few direct
searches for $T$, without $P$ violation ($T$-odd $P$-even, also called ToPe)
have been performed so far [5, 6].

Following previous works [7, 8, 9, 10, 11, 12], we derive possible observables
for measurements of $P$, $T$, and $PT$ -violations in the interaction of
electromagnetic fields with atoms and molecules. We try to be as general as
possible by making minimal assumptions on the atoms/molecules under
consideration, the spatial and temporal shape of the electromagnetic fields,
and the explicit form of the $P$\- and $T$\- odd potentials.

We start by specifying the requirements of the system (sec. 2.1) and then
introduce perturbation through electric and magnetic fields under a low-order
multipole approximation (sec. 2.2). On this basis, we derive the time-
dependent perturbative expressions for the induced electric and magnetic
dipole moments and the electric quadrupole moment (sec. 2.3). We separate
these terms into irreducible tensor components (sec. 3.3.2, and 3.3.4) and
analyze their behavior under $P$ and $T$ transformations. This leads us to
general expressions for the possible (generalized) atomic polarizabilities
(sec. 3.4). In sec. 4, we match some of these polarizabilities to observables
that have been considered in previous works. We further identify terms that
have not been explored so far. This includes, in particular, interactions with
inhomogeneous (sec. 4.2) and time-dependent sec. (4.3) fields. We show that
the former leads to observables sensible to all three kinds of symmetry
violations. In sec. 5 we argue that the latter introduces some difficulties in
the context of $T$-odd observables.

##  2 Derivation of the $P$ and $T$ odd polarizabilities

###  2.1 Requirements on the system

The atomic or molecular system under consideration has to fulfill a few
requirements for the following treatment to be valid. Namely, the Hamiltonian
$H^{0}$ of the unperturbed system needs to commute with Parity $P$, the square
of the total angular momentum $\boldsymbol{J}^{2}$ and its projection on an
arbitrary axis $J_{z}$. Therefore we require $H^{0}$ to be invariant under
$SU(2)$ rotations and spacial reflections. Generally, this will be true for
any free atom or molecule due to the rotational symmetry of their
wavefunctions. Interestingly enough, this holds independent of how precise our
model for the system is. It even stays valid for a full relativistic QED+QCD
description of all its inner processes. Only the inclusion of the weak force
or hypothetical new physics interactions breaks these symmetries. Therefore,
we exclude these from the definition of $H^{0}$ and instead add them later as
small perturbations. From all this, we know that the eigenstates to $H^{0}$
can (without approximation) be written in a basis
$\left|n\right\rangle=\left|aJm\right\rangle$. Here, $J$ is the quantum number
of the total angular momentum, $m$ is the quantum number related to its
projection on a chosen quantization axis, and $a$ includes all the other
quantum numbers of the system.

This comes with a small caveat. To use all this, we need to be able to prepare
a system in these ”true” eigenstates. This will, for example, not be possible
for chiral molecules whose states of different chirality are separated by such
a large energy barrier that they will realistically never be found in a Parity
eigenstate (Hund’s Paradox [13]). Apart from chiral molecules [14, 15],
spontaneous breakings of $P$ and $T$-symmetries can also arise in solid state
systems depending on their crystal symmetries [16, 17]. These do not originate
from the fundamental physics processes we want to consider here. We,
therefore, require that all such spontaneous symmetry breaking are absent.
This is generally fulfilled in free atoms and (small) molecules.

Throughout this paper, we are using natural units:
$c=\hbar=\varepsilon_{0}=1$. In the following, we refer to the situation where
an operator is both odd under $P$ and $T$ as $PT$-violation. We use the
notation $\boldsymbol{J}$ for the total angular momentum, including possible
contributions of the nuclear angular momentum, even though usually the
momentum in such a case would be denoted by $\boldsymbol{F}$.

###  2.2 Interaction with electromagnetic fields

We want to study how the system $H^{0}$ behaves under the influence of applied
electromagnetic fields. To do this, we are using a perturbative treatment in
which we add the semiclassical interaction with an electromagnetic field in
the 2nd-order multipole approximation:

$\displaystyle
V^{EM}=-\boldsymbol{d}\cdot\boldsymbol{\mathcal{E}}-\boldsymbol{\mu}\cdot\boldsymbol{\mathcal{H}}-Q_{ij}\partial_{i}\mathcal{E}_{j}$
(1)

where $\boldsymbol{d},\boldsymbol{\mu}$, and $\boldsymbol{Q}$ are the electric
dipole, the magnetic dipole, and the electric quadrupole (Hermitian)
operators, respectively. We recall that in a simple single electron system,
these moments are defined as follows [18]:

$\displaystyle\boldsymbol{d}$
$\displaystyle=e\boldsymbol{r}-\mu_{B}(\boldsymbol{\sigma}\times\boldsymbol{p})$
$\displaystyle\boldsymbol{\mu}$
$\displaystyle=\mu_{B}\left(g_{s}\boldsymbol{\sigma}+\boldsymbol{r}\times\boldsymbol{p}\right)$
$\displaystyle Q_{ij}$
$\displaystyle=\frac{e}{2}\left(r_{i}r_{j}-\frac{r^{2}}{3}\delta_{ij}\right)$

Here, $e$ is the absolute value of the electron charge, $\boldsymbol{r}$ is
the position- and $\boldsymbol{p}$ the momentum-vector of the electron,
$\mu_{B}$ is the Bohr magneton, $g_{s}$, the electron $g$-factor, and
$\boldsymbol{\sigma}$ is the vector of the three Pauli matrices. In the
definition of $\boldsymbol{d}$, we included the first-order relativistic
correction [19]. In the remainder of this work, we do not refer to the
specific form of these operators and only use their behaviors under parity and
time reversal transformations. That way, the treatment stays valid for multi-
electron atoms and molecules in which $\boldsymbol{d}$, $\boldsymbol{\mu}$,
and $Q_{ij}$ are obtained by summing over all bodies in the system.

The electric and magnetic fields $\boldsymbol{\mathcal{E}}(\boldsymbol{r},t)$
and $\boldsymbol{\mathcal{H}}(\boldsymbol{r},t)$ are defined through the
electromagnetic scalar and vector potentials:

$\displaystyle\boldsymbol{\mathcal{E}}=-\nabla\phi-\partial_{t}\boldsymbol{\mathcal{A}}\,,\;\;\;\boldsymbol{\mathcal{H}}=\nabla\times\boldsymbol{\mathcal{A}}$

For a simple separation between time-independent and time-dependent terms, we
use the gauge freedom to set $\partial_{t}\phi(\boldsymbol{r},t)=0$.
We define $\boldsymbol{\mathcal{A}}(\boldsymbol{r},t)$ in a rather general
way, as a superposition of a static component and plane waves with arbitrary
frequencies and polarizations:

$\displaystyle\boldsymbol{\mathcal{A}}=\boldsymbol{\mathcal{A}}^{0}+\sum_{\ell\neq
0}Re\left(\boldsymbol{\mathcal{A}}^{\ell}e^{i\boldsymbol{k}^{\ell}\cdot\boldsymbol{r}}e^{-i\omega^{\ell}t}\right)$

$\boldsymbol{\mathcal{A}}^{0}$ is the (real) static component of the vector
potential ($\partial_{t}\boldsymbol{\mathcal{A}}^{0}=0$).
$\boldsymbol{k}^{\ell}$ is the wavevector, and $\omega^{\ell}$ is the angular
frequency of the wave. The real part $Re$ is defined as $2Re\,A=A+A^{*}$, with
$A^{*}$ being the complex conjugate to $A$. Throughout this paper, both upper
and lower indices are used for ease of notation without the positioning
carrying a specific meaning. From these choices, the electric field is given
by:

$\displaystyle\boldsymbol{\mathcal{E}}$
$\displaystyle=\boldsymbol{\mathcal{E}}^{0}+\sum_{\ell\neq
0}\boldsymbol{\mathcal{E}}^{\ell}(t)$
$\displaystyle=-\nabla\phi+\sum_{\ell\neq
0}Re\left(i\omega^{\ell}\boldsymbol{\mathcal{A}}^{\ell}e^{i\boldsymbol{k}^{\ell}\cdot\boldsymbol{r}}e^{-i\omega^{\ell}t}\right)$
(2)

with the first term describing the static and the second the time-dependent
field component. Accordingly, for the magnetic field
$\boldsymbol{\mathcal{H}}$:

$\displaystyle\boldsymbol{\mathcal{H}}$
$\displaystyle=\boldsymbol{\mathcal{H}}^{0}+\sum_{\ell\neq
0}\boldsymbol{\mathcal{H}}^{\ell}(t)$
$\displaystyle=\nabla\times\boldsymbol{\mathcal{A}}^{0}+\sum_{\ell\neq
0}Re\left(i\left(\boldsymbol{k}^{\ell}\times\boldsymbol{\mathcal{A}}^{\ell}\right)e^{i\boldsymbol{k}^{\ell}\cdot\boldsymbol{r}}e^{-i\omega^{\ell}t}\right)$
(3)

The Jacobian of $\boldsymbol{\mathcal{E}}$ is given by:

$\displaystyle\partial_{i}\mathcal{E}_{j}$
$\displaystyle=-\partial_{i}\partial_{j}\phi-\sum_{\ell\neq
0}Re\left(\omega^{\ell}k_{i}^{\ell}\mathcal{A}_{j}^{\ell}e^{i\boldsymbol{k}^{\ell}\cdot\boldsymbol{r}}e^{-i\omega^{\ell}t}\right)$
(4)

We solve the system in the inertial center of mass frame of the atom
(molecule). For a moving system, it is necessary to transform the field values
and frequencies accordingly.

In the multipole approximation, only the field values at the center of mass
contribute [18]:

$\displaystyle
V^{EM}=-\boldsymbol{d}\cdot\boldsymbol{\mathcal{E}}|_{\boldsymbol{r}=0}-\boldsymbol{\mu}\cdot\boldsymbol{\mathcal{H}}|_{\boldsymbol{r}=0}-Q_{ij}(\partial_{i}\mathcal{E}_{j})|_{\boldsymbol{r}=0}$

For the time-dependent field components, this implies
$e^{i\boldsymbol{k}^{\ell}\cdot\boldsymbol{r}}=1$. For simplicity of notation,
we will not explicitly indicate this limit in the following.

###  2.3 Perturbative treatment

As usual, the Energy of the system $H^{0}+V^{EM}$ is given by:

$\displaystyle E=$ $\displaystyle\left\langle
N\right|H^{0}+V^{EM}\left|N\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle N\right|H^{0}\left|N\right\rangle-\left\langle
N\right|\boldsymbol{d}\left|N\right\rangle\cdot\boldsymbol{\mathcal{E}}-\left\langle
N\right|\boldsymbol{\mu}\left|N\right\rangle\cdot\boldsymbol{\mathcal{H}}-\left\langle
N\right|Q_{ij}\left|N\right\rangle\partial_{i}\mathcal{E}_{j}$

with $\left|N\right\rangle$ being the eigenvectors of $H=H^{0}+V^{EM}$. In the
following, we want to treat $V^{EM}$ as a perturbation to $H^{0}$. We will
perform this perturbation explicitly for the electric dipole operator
$\boldsymbol{d}$. The perturbations for $\boldsymbol{\mu}$ and $Q_{ij}$ follow
analogously. The general case, valid for both static and varying fields, can
be built from the solutions to the time-independent and dependent problems.
Thus, we do the perturbation in two steps. First, we perform a time-
independent perturbation calculation for the static components of the field
and, afterward, a time-dependent calculation for the time-varying components.
There are multiple, slightly different, formulations for the Dirac-type time-
dependent perturbation theory. One has to be careful in choosing the right one
to avoid the appearance of secular terms and obtain a correct normalization
and convergence in a static limit [20, 21, 22, 23].

A free atom (or molecule) is fully degenerate in the quantum number $m$. We
assume that there are no further (accidental) degeneracies present. For these
assumptions, the states $\left|n\right\rangle=\left|aJm\right\rangle$ will be
”good states”[24] in respect to all the operators involved, which allows us to
perform a treatment analogous to a nondegenerate one. The maximal field
strengths to which such a perturbative treatment will stay valid are highly
dependent on the system and states under consideration.

####  2.3.1 Time-independent perturbation

In a static system, $\boldsymbol{\mathcal{E}}=\boldsymbol{\mathcal{E}}^{0}$,
and $\boldsymbol{\mathcal{H}}=\boldsymbol{\mathcal{H}}^{0}$. The 1st order
(Rayleigh-Schrödinger) perturbation theory for a potential $V$ leads to

$\displaystyle\left|N\right\rangle\simeq\left|n\right\rangle-\sum_{k}\frac{\left\langle
k\right|V\left|n\right\rangle}{\omega_{kn}}\left|k\right\rangle$ (5)

with $\omega_{kn}=E_{k}-E_{n}=E_{a^{\prime}J^{\prime}}-E_{aJ}$. Thus, up to
1st order, the matrix element $\left\langle N\right|d_{i}\left|N\right\rangle$
under the influence of the perturbation $V^{EM}$ can be written as:

$\displaystyle\left\langle N\right|d_{i}\left|N\right\rangle$
$\displaystyle\simeq\left\langle
n\right|d_{i}\left|n\right\rangle+2Re\sum_{k}\Bigg{[}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|d_{j}\left|n\right\rangle}{\omega_{kn}}\mathcal{E}_{j}^{0}$ (6)
$\displaystyle+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{\omega_{kn}}\mathcal{H}_{j}^{0}+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|Q_{j\varkappa}\left|n\right\rangle}{\omega_{kn}}\partial_{j}\mathcal{E}_{\varkappa}^{0}\Bigg{]}$

####  2.3.2 Time-dependent perturbation

For deriving the time-dependent contributions, we start by writing the non
static part of Eq. (1) explicitly under the use of Eqs. (2), (3), and (4):

$\displaystyle V^{EM}(t)$ $\displaystyle\equiv\sum_{\ell\neq
0}V^{\ell}=\sum_{\ell\neq
0}\Big{[}-\boldsymbol{d}\cdot\boldsymbol{\mathcal{E}}^{\ell}-\boldsymbol{\mu}\cdot\boldsymbol{\mathcal{H}}^{\ell}-Q_{ij}\partial_{i}\mathcal{E}^{\ell}\Big{]}$
$\displaystyle=\sum_{\ell\neq
0}2Re\left(\left[-i\omega^{\ell}\boldsymbol{d}\cdot\boldsymbol{\mathcal{A}}^{\ell}-i\boldsymbol{\mu}\cdot\big{(}\boldsymbol{k}^{\ell}\times\boldsymbol{\mathcal{A}}^{\ell}\big{)}+\omega^{\ell}Q_{ij}k_{i}^{\ell}\mathcal{A}_{j}^{\ell}\right]e^{-i\omega^{\ell}t}\right)$
$\displaystyle\equiv\sum_{\ell\neq
0}\Big{[}F^{\ell}e^{-i\omega^{\ell}t}+(F^{\ell})^{\dagger}e^{i\omega^{\ell}t}\Big{]}$

We can see that we have the special situation of a perturbation periodic in
time (see, for example [20, 25]). In this case, the induced dipole moment is
given by:

$\displaystyle\left\langle
N\right|d_{i}\left|N\right\rangle(t)\simeq\left\langle n\right|$
$\displaystyle d_{i}\left|n\right\rangle$ (7) $\displaystyle-\sum_{\ell\neq
0}\sum_{k}$ $\displaystyle\left[\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|F^{\ell}\left|n\right\rangle}{\omega_{kn}-\omega^{\ell}}+\frac{\left\langle
n\right|F^{\ell}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}+\omega^{\ell}}\right]e^{-i\omega^{\ell}t}$
$\displaystyle+$ $\displaystyle\left[\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|(F^{\ell})^{\dagger}\left|n\right\rangle}{\omega_{kn}+\omega^{\ell}}+\frac{\left\langle
n\right|(F^{\ell})^{\dagger}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}-\omega^{\ell}}\right]e^{i\omega^{\ell}t}$

This expression is only true far from resonance
$\left(|\omega^{\ell}|\approx|\omega_{kn}|\right)$. The situation close to
resonance is discussed in sec. 5.
By expanding the fractions and identifying
$-i\omega^{\ell}F^{\ell}e^{-i\omega^{\ell}t}+i\omega^{\ell}(F^{\ell})^{\dagger}e^{i\omega^{\ell}t}$
with the time derivative of $V^{\ell}$, we can rewrite this expression as:

$\displaystyle\left\langle n\right|$ $\displaystyle
d_{i}\left|n\right\rangle-2Re\sum_{\ell\neq 0}\sum_{k}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\big{[}\omega_{kn}V^{\ell}+i\dot{V}^{\ell}\big{]}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}$
(8)

In the static case, $\omega^{\ell}=0$, $\dot{V}^{EM}$ vanishes, and the eq:
returns to the same form as Eq. (6). Therefore, we can combine both. After
reinserting the definition of $V^{\ell}$, we arrive at a general expression
for the electric dipole moment element:

$\displaystyle\langle d_{i}\rangle\simeq\left\langle
n\right|d_{i}\left|n\right\rangle+Re$
$\displaystyle\Big{[}\prescript{dd}{}{\alpha}_{ij}^{\ell}\Big{]}\mathcal{E}_{j}^{\ell}+Im\Big{[}\prescript{dd}{}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{E}}_{j}^{\ell}$
(9) $\displaystyle+Re$
$\displaystyle\Big{[}\prescript{d\mu}{}{\alpha}_{ij}^{\ell}\Big{]}\mathcal{H}_{j}^{\ell}+Im\Big{[}\prescript{d\mu}{}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{H}}_{j}^{\ell}$
$\displaystyle+Re$
$\displaystyle\Big{[}\prescript{dQ}{}{\alpha}_{ij\varkappa}^{\ell}\Big{]}\partial_{j}\mathcal{E}_{\varkappa}^{\ell}+Im\Big{[}\prescript{dQ}{}{\alpha^{\prime}}_{ij\varkappa}^{\ell}\Big{]}\partial_{j}\dot{\mathcal{E}}_{\varkappa}^{\ell}$

Here and in the following, we imply, according to the Einstein convention, the
summation over $\ell$ (including the static case $\ell=0$, $\omega^{0}=0$).
Above, we further introduced the (generalized) polarizabilities:

$\displaystyle\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$
$\displaystyle=2\sum_{k}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\omega_{kn}$
(10) $\displaystyle\prescript{d\mu}{}{\alpha^{\prime}}_{ij}^{\ell}$
$\displaystyle=-2\sum_{k}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}$

with $\prescript{dd}{}{\alpha}_{ij}^{\ell}$,
$\prescript{dd}{}{\alpha^{\prime}}_{ij}^{\ell}$,
$\prescript{dQ}{}{\alpha}_{ij\varkappa}^{\ell}$, and
$\prescript{dQ}{}{\alpha^{\prime}}_{ij\varkappa}^{\ell}$ following
analogously.

Likewise, we can perform the same derivation for the magnetic dipole moment,
which leads to:

$\displaystyle\langle\mu_{i}\rangle\simeq\left\langle
n\right|\mu_{i}\left|n\right\rangle+Re$ $\displaystyle\Big{[}\prescript{\mu
d}{}{\alpha}_{ij}^{\ell}\Big{]}\mathcal{E}_{j}^{\ell}+Im\Big{[}\prescript{\mu
d}{}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{E}}_{j}^{\ell}$ (11)
$\displaystyle+Re$
$\displaystyle\Big{[}\prescript{\mu\mu}{}{\alpha}_{ij}^{\ell}\Big{]}\mathcal{H}_{j}^{\ell}+Im\Big{[}\prescript{\mu\mu}{}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{H}}_{j}^{\ell}$
$\displaystyle+Re$ $\displaystyle\Big{[}\prescript{\mu
Q}{}{\alpha}_{ij\varkappa}^{\ell}\Big{]}\partial_{j}\mathcal{E}_{\varkappa}^{\ell}+Im\Big{[}\prescript{\mu
Q}{}{\alpha^{\prime}}_{ij\varkappa}^{\ell}\Big{]}\partial_{j}\dot{\mathcal{E}}_{\varkappa}^{\ell}$

Finally, for the electric quadrupole moment, we get:

$\displaystyle\langle Q_{ij}\rangle\simeq\left\langle
n\right|Q_{ij}\left|n\right\rangle+Re$
$\displaystyle\Big{[}\prescript{Qd}{}{\alpha}_{ij\varkappa}^{\ell}\Big{]}\mathcal{E}_{\varkappa}^{\ell}+Im\Big{[}\prescript{Qd}{}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{E}}_{\varkappa}^{\ell}$
(12) $\displaystyle+Re$
$\displaystyle\Big{[}\prescript{Q\mu}{}{\alpha}_{ij\varkappa}^{\ell}\Big{]}\mathcal{H}_{\varkappa}^{\ell}+Im\Big{[}\prescript{Q\mu}{}{\alpha^{\prime}}_{ij\varkappa}^{\ell}\Big{]}\dot{\mathcal{H}}_{\varkappa}^{\ell}$
$\displaystyle+Re$ $\displaystyle\Big{[}\prescript{QQ}{}{\alpha}_{ij\varkappa
l}^{\ell}\Big{]}\partial_{\varkappa}\mathcal{E}_{l}^{\ell}+Im\Big{[}\prescript{dQ}{}{\alpha^{\prime}}_{ij\varkappa
l}^{\ell}\Big{]}\partial_{\varkappa}\dot{\mathcal{E}}_{l}^{\ell}$

For simplicity and consistency, we will refer to all $\alpha$ terms as
(generalized) polarizabilities, even if, for specific cases, other names such
as permeability or quadrupolarizability exist. We further even refer to the
field-independent static moments as polarizabilities.

##  3 Analyzing the polarizabilities in terms of parity and time reversal

In usual atomic physics derivations for electric and magnetic dipole moments,
most of the terms in Eqs. (9), (11), and (12) are absent. As we will see in a
moment, this is due to the symmetries of $H^{0}$. These terms can, however,
become observable when $H^{0}$ is additionally perturbed by $P$\- and $T$-odd
potentials $V^{N}$:

$\displaystyle H$ $\displaystyle=H^{0}+V^{EM}+V^{N}$ (13) $\displaystyle
V^{N}$ $\displaystyle=-V^{P}-V^{PT}-V^{T}$

$V^{P}$ is an internal $P$-odd, $T$-even, $V^{PT}$ a $P$-odd, $T$-odd, and
$V^{T}$ a $P$-even, $T$-odd interaction. The global minus sign in $V^{N}$ does
not carry a physical meaning and is just chosen in analogy to Eq (1).
In the following, we will take $\left|N\right\rangle$ to be the eigenstates of
this full Hamiltonian $H$. To make the treatment more general, we do not
specify the structure of these operators but instead only use their
transformation behaviors under $P$ and $T$. For explicit examples of $V^{P}$,
$V^{PT}$, and $V^{T}$ operators that arise in electron-nucleon interactions,
see, for instance, Ref. [26, 27]. We restrict ourselves to symmetry-violating
interactions that take place inside the free atom or molecule and do not
directly couple to the externally applied fields. For many situations, this
assumption makes sense as the internal electric and magnetic fields will
exceed the externally applied ones in strength.

In the following, we want to investigate in detail which terms in Eqs. (9),
(11), and (12) correspond to the breakings of which particular symmetries. The
most important properties of the $P$ and $T$ operators are indicated in table
1.

| $\boldsymbol{d}$ | $\boldsymbol{\mu}$ | $\boldsymbol{Q}$ | $\boldsymbol{J}$ | $V^{P}$ | $V^{PT}$ | $V^{T}$
---|---|---|---|---|---|---|---
$\boldsymbol{P}$ | - | + | + | + | - | - | +
$\boldsymbol{T}$ | + | - | + | - | + | - | -

$\displaystyle\quad P\left|aJm\right\rangle$
$\displaystyle=\pm\left|aJm\right\rangle$ $\displaystyle\quad
T\left|aJm\right\rangle$ $\displaystyle=(-1)^{J-m}\left|aJ-m\right\rangle$

Table 1: Transformation properties of the three multipole operators, the total
angular momentum, the symmetry violating potentials, and the free Hamiltonian
eigenstates $\left|n\right\rangle=\left|aJm\right\rangle$ under the linear
parity $P$, and antilinear time reversal $T$ operator. They satisfy
$PP=T^{\dagger}T=1$.

###  3.1 $P$ and $T$ odd polarizabilities

The effects of $V^{P}$, $V^{T}$, and $V^{PT}$ on
$\langle\boldsymbol{d}\rangle$, $\langle\boldsymbol{\mu}\rangle$, and
$\langle{Q}_{ij}\rangle$ can again be derived using perturbation theory. We
discuss the details of this procedure in appendix B. From it, we get a new,
more general, expression for the induced electric dipole moment $\langle
d_{i}\rangle$ in terms of $P,\;T$-even and -odd polarizabilities:

$\displaystyle\langle d_{i}\rangle\simeq Re$
$\displaystyle\Big{[}\prescript{d}{}{\alpha}_{i}+\prescript{d}{P}{\alpha}_{i}+\prescript{d}{PT}{\alpha}_{i}+\prescript{d}{T}{\alpha}_{i}\Big{]}$
(14) $\displaystyle+Re$
$\displaystyle\Big{[}\prescript{dd}{}{\alpha}_{ij}^{\ell}+\prescript{dd}{P}{\alpha}_{ij}^{\ell}+\prescript{dd}{PT}{\alpha}_{ij}^{\ell}+\prescript{dd}{T}{\alpha}_{ij}^{\ell}\Big{]}\mathcal{E}_{j}^{\ell}$
$\displaystyle+Re$
$\displaystyle\left[\prescript{d\mu}{}{\alpha}_{ij}^{\ell}+\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}+\prescript{d\mu}{PT}{\alpha}_{ij}^{\ell}+\prescript{d\mu}{T}{\alpha}_{ij}^{\ell}\right]\mathcal{H}_{j}^{\ell}$
$\displaystyle+Re$
$\displaystyle\left[\prescript{dQ}{}{\alpha}_{ij}^{\ell}+\prescript{dQ}{P}{\alpha}_{ij}^{\ell}+\prescript{dQ}{PT}{\alpha}_{ij}^{\ell}+\prescript{dQ}{T}{\alpha}_{ij}^{\ell}\right]\partial_{j}\mathcal{E}_{\varkappa}^{\ell}$
$\displaystyle+Im$
$\displaystyle\Big{[}\prescript{dd}{}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{dd}{P}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{dd}{PT}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{dd}{T}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{E}}_{j}^{\ell}$
$\displaystyle+Im$
$\displaystyle\left[\prescript{d\mu}{}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{d\mu}{P}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{d\mu}{PT}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{d\mu}{T}{\alpha^{\prime}}_{ij}^{\ell}\right]\dot{\mathcal{H}}_{j}^{\ell}$
$\displaystyle+Im$
$\displaystyle\left[\prescript{dQ}{}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{dQ}{P}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{dQ}{PT}{\alpha^{\prime}}_{ij}^{\ell}+\prescript{dQ}{T}{\alpha^{\prime}}_{ij}^{\ell}\right]\partial_{j}\dot{\mathcal{E}}_{\varkappa}^{\ell}$

with $\prescript{d}{}{\alpha}_{i}=\left\langle
n\right|d_{i}\left|n\right\rangle$. The first term in each row is, as before,
defined according to eq: (10). As indicated by their lower left indices, the
remaining polarizabilities are proportional to one of the three potentials
$V^{P}$, $V^{T}$, and $V^{PT}$. We define $\prescript{d}{PT}{\alpha}_{i}$ as
(see sec. 3.3):

$\displaystyle\prescript{d}{PT}{\alpha}_{i}=2\sum_{k}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|n\right\rangle}{\omega_{kn}}$ (15)

with $\prescript{d}{P}{\alpha}_{i}$ and $\prescript{d}{T}{\alpha}_{i}$
following accordingly.
The remaining terms are defined analogously to:

$\displaystyle\prescript{d\mu}{PT}{\alpha}_{ij}^{\ell}=2\sum_{k,\,l}\Bigg{[}$
$\displaystyle\frac{\left\langle
n\right|V^{PT}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}$
$\displaystyle+$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})(\omega_{ln}^{2}-(\omega^{\ell})^{2})}\left(\omega_{kn}\omega_{ln}+(\omega^{\ell})^{2}\right)\Bigg{]}$
$\displaystyle\prescript{d\mu}{PT}{\alpha^{\prime}}_{ij}^{\ell}=-2\sum_{k,\,l}\Bigg{[}$
$\displaystyle\frac{\left\langle
n\right|V^{PT}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}$
$\displaystyle+$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})(\omega_{ln}^{2}-(\omega^{\ell})^{2})}\left(\omega_{ln}+\omega_{kn}\right)\Bigg{]}$
(16)

Likewise, expressions for $\langle\mu_{i}\rangle$ and $\langle Q_{ij}\rangle$
in terms of $P$ and $T$ odd polarizabilities can be derived. We will now
analyze these polarizabilities regarding their transformation behavior under
$P$ and $T$. From this, we will see that around three-quarters of all possible
terms in Eq. (14) necessarily vanish in all systems that obey our initial
assumptions made in sec. 2.1.

###  3.2 $P$ and $T$ properties of polarizabilities in zero field: 0th order
in $V^{EM}$

We start by discussing the vector polarizabilities
$\prescript{d}{}{\alpha}_{i}$, $\prescript{d}{P}{\alpha}_{i}$,
$\prescript{d}{PT}{\alpha}_{i}$, and $\prescript{d}{T}{\alpha}_{i}$. Because
these arise in the 0th order perturbation in $V^{EM}$, they are independent of
the externally applied field. From Eq. (14), we can see that their existence
results in a permanent electric dipole moment (EDM).

####  3.2.1 P-violation

It can be easily shown that $\prescript{d}{}{\alpha}_{i}=0$ due to the odd
parity of $\boldsymbol{d}$. Using the properties given in Table 1 we get:

$\displaystyle\prescript{d}{}{\alpha}_{i}=\left\langle
n\right|d_{i}\left|n\right\rangle=\left\langle
n\right|PP\,d_{i}\,PP\left|n\right\rangle=(\pm\left\langle
n\right|)(-d_{i})(\pm\left|n\right\rangle)=-\left\langle
n\right|d_{i}\left|n\right\rangle$ (17)

Thus $\prescript{d}{}{\alpha}_{i}=0$.
To avoid this, we need to mix states of different parity. Effectively, this is
what happens when we introduce the perturbation trough $V^{P}$ or $V^{PT}$. In
this case:

$\displaystyle\prescript{d}{P}{\alpha}_{i}$
$\displaystyle=2\sum_{k}\left\langle
n\right|PPd_{i}PP\left|k\right\rangle\left\langle
k\right|PPV^{P}PP\left|n\right\rangle\,\omega_{kn}^{-1}$
$\displaystyle=2\sum_{k}\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{P}\left|n\right\rangle\,\omega_{kn}^{-1}=\prescript{d}{P}{\alpha}_{i}$

Therefore $\prescript{d}{P}{\alpha}_{i}$ is not forbidden by parity arguments.

####  3.2.2 T-violation

It is further possible to show that $\prescript{d}{}{\alpha}_{i}$ also
vanishes due to reasons of $T$-symmetry. This fact was first demonstrated by
Lee and Yang [28]. Through the Wigner Eckart theorem, we know that the vector
operator $\boldsymbol{d}$ has to be oriented along the total angular momentum
$\boldsymbol{J}$:

$\displaystyle\left\langle
aJm\right|d_{i}\left|aJm\right\rangle=c_{J}^{a}\left\langle
aJm\right|J_{i}\left|aJm\right\rangle$ (18)

with a real $m$-independent constant $c_{J}^{a}$ [18]. We can now introduce
the time reversal operator $T$ to the LHS of Eq. (18). Using properties given
in Table 1 we get:

$\displaystyle\left\langle aJm\right|d_{i}\left|aJm\right\rangle=\left\langle
aJm\right|T^{\dagger}Td_{i}T^{\dagger}T\left|aJm\right\rangle$
$\displaystyle=$ $\displaystyle\left(\left\langle
aJ-m\right|Td_{i}T^{\dagger}\left|aJ-m\right\rangle\right)^{*}=\left(\left\langle
aJ-m\right|d_{i}\left|aJ-m\right\rangle\right)^{*}$

Where the complex conjugation originates from the antilinear properties of $T$
(see appendix A). Analogously, we can show for the RHS of Eq. (18):

$\displaystyle\left\langle
aJm\right|J_{i}\left|aJm\right\rangle=-\left(\left\langle
aJ-m\right|J_{i}\left|aJ-m\right\rangle\right)^{*}$

where we used that $J_{i}$ is $T$-odd. We can now rename $-m\rightarrow m$ and
us the fact that both $d_{i}$, and $J_{i}$ are Hermitian, to arrive at:

$\displaystyle\left\langle
aJm\right|d_{i}\left|aJm\right\rangle=-c_{J}^{a}\left\langle
aJm\right|J_{i}\left|aJm\right\rangle$

Then the comparison with Eq. (18) clearly implies $\left\langle
aJm\right|d_{i}\left|aJm\right\rangle=0$.

We now show how this argument fails under the introduction of a $T$-odd
potential $V^{T}$ or $V^{PT}$:

$\displaystyle Re\left(\prescript{d}{PT}{\alpha}_{i}\right)=$ $\displaystyle
2Re\sum_{k}\left\langle aJm\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|aJm\right\rangle\omega_{nk}^{-1}$ $\displaystyle=$
$\displaystyle\left\langle
aJm\right|\sum_{k}\left[\left(d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\right)+\left(V^{PT}\left|k\right\rangle\left\langle
k\right|d_{i}\right)\right]\left|aJm\right\rangle\omega_{nk}^{-1}$
$\displaystyle\equiv$ $\displaystyle\left\langle
aJm\right|D_{i}\left|aJm\right\rangle$

Where we introduced the Hermitian vector operator $D_{i}$. This operator is
odd under time reversal (and even under Parity):

$\displaystyle TD_{i}T^{\dagger}$
$\displaystyle=\sum_{k}\left[\left(Td_{i}T^{\dagger}T\left|k\right\rangle\left\langle
k\right|T^{\dagger}TV^{PT}T^{\dagger}\right)+\left(TV^{PT}T^{\dagger}T\left|k\right\rangle\left\langle
k\right|T^{\dagger}Td_{i}T^{\dagger}\right)\right]\omega_{nk}^{-1}$
$\displaystyle=-\sum_{k}\left[\left(d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\right)+\left(V^{PT}\left|k\right\rangle\left\langle
k\right|d_{i}\right)\right]\omega_{nk}^{-1}$

where we used that $V^{PT}$ is $T$-odd while $d_{i}$ and
$\sum_{k}\left|k\right\rangle\left\langle k\right|\omega_{nk}^{-1}$ are
$T$-even. Previously, for the Lee-Yang argument, the contradiction arose
because $d$ and $J$, which are proportional to each other, transform
differently under $T$. Now that both $D$ and $J$ are $T$-odd, the
contradiction vanishes.
The combined requirements on parity and time reversal leave
$\prescript{d}{PT}{\alpha}_{i}$ as the only possible source for an EDM and
tells us that such a property is directly proportional to the size of
$V^{PT}$.

####  3.2.3 magnetic dipole moment $\boldsymbol{\mu}$

Contrary to the electric dipole moment, the 0st-order expectation value of the
magnetic dipole moment is already classically allowed. We can also easily see
this from the fact that $D_{i}$ and $\mu_{i}$ transform in the same way under
$P$ and $T$. From this and our derivation above, it directly follows that the
value of the static magnetic dipole moment is insensitive to the effects of
$V^{P}$, $V^{T}$, and $V^{PT}$. Therefore,
$\prescript{\mu}{}{\alpha}_{i}=\left\langle
n\right|\mu_{i}\left|n\right\rangle$ is the only non-zero contribution to the
static magnetic dipole moment. Through the Wigner-Eckart theorem, we can
express it as

$\displaystyle\prescript{\mu}{}{\alpha}_{i}=\prescript{\mu}{}{\alpha}\left\langle
n\right|{J}_{i}\left|n\right\rangle=\prescript{\mu}{}{\alpha}\boldsymbol{\mathcal{J}}_{i}$

where $\prescript{\mu}{}{\alpha}$ is a real constant and equivalent to the
$c_{J}^{a}$ factor that we introduced above. We further define the notation

$\displaystyle\boldsymbol{\mathcal{J}}\equiv\left\langle
n\right|\boldsymbol{J}\left|n\right\rangle$

for the expectation value of the total angular momentum in a given state.

####  3.2.4 Electric quadrupole moment $Q_{ij}$

The electric quadrupole operator ${Q}_{ij}$ is both symmetric and traceless.
The expectation value of such an object can only be proportional to the
angular momentum quadrupole [29]:

$\displaystyle\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\equiv\left\langle
n\right|\tfrac{1}{2}\left[J_{i}J_{j}+J_{j}J_{i}-\tfrac{2}{3}\delta_{ij}\sum_{l}J_{l}J_{l}\right]\left|n\right\rangle$

Therefore, we can write:

$\displaystyle\langle
Q_{ij}\rangle=\prescript{Q}{}{\alpha}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}$

###  3.3 $P$ and $T$ properties of polarizabilities in 1st order in $V^{EM}$

We now continue by analyzing the rank-2 tensor polarizabilities
$\prescript{}{}{\alpha}_{ij}^{\ell}$ by using the techniques that we developed
in the previous section.

####  3.3.1 Restrictions from parity

As previously in Eq. (17), we can analyze the
$\prescript{}{}{\alpha}_{ij}^{\ell}$ that are defined in Eq. (16) by
introducing $P$ into the expression. It can easily be seen that terms that
contain an odd number of $P$-odd operators will vanish
($\prescript{d\mu}{T}{\alpha}_{ij}^{\ell}$ for example contains the $P$-even
operators $\mu_{j}$, and $V^{T}$ and the $P$-odd operator $d_{i}$). Therefore,
half of all possible polarizabilities vanish due to parity reasons.

####  3.3.2 Tensor decomposition of $\prescript{}{}{\alpha}_{ij}^{\ell}$

Before we are able to analyze $\prescript{}{}{\alpha}_{ij}^{\ell}$ in terms of
time reversal, we first need to understand the structure of these (Eucledian)
rank-2 tensors. Such objects can be decomposed into a sum of three components
that transform as scalar, antisymmetric vector, and traceless symmetric
tensors respectively [29]:

$\displaystyle 3\otimes 3$
$\displaystyle=\quad\quad\underset{\text{scalar}}{1}$ $\displaystyle\oplus$
$\displaystyle\quad\quad\;\underset{\text{vector}}{3}$ $\displaystyle\oplus$
$\displaystyle\quad\quad\quad\quad\quad\quad\;\underset{\text{tensor}}{5}$
$\displaystyle\alpha_{ij}^{\ell}$
$\displaystyle=\tfrac{1}{3}\sum_{\varkappa}\alpha_{\varkappa\varkappa}^{\ell}\;\delta_{ij}$
$\displaystyle+$
$\displaystyle\tfrac{1}{2}\left[\alpha_{ij}^{\ell}-\alpha_{ji}^{\ell}\right]$
$\displaystyle+$
$\displaystyle\Big{(}\tfrac{1}{2}\left[\alpha_{ij}^{\ell}+\alpha_{ji}^{\ell}\right]-\tfrac{1}{3}\sum_{\varkappa}(\alpha^{\ell}_{\varkappa\varkappa})\delta_{ij}\Big{)}$

with $\delta_{ij}$ being the Kronecker delta. Angular momentum algebra tells
us that the expectation value of a vector operator is necessarily oriented
along $\boldsymbol{\mathcal{J}}$, and the one of a traceless rank-2 tensor
needs to be proportional to $\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}$ [29]. This
allows us to express the tensor ${\alpha}_{ij}^{\ell}$, with only three
constants ${\bar{\alpha}}_{s}^{\ell}$
$,\prescript{}{}{\bar{\alpha}}_{v}^{\ell}$, and
$\prescript{}{}{\bar{\alpha}}_{t}^{\ell}$:

$\displaystyle\prescript{}{}{\alpha}_{ij}^{\ell}=\delta_{ij}\;\prescript{}{}{\bar{\alpha}}_{s}^{\ell}+\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{}{}{\bar{\alpha}}_{v}^{\ell}+\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{}{}{\bar{\alpha}}_{t}^{\ell}$
(19)

with $\varepsilon_{ijk}$ the Levi-Civita symbol. A non-zero value for the
vector term requires the system to possess $J\geq\tfrac{1}{2}$, while the
tensor term will only exist for systems with $J\geq 1$.
${\bar{\alpha}}_{s}^{\ell}$ $,\prescript{}{}{\bar{\alpha}}_{v}^{\ell}$, and
$\prescript{}{}{\bar{\alpha}}_{t}^{\ell}$ depend on the quantum numbers $a$
and $J$, but not $m$. We use the bar above them to indicate that they are
complex valued. In the following, we will construct expressions for the tensor
decomposition in terms of purely real constants.

####  3.3.3 Restrictions from time reversal

To further eliminate terms, we can again make use of their transformation
under time reversal similar to [30] and to what we did in sec. 3.2.2.
We will perform this procedure on the example of
$\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$ (we ignore for now that this
polarizability vanishes for Parity reasons). Again, we can decompose
$\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$ into its irreducible components:

$\displaystyle\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$
$\displaystyle=\delta_{ij}\;\prescript{d\mu}{}{\bar{\alpha}}_{s}^{\ell}+\varepsilon_{ij\varkappa}\left\langle
aJm\right|J_{\varkappa}\left|aJm\right\rangle\;\prescript{d\mu}{}{\bar{\alpha}}_{v}^{\ell}$
(20) $\displaystyle+\left\langle
aJm\right|\tfrac{1}{2}\left[J_{i}J_{j}+J_{j}J_{i}-\tfrac{2}{3}\delta_{ij}\sum_{l}J_{l}J_{l}\right]\left|aJm\right\rangle\;\prescript{d\mu}{}{\bar{\alpha}}_{t}^{\ell}$

We introduce $1=T^{\dagger}T$ into the LHS that we previously defined trough
Eq. (10) to get:

$\displaystyle\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$
$\displaystyle=2\sum_{k}\frac{\left\langle
aJm\right|T^{\dagger}Td_{i}T^{\dagger}T\left|k\right\rangle\left\langle
k\right|T^{\dagger}T\mu_{j}T^{\dagger}T\left|aJm\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\omega_{kn}$
(21) $\displaystyle=-2\sum_{k}\frac{\left(\left\langle
aJ-m\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|aJ-m\right\rangle\right)^{*}}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\omega_{kn}$

The RHS of Eq. (20) transforms as:

$\displaystyle\prescript{d\mu}{}{\alpha}_{ij}^{\ell}=$
$\displaystyle\delta_{ij}\;\prescript{d\mu}{}{\bar{\alpha}}_{s}^{\ell}-\varepsilon_{ij\varkappa}\left\langle
aJ-m\right|J_{\varkappa}\left|aJ-m\right\rangle\;\prescript{d\mu}{}{\bar{\alpha}}_{v}^{\ell}$
(22) $\displaystyle+\left\langle
aJ-m\right|\tfrac{1}{2}\left[J_{i}J_{j}+J_{j}J_{i}-\tfrac{2}{3}\delta_{ij}\sum_{l}J_{l}J_{l}\right]\left|aJ-m\right\rangle\;\prescript{d\mu}{}{\bar{\alpha}}_{t}^{\ell}$

The important observation here is that the vector part gets a minus sign in
relation to the scalar and tensor parts.
We can now equate Eq. (21) with Eq. (22) and rename $-m\rightarrow m$ to
obtain:

$\displaystyle-\Bigg{[}2\sum_{k}\frac{\left\langle
aJm\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|aJm\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\omega_{kn}\Bigg{]}^{*}$
$\displaystyle=\delta_{ij}\;\prescript{d\mu}{}{\bar{\alpha}}_{s}^{\ell}-\varepsilon_{ij\varkappa}\left\langle
aJm\right|J_{\varkappa}\left|aJm\right\rangle\;\prescript{d\mu}{}{\bar{\alpha}}_{v}^{\ell}$
$\displaystyle\quad\;+\left\langle
aJm\right|\tfrac{1}{2}\left[J_{i}J_{j}+J_{j}J_{i}-\tfrac{2}{3}\delta_{ij}\sum_{l}J_{l}J_{l}\right]\left|aJm\right\rangle\;\prescript{d\mu}{}{\bar{\alpha}}_{t}^{\ell}$
$\displaystyle\Rightarrow$
$\displaystyle\left(\prescript{d\mu}{}{\alpha}_{ij}^{\ell}\right)^{*}=-\delta_{ij}\;\prescript{d\mu}{}{\bar{\alpha}}_{s}^{\ell}+\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{d\mu}{}{\bar{\alpha}}_{v}^{\ell}-\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{d\mu}{}{\bar{\alpha}}_{t}^{\ell}$

The comparison with Eq. (20) finally allows us to express
$\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$ in terms of real coefficients:

$\displaystyle\prescript{d\mu}{}{\alpha}_{ij}^{\ell}=i\delta_{ij}\;\prescript{d\mu}{}{\alpha}_{s}^{\ell}+\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{d\mu}{}{\alpha}_{v}^{\ell}+i\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{d\mu}{}{\alpha}_{t}^{\ell}$

with $\prescript{d\mu}{}{\alpha}_{v}^{\ell}\equiv
Re\left(\prescript{d\mu}{}{\bar{\alpha}}_{v}^{\ell}\right)$. The decomposition
of $\prescript{d\mu}{}{\alpha^{\prime}}_{v}^{\ell}$, that only differs by a
factor of $-\omega_{kn}$ follows in the exact same way.
If we perform an analog treatment on $\prescript{dd}{}{\alpha}_{ij}^{\ell}$,
everything is basically the same except we do not get the global minus sign in
Eq. (21) due to the different $T$-transformation behavior of $d_{j}$ and
$\mu_{j}$. Therefore:

$\displaystyle\left(\prescript{dd}{}{\alpha}_{ij}^{\ell}\right)^{*}$
$\displaystyle=\delta_{ij}\;\prescript{dd}{}{\bar{\alpha}}_{s}^{\ell}-\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{dd}{}{\bar{\alpha}}_{v}^{\ell}+\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{dd}{}{\bar{\alpha}}_{t}^{\ell}$
$\displaystyle\Rightarrow\prescript{dd}{}{\alpha}_{ij}^{\ell}$
$\displaystyle=\delta_{ij}\;\prescript{dd}{}{\alpha}_{s}^{\ell}+i\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{dd}{}{\alpha}_{v}^{\ell}+\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{dd}{}{\alpha}_{t}^{\ell}$
(23)

In appendix C.1, we further show that the argument from above still works for
$P$ and $T$ -odd polarizabilities that are defined according to Eq. (16).

We conclude that polarizabilities containing an even number of $T$-odd
operators possess purely real scalar and tensor components and a purely
imaginary vector component. Polarizabilities with an odd number of $T$-odd
operators, on the other hand, have a real vector component but imaginary
scalar and tensor parts. From the definitions in 3.1, we can see that the
polarizabilities $\alpha$ that couple to the fields need to be necessarily
real while the polarizabilities $\alpha^{\prime}$ that couple to the field
derivatives need to be imaginary.

####  3.3.4 Dipole-quadrupole terms: rank-3 tensor decomposition

Before we can use the statements that we derived above to formulate general
reduced expressions for the induced multipole moments, we still need to
examine the rank-3 polarizability tensors. These arise in Eqs. (9), and (11)
as cross polarizabilities between electric and magnetic dipoles and the
electric quadrupole moment.

As a symmetric tensor, the quadrupole operator can only have 6 independent
elements, leading us to an $\alpha_{ij\varkappa}^{\ell}$ with $3\otimes 6=18$
elements. The decomposition under $SU(2)$ of a rank-3 tensor that is symmetric
in its last two components was derived in Ref. [31]:

$\displaystyle
18=\underset{vector}{3}\oplus\underset{\begin{subarray}{c}tensor\\\
(octupole)\end{subarray}}{7}\oplus\underset{vector}{3}\oplus\underset{\begin{subarray}{c}tensor\\\
(quadrupole)\end{subarray}}{5}$

where the first two tensors are symmetric and the second two are of mixed
symmetry. This allows us to write the rank-3 polarizabilities as:

$\displaystyle\prescript{}{}{\alpha}_{ij\varkappa}^{\ell}=\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}}\,\prescript{}{}{\bar{\alpha}}_{v_{s}}^{\ell}+\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}\,\prescript{}{}{\bar{\alpha}}_{t_{s}}^{\ell}+\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\,\prescript{}{}{\bar{\alpha}}_{v_{m}}^{\ell}+\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\,\prescript{}{}{\bar{\alpha}}_{t_{m}}^{\ell}$

where $S_{ij\varkappa}$ are real symmetric, and $M_{ij\varkappa}$ are real
mixed-symmetry tensors. They depend on the indicated powers of
$\boldsymbol{J}$. From the definitions given in [31], we can derive the
following tensors:

$\displaystyle\mathcal{S}^{\boldsymbol{J}}_{ij\varkappa}$
$\displaystyle=\tfrac{1}{5}\left(\mathcal{J}_{i}\delta_{j\varkappa}+\mathcal{J}_{j}\delta_{i\varkappa}+\mathcal{J}_{\varkappa}\delta_{ij}\right)$
(24) $\displaystyle\mathcal{T}_{ij\varkappa}$
$\displaystyle=\tfrac{1}{6}(J_{i}J_{j}J_{\varkappa}+J_{j}J_{\varkappa}J_{i}+J_{\varkappa}J_{i}J_{j}+J_{j}J_{i}J_{\varkappa}+J_{\varkappa}J_{j}J_{i}+J_{i}J_{\varkappa}J_{j})$
$\displaystyle\mathcal{S}^{\boldsymbol{J}^{3}}_{ij\varkappa}$
$\displaystyle=\left\langle
n\right|\Bigr{[}\mathcal{T}_{ij\varkappa}-\tfrac{1}{5}(\mathcal{T}_{inm}\delta_{nm}\delta_{j\varkappa}+\mathcal{T}_{njm}\delta_{nm}\delta_{i\varkappa}+\mathcal{T}_{nm\varkappa}\delta_{nm}\delta_{ij})\Bigl{]}\left|n\right\rangle$
$\displaystyle\mathcal{M}^{\boldsymbol{J}}_{ij\varkappa}$
$\displaystyle=\tfrac{1}{4}(2\delta_{j\varkappa}\mathcal{J}_{i}-\delta_{ij}\mathcal{J}_{\varkappa}-\delta_{i\varkappa}\mathcal{J}_{j})$
$\displaystyle\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$
$\displaystyle=\tfrac{1}{2}(\delta_{jl}\varepsilon_{im\varkappa}+\delta_{\varkappa
l}\varepsilon_{imj})\mathcal{Q}_{lm}^{\boldsymbol{J}^{2}}$

So far, we have not used the fact that the quadrupole tensor is traceless.
This reduces its number of independent elements by one, leading to $3\otimes
5=15$. For the tensor $\prescript{}{}{\alpha}_{ij\varkappa}^{\ell}$, this
means that the partial trace over the last two components vanishes.
$\mathcal{M}^{\boldsymbol{J}}_{ij\varkappa}$,
$\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$, and
$\mathcal{S}^{\boldsymbol{J}^{3}}_{ij\varkappa}$ fullfill this condition. For
the symmetric vector component $\mathcal{S}^{\boldsymbol{J}}_{ij\varkappa}$,
we have however:

$\displaystyle
0=\prescript{}{}{\bar{\alpha}}_{v_{s}}^{\ell}\,\sum_{l}\mathcal{S}^{\boldsymbol{J}}_{ill}$
$\displaystyle=\prescript{}{}{\bar{\alpha}}_{v_{s}}^{\ell}\,\sum_{l}\tfrac{1}{5}\left(\mathcal{J}_{i}\delta_{ll}+\mathcal{J}_{l}\delta_{il}+\mathcal{J}_{l}\delta_{il}\right)=\prescript{}{}{\bar{\alpha}}_{v_{s}}^{\ell}\,\mathcal{J}_{i}$

This implies $\prescript{}{}{\bar{\alpha}}_{v_{s}}^{\ell}=0$. Therefore the
contribution of $\mathcal{S}^{\boldsymbol{J}}_{ij\varkappa}$ vanishes and we
arrive at the decomposition of the rank-3 tensor into only 3 irreducible
structures:

$\displaystyle\prescript{}{}{\alpha}_{ij\varkappa}^{\ell}=\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}\,\prescript{}{}{\bar{\alpha}}_{t_{s}}^{\ell}+\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\,\prescript{}{}{\bar{\alpha}}_{v_{m}}^{\ell}+\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\,\prescript{}{}{\bar{\alpha}}_{t_{m}}^{\ell}$

Again, the decomposition of ${\alpha^{\prime}}_{ij\varkappa}^{\ell}$ follows
completely analogous.

####  3.3.5 Dipole-quadrupole terms: restrictions under parity and time
reversal

The restrictions from parity still apply as before: only polarizabilities
containing an even number of $P$-odd operators can exist.
To analyze the behavior under time reversal, we can proceed analogously to
sec. 3.3.3. There, we saw that if the polarizability contains an even number
of $T$-odd operators, then the $T$-even scalar and tensor terms are real while
the $T$-odd vector component is purely imaginary. As can be seen by the
indicated powers in $J$, the rank-3 polarizability components proportional to
$\mathcal{M}^{\boldsymbol{J}}_{ij\varkappa}$, and
$\mathcal{S}^{\boldsymbol{J}^{3}}_{ij\varkappa}$ are $T$-odd, while the one
proportional to $\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$ is $T$-even.
Therefore, following the same line of argument as before, we get:

even number of $T$-odd operators:
$\displaystyle\prescript{}{}{\alpha}_{ij\varkappa}^{\ell}=i\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}\,\prescript{}{}{\alpha}_{t_{s}}^{\ell}+i\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\,\prescript{}{}{\alpha}_{v_{m}}^{\ell}+\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\,\prescript{}{}{\alpha}_{t_{m}}^{\ell}$
odd number of $T$-odd operators:
$\displaystyle\prescript{}{}{\alpha}_{ij\varkappa}^{\ell}=\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}\,\prescript{}{}{\alpha}_{t_{s}}^{\ell}+\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\,\prescript{}{}{\alpha}_{v_{m}}^{\ell}+i\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\,\prescript{}{}{\alpha}_{t_{m}}^{\ell}$

with
$\prescript{}{}{\alpha}_{t_{s}}^{\ell},\prescript{}{}{\alpha}_{v_{m}}^{\ell},\prescript{}{}{\alpha}_{t_{m}}^{\ell}\in\mathbb{R}$.

####  3.3.6 Rank-4 tensors

The only rank-4 tensors that appear in our derivation are:
$\prescript{QQ}{}{\alpha}_{ij\varkappa l}^{\ell}$,
$\prescript{QQ}{P}{\alpha}_{ij\varkappa l}^{\ell}$,
$\prescript{QQ}{PT}{\alpha}_{ij\varkappa l}^{\ell}$, and
$\prescript{QQ}{T}{\alpha}_{ij\varkappa l}^{\ell}$. Without decomposing the
tensor, we can show that $\prescript{QQ}{P}{\alpha}_{ij\varkappa l}^{\ell}$,
$\prescript{QQ}{PT}{\alpha}_{ij\varkappa l}^{\ell}$, and
$\prescript{QQ}{T}{\alpha}_{ij\varkappa l}^{\ell}$ vanish (see appendix C.2).
Because we are mainly interested in symmetry-violating polarizabilities, we
will not enter deeper into the structure of
$\prescript{QQ}{}{\alpha}_{ij\varkappa l}^{\ell}$. The decomposition of a
tensor with the same symmetries as $\prescript{QQ}{}{\alpha}_{ij\varkappa
l}^{\ell}$ can be found in [32]. This all applies to the primed tensors
$\prescript{QQ}{}{\alpha^{\prime}}_{ij\varkappa l}^{\ell}$ in the same way.

###  3.4 Full expressions of the induced moments

We can now use the results from sec. 3 to decompose all polarizabilities in
Eq. (14) into their irreducible components and eliminate all terms that vanish
due to reasons of parity and time reversal. From all this, we arrive at a
general expression of the induced electric dipole moment:

$\displaystyle\langle\boldsymbol{d}\rangle$
$\displaystyle\simeq\prescript{d}{PT}{\alpha}\,\boldsymbol{\mathcal{J}}+\prescript{dd}{}{\alpha}_{s}^{\ell}\,\boldsymbol{\mathcal{E}}^{\ell}+\prescript{dd}{}{\alpha^{\prime}}_{v}^{\ell}\,(\dot{\boldsymbol{\mathcal{E}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{dd}{}{\alpha}_{t}^{\ell}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{E}}^{\ell}$
(25)
$\displaystyle+\prescript{d\mu}{P}{\alpha^{\prime}}_{s}^{\ell}\,\dot{\boldsymbol{\mathcal{H}}}^{\ell}+\prescript{d\mu}{P}{\alpha}_{v}^{\ell}\,({\boldsymbol{\mathcal{H}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{d\mu}{P}{\alpha^{\prime}}_{t}^{\ell}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\dot{\boldsymbol{\mathcal{H}}}^{\ell}$
$\displaystyle+\prescript{d\mu}{PT}{\alpha}_{s}^{\ell}\,\boldsymbol{\mathcal{H}}^{\ell}+\prescript{d\mu}{PT}{\alpha^{\prime}}_{v}^{\ell}\,(\dot{\boldsymbol{\mathcal{H}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{d\mu}{PT}{\alpha}_{t}^{\ell}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{H}}^{\ell}$
$\displaystyle+\left(\prescript{dQ}{P}{\alpha^{\prime}}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J^{3}}}+\prescript{dQ}{P}{\alpha^{\prime}}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\hat{\boldsymbol{e}}_{i}\partial_{j}\dot{\mathcal{E}}^{\ell}_{\varkappa}+\prescript{dQ}{P}{\alpha}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}$
$\displaystyle+\left(\prescript{dQ}{PT}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{dQ}{PT}{\alpha}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}+\prescript{dQ}{PT}{\alpha^{\prime}}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\hat{\boldsymbol{e}}_{i}\partial_{j}\dot{\mathcal{E}}^{\ell}_{\varkappa}$

We summarize the notations used in this expression:
$\boldsymbol{\mathcal{E}}^{\ell}$ is the electric and
$\boldsymbol{\mathcal{H}}^{\ell}$ the magnetic field, while
$\dot{\boldsymbol{\mathcal{E}}^{\ell}}$ and
$\dot{\boldsymbol{\mathcal{H}}^{\ell}}$ are their time derivatives.
$\boldsymbol{J}$ is the (conserved) total angular momentum of the system, and
$\boldsymbol{\mathcal{J}}=\left\langle
n\right|\boldsymbol{J}\left|n\right\rangle$.
$\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}$ is the symmetric traceless
matrix defined by $\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}=\left\langle
n\right|\tfrac{1}{2}\left[J_{i}J_{j}+J_{j}J_{i}-\frac{2}{3}\delta_{ij}\sum_{l}J_{l}J_{l}\right]\left|n\right\rangle$.
$\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}$,
$\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}$, and
$\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}$ are rank-3 tensors that have
been defined in sec. 3.3.5. $\hat{\boldsymbol{e}}_{i}$ is a unit vector along
the (Cartesian) coordinate $i$.
The real coefficients $\alpha$, $\alpha^{\prime}$ are the (generalized)
polarizabilities that result from the tensor decomposition of the quantities
defined in sec. 3.1.
As before, the lower right index of the $\alpha$ defines which structure they
belong to ($s$=scalar trace, $v$=antisymmetric vector, $t$=symmetric traceless
quadrupole tensor, $t_{s}$= symmetric traceless octupole tensor, $v_{m}$=mixed
symmetry vector, $t_{m}$=mixed symmetry tensor ).
The lower left index of the $\alpha$ tells if the interaction requires
perturbation by a $P$, $T$, or both $PT$ odd potential.

Equivalently to the electric dipole, we can find the induced magnetic dipole
moment:

$\displaystyle\langle\boldsymbol{\mu}\rangle$
$\displaystyle\simeq\prescript{\mu}{}{\alpha}\,\boldsymbol{\mathcal{J}}+\prescript{\mu\mu}{}{\alpha}_{s}^{\ell}\,\boldsymbol{\mathcal{H}}^{\ell}+\prescript{\mu\mu}{}{\alpha^{\prime}}_{v}^{\ell}\,(\dot{\boldsymbol{\mathcal{H}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{\mu\mu}{}{\alpha}_{t}^{\ell}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{H}}^{\ell}$
(26) $\displaystyle+\prescript{\mu
d}{P}{\alpha^{\prime}}_{s}^{\ell}\,\dot{\boldsymbol{\mathcal{E}}}^{\ell}+\prescript{\mu
d}{P}{\alpha}_{v}^{\ell}\,({\boldsymbol{\mathcal{E}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{\mu
d}{P}{\alpha^{\prime}}_{t}^{\ell}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\dot{\boldsymbol{\mathcal{E}}}^{\ell}$
$\displaystyle+\prescript{\mu
d}{PT}{\alpha}_{s}^{\ell}\,\boldsymbol{\mathcal{E}}^{\ell}+\prescript{\mu
d}{PT}{\alpha^{\prime}}_{v}^{\ell}\,(\dot{\boldsymbol{\mathcal{E}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{\mu
d}{PT}{\alpha}_{t}^{\ell}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{E}}^{\ell}$
$\displaystyle+\left(\prescript{\mu
Q}{T}{\alpha^{\prime}}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{\mu
Q}{T}{\alpha^{\prime}}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\hat{\boldsymbol{e}}_{i}\partial_{j}\dot{\mathcal{E}}^{\ell}_{\varkappa}+\prescript{\mu
Q}{T}{\alpha}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}$
$\displaystyle+\left(\prescript{\mu
Q}{}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{\mu
Q}{}{\alpha}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}+\prescript{\mu
Q}{}{\alpha^{\prime}}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\hat{\boldsymbol{e}}_{i}\partial_{j}\dot{\mathcal{E}}^{\ell}_{\varkappa}$

For the electric quadrupole, we get:

$\displaystyle\langle Q_{ij}\rangle$
$\displaystyle\simeq\prescript{Q}{}{\alpha}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}+\prescript{QQ}{}{\alpha}_{ij\varkappa
l}^{\ell}\,\partial_{\varkappa}\mathcal{E}^{\ell}_{l}$ (27)
$\displaystyle+\left(\prescript{Qd}{P}{\alpha^{\prime}}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{Qd}{P}{\alpha^{\prime}}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\dot{\mathcal{E}}^{\ell}_{\varkappa}+\prescript{Qd}{P}{\alpha}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}{\mathcal{E}}^{\ell}_{\varkappa}$
$\displaystyle+\left(\prescript{Qd}{PT}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{Qd}{PT}{\alpha}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right){\mathcal{E}}^{\ell}_{\varkappa}+\prescript{Qd}{PT}{\alpha^{\prime}}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\dot{\mathcal{E}}^{\ell}_{\varkappa}$
$\displaystyle+\left(\prescript{Q\mu}{T}{\alpha^{\prime}}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{Qd}{T}{\alpha^{\prime}}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\dot{\mathcal{H}}^{\ell}_{\varkappa}+\prescript{Q\mu}{T}{\alpha}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}{\mathcal{H}}^{\ell}_{\varkappa}$
$\displaystyle+\left(\prescript{Q\mu}{}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{Q\mu}{}{\alpha}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right){\mathcal{H}}^{\ell}_{\varkappa}+\prescript{Q\mu}{}{\alpha^{\prime}}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\dot{\mathcal{H}}^{\ell}_{\varkappa}$

We can see that each field configuration of $\boldsymbol{\mathcal{E}}^{\ell}$,
$\boldsymbol{\mathcal{H}}^{\ell}$, $\dot{\boldsymbol{\mathcal{E}}^{\ell}}$,
$\dot{\boldsymbol{\mathcal{H}}^{\ell}}$, and $\boldsymbol{\mathcal{J}}$
uniquely couples to a specific symmetry violation.
The absence of the $T$ odd polarizabilities
$\prescript{dd}{T}{\alpha}^{\ell}$, $\prescript{\mu\mu}{T}{\alpha}^{\ell}$,
and $\prescript{QQ}{T}{\alpha}^{\ell}$ in the expressions above is explained
in appendix C.2.

###  3.5 Induced $\langle V^{P}\rangle$, $\langle V^{PT}\rangle$, and $\langle
V^{T}\rangle$

Additionally to the induced moments that are defined above, also the
expectation values of the symmetry violating potentials $\langle
V^{P}\rangle$, $\langle V^{PT}\rangle$, and $\langle V^{T}\rangle$ themselves
contribute to the full perturbative expressions of $\langle H\rangle$, and
$\left|N\right\rangle$. Obviously, all these potentials vanish in 0st order in
$V^{EM}$:

$\displaystyle\left\langle n\right|V^{P}\left|n\right\rangle=\left\langle
n\right|V^{PT}\left|n\right\rangle=\left\langle
n\right|V^{T}\left|n\right\rangle=0$

If we now consider how $V^{P}$, $V^{PT}$, and $V^{T}$ are perturbed through
$V^{EM}$, one could assume that the situation will be identical to the
discussion above where we perturbed $V^{EM}$ through $V^{P}$, $V^{PT}$, and
$V^{T}$. This is, however, only the case for purely static fields, as we can
see in the following example:

We consider the perturbation of $V^{P}$ through
$-\boldsymbol{d}\cdot\boldsymbol{\mathcal{E}}$. Such a system can be solved
completely analogously to our derivation in sec. 2.3, cf. Eq. (8), with the
replacement $d_{i}\rightarrow V^{P}$:

$\displaystyle\langle V^{P}\rangle\simeq 2Re\sum_{k}$
$\displaystyle\frac{\left\langle n\right|V^{P}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\omega_{kn}\mathcal{E}_{i}^{\ell}$
(28) $\displaystyle-2Im$ $\displaystyle\frac{\left\langle
n\right|V^{P}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\dot{\mathcal{E}}_{i}^{\ell}$

From our discussion in sec. 3.2.2, it follows that $\left\langle
n\right|V^{P}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle$ must be fully imaginary.

While the real part (that is similar to the perturbation of $\boldsymbol{d}$
through $V^{P}$:
$\langle\boldsymbol{d}\rangle\cdot\boldsymbol{\mathcal{E}}^{\ell}=2Re\sum_{k}\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{P}\left|n\right\rangle\omega_{kn}^{-1}\mathcal{E}_{i}^{\ell}$)
vanishes, we still remain with a non-zero imaginary part:

$\displaystyle\langle V^{P}\rangle\simeq-2Im\sum_{k}$
$\displaystyle\frac{\left\langle n\right|V^{P}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\dot{\mathcal{E}}_{i}^{\ell}\equiv\prescript{d}{P}{\beta^{\prime}}^{\ell}\,\boldsymbol{\mathcal{J}}\cdot\dot{\boldsymbol{\mathcal{E}}}^{\ell}$

From this, we can also conclude that even if the Lee-Yang argument (see
sec.3.2.2) forbids an atom to possess a static electric dipole moment without
$T$-violation, this does not mean that there can not be a purely $P$-odd
linear response to an applied (time-dependent) electric field.

If we perform an analogous treatment to calculate how $\langle V^{P}\rangle$,
$\langle V^{PT}\rangle$, and $\langle V^{T}\rangle$ are perturbed in 1st order
through $V^{EM}$, we get only a small number of terms that are not forbidden
by parity and time reversal arguments:

$\displaystyle\langle V^{P}\rangle$
$\displaystyle\simeq\prescript{d}{P}{\beta^{\prime}}^{\ell}\,\boldsymbol{\mathcal{J}}\cdot\dot{\boldsymbol{\mathcal{E}}}^{\ell}$
(29) $\displaystyle\langle V^{PT}\rangle$
$\displaystyle\simeq\prescript{d}{PT}{\beta}^{\ell}\,\boldsymbol{\mathcal{J}}\cdot{\boldsymbol{\mathcal{E}}}^{\ell}$
$\displaystyle\langle V^{T}\rangle$
$\displaystyle\simeq\prescript{\mu}{T}{\beta^{\prime}}^{\ell}\,\boldsymbol{\mathcal{J}}\cdot\dot{\boldsymbol{\mathcal{H}}}^{\ell}+\prescript{Q}{T}{\beta^{\prime}}^{\ell}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\partial_{i}\dot{\mathcal{E}}^{\ell}_{j}$

Regarding the (time-dependent) mixing of states or energy shifts, these
constitute the lowest order (in $V^{EM}$) corrections that violate $P$, $PT$,
and $T$, respectively.

Deriving the 2nd order perturbations through $V^{EM}$ is more complicated.
This is because we have two time-dependent perturbers, which, through
interferences, create a multitude of terms (see, for example, Refs. [20, 33,
23]). For this reason, we limit our discussion of the 2nd order corrections of
$\langle V^{P}\rangle$, $\langle V^{PT}\rangle$, and $\langle V^{T}\rangle$ to
the time-independent case. In appendix B.2, we demonstrate that the resulting
terms coincide with those we derived in sec. 3.3.

##  4 Resulting observables

We now want to connect the terms we derived above to possible experimental
observables. It will become apparent that the derivation presented in this
paper recovers many $P$\- and/or $T$-odd observables that have been considered
in other works. Additionally, we obtain some $P$-, $T$-, and $PT$-odd
observables that have, to our knowledge, not been discussed so far.

###  4.1 Static homogeneous fields

We start by reviewing the situation for static homogeneous fields
($\dot{\mathcal{E}}_{i},\,\dot{\mathcal{H}}_{i}=0,\,\partial_{i}{\mathcal{E}}_{j}=0$).
Under these considerations, most terms in Eqs. (25), (26), and (27) vanish. We
are left with:

$\displaystyle\langle\boldsymbol{d}\rangle$
$\displaystyle\simeq\prescript{d}{PT}{\alpha}\,\boldsymbol{\mathcal{J}}+\prescript{dd}{}{\alpha}_{s}\,\boldsymbol{\mathcal{E}}+\prescript{dd}{}{\alpha}_{t}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{E}}$
(30)
$\displaystyle+\prescript{d\mu}{P}{\alpha}_{v}\,({\boldsymbol{\mathcal{H}}}\times\boldsymbol{\mathcal{J}})+\prescript{d\mu}{PT}{\alpha}_{s}\,\boldsymbol{\mathcal{H}}+\prescript{d\mu}{PT}{\alpha}_{t}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{H}}$
$\displaystyle\langle\boldsymbol{\mu}\rangle$
$\displaystyle\simeq\prescript{\mu}{}{\alpha}\,\boldsymbol{\mathcal{J}}+\prescript{\mu\mu}{}{\alpha}_{s}\,\boldsymbol{\mathcal{H}}+\prescript{\mu\mu}{}{\alpha}_{t}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{H}}$
$\displaystyle+\prescript{\mu
d}{P}{\alpha}_{v}\,({\boldsymbol{\mathcal{E}}}\times\boldsymbol{\mathcal{J}})+\prescript{\mu
d}{PT}{\alpha}_{s}\,\boldsymbol{\mathcal{E}}+\prescript{\mu
d}{PT}{\alpha}_{t}\,\boldsymbol{\mathcal{Q}}^{\boldsymbol{J}^{2}}\cdot\boldsymbol{\mathcal{E}}$
$\displaystyle\langle Q_{ij}\rangle$
$\displaystyle\simeq\prescript{Q}{}{\alpha}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}+\prescript{QQ}{}{\alpha}_{ij\varkappa
l}\,\partial_{\varkappa}\mathcal{E}_{l}$
$\displaystyle+\left(\prescript{Qd}{PT}{\alpha}_{t_{s}}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{Qd}{PT}{\alpha}_{v_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right){\mathcal{E}}_{\varkappa}+\prescript{Qd}{P}{\alpha}_{t_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}{\mathcal{E}}_{\varkappa}$
$\displaystyle+\left(\prescript{Q\mu}{}{\alpha}_{t_{s}}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{Q\mu}{}{\alpha}_{v_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right){\mathcal{H}}_{\varkappa}+\prescript{Q\mu}{T}{\alpha}_{t_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}{\mathcal{H}}_{\varkappa}$

with
$\prescript{dd}{}{\alpha}_{s}\equiv\prescript{dd}{}{\alpha}_{s}^{0}\big{)}$.
In Appendix D, we derive that the induced energy shift of the system can be
expressed through the same polarizabilities:

$\displaystyle\Delta E\simeq$
$\displaystyle-\prescript{\mu}{}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{\mathcal{H}}}-\tfrac{1}{2}\prescript{dd}{}{\alpha}_{s}\,{\mathcal{E}}^{2}-\tfrac{1}{2}\prescript{\mu\mu}{}{\alpha}_{s}\,{\mathcal{H}}^{2}-\tfrac{1}{2}\prescript{dd}{}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}{\mathcal{E}}_{i}{\mathcal{E}}_{j}-\tfrac{1}{2}\prescript{\mu\mu}{}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}{\mathcal{H}}_{i}{\mathcal{H}}_{j}$
$\displaystyle-\prescript{d}{PT}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{E}}-\prescript{d\mu}{PT}{\alpha}_{s}\,\boldsymbol{\mathcal{H}}\cdot\boldsymbol{\mathcal{E}}-\prescript{d\mu}{P}{\alpha}_{v}\,(\boldsymbol{\mathcal{E}}\times\boldsymbol{\mathcal{H}})\cdot\boldsymbol{\mathcal{J}}-\prescript{d\mu}{PT}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\mathcal{E}_{i}{\mathcal{H}}_{j}$
(31)

where the first line shows the terms that are even and the second line the
terms that are odd under either $P$ or $T$.
We start by discussing the former. The first two terms are well-known and
usually have the largest influence on the system. They describe the linear
Zeeman- and the quadratic Stark effect. The third term is the quadratic Zeeman
effect [34]. It is best visible in diamagnetic systems where the linear Zeeman
effect is absent. The fourth term is the Stark tensor-polarizability that can
be relevant for systems with $J>1$ [35]. The final term is an equivalent
Zeeman tensor-interaction. It has been first considered in Ref. [36] as a way
to induce an electric quadrupole moment through an applied magnetic field.

Now, we are going to discuss the symmetry-violating interactions, which are
the main interest of this study.
The term
$\prescript{d}{PT}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{E}}$
is well known in the context of searches for the electric dipole moment of the
electron (eEDM). It allows to restrict $P$\- and $T$-violating physics by
measuring the absence of linear stark shifts in paramagnetic atoms and
molecules. This interaction does not uniquely arise from the eEDM but can also
be caused by a variety of other $PT$-violating interactions, like Schiff
moments, the scalar-pseudoscalar electron-nucleon coupling, or the electron
interaction with the nuclear magnetic quadrupole moment. For a review on the
topic, see Ref. [3]. For purely diamagnetic systems,
$\boldsymbol{\mathcal{J}}$ is zero. Such systems are, therefore, insensitive
to the effects of $\prescript{d}{PT}{\alpha}$.

The remaining three terms in Eq. (31) were first considered in Ref. [8]. The
first of them,
$\prescript{d\mu}{PT}{\alpha}_{s}\,\boldsymbol{\mathcal{H}}\cdot\boldsymbol{\mathcal{E}}$,
describes a direct coupling of the electric and magnetic fields similar to the
magnetoelectric effect that can arise in some solids [16]. In Ref. [12], its
effects were investigated in more detail. Through its scalar nature,
$\prescript{d\mu}{PT}{\alpha}_{s}$ contributes to para- and diamagnetic
systems alike.
The properties of the term
$(\boldsymbol{\mathcal{E}}\times\boldsymbol{\mathcal{H}})\cdot\boldsymbol{\mathcal{J}}$
were studied in Ref. [37], and [38]. Because of its antisymmetric structure,
it is only nonvanishing when the angular momentum is misaligned with the
direction of $\boldsymbol{\mathcal{H}}$ (and also the plane spanned by
$\boldsymbol{\mathcal{E}}$ and $\boldsymbol{\mathcal{H}}$).
The final term
$\prescript{d\mu}{PT}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\mathcal{E}_{i}{\mathcal{H}}_{j}$
again describes a $PT$-odd electric magnetic field coupling. Apart from the
initial proposal of this expression in Ref. [8], we do not know of any other
work discussing this coupling for static fields. This term requires a system
with $J\geq 1$.

To measure these terms, one could directly measure the energy shift (31)
depending on the strength and direction of the applied fields. Alternatively,
one could use a large sample and measure the induced polarization
$\langle\boldsymbol{d}\rangle$, or magnetization
$\langle\boldsymbol{\mu}\rangle$, from Eq. (30). An induced polarization in a
solid can be measured with a precise voltage measurement as it was done in
[39] in the context of an eEDM search. Magnetizations of samples can, for
example, be measured with a SQUID as it was done in another eEDM search in
[40].

####  4.1.1 Alignment of $\boldsymbol{\mathcal{J}}$ in externally applied
fields

The expectation value of the angular momentum
$\boldsymbol{\mathcal{J}}=\left\langle
n\right|\boldsymbol{J}\left|n\right\rangle$ is a vector whose orientation
depends only on the (unperturbed) state of the system. But even though
$\boldsymbol{\mathcal{J}}$ is not directly dependent on the applied
$\boldsymbol{\mathcal{E}}$ and $\boldsymbol{\mathcal{H}}$ fields, its
direction can still be influenced by them. This is because the energy shift in
(31) defines which $m$-substate in the $aJ$-manifold has the lowest energy. If
the system possesses the ability to decay into its ground state, $\left\langle
n\right|\boldsymbol{J}\left|n\right\rangle$ changes. For example, if we apply
a strong magnetic field to a sample, the Zeeman coupling $\Delta
E_{Z}=-\prescript{\mu}{}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{H}}$
dominates all the other terms. It is minimized if $\boldsymbol{\mathcal{J}}$
becomes maximally aligned to the direction of $\boldsymbol{\mathcal{H}}$. In
this case, we can perform the replacement $\boldsymbol{\mathcal{J}}\rightarrow
J\frac{\boldsymbol{\mathcal{H}}}{|\boldsymbol{\mathcal{H}}|}$. Here, $J$ is
the usual angular momentum quantum number.

Alternatively, if magnetic fields are absent, then the application of an
electric field will orient $\boldsymbol{\mathcal{J}}$ directly through the EDM
coupling $\Delta
E_{EDM}=-\prescript{d}{PT}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{E}}$.
We know from sec. 3.2.3 that an oriented angular momentum directly leads to an
oriented magnetic moment $\langle\boldsymbol{\mu}\rangle$. It again could be
picked up through magnetometry. This method of measuring the eEDM was first
proposed in [41].

###  4.2 Static inhomogeneous fields

If we allow the electric field to vary in space
($\partial_{i}{\mathcal{E}}_{j}\neq 0$), then the electric quadrupole terms
will also contribute to the energy of the system. One possibility to apply
strong field gradients to atoms or molecules is to embed them into a
crystalline matrix [42, 43, 44].
In inhomogeneous fields, the dipole moments are extended by:

$\displaystyle\langle\boldsymbol{d}\rangle$
$\displaystyle=\dotsb+\prescript{dQ}{P}{\alpha}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}+\left(\prescript{dQ}{PT}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{dQ}{PT}{\alpha}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}$
$\displaystyle\langle\boldsymbol{\mu}\rangle$
$\displaystyle=\dotsb+\prescript{\mu
Q}{T}{\alpha}_{t_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}+\left(\prescript{\mu
Q}{}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{\mu
Q}{}{\alpha}_{v_{m}}^{\ell}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\hat{\boldsymbol{e}}_{i}\partial_{j}{\mathcal{E}}^{\ell}_{\varkappa}$

The dots represent the terms from in Eq. (30). The inhomogeneous terms lead to
an additional energy shift (see Appendix D):

$\displaystyle\Delta E$
$\displaystyle=\dotsb-\prescript{Q}{}{\alpha}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\partial_{i}\mathcal{E}_{j}-\tfrac{1}{2}\prescript{QQ}{}{\alpha}_{ij\varkappa
l}\,\partial_{i}\mathcal{E}_{j}\partial_{\varkappa}\mathcal{E}_{l}$ (32)
$\displaystyle-\left(\prescript{\mu
Q}{}{\alpha}_{t_{s}}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{\mu
Q}{}{\alpha}_{v_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\mathcal{H}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}$
$\displaystyle-\prescript{dQ}{P}{\alpha}_{t_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\mathcal{E}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}-\prescript{\mu
Q}{T}{\alpha}_{t_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\mathcal{H}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}$
$\displaystyle-\left(\prescript{dQ}{PT}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{dQ}{PT}{\alpha}_{v_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\mathcal{E}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}$

The expressions in the first two lines are classically allowed. The first two
of these terms describe the interactions of static and induced electric
quadrupole moments with the inhomogeneous electric field analogous to the
linear and quadratic Stark and Zeeman effects. The next two terms describe the
induction of an electric quadrupole moment through an applied magnetic field.
That such an interaction, linear in $\boldsymbol{\mathcal{H}}$, exists, was
only considered quite recently [45].

The third and fourth line in Eq. (32) describes $P$, $T$, and $PT$-odd
effects. We proceed by analyzing these terms in the scenario where the angular
momentum structures $\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}$,
$\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}$, and
$\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}$ become aligned through the
externally applied fields.

####  4.2.1 Alignment of $\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}$,
$\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}$, and
$\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}$ in externally applied fields

The terms in the third line of Eq. (32) contain the following tensor, which is
itself dependent on the quadrupole angular momentum structure
$\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}$:

$\displaystyle\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$
$\displaystyle=\tfrac{1}{2}(\delta_{jl}\varepsilon_{im\varkappa}+\delta_{\varkappa
l}\varepsilon_{imj})\mathcal{Q}_{lm}^{\boldsymbol{J}^{2}}$

Similar to the alignment of $\boldsymbol{\mathcal{J}}$ into the direction of
an applied magnetic field discussed in sec. 4.1.1,
$\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}$ as well can be aligned through (large)
$P$-even $T$-even interactions and sufficient field strengths. In this case,
$\boldsymbol{\mathcal{E}}$, $\boldsymbol{\mathcal{H}}$, and
$\partial_{i}\mathcal{E}_{j}$ compete with each other to define the specific
state $\left|aJm\right\rangle$ that minimizes the energy and indirectly
defines the orientation of $\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}$. We can
express this by performing the replacement

$\displaystyle\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\rightarrow
J^{2}\frac{b_{1}\partial_{i}\mathcal{E}_{j}+b_{2}{\mathcal{E}}_{i}{\mathcal{E}}_{j}+b_{3}{\mathcal{H}}_{i}{\mathcal{H}}_{j}}{\sqrt{\sum_{\varkappa,l}\left(b_{1}\partial_{\varkappa}\mathcal{E}_{l}+b_{2}{\mathcal{E}}_{\varkappa}{\mathcal{E}}_{l}+b_{3}{\mathcal{H}}_{\varkappa}{\mathcal{H}}_{l}\right)^{2}}}-\tau\delta_{ij}$
(33)

Here $b_{1}$, $b_{2}$, and $b_{3}$ are real constants that depend on the sizes
of the different classically allowed polarizabilities $\alpha$. The expression
is then normalized through the Frobenius norm. $\tau$ represents the trace of
the fraction. We need to subtract it, to ensure that the new object still
remains traceless.

The first of the two
$\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$-dependent terms in Eq. (32)
describes an interaction that only violates $P$, but not $T$. That such a
quantity could exist in an inhomogeneous electric field was first proposed in
Ref. [38]. Now, we will derive what the resulting observable looks like. The
quadrupole component of an electrostatic field is given by the potential:
$\phi^{Q}=q[(\boldsymbol{r}\cdot\hat{\boldsymbol{n}})^{2}-\tfrac{1}{3}\boldsymbol{r}^{2}]$.
It is oriented along some axis $\hat{\boldsymbol{n}}$ and has an amplitude of
$q$. without loss of generality, we can take $\hat{\boldsymbol{n}}$ in
$z$-direction: $\hat{\boldsymbol{n}}=\hat{e}_{z}$. It follows for the electric
field Jacobian:

$\displaystyle\partial_{i}\mathcal{E}_{j}|_{\boldsymbol{r}=0}=-\partial_{i}\partial_{j}\phi|_{\boldsymbol{r}=0}=-\partial_{i}\partial_{j}\phi^{Q}=-2q[\delta_{iz}\delta_{jz}-\tfrac{1}{3}\delta_{ij}]$

From Eqs. (24), (32), and (33), we can now derive the energy shift that is
induced by the $\prescript{dQ}{P}{\alpha}_{t_{m}}$ term:

$\displaystyle\Delta E$
$\displaystyle=\,\prescript{dQ}{P}{\alpha}_{t_{m}}\,q\,(\delta_{jl}\varepsilon_{im\varkappa}+\delta_{\varkappa
l}\varepsilon_{imj})(\delta_{jz}\delta_{\varkappa
z}-\tfrac{1}{3}\delta_{j\varkappa})\mathcal{Q}_{lm}\mathcal{E}_{i}$
$\displaystyle=2\,\prescript{dQ}{P}{\alpha}_{t_{m}}\,q\,\varepsilon_{imz}\mathcal{Q}_{zm}\mathcal{E}_{i}$
$\displaystyle=2\,\prescript{dQ}{P}{\alpha}_{t_{m}}\,q\mathcal{J}^{2}\,\varepsilon_{imz}\Bigg{[}J^{2}\frac{b_{1}\partial_{i}\mathcal{E}_{j}+b_{2}{\mathcal{E}}_{i}{\mathcal{E}}_{j}+b_{3}{\mathcal{H}}_{i}{\mathcal{H}}_{j}}{\sqrt{\sum_{\varkappa,l}\left(b_{1}\partial_{\varkappa}\partial_{l}\phi^{Q}+b_{2}{\mathcal{E}}_{\varkappa}{\mathcal{E}}_{l}+b_{3}{\mathcal{H}}_{\varkappa}{\mathcal{H}}_{l}\right)^{2}}}-\tau\delta_{ij}\Bigg{]}\mathcal{E}_{i}$

From the antisymmetry of $\varepsilon_{ij\varkappa}$, it follows that
$\varepsilon_{imz}\mathcal{E}_{i}\partial_{z}\partial_{m}\phi^{Q}$,
$\varepsilon_{imz}{\mathcal{E}}_{z}{\mathcal{E}}_{m}\mathcal{E}_{i}$, and
$\varepsilon_{ij\varkappa}\delta_{ij}\tau$ necessarily vanish. We can write
the remaining term as:

$\displaystyle\Delta
E=2J^{2}\frac{\prescript{dQ}{P}{\alpha}_{t_{m}}\,b_{3}\,q\,(\boldsymbol{\mathcal{H}}\cdot\hat{\boldsymbol{n}})(\boldsymbol{\mathcal{E}}\times\boldsymbol{\mathcal{H}})\cdot\hat{\boldsymbol{n}}}{\sqrt{\sum_{\varkappa,l}\left(b_{1}\partial_{\varkappa}\partial_{l}\phi^{Q}+b_{2}{\mathcal{E}}_{\varkappa}{\mathcal{E}}_{l}+b_{3}{\mathcal{H}}_{\varkappa}{\mathcal{H}}_{l}\right)^{2}}}$
(34)

Expressions of a similar form have previously been derived in [46] and [47].
There, it was shown that the energy shift induced by the $P$-odd nuclear
anapole moment is at a range that could realistically be measured with the
current technology.
Analogously to the derivation of $\Delta E$, the same effect also leads
directly to an induced electric dipole moment:

$\displaystyle\langle\boldsymbol{d}\rangle=-2J^{2}\frac{\,\prescript{dQ}{P}{\alpha}_{t_{m}}\,b_{3}\,q\,(\boldsymbol{\mathcal{H}}\cdot\hat{\boldsymbol{n}})(\boldsymbol{\mathcal{H}}\times\hat{\boldsymbol{n}})}{\sqrt{\sum_{\varkappa,l}\left(b_{1}\partial_{\varkappa}\partial_{l}\phi^{Q}+b_{2}{\mathcal{E}}_{\varkappa}{\mathcal{E}}_{l}+b_{3}{\mathcal{H}}_{\varkappa}{\mathcal{H}}_{l}\right)^{2}}}+\dotsb$

In contrast to the $PT$-odd electric dipole that results
from$\prescript{d\mu}{PT}{\alpha}_{s}$, this $P$-odd $T$-even one is oriented
orthogonal to the direction of the applied magnetic field and stays invariant
under $\boldsymbol{\mathcal{H}}\rightarrow-\boldsymbol{\mathcal{H}}$.

The second $\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$-dependent term in
Eq. (32) behaves quite similarly. For it, we can derive:

$\displaystyle\Delta E$ $\displaystyle=2J^{2}\frac{\prescript{\mu
Q}{T}{\alpha}_{t_{m}}\,b_{2}\,q\,(\boldsymbol{\mathcal{E}}\cdot\hat{\boldsymbol{n}})(\boldsymbol{\mathcal{H}}\times\boldsymbol{\mathcal{E}})\cdot\hat{\boldsymbol{n}}}{\sqrt{\sum_{\varkappa,l}\left(b_{1}\partial_{\varkappa}\partial_{l}\phi^{Q}+b_{2}{\mathcal{E}}_{\varkappa}{\mathcal{E}}_{l}+b_{3}{\mathcal{H}}_{\varkappa}{\mathcal{H}}_{l}\right)^{2}}}+\dotsb$
(35) $\displaystyle\text{and }\langle\boldsymbol{\mu}\rangle$
$\displaystyle=-2J^{2}\frac{\prescript{\mu
Q}{T}{\alpha}_{t_{m}}\,b_{2}\,q\,(\boldsymbol{\mathcal{E}}\cdot\hat{\boldsymbol{n}})(\boldsymbol{\mathcal{E}}\times\hat{\boldsymbol{n}})}{\sqrt{\sum_{\varkappa,l}\left(b_{1}\partial_{\varkappa}\partial_{l}\phi^{Q}+b_{2}{\mathcal{E}}_{\varkappa}{\mathcal{E}}_{l}+b_{3}{\mathcal{H}}_{\varkappa}{\mathcal{H}}_{l}\right)^{2}}}+\dotsb$

Unlike all effects discussed so far, this one is $T$-odd, but $P$-even (ToPe).
To our knowledge, all atomic ToPe observables that were previously considered
in the literature depend on dynamic processes. This would make this the first
static mechanism for detecting such a symmetry violation. A review on ToPe
physics can be found in [5, 27]. It could be possible to measure
$\langle\boldsymbol{\mu}\rangle$ in a similar scheme to the one used in [40].

The energy shift $\Delta E$ in Eq. (32) further contains two $PT$-odd terms.
The first of them, couples the electric field $\mathcal{E}_{i}$ and the
Hessian $\partial_{j}\partial_{\varkappa}\phi^{Q}$ to the octupole angular
momentum structure $\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}$. Similar
to the quadrupole in Eq. (33), this octupole as well can be aligned through
the application of external fields.

The other $PT$-odd term in Eq. (32) depends on the symmetric vector quantity:

$\displaystyle\mathcal{M}^{\boldsymbol{J}}_{ij\varkappa}$
$\displaystyle=\tfrac{1}{4}(2\delta_{j\varkappa}\mathcal{J}_{i}-\delta_{ij}\mathcal{J}_{\varkappa}-\delta_{i\varkappa}\mathcal{J}_{j})$

We can derive a simplified form for
$\mathcal{M}^{\boldsymbol{J}}_{ij\varkappa}$ similar to what we did for
$\mathcal{M}^{\boldsymbol{J}^{2}}_{ij\varkappa}$:

$\displaystyle\Delta E$
$\displaystyle=q\,\prescript{dQ}{PT}{\alpha}_{v_{m}}\,\tfrac{1}{2}(2\delta_{j\varkappa}\mathcal{J}_{i}-\delta_{ij}\mathcal{J}_{\varkappa}-\delta_{i\varkappa}\mathcal{J}_{j})(\delta_{iz}\delta_{jz}-\tfrac{1}{3}\delta_{ij})\mathcal{E}_{i}$
$\displaystyle=q\,\prescript{dQ}{PT}{\alpha}_{v_{m}}\,\left[(\boldsymbol{\mathcal{E}}\cdot\boldsymbol{\mathcal{J}})-(\boldsymbol{\mathcal{E}}\cdot\hat{\boldsymbol{n}})(\boldsymbol{\mathcal{J}}\cdot\hat{\boldsymbol{n}})\right]$

As discussed in sec. 4.1.1, $\boldsymbol{\mathcal{J}}$ can align to the
direction of an applied magnetic field, leading to:

$\displaystyle\Delta
E=q\,\prescript{dQ}{PT}{\alpha}_{v_{m}}\,J\left[(\boldsymbol{\mathcal{E}}\cdot\boldsymbol{\mathcal{H}})-(\boldsymbol{\mathcal{E}}\cdot\hat{\boldsymbol{n}})(\boldsymbol{\mathcal{H}}\cdot\hat{\boldsymbol{n}})\right]$

In contrast to the $P$-odd $T$-even, and the $P$-even $T$-odd terms in Eqs.
(34), and (35), this expression is both odd under the inversion of
$\boldsymbol{\mathcal{E}}$ and $\boldsymbol{\mathcal{H}}$. If
$\boldsymbol{\mathcal{E}}$ and $\boldsymbol{\mathcal{H}}$ are parallel, but
orthorgonal to $\hat{\boldsymbol{n}}$, then the contribution from this
interaction becomes very similar to $\prescript{d}{PT}{\alpha}$, and
$\prescript{d\mu}{PT}{\alpha}_{s}$.
Again, we also get an associated induced electric dipole moment:

$\displaystyle\langle\boldsymbol{d}\rangle=-q\,\prescript{dQ}{PT}{\alpha}_{v_{m}}\,J\left[\boldsymbol{\mathcal{H}}-\hat{\boldsymbol{n}}(\boldsymbol{\mathcal{H}}\cdot\hat{\boldsymbol{n}})\right]+\dotsb$

To our knowledge, the $PT$-odd terms described in this section have not been
discussed before.

We can conclude that a static quadrupolar electric field, as it can be present
in crystals, allows one to use a combination of electric and magnetic fields
to search for all three types of symmetry violations. Alternatively, to align
the different angular momentum structures through applied fields, one could,
of course, also prepare the system in a different state by optical pumping.

###  4.3 Time-varying fields

If we allow for time-dependent $\boldsymbol{\mathcal{E}}$\- and
$\boldsymbol{\mathcal{H}}$-fields, all the terms in Eqs. (25), (26), and Eq.
(27) contribute. To measure these time-dependent observables, one could again
directly pick up the induced magnetization $\langle\boldsymbol{\mu}\rangle$ or
polarization $\langle\boldsymbol{d}\rangle$.
For example, $\langle\boldsymbol{\mu}\rangle=\prescript{\mu
d}{P}{\alpha}_{v}^{\ell}\,({\boldsymbol{\mathcal{E}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\prescript{\mu
d}{PT}{\alpha^{\prime}}_{v}^{\ell}\,(\dot{\boldsymbol{\mathcal{E}}}^{\ell}\times\boldsymbol{\mathcal{J}})+\dotsb$
implies that applying an oscillating electric field to a polarized sample will
induce an orthogonal magnetization. Its $P$-odd part oscillates in phase with
the applied field, but its $PT$-odd part is out of phase. Another way to
measure the time-dependent polarizabilities is to prepare the system in a
well-defined state and observe the time evolution of its population when it is
exposed to time-varying fields [48].

Finally, the effects of the $\langle\boldsymbol{d}\rangle$ and
$\langle\boldsymbol{\mu}\rangle$ induced by electromagnetic waves propagating
through a medium can be studied.

Figure 1: Electric and magnetic fields of left ($\sigma^{+}$) and right
($\sigma^{-}$) circular polarized light. $\boldsymbol{\mathcal{S}}$ is the
Poynting vector, defining the propagation direction of the light wave.

Let us consider the terms
$\langle\boldsymbol{d}\rangle=\prescript{dd}{}{\alpha}_{s}^{\ell}\,\boldsymbol{\mathcal{E}}^{\ell}+\prescript{d\mu}{P}{\alpha^{\prime}}_{s}^{\ell}\,\dot{\boldsymbol{\mathcal{H}}}^{\ell}$.
As shown in Fig. 1, for a left circular polarized wave,
$\dot{\boldsymbol{\mathcal{H}}}$ is parallel to $\boldsymbol{\mathcal{E}}$,
while for a right circular polarized wave, they are antiparallel. From this,
it follows that we can rewrite $\langle\boldsymbol{d}\rangle$ for the two
polarizations $\sigma^{\pm}$ as
$\langle\boldsymbol{d}\rangle_{\pm}=\left(\prescript{dd}{}{\alpha}_{s}^{\ell}\,\mp\omega^{\ell}\,\prescript{d\mu}{P}{\alpha^{\prime}}_{s}^{\ell}\,\right)\boldsymbol{\mathcal{E}}^{\ell}$.
For dilute gases, the refractive index $n$ relates to the polarizability
$\alpha$ through $n^{2}=\alpha+1$. This implies that the refractive index of
left and right polarized light differ by an amount that is directly
proportional to the size of the parity-violating potential $V^{P}$. This
difference leads to a small optical rotation of light propagating through an
atomic or (nonchiral) molecular gas. The effect of $P$-odd optical rotation
was successfully utilized to measure atomic parity violations in Bi, Pb, and
Tl ([49] and references within). This is just one example of $P$, $T$
violation-induced birefringence and optical activity. In Refs. [9, 11] some
others are discussed.

However, there is a caveat when dealing with time-dependent fields, which we
discuss in the following section.

##  5 Effects from the natural linewidth

So far, we have neglected the effects from the natural linewidth of the
excited states in our perturbative treatment. These are not naturally
contained in a semiclassical model for the interaction with the
electromagnetic field. However, it is possible to model them by adding a non-
Hermitian term to the Hamiltonian $H^{0}$ [50].

Under these considerations, the time-dependent perturbation expression in Eq.
(7) becomes [51]:

$\displaystyle\left\langle
N\right|d_{i}\left|N\right\rangle(t)\simeq\left\langle n\right|$
$\displaystyle d_{i}\left|n\right\rangle$ (36) $\displaystyle-\sum_{l\neq
0}\sum_{k}$ $\displaystyle\left[\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|F^{\ell}\left|n\right\rangle}{\omega_{kn}-\tfrac{i}{2}\Gamma_{k}-\omega^{\ell}}+\frac{\left\langle
n\right|F^{\ell}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}+\tfrac{i}{2}\Gamma_{k}+\omega^{\ell}}\right]e^{-i\omega^{\ell}t}$
$\displaystyle+$ $\displaystyle\left[\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|(F^{\ell})^{\dagger}\left|n\right\rangle}{\omega_{kn}-\tfrac{i}{2}\Gamma_{k}+\omega^{\ell}}+\frac{\left\langle
n\right|(F^{\ell})^{\dagger}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}+\tfrac{i}{2}\Gamma_{k}-\omega^{\ell}}\right]e^{i\omega^{\ell}t}$

where $\Gamma_{k}$ is the natural linewidth of the state
$\left|k\right\rangle$.
Non-Hermitian Hamiltonians describe dissipative systems. The dissipation leads
to an increase in entropy, which in turn defines an arrow of time. This
effectively breaks $T$ symmetry without the need for a fundamental symmetry
violation. This makes the unambiguous interpretation of $T$-odd observables in
such a system difficult.

###  5.1 Effects on the induced electric dipole moment

We will now illustrate how this effect manifests in the induced moments that
we derived in the paper.

After evaluating Eq. (36) analogously to what we did in sec. 2.3.2, we get:

$\displaystyle\left\langle n\right|d_{i}\left|n\right\rangle-2Re\sum_{\ell\neq
0}\sum_{k}$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\big{[}\omega_{kn}V^{\ell}+i\dot{V}^{\ell}\big{]}\left|n\right\rangle}{(\omega^{\ell})^{2}\Gamma_{k}^{2}+\left(\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}\right)^{2}}\left[i\omega^{\ell}\Gamma_{k}+\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}\right]$

This leads to the following expression for the induced dipole moment:

$\displaystyle\left\langle N\right|d_{i}\left|N\right\rangle(t)\simeq$
$\displaystyle\left\langle n\right|d_{i}\left|n\right\rangle$
$\displaystyle+\sum_{l}\sum_{k}$
$\displaystyle\Big{(}Re\big{[}\prescript{dd}{}{{\alpha}}_{ij}^{\ell
k}\big{]}-G^{\ell k}\,Im\big{[}\prescript{dd}{}{{\alpha}}_{ij}^{\ell
k}\big{]}\Big{)}\mathcal{E}_{j}^{\ell}$ $\displaystyle+\sum_{l}\sum_{k}$
$\displaystyle\Big{(}Re\big{[}\prescript{d\mu}{}{{\alpha}}_{ij}^{\ell
k}\big{]}-G^{\ell k}\,Im\big{[}\prescript{d\mu}{}{{\alpha}}_{ij}^{\ell
k}\big{]}\Big{)}\mathcal{H}_{j}^{\ell}$ $\displaystyle+\sum_{l}\sum_{k}$
$\displaystyle\Big{(}Re\big{[}\prescript{dQ}{}{{\alpha}}_{ij\varkappa}^{\ell
k}\big{]}-G^{\ell k}\,Im\big{[}\prescript{dQ}{}{{\alpha}}_{ij\varkappa}^{\ell
k}\big{]}\Big{)}\partial_{j}\mathcal{E}_{\varkappa}^{\ell}$
$\displaystyle+\sum_{l}\sum_{k}$
$\displaystyle\Big{(}Im\big{[}\prescript{dd}{}{{\alpha^{\prime}}}_{ij}^{\ell
k}\big{]}+G^{\ell k}\,Re\big{[}\prescript{dd}{}{{\alpha^{\prime}}}_{ij}^{\ell
k}\big{]}\Big{)}\dot{\mathcal{E}}_{j}^{\ell}$ $\displaystyle+\sum_{l}\sum_{k}$
$\displaystyle\Big{(}Im\big{[}\prescript{d\mu}{}{{\alpha^{\prime}}}_{ij}^{\ell
k}\big{]}+G^{\ell
k}\,Re\big{[}\prescript{d\mu}{}{{\alpha^{\prime}}}_{ij}^{\ell
k}\big{]}\Big{)}\dot{\mathcal{H}}_{j}^{\ell}$ $\displaystyle+\sum_{l}\sum_{k}$
$\displaystyle\Big{(}Im\big{[}\prescript{dQ}{}{{\alpha^{\prime}}}_{ij\varkappa}^{\ell
k}\big{]}+G^{\ell
k}\,Re\big{[}\prescript{dQ}{}{{\alpha^{\prime}}}_{ij\varkappa}^{\ell
k}\big{]}\Big{)}\partial_{j}\dot{\mathcal{E}}_{\varkappa}^{\ell}$

Here, we defined more general expressions for the polarizabilities that are
valid for non-zero natural linewidths:

$\displaystyle\prescript{d\mu}{}{{\alpha}}_{ij}^{\ell k}$
$\displaystyle=2\frac{\left(\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}\right)\omega_{kn}}{(\omega^{\ell})^{2}\Gamma_{k}^{2}+\left(\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}\right)^{2}}\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle$
$\displaystyle\prescript{d\mu}{}{{\alpha^{\prime}}}_{ij}^{\ell k}$
$\displaystyle=-2\frac{\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}}{(\omega^{\ell})^{2}\Gamma_{k}^{2}+\left(\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}\right)^{2}}\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle$ $\displaystyle G^{\ell k}$
$\displaystyle=\frac{\omega^{\ell}\Gamma_{k}}{\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}-(\omega^{\ell})^{2}}$

If we take the static limit $\ell=0$, we get:

$\displaystyle\sum_{k}\prescript{d\mu}{}{{\alpha}}_{ij}^{0k}$
$\displaystyle=2\sum_{k}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{\omega_{kn}^{2}-\tfrac{1}{4}\Gamma_{k}^{2}}\omega_{kn}\quad\text{and}\quad
G^{0k}=0$

Therefore, in the case of static fields, everything considered in previous
sections still holds. The only difference is that the size of all 1st order
terms increases (slightly) through the difference in the denominator.

###  5.2 Implication for time-dependent fields

Going back to the general case $\omega_{\ell}\neq 0$, both the imaginary and
real part of the polarizabilities couple to the fields and their derivatives.
We saw in previous sections that the observables for $T$-even and $T$-odd
terms are different in that if one of them contains a field, the other
necessarily contains its derivative. Now, with the inclusion of the natural
linewidth, this is not true anymore. Measuring a certain electromagnetic field
response of a system no longer allows us to uniquely distinguish if it was
caused by a $P$-odd $T$-even or by a $P$-odd $T$-odd effect because the same
field configuration couples both to ${}_{P}{{\alpha}}^{\ell}$, and
${}_{PT}{{\alpha}}^{\ell}$. In the known standard model interactions, the
strength of $P$-violating processes far exceeds that of $P$\- and
$T$-violating ones [26]. Therefore, it is natural to assume the hierarchy
$V^{P}\gg V^{PT}$. $P$-violating interactions can, therefore, pose a large
background for experimental detections of $PT$-violating ones. To avoid this,
we need the $G^{\ell k}$ factors to become small. This happens when
$\omega^{\ell}$ is far detuned from all the resonances contributing to the
polarizability. As we have seen before, this can be achieved by choosing a low
frequency. This has the potential disadvantage that also the field derivatives
$\dot{\boldsymbol{\mathcal{H}}}^{\ell}$, and
$\dot{\boldsymbol{\mathcal{E}}}^{\ell}$ become small in this limit. In the
other case of very high frequencies, the polarizabilities
$\prescript{}{}{{\alpha}}^{\ell k}\sim\left(\omega^{\ell}\right)^{-2}$
themselves decrease rapidly.

In Refs. [52, 53], it was shown that oscillating electric fields in neutral
atoms and molecules are not well shielded at the nucleus. It was proposed to
use this effect to measure nuclear EDMs. The here presented effect of $P$\-
violation ”mimicking” as $PT$-violation through the effects of the natural
linewidths could pose an important background for these kinds of measurements.
The situation is even worse for $P$-even $T$-odd observables. Through the same
effect, these mix with classical $P$-even, $T$-even potentials, whose size
will be many orders of magnitude above the expected size of $T$-odd physics.
It is, therefore, advantageous to use static polarizabilities such as
$\prescript{\mu Q}{T}{\alpha}_{t_{m}}$ to study ToPe effects.

##  6 Conclusion and outlook

In this work, we investigated how the violation of the symmetries of parity
$P$ and time reversal $T$ leads to atomic and molecular responses to
externally applied electric and magnetic fields that are classically
forbidden. This led us to general expressions for the induced electric dipole
(25), magnetic dipole (26) and electric quadrupole moment (27). We
specifically showed how atoms or molecules can obtain electric dipole moments,
not only through $PT$-violation (sec. 3.2.2), but also through $P$ without
$T$-violation in time-dependent (sec. 3.3.3, 3.5, 5), and even purely static
fields 4.2. Additionally, we derived Eqs. (31), (32) for the energy shift in
static fields and Eq. (29) for the effects in time-varying ones. We connected
most of these terms to effects that have previously been considered or used in
experimental searches for atomic $P$\- and $T$\- violations (sec. 4.1, 4.2,
4.3). Finally, we argued how the effects of natural linewidths could cause
problems in searches of $T$\- and $PT$\- odd effects (sec. 5).

To assess the feasibility of detecting the various terms presented in this
work, it is, of course, necessary to evaluate their size as well as the
signal-to-noise ratio that can be reached in a given experiment. However,
making general statements about them is difficult as they depend heavily on
the specific system under consideration. This is the level structure of the
atom or molecule, which governs the size of $\omega_{nk}$, as well as the
specific symmetry-violating effects that one wants to measure. Usually, one
considers potentials $V^{P}$, $V^{PT}$, and $V^{T}$ that involve electron-
nucleon contact interactions and, therefore, scale steeply with nucleon
numbers, and the overlap of the electrons with the nucleus [49, 27].
Therefore, searches involving very heavy atoms or molecules are usually the
best choice [54]. However, there also exist models introducing long-range
interactions $V^{P}$, $V^{PT}$, and $V^{T}$ through new light Bosons that may
be easier to restrict in lighter systems [55, 56, 57, 27].
To maximize the size of the polarizabilities, one also wants large
$\boldsymbol{d}$, $\boldsymbol{\mu}$, and $\boldsymbol{Q}$ transition elements
between close-lying states. Such configurations are usually easier to achieve
in molecules than in atoms.

We hope that the treatment we presented in this paper can also provide a
starting point for further investigations of other symmetry-violating effects
that are not captured in our model due to its initial assumptions. For
example, with some modifications, one can study $P$ or $T$-forbidden dipole
transitions. One can also extend the multipole expansion to include the
magnetic quadrupole and toroidal moments. One could also include a different
class of new physics interactions by introducing $P$, $T$ odd potentials that
are time-dependent or that couple directly to external fields. Finally, it
would also be interesting to understand what happens if the field strength
becomes too strong to justify the nondegenerate perturbative treatment that we
used.

##  7 Acknowledgments

This research was financed in whole or in part by Agence Nationale de la
Recherche (ANR) under the project ANR-21-CE30 -0028-01.
A CC-BY public copyright license has been applied by the authors to the
present document and will be applied to all subsequent versions up to the
Author Accepted Manuscript arising from this submission, in accordance with
the grant’s open access conditions.
We thank O. Dulieu for fruitful discussions.

##  Appendix A The time-reversal operator $T$

As an antiunitary operator, the time reversal operator $T$ has some
unintuitive properties with respect to the usual bra-ket notation (see, for
example, Ref. [58]). For linear Operators $L$, the bra-ket notation is
associative in the sense that:

$\displaystyle\left\langle\varphi\right|(L\left|\psi\right\rangle)=(\left\langle\varphi\right|L)\left|\psi\right\rangle\equiv\left\langle\varphi\right|L\left|\psi\right\rangle$

For an antiunitary operator, this is not true. There we have:

$\displaystyle\left\langle\varphi\right|(A\left|\psi\right\rangle)=\left[(\left\langle\varphi\right|A)\left|\psi\right\rangle\right]^{*}$

For ease of notation, we do not indicate the parentheses explicitly in the
rest of the paper. Instead, we define:

$\displaystyle\left\langle\varphi\right|A\left|\psi\right\rangle\equiv\left\langle\varphi\right|(A\left|\psi\right\rangle)$

For the action of the inverse of the antiunitary time reversal operator
$T^{\dagger}$ onto a $\left\langle\text{bra}\right|$ follows:

$\displaystyle\left\langle aJm\right|T^{\dagger}\left|k\right\rangle$
$\displaystyle\equiv\left\langle
aJm\right|(T^{\dagger}\left|k\right\rangle)=\big{[}(\left\langle
k\right|T)\left|aJm\right\rangle\big{]}^{*}=\left\langle
k\right|(T\left|aJm\right\rangle)$ $\displaystyle=(-1)^{J-m}\langle
k|aJ-m\rangle=(-1)^{J-m}\big{[}\langle aJ-m|k\rangle\big{]}^{*}$

##  Appendix B Derivation of the $P$, $T$ violating polarizability expressions

We now derive the expressions for the polarizabilities that originate from the
simultaneous perturbation through the electromagnetic potential $V^{EM}$ and
the symmetry-violating potential $V^{N}$.

###  B.1 0st and 1st order in $V^{EM}$

Because of $V^{EM}$, we need to solve the system using time-dependent
perturbation theory. We can, for example, define the perturbation potential
$V=-\mu_{i}\mathcal{H}_{i}-V^{PT}$ and solve the 2nd-order correction
according to Ref. [23]. We ignore here and throughout the paper terms that are
quadratic in $V^{N}$ due to their small size. If we, for now, also ignore
terms that are quadratic in $V^{EM}$, we obtain the following expression for
the perturbed time-dependent wavefunction:

$\displaystyle\left|N(t)\right\rangle\simeq\left|n\right\rangle+\sum_{k}$
$\displaystyle\frac{\left\langle
k\right|\mu_{i}\left|n\right\rangle}{\omega_{kn}^{2}-(\omega^{\ell})^{2}}\left[\omega_{kn}\mathcal{H}_{i}^{\ell}+i\dot{\mathcal{H}}_{i}^{\ell}\right]\left|k\right\rangle+\sum_{k}\frac{\left\langle
k\right|V^{PT}\left|n\right\rangle}{\omega_{kn}}\left|k\right\rangle$
$\displaystyle+\sum_{k,\,l}$ $\displaystyle\frac{\left\langle
k\right|\mu_{i}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\left[\omega_{kn}\mathcal{H}_{i}^{\ell}+i\dot{\mathcal{H}}_{i}^{\ell}\right]\left|k\right\rangle$
$\displaystyle+\sum_{k,\,l}$ $\displaystyle\frac{\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|\mu_{i}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\left|k\right\rangle\Bigg{]}\left[\left(\omega_{kn}\omega_{ln}+(\omega^{\ell})^{2}\right)\mathcal{H}_{i}^{\ell}+\left(\omega_{kn}+\omega_{ln}\right)\dot{\mathcal{H}}_{i}^{\ell}\right]$
$\displaystyle-\sum_{k}$ $\displaystyle\frac{\left\langle
k\right|V^{PT}\left|n\right\rangle\left\langle
n\right|\mu_{i}\left|n\right\rangle}{\omega_{kn}^{2}}\left|k\right\rangle\,\mathcal{H}_{i}^{\ell}$

where we already removed a term that is proportional to $\left\langle
n\right|V^{PT}\left|n\right\rangle=0$.
From this, we can now calculate the 1st order (in $V^{EM}$) perturbation of
the $P$, $T$ odd induced electric dipole moment:

$\displaystyle\langle d_{i}\rangle=\left\langle N(t)\right|$ $\displaystyle
d_{i}\left|N(t)\right\rangle\simeq 2Re\sum_{k}\Bigg{[}\frac{\left\langle
n\right|V^{PT}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}}\Bigg{]}$ (37)
$\displaystyle+2Re\sum_{k,\,l}\Bigg{[}$ $\displaystyle\frac{\left\langle
n\right|V^{PT}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}$
$\displaystyle+$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})(\omega_{ln}^{2}-(\omega^{\ell})^{2})}\left(\omega_{kn}\omega_{ln}+(\omega^{\ell})^{2}\right)\Bigg{]}\mathcal{H}_{j}^{\ell}$
$\displaystyle-2Im\sum_{k,\,l}\Bigg{[}$ $\displaystyle\frac{\left\langle
n\right|V^{PT}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}$
$\displaystyle+$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})(\omega_{ln}^{2}-(\omega^{\ell})^{2})}\left(\omega_{ln}+\omega_{kn}\right)\Bigg{]}\dot{\mathcal{H}}_{j}^{\ell}$
$\displaystyle-2Re\sum_{k}$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|n\right\rangle}{\omega_{kn}^{2}}\left\langle
n\right|\mu_{i}\left|n\right\rangle\,\mathcal{H}_{j}^{\ell}$
$\displaystyle\equiv$ $\displaystyle
Re\Big{[}\prescript{d}{PT}{\alpha}_{i}\Big{]}+Re\Big{[}\prescript{d\mu}{PT}{\alpha}_{ij}^{\ell}\Big{]}\mathcal{H}_{j}^{\ell}+Im\Big{[}\prescript{d\mu}{PT}{\alpha^{\prime}}_{ij}^{\ell}\Big{]}\dot{\mathcal{H}}_{j}^{\ell}+R$

Here, we ignored the term $\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$, which
vanishes due to parity.
We can see that the solution to this problem not only gives us expressions for
the $P$, $T$ odd polarizabilities but also leads to a certain type of rest
term:

$\displaystyle R=-2Re\sum_{k}$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|n\right\rangle}{\omega_{kn}^{2}}\left\langle
n\right|\mu_{j}\left|n\right\rangle\,\mathcal{H}_{j}^{\ell}$

Depending on the time-dependent perturbation treatment, these kinds of rest
terms can be absent (see, for example, Ref. [33]). They are, however,
necessary for a proper normalization and convergence in the limit of static
fields [20, 23].

It is apparent that in the rest term, $\boldsymbol{d}$ and $\boldsymbol{\mu}$
are uncorrelated in the sense that their expectation values factorize. For
example, $\mu_{j}$ in the expression above does not impose any restrictions on
the $\left|k\right\rangle$-states that $d_{i}$ and $V^{PT}$ connect to. For
these reasons, it makes sense to view these as products of lower-order terms.
As such, they only provide small corrections to already considered observables
without introducing the new tensor structures that we saw in the
$\prescript{}{}{\alpha}_{ij}^{\ell}$. For these reasons, we ignore these
$R$-terms in our treatment.

###  B.2 2nd order in $V^{EM}$

In this order of perturbation, we will only consider the expectation values of
the $P$, $T$ odd Potentials $V^{P}$, $V^{PT}$, and $V^{T}$. These terms that
are linear in $V^{N}$ but quadratic in $V^{EM}$ will, therefore, be of the
same size as the ones we discussed above. As mentioned before, a time-
dependent treatment in this order is more involved and beyond the scope of
this paper. We will only discuss the time-independent case. We get, for
example, for the perturbation of $\langle V^{P}\rangle$ through $d$, and
$\mu$:

$\displaystyle 2Re\sum_{k,\,l}\Bigg{[}$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|l\right\rangle\left\langle
l\right|V^{P}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{\omega_{kn}\omega_{ln}}+\frac{\left\langle
n\right|V^{PT}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|n\right\rangle}{\omega_{kn}\omega_{ln}}$ (38)
$\displaystyle+$ $\displaystyle\frac{\left\langle
n\right|V^{PT}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|d_{i}\left|n\right\rangle}{\omega_{kn}\omega_{ln}}\Bigg{]}\mathcal{E}_{i}\mathcal{H}_{j}$
$\displaystyle\equiv
Re\Big{[}\prescript{d\mu}{P}{\beta}_{ij}\Big{]}\mathcal{E}_{i}\mathcal{H}_{j}$

This is identical to (16) in the limit $\ell=0$. Therefore it follows
$\prescript{d\mu}{P}{\beta}_{ij}=\prescript{d\mu}{P}{\alpha}_{ij}^{0}\equiv\prescript{d\mu}{P}{\alpha}_{ij}$.

##  Appendix C Properties of the $P$, $T$ violating polarizability expressions

In this appendix, we prove some of the properties of the $P$, $T$ odd
polarizabilities that we use throughout the paper.

###  C.1 Behavior of the $d\mu$-terms under time reversal

We now show, on the example of $\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}$, that
the argument presented in sec. 3.3.3 regarding transformation under time
reversal of $\prescript{d\mu}{}{\alpha}_{ij}^{\ell}$ still applies for $P$,
$T$ odd polarizabilities like $\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}$.
We start by decomposing $\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}$ into its
irreducible components:

$\displaystyle\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}$
$\displaystyle=\delta_{ij}\;\prescript{d\mu}{P}{\bar{\alpha}}_{s}^{\ell}+\varepsilon_{ij\varkappa}\left\langle
aJm\right|J_{\varkappa}\left|aJm\right\rangle\;\prescript{d\mu}{P}{\bar{\alpha}}_{v}^{\ell}$
(39) $\displaystyle+\left\langle
aJm\right|\tfrac{1}{2}\left[J_{i}J_{j}+J_{j}J_{i}-\tfrac{2}{3}\delta_{ij}\sum_{l}J_{l}J_{l}\right]\left|aJm\right\rangle\;\prescript{d\mu}{P}{\bar{\alpha}}_{t}^{\ell}$

It is clear that the RHS will transform exactly the same as in Eq. (22). For
$\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}$ on the LHS, we can introduce
$1=T^{\dagger}T$ into Eq. (37). After using the transformation behaviors of
the different operators given in table 1, we get:

$\displaystyle\prescript{d\mu}{P}{\alpha}_{ji}^{\ell}=-2\sum_{k,\,l}\Bigg{[}$
$\displaystyle\frac{\left\langle
aJ-m\right|V^{P}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|aJ-m\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}$
(40) $\displaystyle+$ $\displaystyle\frac{\left\langle
aJ-m\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|V^{P}\left|aJ-m\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}$
$\displaystyle+$ $\displaystyle\frac{\left\langle
aJ-m\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{P}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|aJ-m\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})(\omega_{ln}^{2}-(\omega^{\ell})^{2})}\left(\omega_{kn}\omega_{ln}+(\omega^{\ell})^{2}\right)\Bigg{]}^{*}$

Equating this with the transformed version of the RHS of Eq. (39) and renaming
$-m\rightarrow m$, we get:

$\displaystyle\left(\prescript{d\mu}{P}{\alpha}_{ij}^{\ell}\right)^{*}=-\delta_{ij}\;\prescript{d\mu}{P}{\bar{\alpha}}_{s}^{\ell}+\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{d\mu}{P}{\bar{\alpha}}_{v}^{\ell}-\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{d\mu}{P}{\bar{\alpha}}_{t}^{\ell}$

This tells us that the symmetric part is purely imaginary while the
antisymmetric vector component is real.
It is now also clear that when considering
$\prescript{d\mu}{PT}{\alpha}_{ij}^{\ell}$ instead of
$\prescript{d\mu}{P}{\alpha}_{ji}^{\ell}$, the RHS of Eq. (40) will change by
a global minus sign due to the fact that $V^{PT}$ is $T$-odd. That way, we
have:

$\displaystyle\left(\prescript{d\mu}{PT}{\alpha}_{ij}^{\ell}\right)^{*}=\delta_{ij}\;\prescript{d\mu}{PT}{\bar{\alpha}}_{s}^{\ell}-\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{d\mu}{PT}{\bar{\alpha}}_{v}^{\ell}+\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{d\mu}{PT}{\bar{\alpha}}_{t}^{\ell}$
(41)

###  C.2 Insensitivity of $dd$, $\mu\mu$, and $QQ$ terms to $V^{N}$

In Eq. (25), that describes the full induced electric dipole moment
$\langle\boldsymbol{d}\rangle$, the only polarizabilities that are quadratic
in $d$ are classically allowed ones. It is clear from parity reasons that
$\prescript{dd}{P}{\alpha}_{ij}^{\ell}$, and
$\prescript{dd}{PT}{\alpha}_{ij}^{\ell}$ vanish. But from the arguments
presented in sec. 3.3.3 one would still expect
$\prescript{dd}{T}{\alpha}_{ij}^{\ell}$, and
$\prescript{dd}{T}{\alpha^{\prime}}_{ij}^{\ell}$ to have nonvanishing
components. In this section, we show that the terms that are quadratic in a
multipole moment possess an additional symmetry that prevents the existence of
$T$-odd terms. We demonstrate this on the example of
$\prescript{dd}{T}{\alpha}_{ij}^{\ell}$:

$\displaystyle\prescript{dd}{T}{\alpha}_{ij}^{\ell}=2\sum_{k,\,l}\Bigg{[}$
$\displaystyle\frac{\left\langle n\right|V^{T}\left|l\right\rangle\left\langle
l\right|d_{i}\left|k\right\rangle\left\langle
k\right|d_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}+\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|d_{j}\left|l\right\rangle\left\langle
l\right|V^{T}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})\omega_{ln}}\omega_{kn}$
$\displaystyle+$ $\displaystyle\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{T}\left|l\right\rangle\left\langle
l\right|d_{j}\left|n\right\rangle}{(\omega_{kn}^{2}-(\omega^{\ell})^{2})(\omega_{ln}^{2}-(\omega^{\ell})^{2})}\left(\omega_{kn}\omega_{ln}+(\omega^{\ell})^{2}\right)\Bigg{]}$

From this definition, it follows that
$\big{(}\prescript{dd}{T}{\alpha}_{ij}^{\ell}\big{)}^{*}=\prescript{dd}{T}{\alpha}_{ji}^{\ell}$.
Regarding the tensor decomposition, this relation implies:

$\displaystyle\left(\prescript{dd}{T}{\alpha}_{ij}^{\ell
k}\right)^{*}=\delta_{ij}\;\prescript{dd}{T}{\bar{\alpha}}_{s}^{\ell
k}-\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{dd}{T}{\bar{\alpha}}_{v}^{\ell
k}+\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{dd}{T}{\bar{\alpha}}_{t}^{\ell
k}$ (42)

where we used that $\varepsilon_{ij\varkappa}$ is antisymmetric. However, from
its transformation under time reversal, we know that
$\prescript{dd}{T}{\alpha}_{ij}^{\ell}$, which contains an odd number of
$T$-odd operators, must fulfill the relation:

$\displaystyle\left(\prescript{dd}{T}{\alpha}_{ij}^{\ell
k}\right)^{*}=-\delta_{ij}\;\prescript{dd}{T}{\bar{\alpha}}_{s}^{\ell
k}+\varepsilon_{ij\varkappa}\mathcal{J}_{\varkappa}\;\prescript{dd}{T}{\bar{\alpha}}_{v}^{\ell
k}-\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\;\prescript{dd}{T}{\bar{\alpha}}_{t}^{\ell
k}$ (43)

Together, these relations can only be true if all three components are zero.
We can make analogous arguments for the polarizabilities that are quadratic in
$\mu$.
In the case of $\prescript{QQ}{T}{\alpha}_{ij\varkappa l}^{\ell k}$, we are
dealing with a rank-4 tensor. Because the quadrupole operator is symmetric, it
obeys the relation $\left(\prescript{QQ}{}{\alpha}_{ij\varkappa l}^{\ell
k}\right)^{*}=\prescript{QQ}{}{\alpha}_{\varkappa lij}^{\ell k}$. One can
again see that this relation is in conflict with the relations that follow
from time reversal.

In Ref. [12], macroscopic equations describing the possible $P$, $PT$, and $T$
odd responses of systems to homogeneous time-dependent electromagnetic fields
were formulated. From the results of this section, we can see that (under our
initial assumptions) there are no microscopic processes that could generate
the proposed ToPe responses.

##  Appendix D Energy shift from static electromagnetic fields

We now derive how the polarizations are related to the energy shift that is
induced when static electromagnetic fields are applied to the system. As
before, our system is described by $H=H^{0}+V^{EM}+V^{N}$. The perturbative
energy corrections in such a situation are well known [25, 23] and given by:

$\displaystyle\Delta E\;$ $\displaystyle\simeq\Delta E_{1}+\Delta E_{2}+\Delta
E_{3}$ $\displaystyle\Delta E_{1}$ $\displaystyle=\left\langle
n\right|\left(V^{EM}+V^{N}\right)\left|n\right\rangle$ $\displaystyle\Delta
E_{2}$ $\displaystyle=-\sum_{k}\frac{|\left\langle
n\right|\left(V^{EM}+V^{N}\right)\left|k\right\rangle|^{2}}{\omega_{kn}}$
$\displaystyle\Delta E_{3}$ $\displaystyle=\sum_{k,\,l}\frac{\left\langle
n\right|\left(V^{EM}+V^{N}\right)\left|k\right\rangle\left\langle
k\right|\left(V^{EM}+V^{N}\right)\left|l\right\rangle\left\langle
l\right|\left(V^{EM}+V^{N}\right)\left|n\right\rangle}{\omega_{kn}\,\omega_{ln}}$
$\displaystyle\quad-\left\langle
n\right|\left(V^{EM}+V^{N}\right)\left|n\right\rangle\sum_{k}\frac{|\left\langle
n\right|\left(V^{EM}+V^{N}\right)\left|k\right\rangle|^{2}}{\omega_{kn}^{2}}$

As discussed in sec. 3.2, most terms in $\Delta E_{1}$ vanish:

$\displaystyle\Delta E_{1}=-\left\langle
n\right|\mu_{i}\left|n\right\rangle\mathcal{H}_{i}-\left\langle
n\right|Q_{ij}\left|n\right\rangle\partial_{i}\mathcal{E}_{j}=-\prescript{\mu}{}{\alpha}\,\mathcal{J}_{i}\mathcal{H}_{i}-\prescript{Q}{}{\alpha}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\partial_{i}\mathcal{E}_{j}$

In the next order, we get both the $P$, $T$ odd polarizabilities that are
linear in $V^{EM}$ (see sec. 3.5) and the $P$, $T$ even polarizabilities that
are quadratic in $V^{EM}$ (see sec. 2.3.2). For example for
$V^{EM^{\prime}}=-\boldsymbol{d}\cdot\boldsymbol{\mathcal{E}}$, and
$V^{N^{\prime}}=-V^{PT}$, we get:

$\displaystyle\Delta E_{2}$ $\displaystyle=-\sum_{k}\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|d_{j}\left|n\right\rangle}{\omega_{kn}}\mathcal{E}_{i}\mathcal{E}_{j}-\frac{\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|n\right\rangle}{\omega_{kn}}\mathcal{E}_{i}-\frac{\left\langle
n\right|V^{PT}\left|k\right\rangle\left\langle
k\right|d_{i}\left|n\right\rangle}{\omega_{kn}}\mathcal{E}_{i}$
$\displaystyle=-\tfrac{1}{2}\prescript{dd}{}{\alpha}_{s}\,\mathcal{E}^{2}-\tfrac{1}{2}\prescript{dd}{}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\,\mathcal{E}_{i}\mathcal{E}_{j}-\prescript{d}{PT}{\alpha}\,\mathcal{J}_{i}\mathcal{E}_{i}$

The 3rd energy correction $\Delta E_{3}$ is a sum of two terms. The second one
describes the shift from the uncorrelated terms that we discussed in appendix
B.1. For the reasons stated before, we neglect these. The first term in
$\Delta E_{3}$ gives us the $P$, $T$-odd polarizabilities quadratic in
$V^{EM}$. From the example
$V^{EM^{\prime}}=-\boldsymbol{d}\cdot\boldsymbol{\mathcal{E}}-\boldsymbol{\mu}\cdot\boldsymbol{\mathcal{H}}$,
and $V^{N^{\prime}}=-V^{PT}$, we get:

$\displaystyle\Delta E_{3}=-\sum_{k,\,l}[$
$\displaystyle(\omega_{kn}\omega_{ln})^{-1}\,\mathcal{E}_{i}\mathcal{H}_{j}]$
$\displaystyle\cdot\Big{[}$ $\displaystyle\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle+\left\langle
n\right|d_{i}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|n\right\rangle$ $\displaystyle+$
$\displaystyle\left\langle n\right|\mu_{j}\left|k\right\rangle\left\langle
k\right|d_{i}\left|l\right\rangle\left\langle
l\right|V^{PT}\left|n\right\rangle+\left\langle
n\right|\mu_{j}\left|k\right\rangle\left\langle
k\right|V^{PT}\left|l\right\rangle\left\langle
l\right|d_{i}\left|n\right\rangle$ $\displaystyle+$ $\displaystyle\left\langle
n\right|V^{PT}\left|k\right\rangle\left\langle
k\right|d_{i}\left|l\right\rangle\left\langle
l\right|\mu_{j}\left|n\right\rangle+\left\langle
n\right|V^{PT}\left|k\right\rangle\left\langle
k\right|\mu_{j}\left|l\right\rangle\left\langle
l\right|d_{i}\left|n\right\rangle\Big{]}$
$\displaystyle=-\prescript{d\mu}{PT}{\alpha}_{s}\,\mathcal{E}_{i}$
$\displaystyle\mathcal{H}_{i}-\prescript{d\mu}{PT}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\,\mathcal{E}_{i}\mathcal{H}_{j}$

From this, we can also see: $\prescript{d\mu}{PT}{\alpha}_{s}=\prescript{\mu
d}{PT}{\alpha}_{s}$. If we perform the treatment for the full $V^{N}$ and
$V^{EM}$, we arrive at a general expression for the energy shift of an atom or
molecule exposed to static electromagnetic fields:

$\displaystyle\Delta E\simeq$
$\displaystyle-\prescript{\mu}{}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{\mathcal{H}}}-\tfrac{1}{2}\prescript{dd}{}{\alpha}_{s}\,{\mathcal{E}}^{2}-\tfrac{1}{2}\prescript{\mu\mu}{}{\alpha}_{s}\,{\mathcal{H}}^{2}-\tfrac{1}{2}\prescript{dd}{}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}{\mathcal{E}}_{i}{\mathcal{E}}_{j}-\tfrac{1}{2}\prescript{\mu\mu}{}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}{\mathcal{H}}_{i}{\mathcal{H}}_{j}$
$\displaystyle-\prescript{d}{PT}{\alpha}\,\boldsymbol{\mathcal{J}}\cdot\boldsymbol{\mathcal{E}}-\prescript{d\mu}{PT}{\alpha}_{s}\,\boldsymbol{\mathcal{H}}\cdot\boldsymbol{\mathcal{E}}-\prescript{d\mu}{P}{\alpha}_{v}\,(\boldsymbol{\mathcal{E}}\times\boldsymbol{\mathcal{H}})\cdot\boldsymbol{\mathcal{J}}-\prescript{d\mu}{PT}{\alpha}_{t}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\mathcal{E}_{i}{\mathcal{H}}_{j}$
$\displaystyle-\prescript{Q}{}{\alpha}\,\mathcal{Q}^{\boldsymbol{J}^{2}}_{ij}\partial_{i}\mathcal{E}_{j}-\tfrac{1}{2}\prescript{QQ}{}{\alpha}_{ij\varkappa
l}\,\partial_{i}\mathcal{E}_{j}\partial_{\varkappa}\mathcal{E}_{l}$
$\displaystyle-\left(\prescript{\mu
Q}{}{\alpha}_{t_{s}}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{\mu
Q}{}{\alpha}_{v_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\mathcal{H}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}$
$\displaystyle-\prescript{dQ}{P}{\alpha}_{t_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\mathcal{E}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}-\prescript{\mu
Q}{T}{\alpha}_{t_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}^{2}}\mathcal{H}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}$
$\displaystyle-\left(\prescript{dQ}{PT}{\alpha}_{t_{s}}^{\ell}\,\mathcal{S}_{ij\varkappa}^{\boldsymbol{J}^{3}}+\prescript{dQ}{PT}{\alpha}_{v_{m}}\,\mathcal{M}_{ij\varkappa}^{\boldsymbol{J}}\right)\mathcal{E}_{i}\partial_{j}{\mathcal{E}}_{\varkappa}$

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