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# Microwave spectroscopy and Zeeman effect of cesium $(n+2)D_{5/2}\rightarrow
nF_{J}$ Rydberg transitions

Jingxu Bai1 Rong Song1 Zhenhua Li1 Yuechun Jiao1,2 Georg Raithel3 Jianming
Zhao1,2 Suotang Jia1,2 1State Key Laboratory of Quantum Optics and Quantum
Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan
030006, People’s Republic of China
2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan
030006, People’s Republic of China
3Department of Physics, University of Michigan, Ann Arbor, Michigan
48109-1120, USA

###### Abstract

We report on high-resolution microwave spectroscopy of cesium Rydberg
$(n+2)D_{5/2}\rightarrow nF_{J}$ transitions in a cold atomic gas. Atoms
laser-cooled and trapped in a magnetic-optical trap are prepared in the $D$
Rydberg state using a two-photon laser excitation scheme. A microwave field
transmitted into the chamber with a microwave horn drives the Rydberg
transitions, which are probed via state selective field ionization. Varying
duration and power of the microwave pulse, we observe Fourier side-band
spectra as well as damped, on-resonant Rabi oscillations with pulse areas up
to $\gtrsim 3\pi$. Furthermore, we investigate the Zeeman effect of the
clearly resolved $nF_{J}$ fine-structure levels in fields up to 120 mG, where
the transition into $nF_{7/2}$ displays a thee-peak Zeeman pattern, while
$nF_{5/2}$ shows a two-peak pattern. Our theoretical models explain all
observed spectral characteristics, showing good agreement with the experiment.
Our measurements provide a pathway for the study of high-angular-momentum
Rydberg states, initialization and coherent manipulation of such states,
Rydberg-atom macrodimers, and other Rydberg-atom interactions. Furthermore,
the presented methods are suitable for calibration of microwave radiation as
well as for nulling and calibration of DC magnetic fields in experimental
chambers for cold atoms.

##  I Introduction

Rydberg atoms with principal quantum numbers $n\gtrsim 10$ have exaggerated
properties [1], such as long lifetimes ($\sim n^{3}$), large microwave
electric-dipole transition matrix elements ($\sim n^{2}$), enormous electric
polarizabilities ($\sim n^{7}$), and strong van der Waals ($\sim n^{11}$) and
other Rydberg-atom interactions [2, 3]. These properties are the foundation
for recent developments in single-atom manipulation [2, 4], single-photon
sources [5, 6, 7, 8], quantum information and simulation [9, 10, 11], as well
as precise quantum measurements [12, 13, 14, 15, 16, 17]. The coupling of
nuclear, $\bf{I}$, and electronic angular momenta, $\bf{J}=\bf{L}+\bf{S}$,
causes hyperfine and fine structure splittings and shifts [18, 19] [$\bf{L}$
and $\bf{S}$ denote orbital and spin angular momentum of the electron(s)]. The
high sensitivity of Rydberg atoms to microwave fields, combined with their
Zeeman and Paschen-Back effects, leads to rich spectroscopic structures. In
room temperature cells, Zhang $et~{}al.$ [20, 21] have observed the Zeeman
effect of Cs Rydberg atoms in a magnetic field. Cheng $et~{}al.$ [22] have
conducted high-precision spectroscopy of Rydberg-level splittings in an
external magnetic field with two different optical polarization combinations
of $\sigma^{+}\pm\sigma^{-}$ for probe and coupling laser beams in ladder-type
electromagnetically induced transparency (EIT). In a cesium cold-atom
experiment with a controllable magnetic field, Xue $et~{}al.$ [23] have
investigated the influence of Zeeman splittings, optical pumping, and probe-
light-induced radiation pressure in cold-atom Rydberg-EIT spectra. Recently,
laser frequencies could be stabilized to Rydberg transitions using Zeeman-
modulation Rydberg-EIT spectroscopy [24, 25]. This method is applicable to a
variety of applications in modern-physics research, including quantum
information processing [10, 11], many-body dynamics [26], and Rydberg
molecules [27].

Microwave spectroscopy of Rydberg atoms, which typically has a resolution
limit in the range of about 10 kHz, is well-suited to measure energy levels
and quantum defects of Rydberg states for which no convenient laser excitation
schemes exist, such as high-$\ell$ ($\ell$=3,4,and 5) states of rubidium [28,
29, 30, 31] and cesium [32, 33, 34]. Recently, Cardman $et~{}al.$ [35] have
measured the hyperfine structure (HFS) and Zeeman effect of $nP_{1/2}$ Rydberg
states using narrow-linewidth mm-wave spectroscopy in a cold 85Rb ensemble.

In the present work, we report on high-resolution microwave spectroscopy of
$(n+2)D_{5/2}\rightarrow nF_{J}$ transitions by scanning a weak microwave
field in a cold Cs Rydberg atom sample. In our first study we change the
microwave pulse duration and power at zero magnetic field to establish the
spectral resolution. We observe symmetric, high-contrast Fourier-sideband
spectra that demonstrate coherent excitation for microwave pulses of up to
several microseconds in duration, while at $20~{}\mu$s we reach our linewidth
limit of $\sim$140 kHz. The fine structure (FS) splitting of the $nF_{J}$
levels is clearly resolved in all cases. In the second part of our work, we
apply a controlled homogeneous magnetic field using a pair of Helmholtz coils,
and perform high-precision Zeeman spectroscopy for both FS components
$nF_{5/2}$ and $nF_{7/2}$. We observe two-peak and three-peak Zeeman splitting
patterns, respectively, in agreement with calculations. The high-resolution
Zeeman spectroscopy of high angular momentum ($\ell$=3) Rydberg states,
presented in our work, will be helpful for in-situ calibration of magnetic
fields in cold-atom and cold-molecule research, it may enhance the accuracy of
future studies of high-$\ell$ quantum defects, and it may serve well in future
quantum-control and -simulation studies involving high-$\ell$ Rydberg states.

##  II Experimental setup

Figure 1: (color online). (a) Schematic of the experimental setup. Cesium
atoms are cooled and trapped at the center of a MOT. Two excitation lasers
(wavelengths 852 nm and 510 nm) are overlapped and counter-propagated through
the MOT to prepare $(n+2)D_{5/2}$ Rydberg atoms. A tunable microwave field,
applied via a microwave horn, allows us to perform high-resolution
spectroscopy of $(n+2)D_{5/2}\to nF_{J}$ transitions. Three pairs of
orthogonal electric grids and Helmholtz coils are employed to compensate stray
electric and magnetic fields, respectively, and to scan the magnetic field in
$z$-direction for Zeeman-effect studies. Rydberg-atom electric-field
ionization and time-resolved ion detection allow us to measure Rydberg-
transition probabilities. See text for details. (b) Energy level diagram. Two-
photon excitation using the 852 nm and 510 nm lasers (respective single-photon
Rabi frequencies $\Omega_{852}$ and $\Omega_{510}$ on the $6S_{1/2}\rightarrow
6P_{3/2}$ and $6P_{3/2}\rightarrow(n+2)D_{5/2}$ transitions, and intermediate-
state detuning $\delta=330$ MHz) yields $(n+2)D_{5/2}$ Rydberg atoms. The
tunable microwave field drives $(n+2)D_{5/2}$ to $nF_{J}$ transitions.

The experimental setup is shown in Fig. 1(a). Cesium atoms are cooled and
trapped in a magneto-optical trap (MOT) with a density of $\approx
10^{10}~{}\mathrm{cm}^{-3}$ and a temperature of $\approx 100~{}\mu$K. After
switching off the MOT and waiting for a delay time of 1 ms, we apply the
852-nm and 510-nm Rydberg-excitation laser pulses, which have respective
Gaussian beam-waist parameters of $\omega_{852}$ = 750 $\mu$m and
$\omega_{510}$ = 1000 $\mu$m. The two-photon optical Rydberg-excitation Rabi
frequency in units Hz, at the beam centers, is kept at
$\Omega_{2Ph}=\Omega_{852}\Omega_{510}/(2\delta)\approx 9$ kHz, which for our
MOT atom density and optical excitation pulse length of $t=1~{}\mu$s results
in a Rydberg-atom excitation probability of
$(2\pi\Omega_{2Ph}\,t)^{2}/4\approx 0.8\times 10^{-3}$. This is sufficiently
low that Rydberg-atom interactions play no predominant role in our experiment.
The microwave field ranges in frequency from 22 GHz to 33 GHz and is emitted
by a small antenna into the chamber. Since the chamber is largely made from
metal, the calibration factor between microwave field strength and square-root
of power must be determined experimentally based upon the measured atomic
response (see Sec. III.1).

Three pairs of electric grids with symmetry points centered at the MOT are
placed on the $x$-, $y$\- and $z$-axes (in Fig. 1 we only show the pair of
grids on the $x$-axis). The grids are used to compensate stray electric fields
and to apply an electric-field ramp for state-selective electric-field
ionization and detection of the Rydberg atoms. After field compensation, the
stray electric field during laser and microwave excitation is reduced less
than 2 mV/cm [34]. The liberated ions are accelerated to reach the MCP
detector. Time-resolved ion signals, recorded with a boxcar integrator and
computer, yield the Rydberg-state populations. Furthermore, three orthogonal
pairs of Helmholtz coils are placed around the chamber to compensate stray
magnetic fields and to apply controlled magnetic fields to investigate the
Zeeman splitting of the $(n+2)D_{5/2}$ to $nF_{J}$ microwave transitions. The
stray magnetic field is less than 5 mG after compensation [34].

Fig. 1(b) shows the level diagram. The $(n+2)D_{5/2}$ state is populated via
two-photon laser excitation from $6S_{1/2}$ into $(n+2)D_{5/2}$. The two-
photon transition is blue-detuned from the intermediate $6P_{3/2}$ state by
$\delta=330$ MHz in order to reduce photon scattering and light radiation
pressure. In the experiment, the 852-nm laser is locked with modulation-free
polarization spectroscopy [36], and the 510-nm laser is locked to a Fabry-
Perot cavity with a finesse of $15\times 10^{3}$ using the Pound-Drever-Hall
technique [37]. The linewidths of both locked lasers are less than 100 kHz.
The frequency of the microwave field, which is radiated into the chamber from
a microwave horn, is scanned across selected $(n+2)D_{5/2}$ to $nF_{J}$
transitions to obtain narrow-linewidth Rydberg microwave spectra.

##  III Results and discussions

###  III.1 Fourier sideband spectroscopy

Figure 2: (color online). Measurements of microwave spectra of the $48D_{5/2}$
$\to$ $46F_{7/2}$ transition with the power of the signal generator set to
$-50$ dBm, a scan step size 100 kHz, and pulse durations of (a) 1 $\mu$s and
(b) 2 $\mu$s. Black lines show smoothed curves obtained with the Savitzky-
Golay method. We clearly observe Fourier sidebands. The spectrum in (a) is
below saturation, while the one in (b) is somewhat saturated (details see
text). (c) Spectrum at $-72$ dBm, scan step size 5 kHz, and pulse duration
$\tau=20~{}\mu$s. The Lorentzian fit (black curve) has a full width at half
maximum (FWHM) of 140 kHz. Error bars show the standard error of the mean
(SEM) of the samples.

In our first experiment, the frequency of the 510-nm laser is tuned to excite
the $48D_{5/2}$ state. Electric and magnetic fields are set to zero, within
the uncertainties of the field compensation routine. The microwave field then
drives the transition $48D_{5/2}$ $\to$ $46F_{J}$. Microwave spectra are
obtained by state-selective field ionization and gated ion detection, as the
microwave frequency is scanned across the $48D_{5/2}$ $\to$ $46F_{7/2}$
transition (center frequency 29.74323 GHz and scan ranges from 1.6 MHz to 8
MHz). In Fig. 2(a), we present a typical microwave spectrum of the transition
for a microwave pulse duration of $\tau$=1 $\mu$s. The spectrum in Fig. 2(a)
has a FWHM linewidth of 705 kHz. This approximately accords with the Fourier
limit of 890 kHz for the case of non-saturated excitation, where the
excitation spectrum is the square of a sinc-function with a FWHM of
$0.89/\tau$.

In order to reduce the linewidth, in Figs. 2(b) and (c) we increase the pulse
duration $\tau$ to $2~{}\mu$s and 20 $\mu$s, respectively. The width of the
central lobe in Fig. 2(b) agrees well with the (unsaturated) Fourier limit of
445 kHz; however, the overall width of the spectrum, including its side lobes,
is considerably wider due to saturation. The experimental resolution limit of
140 kHz is reached in Fig. 2(c), where we have reduced the Rabi frequency by
about a factor of 12.5 and increased $\tau$ by a factor of 10 relative to Fig.
2(b). The FWHM width in Fig. 2(c) is about three times larger than the Fourier
limit (which is 45 kHz for a $\tau$= 20 $\mu$s pulse). The excess width is
attributed to dipolar Rydberg-atom interactions and residual Zeeman shifts of
the different sublevels in the residual stray magnetic field (for more details
see our previous work [34]).

Figure 3: (color online) Measurements of microwave spectra of
$48D_{5/2}\rightarrow 46F_{J}$ transition as a function of microwave frequency
for the indicated microwave powers and estimated pulse areas for resonant
excitation of $46F_{7/2}$ and microwave pulse duration of 1 $\mu$s. The two
vertical gray dash-dotted lines at 29.743 GHz and 29.753 GHz mark the resonant
frequencies for $46F_{7/2}$ (left) and $46F_{5/2}$ (right), respectively. Each
spectrum is an average over 20 measurements. (e1) Contour plot of microwave
spectra as a function of microwave detuning $\Delta$ and for the power ranging
from $-62.5~{}\mathrm{dBm}$ to $-35~{}\mathrm{dBm}$ in steps of 2.5 dBm. The
four horizontal dashed lines mark the spectra of (a) (orange), (b) (blue), (c)
(red) and (d) (gray), respectively. (e2) Simulation result for (e1). (f)
Measured signal strength of the resonant transition $48D_{5/2}\to 46F_{7/2}$
as a function of microwave power, showing a damped Rabi oscillation, and fit
(see text).

Next we comment on the Fourier-sideband strengths. In the case of weak
saturation, where the microwave Rabi frequency $\Omega<|\Delta|$ on the
Fourier sidebands, the amplitude of the sidebands rapidly diminishes with
order (the first sidebands have a strength of 4.7% of the central peak, the
second sidebands have 1.6%, etc). Fig. 2(a) is near the weak-saturation limit.
With increasing saturation, where $|\Delta|\lesssim\Omega$ within at least the
first sideband, the sidebands rise in strength relative to the central peak,
as is evident in Fig. 2(b).

In order to exhibit the spectral dependence on both microwave power and
$\Delta$, as well as to estimate the degree of coherence in the excitation
process, in Fig. 3 we show a series of measured microwave spectra for
microwave powers in the range between $-35$ dBm and $-62.5$ dBm and over a
detuning range $|\Delta|\lesssim$15 MHz, for fixed $\tau=1~{}\mu$s. In Fig.
3(a-d), we present selected microwave spectra of $48D_{5/2}\rightarrow
46F_{J}$ for microwave powers as indicated. Gray vertical dashed lines mark
the resonant transition frequencies of 29.743 GHz and 29.753 GHz for
$46F_{7/2}$ and $46F_{5/2}$, respectively. Due to the interaction between the
atomic core and the valence electron, the FS of the Rydberg $F$-state is
inverted, resulting in a greater transition frequency for $46F_{5/2}$ than for
$46F_{7/2}$. Also, since angular transition matrix elements for $46F_{7/2}$
are much greater than those for $48F_{5/2}$, the peak intensity of the
$46F_{7/2}$ exceeds that of the $46F_{5/2}$ signal. The $46F_{7/2}$ line also
is considerably more susceptible to broadening due to dipole-dipole
interactions than the $46F_{5/2}$ line [34].

While the radial transition matrix elements for the $48D_{5/2}\rightarrow
46F_{7/2}$ and the $48D_{5/2}\rightarrow 46F_{5/2}$ transitions are nearly
identical, the squares of the angular matrix elements averaged over magnetic
quantum numbers differ by a factor of 20, equivalent to 13 dB in microwave
power. As a result, the onset of the $F_{5/2}$ line in Fig. 3 is shifted to
higher power relative to that of the $F_{7/2}$ line. The $48D_{5/2}\rightarrow
46F_{5/2}$ transition remains un-saturated throughout the investigated power
range and only becomes visible at powers above $-47.5$ dBm, while the
$48D_{5/2}\rightarrow 46F_{7/2}$ transition becomes saturated. Due to its
larger Rabi frequency, the $46F_{7/2}$ line develops visible Fourier side
bands at powers above about $-50$ dBm and becomes saturation-broadened [see
Figs. 3(a-e2)]. In their respective unsaturated regimes, both lines have a
FWHM of about 800 kHz, which is at the Fourier limit for $\tau=1~{}\mu$s.

The cuts through the spectrum shown in Fig. 3(a-d) correspond with approximate
resonant pulse areas of $\pi/2$, $\pi$, $2\pi$ and $3\pi$ for the
$48D_{5/2}\rightarrow 46F_{7/2}$ transition, averaged over $m_{j}$. At the
resonant frequency of the $48D_{5/2}\rightarrow 46F_{7/2}$ transition we
observe a damped Rabi oscillation, as is evident in Fig. 3(f).

In our model for the spectra in Figs. 2 and 3, we use the function,

$\sum_{m_{j}}AP(m_{j})\frac{\Omega_{j}^{2}}{\Delta^{2}+\Omega_{j}^{2}}\big{[}1-\exp(-2\pi\tau\,\gamma\,\sqrt{\Delta^{2}+\Omega_{j}^{2}}\,)\cos(2\pi\tau\,\sqrt{\Delta^{2}+\Omega_{j}^{2}})\big{]},$
(1)

where $\Delta$ is the microwave detuning in Hz, $A$ is an amplitude fit
parameter, $P(m_{j})$ are the probabilities for the $m_{j}$ sub-levels of the
$nD_{5/2}$-state (which we set equal), $\Omega_{j}$ are the $m_{j}$-dependent
microwave Rabi frequencies, and $\gamma$ is a damping parameter. For linearly
polarized microwaves and adopting the microwave electric-field direction as
quantization axis, the microwave Rabi frequencies are

$\Omega_{j}=\frac{ea_{0}R_{(n+2)D_{5/2}}^{nF_{j}}w_{D_{5/2},m_{j}}^{F_{J},m_{j}}E_{MW}}{h},$
(2)

in units Hz. There, $R$ is the radial matrix element, which equals 2564 in
atomic units for both $J$-values for $n=46$, $w$ is the angular matrix
element, and $E_{MW}$ is the microwave electric field. The $w$ for $F_{7/2}$
are 0.3499, 0.4518, 0.4949 for $|m_{j}|$= 5/2, 3/2 and 1/2, respectively.

We have modeled the spectra in Fig. 3(e) as $\Delta$\- and power-dependent
Rabi oscillations according to Eqs. (1) and (2). The exponential damping term
has a form $\propto\exp(-\Phi\,\gamma)$, where $\Phi$ is the pulse area in
rad/s, $2\pi\tau\sqrt{\Omega_{j}^{2}+\Delta^{2}}$. Good agreement is found for
a damping constant $\gamma\approx 0.088~{}$s/rad, i.e., we are able to observe
several Rabi cycles. The visibility reduction of the Rabi oscillation observed
for $\Phi\gtrsim 2\pi$ is, in part, due to the 40-$\%$ variation of the Rabi
frequencies $\Omega_{j}$ as a function of $m_{j}$. The $m_{j}$-inhomogeneity
could be addressed by optical pumping of the sample and by using proper
optical and microwave polarizations. The level of coherence in Fig. 3 would
suffice for applications in quantum control and quantum simulations using
ensembles of $\ell=2$ and $\ell=3$ Rydberg atoms.

A comparison of measured and simulated spectral maps in Fig. 3(e) allows us to
calibrate the microwave electric field at 29.743 GHz. Averaging our
calibrations performed for three pulse areas ($\pi$, $2\pi$ and $3\pi$), we
find $E_{MW}=10^{P_{MW}/20}\times 7.5~{}$V/m$~{}\pm 10\%$, where $P_{MW}$ is
the microwave power in dBm. The uncertainty reflects experimental noise and
half of our power step size of 2.5 dBm. Our simulation for the
$46F_{7/2}$-spectrum, plotted for this calibration in Fig. 3(e2), agrees very
well with the measurement, lending credence to the calibration. Similar
calibrations can be performed for other transitions. It is noted that the
calibration generally depends on the microwave frequency, because the mode
pattern inside the metal vacuum chamber must be assumed to have a complex
frequency dependence. However, the calibration is assumed to be constant over
the relatively short frequency ranges used for each set of
$(n+2)D_{5/2}\rightarrow nF_{J}$ transitions.

Figure 4: (color online). Measurements and simulations of Zeeman spectra of
the $50D_{5/2}\rightarrow 48F_{J}$ microwave transitions. In (a) and (b) we
present Zeeman spectra for $J=7/2$ at $B=$83 mG and for $J=5/2$ at 58 mG,
respectively. Red curves display the results of multi-peak Lorentzian fits.
Contour plots of larger sets of measurement results are shown in (c) and (d)
for $J=7/2$ and $J=5/2$, respectively. In panels (e) and (f) we present the
results of corresponding simulations, in which the angle between the
polarizations of the microwave and magnetic fields is set to $55^{\circ}$,
which leads to good agreement between experiment and simulations.

###  III.2 Zeeman spectroscopy

In order to obtain narrow-linewidth microwave spectra in a weak magnetic
field, we employ a microwave pulse duration $\tau$ = 20 $\mu$s in the Zeeman-
splitting measurements presented in this section. After compensation of
electric and magnetic stray fields, we apply a small current in the Helmholtz
coils on the $z$-axis to generate an uniform magnetic field at the MOT center,
which allows us to observe the Zeeman splitting of the $50D_{5/2}\to 48F_{J}$
transitions.

Figures 4(a) and (b) show Zeeman spectra for microwave fields driving the
$50D_{5/2}\rightarrow 48F_{7/2}$ and $50D_{5/2}\rightarrow 48F_{5/2}$
transitions at magnetic fields of 83 and 58 mG, respectively. It is seen that
the microwave spectra for $48F_{7/2}$ and $48F_{5/2}$ display different line
profiles, namely three Zeeman peaks for $48F_{7/2}$ and two for $48F_{5/2}$.
In the color-scale plots in Figs. 4(c) and (d) we present a wider set of
measurements for $z$-magnetic fields covering ranges of $\pm 120~{}$ mG and
$\pm 90$ mG for $48F_{7/2}$ and $48F_{5/2}$, respectively. It is confirmed
that at fields exceeding several tens of mG the $48F_{7/2}$ and $48F_{5/2}$
spectra smoothly split into three and two lines, respectively.

###  III.3 Theoretical analysis of Zeeman spectra

To understand the measured Zeeman spectra and the difference between the line
profiles for the $nF_{7/2}$ and $nF_{5/2}$ upper states, we first plot all
microwave-coupled Zeeman levels in Fig. 5(a) and calculate Zeeman shifts and
transition matrix elements for all $m_{j}$ levels [ignoring the weak Rydberg
HFS of the $(n+2)D_{5/2}$ lower state]. In a weak magnetic field, the
$(n+2)D_{5/2}$ and $nF_{J}$ states are in the Zeeman regime of the FS, and the
Zeeman shifts exhibit a linear dependence on the magnetic field. The weak
magnetic field lifts the degeneracy of the $m_{j}$ sublevels, as shown in Fig.
5(a). According to electric-dipole selection rules, there are three transition
types for changes in $m_{j}$, namely $\Delta m_{j}=-1,0,+1$, corresponding to
$\sigma^{-}$, $\pi$ and $\sigma^{+}$ microwave polarizations. In Fig. 5(a),
these are marked with red, purple and blue arrows, respectively.

Figure 5: (color online). (a) Zeeman sublevels for the $(n+2)D_{5/2}\to
nF_{7/2}$ transition in the presence of a weak magnetic field (main separation
not to scale). Red, purple and blue arrows represent the transitions $\Delta
m_{j}=-1$ ($\sigma^{-}$), $\Delta m_{j}=0$ ($\pi$) and $\Delta m_{j}=+1$
($\sigma^{+}$), respectively. The anomalous Zeeman effect for $(n+2)D_{5/2}$
$\to$ $nF_{7/2}$ and $(n+2)D_{5/2}$ $\to$ $nF_{5/2}$ produces the line patters
shown in (b) and (c), respectively. For the line strengths we have assumed
$x$\- or $z$-polarized microwave electric fields; the former drives the
$\sigma^{\pm 1}$-transitions with equal strengths, while the latter drives the
$\pi$-transitions. For the $(n+2)D_{5/2}$ $\to$ $nF_{7/2}$ transition, the six
transition shifts for each $\Delta m_{j}=0,\pm 1$ are tightly grouped and well
separated, forming a three-peak Zeeman spectrum for cases of mixed microwave
polarization. For the $(n+2)D_{5/2}$ $\to$ $nF_{5/2}$ transition, the six
$\Delta m_{j}=0$ lines blend with the five-each $\Delta m_{j}=-1$ and $\Delta
m_{j}=+1$ lines, generally forming a two-peak Zeeman spectrum, regardless of
microwave polarization. The labels $[m_{j}$,$m^{\prime}_{j}]$ mark the
transitions with the largest transition probabilities.

Since in our work the magnetic field is in the Paschen-Back regime of the
Rydberg HFS and the linear Zeeman regime of the FS, the level shifts from the
field-free positions, visualized in Fig. 5(a) are,

$\begin{split}\Delta_{j}=\mu_{B}B(m_{j}g_{j}+g_{I}m_{I}),\\\ \end{split}$ (3)

where $\mu_{B}$ is the Bohr magneton, $g_{j}$ and $g_{I}$ are the applicable
electronic and nuclear $g$ factors, and $m_{j}$ and $m_{I}$ the corresponding
magnetic quantum numbers. For our $(n+2)D_{5/2}$ and $nF_{J}$ Rydberg states
of cesium the nuclear term is negligible. The Landé $g$-factors for the
electronic Zeeman shifts of the relevant states,

$g_{j}\simeq 1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)},$ (4)

are $g_{D_{5/2}}$=1.2, $g_{F_{7/2}}$=1.1433, and $g_{F_{5/2}}$=0.8572. The
transition shifts for $\Delta m_{j}=-1,0,+1$ can then be expressed as,

$\begin{split}\Delta(\sigma^{-})&=\mu_{B}B\,[m_{j}(g_{F_{J}}-g_{D_{5/2}})-g_{F_{J}}]\\\
\Delta(\pi)&=\mu_{B}B\,m_{j}(g_{F_{J}}-g_{D_{5/2}})\\\
\Delta(\sigma^{+})&=\mu_{B}B\,[m_{j}(g_{F_{J}}-g_{D_{5/2}})+g_{F_{J}}].\end{split}$
(5)

In Fig. 5(a) we plot the level scheme, and in Figs. 5(b) and (c) the relative
transition strengths, $|w_{D_{5/2},m_{j}}^{F_{J},m^{\prime}_{j}}|^{2}$, versus
the line positions from Eq. (5). As seen in Fig. 5(a), for $(n+2)D_{5/2}\to
nF_{7/2}$ there are six transitions for every $\Delta m_{j}$. Due to the small
difference of $0.07$ between the lower- and upper-state $g_{j}$-factors,
transitions with same $\Delta m_{j}$ cluster into tight groups, as is evident
from Eq. (3) and seen in Fig. 5(b). For mixed microwave polarization, this
leads to three peaks in the microwave spectra, in agreement with Figs. 4(a,
c). The small difference in $g_{j}$-factors in combination with residual stray
magnetic fields, as well as quite strong dipolar “flip” interactions between
atom pairs in $(n+2)D_{5/2}$ and $nF_{7/2}$ [34], lead to line broadening on
the order of 100 kHz that keeps us from reaching the Fourier limit of the
linewidth at zero $B$ field in Fig. 4(c). Notably, the three $\Delta
m_{j}$-resolved lines at non-zero $B$-field are considerably narrower than the
(single) line at $B$ near zero, pointing to dipolar interactions as the main
cause for the line broadening at $B$ near zero.

Next, we consider the transition $(n+2)D_{5/2}\to nF_{5/2}$. In this case, the
$g_{J}$-factors between the lower and upper levels differ by $0.343$, and the
five $\Delta m_{j}=\pm 1$ and six $\Delta m_{j}=\pm 0$ transitions split by
considerably more than in the transition $(n+2)D_{5/2}\to nF_{7/2}$, leading
to a larger anomality of the Zeeman effect. Factoring in the distribution of
oscillator strength, the overall line pattern features two lines, regardless
of microwave polarization, and only the width of the two lines depends on the
polarization. The insensitivity of the $(n+2)D_{5/2}\to nF_{5/2}$ Zeeman
pattern to microwave polarization strongly differs from $(n+2)D_{5/2}\to
nF_{7/2}$, where the Zeeman pattern strongly depends on microwave
polarization. This behavior, which is evident from Figs. 5(b) and (c), is a
result of the symmetry between line shifts and transition strengths, seen
within the $\Delta m_{j}=0$ group as well as between the $\Delta m_{j}=\pm 1$
groups. In addition, for the case of $(n+2)D_{5/2}\to nF_{5/2}$ it also is
important that $\pi$-transitions are strongest for large $|m_{j}|$, while the
$\sigma^{\pm}$-transitions are strongest for small $|m_{j}|$ [see transition
labels in Figs. 5(b) and (c)]. Further, the line broadening of the transition
$(n+2)D_{5/2}\to nF_{5/2}$ near zero $B$-field is considerably less than that
of the transition $(n+2)D_{5/2}\to nF_{7/2}$. We attribute this difference to
the relatively low strength of dipolar “flip” interactions between atom pairs
in $(n+2)D_{5/2}$ and $nF_{5/2}$ [34].

For a comprehensive comparison between theory and measurement, we have
simulated the Zeeman spectrum using the following principles. (1) Ground-state
$6S_{1/2},m_{j}$ populations are isotropic. (2) Initial Rydberg-state
$m_{j}$-populations in $(n+2)D_{5/2}$ are proportional to the squares of two-
photon matrix elements for the $6S_{1/2},m_{0,j}$ to $(n+2)D_{5/2},m_{j}$ two-
photon transitions for the given optical polarizations. This is justified by
the small, far-from-saturation Rydberg excitation probabilities in our
experiment. (3) Microwave Rydberg-transition probabilities are proportional to
$|w_{D_{5/2},m_{j}}^{F_{J},m^{\prime}_{j}}|^{2}$. In Fig. 4 this is justified
because microwave powers were chosen well below saturating the Rydberg
transitions. (4) All Zeeman Rydberg lines are convoluted with Gaussians of 100
kHz FWHM and are summed up to arrive at a net spectrum. (5) For the optical
transitions, all HFS and FS effects are included. (6) For the microwave
transitions, all HFS and FS effects are also included. However, the HFS of the
$(n+2)D_{5/2}$ only has a minimal effect in the range of 10 kHz shifts, while
the HFS of the $nF_{J}$ levels is entirely negligible. The simulated microwave
Zeeman spectra are displayed in Figs. 4(e) and (f) for the
$50D_{5/2}\rightarrow 48F_{7/2}$ and $50D_{5/2}\rightarrow 48F_{5/2}$
transitions, respectively. The simulation reproduces our experimental
measurements well, including the two-peak profile for the
$50D_{5/2}\rightarrow 48F_{5/2}$ transition and the three-peak profile for the
$50D_{5/2}\rightarrow 48F_{7/2}$ transition. For the latter, it was necessary
to assume an $\approx 55^{\circ}$ microwave polarization angle relative to the
magnetic-field direction to achieve agreement. Such an angle is quite
plausible due to the fact that the microwave is radiated into a complex metal
chamber. Comparing measured data and simulated data for a range of
polarization angles (not shown), the uncertainty of the polarization angle is
determined to be about $\pm 5^{\circ}$. Further, there is a small shift in the
simulation at zero magnetic field and near zero microwave detuning. This shift
may be attributed to the HFS of $nD_{5/2}$, which is included in the
simulation but is otherwise beyond the scope of our present work.

##  IV Summary

We have reported the observation of high-resolution microwave spectra of
$(n+2)D_{5/2}\rightarrow nF_{J}$ transitions in cold cesium atom clouds. For
20-$\mu$s long microwave pulses the observed spectral linewidth was 140 kHz,
while for 1-$\mu$s pulses coherent, Fourier-limited spectra with pulse areas
up to about $3\pi$ were achieved, in agreement with a model that takes damping
into account. We have further studied the Zeeman effect of the fine-structure-
split $(n+2)D_{5/2}\rightarrow nF_{J}$ transitions in fields up to about 100
mG. Results agree well with our theoretical models, which have provided
detailed insight into the spectroscopic structure and the microwave electric-
field strength and polarization within the atom-field interaction region.

In the future, Rydberg quantum simulators may benefit from our demonstrated
coherent drive of relatively strong, high-$\ell$ transitions. Also, the method
allows for microwave field-strength calibration and polarization estimation,
which is useful for metal chambers in which computational approaches are
prohibitive for field-mode estimation. Furthermore, the work is helpful to
null and to calibrate DC magnetic fields in complex vacuum chambers utilizing
optical methods and calibration-free atomic spectroscopy. Future simulation
work for higher optical and microwave powers will likely require sophisticated
methods to solve quantum master equations suitable to treat problems with many
quantum states, such as quantum Monte Carlo wave-function simulations (QMCWF;
see our previous work in [23]).

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grants No. 61835007, 12120101004, 62175136, 12241408); the Scientific
Cooperation Exchanges Project of Shanxi Province (Grant No. 202104041101015);
the Changjiang Scholars and Innovative Research Team in Universities of the
Ministry of Education of China (Grant No. IRT 17R70); and the 1331 project of
Shanxi Province. G.R. acknowledges support by the University of Michigan.

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