††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Enantioselective Orientation of Chiral Molecules Induced by Terahertz Pulses
with Twisted Polarization

Ilia Tutunnikov  AMOS and Department of Chemical and Biological Physics, The
Weizmann Institute of Science, Rehovot 7610001, Israel  Long Xu  AMOS and
Department of Chemical and Biological Physics, The Weizmann Institute of
Science, Rehovot 7610001, Israel  Robert W. Field<EMAIL_ADDRESS>Department
of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139,USA
Keith A. Nelson<EMAIL_ADDRESS>Department of Chemistry, Massachusetts
Institute of Technology, Cambridge, MA 02139,USA  Yehiam Prior
<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological
Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel  Ilya Sh.
Averbukh<EMAIL_ADDRESS>AMOS and Department of Chemical and
Biological Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

Chirality and chiral molecules are key elements in modern chemical and
biochemical industries. Individual addressing, and the eventual separation of
chiral enantiomers has been and still is an important elusive task in
molecular physics and chemistry, and a variety of methods has been introduced
over the years to achieve this goal. Here, we theoretically demonstrate that a
pair of cross-polarized THz pulses interacting with chiral molecules through
their permanent dipole moments induces an enantioselective orientation of
these molecules. This orientation persists for a long time, exceeding the
duration of the THz pulses by several orders of magnitude, and its dependency
on temperature and pulses’ parameters is investigated. The persistent
orientation may enhance the deflection of the molecules in inhomogeneous
electromagnetic fields, potentially leading to viable separation techniques.

##  I Introduction

Molecular chirality was discovered in the 19th century by Louis Pasteur [1]
(for a historical excursion, see e.g. [2]), and since then chirality has
attracted considerable interest owing to its importance in physics, chemistry,
biology, and medicine. Chiral molecules exist in two forms, called left-, and
right-handed enantiomers, which are mirror images of each other and cannot be
superimposed by translations or rotations [3]. Even though the two enantiomers
have many identical chemical and physical properties, e.g. boiling points and
moments of inertia, they often differ in their biological activities. This is
only one of the reasons why the abilities to differentiate, selectively
manipulate, and separate the enantiomers in mixtures containing both of them
are of great practical importance. From the point of view of fundamental
physics, chiral molecules seem to be promising candidates for experiments
aimed at measuring parity violation effects [4].

Some of the physical properties of the two enantiomers mimic their reflection
relation. For example, the product of three Cartesian components of the
molecular dipole moment has the opposite sign for the two enantiomers [5].
Utilizing the distinct properties of the enantiomers, a variety of methods for
chiral discrimination using electromagnetic fields has been developed over the
years, including: microwave three-wave mixing spectroscopy [5, 6, 7, 8, 9,
10], photoelectron circular dichroism [11, 12, 13, 14, 15], Coulomb explosion
imaging [16, 17, 18], and high-order harmonic generation [19]. Most recently,
a generic purely optical method for enantioselective _orientation_ has been
proposed [20, 21, 22, 23] and experimentally demonstrated on propylene oxide
molecule spun by an optical centrifuge [24] in which the polarization vector
rotates unidirectionally with increasing angular velocity. This last method
utilizes laser fields with “twisted” polarization [see Fig. 1(a)] and relies
on the off-diagonal elements of the polarizability tensor which, in chiral
molecules, have opposite signs for the two enantiomers. In related works, it
was demonstrated that the optically induced orientation persists long after
the end of the laser pulses [23, 25]. For the purpose of the current work,
fields with twisted polarization are understood to include: pairs of delayed
cross-polarized pulses [26, 27, 28] [see Fig. 1(b)], chiral pulse trains [29,
30], polarization-shaped pulses [31, 32, 33, 34, 35], and the aforementioned
optical centrifuge for molecules [36, 37, 38, 39, 40].

In this work, we study the orientation of chiral molecules induced by
terahertz (THz) pulses with twisted polarization. Since the advent of THz
pulse technology, intense THz pulses have been exploited for producing
transient field-free orientation of polar molecules of various complexity [41,
42, 43, 44, 45, 46, 47]. Although transient orientation revival spikes may
periodically appear on long time scales, the orientation signature generally
rides on a zero baseline and its time average is exactly zero. Recently, it
was shown that a single linearly polarized THz pulse induces persistent
orientation in symmetric- and asymmetric-top molecules, including chiral
molecules [48]. In contrast to the transient signal, persistent orientation
means that the time-averaged post-pulse orientation degree differs from zero
on a long time scale, in the case discussed in this paper - exceeding the
duration of the THz pulse by several orders of magnitude.

Here, we theoretically demonstrate that when applied to _chiral molecules_ , a
_pair of time-delayed cross-polarized THz pulses_ induces orientation in a
direction perpendicular to the plane spanned by the polarizations. The
orientation is enantioselective, meaning that the two enantiomers are oriented
in opposite directions relative to the plane. We show that the time-averaged
projections of the molecular dipole moment onto all three laboratory axes
remains nonzero on a nanosecond timescale. The paper is organized as follows.
In Sec. II, we briefly summarize our theoretical methods. In Sec. III, we
present and analyze the results of classical and fully quantum simulations of
THz field-driven molecular rotational dynamics, and Section IV concludes the
presentation.

##  II Methods

We carried out classical as well as fully quantum mechanical simulations of
the THz field-driven rotational dynamics of chiral molecules. This section
outlines the theoretical approaches used in both cases.

Quantum simulation. The Hamiltonian describing the molecular rotation driven
by a THz field interacting with the molecular dipole moment is given by [49,
50]

$\hat{H}(t)=\hat{H}_{r}+\hat{H}_{\mathrm{int}}(t),$ (1)

where $\hat{H}_{r}$ is the rotational kinetic energy Hamiltonian and
$\hat{H}_{\mathrm{int}}(t)=-\hat{\bm{\mu}}\cdot\mathbf{E}(t)$ is the molecule-
field interaction. Here $\hat{\bm{\mu}}$ is the molecular dipole moment
operator and $\mathbf{E}(t)$ is the external electric field. In this work, the
contributions of higher order interaction terms are small, and are not
included. For the quantum mechanical treatment, it is convenient to express
the Hamiltonian in the basis of field-free symmetric-top wave functions
$|JKM\rangle$ [51]. Here $J$ is the value of the total angular momentum (in
units of $\hbar$), while $K$ and $M$ are the values of the projections on the
molecule-fixed axis (here the axis with smallest moment of inertia) and the
laboratory-fixed $Z$ axis, respectively. The nonzero matrix elements of the
asymmetric-top kinetic energy operator are given by [51]

$\displaystyle\langle JKM|\hat{H}_{r}|JKM\rangle=\frac{B+C}{2}$
$\displaystyle\left[J(J+1)-K^{2}\right]+AK^{2},$ (2) $\displaystyle\langle
JKM|\hat{H}_{r}|J,K\pm 2,M\rangle$ $\displaystyle=\frac{B-C}{4}f(J,K\pm 1),$
(3)

where $f(J,K)=\sqrt{(J^{2}-K^{2})[(J+1)^{2}-K^{2}]},$
$A=\hbar^{2}/2I_{a},\,B=\hbar^{2}/2I_{b},\,C=\hbar^{2}/2I_{c}$ are the
rotational constants ($A>B>C$), and $I_{a}<I_{b}<I_{c}$ are the molecular
moments of inertia. The time-dependent Schrödinger equation
$i\hbar\partial_{t}|\Psi(t)\rangle=\hat{H}(t)|\Psi(t)\rangle$ is solved by
numerical exponentiation of the Hamiltonian matrix (see Expokit [52]), and a
detailed description of our numerical scheme can be found in [48]. In our
simulations, the computational basis included all the rotational states with
an angular momentum $J\leq 44$. For the case of propylene oxide molecules
which is used as an example, at a temperature of $T=5\,\mathrm{K}$, initial
states with $J\leq 8$ are included in the thermal averaging.

Classical simulation. In the classical limit, chiral molecules are modeled as
asymmetric tops driven by an external torque. The classical equations of
motion for the angular velocities (written in the frame of principal axes of
inertia tensor) are the Euler’s equations [53]

$\mathbf{I}\bm{\dot{\Omega}}=(\mathbf{I}\bm{\Omega})\times\bm{\Omega}+\mathbf{T},$
(4)

where $\bm{\Omega}=(\Omega_{a},\Omega_{b},\Omega_{c})$ is the angular velocity
vector, $\mathbf{I}=\mathrm{diag}(I_{a},I_{b},I_{c})$ is the moment of inertia
tensor, and $\mathbf{T}=(T_{a},T_{b},T_{c})$ is the external torque vector.
The external torque originates from the electric field, which is defined in
the laboratory frame. In order to solve Eq. (4), a time-dependent relation
between the molecular frame (the frame of principal axes of inertia) and the
laboratory frame is required.

Such a relation can be established with the help of a time-dependent unit
quaternion, which is used to parametrize the rotation of the rigid body.
Quaternions extend complex numbers, and are defined as quadruples of real
numbers, $q=(q_{0},q_{1},q_{2},q_{3})$. The relation between a vector
$\mathbf{x}$ in the molecular frame and a vector $\mathbf{X}$ in the
laboratory frame is given by a simple linear transformation
$\mathbf{x}=Q(t)\mathbf{X},$ where $Q(t)$ is a $3\times 3$ matrix composed of
the quaternion’s elements [54, 55]. The quaternion obeys the following
equation of motion

$\dot{q}=\frac{1}{2}q\Omega,$ (5)

where $\Omega=(0,\bm{\Omega})$ is a pure quaternion and the quaternion
multiplication rule is implied [54, 55]. Equations (4) and (5) are coupled via
the torque, $\mathbf{T}=\boldsymbol{\mu}\times Q\mathbf{E}$, where
$\boldsymbol{\mu}$ is the molecular dipole moment vector.

To simulate the behavior of a classical ensemble, we use the Monte Carlo
approach. For each individual asymmetric top, we numerically solve the system
of Eqs. (4) and (5). We use ensembles consisting of $N=10^{7}$ molecules,
which are initially isotropically distributed in space, and the initial
angular velocities are given by the Boltzmann distribution

$P(\bm{\Omega})\propto\exp\left(-\frac{\bm{\Omega}^{T}\mathbf{I}\bm{\Omega}}{2k_{B}T}\right)=\prod_{i}\exp\left(-\frac{I_{i}\Omega_{i}^{2}}{2k_{B}T}\right),$
(6)

where $T$ is the temperature and $k_{B}$ is the Boltzmann constant. The
initial uniform random quaternions were generated using the recipe described
in [56].

##  III Results

Figure 1: (a) Illustration of the electric field with twisted polarization
composed of two delayed cross-polarized THz pulses [see Eq. (7)]. The vertical
blue arrows represent the $X$-projection of the field. The gray line in the
$YZ$ plane represents the $Y$-projection of the field. (b) Amplitude of the
$X$-polarized (green) and the $Y$-polarized (red) pulses. Time dependence of
the first pulse is given by $f(t)=(1-2at^{2})e^{-at^{2}}$, while that of the
second pulse by $f(t-\tau)$. Here, $a=3.06\,\mathrm{ps^{-2}}$ and
$\tau=0.80\,\mathrm{ps}$. Time integral of the electric field is identically
zero.

Molecule
---

& Moments of inertia Molecular dipole components (_R_)-propylene oxide
$I_{a}=180386$ $\mu_{a}=0.965$ $I_{b}=493185$ $\mu_{b}=-1.733$ $I_{c}=553513$
$\mu_{c}=0.489$

Table 1: Molecular properties of (_R_)-PPO: eigenvalues of the moment of
inertia tensor (in atomic units) and components of dipole moment (in Debye) in
the frame of molecular principal axes of inertia.

We consider propylene oxide (referred to as PPO hereafter) as a typical
example of a chiral molecule. Table III summarizes the molecular properties of
the right-handed enantiomer, (_R_)-PPO. Molecular moments of inertia and the
components of molecular dipole moments were computed with the help of the
GAUSSIAN software package (CAM-B3LYP/aug-cc-pVTZ method) [57].

The molecules are excited by a pair of delayed cross-polarized THz pulses. The
combined electric field of the pulses is modeled using [45]

$\mathbf{E}(t)=E_{0}\left[f(t)\mathbf{e}_{X}+f(t-\tau)\mathbf{e}_{Y}\right],$
(7)

where $E_{0}$ is the peak amplitude, $f(t)=(1-2at^{2})e^{-at^{2}}$ defines the
time dependence of each pulse, $a$ determines the temporal width of the pulse,
$\tau$ is the time delay between the peaks of the two pulses, and
$\mathbf{e}_{X}$ and $\mathbf{e}_{Y}$ are the unit vectors along the
laboratory $X$ and $Y$ axes, respectively. The pulses propagate along the
laboratory $Z$ axis, while $\mathbf{E}$ twists in the $XY$ plane, as shown in
Fig. 1.

Figure 2: Projections of the molecular dipole moment on the laboratory-fixed
axes. The classical results (red and green) are ensemble averages, while
quantum results (blue and purple) are incoherent averages over the initial
thermally populated rotational states. Here $T=5\,\mathrm{K}$, and the THz
field parameters are similar to Fig. 1, with $E_{0}=8.0\,\mathrm{MV/cm}$. The
insets show the magnified portions of the plots during the first 30 ps.

Figure 3: Long-term behavior of the dipole projections shown in Fig. 2. Open
circles represent sliding time average of the dipole signals
$\braket{\mu_{X}}$ (green), $\braket{\mu_{Y}}$ (red), and $\braket{\mu_{Z}}$
(blue and purple) calculated quantum mechanically. The sliding time average is
evaluated according to $\overline{\langle\mu_{i}\rangle(t)}=(\Delta
t)^{-1}\int_{t-\Delta t/2}^{t+\Delta
t/2}\mathrm{d}t^{\prime}\langle\mu_{i}\rangle(t^{\prime})$, where $\Delta
t=100$ ps. For comparison, the correspondingly colored solid lines are the
classical results.  Figure 4: Classically calculated permanent values of the
dipole signals (a) $\langle\mu_{X}\rangle_{p}$, (b)
$\langle\mu_{Y}\rangle_{p}$, and (c) $\langle\mu_{Z}\rangle_{p}$ as functions
of temperature, $T$, and time delay between the two THz pulses, $\tau$. The
color scales are in the units of millidebye (mD).

Figure 2 shows the dipole moment projections along the three laboratory axes,
$\braket{\mu_{X}}$, $\braket{\mu_{Y}}$, and $\braket{\mu_{Z}}$, as functions
of time. The angle brackets $\braket{\cdot}$ denote the incoherent average of
initial thermally populated rotational states, or the ensemble average in the
classical case. The parameters of the field are $a=3.06\,\mathrm{ps^{-2}}$,
$\tau=0.80\,\mathrm{ps}$, and $E_{0}=8.0\,\mathrm{MV/cm}$ (corresponding to
the peak intensity of $8.5\times 10^{10}\,\mathrm{W/cm^{2}}$), see Eq. (7) and
Fig. 1. Note that THz pulses with peak amplitudes of tens of $\mathrm{MV/cm}$,
especially with the use of field enhancement structures [58, 59, 60], are
experimentally available. It is evident from the insets in Fig. 2 that on the
short time scale the classical and quantum results are in excellent agreement.
Each of the THz pulses is followed by a splash of dipole signal in the
direction of the pulse polarization, i.e. initially along the $X$ axis, and
then along the $Y$ axis [see the minima in the insets of Figs. 2(a) and 2(b),
which are before and after 0 ps, respectively]. This transient orientation
induced by single THz pulses is expected and has been observed before [41, 42,
43, 44, 45, 46, 47, 48].

However, an unexpected result emerges: a dipole projection along the $Z$ axis
(perpendicular to the plane of polarization twisting) appears after the second
pulse [see Fig. 2(c)]. This orientation which is unique to chiral molecules is
enantioselective, in the sense that the sign of the projection is opposite for
the two enantiomers, positive for (_S_)-PPO and negative for (_R_)-PPO. The
enantioselectivity of the orientation in the $Z$ direction can be formally
derived as well [for details, see Appendix A]. Similar enantioselective
orientation was observed in chiral molecules optically excited by laser fields
with twisted polarization acting on the molecular polarizability [21, 22, 24].

Furthermore, the classical results clearly show another remarkable feature of
the induced orientation. After the field is switched off [$t>2.5$ ps, see Fig.
1(b)], the dipole projections along all three laboratory axes persist on the
nanosecond timescale. The direct quantum simulation deviates from the
classical one on the long time scale, exhibiting quantum beats/revivals [61,
62, 63]. Nevertheless, as can be seen from Fig. 3, on a timescale of 0.5 ns
the time-averaged quantum signals reproduce well the steady state dipole
signals obtained by the classical calculation.

Classically, the persistent long-term orientation shown in Figs. 2 and 3 is in
fact permanent. In the absence of external torques (external fields), in the
laboratory frame the angular momentum vector is conserved, while in the
molecular frame the angular momentum follows a fixed trajectory, which can be
visualized using the Binet construction [53]. Although the absolute
orientation of an asymmetric top in the laboratory frame never recurs,
depending on the energy and magnitude of the angular momentum, the projections
of the molecular principal axes $a$ or $c$ on the conserved angular momentum
vector have a constant sign [53]. As a result, the attained asymptotic values
of the orientation factors do not change after the initial dephasing of the
ensemble which, according to Fig. 2, takes about 30 ps. On the other hand,
quantum mechanically, the notion of well-defined trajectories is invalid,
which means that the orientation is simply _long-lived_ and eventually will
change its sign. Since the kinetic energy Hamiltonian [see Eq. (3)] couples
rotational states with different $K$ quantum number, the quantum-mechanical
asymmetric top does not have permanently oriented eigenstates. Any initially
oriented state will eventually oscillate between being oriented and anti-
oriented, an effect known as dynamical tunneling [64]. Formally, for a quantum
mechanical chiral rotor, one would expect no permanent orientation after the
turn-off of all external fields. However, as we show here and as was shown in
[23, 25], the tunneling timescale may exceed the excitation timescale by
orders of magnitude.

Notice, the persistent orientation appearing in the directions of each of the
pulses (along the $X$ and $Y$ axes) does not rely on chirality. It was
recently shown that single THz pulses applied to symmetric- and asymmetric-top
molecules also induce persistent orientation [48]. In contrast, both the
appearance and permanency of the orientation along the propagation direction
which is perpendicular to both the $X$ and $Y$ axes depend on the chirality of
the molecule. Specifically, these effects rely on the lack of molecular
symmetry, i.e. all three molecular dipole moment components must differ from
zero, $\mu_{a},\,\mu_{b},\,\mu_{c}\neq 0$ [see Appendix B]. For comparison, in
the case of optical excitation by the laser fields with twisted polarization,
the orientation relies on the existence of the off-diagonal elements of the
polarizability tensor, which is a property of chiral molecules as well. In
that case, the _induced dipole_ has nonzero projections along all three
principal axes, even when the laser field is polarized along only one of the
molecular principal axes.

The magnitude of the THz-induced orientation is sensitive to the initial
temperature and external field parameters [65, 66, 47, 48]. Through our
classical simulations, we carried out an extensive study of the permanent
orientation dependence on temperature, $T$, and the time delay between the two
cross-polarized pulses, $\tau$ [see Eq. (7)]. The results are summarized in
Fig. 4. The permanent values of the dipole projections (denoted by
$\langle\mu_{X}\rangle_{p}$, $\langle\mu_{Y}\rangle_{p}$, and
$\langle\mu_{Z}\rangle_{p}$) were obtained by following the field-free
dynamics for a sufficiently long time until a steady-state is reached
(typically $t>100$ ps). Figure 4 shows that for a given rotational
temperature, there are one or several disjoint ranges of $\tau$ resulting in
optimal (largest absolute value) orientation. In general, the optimal time
delay between the pulses is shorter for higher temperatures. Note, however,
that the temperature dependence of the individual projections
$\langle\mu_{X}\rangle_{p}$, $\langle\mu_{Y}\rangle_{p}$, and
$\langle\mu_{Z}\rangle_{p}$ is non-monotonic. For example, at a fixed time
delay $\tau\approx 0.5\,\mathrm{ps}$, $|\langle\mu_{Z}\rangle_{p}|$ increases
with temperature up to $T\approx 120\,\mathrm{K}$, after which
$|\langle\mu_{Z}\rangle_{p}|$ begins to decrease.

##  IV conclusions

We theoretically demonstrated a qualitatively new phenomenon of field-free
enantioselective orientation of chiral molecules induced by THz pulses with
twisted polarization. The twisted pulse induces orientations along the
polarization directions of the two pulses forming the twisted pulse, and this
orientation is of the same sign for both enantiomers. In the direction
perpendicular to the polarization direction of both pulses, we find that the
orientation is of opposite sign for the two enantiomers. The latter effect
relies on the molecular chirality, namely on the lack of molecular symmetry,
such that the molecular dipole has nonzero projections on all three molecular
principal axes. The orientation was shown to persist long after the end of the
pulses. We studied the dependence of persistent orientation values on the
temperature and the time delay between the two cross-polarized THz pulses. The
orientation factors were found to be quite robust against the detrimental
effects of temperature provided that the time delay is adjusted appropriately.
The orientation dynamics on timescales beyond nanoseconds requires a more
detailed analysis, as other effects such as collisions and fine structure
effects [67] become important in addition to dynamical tunneling. The relative
importance of such effects should be assessed in future works. The orientation
persisting on the nanosecond timescale may be measured by means of second (or
higher order) harmonic generation [68], and could be used for deflection by
inhomogeneous electric fields [69, 70] (for an extensive review, see [71]).
The enantioselective orientation along the propagation direction may be useful
for fast and precise analysis of enentiomeric excess, and may facilitate
_enantioselective_ separation using inhomogeneous fields [72]. In the past
[20, 21, 22, 23, 24, 25], related effects induced by optical pulses with
twisted polarizations have been reported, and further exploration will examine
the combined effect of THz and optical fields together that could maximize the
difference in orientations between the two enantiomers.

## Acknowledgments

I.A. gratefully acknowledges support by the Israel Science Foundation (Grant
No. 746/15). K.A.N. acknowledges support by the U.S. National Science
Foundation Grant CHE-1665383. I.A. acknowledges support as the Patricia Elman
Bildner Professorial Chair. This research was made possible in part by the
historic generosity of the Harold Perlman Family.

##  Appendix A Qualitative demonstration of enantioselective orientation

Figure 5: Thermally averaged $X,Y,Z$-projections of the dipole moment in the
laboratory frame, $\braket{\mu_{X}}$, $\braket{\mu_{Y}}$, and
$\braket{\mu_{Z}}$ as functions of time for (a-c) $\mathrm{CH_{3}Cl}$ and
(d-f) $\mathrm{SO_{2}}$ molecules. The results of quantum and classical
simulations are shown in blue and red lines, respectively. The green curve
represents the sliding time average defined by
$\overline{\langle\mu_{i}\rangle(t)}=(\Delta t)^{-1}\int_{t-\Delta
t/2}^{t+\Delta t/2}\mathrm{d}t^{\prime}\langle\mu_{i}\rangle(t^{\prime})$,
where (a-c) $\Delta t=38.2$ ps and (d-f) $\Delta t=100$ ps, respectively. The
insets show a magnified portion of the signals.

Consider two overlapping THz pulses propagating along the laboratory $Z$ axis,
and which are polarized along $X$ and $Y$ axes, respectively. The nonzero
matrix elements of the interaction potential can be written as

$\displaystyle\langle JKM|\hat{H}_{\mathrm{int}}(t)|J^{\prime},K-s,M-p\rangle$
$\displaystyle=$ $\displaystyle\mu_{s}^{(1)}E_{-p}^{(1)}(t)\langle
JKM|{D_{p,s}^{1}}^{*}(R)|J^{\prime},K-s,M-p\rangle,$ (8)

where $s=0,\pm 1$, and $p=\pm 1$ (since $E_{0}^{(1)}=0$). Here $\mu_{s}^{(1)}$
and $E_{p}^{(1)}$ are the dipole moment and electric field, respectively,
expressed as spherical tensors of rank 1, with components $\mu_{\pm
1}^{(1)}=\mp(\mu_{b}\pm i\mu_{c})/\sqrt{2}$, $\mu_{0}^{(1)}=\mu_{a}$, and
$E_{\pm 1}^{(1)}=\mp(E_{X}\pm iE_{Y})/\sqrt{2}$, $E_{0}^{(1)}=E_{Z}$.
${D_{p,s}^{1}}^{*}(R)$ is the complex conjugate of the Wigner D-matrix, where
$R$ denotes the set of the three Euler angles, $(\theta,\phi,\chi)$.

The THz pulses induce dynamics of the $Z$-component of the polarization,
defined by

$P_{Z}(t)=\sum\limits_{s=-1}^{+1}\mu_{s}^{(1)}\langle\Psi(t)|{D_{0,s}^{1}}^{*}(R)|\Psi(t)\rangle,$
(9)

where the wave function is given by
$|\Psi(t)\rangle=\hat{U}(t,0)|\Psi(0)\rangle,$ with $\hat{U}(t,0)$ being the
evolution operator. Since the discussion here is qualitative, for simplicity
we assume that initially the rotor is in the ground rotational state
$\ket{JKM}=\ket{000}$, such that $\ket{\Psi(t)}=\hat{U}(t,0)|000\rangle$.

The evolution operator, $\hat{U}(t,0)$ can be expanded in a Dyson series

$\hat{U}(t,0)=\sum\limits_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\cdots\int_{0}^{t}\mathrm{d}t_{n}\hat{\mathcal{T}}\hat{H}_{\mathrm{int}}(t_{1})\hat{H}_{\mathrm{int}}(t_{2})\cdots\hat{H}_{\mathrm{int}}(t_{n}),$
(10)

where $\hat{\mathcal{T}}$ is the time-ordering operator. From the properties
of Wigner 3-j symbols it follows that the only non-zero matrix elements,
$\langle JKM|{D_{0,s}^{1}}^{*}(R)|J^{\prime},K-s,M\rangle\neq 0$, are those
satisfying $|J-1|\leq J^{\prime}\leq J+1$. Hence, the nonzero components of
polarization are

$\displaystyle V_{s}$
$\displaystyle\sim\mu_{s}^{(1)}\langle\prod\limits_{n}\hat{H}_{\mathrm{int}}(t_{n})000|{D_{0,s}^{1}}^{*}(R)|\prod\limits_{n^{\prime}}\hat{H}_{\mathrm{int}}(t_{n^{\prime}}^{\prime})000\rangle$
$\displaystyle=\mu_{s}^{(1)}\sum\limits_{J_{1}^{\prime}J_{1}K_{1}M_{1}}\langle\prod\limits_{n}\hat{H}_{\mathrm{int}}(t_{n})000|J_{1}K_{1}M_{1}\rangle\langle
J_{1}K_{1}M_{1}|{D_{0,s}^{(1)*}}|J_{1}^{\prime},K_{1}-s,M_{1}\rangle\langle
J_{1}^{\prime},K_{1}-s,M_{1}|\prod\limits_{n^{\prime}}\hat{H}_{\mathrm{int}}(t_{n^{\prime}}^{\prime})000\rangle.$
(11)

Note that for the qualitative argument here, the evolution operator in Eq.
(11) is considered as
$\hat{U}(t,0)\sim\prod\limits_{n}\hat{H}_{\mathrm{int}}(t_{n})$. From Eq.
(11), we know that $V_{s}$ differs from zero only if $\langle
J_{1}+J_{1}^{\prime},2K_{1}-s,2M_{1}|\prod\limits_{n}\hat{H}_{\mathrm{int}}(t_{n})\prod\limits_{n^{\prime}}\hat{H}_{\mathrm{int}}(t_{n^{\prime}}^{\prime})|000\rangle\neq
0$. Also, according to Eq. (8), the allowed transitions are between the states
with $M$ and $M\pm 1$, because $p=\pm 1$. Hence, it follows from Eq. (11) that
the state changing from $0$ to $2M_{1}$ involves an even number of interaction
terms ($\hat{H}_{\mathrm{int}}$). Similarly, transitions between states with
$K$ quantum number equals $0$ and $2K_{1}-s$ involve $2l+s$ interaction terms
with $s=0$ [see Eq. (8), with $s=0$] and $2L+s$ interaction terms with $s=\pm
1$ [see Eq. (8), with $s=\pm 1$], where $L$ and $l$ are integers. The dipole
moments of two enantiomers of the chiral molecule satisfy $\mu_{\pm
1}^{(S)}=\mu_{\pm 1}^{(R)}$ and $\mu_{0}^{(S)}=-\mu_{0}^{(R)}$, such that the
non-zero polarization components [see Eq. (11)] satisfy

$\displaystyle\frac{V_{s}^{(S)}}{V_{s}^{(R)}}$
$\displaystyle=\frac{\mu_{s}^{(S)}}{\mu_{s}^{(R)}}\left(\frac{\mu_{-1}^{(S)}}{\mu_{-1}^{(R)}}\right)^{L^{\prime}}\left(\frac{\mu_{0}^{(S)}}{\mu_{0}^{(R)}}\right)^{2l+s}\left(\frac{\mu_{1}^{(S)}}{\mu_{1}^{(R)}}\right)^{2L+s-L^{\prime}}$
$\displaystyle=\frac{\mu_{s}^{(S)}}{\mu_{s}^{(R)}}\left(\frac{\mu_{0}^{(S)}}{\mu_{0}^{(R)}}\right)^{s}=-1,$
(12)

where $s=0,\pm 1$, and $L^{\prime}$ is an integer. As one can see, the
components of polarization of the two enantiomers are of opposite signs, and
thereby the polarizations satisfy $P_{Z}^{(S)}(t)=-P_{Z}^{(R)}(t)$.

##  Appendix B Orientation of non-chiral molecules

& Moments of inertia Molecular dipole components methyl chloride $I_{a}=20910$
$\mu_{a}=1.986$ $I_{b}=251506$ $\mu_{b}=0$ $I_{c}=251506$ $\mu_{c}=0$ sulfur
dioxide $I_{a}=57812$ $\mu_{a}=0$ $I_{b}=323141$ $\mu_{b}=-1.805$
$I_{c}=380953$ $\mu_{c}=0$

Table 2: Molecular properties of methyl chloride ($\mathrm{CH_{3}Cl}$) and
sulfur dioxide ($\mathrm{SO_{2}}$) molecules: eigenvalues of the moment of
inertia tensor (in atomic units) and components of dipole moment (in Debye) in
the frame of molecular principal axes of inertia.

Figure 5 shows the orientation signals along the three laboratory axes,
$\braket{\mu_{X}}(t)$, $\braket{\mu_{Y}}(t)$, and $\braket{\mu_{Z}}(t)$, for
two non-chiral molecules. The first one is the symmetric-top methyl chloride
($\mathrm{CH_{3}Cl}$), in which the molecular dipole moment is along the
symmetry axis, $a$ axis ($\mu_{b}=\mu_{c}=0$, see Table III). The second
molecule, sulfur dioxide ($\mathrm{SO_{2}}$) is an asymmetric-top molecule in
which the molecular dipole moment is along the molecular $b$ axis
($\mu_{a}=\mu_{c}=0$, see Table III). The parameters of the THz pulses are
similar to Fig. 2. The initial temperature is $T=5\,\mathrm{K}$. The quantum
and classical results are in good agreement. As expected, both the symmetric-
[Figs. 5(a) and 5(b)] and non-chiral asymmetric-top [Figs. 5(d) and 5(e)]
molecules immediately respond to the $X$-polarized and $Y$-polarized pulses.
However, in contrast to chiral molecules [see Fig. 2], the $Z$-projection of
the dipole moment remains zero [see Figs. 5(c) and 5(f)]. Note that
$\braket{\mu_{Z}}$ calculated quantum mechanically is identically zero, while
the small amplitude oscillations appearing in the classical results are due to
the finite number ($N$) of molecules in the ensemble. The lack of orientation
in the cases of symmetric- or non-chiral asymmetric-top molecules indicates
that all three molecular dipole components ($\mu_{a},\,\mu_{b},\,\mu_{c}$) are
indeed required for inducing the perpendicular (in the $Z$ direction)
orientation. Other combinations were considered numerically as well (not
shown), e.g. asymmetric top molecule having two non-zero molecular dipole
components. In all cases, there is no perpendicular orientation, i.e.
$\braket{\mu_{Z}}=0$.

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# Injection of Positrons into a Dense Electron Cloud in a Magnetic Dipole Trap

M. Singer  Technische Universität München, 85748 Garching, Germany  M. R.
Stoneking  Lawrence University, Appleton, WI 54911, USA  Max Planck Institute
for Plasma Physics, 17491 Greifswald, 85748 Garching, Germany  E.V. Stenson
Max Planck Institute for Plasma Physics, 17491 Greifswald, 85748 Garching,
Germany  A. Deller  Max Planck Institute for Plasma Physics, 17491 Greifswald,
85748 Garching, Germany  A. Card  Max Planck Institute for Plasma Physics,
17491 Greifswald, 85748 Garching, Germany  S. Nißl  Technische Universität
München, 85748 Garching, Germany  Max Planck Institute for Plasma Physics,
17491 Greifswald, 85748 Garching, Germany  J. Horn-Stanja  Max Planck
Institute for Plasma Physics, 17491 Greifswald, 85748 Garching, Germany  T.
Sunn Pedersen  Max Planck Institute for Plasma Physics, 17491 Greifswald,
85748 Garching, Germany  University of Greifswald, 17489 Greifswald, Germany
H. Saitoh  The University of Tokyo, Kashiwa 277-8561, Japan  C. Hugenschmidt
Technische Universität München, 85748 Garching, Germany

(August 29, 2024)

###### Abstract

The creation of an electron space charge in a dipole magnetic trap and the
subsequent injection of positrons has been experimentally demonstrated.
Positrons ($5\text{\,}\mathrm{eV}$) were magnetically guided from their source
and injected into the trapping field generated by a permanent magnet
($0.6\text{\,}\mathrm{T}$ at the poles) using a cross field E$\times$B drift,
requiring tailored electrostatic and magnetic fields. The electron cloud is
created by thermionic emission from a tungsten filament. The maximum space
charge potential of the electron cloud reaches $-42\text{\,}\mathrm{V}$, which
is consistent with an average electron density of $(4\pm
2)\text{\times}{10}^{12}$${\mathrm{m}}^{-3}$ and a Debye length of $2\pm
1\text{\,}\mathrm{cm}$. We demonstrate that the presence of this space
potential does not hamper efficient positron injection. Understanding the
effects of the negative space charge on the injection and confinement of
positrons represents an important intermediate step towards the production of
a confined electron–positron pair plasma.

††preprint: AIP/123-QED

##  I Introduction

Efforts to create pair plasmas consisting of quasi-neutral combinations of
particles of opposite charge and equal mass are being pursued across a wide
range of regimes Sarri  _et al._ (2015); Chen  _et al._ (2015); Higaki  _et
al._ (2017); Oohara and Hatakeyama (2003); Stoneking  _et al._ (2020a).
Successful trapping of pair plasma would create a novel and very likely stable
system Tsytovich and Wharton (1978); Iwamoto (1993); Helander and Connor
(2016); Zank and Greaves (1995). In particular, production of a low-
temperature magnetically well-confined electron-positron plasma would offer
numerous opportunities for experimental tests of the theoretical predictions.
It is, however, a significant experimental challenge to achieve the density
threshold for plasma phenomena in the laboratory with a collection of
electrons and positrons. In a traditional quasi-neutral plasma, positive ions
and electrons are born together via the ionization of neutral gas. While one
approach to creating electron-positron plasma is to first produce copious
quantities of positronium Deller  _et al._ (2016) which can then be ionized in
a manner similar to the ionization of neutral gas, there are significant
technical challenges to this method. For our purposes, a more feasible
approach is to assemble a pair plasma by mixing initially separate populations
of positive and negative charges. Established devices for confining either
positrons or electrons are called Penning-Malmberg traps, and nested
combinations of such devices have been used to bring particles of opposite
charge into interaction with one another Greaves and Surko (1995); Gilbert
_et al._ (2001); Hall and Gabrielse (1996); Gabrielse  _et al._ (1999); Amole
_et al._ (2013); Kabantsev and Driscoll (2007). However, such traps are not
suitable for long confinement of quasi-neutral pair plasmas with small Debye
lengths. Other efforts have employed non-neutral plasmas or dense beams as
electrostatic traps for particles of the opposite sign of charge Oshima  _et
al._ (2004); Levine  _et al._ (1988). However, these experiments did not seek
to produce quasi-neutral combinations of particles that are of interest here.

The APEX (A Positron-Electron eXperiment) collaboration aims to confine a
magnetized electron-positron plasma in toroidal traps with closed field lines
that allow the simultaneous trapping of both species. Since 2014, we have been
working toward achieving this in a levitated dipole trap, a geometry that
provides excellent confinement properties for both quasi-neutral and non-
neutral plasmas Yoshida  _et al._ (2012); Baitha  _et al._ (2020); Boxer  _et
al._ (2010); Saitoh  _et al._ (2016). We have also recently started to pursue
development of an optimized stellarator as a second type of confinement device
with suitable and complementary properties. Meanwhile, magnetic mirror traps
are being explored by Higaki, _et al._ Higaki  _et al._ (2017). In the
ultimate realization of the dipole branch of our project, positrons from an
intense source will be accumulated and delivered in large pulses (>
${10}^{8}$) to a trap consisting of a levitated superconducting coil that is
pre-loaded with an electron plasma. In a series of important preliminary
experiments we employed a prototype dipole trap with a supported rare earth
magnet to develop the required methods for positron injection, confinement and
diagnosis. Important milestones, such as establishing highly efficient
positron injectionStenson  _et al._ (2018) and long confinement timesHorn-
Stanja  _et al._ (2018), have been achieved. We report here on the next
experiments in this series in which we create a dense electron cloud in the
prototype trap and measure its impact on our ability to inject positrons. We
find that the presence of an electron cloud with densities and space charge
potentials relevant to future experiments does not negatively impact the
efficiency of positron injection. This result represents an important
validation of our planned approach to creating electron-positron plasma.

##  II Experimental Setup

One major challenge for establishing a pair plasma is the acces to the
requisite number of positrons. These are provided by the NEPOMUC (NEutron
induced POsitron Source MUniCh) positron facility Hugenschmidt  _et al._
(2012), situated at the FRM II neutron source. The positrons are created in
the NEPOMUC beam tube located within the reactor’s heavy water moderator tank
in close proximity to the fuel element. Thermal neutrons are captured in
enriched ${}^{113}\mathrm{C}\mathrm{d}$, generating high-energy gamma
radiation. These photons subsequently interact with platinum foils to produce
electrons and positrons through pair production. These foils are designed as a
system of electrostatic lenses that extract the positrons and establish the
positron beam. This primary beam provides up to
${10}^{9}\text{\,}{\mathrm{\text{ $e$ }}}^{+}\text{\,}{\mathrm{s}}^{-1}$ when
set to an accelerating energy of $1\text{\,}\mathrm{keV}$ and, beam energies
as low as $5\text{\,}\mathrm{eV}$ are feasible. A narrower spatial and energy
distribution is achieved by implanting the $1\text{\,}\mathrm{keV}$ beam into
a tungsten single-crystal film and extracting the remoderated positrons Stanja
_et al._ (2016). For the work described here, we used the remoderated positron
beam with an intensity of $(4.5\pm 0.3)\text{\times}{10}^{7}$ ${\mathrm{\text{
$e$ }}}^{+}\text{\,}{\mathrm{s}}^{-1}$ and a mean longitudinal energy of
$4.9\pm 0.7$ $\mathrm{eV}$.

The positron beam is adiabatically guided by $5\text{\,}\mathrm{mT}$ magnetic
fields to the prototype APEX setup Saitoh  _et al._ (2015) (Figure 2). It
consists of two vertically stacked vacuum chambers. Immediately upstream of
this setup (but not shown here) are a set of diagnostics, including an
annihilation target / charge collector, microchannel plate and phosphor
screen, and a retarding field analyzer. When the diagnostics are retracted,
the beam continues to the experiment setup, passing through two pairs of
steering coils.

At the center of the experiment, enclosed in a copper case, is a cylindrical
permanent magnet that provides the dipole magnetic field to confined charged
particles. The magnet has a diameter of $28\text{\,}\mathrm{mm}$, a height of
$40\text{\,}\mathrm{mm}$ and a magnetic field strength of
$0.6\text{\,}\mathrm{T}$ at the pole faces. Trapped particles gyrate around
the field lines, are magnetically mirrored between the two poles, and drift
toroidally around the magnet, due to curvature, $\nabla$ B and E$\times$B
drifts. Particles with opposite signs of charge experience opposite toroidal
drift directions when the magnetic drifts are dominant. In our configuration,
positrons drift clockwise—when viewed from above (Figure 2)—whereas electrons
drift counter-clockwise.

Near the wall of the chamber, electrodes form a cylindrical boundary in two
layers. The bottom layer consists of eight equally sized sections (shown in
Figure 2), while the top layer has a 1/8 (above sector 1 in Figure 2) to 7/8
division. In the vicinity of this 1/8 electrode, between the magnet and the
beam line, two oppositely charged, vertically oriented electrodes provide the
electric field that generates E$\times$B drift to transport positrons off
magnetic field lines that connect to the beam line, onto magnetic field lines
that intersect only the permanent magnet (i.e., the confinement region). To
minimize the effect of the injection potentials on positrons elsewhere in the
trap, a third vertical plate, which is grounded, serves as a shield plate.

Three instruments can be inserted into the confinement region on its
equatorial plane. The positron injection takes place in the $None$ area (right
side in Figure 2). The injected positron beam propagates clockwise. After a
$None$ drift, a target probe can intercept the positron beam to measure its
intensity. At the $None$ position, a filament serves both as an annihilation
target for positrons and as an electron source. Electrons propagate
counterclockwise, therefore the injected electron cloud fills the region
through which the injected positrons drift.

The main diagnostics for observing the positron injection are two BGO (bismuth
germanate) gamma detectors. Their fields of view are both collimated to
monitor annihilation events on the filament assembly. One detector (hereafter
referred to as detector A) is placed in the equatorial plane, the other
(hereafter referred to as detector B) is placed underneath the chamber. The
yellow triangle (Figure 2) depicts the approximate line of sight of detector A
placed in the equatorial plane of the magnet. The blue rectangle shows the
line of sight of detector B placed underneath the chamber.

Figure 1: Positron injection into the dipole trap: the positron beam passes
the steering coils which are used, to adjust the beam position immediately
upstream of the E$\times$B plates. The red line depicts a sample trajectory of
a $5\text{\,}\mathrm{eV}$ positron.

Figure 2: Top-down view of the chamber. The yellow triangle represents the
field of view of the side BGO detector (A) placed in the equatorial plane of
the magnet and the space charge filament assembly. The blue rectangle shows
the field of view of the bottom detector (B) in the same plane. The E$\times$B
plates are located above the confining area in the region near sector 1
($None$).

##  III Establishing a Space Charge

In this section, we describe the methods used to create and characterize the
electron cloud into which the positron beam was injected.

###  III.1 Electron Source

The electron source used for this work emits electrons through thermionic
emission from a 40–turn coil made of $0.25\text{\,}\mathrm{mm}$ diameter
thoriated tungsten wire with a coil length of $33\text{\,}\mathrm{mm}$ and an
outer coil diameter of $2.8\text{\,}\mathrm{mm}$ (Figure 3). It is positioned
on the equatorial plane of the permanent magnet. The distance between the tip
of its assembly and the copper case of the magnet is $10\text{\,}\mathrm{mm}$.
A current in the range of $\mathrm{I}_{fil}\mathrm{=}$
$3.4\text{\,}\mathrm{A}4.0\text{\,}\mathrm{A}$ heats the filament, leading to
a voltage drop of $\mathrm{V}_{fil}\mathrm{=}$
$13.5\text{\,}\mathrm{V}17.5\text{\,}\mathrm{V}$ across it. The filament is
biased negatively to accelerate electrons away from it
($-60\text{\,}\mathrm{[}$V_acc=]); the end of the filament closest to the
magnet is set to $\mathrm{V}_{acc}$, and the end closest to the wall is
typically between $-73.5\text{\,}\mathrm{V}-77.5\text{\,}\mathrm{V}$. The
emission current for the configuration with $4.0\text{\,}\mathrm{[}$I_fil=] is
$0.80\pm 0.05\text{\,}\mathrm{mA}$.

Since the filament is placed in the dipole field of the magnet, emitted
electrons follow field lines. Electron current measurements on the wall
electrodes, the magnet and a target probe inserted at $None$ (instead of the
emissive probe) indicate that the amount of electrons hitting the wall
electrodes is negligible. Approximately 90% of electrons are immediately lost
on the (grounded) magnet. The rest are mirror-trapped and contribute establish
a space charge by drifting toroidally (counterclockwise).

Figure 3: Electron source assembly. Two insulated rods provide the electrical
connections to heat and bias the tungsten filament. The plate on the right
serves as a heat radiation shield to protect the magnet, which is placed with
a distance of $10\text{\,}\mathrm{mm}$ to its right side.

###  III.2 Measuring the Space Potential

The space potential of the electron cloud established from the filament source
was measured using the emissive probe Sheehan and Hershkowitz (2011), inserted
into the equatorial plane at the $None$ position. Its working principle can be
described using a simple model: when the filament has a more negative
potential than the surrounding potential, electrons are emitted. In contrast,
when the filament is more positive than the surrounding region, electrons are
accelerated towards it and collected. This results in an equilibration of the
potentials between filament and its surroundings. If the temperature of the
tungsten wire is high enough to enable sufficient emission current, the
potential of the electrically floating filament corresponds to the space
potential.

The supporting probe shaft is covered with insulating Kapton tape and is
therefore expected to become negatively charged. This represents a smaller
perturbation to the space potential in the vicinity of the probe tip than a
grounded, conducting shaft Kremer  _et al._ (2007). The radial profile of the
space potential was measured by step-wise retracting the heated filament from
the vicinity of the magnet to the outer electrode and measuring the floating
potential difference at each step (Figure 4). For all applied heating
currents, the most negative potential was measured around position
$50\text{\,}\mathrm{[}$r=]. It can also be seen, that at the ends of the
range, in close proximity to the positively biased magnet
($8\text{\,}\mathrm{[}$+]) and wall electrode ($10\text{\,}\mathrm{[}$+]), a
positive potential was measured. At radii which are populated by electrons,
the positive applied potentials are shielded by the electron cloud. This
shielding effect also results in nearly identical potential profiles for the
configurations with biased ($150\text{\,}\mathrm{[}$V_E ×B=]) and unbiased
E$\times$B plates.

There is significant radial structure in the space potential profiles shown in
Figure 4, as well as a significant mismatch between the linear potential
variation across the filament ($-60\text{\,}\mathrm{V}$ at
$28\text{\,}\mathrm{mm}$ to $-77.5\text{\,}\mathrm{V}$ at
$66\text{\,}\mathrm{mm}$). Without more complete measurement of the three-
dimensional structure of the potential we cannot explain this structure in
detail. However, one effect that may be at play is that the portion of the
outer wall electrode adjacent to where these measurements took place is biased
to $10\text{\,}\mathrm{[}$+]. As the electrons drift into this region they
will experience an inward E$\times$B drift that may be responsible for the
location of the steepest gradient in the potential occurring closer to the
radial location of the middle of the filament rather than its outermost end.

Figure 4: Space charge potential profiles for different emission currents,
measured with the emissive probe at $None$. The space charge used for the data
presented in this work ($0.8\text{\,}\mathrm{[}$I_em=]), is represented by the
blue dots, while the other colors show the space potentials established by
lower heating currents. The vertical line at $16\text{\,}\mathrm{mm}$ shows
the radius of the magnet case, while the line at $86\text{\,}\mathrm{mm}$
shows the radius of the outer wall electrode set. The orange bar illustrates
the radial extent of the emitting section of the filament assembly (on the
other side of the magnet).

###  III.3 Estimating the Electron Density and Debye Length

We aim to create an electron cloud that is dense enough to constitute a
reasonable test of our strategy for creating electron–positron plasma by
injection of positrons into a pre-formed pure electron plasma. Additionally,
if the electron cloud in these tests is cold enough, it would qualify as a
non-neutral plasma if the Debye length is significantly smaller than the
system size. The Debye length is defined as:

$\lambda_{d}=\sqrt{\frac{\epsilon_{0}k_{B}T}{n_{e}e^{2}}},$ (1)

where $\epsilon_{0}$ is the vacuum permittivity, $k_{B}$ the Boltzmann
constant, $T$ the electron temperature, $n_{e}$ the electron density and $e$
the elementary charge.

An effective electron temperature can be defined in relation to the mean
kinetic energy by:

$E_{e}=\frac{3}{2}k_{B}T_{\mathrm{eff}}.$ (2)

A value for the mean kinetic energy is determined from a volume average of the
difference between the filament potential (linearly varying from
$-60\text{\,}\mathrm{V}-77.5\text{\,}\mathrm{V}$) and the space charge
potential (Figure 4). For the highest emission current case, this yields a
mean energy of $44\text{\,}\mathrm{eV}$ or an effective temperature of about
$30\text{\,}\mathrm{eV}$.

There are two (nearly) independent ways we can estimate the order of magnitude
of the electron density. First, we observe that 90% of the emitted electrons
are promptly lost on the magnet. The remaining 10% are mirror trapped and
drift toroidally around the magnet. Therefore, for an emission current of
$0.80\pm 0.05\text{\,}\mathrm{[}$I_emission=], $0.08\pm
0.01\text{\,}\mathrm{mA}$ contributes to establishing the space charge. Due to
the curvature of the magnetic field lines, the shape of the volume occupied by
electrons is a torus with a C-shaped cross section. The volume of this torus
is determined by modeling this cross section in a CAD program, based on
simulated single–particle trajectories. By revolving this shape around the
axis of the magnet, a volume of approximately $0.5\text{\,}\mathrm{l}$
results. Trajectory calculations Saitoh  _et al._ (2015) for
$30\text{\,}\mathrm{eV}$ electrons with an initial radial position of
$50\text{\,}\mathrm{mm}$ indicate that the toroidal drift period is about
$3\text{\,}\mathrm{\SIUnitSymbolMicro s}$. Accordingly, the mean electron
density, consistent with these numbers, is $(3\pm
1)\text{\times}{10}^{12}\text{\,}\mathrm{[}$n_e=].

The second method for estimating the electron density uses the measured space
charge potential and Poisson’s equation, which describes the relation between
the electrostatic potential $\varphi$ and charge density.

$\nabla^{2}\varphi=\frac{n_{e}e}{\epsilon_{0}},$ (3)

To obtain an approximation for the radial profile of the space charge
potential, we used several polynomial fits to the data in Figure 4 and
calculated the Laplacian, assuming cylindrical and up-down symmetry. The
volume–averaged density that results is $(4\pm
2)\text{\times}{10}^{12}\text{\,}{\mathrm{m}}^{-3}$, which is consistent with
the first method.

Importantly, these density estimates are in the target range for planned pair
plasma experiments. The estimated temperature, meanwhile, is about an order of
magnitude larger than what is ultimately foreseen. The Debye length computed
using the estimates above yields values between $1\text{\,}\mathrm{cm}$ and
$3\text{\,}\mathrm{cm}$. These calculations suggest that the Debye length is
somewhat smaller than the size of the system
($5\text{\,}\mathrm{cm}10\text{\,}\mathrm{cm}$). It was not the primary goal
of these experiments to create an electron plasma, but rather to establish a
space charge and electron density comparable to the target for pair plasma
studies. Nevertheless, we have created an electron cloud that marginally
qualifies as being a non–neutral plasma.

##  IV Injection of Positrons Without an Electron Space Charge

The initial position of the positron beam prior to the E$\times$B drift
injection is important for achieving lossless injection. To optimize this, the
currents applied to the two pairs of steering coils were scanned, while the
positron current was measured on the target probe, which was fully inserted at
the $None$ position. This was done without the electron space charge. The
current applied to one set of coils ($\mathrm{I}_{\mathrm{r}}$) moves the beam
in the radial direction, and other moves the beam in the azimuthal direction
($\mathrm{I}_{\theta}$), in the cylindrical coordinate system centered at and
coaxial with the magnet. Figure 5 a shows $\mathrm{I}_{\mathrm{r}}$ on the
horizontal axis, $\mathrm{I}_{\theta}$ on the vertical axis and the injected
positron intensity with a color scale. A similar measurement was made by
retracting the $None$ target probe and recording the annihilation radiation
originating from positrons hitting the filament assembly (Figure 5 b) using
two gamma detectors (A and B) with collimated fields of view. This set of data
was acquired when the E$\times$B plates were set to $\pm
150\text{\,}\mathrm{[}$V_E ×B=], $12\text{\,}\mathrm{[}$V_Top 1=+],
$10\text{\,}\mathrm{[}$V_sec 1=+] and $8\text{\,}\mathrm{[}$V_mag=+], and a
total counting time of $1\text{\,}\mathrm{s}$. The remaining electrodes were
grounded. It has been shown that these potentials provide lossless injection
of the $5\text{\,}\mathrm{eV}$ positron beam if the steering coils have been
adequately adjusted Stenson  _et al._ (2018). Data from detector A is shown in
Figure 5 b-d. The counts observed by detector B are very similar, but the
count rate is lower due to narrower collimation and the smaller size of the
BGO crystal.

Figure 5 a) demonstrates that, over a large parameter space, the injection
efficiency is homogeneous within the accuracy of the measurements. The
differences between 5a) and 5b) can be explained by the angular offset between
the current probe ($None$) and the field of view of the detector A ($None$). A
large positive $\mathrm{I}_{\mathrm{r}}$ value corresponds to injection orbits
close to the outer wall (Point 1). Such injection radii likely result in some
positrons drifting out to annihilate on the outer wall before reaching the
$None$ position Stenson  _et al._ (2018); Nißl  _et al._ (2020). Consequently,
two effects can occur. First, these particles cannot be measured if they
annihilate outside the detector’s field of view or second, even if they are
within the field of view, the count rate is lower due to the larger distance
to the detector.

Figure 5: $\mathrm{I}_{\mathrm{r}}$ vs $\mathrm{I}_{\theta}$ maps with color-
coded positron data. The black line represents the 450 count (in
$1\text{\,}\mathrm{s}$) contour line extracted from b). The numbered points
are parameter values where current profiles are measured, as explained in the
text: a) positron current (${\mathrm{\text{ $e$
}}}^{+}\text{\,}{\mathrm{s}}^{-1}$) measured at the target probe by a charge
integrator, b) annihilation counts (in $1\text{\,}\mathrm{s}$) originating
from the filament assembly measured by detector A with the filament unheated
and unbiased. c) annihilation counts with the unheated but biased
($-60\text{\,}\mathrm{[}$V_acc=]) filament. d) annihilation counts for the
heated $4.0\text{\,}\mathrm{[}$I_fil=] and biased filament. The area of
injection is extended towards positive $\mathrm{I}_{\mathrm{r}}$ values,
meaning that the electron cloud allows a successful injection of positrons
from parameters that are not feasible otherwise, while preserving the
injection efficiency.

While this type of measurement illustrates the total intensity of positrons
for different injection conditions, it does not provide information about the
radial position of trapped positrons. Radial profiles are measured by step-
wise retracting the target probe at $None$ while measuring the positron
intensity (Figure 6). Consider several profiles with the same azimuthal
steering coil current $1.0\text{\,}\mathrm{[}$I_θ=]. Changing the radial
steering current $\mathrm{I}_{\mathrm{r}}$ from negative to positive values,
the positron beam moves onto orbits that are closer or farther away from the
magnet. The profiles shown in Figure 6 illustrate this: at Point 3 positrons
are injected onto orbits closest to the magnet, the profile is shifted
gradually towards the midradius for point 2. Point 1 does not show a
measurable positron current.

Figure 6: Injected positron intensity profiles measured at various parameters
(points shown in figure 5) for $150\text{\,}\mathrm{[}$V_E ×B=]. The surface
of the magnet is positioned at $16\text{\,}\mathrm{mm}$. A low
$\mathrm{I}_{\mathrm{r}}$ value results in positron profiles close to the
magnet. The dots show the measured values, fitted with error function (lines)
and the corresponding derivative (dashed lines). These gaussians illustrate
the radial distribution of the injected positrons.

##  V Injection of Positrons into an Electron Space Charge

When establishing the electron space potential, two distinctive effects occur.
One is the effect of the additional potential created by the negatively biased
filament, while the other is the effect of the electron cloud. It is important
to note that the rate of annihilation between the injected positrons and the
electrons making up the space charge is expected to be entirely negligible in
the parameter regime of these and future experiments Stoneking  _et al._
(2020b).

###  V.1 Effect of the Negative Filament Bias

The effect of the filament potential can be seen when comparing Figures 5 b)
and 5 c). By applying the negative potential, without heating the filament,
the formerly uniform parameter regions of efficient injection shows a hollow
feature. Particle trajectory simulations Nißl  _et al._ (2020) indicate that
the additional potential applied to the filament of $-60\text{\,}\mathrm{V}$
compresses the positron orbits towards the magnet, allowing a large fraction
to pass through the $1\text{\,}\mathrm{cm}$ gap between the filament assembly
and the magnet. Consider points 3 and 4 in Figures 5 and 6. When positrons are
injected with these steering coil current settings, their orbits are closest
to the magnet (Figure 6) and are located inside the hollow feature in Figure 5
c). Figure 7 depicts particle simulations for the experimental injection
scenario where the positrons are injected at radial positions close to the
magnet, with the filament assembly biased to $0\text{\,}\mathrm{V}$ and
$-60\text{\,}\mathrm{V}$, respectively. When grounded, positrons annihilate on
the filament assembly. When biased negatively, the E$\times$B drift moves the
beam inwards, allowing some of the positrons to pass it Nißl  _et al._ (2020).
As these positrons continue to toroidally drift around the magnet, they leave
the collimated fields of view of the two gamma detectors and annihilate at an
unobserved position.

Figure 7: Simulations of injected positron trajectories. a) filament assembly
grounded b) filament assembly biased to $-60\text{\,}\mathrm{V}$. Positrons
pass through the gap between magnet and filament when it is biased.

Further support for this explanation for the hollow feature comes from a
similar experimental campaign, during which a third detector C was installed
on the equatorial plane of the magnet behind wall sector 8 aimed at observing
gammas from positrons annihilating on the target probe at $None$, when it was
inserted. Due to space constraints, however, detector C’s field of view was
wider than that for detector A and also included the magnet and space charge
filament (Figure 8 a. The field of view of this detector included the magnet,
and the sector 5 filament. The positron intensity during this campaign was
lower ($1.5\text{\times}{10}^{7}\text{\,}{\mathrm{\text{ $e$
}}}^{+}\text{\,}{\mathrm{s}}^{-1}$) and the counting time was
$1\text{\,}\mathrm{s}$. Figure 8 compares the annihilation counts from the two
detectors (A and C), acquired simultaneously while $\mathrm{I}_{\mathrm{R}}$
and $\mathrm{I}_{\theta}$ were scanned. The data from detector A (Fig. 8a)
again shows the hollow region when the space charge filament is biased (but
not emitting). The data from detector C, however, shows an approximately
inverted pattern; the highest count rates at this detector correspond to
regions of suppressed counts at detector A. This means that the injection
process for positrons remains efficient when the filament is biased
negatively, but for positron orbits close to the magnet, annihilation takes
place outside the field of view of detector A and most likely on the magnet or
near the E$\times$B plates, where they are observed with detector C.

Figure 8: Annihilation data acquired in a similar campaign with lower positron
intensity. a) detector A. The "hollow" area in the top left corner is also
visible here ($-60\text{\,}\mathrm{[}$V_acc=], $0\text{\,}\mathrm{[}$I_fil=],
$150\text{\,}\mathrm{[}$V_E ×B=]). b) detector B: The histogram appears like
an inverted version, indicating that the injection process is not halted by
the negative filament, but rather positrons annihilate at different locations
($-60\text{\,}\mathrm{[}$V_acc=], $0\text{\,}\mathrm{[}$I_fil=],
$150\text{\,}\mathrm{[}$V_E ×B=]).

###  V.2 Effect of the Electron Space Charge

When applying the heating current of $4.0\text{\,}\mathrm{[}$I_fil=] to the
electron source (Figure 5 d), the range of steering coil currents resulting in
successful injection and $None$ transit around the magnet is extended towards
positive $\mathrm{I}_{\mathrm{r}}$ values. Further, the maximum number of
annihilation counts observed is not altered, indicating that even though the
two additional potentials created by the bias on the filament and by the space
charge are significantly higher than the positron beam energy, the injection
efficiency remains nearly lossless. This finding is of major importance for
the future APEX project, as it demonstrates experimentally that the presence
of an electron cloud affects but does not prohibit the injection of positrons;
if anything, it improves it for some injection conditions.

To further verify that this effect truly originates from the presence of
electrons, a measurement investigating the correlation between the emission
current of the filament and the number of registered annihilation events on
the filament probe was conducted. The points 1 and 6 (Figure 5 a,b), are
selected as steering coil current settings. The filament temperature threshold
for electron emission is reached for a heating current above
$3.0\text{\,}\mathrm{[}$I_fil=]A; even though the filament is biased and
heated, significant electron emission does not take place for heating currents
less than this value. Increasing the heating current from
$3.0\text{\,}\mathrm{A}$ to $4.0\text{\,}\mathrm{A}$ in steps of
$0.05\text{\,}\mathrm{A}$ increases the emission current from
$0.0\text{\,}\mathrm{mA}$ to $0.7\text{\,}\mathrm{mA}$. By measuring the
correlation between the emission current (which establishes the space charge)
and the annihilation signal from the filament probe, we can largely separate
the effects of the bias on the filament (which changes very little as the
heating current varies from $3.0\text{\,}\mathrm{A}4.0\text{\,}\mathrm{A}$)
from the effects due to the presence of the space charge.

Most injection parameter settings are not affected by the presence of the
electron cloud (inside the black contour in Figure 5 d)) such as Point 6,
where the count rates remain almost constant as the emission current is
increased (Figure 10). By contrast, without an electron cloud, positron
injection is ineffective for the steering coil settings represented by Point
1. Increasing the emission current and therefore the negative space potential
increases the annihilation signal (Figure 10). This effect saturates for
emission currents larger than $0.15\text{\,}\mathrm{[}$I_emission=],
corresponding to a filament heating current of
$3.65\text{\,}\mathrm{[}$I_fil=] and a maximum space potential of
approximately $-20\text{\,}\mathrm{V}$ (Figure 4).

Figure 9: Annihilation count rate as function of emission current at Point 6
(in Figure 5a), $150\text{\,}\mathrm{[}$V_E ×B=] (detector A (blue) and
detector B (red)). The registered counts remain roughly constant, as injection
for this steering current configuration is not affected by the electron cloud.

Figure 10: Annihilation count rate as function of emission current at Point 1,
$150\text{\,}\mathrm{[}$V_E ×B=]. $150\text{\,}\mathrm{[}$V_E ×B=] (detector A
(blue) and detector B (red)). The rising countrate shows that the electron
cloud enhances the injection efficiency for these injection parameters.

##  VI Conclusions

The prototype permanent magnet trap was used to demonstrate on of the
techniques required to generate an electron–positron pair plasma in a
levitated dipole trap in the near future. An important step towards this goal
is the injection of low–energy positrons into an electron cloud. The magnetic
drifts of the two species propagate in opposite directions around the magnet.
As their starting points in the trap are $None$ apart from each other, their
orbits overlap, and the influence of the electron cloud on the injection of
positrons is investigated. The thermionic electron source generates a negative
space potential up to $-42\text{\,}\mathrm{V}$. The electrostatic potentials,
produced by the bias voltage on the electron source and the space charge
exceed the longitudinal positron energy and the remaining electrostatic
potentials in the trap significantly. Nevertheless, 100% injection efficiency
is preserved. What initially appeared to be a ’hole’ in the region of
parameter space for lossless injection is instead found to be caused by deeply
injected positrons that drift between the filament assembly (doubling as an
annihilation target) and the magnet, then annihilating outside the field of
view of the detectors. Further, the parameter range of efficient positron
injection is extended noticeably. The method used here for creating a dense
electron cloud is not compatible with our ultimate plans for creating
electron-positron plasma. At this time, we are considering at least two
approaches which will be tested in future experiments: 1) injecting electrons
from a source placed at the edge of the confinement region as in Saitoh, _et
al._ Saitoh  _et al._ (2010), and 2) injection from an upstream source that
brings electrons into the trap along a path similar to that of the positrons.

In summary, we have demonstrated successful injection of positrons in the
prototype dipole trap in the presence of an electron space charge. The
measured electron density is comparable to the target density of the APEX
electron–positron pair plasma. Such a result was by no means assured, given
the E$\times$B drift technique we employ to inject positrons. It might have
been the case that the electron plasma shielded the fringing fields from the
E$\times$B plates in a way that prevented positron injection. The results
therefore represent an important validation of our approach to achieving a
pair plasma.

##  VII Acknowledgments

The measurements described in this work were performed at the NEPOMUC positron
beam facility installed at the FRM II at the Heinz-Maier-Leibnitz Zentrum
(MLZ), Garching, Germany. Funding was received from the European Research
Council (ERC-2016-ADG no. 741322), the Deutsche Forschungs Gesellschaft (Hu
978/15, Hu 978/16, Sa 2788/2), the Max-Planck-Institute for Plasma Physics
(IPP), the Helmholtz Association and the UCSD Foundation.

##  VIII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  IX References

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