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

Also at ]State Key Laboratory of Precision Measurement Technology and
Instruments, Department of Precision Instrument, Tsinghua University, Haidian,
Beijing 100084, China

# An isotropic antenna based on Rydberg atoms

Shaoxin Yuan  Mingyong Jing<EMAIL_ADDRESS>[  Hao Zhang  Linjie Zhang
<EMAIL_ADDRESS>Liantuan Xiao  Suotang Jia  State Key Laboratory of Quantum
Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi
University, Taiyuan, Shanxi 030006, China  Collaborative Innovation Center of
Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China

###### Abstract

Governed by the hairy ball theorem, classical antennas with isotropic
responses to linearly polarized radio waves are unrealizable. This work shows
that the antenna based on Rydberg atoms can theoretically achieve an ideal
isotropic response to linearly polarized radio waves; that is, it has zero
isotropic deviation. Experimental results of isotropic deviation within 5 dB,
and 0.3 dB achievable after optimization, in microwave and terahertz wave
measurements support the theory and are at least 15 dB improvement than the
classical omnidirectional antenna. Combined with the SI traceable and
ultrawideband property, the ideal isotropic response will make radio wave
measurement based on atomic antenna much more accurate and reliable than the
traditional method. This isotropic atomic antenna is an excellent example of
what a tailored quantum sensor can realize, but a classical sensor cannot. It
has crucial applications in fields such as radio wave electrometry.

##  I Introduction

Metrology is the mother of science; the advancement of technology is
inseparable from precision measurement. The precision measurement of many
physical quantities has rapidly developed in recent years, and a typical
example is the measurement of time/frequency [1]. However, things go awry when
measuring the radio frequency (RF) electric field (E-field). An ideal antenna
with an isotropic reception pattern is crucial in the precision
measurement/calibration of E-field [2, 3]; otherwise, even a mere change in
how they are held can introduce significant measurement errors.

Unfortunately, isotropic antennas that can provide full spatial coverage are
currently fictional and theoretically cannot be realized based on the
classical metal antenna. This is because the transverse electric field in the
far field region cannot be uniform over a sphere if the field is linearly
polarized everywhere, or more basically, is governed by the hairy ball
theorem: An even dimensional topological sphere $\mathbb{S}^{n}$ does not
admit any continuous vector field of non-zero tangent vectors [4, 5, 6].
According to the antenna reciprocity theorem [7], the hairy ball theorem
applies to both receiving and radiating antennas. Scientists have conducted
many studies based on classical techniques to realize null-free quasi-
isotropic antennas, including folded dipoles [8, 9, 10], magnetic dipoles
[11], orthogonal dipoles [12, 13, 14], etc. However, they can only respond
approximately isotropically, are complex in structure [15], and are sensitive
to wavelength [9, 11]. In the classical case, the only way to reduce the error
due to non-isotropic response is to calibrate the antenna; however, the
calibration faces a causal dilemma [16, 17].

Here we introduce another alternative path: an isotropic antenna based on
Rydberg atoms [18, 19, 20], which theoretically allows an ideal isotropic
response to a linearly polarized field to be measured. The most significant
difference between the classical antenna and the atomic antenna is that the
classical antenna is made by conducted material, in which electrons are
delocalized and can move or distribute freely in the three-dimensional
Euclidean space to form a macroscopic charged topological 2-sphere
$\mathbb{S}^{2}$. In contrast, the atomic antenna is constructed by many
Rydberg atoms. The electron of Rydberg atoms is gravitated and localized
around the atom; thus, it can only be regarded as a single point and cannot
form a topological surface. Therefore, the hairy ball theorem does not apply
to atomic antennas, making the theoretical isotropic response possible. The
following sections will give more rigorous theoretical analysis and
experimental verification. This ideal isotropic antenna has essential
applications in electric field calibration [21, 22, 23] and some modern
communications where full spatial coverage and/or uniform signal reception is
required [12, 24].

##  II Intuitive Theory of Isotropic Atomic antenna

We use an intuitive theory to illustrate the working principle of the
isotropic atomic antenna. This intuitive theory only considers that the RF
field couples two Rydberg energy levels and constructs the Hamiltonian of the
light-atom interaction. The eigenanalysis of the interaction Hamiltonian shows
that the eigenvalue of this Hamiltonian is unaffected by the polarization
direction (equivalent, direction of propagation) of the RF field under the
configuration that meets the requirements of the application scenario.

Figure 1: Schematic of intuitive theory. a). Schematic of RF E-field vector
(polarization) orientation. The Z-axis is chosen as the quantum axis. $\chi$
(inclination) is the angle between the RF field polarization and the quantum
axis. $\theta$ (azimuth) is the angle between the X-axis and the orthogonal
projection of the RF field polarization on the XY plane. b). Level diagram of
intuitive theory, $\ket{g}$ refers to the first Rydberg state with an angular
momentum of $J=1/2$, and $\ket{e}$ refers to the second Rydberg state with an
angular momentum of $J=3/2$. The RF fields couple each sublevel of $\ket{g}$
and $\ket{e}$ through $\sigma^{\pm}$ or $\pi$ transition.

As shown in Fig. 1(b), the two energy levels of the atom have angular momentum
of $J=1/2$ and $J=3/2$, may correspond to actual Rydberg levels of $nS_{1/2}$
and $n^{\prime}P_{3/2}$, respectively. The atomic state with $J=1/2$ allows
two sub-energy levels with $m_{j}=-1/2$ and $1/2$, and the atomic state with
$J=3/2$ allows four sub-energy levels with $m_{j}=-3/2$, $-1/2$, $1/2$, and
$3/2$. According to the transition selection rules, the RF field couple these
sub-levels through $\pi$ and $\sigma^{\pm}$ transitions with $\delta m_{j}=0$
and $\pm 1$. The strength of the individual transitions depends on the
orientation of RF field polarization. Figure 1(a) shows the orientation of the
RF field polarization vector. We chose Z-axis as the quantum axis for
convenience, and the orientation of field polarization can be fully described
by the inclination $\chi$ and azimuth $\theta$ in the spherical coordinate.
According to this configuration, the field-atom interaction Hamiltonian in
Cartesian coordinate has the following form:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\begin{bmatrix}M_{g}&M^{\dagger}_{I}\\\
M_{I}&M_{e}\end{bmatrix},$ (1)

where $M_{g}$ represents the Hamiltonian of bare $\ket{g}$ state, $M_{e}$
represents the Hamiltonian of bare $\ket{e}$ state and $M_{I}$ represents the
Hamiltonian of the interaction. $M_{g}$, $M_{e}$ and $M_{I}$ take the form of:

$\displaystyle M_{g}$ $\displaystyle=$ $\displaystyle\begin{bmatrix}0&0\\\
0&0\end{bmatrix},$ (2)

$\displaystyle M_{e}=\begin{bmatrix}-\Delta&0&0&0\\\ 0&-\Delta&0&0\\\
0&0&-\Delta&0\\\ 0&0&0&-\Delta\end{bmatrix},$ (3)

$\displaystyle
M_{I}=\frac{\Omega}{4}\begin{bmatrix}-\sqrt{3}e^{i(\theta+\phi)}\sin\chi&0\\\
2\cos\chi&-e^{i(\theta+\phi)}\sin\chi\\\
e^{-i(\theta-\phi)}\sin\chi&2\cos\chi\\\
0&\sqrt{3}e^{-i(\theta-\phi)}\sin\chi\\\ \end{bmatrix},$ (4)

where $\Delta$ is the frequency detuning, $\sqrt{6}\Omega$ is the reduced Rabi
frequency of the RF field, and $\phi$ is the relative phase angle between two
field polarization components, one of which along the quantum axis and the
other along the XY plane.

This Hamiltonian gives six eigenvalues ($\lambda_{1}$ to $\lambda_{6}$) of

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle\lambda_{2}=-\Delta$
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\Delta+\sqrt{\Delta^{2}+\Omega^{2}(1+\sin\chi\cos\chi\sin\phi)})$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\Delta-\sqrt{\Delta^{2}+\Omega^{2}(1+\sin\chi\cos\chi\sin\phi)}).$
(5) $\displaystyle\lambda_{5}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\Delta+\sqrt{\Delta^{2}+\Omega^{2}(1-\sin\chi\cos\chi\sin\phi)})$
$\displaystyle\lambda_{6}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\Delta-\sqrt{\Delta^{2}+\Omega^{2}(1-\sin\chi\cos\chi\sin\phi)})$

Consider the specific application scenario: measuring the amplitude of a
linear polarized RF field with known frequency, the $\Delta$ will be known,
and the relative phase angle between different polarization components should
be zero, i.e., $\phi=0$. Then the eigenvalues will reduce to

$\displaystyle\lambda_{1}$ $\displaystyle=\lambda_{2}$ $\displaystyle=-\Delta$
$\displaystyle\lambda_{3}$ $\displaystyle=\lambda_{5}$
$\displaystyle=-\frac{1}{2}(\Delta+\sqrt{\Delta^{2}+\Omega^{2}}),$ (6)
$\displaystyle\lambda_{4}$ $\displaystyle=\lambda_{6}$
$\displaystyle=-\frac{1}{2}(\Delta-\sqrt{\Delta^{2}+\Omega^{2}})$

and the energy difference between $\lambda_{3(5)}$ and $\lambda_{4(6)}$ is

$\Delta_{\rm AT}=\sqrt{\Delta^{2}+\Omega^{2}},$ (7)

where $\Delta_{\rm AT}$ can be direct measured by Autler-Townes (A-T)
splitting spectrum in experiments [25, 26]. With known of $\Delta$ and
$\Delta_{\rm AT}$, the following measurement equation can obtain the field
amplitude:

$E_{\rm MW}=\frac{\hbar}{\mu}\sqrt{\Delta_{\rm AT}^{2}-\Delta^{2}}.$ (8)

Equation 8 shows that $\chi$ and $\theta$ do not affect measured results; that
is, the incident direction of the RF field does not affect the measured field
amplitude by the atomic antenna and the atomic antenna has an isotropic
response. Equation 8 also shows the SI-traceable feature unique to the atomic
antenna [18]: Its measurement equation only involves frequency measurement and
can be directly traced to the Planck’s constant through the transition dipole
moment, which can be accurately calculated through the Alkali.ne Rydberg
Calculator (ARC) package [27]. Here we define the normalized receive-gain
($G_{\rm norm}$) of atomic antenna in decibels as

$G_{\rm norm}=20\log\frac{\Delta_{\rm AT}/E_{\rm MW}}{{\rm max}(\Delta_{\rm
AT}/{E_{\rm MW}})},$ (9)

where ${\rm max}(x)$ is the the largest value of set $x$. Theoretically, for
the atomic antenna, the normalized gain is a constant of 0 dB.

##  III Experimental Results

We then perform a demonstrative experiment to further illustrate the atomic
antenna’s isotropic response. In the experiment, the energy differences
between dressed states formed by the interaction of the RF field and Rydberg
states, i.e., $\Delta_{\rm AT}$, are obtained by measuring the transmission
spectrum of the probe laser. Figure 2(a) shows the experimental setup. The
linearly polarized probe laser and coupling laser coincide inside a room-
temperature cesium vapor cell to prepare the cesium atoms into the Rydberg
state by two-step excitation. The probe laser is finally incident into a
photodetector so that its transmission spectrum can be recorded when the
frequency of the coupling laser is scanned. The probe laser and the coupling
laser have opposite propagation directions along X-axis to minimize the effect
of Doppler mismatch on the probe laser’s transmission spectrum. The linear
polarized RF field is irradiated by a horn antenna into the vapor cell, and
its incident or polarization direction is adjustable by changing the position
and angle of the antenna.

Figure 2: Overall experimental configuration. a). Experimental setup. The red
beam represents the 852 nm probe laser, and the green beam represents the 510
nm coupling laser. HR: High Reflection dielectric mirror. b). Energy level of
terahertz measurement. c). Energy level of MW measurement. d). An example of
EIT spectrum and A-T splitting spectrum, measured in terahertz measurement
configuration.

We use two energy level configurations, one enables the measurement of RF
fields in the terahertz band, and the other enables the measurement of RF
fields in the microwave band, as shown in Fig. 2(b) and 2(c), respectively. In
both configurations, we use a probe laser with a wavelength of around 852.35
nm to excite ground state atoms into the excited state of cesium D2 transition
($6S_{1/2},F=4\to 6P_{3/2},F=5$). For terahertz (microwave) measurement, a
coupling laser with a wavelength of around 511.69 (508.64) nm is used to
prepare atoms into $33S_{1/2}$ ($93S_{1/2}$) Rydberg state, and a terahertz
(microwave) field with a frequency of around 0.1296 THz (4.8 GHz) couples
Rydberg transition of $33S_{1/2}\to 33P_{3/2}$ ($93S_{1/2}\to 92P_{3/2}$).
When measuring the terahertz field, we use a borosilicate glass vapor cell
with an inner size of 20 mm cubic, and the thickness of the cell wall is 2 mm.
The inner size of the vapor cell is changed to 20*20*80 mm3 when measuring the
microwave field to increase the atomic sample’s length, which can compensate
for the reduction of the spectral brightness caused by the decrease of the
coupling laser Rabi frequency for such highly principal quantum number Rydberg
state excitation. The laser and energy level configurations in this
experimental setup enable a phenomenon of electromagnetically induced
transparency (EIT) [28, 29], and the degenerate EIT dark states will split
into two states with different energy when Rydberg levels are dressed by RF
field coupling, i.e., Autler-Townes (A-T) splitting phenomenon. Figure 2(d)
shows a typical spectrum example of EIT (red) and A-T splitting (black). The
frequency interval between A-T peaks reads out $\Delta_{\rm AT}$, and with the
known of $\Delta_{\rm AT}$ and $\Delta$, the measurement output of atomic
antenna ($E_{\rm MW}$) can be deduced according to Eq. 8.

Figure 3: The normalized receive-gain patterns of the atomic antenna and
classical omnidirectional antenna. a). Gain patterns of atomic antenna (solid
line) and classical antenna (dashed line) measured at 4.8 GHz and 4.9 GHz,
respectively. The pink shadow shows the simulation measurement uncertainty of
the 4.8 GHz field caused by cell geometry’s disturbance of field distribution.
b). Gain patterns of atomic antenna measured at 0.1296 THz (solid line). The
same gain patterns as Fig. 3(a) of classical antenna measured at 4.9 GHz is
also shown as a reference. The pink shadow shows the theoretical measurement
uncertainty of the 0.1296 THz field caused by cell geometry’s disturbance of
field distribution.

Whether the atomic antenna has an isotropic response can be verified by
measuring normalized gain patterns on three independent orthogonal planes in
space, i.e., XY, XZ, and YZ planes. Throughout the overall measurement, we
keep the distance of the radiating antenna from the cell, the power injected
into the radiating antenna the same, and keep the antenna always pointing to
the center of the cell. These ensure that the electric field strength felt by
the atoms is always consistent. When measuring the pattern on the XY plane, we
move the antenna around the cell on the XY plane for a circle. When measuring
the pattern on the XZ plane, for the convenience of the experiment, we point
the radiating antenna to the cell along the Y-axis, then rotate the antenna
around the Y-axis so that the RF field polarization at the location of the
cell rotates around the Y-axis. Since the outcome of atom-field interaction
depends only on the polarization direction of the field, this is equivalent to
moving and rotating the antenna around the cell in the XZ plane. For the same
reason and principle, when measuring the pattern on the YZ plane, instead of
moving and rotating the antenna in the YZ plane, we rotate the polarization
angle of the linearly polarized excitation laser around the X-axis.

Figure 3 shows the experimental measured normalized receive-gain patterns of
the atomic antenna. We use the isotropic deviation (gain deviation) [30] to
quantitatively describe the quality of isotropic properties of a receive-
sensor, where the isotropic deviation is defined as the difference between the
maximum and minimum values of the normalized gain in the same pattern.
According to Fig. 3(a), the isotropic deviation of the atomic antenna in XY
(blue solid line), XZ (green solid line), and YZ planes (red solid line) for
microwave field measurement is about 3.6 dB, 4.5 dB, and 2.5 dB, respectively.
The corresponding value for terahertz measurement is about 4.4 dB, 1.4 dB, and
0.28 dB according to Fig. 3(b), respectively. Both isotropic deviations for
microwave and terahertz measurements have a minimal value on the YZ plane and
are sourced from the measurement uncertainty of $\Delta_{\rm AT}$. The A-T
splitting spectrum for microwave measurement has less signal-to-noise ratio
than the terahertz measurement spectrum, which leads to much more uncertainty
in $\Delta_{\rm AT}$, thus a larger isotropic deviation in YZ plane. The
isotropic deviation on other planes is significantly increased relative to the
YZ plane, which is mainly due to two reasons: one is that the antenna is
artificially moved or rotated in these measurements, which is difficult to
guarantee high accuracy, resulting in additional measurement uncertainties;
the other is that the field is not uniformly distributed inside the vapor
cell.

To characterize the effect of field inhomogeneity, we used CST software to
simulate the distribution of RF fields inside the cell with different field
incident angles. The simulation results of field distribution inside the cubic
cell of the terahertz field are shown in Fig. 4, which show that the terahertz
field forms a standing-wave field distribution inside the 20 mm cubic cell.
The distribution pattern of the standing wave field has a pronounced change
with the incident direction of the terahertz field. Since the atomic antenna
reads out the averaged field along the excitation laser path (black line in
Fig. 4(a)), we also averaged the field amplitude along the laser propagation
path in the simulation results. After counting this average value at different
field incident angles, the deviation of the field strength can be obtained,
shown as the pink shadow in Fig. 3(a). Since the simulation and data
processing process is the same as that of terahertz, the simulation results of
the microwave distribution pattern are not shown, and only the expected
deviation caused by the inhomogeneity of field distribution is shown as the
pink shadow in Fig. 3(b).

For the current experiment, the maximum isotropic deviation of the atomic
antenna in whole space is within 5 dB, and the isotropic deviation in a single
plane can achieve as minimum as 0.28 dB. The largest isotropic deviation is
roughly within the error range due to microwave inhomogeneity given by the
simulation. At the same time, the minimal isotropic deviation in a single
plane proves that with the improvement of experimental accuracy, the measured
isotropic deviation can be further reduced and will constantly approach the
theoretically predicted value of 0 dB. It shows significant advantages over
classical antennas, of which isotropic deviation in whole space is typically
larger than 20 dB, as shown in dash lines in Fig. 3. The dash lines in Fig. 3
show measurement results of the normalized gain pattern of a classical
omnidirectional antenna. This antenna works around 4.9 GHz, and its gain
pattern is measured in a microwave anechoic chamber by rotating its direction
at a fixed receive position.

Figure 4:  Simulation results of E-field distribution inside a borosilicate
glass cell at different incident angles. The field has a frequency of 0.1296
THz. a)-j). The incident angle of the terahertz field is from 0 to 90 degrees,
with a stepping of 10 degrees.

##  IV Conclusion

In summary, we have proved that the atomic antenna can achieve perfect
isotropic response in principle. This work has conducted a rigorous analysis
of the conclusion based on intuitive theory and verified it experimentally.
Although the intuitive theory considers RF fields coupling two Rydberg states
with the angular momentum of 1/2 and 3/2, the conclusion also applies to other
energy-level structures. We verified the theoretical prediction
experimentally, achieved an isotropic deviation of less than 5 dB in a
demonstration experiment and proved that the experimental isotropic deviation
mainly comes from some solvable technical reasons, such as imperfect
experimental conditions and the non-uniform field distribution inside the
vapor cell. It can further improve to within 0.3 dB by performing the
measurement in a better environment and carefully designing the cell geometry.
Compared with the isotropic deviation of at least 20 dB of the classical
antenna, the isotropic deviation of the atomic antenna realized in this work
has an improvement of at least 15 dB. This isotropic atomic antenna is an
excellent example of what a tailored quantum sensor can realize but a
classical sensor cannot. It will have an essential application in radio
frequency electric field metrology.

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