2021

[1]Sergej Markmann

1]Institute for Quantum Electronics, ETH Zürich, Auguste-Piccard-Hof 1,
Zürich, 8093, Zürich, Switzerland

# Frequency chirped Fourier-Transform spectroscopy

<EMAIL_ADDRESS>Martin Franckié  Mathieu Bertrand  Mehran Shahmohammadi
Andres Forrer  Pierre Jouy  Mattias Beck  Jérôme Faist  Giacomo Scalari  [

###### Abstract

Fast (sub-second) spectroscopy with high spectral resolution is of vital
importance for revealing quantum chemistry kinetics of complex chemical and
biological reactions. Fourier transform (FT) spectrometers can achieve high
spectral resolution and operate at hundreds of ms time scales in rapid-scan
mode. However, the linear translation of a scanning mirror imposes stringent
time-resolution limitations to these systems, which makes simultaneous high
spectral and temporal resolution impossible. Here, we demonstrate an FT
spectrometer whose operational principle is based on continuous rotational,
rather than linear, motion of the scanning mirror, decoupling the spectral
resolution from the temporal one. This enables 0.5 cm-1 resolution on sub-ms
time scales. Furthermore, we show that such rotational FT spectrometers can
perform dual-comb spectroscopy with a single comb source, since the Doppler-
shifted version of the comb serves as the second comb. In this way, we combine
the advantages of dual-comb and FT spectroscopy using a single quantum cascade
laser frequency comb as a light source. Our technique does not require any
diffractive or dispersive optical elements and hence preserve the
Jacquinot’s-, Fellgett’s-, and Connes’-advantages of FT spectrometers. The
system supports a large optical bandwidth from visible to THz frequencies. The
combination of a rotational delay line with collimated coherent or non-
coherent light sources pave the way for FT spectrometers in applications where
high speed, large optical bandwidth, and high spectral resolution are desired.

###### keywords:

Fourier-Transform spectroscopy, frequency combs, Laser spectroscopy, High
Resolution Spectroscopy, rotational optics

##  1 Introduction

Real-time monitoring of non-repetitive events is currently a rapidly
developing area in the field of instrumentation schmidt_chapter_2020 . Rapid
optical analysis in applications such as chemical reaction monitoring
fleisher_mid-infrared_2014 , protein dynamics norahan_microsecond-
resolved_2021 , micromachining cheng_review_2013  or phase transitions in
novel material systems ratri_phase-transition_2013  is of uttermost importance
to understand the underlying complex dynamics. A general purpose instrument
for spectroscopic characterisation is a Fourier-Transform (FT) spectrometer.
However, the scan rate of the traditional FT spectrometer is often too slow
when compared to the timescale of the physical processes of interest
wagner_time-resolved_2011 . This is due to the fact that the information is
acquired via the linear displacement of a mirror, which results in a trade-off
between the temporal and spectral resolutions given that the system is not
limited by the signal-to-noise ratio. In the last two decades two different
directions appeared in the technological development of spectrometers. On one
hand, the miniaturisation of spectrometers yang_miniaturization_2021  which is
very promising for cost-effective lab-on chip systems souza_fourier_2018 ;
zheng_microring_2019 ; yang_miniaturization_2021 ; fathy_-chip_2020 ; li_-
chip_2021 ; zhang_parametric_2013  for specialized industrial applications. On
the other hand, several new operational principles have been proposed and
demonstrated with the goal of eliminating the above-described disadvantages of
traditional benchtop FT spectrometers. Recently, successful proof-of-concept
of phase-controlled FT spectroscopy hashimoto_phase-controlled_2018-1  and
time-strech spectroscopy kawai_time-stretch_2020 ; goda_dispersive_2013 ;
mahjoubfar_time_2017  have been demonstrated, which outperform the acquisition
speed of traditional FT spectrometer by several orders of magnitude, while
maintaining a relatively high spectral resolution. However, these newly
developed methods rely on dispersive elements such as gratings, dispersive
fibers, or prisms. Hence, they can be operated only over a limited optical
bandwidth, mainly at visible and near-infrared frequencies. Although these
spectral regions are of interest for certain applications, the most relevant
one for molecular spectroscopy is the so-called molecular fingerprint region
which covers mid-infrared (mid-IR) wavelengths from 4 to 12 $\mu$m. Recent
progress in dual-comb spectrometers with mid-IR quantum cascade lasers (QCLs)
hugi_mid-infrared_2012  have gained a lot of attraction. Such a system
acquires information not only on microsecond times scales, but can also be
used for high-resolution spectroscopy via spectral interleaving villares_dual-
comb_2014  with high brightness which is very important for highly absorbing
samples. However, there is a fundamental issue, which is the requirement on
coherence for both frequency combs. The lack of mutual coherence of two free
running lasers results in additive noise and the absence of an absolute
frequency reference coddington_dual-comb_2016 . The latter can be overcome, as
has been recently shown in gianella_frequency_2022 , at the cost of additional
spectrometer complexity. These technological hurdles raise the question of
whether it is possible to perform dual-comb spectroscopy with a single
frequency comb source, without sacrificing spectral resolution and still
keeping an acquisition speed sufficient for the majority of high-speed
applications (typically 0.1-100’s of ms).
This question naturally leads one to consider the incorporation of the
frequency comb source into a FT spectrometer with the goal to preserve the
advantages of a dual-comb system and the flexibility of the FT spectrometer.
Such integration has been realized in ref. mandon_fourier_2009 . Even though
the comb modes (mode spacing of 140 MHz) were not resolved in this work due to
the limitation of the mechanical delay line (resolution of 1.5 GHz), and the
interferogram acquisition was only slightly below 5 minutes, it demonstrated
the high potential of this approach.
To address the growing demand for real-time spectroscopy in the NIR and mid-IR
(2um-14um), the temporal and spectral performance of existing FT spectrometers
has to be improved. Several optical delay line systems have been developed
which are based on a rotational delay line, such as a circular involute stage
xu_circular_2004  and curvilinear reflectors skorobogatiy_linear_2014 ,
kim_high_2008 . These two systems have in common that the optical path
difference is angle dependent, obtained via rotational optics which guide the
incoming beam to a static reflector which self reflects the beam back. Hence,
in such systems the incoming and out going beam are anti-parallel, which will
result in large undesired feedback into the laser source. Moreover, these
systems do not support extended light beams and hence posse a phase-dependent-
delay across the beam diameter, which make them impractical for usage in the
FT spectrometer. On the other hand, a compact flexible multi-pass rotary delay
line using spinning micro-machined mirrors have been developed
fleddermann_compact_2017 . The big disadvantage of these systems is relatively
small achievable optical path difference, which is typically less than 1 mm,
and the requirement of fiber optics and optical circulators, thus limiting
this system to a narrow optical bandwidth. Another interesting approach was
shown in ref. kim_novel_2010  where a rotating retro-reflector in combination
with a static plain mirror is used. The incoming beam which is guided via a
rotational retro-reflector to the plane mirror is back reflected. The
disadvantage of this system lies in a not separable incoming and outgoing
beams. Moreover, the system size grows linearly with the increase of the
optical achievable delay path. This make the system impractical for usage of
desired delay lengths of 3 cm and more at large acquisition speeds.
By taking the advantages and disadvantages of the above described optical
delay systems into account, we can define criteria which the desired system
should fulfill: (I) Support of a large optical bandwidth; hence dispersive
elements have to be avoided. (II) High spectral resolution; 0.5 cm-1 or lower.
(III) Preserved polarisation of the light source. (IV) Fast acquisition
speeds: 0.1-100’s of ms. (V) Easy integration into existing systems and hence
small footprint. (VI) Decoupled temporal and spectral resolution as well the
decoupling of the incoming and outgoing beam. (VII) Support of extended beam
diameter size.
In this work we demonstrate an FT spectrometer based on a rotational delay
line and a single frequency comb light source, which fulfills the conditions
listed above and overcomes many of the difficulties of dual-comb and linear FT
spectroscopy. In conventional FT spectrometers, which are based on linear or
pendulum motion of a scanning mirror, the direction of the moving mirror has
to be reversed. This sets the maximum achievable acquisition speed for a given
spectral resolution, due to the mirror inertia which would require large
external forces that scale quadratically with the acquisition time. The use of
a rotational element decouples the spectral resolution from the interferogram
acquisition time. This is due to the fact, that the spectral resolution is
fixed and is given by the total optical path length of the rotational delay
line. The temporal resolution is given by the angular velocity of the
rotational optical element and is independent on the total optical path
length. With our rotational FT spectrometers we can reach ms acquisition
speeds which can be improved by further technological development as will be
discussed later. Such a high acquisition speed results in a high signal-to-
noise-ratio (SNR), due to minimization of 1/f noise within the signal
bandwidth which is crucial for a free running mid-IR QCL
cappelli_intrinsic_2015 , tombez_linewidth_2012 . The combination of our
rotational FT spectrometer with the frequency comb as a light source does not
limit our system in spectral resolution given by the total optical path
length. As a proof-of-principle, by exploiting a quantum cascade laser
frequency comb as a light source, we demonstrate high spectral resolution
measurements via interleaving gianella_high-resolution_2020 , resolving the
transmission resonances of a silicon etalon and the absorption lines of a
Doppler-broadened low-pressure methane sample (CH4) with a resolution $<$ 250
MHz. Moreover, we demonstrate combined high spectral and temporal spectrometer
capabilities by monitoring the absorbance lines of a low pressure methane
during evacuation of a gas cell.

##  2 Results

###  2.1 Rotational FT-spectrometer

A diagram of the rotational-based FT spectrometer is shown in Fig. 1. In this
first demonstration we employ a mid-IR QCL frequency comb hugi_mid-
infrared_2012  as a light source emitting around 8 $\mu$m wavelength. However
any collimated light source with beam diameter below 7 mm could be used in our
spectrometer. The QCL is 2.5 mm long (free spectral range of $\approx$ 19.7
GHz) with high-reflection coating on the back facet and with maximum emission
power of 60 mW at T=-20∘C. In contrast to amplitude modulated frequency comb
light sources, which emit light pulses spaced by the cavity round-trip time
($t_{rep}$=$\frac{1}{f_{rep}}$), the QCL emits a frequency modulated comb (FM-
comb). This type of a frequency comb exhibits a linear phase difference
between neighboring longitudinal laser modes, resulting in a linearly chirped
instantaneous frequency with nearly constant intensity singleton_evidence_2018
. The absolute frequencies $f_{m}$ of the comb teeth in the optical domain are
described by $f_{m}=f_{ceo}+mf_{rep}$, where $f_{ceo}$ is the carrier envelope
offset frequency, $f_{rep}$ the repetition rate and $m$ an integer
representing the mode number. The emitted and collimated QCL light beam is
divided by a beam splitter (70/30) into two paths; one static beam path and
one which passes through the rotational delay line (RDL). The two beams are
then recombined on a second beam splitter (50/50), creating a sample and a
normalization beam path. Since the beam is going through the RDL, it acquires
an optical path delay (OPD) with respect to the static beam, two separate
interferograms can be recorded at the sample ($D_{S}$) and normalization
($D_{N}$) detectors, respectively. We use thermo-electrically cooled HgCdTe
detectors from Vigo system (model:PVMI-4TE-10.6-1x1-MIP-10k-100M-TO8-wZnSeAR)
with a peak sensitivity at 8 $\mu$m and 100 MHz bandwidth. As for dual-comb
spectroscopy setups, the normalization detector is required for canceling
common mode fluctuations which can originate from power and frequency
instabilities of the QCL comb. The key element of our setup is the RDL which
is schematically shown in Fig. 2 (a). It consists of a rotational retro-
reflecting (RR) octagrammic prism and a static retro-reflecting (SR) system,
which are arranged so that the incoming beam towards the SR is redirected
along the rotation axis (z-axis) and reflected back towards the RR. The RR can
in principle consist of any number $M\geq 2$ retro-reflectors, symmetrically
arranged around the rotational axis z. The optimal $M$ will depend on the
repetition rate of the laser, the speed of the motor, the tolerable losses due
to number of reflection in the delay line, and the desired acquisition rate.
In our case, $M=8$ is optimal for the target of recording a single IFG in
$\sim$ 1 ms and faster. The SR consists of two retro reflectors arranged next
to each other and shifted along the z-axis, so that the incoming beam is
reflected several times at different height of the RR. In this configuration,
the incoming and out-going beams on the same side of the RR have opposite
propagation directions and are separated in height. Moreover, the retro-
reflecting arrangement preserves the polarisation of the laser beam. The out-
going beam can be guided via mirrors to the opposing side of the RR in order
to bring it to the height of the initial incoming beam. This way of lowering
the beam has the additional advantage of doubling the optical path difference,
which allows for resolving the comb modes. Fig. 2 (b)-(c) illustrate the beam
propagation at zero degree and at an angle $\alpha$. The general expression
for the total optical path length in the RDL is given by

$\begin{split}L(\alpha)=&2N(D+S_{1}x_{p}-C_{1}+\sqrt{(T-x_{p})^{2}+(S_{3}T+C_{3}-S_{1}x_{p}-C_{1})^{2}}\\\
&-S_{3}T-C_{3})+2(N-1)l_{4},\end{split}$ (1)

where $S_{1}=\tan(\alpha-\pi/4)$, $C_{1}=\cos(\alpha)b+S_{1}\sin(\alpha)b$,
$S_{2}=\tan(\alpha+\pi/4)$, $C_{2}=\cos(\alpha)b+S_{2}\sin(\alpha)b$,
$S_{3}=\tan(2\alpha)$ and $C_{3}=C_{1}+S_{1}x_{p}-S_{3}x_{p}$. Here,
$\alpha(t)=\omega t$ is the rotation angle, $\omega$ the angular velocity of
the RR, $t$ is the laboratory time, and $N$ is the number of times the beam is
injected onto the RR (in our case $N=4$). The derivation of Eq. (1) can be
found in the Supplementary Materials. The OPD is then $\Delta
L=L(\alpha)-L(\alpha=0)$. Fig. 3 (a) shows the optical path length according
to Eq. (1) when the incoming beam enters the RDL at $x=x_{p}$ and at
$x=x_{p}+\frac{d}{2}$ at $\alpha(t=0)=0$), where $d$ is the beam diameter. It
shows that the incoming position of the beam on the $x$-axis (Fig. 2 (b) and
(c)) determines the maximum achievable OPD. It is important to note, that the
phase delay is independent on the laser beam diameter $d$, which is crucial
for spectroscopy applications. The beam diameter size only influences a
maximum achievable OPD. The maximum achievable OPD is given for $d=0$ and the
entry point $x=x_{p}$. Moreover, the phase delay is nonlinear in the angle
$\alpha$, and hence in the laboratory time t.
In order to take the nonlinear OPD into account, as well as to provide an
absolute frequency reference for the QCL spectrum, a single mode continuous-
wave (CW) reference laser (Rio GRANDE 1550 nm High Power Laser Module) is
spatially superimposed on the QCL beam. In this way, the two beams are co-
propagating through the RDL and experience the same nonlinear OPD. After
propagation through the RDL the reference beam is spatially separated from the
QCL one by means of an optical low-pass filter, which acts as a reflective
element for the reference laser and transmissive for mid-IR QCL beam. As in
the case of the QCL, a static and a non-static CW reference laser beam are
combined on a beam splitter and the resulting interferogram is recorded on the
reference detector $D_{r}$.
In addition to the detection of the optical signals, we exploit the fact that
the beating of the linearly spaced frequency comb modes of the QCL can be
directly measured in the radio-frequency domain. The beating gives rise to a
modulation of the laser bias and can be extracted from the device through a
bias-tee (Fig.1). We record this inter-modal beat note signal by down-mixing
(DM) it with a local oscillator LO with the frequency $f_{LO}$ to the
frequency $f_{DM}\approx$ 100 MHz. All signals are then digitized with the
acquisition card (model: Spectrum M4i.445x-x4) with 500 MS/s and 14 bit
resolution. The digitized data is then band pass filtered with the goal to
suppress the noise outside the signal bandwidth. With the help of the LO and
the recorded $f_{DM}$ signal, the repetition rate $f_{rep}=\lvert
f_{LO}-f_{DM}\rvert$ of the QCL is determined. The optical frequency of the
reference CW laser was determined with precision better than few 100s of MHz
using an optical spectrum analyser. This reference laser features frequency
stability of $<$ 15 kHz. Since the zero crossings of the CW reference
interferogram are used for resampling of the QCL interferogram, the frequency
accuracy of the resampled QCL spectrum is limited by the optical frequency
resolution of the CW reference laser which is several order of magnitudes
larger than the laser stability of $<$ 15 kHz. However, the frequency of the
reference laser can be calibrated more precisely (several MHz) with a narrow
absorption line of a known reference gas, as also realized in this work in the
interleaving section. Please note, that a laser with a frequency stabilty of
1-2 MHz would be also sufficient for our experiment since we perform a
calibration on narrow absorption line. The developed RDL provide around 3 cm
of OPD. A maximal RR angular frequency of 85 Hz (limited by the employed
motor’s stability) results in relatively high acquisition rates, with a duty
cycle of 50$\%$ as shown in Fig. 2 (d), where each interferogram is recorded
on one of the retro-reflectors of the octogram. Moreover, as we will discuss
later any imperfections in the fabrication of the RR might result in tiny
differences in the interferogram acquired on each retro (as can be seen in
Fig. 2 (d) (1) and (8)). However, since the normalization detector is used,
the important measure is the spectral ratio of the sample and the
normalization detector which will eliminate any fabrication imperfections of
the RR.

###  2.2 Operational principle of rotational FT spectrometers

Since the individual interferogram is acquired within a ms or even faster,
this has implications on the signal itself due to the Doppler effect
benirschke_frequency_2021 . The relativistic Doppler effect shifts the
frequency $f_{m}$ of an electromagnetic wave to the frequency $f_{D_{m}}$
according to

$f_{D_{m}}=f_{m}\sqrt{\frac{1-v/c}{1+v/c}}\approx f_{m}(1-v/c),$ (2)

where $v=\frac{dL_{tot}}{dt}<<c$ is the rate of change of the optical path
length $L_{tot}$ and $c$ is the speed of light. The frequency shift depends on
the speed of the motor and the OPD which is given by the radius of the
rotating disc, and the number of reflections onto the rotating disc (see
Supplementary materials), which in our case gives an average $v$ of about 40
m/s. The Doppler effect of mid-IR frequencies at these speeds is significant
and amounts to several MHz. Since the Doppler-shifted and non-Doppler shifted
light beams are superimposed onto a detector, the neighboring optical
frequencies of the beams will beat and generate a heterodyne signal in the
radio frequency (RF) domain given by

$f_{RF_{m}}(t)=f_{m}\frac{1}{c}\frac{\text{d}L_{\text{tot}}(\alpha(t))}{\text{d}t}=f_{m}\frac{\partial
L_{\text{tot}}(\alpha)}{\partial\alpha}\frac{\omega}{c}.$ (3)

The frequency chirp induced by the nonlinear OPD 3 (a) is visualised in Fig. 3
(b) by recording an interferogram of the single-mode reference laser. Here,
the first 5 $\mu$s of the recorded interferogram and the same time interval
after 50 $\mu$s is shown, while the chirped Doppler shift of the entire
interferogram is shown by the spectrogram in Fig. 3 (c).

In the case of a frequency comb source, the signal recorded on detector
$D_{S}$ and $D_{N}$ without a sample is given by

$S=\Re\left(\sum_{m=0}^{\infty}A_{m}\exp\left(\text{j}\left(2\pi
f_{RF_{m}}(t)t+\phi_{m}\right)\right)\right)$ (4)

where $A_{m}$ is the amplitude and $\phi_{m}$ the phase of mode $m$. The
nonlinear OPD will not only induce a chirp to all modes $m$, resulting in
absolute frequency shifts of the mode frequencies $f_{RF_{m}}$, but will also
directly modify $f_{rep}$ of the comb during the interferogram acquisition.
This effect is illustrated in Fig. 3 (d) at three different time points,
corresponding to the same color-coded points on the curve in Fig. 3 (a). The
first orange point at $\alpha(t=0)=0$ is related to Fig. 3 (d) (1) and
corresponds to the case with zero OPD and no Doppler shift. Increasing
$\alpha$ to 20 degrees impacts the repetition rate of the comb
($f_{rep_{2}}=f_{rep_{1}}\cdot(1+v(\alpha=20^{\circ})/c)$) as well as the
absolute frequencies of the individual optical modes, which acquire a Doppler
shift as well (Fig. 3 (d) (2)). A further angle increase of the RR results in
a larger Doppler shift and hence also larger $f_{rep_{3}}$ (Fig. 3 (d) (3)).
Such a frequency-chirped interferogram of a QCL frequency comb is shown in
Fig. 3 (e). In order to obtain a chirp-free interferogram from which the
spectrum can be retrieved, the chirped QCL interferogram is resampled using
the zero crossings of the reference laser interferogram. This is possible
since both beams undergo identical nonlinear delay and the frequency response
of the electronic is flat. The resampled interferogram of the QCL frequency
comb with improved SNR by coherently averaging 20 interferograms, which
averages out uncorrelated noise present in individual interferograms is shown
in Fig. 3 (f). The resolution of the system is 15 GHz and is given by the
maximum OPD of 3 cm, while the single burst acquisition speed on one of the
octogram retro-reflectors is 1 ms which is given by the maximum angular
velocity of the motor and the geometrical realisation of the rotating and
static retro-reflectors via Eq. (1).

###  2.3 Spectroscopy with rotational FT spectrometer

In order to perform absorption spectroscopy on a sample, a sequence of two
measurements is needed: a background spectrum and sample spectrum. The
background spectrum is acquired on the sample $D_{S}$ and normalization
detector $D_{N}$ without a sample present. Since the QCL spectrum is extracted
from the acquired single-burst interferogram with the knowledge of the
computed $f_{rep}$, we employ sub-nominal resolution method described in ref.
rutkowski_optical_2018 , in which precisely one repetition rate of the laser
is acquired, in order to eliminate spectral leakage. As pointed out earlier,
the normalization detector $D_{N}$ is used to suppress common noise on the two
beam paths, stemming from laser relative intensity noise (RIN). Please note,
that the laser RIN of a free running QCL significantly reduces for acquisition
speed of 1ms for an interferogram tombez_linewidth_2012 . This is also where
the speed of our rotational FT spectrometer plays an important role. Moreover,
we have also verified that the light scattering due to surface quality
(diamond polish aluminum with surface roughness $<$10 nm Ra, arithmetical mean
deviation) of the RDL (RR and RS) does not contribute to the intensity noise.
This is realized by computing the background ratio

$R_{BG}=\frac{S^{BG}_{S}}{S^{BG}_{{N}}},$ (5)

where $S^{BG}_{{S}}$ and $S^{BG}_{{N}}$ are the spectra recorded by detector
$D_{S}$ and $D_{N}$, respectively. Placing a sample into the sample beam path
(Fig. 1) and repeating the spectral measurements the sample transmittance can
be computed as

$T=\frac{R_{S}}{R_{BG}}$ (6)

where

$R_{S}=\frac{S^{S}_{S}}{S^{S}_{{N}}}$ (7)

and $S^{S}_{{S}}$ and $S^{S}_{{N}}$ are the spectra recorded with detector
$D_{S}$ and $D_{N}$, respectively. We note that in the system configuration
shown in Fig. 1 only the magnitude information, i. e. transmittance or
absorbance, can be measured. However, arranging the system geometry as in the
case for Dispersive FT spectroscopy parker_dispersive_1990 , where a sample is
place in one of the arms of the interferometer, the phase information can be
obtained as well. Since in our case we utilize a normalisation detector for
common intensity noise suppression of the QCL, a scheme as proposed by
Schiller schiller_spectrometry_2002  for dual-comb spectroscopy, where the
second comb corresponds in our case to the Doppler shifted beam path, can be
exploited as well.

###  2.4 Amplitude noise of the system

Before demonstrating high-resolution spectroscopy we analyse the system
sensitivity and discuss possible noise sources and their origin. The SNR of
the system is given by the amplitude noise of the heterodyne beat note signal
in the RF domain. For this purpose we analyse the amplitude noise fluctuations
of the spectrum shown in Fig. 4 (a). Fig. 4 (b) shows a time trace of the
normalized amplitude ratio of the strong mode, with an amplitude fluctuations
$<1\%$, showing excellent stability over several minutes. To quantify
different noise sources we compute the Allan deviation $\sigma(\tau)$ of comb
teeth with high and low amplitudes, indicated in Fig. 4 (a) with red and blue
arrows. Fig. 4 (b) shows the Allan deviations as functions of the integration
time $\tau$. Additionally, we show the computed noise floor originating from
the detector noise-equivalent power (NEP) computed from the spectrum for a
power of a strong mode. Both the high and low amplitude peaks show that the
noise of the system decreases with the square root of the integration time
$\tau$ up to 40 s and the system SNR is close to the detector NEP which
deviates by a factor of 2.8 from the NEP limit. This deviation might
originated from additional thermal and electrical noise sources which are not
take into account by computing NEP. Thus, with the employed detectors and the
laser, the SNR can be improved by longer integration times or by more powerful
laser, since the detector saturation power is around 8 mW, whereby the average
power on the detector is 0.5 mW. With higher power close to the saturation
power we would expect improvement of the SNR by factor 10. At longer
integration times ($\tau>$10 s) we can reach SNR of 104 and 103 for strong and
weak modes, respectively. We would like to point out that the 1/f noise in the
system is minimized, due to fast acquisition, since we observe a slope of
$-1/2$ for entire Allan plot which is characteristic of NEP-noise-limited
systems. In order to perform long-term measurements, we have to quantify the
background stability. For this reason we record the system background on
minutes time scales. Fig. 4 (d) shows a normalized system background
($R_{BG}$) spectrum, so called 100$\%$ transmission line, for a single
interferogram over the entire comb bandwidth. Since the comb spectrum consist
out of two lobs with high power on the sides and with low power in the center
of the spectrum, the computed 100% line also reflects this spectral power
distribution with 2% deviation in the weak spectral region. For demonstration
of the system and hence background stability we show the computed standard
deviation (STD) of the 100% line over the entire spectrum in Fig. 4 (e). As we
can see from the Fig. 4 (e) the system background remains stable over 100
seconds with deviation below 1.5%. This ensures for performing long-term
measurements which are characteristic for high resolution spectroscopy via
interleaving technique gianella_high-resolution_2020 .

###  2.5 Interleaving spectroscopy

#### Interleaving on silicon etalon

In order to demonstrate the system capabilities in high-resolution
spectroscopy, we use spectral interleaving similar to what described in Ref.
gianella_high-resolution_2020 . In the case of interleaving, the resulted
point spacing resolution is smaller than the free spectral range of the laser
given by the QCL cavity length, which in our case corresponds to a resolution
of $\approx$ 19.7 GHz. In contrast to dual-comb spectroscopy, where two
synchronised current ramps with different slopes are required for the laser
tuning gianella_high-resolution_2020 , we apply a single triangular current
modulation ramp (Fig. 1) to the QCL current driver (Wavelength electronics
QCL1000) which is controlled by means of a wave function generator. The
interferogram acquisition trigger (the Hall sensor of the motor) does not need
to be synchronized with the wave function generator, since the extracted
frequencies from individual spectra can be sorted in post-processing. The
triangular current ramp of the function generator is set so that the overall
QCL spectrum can be tuned by one $f_{rep}$. In this way, we can access
frequencies between modes and even provide a gap-less spectral coverage over
the entire optical bandwidth of the QCL comb. Firstly, we demonstrate the
high-resolution capability of the system on $\approx$500 $\mu$m thick silicon
etalon. For this measurement, we electrically tune the QCL spectrum over one
$f_{rep}$ in about 7 seconds. During this time, the measurements are acquired
continuously at a rate of 50 Hz. Due to the slow current ramp rate and fast
interferogram acquisition rate (1 ms) with 20 ms periodicity (acquisition on
the same single retro-reflector), the information within the recorded single-
burst interferogram is not influenced by the slow current changes induced by
the current ramp. As explained earlier, we record a background and a sample
measurement, and extract information from a single-burst interferogram via the
sub-nominal resolution technique rutkowski_optical_2018 . The corresponding
interleaving data is shown in Fig. 5 (a), where 20 consecutive interferograms
are coherently co-added before a spectrum is computed. The resulting spectra
are then frequency binned to 5 GHz resolution. As it can be seen in Fig. 5 (a)
and the zoomed region (Fig. 5 (b)) the recorded interleaved spectra fit well
the computed etalon spectrum. Measurements deviation from the model,
especially at frequencies with high transmission can be attributed to losses
in the silicon etalon which were not taken into account in the model. Also
small angle deviation from perpendicular incidence on the etalon will result
in deviation from the model. This measurement shows that in a relatively short
time period of 7 s, we can clearly resolve etalon features with resolution
$<$0.15cm -1. This addresses the question whether sub-GHz spectroscopy is
possible with our spectrometer.

#### Interleaving on methane

A more challenging and practically relevant scenario is high-resolution
spectroscopy on low-pressure Doppler-broadened methane lines. For this
experiment, the background measurements were carried out with a 10 cm-long
evacuated gas cell inserted into the sample beam path and the sample
measurements were performed on the filled gas cell with CH4 at a pressure of
200 hPa. Contrary to the low-finesse etalon in the previous section, the low
pressure methane lines have sub-GHz linewidths. The high-resolution
measurements were performed in the same way as for the etalon, but with the
QCL current ramp period set to 25 seconds. In this way, thousands of
interferograms are recorded, whereby each interferogram is recorded on ms time
scale, and 10 consecutive ones are co-added before computing each spectra.
With such a slow current ramp period we can ensure a high frequency resolution
and resolve low pressure Doppler broadened methane lines. The resulting
interleaved spectrum of methane is shown in Fig. 5 (c), where the resolution
has been reduced by frequency binning to 250 MHz. There are two reasons for
frequency binning to 250 MHz. First, it results in improved SNR and secondly
the interleaved spectrum is equidistant. The zoom into the spectrum of Fig. 5
(c) is shown in Fig. 5 (d) and shows that the system is capable of resolving
Doppler broadened molecular absorption lines with sub-GHz linewidths. However,
the resolved linewidths of some methane lines are broadened by up to 5%
compared to the bare HITRAN 2020 data base spectrum gordon_hitran2020_2022 .
For the computed HITRAN data spectrum no instrumental linewidth has been taken
into account, because we do not expect any line broadening from our rotational
FT spectrometer in combination with the QCL frequency comb. Additionally, we
observe that the 100% transmission line is not entirely flat. This measurement
deviations can be explained with contributions from two different origins.
Firstly, the QCL comb short-term stability can be affected by back-reflections
in the system. Secondly, high-frequency electrical instabilities of the laser
can occur while interleaving, since it is challenging to identify a current
range where the QCL tunes continuously over one $f_{\text{rep}}$ in the comb
regime. This will result in slightly different QCL states for background and
sample measurement. We note that this noise source could be further reduced by
using low-noise current sources and a more stable comb source. However, Fig. 5
(d) shows clearly that all sub-GHz CH4 absorption lines can be well resolved.
Providing sub-GHz spectral resolution shows a clear advantage of our approach,
since such a high resolution is challenging to achieve with the sate-of-the-
art FT spectrometers which are typically on the order of 0.2 cm-1 (6 GHz) or
0.06 cm-1 (1.8 GHz) for high resolution spectroscopy.

##  3 Time-resolved measurements on methane

Replacing the translational delay stage used in state-of-the-art FTIR
spectrometers by our rotational delay line enables much higher acquisition
speeds for a given spectral point spacing. In order to demonstrate this
capability of our system, we perform time-resolved spectroscopy on methane. We
monitor multiple methane absorption lines simultaneously while reducing the
pressure of the methane in the cell over a few ($<$6) seconds. Initially, a
gas cell is filled with methane at a low pressure of 200 hPa via a gas inlet
attached to a pump via mechanical valve, while the pressure is monitored with
a vacuum gauge (Pfeiffer vacuum gauge controller TPG 361) in 100 ms time steps
via a second connection. At $t=0$ the valve is opened while simultaneously the
pressure and the absorbance are monitored. The absorbance spectra are acquired
in $\sim$10 ms intervals by acquiring single interferograms in 625 $\mu$s with
the QCL operated at a constant current in the comb regime. From the recorded
transmittance ($I/I_{0}$,) and the background measurement (empty cell), we
compute the absorbance

$A=-\log_{10}(I/I_{0}),$ (8)

which is shown in Fig. 6 (a). For $t<0$ s, where the valve is closed, we
observe 10 methane absorption lines within the optical bandwidth of the comb.
By opening the valve ($t=0$ s) we observe a rapid absorbance decrease within 2
seconds. The absorbance is given by the Beer–Lambert law as

$A=\epsilon_{\lambda}\cdot\tilde{c}\cdot l,$ (9)

where $\epsilon_{\lambda}$ is the molar attenuation coefficient at wavelength
$\lambda$, $\tilde{c}$ the amount concentration, and $l$ the length of the gas
cell. Both $\epsilon_{\lambda}$ and $\tilde{c}$ are functions of the gas
pressure $p$ and temperature. Since the linewidth of methane absorption lines
(sub GHz) at this low pressure is much smaller than the spectral point spacing
of $\sim 19.7$ GHz of the particular QCL comb used in this experiment, the
absorbance is measured off the center of the absorption lines. This results in
a non-trivial time-dependence of the absorbance as the molar attenuation
coefficient is both wavelength and pressure dependent; namely, a linewidth
narrowing of the Doppler broaden methane absorption lines and hence in the
time-dependent change of $\epsilon_{\lambda}$, as well as a decrease in the
amount concentration $c(p)$. We demonstrate this effect by showing a cut in
Fig. 6 (a) (cut along the marked dotted line) along the temporal-evolution of
the strongest absorbance line in Fig. 6 (b), and compare it to the monitored
pressure inside the gas cell. The absorbance and the pressure decrease at the
same rate down to a pressure of about 100 hPa. Moreover, for $t>6$ s, which
corresponds to a pressure of 7 hPa, the absorbance is zero, which can be
explained by the fact that the QCL mode is either off from the resonance of
the methane absorbance line, or it is located in the absorption tail, thus
resulting in a weak absorption. We show the pressure related linewidth
narrowing of methane in Fig. 6 (c) where we compute the methane absorbance
exemplary for three different pressures and overlap the measured absorbance
points which corresponds to measurements times when these pressure are
reached. We observe a good agreement of spectral and temporal resolved methane
absorbance lines. Please note, that each absorbance line can be seen as
independent barometer, hence by calibrating each line one can increase the
accuracy of the measurement. Our spectrometer demonstrates high temporal and
spectral resolution which is predominant over the state-of-the-art FT
spectrometers, which are not able to provide such simultaneous temporal and
spectral resolution.

##  4 Discussion

In this work we have demonstrated a proof-of-principle of the rotational FT
spectrometer which decouples the spectral resolution from the temporal one.
Moreover, we have demonstrated millisecond dual-comb spectroscopy using a
single QCL frequency comb, enabled by the fast acquisition speed of the
system. A further benefit of a fast acquisition is the minimized 1/f noise
within the signal bandwidth, which is inevitable in the state-of-the-art
FTIRs. This is due to the fact, that the RIN of a free running laser is
reduced at least by two to three orders of magnitude cappelli_intrinsic_2015 .
Additionally, we have demonstrated high resolution, broadband spectroscopy
capabilities on a low-finesse etalon and low-pressure, Doppler-broadened
methane lines via the interleaving technique. Furthermore, since the geometry
of the rotational delay line allows incorporation of multiple sources, either
via time-multiplexing scheme or under simultaneous operation, a broader
optical bandwidth can be achieved straight-forwardly. We would like to point
out that our developed system may be exploited for the recently proposed
frequency comb ptychoscopy scheme benirschke_frequency_2021 , where an
external signal is analysed with a frequency comb and a Doppler-shifted
version of it. Since the frequency shift of the Doppler shifted comb can be
adjusted to acoustical frequencies (sub MHz), the system can also be employed
for photo-acoustic dual-comb spectroscopy wildi_photo-acoustic_2020 .
Moreover, the geometrical structure of the developed rotational delay line is
attractive for the implementation of multidimensional FT spectroscopy
lomsadze_frequency_2017  with a single frequency comb source. The compact
system demonstrated in this work is of high interest for performing high
spectral resolution and time-resolved spectroscopy, especially when combining
with ultra broad band white, NIR and mid-IR supercontinuum laser sources
hoghooghi_1-ghz_2022 ; alfano_supercontinuum_2016 ; chang_ultra-broadband_2021
; diouf_ultra-broadband_2017 . Moreover, in the case of light sources with
stable temporal intensity, the complexity of the system can be reduces by
eliminating the normalization detector and reducing the acquisition system to
two channels. The nominal spectral resolution can be significantly improved by
a factor 2 to 4 by increasing the size of the RDL or by increasing the number
of reflections ($N$ in Eqn. (1)). An increased temporal resolution is also
readily achievable by replacing the existing motor with a magnetically
levitated one. This would increase the single-spectrum acquisition speed up to
30 kHz (33$\mu$s), which is sufficient for the majority of spectroscopy
applications, such as reaction monitoring lehman_optical_2021 , leakage
monitoring alden_single-blind_2019  and in-line process analytical technology
(PAT) jin_near_2013 .

Supplementary information Supplementary information is provided additionally.
Acknowledgments We acknowledge the NCCR QSIT, the ERC Grant CHIC (No. 724344),
the quantum Flagship project QOMBS and the EU project FLASH for financial
support. G.S. would like to thank J.Hillbrand for help and A. Mayer for
discussions.

Competing Interests The authors declare no competing interests.

Author contributionsG.S. conceived the idea for Doppler-based spectroscopy.
S.M. invented, designed, and constructed the rotational FT spectrometer, and
carried out the measurements. S.M and M.F performed the data analysis. M.F.
implemented a real-time data acquisition on a graphical processor unit (GPU).
Early prototypes for rotational delay lines were investigated by S.M., M.S.
and A.F. Initial laser characterisation were performed by M.Bertrand. Laser
was processed by P.J. and grown by M.Beck. The manuscript was written by S.M
and M.F. All authors provided critical feedback and contributed to the final
shape of the manuscript. The work was supervised by J.F.and G.S.

## References

  *   * (1) Schmidt, F.M., Metsälä, M., Apolonski, A., Cristescu, S.M.: Chapter 14 - Optical spectroscopy. In: Beauchamp, J., Davis, C., Pleil, J. (eds.) Breathborne Biomarkers and the Human Volatilome (Second Edition), pp. 221–238. Elsevier, Boston (2020)
  * (2) Fleisher, A.J., Bjork, B.J., Bui, T.Q., Cossel, K.C., Okumura, M., Ye, J.: Mid-Infrared Time-Resolved Frequency Comb Spectroscopy of Transient Free Radicals. The Journal of Physical Chemistry Letters 5, 2241–2246 (2014)
  * (3) Norahan, M.J., Horvath, R., Woitzik, N., Jouy, P., Eigenmann, F., Gerwert, K., Kötting, C.: Microsecond-Resolved Infrared Spectroscopy on Nonrepetitive Protein Reactions by Applying Caged Compounds and Quantum Cascade Laser Frequency Combs. Analytical Chemistry 93, 6779–6783 (2021)
  * (4) Cheng, J., Liu, C.-s., Shang, S., Liu, D., Perrie, W., Dearden, G., Watkins, K.: A review of ultrafast laser materials micromachining. Optics & Laser Technology 46, 88–102 (2013)
  * (5) Ratri, P.J., Tashiro, K.: Phase-transition behavior of a crystalline polymer near the melting point: case studies of the ferroelectric phase transition of poly(vinylidene fluoride) and the $\beta$-to-$\alpha$ transition of trans-1,4-polyisoprene. Polymer Journal 45, 1107–1114 (2013)
  * (6) Wagner, C., Buchegger, W., Vellekoop, M., Kraft, M., Lendl, B.: Time-resolved mid-IR spectroscopy of (bio)chemical reactions in solution utilizing a new generation of continuous-flow micro-mixers. Analytical and Bioanalytical Chemistry 400, 2487–2497 (2011)
  * (7) Yang, Z., Albrow-Owen, T., Cai, W., Hasan, T.: Miniaturization of optical spectrometers. Science 371, 0722 (2021)
  * (8) Souza, M.C.M.M., Grieco, A., Frateschi, N.C., Fainman, Y.: Fourier transform spectrometer on silicon with thermo-optic non-linearity and dispersion correction. Nature Communications 9, 665 (2018)
  * (9) Zheng, S.N., Zou, J., Cai, H., Song, J.F., Chin, L.K., Liu, P.Y., Lin, Z.P., Kwong, D.L., Liu, A.Q.: Microring resonator-assisted Fourier transform spectrometer with enhanced resolution and large bandwidth in single chip solution. Nature Communications 10, 2349 (2019)
  * (10) Fathy, A., Sabry, Y.M., Nazeer, S., Bourouina, T., Khalil, D.A.: On-chip parallel Fourier transform spectrometer for broadband selective infrared spectral sensing. Microsystems & Nanoengineering 6, 1–9 (2020)
  * (11) Li, A., Fainman, Y.: On-chip spectrometers using stratified waveguide filters. Nature Communications 12, 2704 (2021)
  * (12) Zhang, C., Xu, J., Chui, P.C., Wong, K.K.Y.: Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation. Scientific Reports 3, 2064 (2013)
  * (13) Hashimoto, K., Ideguchi, T.: Phase-controlled Fourier-transform spectroscopy. Nature Communications 9, 4448 (2018)
  * (14) Kawai, A., Hashimoto, K., Dougakiuchi, T., Badarla, V.R., Imamura, T., Edamura, T., Ideguchi, T.: Time-stretch infrared spectroscopy. Communications Physics 3, 1–6 (2020)
  * (15) Goda, K., Jalali, B.: Dispersive Fourier transformation for fast continuous single-shot measurements. Nature Photonics 7, 102–112 (2013)
  * (16) Mahjoubfar, A., Churkin, D.V., Barland, S., Broderick, N., Turitsyn, S.K., Jalali, B.: Time stretch and its applications. Nature Photonics 11, 341–351 (2017)
  * (17) Hugi, A., Villares, G., Blaser, S., Liu, H.C., Faist, J.: Mid-infrared frequency comb based on a quantum cascade laser. Nature 492, 229–233 (2012)
  * (18) Villares, G., Hugi, A., Blaser, S., Faist, J.: Dual-comb spectroscopy based on quantum-cascade-laser frequency combs. Nature Communications 5, 5192 (2014)
  * (19) Coddington, I., Newbury, N., Swann, W.: Dual-comb spectroscopy. Optica 3, 414–426 (2016)
  * (20) Gianella, M., Vogel, S., Wittwer, V.J., Südmeyer, T., Faist, J., Emmenegger, L.: Frequency axis for swept dual-comb spectroscopy with quantum cascade lasers. Optics Letters 47, 625 (2022)
  * (21) Mandon, J., Guelachvili, G., Picqué, N.: Fourier transform spectroscopy with a laser frequency comb. Nature Photonics 3, 99–102 (2009)
  * (22) Xu, J., Zhang, X.-C.: Circular involute stage. Optics Letters 29, 2082–2084 (2004)
  * (23) Skorobogatiy, M.: Linear rotary optical delay lines. Optics Express 22, 11812–11833 (2014)
  * (24) Kim, G.-J., Jeon, S.-G., Kim, J.-I., Jin, Y.-S.: High speed scanning of terahertz pulse by a rotary optical delay line. Review of Scientific Instruments 79, 106102 (2008)
  * (25) Fleddermann, R., Lee, W.M., Huang, K., Campbell, G.T., Lam, P.K., Chow, J.H.: Compact flexible multi-pass rotary delay line using spinning micro-machined mirrors. Scientific Reports 7, 9299 (2017)
  * (26) Kim, G.-J., Jeon, S.-G., Kim, J.-I., Han, S.-T.: A Novel Optical Delay Line Using a Rotating Planar Reflector for Fast Measurement of a Terahertz Pulse. Journal of the Korean Physical Society 56, 1763–1766 (2010)
  * (27) Cappelli, F., Villares, G., Riedi, S., Faist, J.: Intrinsic linewidth of quantum cascade laser frequency combs. Optica 2, 836 (2015)
  * (28) Tombez, L., Schilt, S., Di Francesco, J., Führer, T., Rein, B., Walther, T., Di Domenico, G., Hofstetter, D., Thomann, P.: Linewidth of a quantum-cascade laser assessed from its frequency noise spectrum and impact of the current driver. Applied Physics B 109, 407–414 (2012)
  * (29) Gianella, M., Nataraj, A., Tuzson, B., Jouy, P., Kapsalidis, F., Beck, M., Mangold, M., Hugi, A., Faist, J., Emmenegger, L.: High-resolution and gapless dual comb spectroscopy with current-tuned quantum cascade lasers. Optics Express 28, 6197–6208 (2020)
  * (30) Singleton, M., Jouy, P., Beck, M., Faist, J.: Evidence of linear chirp in mid-infrared quantum cascade lasers. Optica 5, 948–953 (2018)
  * (31) Benirschke, D.J., Han, N., Burghoff, D.: Frequency comb ptychoscopy. Nature Communications 12, 4244 (2021)
  * (32) Rutkowski, L., Masłowski, P., Johansson, A.C., Khodabakhsh, A., Foltynowicz, A.: Optical frequency comb Fourier transform spectroscopy with sub-nominal resolution and precision beyond the Voigt profile. Journal of Quantitative Spectroscopy and Radiative Transfer 204, 63–73 (2018)
  * (33) Parker, T.J.: Dispersive Fourier transform spectroscopy. Contemporary Physics 31 (1990)
  * (34) Schiller, S.: Spectrometry with frequency combs. Optics Letters 27, 766–768 (2002)
  * (35) Gordon, I.E., et al.: The HITRAN2020 molecular spectroscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer 277, 107949 (2022)
  * (36) Wildi, T., Voumard, T., Brasch, V., Yilmaz, G., Herr, T.: Photo-acoustic dual-frequency comb spectroscopy. Nature Communications 11, 4164 (2020)
  * (37) Lomsadze, B., Cundiff, S.T.: Frequency combs enable rapid and high-resolution multidimensional coherent spectroscopy. Science 357, 1389–1391 (2017)
  * (38) Hoghooghi, N., Xing, S., Chang, P., Lesko, D., Lind, A., Rieker, G., Diddams, S.: 1-GHz mid-infrared frequency comb spanning 3 to 13 {\mu}m. arXiv:2201.07134 [physics] (2022)
  * (39) Alfano, R.R.: The Supercontinuum Laser Source: The Ultimate White Light. Springer, ??? (2016)
  * (40) Chang, K.-Y., Wang, R.-C., Yu, H.-C., Liu, J.-M., Liu, J.-M.: Ultra-broadband supercontinuum covering a spectrum from visible to mid-infrared generated by high-power and ultrashort noise-like pulses. Optics Express 29, 26775–26786 (2021)
  * (41) Diouf, M., Cherif, R., Ben Salem, A., Wague, A.: Ultra-broadband, coherent mid-IR supercontinuum expanding from 1.5 to 12.2 $\mu$m in new design of AsSe2 photonic crystal fibre. Journal of Modern Optics 64, 1335–1341 (2017)
  * (42) Lehman, J.H., Weichman, M.L.: Optical Frequency Combs for Molecular Spectroscopy, Kinetics, and Sensing. In: Emerging Trends in Chemical Applications Of Lasers. ACS Symposium Series, vol. 1398, pp. 61–88. American Chemical Society, ??? (2021)
  * (43) Alden, C.B., Coburn, S.C., Wright, R.J., Baumann, E., Cossel, K., Perez, E., Hoenig, E., Prasad, K., Coddington, I., Rieker, G.B.: Single-Blind Quantification of Natural Gas Leaks from 1 km Distance Using Frequency Combs. Environmental Science & Technology 53, 2908–2917 (2019)
  * (44) Jin, Y., Wu, Z., Liu, X., Wu, Y.: Near infrared spectroscopy in combination with chemometrics as a process analytical technology (PAT) tool for on-line quantitative monitoring of alcohol precipitation. Journal of Pharmaceutical and Biomedical Analysis 77, 32–39 (2013)

Figure 1: Schematic of the constructed rotational FT spectrometer with a QCL
as a light source. A CW reference laser which is used to provide a frequency
reference and serves also as a tool for mapping out and removing the induced
frequency chirp imposed by the nonlinear rotational delay line. QCL and CW
laser beam are spatially superimposed via a beam splitter (BS) and also
separated via the optical low-pass filter (LP). Interferograms of the QCL are
acquired with the sample (DS) and normalization (DN) detector, where the
reference interferogram of the CW laser is recorded with detector DR. All
measurements are synchronized with the trigger from the rotational delay line.
The wave function generator (FWG) can be used for continuously tuning of the
QCL operation point by applying a ramp to the current driver of the QCL. The
inter-modal beat note signal can be extracted directly from the waveguide via
bias-tee and the down-mixed version of it $f_{DM}$ can be recorded with the
help of the local oscillator LO. Figure 2: Rotational delay line. (a)
Realization of the rotational delay line (RDL) consisting out of three-
dimensional rotational retro-reflecting (RR) octagrammic prism and a static
retro-reflecting system (SR). The SR is arrange in such a way with respect to
RR, that multiple reflections between SR and RR are possible. By exploiting
the point symmetry of the system, the optical path delay can be doubled with
two SR on opposite sides of the RR. (b)/(c) Top view of a beam propagation in
the RDL at rotation angle 0∘ and $\alpha$. (d) Recorded interferogram on each
rotational retro-reflector of delay line. Figure 3: Operational principal. (a)
Induced nonlinear optical path length difference $\Delta$L by rotational delay
line as a function of rotation angle $\alpha$. The maximal achievable path
length difference depends on the entry point of the light beam along the
x-axis (Fig. 2(b)/(c)). The orientation of the rotational delay line with
respect to the incoming beam is shown for the selected point. (b)
Interferogram time slices of CW reference laser on detector DR. (c)
Spectrogram of CW reference laser which shows almost linear induced frequency
chirp by rotational delay line. (d) Illustration of induced frequency chirp to
a frequency comb. Three different color coded time points of comb spectrum,
which corresponds to the same color-coded points on the curve in (a).(e)
Recorded QCL interferogram on one of eight retro-reflectors of the octogram.
Increasing frequency chirp is clearly visible with increasing time. (f)
Resampled QCL inteferogram of (e) on zero crossings of a CW reference laser
from (b) with a coherently co-added 20 consecutive interferograms. Figure 4:
Amplitude noise investigation. (a) QCL spectrum on which Allan deviation is
performed. (b) Temporal evolution of normalized amplitude mode ratio of the
strongest mode, which is marked with blue arrow in (a). It shows the system
stability over the time period of 100s with amplitude fluctuations below 1%.
(c) Computed Allan variance of high (blue) and low (red) spectral amplitudes
of the spectrum in (a) which is indicated with arrows in the same color code.
The black curve shows the computed Allan variance taking the background (Noise
equivalent power (NEP) of a detector) into account for a power of strong mode.
(d) Background, or so called 100% line transmission, of a system for a single
interferogram and its temporal stability expressed as a standard deviation
(STD) in % over the entire optical bandwidth over 100s Figure 5: High
resolution spectroscopy. (a)/(b) Interleaving spectrum (blue) of $\approx$
500$\mu$m thick silicon etalon with a binned frequency resolution to 5 GHz and
the computed etalon transmission (red) and the zoom in recorded in 7 seconds.
(c) Doppler broaden methane (CH4) spectrum recorded via interleaving (blue) at
a pressure of 200 mbar in 25 seconds with frequency binned resolution to 250
MHz and the HITRAN data base reference (red). (d) Zoom in of (c) with sub-GHz
resolved methane absorption lines. Figure 6: Time-resolved spectroscopy on
methane. (a) Absorbance is computed from time resolved-transmission
measurements on the filled gas cell to a pressure of 200 hPa which is
evacuated via a pump. The computed methane absorbance for a pressure of 200
hPa is shown on the right side with white lines. Temporal evolution is
monitored by initializing the evacuation of the gas cell a $t=0$ s. The
acquisition is performed with periodicity of 10 ms in the comb regime at a
constant current mode. Simultaneously several low pressure methane lines are
observed for $t<0$ s. (b) Visualization of the temporal evolution of the
strongest absorption line (marked with a dotted black line) in Fig. (a) and
the corresponding pressure of the gas cell. A zoom in is shown in the top
right corner with 500 ms time duration starting 1s after opening the valve.
(c) Computed HITRAN absorbance of methane at a pressure of 200 hPa, 100 hPa
and 50 hPa and the measured absorbance close to the strongest absorbance peak
at these pressure and hence corresponding laboratory time.


# Thermal radiative corrections to hyperfine structure of light hydrogen-like
systems

T. Zalialiutdinov1,2, D. Glazov1 and D. Solovyev1 1 Department of Physics, St.
Petersburg State University, Petrodvorets, Oulianovskaya 1, 198504, St.
Petersburg, Russia
2 Petersburg Nuclear Physics Institute named by B.P. Konstantinov of National
Research Centre ”Kurchatov Institut”, St. Petersburg, Gatchina 188300, Russia

###### Abstract

In this work, we consider the thermal correction to the hyperfine interaction
in hydrogen, deuterium, and the 3He+ ion. This correction is effectively
described by one-loop Feynman graphs in the framework of the quantum
electrodynamics theory for bound states at a finite temperature. A simple
analysis shows the importance of the obtained results for future prospects for
measuring hyperfine splitting. In addition, the application for testing the
time variation of fundamental constants is briefly discussed.

##  I Introduction

The formation and development of quantum theory for bound states is closely
connected with precision spectroscopic experiments performed on the hydrogen
atom and a number of hydrogen-like ions. Having their own methodological
features, such systems are most accurately theoretically described within the
framework of quantum electrodynamics (QED). In the last decades, experimental
measurements and theoretical calculations of the hyperfine structure (HFS)
Seelig  _et al._ (1998); Beiersdorfer  _et al._ (2001); Shabaev (1994);
Shabaev  _et al._ (1997); Blundell  _et al._ (1997); Sunnergren  _et al._
(1998); Rothery and Hessels (2000); Karshenboim (2001); Volotka  _et al._
(2008, 2012); Puchalski and Pachucki (2013); Diermaier  _et al._ (2017);
Puchalski  _et al._ (2020, 2021) and g-factors Häffner  _et al._ (2000); Sturm
_et al._ (2013); Wagner  _et al._ (2013); Sturm  _et al._ (2014); Köhler  _et
al._ (2016); Harman  _et al._ (2018); Arapoglou  _et al._ (2019); Glazov  _et
al._ (2019); Sailer  _et al._ (2022) in simple atomic systems have attracted
particular interest. Significant progress in evaluation of QED radiative and
nuclear-structure corrections (see, e.g., Volotka  _et al._ (2013); Glazov
_et al._ (2014); Shabaev  _et al._ (2015); Indelicato (2019) and references
therein), as well as the improvement of experimental techniques, make it
possible to identify the fundamental parameters of the theory (the electron to
proton mass ratio $m_{e}/m_{p}$ Jentschura  _et al._ (2005); Sturm  _et al._
(2014), the Rydberg constant Grinin  _et al._ (2020), nuclear parameters
Köhler  _et al._ (2016); Sailer  _et al._ (2022), etc.), to set constraints on
the variation of the fine structure constant Kaplinghat  _et al._ (1999);
Safronova (2019) and to look for manifestations of new physics Debierre  _et
al._ (2020).

To date, the hyperfine splitting energies of atomic states are measured with
extremely high accuracy and also serve as a tool for building atomic clocks,
metrological frequency standards, tests of perturbative quantum chromodynamics
(QCD) Appelquist and Politzer (1975) and others. In particular, the relative
error in measuring the energy of the hyperfine splitting of the ground state
in hydrogen is about $10^{-13}$ Hellwig  _et al._ (1970).

In view of the achieved experimental and theoretical accuracy, studies of such
quantities as, e.g., hyperfine splitting, should necessarily take into account
the influence of thermal radiation, the blackbody radiation (BBR) field,
represented by the Planck equilibrium distribution function Riehle (2006). The
most revealing manifestations of the effects induced by BBR are the quadratic
ac-Stark shift of atomic energy levels and dynamic corrections to it, as well
as line broadening due to electron transitions between atomic states
stimulated by BBR Farley and Wing (1981); Gallagher and Cooke (1979). These
effects are extremely important for the spectroscopy of Rydberg atoms, the
construction of atomic clocks, and the determination of frequency standards
Hollberg and Hall (1984); Itano  _et al._ (1982); Safronova  _et al._ (2011).

The study of the effect of equilibrium thermal radiation on the
characteristics of atomic systems is usually limited to the quantum mechanical
(QM) description, in which the root-mean-squared field induced by BBR is
considered as a perturbation Farley and Wing (1981). However, the QED theory
makes it possible to reveal the effects that go beyond the QM approach. In our
recent works, the quantum electrodynamics theory for bound states at finite
temperature (BS-TQED) was developed to calculate thermal effects in atomic
systems Solovyev  _et al._ (2015); Solovyev (2020); Solovyev  _et al._ (2020,
2021). Within the framework of this theory, based on the $S$-matrix formalism,
various corrections to transition probabilities and ionization/recombination
cross sections were also considered Solovyev  _et al._ (2015); Zalialiutdinov
_et al._ (2017, 2020a, 2020b); Solovyev  _et al._ (2019); Zalialiutdinov  _et
al._ (2022). In this work, using previously developed methods for calculating
finite temperature effects Solovyev (2020); Zalialiutdinov  _et al._ (2022),
we study thermal one-loop radiative corrections to the hyperfine splitting
interval of $s$-states in hydrogen, deuterium and singly ionized helium-3
isotope.

The paper is organized as follows. In section II within the framework of
rigorous quantum electrodynamics at finite temperature and the adiabatic
$S$-matrix formalism, equations for radiative corrections of thermal self-
energy to hyperfine splitting in an one-electron ion are analytically derived.
Then the results of the numerical evaluation are discussed in section III. The
relativistic units $\hbar=c=m_{e}=1$ ($\hbar$ is the Planck constant, $c$ is
the speed of light, $m_{e}$ is the electron mass, it is written explicitly in
some places for clarity) and charge units $\alpha=e^{2}$ ($\alpha$ is the fine
structure constant) are used throughout the paper. The product of the
Boltzmann constant $k_{B}$ and the temperature $T$ is written in relativistic
units.

##  II Basic equations for thermal loop correction to HFS splitting

The magnetic dipole moment of the nucleus is

$\displaystyle\bm{\mu}=g_{I}\mu_{N}\bm{I}$ (1)

where $g_{I}$ denotes the nuclear g-factor,
$\mu_{N}=|e|/(2m_{p})=\mu_{B}(m_{e}/m_{p})$ is the nuclear magneton,
$\mu_{B}=|e|/(2m_{e})$ is the Bohr magneton, and $m_{e}$ and $m_{p}$ are the
electron and proton masses, respectively. The vector potential generated by
the nuclear dipole moment given by Eq. (1) is

$\displaystyle\bm{A}=\frac{\bm{\mu}\times\bm{r}}{r^{3}}.$ (2)

Then the interaction of the bound electron with the dipole nuclear magnetic
field is given by the Fermi-Breit operator

$\displaystyle\hat{V}_{\mathrm{hfs}}=-e\bm{\alpha}\bm{A}=|e|\frac{\bm{\alpha}\cdot(\bm{\mu}\times\bm{r})}{r^{3}},$
(3)

where $\bm{\alpha}$ is the Dirac alpha matrix.

The corresponding expectation value of Eq. (3) on the Dirac point-nucleus wave
functions of a hydrogen-like ion looks as follows

$\displaystyle E_{\mathrm{hfs}}=\alpha(\alpha
Z)^{3}\frac{g_{I}}{2}\frac{m_{e}^{2}}{m_{N}}\frac{\kappa}{|\kappa|}\frac{1}{n^{3}(2\kappa+1)(\kappa^{2}-1/4)}$
(4) $\displaystyle\times A(Z\alpha)[F(F+1)-I(I+1)-j(j+1)],$

where $A(Z\alpha)$ is the relativistic factor

$\displaystyle
A(Z\alpha)=n^{3}|\kappa|(2\kappa+1)\frac{2\kappa(2\gamma+n_{r})-N}{N^{4}\gamma(4\gamma^{2}-1)}.$
(5)

Here, $N=\sqrt{n_{r}^{2}+2n_{r}\gamma+\kappa^{2}}$, $n_{r}=n-|\kappa|$,
$\gamma=\sqrt{\kappa^{2}-(\alpha Z)^{2}}$, $n$ is the principal quantum number
of the electron, $\kappa$ is its Dirac angular quantum number,
$j=|\kappa|-1/2$, $Z$ is the nuclear charge, $F$ is the total angular momentum
of atom and $I$ is the nuclear spin.

In the nonrelativistic limit, the hyperfine Hamiltonian is given by the sum of
two contributions. The first one is proportional to $\bm{s}\cdot\bm{B}$, where
$\bm{s}$ is the electron spin and $\bm{B}$ is the magnetic field corresponding
to the vector potential Eq. (2):

$\displaystyle\bm{B}=\nabla\times\bm{A}=\frac{8\pi}{3}\bm{\mu}\delta^{(3)}(\bm{r})+\frac{1}{r^{3}}\left(3\left(\bm{\mu}\cdot\bm{n}\right)\bm{n}-\bm{\mu}\right).$
(6)

The second contribution corresponds to the interaction of the nuclear moment
with the magnetic field of the moving electron, which in turn is proportional
to the orbital angular momentum $\bm{l}$. Then in the nonrelativistic limit we
have

$\displaystyle\hat{V}_{\mathrm{hfs}}^{\mathrm{nr}}=-2\mu_{B}\left[-\frac{8\pi}{3}(\bm{s}\cdot\bm{\mu})\delta^{(3)}(\bm{r})\right.$
(7)
$\displaystyle-\frac{\bm{l}\cdot\bm{\mu}}{r^{3}}\left.-\frac{1}{r^{3}}\left(3\left(\bm{s}\cdot\bm{n}\right)\bm{n}-\bm{s}\right)\cdot\bm{\mu}\right].$

The evaluation of matrix elements of Eq. (7) with Schrödinger point-nucleus
wave functions in the $lsj(I)FM_{F}$ coupling scheme is presented in Appendix
A. For the hyperfine splitting of the ground state in hydrogen, using the
expression (7), we find the known wavelength value 21 cm, or 1420 MHz for the
frequency.

Figure 1: Feynman diagrams describing the TQED self-energy corrections to the
hyperfine interaction vertex. The tiny line with the cross indicates
interaction with vector potential generated by the nuclear dipole moment. The
double solid line denotes the bound electron in the Furry picture. The bold
wavy line represents the thermal photon propagator.

The thermal self-energy (TSE) correction in the presence of a binding Coulomb
field and an additional perturbing potential defined by Eq. (3) is graphically
represented by the Feynman diagrams in Fig. 1. The total energy shift of the
third order in $e$ can be written as a sum of vertex (Fig. 1 (a)) and wave-
function (Fig. 1 (b), (c)) contributions, $\Delta
E_{a}^{\mathrm{total}}=\Delta E_{a}^{\mathrm{ver}}+\Delta
E_{a}^{\mathrm{wf}}$, see e.g. Zalialiutdinov  _et al._ (2022) for details, as
follows

$\displaystyle\Delta
E_{a}^{\mathrm{ver}}=-\frac{e^{3}}{\pi}\mathrm{Re}\sum\limits_{\pm}\sum\limits_{n,m}\int\limits_{0}^{\infty}d\omega
n_{\beta}(\omega)\qquad$ (8) $\displaystyle\times\frac{\langle
am|\frac{\alpha^{\mu}_{1}\alpha_{2\mu}}{r_{12}}\sin(\omega
r_{12})|na\rangle\langle n|\bm{\alpha}\bm{A}|m\rangle}{(E_{a}\pm\omega-
E_{n}(1-\mathrm{i}0))(E_{a}\pm\omega-E_{m}(1-\mathrm{i}0))}\,,$
$\displaystyle\Delta
E_{a}^{\mathrm{wf}}=-\frac{2e^{3}}{\pi}\mathrm{Re}\sum\limits_{\pm}\int\limits_{0}^{\infty}d\omega
n_{\beta}(\omega)\qquad$ (9)
$\displaystyle\times\left[\sum\limits_{\begin{subarray}{c}n,m\\\ m\neq
a\end{subarray}}\frac{\langle
an|\frac{\alpha^{\mu}_{1}\alpha_{2\mu}}{r_{12}}\sin(\omega
r_{12})|nm\rangle\langle m|\bm{\alpha}\bm{A}|a\rangle}{(E_{a}\pm\omega-
E_{n}(1-\mathrm{i}0))(E_{a}-E_{m})}\right.$
$\displaystyle\left.-\frac{1}{2}\sum\limits_{\begin{subarray}{c}n\end{subarray}}\frac{\langle
an|\frac{\alpha^{\mu}_{1}\alpha_{2\mu}}{r_{12}}\sin(\omega
r_{12})|na\rangle\langle a|\bm{\alpha}\textbf{A}|a\rangle}{(E_{a}\pm\omega-
E_{n}(1-\mathrm{i}0))^{2}}\right]\,.$

Here $\sum\limits_{\pm}$ denotes the sum of two contributions with $+$ and $-$
in energy denominators, vector potential $\bm{A}$ is given by Eq. (2),
$n_{\beta}(\omega)=(\exp(\omega/(k_{B}T)-1)^{-1}$, $k_{B}$ is the Boltzmann
constant, $T$ is the temperature (in Kelvin). In contrast to the ordinary
’zero-vacuum’ corrections, the expressions (8) and (9) converge in the
ultraviolet limit due to the natural cut-off provided by the Planckian
distribution function $n_{\beta}(\omega)$. The second term in Eq. (9)
corresponds to the ”reference-state” contribution, $m=a$.

Then substituting Eq. (2) into Eqs. (8) and (9) and taking the nonrelativistic
limit (see details in Zalialiutdinov  _et al._ (2022)), we find

$\displaystyle\Delta
E_{a}^{\mathrm{ver}}=-\frac{2e^{2}}{3\pi}\mathrm{Re}\sum\limits_{\pm}\sum\limits_{n,m}\int\limits_{0}^{\infty}d\omega\omega^{3}n_{\beta}(\omega)\qquad$
(10) $\displaystyle\times\frac{\langle a|\textbf{r}|n\rangle\langle
n|\hat{V}_{\mathrm{hfs}}^{\mathrm{nr}}|m\rangle\langle
m|\textbf{r}|a\rangle}{(E_{a}\pm\omega-E_{n}(1-\mathrm{i}0))(E_{a}\pm\omega-
E_{m}(1-\mathrm{i}0))},$ $\displaystyle\Delta
E_{a}^{\mathrm{wf}}=-\frac{4e^{2}}{3\pi}\mathrm{Re}\sum\limits_{\pm}\int\limits_{0}^{\infty}d\omega
n_{\beta}(\omega)\omega^{3}\qquad$ (11)
$\displaystyle\times\left[\sum\limits_{\begin{subarray}{c}n,m\\\ m\neq
a\end{subarray}}\frac{\langle a|\textbf{r}|n\rangle\langle
n|\textbf{r}|m\rangle\langle
m|\hat{V}_{\mathrm{hfs}}^{\mathrm{nr}}|a\rangle}{(E_{a}\pm\omega-
E_{n}(1-\mathrm{i}0))(E_{a}-E_{m})}\right.$
$\displaystyle-\left.\frac{1}{2}\sum\limits_{n}\frac{\langle
a|\textbf{r}|n\rangle\langle n|\textbf{r}|a\rangle\langle
a|\hat{V}_{\mathrm{hfs}}^{\mathrm{nr}}|a\rangle}{(E_{a}\pm\omega-
E_{n}(1-\mathrm{i}0))^{2}}\right],$

where $\hat{V}_{\mathrm{hfs}}^{\mathrm{nr}}$ is given by Eq. (7) and the
summation over the $n$ spectrum is carried out over the discrete and continuum
solutions of the Schrödinger equation for an electron in the Coulomb field of
the nucleus.

The corresponding total shift for the transition energy between hyperfine
components $F_{a}^{\prime}$ and $F_{a}$ of the same state $n_{a}l_{a}j_{a}$ is

$\displaystyle\Delta\nu^{\mathrm{hfs}}=\Delta
E_{n_{a}l_{a}s_{a}j_{a}F^{\prime}_{a}}^{\mathrm{total}}-\Delta
E_{n_{a}l_{a}s_{a}j_{a}F_{a}}^{\mathrm{total}}.$ (12)

For hydrogen-like ions with a nuclear charge $Z$, a parametric estimates for
Eqs. (10) and (11) can be found taking into account that in relativistic units
$r\sim(m_{e}\alpha Z)^{-1}$, $E_{a}\sim m_{e}(\alpha Z)^{2}$ and
$\int\limits_{0}^{\infty}d\omega\,\omega^{k}n_{\beta}(\omega)\sim(k_{B}^{\mathrm{\;r.u.}}T)^{k+1}$.
Then $\Delta\nu^{\mathrm{hfs}}$ given by Eq. (12) is parametrized as follows

$\displaystyle\Delta\nu^{\mathrm{hfs}}\sim\frac{(k_{B}T)^{4}_{\mathrm{r.u.}}}{\alpha
m_{p}m_{e}^{2}Z^{3}}\,,$ (13)

where $m_{p}$ is the proton mass.

In particular, the estimate Eq. (13) is valid for the state $a=1s$, when the
summation runs over the $np$ states and the energy difference in the
denominators of Eqs. (10) and (11) is always of the order $m_{e}(\alpha
Z)^{2}$. However, for $n_{a}l_{a}$ states with $n_{a}\geq 2$ (for example,
$a=2s$), the dominant contribution in the sum over $n$ corresponds to
$E_{n}=E_{n_{a}}$. In this case, the following parametrization is valid

$\displaystyle\Delta\nu^{\mathrm{hfs}}\sim\frac{Z\alpha^{3}(k_{B}T)^{2}_{\mathrm{r.u.}}}{m_{p}}\,.$
(14)

##  III Results and discussion

Numerical results of $\Delta\nu^{\mathrm{hfs}}$ calculation for $ns_{1/2}$
($n=1,\,2,\,3,\,4$) states in hydrogen, deuterium and 3He+ ion are collected
in Tables 1, 2, and 3, respectively, at different temperatures. Numerical
evaluation of radial integrals and summation over the entire spectrum in Eqs.
(10) and (11) was performed using the B-spline method for the solution of
Schrödinger equation.

Table 1: Thermal self-energy correction to the hyperfine splitting $\Delta\nu^{\mathrm{hfs}}_{a}$ of $ns$, $np$ and $nd$ states in hydrogen $e^{-}p^{+}$ ($I=1/2$, $g_{I}=5.5856946893(16)$) at different temperatures (in kelvin). All values are in hertz. Energy shift | T=300 | T=1000 | T=3000
---|---|---|---
$1s_{1/2}^{F=1-F=0}$ | $2.00\times 10^{-8}$ | $2.48\times 10^{-6}$ | $2.07\times 10^{-4}$
$2s_{1/2}^{F=1-F=0}$ | $1.17\times 10^{-3}$ | $1.31\times 10^{-2}$ | $1.23\times 10^{-1}$
$2p_{1/2}^{F=1-F=0}$ | $2.61\times 10^{-4}$ | $2.93\times 10^{-3}$ | $2.88\times 10^{-2}$
$2p_{3/2}^{F=2-F=1}$ | $-2.09\times 10^{-4}$ | $-2.32\times 10^{-3}$ | $-1.96\times 10^{-2}$
$3s_{1/2}^{F=1-F=0}$ | $2.10\times 10^{-3}$ | $2.39\times 10^{-2}$ | $1.85\times 10^{-1}$
$3p_{1/2}^{F=1-F=0}$ | $6.97\times 10^{-4}$ | $7.90\times 10^{-3}$ | $6.15\times 10^{-1}$
$3p_{3/2}^{F=2-F=1}$ | $-3.06\times 10^{-5}$ | $3.52\times 10^{-4}$ | $-7.71\times 10^{-5}$
$3d_{3/2}^{F=2-F=1}$ | $-8.72\times 10^{-5}$ | $-8.75\times 10^{-4}$ | $-1.42\times 10^{-2}$
Table 2: Thermal self-energy correction to the hyperfine splitting $\Delta\nu^{\mathrm{hfs}}$ of $ns$, $np$ and $nd$ states in deuterium $e^{-}d^{+}$ ($I=1$, $g_{I}=0.8574382338(22)$) at different temperatures (in kelvin). All values are in hertz. Energy shift | T=300 | T=1000 | T=3000
---|---|---|---
$1s_{1/2}^{F=3/2-F=1/2}$ | $4.61\times 10^{-9}$ | $5.70\times 10^{-7}$ | $4.76\times 10^{-5}$
$2s_{1/2}^{F=3/2-F=1/2}$ | $2.71\times 10^{-4}$ | $3.02\times 10^{-3}$ | $2.82\times 10^{-2}$
$2p_{1/2}^{F=3/2-F=1/2}$ | $6.01\times 10^{-5}$ | $6.74\times 10^{-4}$ | $6.65\times 10^{-3}$
$2p_{3/2}^{F=3/2-F=1/2}$ | $-2.40\times 10^{-5}$ | $-2.67\times 10^{-4}$ | $-2.36\times 10^{-3}$
$2p_{3/2}^{F=5/2-F=3/2}$ | $-4.01\times 10^{-5}$ | $-4.44\times 10^{-4}$ | $-3.60\times 10^{-3}$
$3s_{1/2}^{F=3/2-F=1/2}$ | $4.81\times 10^{-4}$ | $5.56\times 10^{-3}$ | $4.24\times 10^{-2}$
$3p_{1/2}^{F=3/2-F=1/2}$ | $1.61\times 10^{-4}$ | $5.76\times 10^{-3}$ | $1.41\times 10^{-4}$
$3p_{3/2}^{F=3/2-F=1/2}$ | $-3.41\times 10^{-5}$ | $-3.66\times 10^{-4}$ | $-4.48\times 10^{-3}$
$3p_{3/2}^{F=5/2-F=3/2}$ | $-5.69\times 10^{-5}$ | $-6.05\times 10^{-4}$ | $-7.51\times 10^{-3}$
$3d_{3/2}^{F=3/2-F=1/2}$ | $-8.45\times 10^{-6}$ | $-1.01\times 10^{-4}$ | $-9.58\times 10^{-4}$
$3d_{3/2}^{F=5/2-F=3/2}$ | $-1.41\times 10^{-5}$ | $-1.68\times 10^{-4}$ | $-1.69\times 10^{-3}$
Table 3: Thermal self-energy correction to the hyperfine splitting $\Delta\nu^{\mathrm{hfs}}$ of $ns$, $np$ and $nd$ states in 3He+ ($I=1/2$, $g_{I}=-4.255250615(50)$) at different temperatures (in kelvin). All values are in hertz. Energy shift | T=300 | T=1000 | T=3000
---|---|---|---
$1s_{1/2}^{F=1-F=0}$ | $-1.90\times 10^{-9}$ | $-2.35\times 10^{-7}$ | $-1.91\times 10^{-5}$
$2s_{1/2}^{F=1-F=0}$ | $-1.79\times 10^{-3}$ | $-1.99\times 10^{-2}$ | $-1.79\times 10^{-1}$
$2p_{1/2}^{F=1-F=0}$ | $-3.98\times 10^{-4}$ | $-4.42\times 10^{-3}$ | $-4.00\times 10^{-2}$
$2p_{3/2}^{F=2-F=1}$ | $3.18\times 10^{-4}$ | $2.33\times 10^{-3}$ | $3.19\times 10^{-2}$
$3s_{1/2}^{F=1-F=0}$ | $-3.18\times 10^{-3}$ | $-3.54\times 10^{-2}$ | $-3.23\times 10^{-1}$
$3p_{1/2}^{F=1-F=0}$ | $-1.06\times 10^{-3}$ | $-1.18\times 10^{-2}$ | $-1.09\times 10^{-1}$
$3p_{3/2}^{F=2-F=1}$ | $1.30\times 10^{-4}$ | $1.45\times 10^{-3}$ | $1.36\times 10^{-2}$
$3d_{3/2}^{F=2-F=1}$ | $1.33\times 10^{-4}$ | $1.48\times 10^{-3}$ | $1.31\times 10^{-2}$

Due to the uncertainty in determining the size of nuclei, the possibility of
carrying out QED tests in atomic systems based on the analysis of a specific
energy difference has been proposed Zwanziger (1961); Shabaev  _et al._
(2001). A simple analogue of this difference for light systems, which is
weakly sensitive to the contributions arising from the nuclear structure, is

$\displaystyle
D_{21}=8\Delta\nu^{\mathrm{hfs}}_{2s}-\Delta\nu^{\mathrm{hfs}}_{1s}\,.$ (15)

Over the past few years, the accuracy of calculating corrections to the energy
of hyperfine splitting and the value of $D_{21}$ has increased significantly.
A number of corrections have been calculated, including those for the
electroweak interaction, the hadronic vacuum polarization, and the structure
of the nucleus Alcorta and Grifols (1994); Asaka  _et al._ (2018); Faustov
_et al._ (1999); Bég and Feinberg (1974); Kalinowski  _et al._ (2018). At
present, the inaccuracy of theoretical calculations is much lower than the
experimental error in measuring the parameter $D_{21}$, which is mainly
determined by the error in measuring the hyperfine splitting of the $2s$
metastable state. For hydrogen and deuterium atoms, the experimental error in
measuring the $2s$ hyperfine interval is about ten hertz Heberle  _et al._
(1956); Kolachevsky  _et al._ (2004), while for the ground state it reaches
several millihertz Hellwig  _et al._ (1970); Wineland and Ramsey (1972);
Mathur  _et al._ (1967).

According to the results listed in Tables 1, 2 and 3, the correction Eq. (12)
for the ground state can be excluded from consideration for the $D_{21}$
difference in hydrogen, deuterium and 3He+ ion. However, a decrease in the
experimental error in determining the hyperfine splitting of the $2s$ state to
the accuracy level of the ground state splitting will be sensitive to this
correction.

Another application of the correction considered in this paper concerns the
search for time variation of fundamental physical constants, see, for example,
Webb  _et al._ (2001); Kaplinghat  _et al._ (1999). Verification of the fine
structure constant variation can be performed by comparing in detail the
spectral data of quasars and laboratory results for the hyperfine splitting of
the ground state in the hydrogen atom. In this case, the higher temperatures
can play a role. According to the results listed in Table 1, the
$\Delta\nu^{\mathrm{hfs}}_{a}$ reaches the accuracy level of laboratory
experiments only at sufficiently high $3000$ K and become larger at higher
temperatures. Thus, corrections (10) and (11) impose additional constrains on
this type of research. Simple analysis of calculated shifts in hydrogen,
deuterium and 3He+ presented in Tables 1, 2 and 3 shows that hyperfine
splitting interval is weakly sensitive to the considered thermal correction.

Despite the obtained values are so small, it is important to note that the
thermal one-loop correction to the hyperfine structure considered in this
paper is of the order of Zeeman shifts induced by blackbody radiation (BBRZ)
Han  _et al._ (2019); Itano  _et al._ (1982). For alkali metals and alkali-
like ions the relative value of the BBRZ shift for the ground state reaches
the same order of magnitude as for hydrogen (see Table 2 in Han  _et al._
(2019)). For many-electron atoms and ions with one valence electron, rough
estimates can be given with the value of $Z_{eff}$ little different from unity
Clementi  _et al._ (1967). Thus, using estimates (10), (11) when $Z$ is
replaced by $Z_{eff}$, we can expect a contribution of the same order for
alkali metals. This can be especially important when elaborating frequency
standards operating on hyperfine transitions of the outermost $s$-electron. We
leave detailed calculations of thermal one-loop corrections in alkali metals
for future work.

##  IV Acknowledgements

The calculations for hydrogen and deuterium were supported by the Russian
Scientific Foundation under grant No. 22-12-00043. The calculations for helium
were supported by the Russian Scientific Foundation under grant No.
22-12-00258.

## Appendix A: Evaluation of matrix elements

Taking into account Eq. (1) and performing angular algebra with the use of
Eckart-Wigner theorem the first term in Eq. (7) can be evaluated with the help
of following equality

$\displaystyle\langle
n^{\prime}l^{\prime}s^{\prime}j^{\prime}(I)F^{\prime}M_{F^{\prime}}|(\bm{s}\cdot\bm{I})\delta^{3}(\bm{r})|nlsj(I)FM_{F}\rangle$
$\displaystyle=\delta_{F^{\prime}F}\delta_{M_{F^{\prime}}M_{F}}\delta_{l^{\prime}l}\delta_{s^{\prime}s}(-1)^{F+j+I+l^{\prime}+s+j^{\prime}+1}$
$\displaystyle\times\sqrt{\frac{3}{2}}\begin{Bmatrix}j^{\prime}&j&1\\\
I&I&F\end{Bmatrix}\begin{Bmatrix}s^{\prime}&j^{\prime}&l\\\
j&s&1\end{Bmatrix}$
$\displaystyle\times\left[(2j^{\prime}+1)(2j+1)I(I+1)(2I+1)\right]^{1/2}\frac{1}{4\pi}R_{n^{\prime}l^{\prime}}(0)R_{nl}(0).$

The radial nonrelativistic wave-functions $R_{nl}$ taken at zero are
nonvanishing only for $s$-states. Then for discrete and continuum solutions of
Schrödinger equation we have

$\displaystyle R_{n0}(0)=\frac{2}{n^{3/2}},$ (A1) $\displaystyle
R_{x0}(0)=\frac{2\sqrt{x}}{\sqrt{1-e^{-2\pi/x}}},$

respectively.

In a similar manner, the second term in Eq. (7) can be evaluated with the use
of equality

$\displaystyle\langle
n^{\prime}l^{\prime}s^{\prime}j^{\prime}(I)F^{\prime}M_{F^{\prime}}|\frac{\bm{l}\cdot\bm{I}}{r^{3}}|nlsj(I)FM_{F}\rangle$
$\displaystyle=\delta_{F^{\prime}F}\delta_{M_{F^{\prime}}M_{F}}\delta_{l^{\prime}l}\delta_{s^{\prime}s}(-1)^{2j+l^{\prime}+s^{\prime}+I+F+1}$
$\displaystyle\times\left[(2j^{\prime}+1)(2j+1)I(I+1)(2I+1)\right]^{1/2}$
$\displaystyle\times\begin{Bmatrix}F&I&j^{\prime}\\\
1&j&I\end{Bmatrix}\begin{Bmatrix}l^{\prime}&j^{\prime}&s^{\prime}\\\
j&l&1\end{Bmatrix}\sqrt{l(l+1)(2l+1)}$
$\displaystyle\times\int\limits_{0}^{\infty}drr^{2}R_{n^{\prime}l^{\prime}}(r)\left(\frac{1}{r^{3}}\right)R_{nl}(r).$

The third term in Eq. (7) can be easily evaluated with the use of following
relation

$\displaystyle\bm{K}\cdot\bm{I}\equiv
r^{-3}\left(3\left(\bm{s}\cdot\bm{n}\right)\bm{n}-\bm{s}\right)\cdot\bm{I}$
(A2)
$\displaystyle=r^{-3}\sqrt{10}\sum\limits_{q}(-1)^{q}[\mathrm{C}^{2}\times\mathrm{S}^{1}]_{1q}I_{1-q}.$

Then the reduction of the corresponding matrix elements yields

$\displaystyle\langle
n^{\prime}l^{\prime}s^{\prime}j^{\prime}(I)F^{\prime}M_{F^{\prime}}|\bm{K}\cdot\bm{I}|nlsj(I)FM_{F}\rangle\qquad$
(A3)
$\displaystyle=\delta_{F^{\prime}F}\delta_{M_{F^{\prime}}M_{F}}\delta_{s^{\prime}s}\sqrt{45}(-1)^{j+I+F+l^{\prime}}\begin{Bmatrix}F&I&j^{\prime}\\\
1&j&I\end{Bmatrix}$
$\displaystyle\times[I(I+1)(2I+1)(2l^{\prime}+1)(2l+1)(2j^{\prime}+1)(2j+1)]^{1/2}$
$\displaystyle\times\begin{Bmatrix}l^{\prime}&l&2\\\ s^{\prime}&s&1\\\
j^{\prime}&j&1\end{Bmatrix}\begin{pmatrix}l^{\prime}&2&l\\\
0&0&0\end{pmatrix}\int\limits_{0}^{\infty}drr^{2}R_{n^{\prime}l^{\prime}}(r)\left(\frac{1}{r^{3}}\right)R_{nl}(r).$

## References

  * Seelig  _et al._ (1998) P. Seelig, S. Borneis, A. Dax, T. Engel, S. Faber, M. Gerlach, C. Holbrow, G. Huber, T. Kühl, D. Marx, K. Meier, P. Merz, W. Quint, F. Schmitt, M. Tomaselli, L. Völker, H. Winter, M. Würtz, K. Beckert, B. Franzke, F. Nolden, H. Reich, M. Steck, and T. Winkler, Phys. Rev. Lett. 81, 4824 (1998).
  * Beiersdorfer  _et al._ (2001) P. Beiersdorfer, S. B. Utter, K. L. Wong, J. R. Crespo López-Urrutia, J. A. Britten, H. Chen, C. L. Harris, R. S. Thoe, D. B. Thorn, E. Träbert, M. G. H. Gustavsson, C. Forssén, and A.-M. Mårtensson-Pendrill, Phys. Rev. A 64, 032506 (2001).
  * Shabaev (1994) V. M. Shabaev, J. Phys. B 27, 5825 (1994).
  * Shabaev  _et al._ (1997) V. M. Shabaev, M. Tomaselli, T. Kühl, A. N. Artemyev, and V. A. Yerokhin, Phys. Rev. A 56, 252 (1997).
  * Blundell  _et al._ (1997) S. A. Blundell, K. T. Cheng, and J. Sapirstein, Phys. Rev. A 55, 1857 (1997).
  * Sunnergren  _et al._ (1998) P. Sunnergren, H. Persson, S. Salomonson, S. M. Schneider, I. Lindgren, and G. Soff, Phys. Rev. A 58, 1055 (1998).
  * Rothery and Hessels (2000) N. E. Rothery and E. A. Hessels, Phys. Rev. A 61, 044501 (2000).
  * Karshenboim (2001) S. G. Karshenboim, “2s hyperfine structure in hydrogen atom and helium-3 ion,” in  _The Hydrogen Atom: Precision Physics of Simple Atomic Systems_ , edited by S. G. Karshenboim, F. Bassani, F. Pavone, M. Inguscio, and T. Hänsch (2001) pp. 335–343.
  * Volotka  _et al._ (2008) A. V. Volotka, D. A. Glazov, I. I. Tupitsyn, N. S. Oreshkina, G. Plunien, and V. M. Shabaev, Phys. Rev. A 78, 062507 (2008).
  * Volotka  _et al._ (2012) A. V. Volotka, D. A. Glazov, O. V. Andreev, V. M. Shabaev, I. I. Tupitsyn, and G. Plunien, Phys. Rev. Lett. 108, 073001 (2012).
  * Puchalski and Pachucki (2013) M. Puchalski and K. Pachucki, Phys. Rev. Lett. 111, 243001 (2013).
  * Diermaier  _et al._ (2017) M. Diermaier, C. B. Jepsen, B. Kolbinger, C. Malbrunot, O. Massiczek, C. Sauerzopf, M. C. Simon, J. Zmeskal, and E. Widmann, Nature Communications 8, 15749 (2017).
  * Puchalski  _et al._ (2020) M. Puchalski, J. Komasa, and K. Pachucki, Phys. Rev. Lett. 125, 253001 (2020).
  * Puchalski  _et al._ (2021) M. Puchalski, J. Komasa, and K. Pachucki, Phys. Rev. Research 3, 013293 (2021).
  * Häffner  _et al._ (2000) H. Häffner, T. Beier, N. Hermanspahn, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú, and G. Werth, Phys. Rev. Lett. 85, 5308 (2000).
  * Sturm  _et al._ (2013) S. Sturm, A. Wagner, M. Kretzschmar, W. Quint, G. Werth, and K. Blaum, Phys. Rev. A 87, 030501 (2013).
  * Wagner  _et al._ (2013) A. Wagner, S. Sturm, F. Köhler, D. A. Glazov, A. V. Volotka, G. Plunien, W. Quint, G. Werth, V. M. Shabaev, and K. Blaum, Phys. Rev. Lett. 110, 033003 (2013).
  * Sturm  _et al._ (2014) S. Sturm, F. Köhler, J. Zatorski, A. Wagner, Z. Harman, G. Werth, W. Quint, C. Keitel, and K. Blaum, Nature 506 (2014), 10.1038/nature13026.
  * Köhler  _et al._ (2016) F. Köhler, K. Blaum, M. Block, S. Chenmarev, S. Eliseev, D. A. Glazov, M. Goncharov, J. Hou, A. Kracke, D. A. Nesterenko, Y. N. Novikov, W. Quint, E. Minaya Ramirez, V. M. Shabaev, S. Sturm, A. V. Volotka, and G. Werth, Nat. Commun. 7, 10246 (2016).
  * Harman  _et al._ (2018) Z. Harman, B. Sikora, V. A. Yerokhin, H. Cakir, V. Debierre, N. Michel, N. S. Oreshkina, N. A. Belov, J. Zatorski, and C. H. Keitel, J. Phys.: Conf. Ser. 1138, 012002 (2018).
  * Arapoglou  _et al._ (2019) I. Arapoglou, A. Egl, M. Höcker, T. Sailer, B. Tu, A. Weigel, R. Wolf, H. Cakir, V. A. Yerokhin, N. S. Oreshkina, V. A. Agababaev, A. V. Volotka, D. V. Zinenko, D. A. Glazov, Z. Harman, C. H. Keitel, S. Sturm, and K. Blaum, Phys. Rev. Lett. 122, 253001 (2019).
  * Glazov  _et al._ (2019) D. A. Glazov, F. Köhler-Langes, A. V. Volotka, K. Blaum, F. Heiße, G. Plunien, W. Quint, S. Rau, V. M. Shabaev, S. Sturm, and G. Werth, Phys. Rev. Lett. 123, 173001 (2019).
  * Sailer  _et al._ (2022) T. Sailer, V. Debierre, Z. Harman, F. Heiße, C. König, J. Morgner, B. Tu, A. V. Volotka, C. H. Keitel, K. Blaum, and S. Sturm, Nature 606, 479 (2022).
  * Volotka  _et al._ (2013) A. V. Volotka, D. A. Glazov, G. Plunien, and V. M. Shabaev, Annalen der Physik 525, 636 (2013), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.201300079 .
  * Glazov  _et al._ (2014) D. A. Glazov, A. V. Volotka, V. M. Shabaev, and G. Plunien, “Qed theory of the bound-electron magnetic moment,” in  _Fundamental Physics in Particle Traps_, edited by W. Quint and M. Vogel (Springer Berlin Heidelberg, Berlin, Heidelberg, 2014) pp. 137–163.
  * Shabaev  _et al._ (2015) V. M. Shabaev, D. A. Glazov, G. Plunien, and A. V. Volotka, J. Phys. Chem. Ref. Data 44, 031205 (2015).
  * Indelicato (2019) P. Indelicato, Journal of Physics B: Atomic, Molecular and Optical Physics 52, 232001 (2019).
  * Jentschura  _et al._ (2005) U. Jentschura, A. Czarnecki, K. Pachucki, and V. Yerokhin, International Journal of Mass Spectrometry 251, 102 (2005).
  * Grinin  _et al._ (2020) A. Grinin, A. Matveev, D. C. Yost, L. Maisenbacher, V. Wirthl, R. Pohl, T. W. Hänsch, and T. Udem, Science 370, 1061 (2020), https://science.sciencemag.org/content/370/6520/1061.full.pdf .
  * Kaplinghat  _et al._ (1999) M. Kaplinghat, R. J. Scherrer, and M. S. Turner, Phys. Rev. D 60, 023516 (1999).
  * Safronova (2019) M. S. Safronova, Annalen der Physik 531, 1800364 (2019).
  * Debierre  _et al._ (2020) V. Debierre, C. Keitel, and Z. Harman, Phys. Lett. B 807, 135527 (2020).
  * Appelquist and Politzer (1975) T. Appelquist and H. D. Politzer, Phys. Rev. Lett. 34, 43 (1975).
  * Hellwig  _et al._ (1970) H. Hellwig, R. F. C. Vessot, M. W. Levine, P. W. Zitzewitz, D. W. Allan, and D. J. Glaze, IEEE Transactions on Instrumentation and Measurement 19, 200 (1970).
  * Riehle (2006) F. Riehle,  _Frequency standards: basics and applications_ (John Wiley & Sons, 2006).
  * Farley and Wing (1981) J. W. Farley and W. H. Wing, Phys. Rev. A 23, 2397 (1981).
  * Gallagher and Cooke (1979) T. F. Gallagher and W. E. Cooke, Phys. Rev. Lett. 42, 835 (1979).
  * Hollberg and Hall (1984) L. Hollberg and J. L. Hall, Phys. Rev. Lett. 53, 230 (1984).
  * Itano  _et al._ (1982) W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233 (1982).
  * Safronova  _et al._ (2011) M. S. Safronova, M. G. Kozlov, and C. W. Clark, Phys. Rev. Lett. 107, 143006 (2011).
  * Solovyev  _et al._ (2015) D. Solovyev, L. Labzowsky, and G. Plunien, Phys. Rev. A 92, 022508 (2015).
  * Solovyev (2020) D. Solovyev, Annals of Physics 415, 168128 (2020).
  * Solovyev  _et al._ (2020) D. Solovyev, T. Zalialiutdinov, and A. Anikin, Phys. Rev. A 101, 052501 (2020).
  * Solovyev  _et al._ (2021) D. Solovyev, T. Zalialiutdinov, and A. Anikin, Phys. Rev. Research 3, 023102 (2021).
  * Zalialiutdinov  _et al._ (2017) T. Zalialiutdinov, D. Solovyev, and L. Labzowsky, J. Phys. B: At. Mol. Opt. Phys. 51, 015003 (2017).
  * Zalialiutdinov  _et al._ (2020a) T. Zalialiutdinov, D. Solovyev, and L. Labzowsky, Phys. Rev. A 101, 052503 (2020a).
  * Zalialiutdinov  _et al._ (2020b) T. Zalialiutdinov, A. Anikin, and D. Solovyev, arXiv:2006.03928 [physics.atom-ph] (6 Jun 2020b).
  * Solovyev  _et al._ (2019) D. Solovyev, T. Zalialiutdinov, A. Anikin, J. Triaskin, and L. Labzowsky, Phys. Rev. A 100, 012506 (2019).
  * Zalialiutdinov  _et al._ (2022) T. Zalialiutdinov, D. Glazov, and D. Solovyev, Phys. Rev. A 105, 012804 (2022).
  * Zwanziger (1961) D. E. Zwanziger, Phys. Rev. 121, 1128 (1961).
  * Shabaev  _et al._ (2001) V. M. Shabaev, A. N. Artemyev, V. A. Yerokhin, O. M. Zherebtsov, and G. Soff, Phys. Rev. Lett. 86, 3959 (2001).
  * Alcorta and Grifols (1994) R. Alcorta and J. Grifols, Annals of Physics 229, 109 (1994).
  * Asaka  _et al._ (2018) T. Asaka, M. Tanaka, K. Tsumura, and M. Yoshimura, “Precision electroweak shift of muonium hyperfine splitting,”  (2018).
  * Faustov  _et al._ (1999) R. N. Faustov, A. Karimkhodzhaev, and A. P. Martynenko, Phys. Rev. A 59, 2498 (1999).
  * Bég and Feinberg (1974) M. A. B. Bég and G. Feinberg, Phys. Rev. Lett. 33, 606 (1974).
  * Kalinowski  _et al._ (2018) M. Kalinowski, K. Pachucki, and V. A. Yerokhin, Phys. Rev. A 98, 062513 (2018).
  * Heberle  _et al._ (1956) J. W. Heberle, H. A. Reich, and P. Kusch, Phys. Rev. 101, 612 (1956).
  * Kolachevsky  _et al._ (2004) N. Kolachevsky, P. Fendel, S. G. Karshenboim, and T. W. Hänsch, Phys. Rev. A 70, 062503 (2004).
  * Wineland and Ramsey (1972) D. J. Wineland and N. F. Ramsey, Phys. Rev. A 5, 821 (1972).
  * Mathur  _et al._ (1967) B. S. Mathur, S. B. Crampton, D. Kleppner, and N. F. Ramsey, Phys. Rev. 158, 14 (1967).
  * Webb  _et al._ (2001) J. K. Webb, M. T. Murphy, V. V. Flambaum, V. A. Dzuba, J. D. Barrow, C. W. Churchill, J. X. Prochaska, and A. M. Wolfe, Phys. Rev. Lett. 87, 091301 (2001).
  * Han  _et al._ (2019) J. Han, Y. Zuo, J. Zhang, and L. Wang, Eur. Phys. J. D 73, 020502(R) (2019).
  * Clementi  _et al._ (1967) E. Clementi, D. L. Raimondi, and W. P. Reinhardt, The Journal of Chemical Physics 47, 1300 (1967), https://doi.org/10.1063/1.1712084 .