11institutetext: Center for Theoretical Physics, Sloane Physics Laboratory,
Yale University, New Haven, Connecticut 06520, USA 22institutetext: Department
of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana
70803, USA

# Magnetic dipole $\gamma$-ray strength functions of heavy nuclei in the
configuration-interaction shell model

Y. Alhassid 11<EMAIL_ADDRESS>P. Fanto 11<EMAIL_ADDRESS>A.
Mercenne 1122<EMAIL_ADDRESS>
###### Abstract

A low-energy enhancement (LEE) has been observed in the deexcitation
$\gamma$-ray strength function ($\gamma$SF) of compound nuclei. The LEE has
been a subject of intense experimental and theoretical interest since its
discovery, and, if the LEE persists in heavy neutron-rich nuclei, it would
have significant effects on calculations of r-process nucleosynthesis.
Standard configuration-interaction (CI) shell-model calculations in medium-
mass nuclei have attributed the LEE to the magnetic dipole $\gamma$SF but such
calculations are computationally intractable in heavy nuclei. We review a
combination of beyond-mean-field many-body methods within the framework of the
CI shell model that enables the calculation of $\gamma$SF in heavy nuclei, and
discuss the recent theoretical identification of a LEE in the magnetic dipole
$\gamma$SF of lanthanide isotopes.

##  1 Introduction

$\gamma$-ray strength functions ($\gamma$SF) barth1973 , along with level
densities, are necessary input to the Hauser-Feshbach theory hauser1952  of
compound nucleus reactions. In recent years, a low-energy enhancement (LEE)
was observed in the $\gamma$SFs of mid-mass nuclei voinov2004 ; larsen2013 ;
larsen2018  and in several rare-earth nuclei simon2016 ; naqvi2019 ;
guttormsen2022 . If the LEE persists in heavy neutron-rich nuclei, it would
have significant effects on the calculations of $r$-process nucleosynthesis by
enhancing radiative neutron capture rates near the neutron drip line
larsen2010 .

Conventional configuration-interaction (CI) shell-model calculations have
attributed the LEE to the magnetic dipole ($M1$) $\gamma$SF schwengner2013 ;
brown2014 ; karampagia2017 ; schwengner2017 ; midtbo2018 . However, CI shell
model calculations are prohibited in heavy nuclei because of the combinatorial
increase in the dimensionality of the many-particle model space with
increasing number of valence orbitals and/or number of valence nucleons.

The shell-model Monte Carlo (SMMC) method johnson1992 ; alhassid1994 ;
koonin1997 ; alhassid2008  enables microscopic calculations in model spaces
that are many orders of magnitude larger than those that can be treated by
conventional diagonalization methods; see Ref. alhassid_rev  for a recent
review. It has been established as the state-of-the-art method for the
microscopic calculation of level densities.

The microscopic calculation of $\gamma$SFs in the framework of SMMC has been a
major challenge. A strength function is usually calculated from the Fourier
transform of the real-time response function. However, in SMMC we can only
calculate imaginary-time response functions whose inverse Laplace transform is
the strength function. This inverse Laplace transform is numerically an ill-
posed problem and requires the analytic continuation to real time. This can be
done numerically using the maximum entropy method gubernatis1991 ; jarrell1996
; gubernatis_book , but its success depends crucially on a good choice of a
prior strength function. We found that the static-path approximation plus
random-phase approximation (SPA+RPA) puddu1991 ; attias1997 ; rossignoli1998b
provides a good prior strength function fanto_arXiv . The SPA itself also
provides a good prior strength at not too low temperatures mercenne2023 . The
SPA+RPA includes large-amplitude static fluctuations lauritzen1988  of the
auxiliary fields plus small time-dependent fluctuations around each static
configuration. It was recently applied to calculate state densities in heavy
nuclei in close agreement with SMMC state densities fanto2021 .

##  2 The shell model Monte Carlo (SMMC) method

SMMC is based on the Hubbard-Stratonovich transformation stratonovich1957 ;
hubbard1959 , in which the Gibbs ensemble $e^{-\beta\hat{H}}$ is written as a
superposition of ensembles $U_{\sigma}$ of non-interacting nucleons that move
in time-dependent fields $\sigma(\tau)$

$e^{-\beta H}=\int D[\sigma]G_{\sigma}U_{\sigma}\;,$ (1)

where $G_{\sigma}$ is a Gaussian weight and $D[\sigma]$ is an integration
measure.

The calculation of the integrand in Eq. (1) reduces to matrix algebra in the
single-particle space of typical dimension $\sim 50-100$ for heavy nuclei.
However, the main challenge is carrying out the integration over the large
number of auxiliary fields. This is done through the use of Monte Carlo
methods to select the important configurations of the auxiliary fields.

SMMC is a powerful method to calculate thermal observables in the presence of
correlations but it cannot be used to calculate directly $\gamma$SFs. In SMMC,
we can calculate the imaginary-time response function of the respective
electromagnetic operator $\hat{O}_{\lambda}$, and it is necessary to carry out
the analytic continuation to real time.

##  3 Finite-temperature $\gamma$SF

The finite-temperature strength function of an hermitean tensor operator
$O_{\lambda}$ of rank $\lambda$ (e.g., $E1$, $M1$,…) is defined by

$\displaystyle
S_{\\!\mathcal{O}_{\lambda}}(\omega)=\sum_{\begin{subarray}{c}\alpha_{i}J_{i}\\\
\alpha_{f}J_{f}\end{subarray}}\frac{e^{-\beta
E_{\alpha_{i}J_{i}}}}{Z}|(\alpha_{f}J_{f}||\hat{\mathcal{O}}_{\lambda}||\alpha_{i}J_{i})|^{2}\delta(\omega-
E_{\alpha_{f}J_{f}}+E_{\alpha_{i}J_{i}})\,,$ (2)

where $(\alpha J)$ denotes eigenstates with energy $E_{\alpha J}$ and spin
$J$, and $Z=\sum_{\alpha J}(2J+1)e^{-\beta E_{\alpha J}}$ is the canonical
partition function. The strength function can be written as a Fourier
transform of the real-time response function of $\hat{O}_{\lambda}$. However,
in SMMC it is only possible to calculate the imaginary-time response function

$R_{\mathcal{O}_{\lambda}}(\tau)=\left\langle\sum_{\lambda\mu}(-)^{\mu}\hat{O}_{\lambda\mu}(\tau)\hat{O}_{\lambda-\mu}(0)\right\rangle\;,$
(3)

where the expectation value is over the canonical ensemble at temperature $T$
and
$\hat{O}_{\lambda\mu}(\tau)=e^{\tau\hat{H}}\hat{O}_{\lambda\mu}e^{-\tau\hat{H}}$.
The strength function is then the Laplace transform of the imaginary-time
response function. The $\gamma$SF satisfies the symmetry relation
$S_{\\!\mathcal{O}_{\lambda}}(-\omega)=e^{-\beta\omega}S_{\\!\mathcal{O}_{\lambda}}(\omega)$
and the Laplace transform can be rewritten in the form

$R_{\mathcal{O}_{\lambda}}(\tau)=\int_{0}^{\infty}d\omega\,K(\tau,\omega)S_{\\!\mathcal{O}_{\lambda}}(\omega)\,,$
(4)

where $K(\tau,\omega)=e^{-\tau\omega}+e^{-(\beta-\tau)\omega}$ is a
symmetrized kernel.

##  4 Analytic continuation: maximum-entropy method

The inversion of (4) to calculate the strength function from the imaginary-
time response function requires analytic continuation to real time and is
numerically an ill-defined problem that does not have a unique solution. We
use the maximum entropy method (MEM) jarrell1996  to carry out this analytic
continuation numerically. In this method we choose the strength function that
maximizes the objective function

$\mathcal{Q}(S_{\\!\mathcal{O}_{\lambda}};\alpha)=\alpha\mathcal{S}-\frac{1}{2}\chi^{2}\,$
(5)

at a given value of $\alpha$. Here $\chi^{2}$ is given by

$\chi^{2}=(\overline{R}_{\mathcal{O}_{\lambda}}-R_{\mathcal{O}_{\lambda}})^{T}\mathcal{C}^{-1}(\overline{R}_{\mathcal{O}_{\lambda}}-R_{\mathcal{O}_{\lambda}})\,,$
(6)

where ${\bar{R}}$ and ${\mathcal{C}}$ are the SMMC response function and its
covariance matrix, respectively. $\mathcal{S}$ is the entropy function

$\mathcal{S}=-\int
d\omega\,\bigg{(}S_{\\!\mathcal{O}_{\lambda}}(\omega)-S^{\rm
prior}_{\\!\mathcal{O}_{\lambda}}(\omega)-S_{\\!\mathcal{O}_{\lambda}}(\omega)\ln\left[S_{\\!\mathcal{O}_{\lambda}}(\omega)/S^{\rm
prior}_{\\!\mathcal{O}_{\lambda}}(\omega)\right]\bigg{)}\,,$ (7)

where $S^{\rm prior}_{\\!\mathcal{O}_{\lambda}}(\omega)$ a prior strength
function. The coefficient $\alpha$ controls the relative weight between the
entropy and the $\chi^{2}$ terms. We use Bryan’s method bryan1990 , in which
the final strength function is obtained by averaging over $\alpha$ with a
certain probability distribution.

##  5 Prior strength function: SPA + RPA

The choice of a good prior strength is key to the success of the MEM. We
evaluate the prior strength function in the SPA+RPA puddu1991 ; attias1997 ;
rossignoli1998b

$S^{\rm prior}_{\\!\mathcal{O}_{\lambda}}(\omega)\approx\frac{\int d\sigma
M(\sigma)Z_{\eta}(\sigma)C_{\eta}(\sigma)S_{\\!\mathcal{O}_{\lambda},\eta}(\sigma;\omega)}{\int
d\sigma M(\sigma)Z_{\eta}(\sigma)C_{\eta}(\sigma)}\,,$ (8)

where $\sigma$ are static auxiliary fields, and $M(\sigma)$ is a measure
fanto2021 . The quantity $Z_{\eta}(\sigma)={\rm
Tr}\left[\hat{P}_{\eta}e^{-\beta\left(\hat{h}_{\sigma}-\sum_{\lambda=p,n}\mu_{\lambda}\hat{N}_{\lambda}\right)}\right]$
is the number-parity projected one-body partition function, where
$\hat{h}_{\sigma}$ is a one-body Hamiltonian for a given static configuration
of the fields, and $P_{\eta}=(1+\eta e^{i\pi\hat{N}})/2$ is the number-parity
projection with $\eta=+1(-1)$ for even (odd) number parity. $C_{\eta}(\sigma)$
is the RPA correction factor that accounts for the Gaussian integration over
small amplitude time-dependent auxiliary-field fluctuations puddu1991 ;
attias1997 ; rossignoli1998a ; nesterov2013 ; fanto2021 .
$S_{\\!\mathcal{O}_{\lambda},\eta}(\sigma;\omega)$ is a $\sigma$-dependent
QRPA strength function at temperature $T$; see Eqs. (5) and (6) in Ref.
fanto_arXiv . The integration in Eq. (8) is carried out using the Metropolis-
Hastings algorithm with a weight function of
$\mathcal{W}_{\eta}(\sigma)=M(\sigma)Z_{\eta}(\sigma)$.

The above SPA+RPA strength function was used as prior for a pairing plus
quadrupole interaction fanto_arXiv . The complete SMMC interaction includes
additional components, and the SPA+RPA, which requires the diagonalization of
the QRPA matrix at many static configurations of the auxiliary field, becomes
time consuming. Instead we apply the SPA alone mercenne2023 , for which
$C_{\eta}(\sigma)=1$ and the strength function in the integrand is the
$\sigma$-dependent SPA strength function.

##  6 $M1$ $\gamma$SFs in lanthanide isotopes

We applied the methods discussed above to isotopic chains of lanthanides. The
single-particle valence model space (using Woods-Saxon central potential plus
spin-orbit interaction) is the 50-82 shell plus the $1f_{7/2}$ orbital for
protons, and the 82-126 shell plus the $0h_{11/2}$ and $1g_{9/2}$ orbitals for
neutrons. For the samarium isotopes we used a pairing plus quadrupole
interaction and for the neodymium isotopes we used monopole pairing and
multipole-multipole interaction that includes quadrupole, octupole and
hexadecupole components ozen2013 .

For the magnetic dipole transition operator $M1$ we used

$\hat{\mathcal{O}}_{M1}=\sqrt{\frac{3}{4\pi}}\frac{\mu_{N}}{\hbar
c}\left(g_{l}\mathbf{l}+g_{s}\mathbf{s}\right)\,,$ (9)

where $\mathbf{l}$ and $\mathbf{s}$ are the orbital and spin angular momentum
operators, respectively. We used the free-nucleon $g$ factors $g_{l,p}=1$,
$g_{l,n}=0$, $g_{s,p}=5.5857$, and $g_{s,n}=-3.8263$.

###  6.1 Samarium isotopes

In Fig. 1 we show the imaginary-time $M1$ response function for chain of
samarium isotopes at a low temperature of $T=0.22$ MeV. The SPA+RPA response
function (orange dots) is close to the exact SMMC response function (black
squares), indicating that the SPA+RPA is a good choice for a prior $M1$
$\gamma$SF. For comparison we also show the SPA response function (blue
circles).

Figure 1: The $M1$ imaginary-time response functions in a chain of even-mass
samarium isotopes. The SPA +RPA results (orange dots) are compared with the
SMMC results (black dots). Also shown are the SPA (blue circles) and MEM
(green dashed lines) response functions. Adapted from Ref. fanto_arXiv .

The corresponding $M1$ $\gamma$SFs are shown in Fig. 2 with positive values of
$\omega$ describing absorption of $\gamma$-rays. We apply the MEM discussed in
Sec. 4 with the SPA+RPA strength as the prior strength to find the dashed
green lines. These provide us with the $M1$ $\gamma$SFs near the ground state.
We observe a peak around 6 MeV, which we interpret as the spin-flip mode. As
we add neutrons and the nucleus becomes more deformed, some of the strength in
this peak is transferred to a structure around 1-3 MeV, which is consistent
with the scissors mode known in heavy deformed nuclei bohle1984 ; heyde2010 .

Figure 2: $M1$ $\gamma$SFs near the ground state in a chain of even-mass
samarium isotopes. The MEM $M1$ $\gamma$SFs (dashed green lines) are compared
with the SPA (blue circles) and the SPA+RPA (orange dots) strengths. Adapted
from Ref. fanto_arXiv .

In Fig. 3 we show the corresponding $M1$ $\gamma$SFs for the samarium isotopes
but at temperatures that correspond to the neutron resonance energies. We
observe a pronounced peak at $\omega\approx 0$ that, as shown below, leads to
a low-energy enhancement in the $\gamma$-ray decay. As neutron number
increases, the strength of this peak reduces and a small peak develops at
$\omega\approx 2$ MeV, which we interpret as a scissors mode built on top of
excited states krticka2011 . We still observe the spin-flip mode at
$\omega\approx 6$ MeV.

Figure 3: Same as in Fig. 2 but at temperatures that correspond to the neutron
resonance energies. Adapted from Ref. fanto_arXiv .

###  6.2 Neodymium isotopes

We calculated the $M1$ $\gamma$SFs for a chain of even-mass neodymium
isotopes, for which recent experimental data are available. We used the SPA
strength as the prior strength for the MEM. In Fig. 4 we show the
corresponding $M1$ $\gamma$SFs for the neodymium isotopes at temperatures that
correspond to the neutron resonance energies. The MEM results (green lines)
are compared with the SPA strengths (dashed orange lines).

We observe similar features as for the samarium isotopes, and in particular a
peak at $\omega\approx 0$. We also identify a spin-flip mode close to 6 MeV,
and with the addition of neutrons a scissors mode around 1-2 MeV, both of
which are built on top of excited states.

Figure 4: $M1$ $\gamma$SFs at the corresponding neutron resonance energies for
a chain of even-mass neodymium isotopes. The MEM $\gamma$SFs (green lines) are
compared with the SPA $\gamma$SFs (dashed orange lines). Adapted from Ref.
mercenne2023 .

Experimental studies often use the deexcitation strength function $f_{M1}$
which can be related to $S_{\\!M1}$ by private

${f}_{M1}({E}_{\gamma},{E}_{i})\approx\frac{1}{3}a\frac{\tilde{\rho}(E_{i})}{\tilde{\rho}(E_{i}-E_{\gamma})}{S}_{M1}(\omega=-{E}_{\gamma})\;,$
(10)

where $a=\frac{16\pi}{9(\hbar c)^{3}}$ and ${E}_{\gamma}$ is the emitted
$\gamma$-ray energy. $\tilde{\rho}(E_{i})$ and
$\tilde{\rho}(E_{i}-E_{\gamma})$ are the initial and final level densities,
respectively.

In Fig. 5 we show $f_{M1}$ for the even-mass neodymium isotopes as a function
of the emitted $\gamma$-ray energy $E_{\gamma}$. We used Eq. (10), where the
level densities of the neodymium isotopes were computed within SMMC using spin
projection methods guttormsen2021 . We clearly observe a LEE at low
$\gamma$-ray energies that originates from the $\omega\approx 0$ peak in
$S_{\\!M1}(\omega)$.

We find that the LEE follows an exponential decay $\sim e^{-E_{\gamma}/T_{B}}$
as was observed in Refs. schwengner2013 ; karampagia2017 . The fitted values
of $T_{B}$ are approximately constant over a range of initial energies.

Figure 5: $f_{M1}$ vs. $E_{\gamma}$ in a chain of even-mass neodymium
isotopes. The MEM results (green lines) are compared with the SPA results
(dashed orange lines). Adapted from Ref. mercenne2023 .

##  7 Conclusion

We reviewed a method to calculate $\gamma$-ray strength functions for heavy
nuclei in the the framework of the configuration-interaction shell model by
combining two many-body methods: SMMC and the SPA+RPA or SPA fanto_arXiv ;
mercenne2023 . Using this method, we made the first theoretical identification
of a low-energy enhancement (LEE) in heavy nuclei. In particular, we observed
a LEE in the $M1$ $\gamma$SFs in isotopic chains of samarium fanto_arXiv  and
neodymium mercenne2023  nuclei. We also observed the emergence of a structure
consistent with a scissors mode built on top of excited states as neutron
number increases within each chain.

The LEE was observed experimentally in lanthanide isotopes simon2016 ;
naqvi2019 ; guttormsen2022  in experiments based on the Oslo method, but these
experiments cannot separate the $E1$ and $M1$ components of the $\gamma$SF. A
detailed comparison with such experiments requires the calculation of the $E1$
$\gamma$SF.

##  8 Acknowledgments

This work was supported in part by the U.S. DOE grant No. DE-SC0019521, and by
the U.S. DOE NNSA Stewardship Science Graduate Fellowship under cooperative
agreement No. NA-0003960. The calculations used resources of the National
Energy Research Scientific Computing Center (NERSC), a U.S. Department of
Energy Office of Science User Facility operated under Contract No. DE-
AC02-05CH11231. We thank the Yale Center for Research Computing for guidance
and use of the research computing infrastructure.

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[1,2]Hanieh Fattahi

1]Max Planck Institute for the Science of Light, Staudstrasse 2, Erlangen,
91058, Germany

2]Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Staudstrasse 7, Erlangen, 91058, Germany

# Near-Petahertz Fieldoscopy of Liquid

Anchit Srivastava  Andreas Herbst  Mahdi M. Bidhendi  Max Kieker  Francesco
Tani<EMAIL_ADDRESS>[  [

###### Abstract

Measuring transient optical field is pivotal not only for understanding
ultrafast phenomena but also for quantitative detection of various molecular
species in a sample. In this work, we demonstrate near-petahertz electric
field detection of a few femtosecond pulses with 2oo attosecond temporal
resolution, 108 detection dynamic range in electric field and sub-femtojoule
detection sensitivity, exceeding those reported by the current methods. By
field-resolved detection of the impulsively excited molecules in the liquid
phase, termed “femtosecond fieldoscopy”, we demonstrate temporal isolation of
the response of the target molecules from those of the environment and the
excitation pulse. In a proof-of-concept analysis of aqueous and liquid
samples, we demonstrate field-sensitive detection of combination bands of 4.13
$\mu$mol ethanol for the first time. This method expands the scope of aqueous
sample analysis to higher detection sensitivity and dynamic range, while the
simultaneous direct measurements of phase and intensity information pave the
path towards high-resolution biological spectro-microscopy.

###### keywords:

Near-infrared spectroscopy, Field-resolved spectroscopy, ultrashort pulses,
time-domain spectroscopy

##  1 Introduction

Laser-based, label-free quantitative determination of sample composition has
proven to be a potent tool across a wide spectrum from fundamental research to
real-life applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For accurate and
delicate spectroscopic measurements, it has been crucial to isolate the sample
from environmental interferences. For instance, water comprises approximately
60 % of the human body, envelops 70 % of the Earth’s surface, and permeates
our surroundings through the atmosphere. Water has a broad absorption spectrum
spanning from the visible to mid-infrared (MIR) and is a persistent component
on our detectors. Due to its strong absorption cross-section at MIR, sensitive
spectroscopy of samples at their resonance frequencies is challenging, as
water dominates other, more subtle, absorbance features arising from other
molecules. Moreover, the excessive absorbed energy by water at MIR is left in
the sample as thermal energy, limiting non-invasive analysis. On the other
hand, NIR spectroscopy provides a fingerprint of sample constituents similar
to MIR spectroscopy. It distinguishes itself by offering higher spatial
resolution and enhanced penetration depth, afforded by the lower absorption
cross-section of water in the NIR region [12, 13, 14]. This feature makes NIR
spectroscopy particularly suitable for the non-invasive and label-free
examination of soft matter and large-volume aqueous samples [15, 16].

Overtone and combination vibrations of molecules, which are primarily detected
in NIR spectroscopy extend beyond 0.12 petahertz (PHz). Spectrometers have
been used for frequency domain detection in this range. However, their
detection sensitivity is constrained by the excitation light, which manifests
as a background within the same spectral range [17, 18, 19]. Since the
resonance frequencies of overtone and combination bands surpass the
sensitivity of silicon-based detectors, employing a second detector, generally
more prone to noise in this range, is necessary to capture the system’s entire
response. Fourier transform spectroscopy is an alternative method allowing for
precise spectroscopic detection. Recent advancements in NIR dual-comb
spectroscopy have paved the way for precision Fourier transform spectroscopy,
albeit primarily in the gas phase [20, 21, 22, 23, 24]. Nonetheless, the
technique remains constrained by detectors’ spectral response and the
existence of a background signal analogous to frequency domain detection [25,
13].

In contrast, field-resolved detection allows for the direct measurement of
light-matter interactions with attosecond precision in a sub-cycle regime,
capturing both amplitude and phase information [26, 27]. For decades
attosecond streaking was the sole method to probe the electric field of light
with a bandwidth approaching the PHz [28, 29]. A significant drawback was its
confinement to vacuum operations. Over the past decade, various techniques
have been developed that enable the near-PHz field-resolved detection of light
in ambient air [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46]. Among these techniques electro-optic sampling (EOS) stands out for
its unparalleled detection sensitivity [47, 48, 27]. In EOS a short probe
pulse is employed to resolve the cycles of the electric field of light by up-
converting its spectral bandwidth to higher frequencies, making it possible to
apply silicon detectors for broadband NIR detection [49]. Moreover, the
combination of bright, ultra-short pulses [50, 51, 52, 53] and heterodyne
detection allows for higher detection signal-to-noise ratio and higher
detection sensitivity, leaving the shot noise of the probe pulse the primary
source of noise [54].

In this work, we report on the direct detection of the electric field of light
at near-PHz frequencies with unparalleled sensitivity and dynamic range. This
has been enabled by developing a unique laser source delivering broadband
pulses with carrier-to-envelope phase (CEP) stability, which were
intrinsically synchronized to near-single-cycle pulses at megahertz (MHz)
repetition rates. The unique frontend enabled direct electric-field detection
of CEP-stable, few-cycle pulses with unprecedented detection sensitivity and
dynamic range via EOS with attosecond temporal resolution.

Employing the bright ultrashort pulses, we report on the field-sensitive
detection of molecular response at the NIR region for the first time. Few-
femtosecond, phase-coherent pulses were utilized for both broadband molecular
excitation and the near-PHz electric-field detection of their response. Here,
the confinement of the excitation pulses allows for temporal gating of the
molecular response, while accessing the electric field enables the precise
detection of the response of the target molecules from those of the
environment. We evaluated our approach by conducting field-resolved detection
of water vibration modes in both gas and liquid phases in the NIR region.
Additionally, we detected the subtle combination bands of ethanol in the
liquid phase. These results show to the best of our knowledge the first field-
detection of the NIR molecular response in ambient air, paving the path for
the emergence of innovative, field-sensitive, label-free spectroscopy and
microscopy techniques.

Figure 1: Near-Petahertz fieldoscopy. (a) An ultrashort pulse excites
molecules at their NIR resonances. Here, the molecules inside a cuvette
represent the sample under scrutiny, while the surrounding molecules represent
atmospheric water vapor molecules. The transmitted field contains the global
molecular response of both the sample and the environment. A second short
pulse at higher frequencies is used for up-conversion and generation of a
delay-dependent signal in a nonlinear crystal, where the correlation signal is
directly proportional to the electric field of the excitation pulse. The
measured electric contains the ultrashort excitation pulse, the delayed
response of the liquid spanning over several picoseconds, and a long-lasting
response of atmospheric gases lasting for hundreds of nanoseconds. By time
filtering and subsequent data analysis, the molecular response can be
decomposed to the short-lived liquid and long-lived gas responses. (b) The
biological relevant vibrational modes at NIR spectral range. Different
compounds like proteins, carbohydrates, lipids, polyphenols, and alcohols are
associated with the shown bands. Associated bonds are described in the legend
below the plot where “str” refers to stretching vibration, whereas “bend”
refers to bending vibration. The numbers one and two indicate the first and
second overtones, while the plus sign (+) indicates combination bands. The
values were taken from [16].

##  2 Results

Fig. 1 illustrates the measurement concept. Few-cycle, CEP-stable, NIR pulses,
are utilized to extract sensitive spectroscopic information from liquid
samples under atmospheric conditions. The electric field of the molecular
response, in the wake of the excitation pulse, is resolved via EOS. Due to the
high detection sensitivity, we can resolve not only the response of the
molecular vibrations of the sample at their overtone and combination
resonances but also the response of atmospheric molecules along the beam path.
The temporal confinement of the excitation pulses ensures that the molecular
responses from both the sample and ambient air are temporally separated from
the excitation pulse. Moreover, the response of the liquid sample is
temporally distinguished from the ambient air’s fingerprint due to the faster
dephasing in the liquid phase. Realizing such a concept requires the
generation of intrinsically synchronized bright few-cycle pulses with at least
one-octave separation and sub-cycle temporal synchronization.

Figure 2: Experimental setup. (a) The shaded regions highlight different parts
of the optical setup: nonlinear fiber stages in pink, IPDFG in blue, and EOS
in green. (b) Spectrum of the laser (red), the first fiber stage (blue), and
the second fiber stage (black). (c) Measured (top) and retrieved (bottom)
spectrograms. The near-single-cycle pulses were measured after the nonlinear
compression stages via second-harmonic generation frequency-resolved optical
gating (SHG-FROG). (d) The retrieved temporal pulse duration at the output of
the second fiber at a 1 MHz repetition rate. (e) Retrieved spectral intensity
and phase. SR-PCF: single-ring hollow-core photonic crystal fiber; DM:
dichroic mirror; WP: wedge pair; BS: beam splitter; WGP: wire grid polarizer;
FEL: long pass filter; FES: short pass filter; QWP: quarter waveplate; BD:
balanced photodiode; BBO: beta barium borate; BiBO: bismuth borate; EDFA:
erbium-doped fiber amplifier.

The pink-shaded region in Fig. 2(a) shows the optical setup for near single-
cycle pulse generation at MHz repetition rates. Single-ring hollow-core
photonic crystal fibers (SR-PCF) are used due to their relatively low-loss,
broadband guidance, and tunable dispersion [55, 56], allowing the generation
of ultrashort, bright pulses containing tens of microjoules of energy [53,
57]. In the first stage, 20 $\mu$J, 1 MHz laser pulses were compressed from
255 fs to 25 fs at full width at half maximum (FWHM), by self-phase-modulation
based spectral broadening in Argon-filled SR-PCF (see supplementary
information (SI) Fig. 5 and Fig. 6), followed by group-delay dispersion
compensation by a chirped mirror (CM) compressor. Subsequently, in the second
stage, the 25 fs pulses were compressed to a near single-cycle duration via
soliton-effect self-compression in a similar fiber (Fig. 2(b)). In both
stages, the gas species used as a medium were selected to minimize
photoionization and subsequent long-lived effects occurring at MHz repetition
rates [58]. The accumulated dispersion on the near single-cycle pulses due to
propagation in media after the fiber was compensated in a CM compressor to 4.8
fs at FWHM, limited by the bandwidth of the CM compressor (Fig. 2(c) and Fig.
2(d)). This corresponds to 3.5 GW of peak power with 69 % of the energy in the
main pulse. The retrieved spectrum shown in Fig. 2(e) spans over 300 THz
bandwidth supporting 3 fs pulses at FWHM.

Figure 3: NIR electric field sampling. (a) The measured electric field of CEP-
stable pulses via EOS. The inset displays two low-energy fields in the
presence of ND4 (magenta) and ND8 filters (yellow). Both fields are longer
than the blue curve due to additional material dispersion caused by the ND
filters. (b) The temporal profile of the CEP-stable pulses with a pulse
duration of 15 fs FWHM. (c) Retrieved spectrum and phase of the CEP-stable
pulses. (d) Retrieved spectrum of the CEP-stable pulses after attenuation with
the ND4 and ND8 filters. The spectrum of the unattenuated pulse is shown in
blue for comparison. The dashed black line represents the measured noise floor
in the absence of the CEP-stable pulses on the balanced detector. The legend
displays the acquisition time for each field, along with the ND filter label.
This highlights the system’s ability to measure fields with a dynamic range
greater than 108 in the electric field. ND: neutral density. The retrieved
spectra are not corrected for the spectral response of the filters.

Intrapulse difference frequency generation (IPDFG) was employed to generate
broadband NIR pulses with a passive CEP-stability (blue region in Fig. 2(a))
[59]. A 500 $\mu$m-thick bismuth borate (BiBO) crystal was pumped by 4.8 fs
pulses at 1 $\mu$m to generate a broadband spectrum spanning from 0.1 PHz to
0.23 PHz and 160 mW of average power. A custom-made dichroic beam splitter
with the cut-off at 0.2 PHz was used to separate the residual pump from the
CEP-stable pulses. After dispersion management with a custom-made CM
compressor, the NIR pulses along with the fraction of 4.8 fs pulses were sent
to a 20 $\mu$m beta barium borate (BBO) crystal for electric-field sampling.
The green-shaded region in Fig. 2(a) highlights the schematic of the NIR EOS.
The up-converted signal is spectrally filtered with a pass band filter from
0.425 PHz to 0.5 PHz to enhance the detection sensitivity by eliminating the
spectral components that do not carry specific field information. The measured
electric field of the broadband CEP-stable pulses and its temporal intensity
profile with a 15 fs pulse duration at FWHM are shown in Fig. 3(a) and Fig.
3(b), respectively. The corresponding spectral intensity and phase obtained
through the Fourier transform shown in Fig. 3(c) reveals residual higher-order
dispersion, which can be compensated by optimizing the design of the CM
compressor. While the high-frequency cut-off of the spectrum is limited by the
beam splitter roll-off, the crystal absorption constrains the low-frequency
cut-off to 0.1 PHz. To verify the detection sensitivity and dynamic range of
the detector, the energy of the IPDFG pulses before EOS crystal was reduced
from 80 nJ to sub-femto joule energies by using a series of neutral density
filters. Fig. 3(a) inset shows two field-resolved measurements at 8 pJ
(magenta) and 0.7 fJ (in yellow), respectively. The spectral intensity
counterpart at the three different pulse energies is shown in Fig. 3(d)
corresponding to a 110 dB dynamic range. To establish the ability of the
system to detect the response of minute quantities of molecules, we resolved
the electric field of the atmospheric water vapor molecules, liquid water, and
ethanol in ambient air after excitation by femtosecond NIR pulses. The
temporally gated molecular response and their frequency counterparts are shown
in Fig. 4.

##  3 Discussion

Figure 4: Benchmarking measurements. Each row shows the time-gated electric
field on the left and its Fourier-transformed spectrum on the right. (a) The
light grey field represents atmospheric water molecules at 50 % RH (16.5
$\mu$mol), while the black curve represents 8 % RH (2.64 $\mu$mol). (b) Two
absorption modes are visible: a fundamental mode at 0.115 PHz and a
combination band at 0.16 PHz. (c) Liquid water molecules at 5.23 $\mu$mol
(red) and 2.64 $\mu$mol (pink) along with pure acetic acid (brown) as a
reference measurement. (d) The molecular response of pure acetic acid is
compared to that of aqueous solutions. (e) Pure liquid ethanol at different
volumes of 4.13 $\mu$mol, 16.5 $\mu$mol, and 82.5 $\mu$mol. (f) The weak
combination peak centered at 0.130 PHz is observed at all three volumes.

Water has two prominent absorption peaks within the spectral coverage of the
excitation pulses: i) an asymmetric stretch centered at 0.115 PHz (3836 cm-1)
and ii) a combination resonance of bending and asymmetrical stretch centered
at 0.16 PHz (5337 cm-1) [60]. Fig. 4(a) shows the electric field of
atmospheric water vapor molecules at two different laboratory relative
humidity (RH). The measurements at 50 % RH and 8 % RH, correspond to 16.5
$\mu$mol and 2.64 $\mu$mol of atmospheric water vapour molecules interacting
with the broadband 15 fs excitation pulses, respectively (see SI). The minimum
laboratory achievable RH of 8 % was reached by purging the beam path with dry
air and nitrogen. Fig. 4(b) shows the corresponding absorption frequencies for
both concentrations. The spectra are achieved by Fourier-transformation of the
temporally gated molecular response at 1 ps after the excitation pulse for a
temporal window of 30 ps and agree with the HITRAN database [61] (see SI
Fig.7). At the 2.64 $\mu$mol level, the absorption peak at 0.16 PHz is barely
resolved due to the low absorption cross-section of the water’s combination
band (65$\times$10-21cm2/molecule) compared to its counterpart at the
fundamental resonance (600$\times$10-21cm2/molecule) [61].

Fig. 4(c) shows the measured molecular response of 5.23 $\mu$mol and 2.64
$\mu$mol diluted water in the liquid phase, which were prepared by mixing 20
$\mu$L and 10 $\mu$L of deionized water in 1 mL acetic acid. We examined the
molecular response in the liquid phase in the temporal window of 0.3 ps to 1
ps, as dephasing occurs faster (Fig. 4(d)) [62, 63]. The absorption amplitude
of both concentrations was normalized to 0.2 PHz peak, which is present at
this time scale due to the cut-off of the dichroic beam splitter used in our
setup. Water’s asymmetric stretch resonance at 0.115 PHz (3836 $cm^{-1}$) in
Fig. 4(d) is distinct for 5.23 $\mu$mol (red curve) and 2.64 $\mu$mol (magenta
curve). The pure acetic acid response in the presence of water vapor molecules
at RH of 8 % is shown in Fig. 4(d) (black curve). To evaluate the detection
sensitivity of our setup in environmental conditions, ethanol was measured due
to its distinct resonance frequency in comparison to water and its low
absorption cross-section. Fig. 4(e) and Fig. 4(f) show the resonance of pure
liquid ethanol at different concentrations and 8 % RH, in time and frequency
domain, respectively. The weak absorption at 0.13 PHz (4336 $cm^{-1}$) is due
to the combination band resonance arising from C-H stretch and C-H bend mode,
which was resolved very clearly in our measurement at the minimum detectable
amount of 4.13 $\mu$mol [64].

We define a figure of merit (FOM) to allow detection sensitivity comparison of
absorption bands between different species and various absorption cross-
sections. The FOM is defined as:

$FOM=n\times\sigma$ (1)

Where n is the amount of substance in mol and $\sigma$ is the absorption
cross-section of the absorption band in cm2/molecule. Given the absorption
cross-section values of water (600$\times$10-21 cm2/molecule) and ethanol
(3.2$\times$10-21 cm2/molecule) [61, 65], the calculated FOM value for water
at 2.64 $\mu$mol is $9\times 10^{-28}$ m2 mol/molecule, while FOM for
ethanol’s combination band at 4.13 $\mu$mol is $1.322\times 10^{-30}$ m2
mol/molecule with merely 25 $\mu$m path length.

In conclusion, we report on field-sensitive, near-PHz detection of molecular
fingerprints in the liquid phase for the first time. To this end bright,
intrinsically synchronized CEP-stable, 15 fs pulses at 2 $\mu$m, and 4.8 fs
pulses at 1 $\mu$m were generated for impulsive excitation and probing of the
molecular response via EOS. Using the MHz ultrashort laser pulses, we
demonstrated the ambient air field-resolved detection of femtosecond pulses
with sub-femtojoule energy, and $10^{8}$ detection dynamic range in the
electric field, enhancing the near-PHz field-detection sensitivity by three
orders of magnitude compared to other field-resolved techniques [47]. The
source’s MHz repetition rate augments both the signal-to-noise ratio and
detection sensitivity, while the signal upconversion in our scheme alleviates
the bandwidth constraint inherent in silicon-based detectors. In a proof of
concept, we reported on the sensitive detection of the vibration modes of
atmospheric and aqueous water molecules at 2.64 $\mu$mol, and ethanol
combination band at 4.13 $\mu$mol. To the best of our understanding, these
measurements mark the first field-resolved detection of both fundamental and
combination bands in the liquid phase.

Femtosecond pump-probe spectroscopy has provided evidence that rapid dynamics
occurring within a time scale of fewer than 100 fs of liquid water have a
significant impact on chemical reactions taking place in the aqueous phase
[66]. This underscores the vital importance of ultrafast processes in
comprehending aqueous phase chemistry for example on grasping how water
molecules dissipate energy [67]. The electric field measurements presented in
Fig. 4 thus establish a foundation for studying aqueous solutions with
enhanced sensitivity and dynamic range compared to femtosecond intensity pump-
probe techniques [68, 69], rooted in the higher amplitude of the molecular
response relative to the excitation pulses in field-resolved detection (see SI
Fig. 8) [70, 71]. Furthermore, the high repetition rate of near-single-cycle
pulses not only lays the groundwork for single-shot monitoring of chemical
reactions in liquids [72] but also presents intriguing possibilities for
exploring nonlinear interactions due to the unique combination of peak and
average power in the near-single-cycle domain.

Stimulating the molecular composition of a sample with phase-coherent
femtosecond excitation pulses leads to temporal gating between the molecular
response from the excitation pulses. Moreover, accessing the sub-cycle
electric field of light allows for decomposing the short-lived liquid
molecular response from the long-lived ambient gas responses (see SI Fig. 9).
Femtosecond fieldoscopy expands the scope of aqueous sample analysis and paves
the path toward novel methods for multi-dimensional spectroscopy and high-
resolution biological spectro-microscopy.

##  4 Materials and Methods

###  4.1 Near single-cycle pulse generation

A commercially available Yb:KGW amplifier (CARBIDE Light Conversion)
delivering 255 fs pulses at 1030 nm with 20 W of average power, and at 1 MHz
repetition rate, is used as the source laser. In the first nonlinear fiber
stage, we used a 100 mm focal length (Thorlabs LA1509-B-ML) lens to couple 20
$\mu$J circularly polarized pulses into a 50 cm long SR-PCF with a core
diameter of 55 $\mu$m (see SI Fig. 10) filled with 15 bar of Argon. The
spectrally broadened laser pulses were compressed to 25 fs (FWHM) using a CM
compressor (UltraFast Innovations GmBH PC1611) with 12 bounces (see SI Fig.
11). The total group delay dispersion compensated by the CM compressor was
-1800 fs2. Subsequently, the 25 fs compressed pulses were coupled (via
Thorlabs LA1509-BML) to a 31 cm long SR-PCF (parameter same as before), filled
with 15 bar of Helium. After the second stage, a 2-inch off-axis silver
parabola (Edmund 36-598) with an effective focal length of 177.8 mm was used
to collimate the beam. The two gas cells were mounted on a 3-axis stage
(MDE122) from Elliot Scientific. The two-stage fiber system achieves a total
throughput of 85 % before the collimating parabola. Afterward, the soliton-
compressed pulses were sent to a CM compressor (UFI PC105) consisting of four
double-angle CM (total group delay dispersion of -160 fs2) and a pair of
wedges (Altechna M0067705) to compensate for the dispersion caused by the two
mm-thick MgF2 output window of the second gas cell and to pre-compensate for
the accumulated dispersion on the pulse before reaching the IPDFG crystal.
After the compressor, we placed a 1 mm AR-coated window (UFI AR7203) that
reflects 5 % of the beam, to separate the probe pulses for EOS. We employed a
home-built second-harmonic generation frequency-resolved optical gating (SHG-
FROG) for temporal characterization. To ensure accurate measurements, the
device utilized all-reflective dispersion-free optics in a non-collinear
geometry with 10 $\mu$m-thick BBO crystal (Castech) cut for type I phase
matching.

###  4.2 CEP-stable, NIR pulse generation

For the IPDFG, we focused 12 $\mu$J of the compressed pulses from the fiber
stages to 40 $\mu$m by using a 1-inch off-axis parabola. A type II BiBO
crystal with a phase matching angle of 12° (Castech) was placed a few
millimeters behind the focus to avoid white light generation in the crystal. A
half-wave plate was introduced into the beam path before the second fiber
stage to rotate 1 % of the input ‘p’ polarisation state to ‘s’ polarisation
state. The generated CEP-stable pulse was collimated to a beam size of 3.2 mm
at 1/e2 employing a 4-inch focal length parabola (Thorlabs MPD254508-90-P01).
A custom-built (UFI BS2214-RC2) broadband dichroic beam splitter separated the
pump and the NIR beam. The measured power after the beam splitter was 160 mW.
A custom-built double-angle CM compressor (UFI IR7202) (see SI Fig. 12) with
four bounces was used for temporal compression to 15 fs.

###  4.3 Field-resolved detection

In EOS, probe and excitation pulses propagate collinearly in a nonlinear
crystal, generating spectral components at sum and difference frequencies. The
up-converted field-sensitive signal arises from the interference between
partially overlapping spectra of the probe pulse and the sum or difference
frequency pulse. Through an ellipsometer, direct access is obtained to the
electric field of the sampled pulse [37]. By utilizing a lock-in amplifier and
balanced detection, the technical noise surplus of the gate pulse was
mitigated, thereby making the shot noise of the probe pulse the primary
limitation on detection sensitivity. In the EOS, the IPDFG pulses were used as
an excitation pulse, whereas a 5 % reflection from the second fiber stage’s
output was used as a probe pulse. A wire grid polarizer combined the probe and
the excitation pulses with orthogonal polarization, collinearly in the EOS
crystal. An off-axis parabolic mirror of 3-inch focal length was used to focus
the beams in a 20-$\mu$m-thick, type II BBO crystal to generate the sum
frequency signal. The sum frequency signal interferes with the high-frequency
portion of the probe pulse, which acts as a local oscillator for a heterodyne
detection. Appropriate filters (Thorlabs FEL 600 and FES 700) were placed
after the EOS crystal. The resulting polarization rotation was measured by an
ellipsometer, which included a Wollaston prism, a quarter waveplate, and
balanced photodiodes. The quarter-wave plate is adjusted to ensure that both
photodiodes receive the same intensity in the presence of the probe pulses. A
mechanical chopper modulated the excitation pulses at 5.8 kHz to enable
heterodyne lock-in detection. The delay line was based on the linear motorized
stage (Physik Instrument V-528.1AA) with a scanning range of 20 mm,
corresponding to a scanning delay of 132 ps. An interferometric delay tracking
system [73] was employed to precisely track the delay line and any timing
jitter artifacts (see SI).

In EOS, the measured interference of the sampling field with sum frequency
field components is convoluted with the detector response. Consequently, it is
subject to a complex response function comprising both amplitude and phase
components. The response function can be calculated using the methods
described in [30]. Based on the wavevector mismatch calculation (see SI Fig.
13), it can be seen that the post-processing of the measured field can be
neglected for our spectral range as the nonlinear response remains constant
throughout the EOS detection range.

###  4.4 Sample preparation

A Pike Technologies liquid cell (162-1200) with two 3 mm-thick barium fluoride
windows and a spacer was used to hold the liquid samples. To examine the
liquid water, 10 $\mu$L and 20 $\mu$L of deionized water were mixed with 1 mL
acetic acid buffer solution (chemlab - CL00.0119). The samples were placed
between two windows using a 0.5 mm Teflon O-ring, corresponding to an
irradiation volume of 4.81 $\mu$L. Three cells with different concentrations
were mounted side to side on a translation stage to reduce the systematic
error. For ethanol measurements, pure ethanol (VWR chemicals - 85033.360) was
filled into three liquid cells with Teflon O-ring spacers of different
thicknesses (0.5 mm, 0.1 mm, and 0.025 mm). The spacers corresponded to
irradiation volumes of 4.81 $\mu$L, 0.962 $\mu$L, and 0.241 $\mu$L,
respectively.

## Acknowledgements

We thank PSJ Russell, MH Frosz, and their team for the production of the
fibers used in this experiment. We want to express our deep gratitude to
Daniel Schade, Wolfgang Schweinberger, Gunnar Arisholm, Nicholas Karpowicz,
and Mallika I Suresh for their invaluable guidance and support.

## Declarations

  * •

This work was supported by research funding from the Max Planck Society.

  * •

Conflict of interest/Competing interests: The authors do not declare any
competing interests.

  * •

Authors’ contribution: H.F. envisioned and designed the experiment. A.S.,
A.H., F.T., M.K. implemented fiber stages for short pulse generation. A.S,
A.H. M.B implemented the data acquisition system. A.S. performed the
fieldoscopy measurements. A.S. and H.F performed the data analysis and wrote
the manuscript. All authors proofread the manuscript.

## Supplementary Information

### Data acquisition

The experimental configuration incorporates custom-developed software tailored
to comprehensively control various components involved in the measurement
process, namely the delay stage, PicoScale, and lock-in amplifier. The delay
stage employed is a Physik Instrument V-528.1AA, specifically chosen for its
suitability within the experimental framework. The PicoScale, an integral
setup component, is a commercially available interferometer manufactured by
SmarAct. It operates on the principles of sinusoidal phase modulation for
precise interferometric displacement measurements. Notably, the PicoScale
measures temporal delays and detects any mechanical jitter introduced within
the beam path. Integrating the interferometer into the experimental
arrangement involves amplifying the PicoScale laser (1550 nm, CW) through an
erbium-doped fiber amplifier and directing it parallel to the optical setup.
This enables accurate tracking of the optical path difference between the
pulses with attosecond precision. To facilitate lock-in measurements, an
optical chopper modulates the NIR beam at a frequency of 5.882 kHz. For the
presented measurements, the stage velocity is 0.4 mm/s.

### Concentration Calculations

The number of molecules is calculated by the ideal gas law:

$PV=nRT$ (2)

where, P = pressure, V = volume, n = Number of moles, R = ideal gas constant
(8.314 J/mol K) and T = temperature in Kelvin. 3.2 m of beam propagation in
the air with a beam diameter of 3.5 mm corresponds to a volume of 3.08
$\times$ 10-5 m3 (0.0308 L). At 22° C (295.15 K), the saturation vapor
pressure of water ($P_{saturated}$) is 2.64 kPa. To calculate the partial
pressure of water vapor $P_{water}$, we use:

$P_{water}=RH\,\times\,P_{saturated}$ (3)

For RHs of 50 % and 8 %, the corresponding values for $P_{water}$ are 1.32 kPa
and 0.211 KPa respectively. Using the equation 2, we have
$n_{RH50\,\%}=16.5\,\mu$mol and $n_{RH8\%}=2.64\,\mu$mol.
For liquid water samples, we dissolved 10 $\mu L$ and 20 $\mu$L of deionized
water in 1 mL of acetic acid buffer, corresponding to a Molarity of 0.555 M
and 1.088 M, respectively (taking into account the density of water as 1 g/mL
and the molar mass of water as 18.02 g/mol). For a beam diameter of 3.5 mm and
an O-ring spacer of 0.5 mm, irradiated volume is calculated to be 4.81 $\mu$L.
To determine the number of moles in the irradiation volume, the irradiation
volume is multiplied by the molarity of the corresponding solution. For 10
$\mu L$ and 20 $\mu L$ dissolved water, we obtain 2.64 $\mu$mol and 5.23
$\mu$mol, respectively.

For calculating the number of moles of ethanol at three different volumes of
4.81 $\mu L$, 0.55 $\mu L$, and 0.137 $\mu L$: firstly the corresponding mass
was calculated by multiplying the volume and density. Afterward, the obtained
values were divided by ethanol’s molar mass. Obtaining 0.79 g/ml as the
density of ethanol, and 46.068 g/mol for its molar mass, the number of moles
was calculated to be 82.5 $\mu$mol, 16.5 $\mu$mol, and 4.13 $\mu$mol,
respectively. Molar mass and density information for all compounds were
retrieved from PubChem.[74].

### Nonlinear propagation in both fibers

Figure 5: Numerical simulation for nonlinear pulse propagation in the fiber
stages. (a) The spectral evolution of the first fiber stage based on self-
phase modulation. (b) The corresponding time domain propagation. External
methods are required for pulse compression to its Fourier transform. (c)
Spectral evolution of the second fiber stage. (d) The time evolution indicates
the temporal compression of the pulses to 3 fs at the fiber output, due to
soliton self-effect compression. These values correspond to 317 TW/cm2 peak
intensity. The numerical simulations used a model described in [75]. Figure 6:
Fiber dispersion tuning. (a) Fiber dispersion for SR-PCF when filled with 15
bar of Argon. (b) Fiber dispersion for SR-PCF when filled with 15 bar of
Helium. Both curves are calculated using the model described in this paper
[76]. Two discontinuities stretching longitudinally indicate first and second-
order resonances inside the fiber due to core-wall capillary thickness. The
intersection of the blue curve and the dotted line indicates the zero
dispersion wavelength (ZDW). The region to the right of ZDW has anomalous
dispersion, whereas the region to the left has normal dispersion.

### Field-resolved detection

Figure 7: Comparison with HITRAN. The black spectra obtained from the field-
resolved water measurement at 50 % relative humidity, shown in Figure 4(a),
match the red HITRAN database spectra. The inset figure displays the
combination band at 0.160 PHz, in agreement with the HITRAN spectrum. At 0.10
PHz and 0.12 PHz spectral range, the measured spectrum appears to be blue-
sifted compared to the HITRAN reference spectrum, which is caused by the
narrower bandwidth of our excitation spectrum. Figure 8: Molecular information
in the electric field vs intensity. (a) Electric field of the molecular
response when excited impulsively. (b) The detected intensity counterpart in
pump-probe intensity techniques. In field-resolved detection the measurement
signal scales linearly with the electric field. Therefore molecular response
is recorded by higher sensitivity than the pump-probe techniques.

### Effectiveness of temporal gating in isolating discrete responses

Figure 9: Temporal gating in ethanol sample. (a) The sampled electric field
with 82.5 $\mu$mol of liquid ethanol in the beam path. Inset a1 and a2 show
two distinct temporal gating times. (b) The black color represents the Fourier
transform of the entire waveform, while the blue and red colors denote the
Fourier transform of the inset a1 and a2, respectively. The legend illustrates
the time frame for which the Fourier transform is applied.

The dephasing duration of molecular response is subject to the molecular phase
and varies from hundreds of femtoseconds in the liquid state to a few
nanoseconds in the gas phase. Therefore, effective time filtering enables
relative isolation of phase responses. Fig. 9 (a) shows the full trace of the
measured electric field for ethanol, while a1 and a2 in the inset show two
magnified regions of the trace at the trailing edge of the femtosecond
excitation pulse corresponding to the short-lived liquid response, and long-
lived gas response, respectively. The Fourier transform of different
temporally gated areas of panel (a) is shown in panel (b). The black spectrum
shows the Fourier Transform of the entire sampled field, while the blue
spectrum shows the absorption peaks of ethanol corresponding to panel a1, and
the red spectrum shows the absorption of atmospheric water vapor corresponding
to Figure a2. The absorption band for the O-H stretch in ethanol and water
overlaps.

Figure 10: SEM fiber images. Both nonlinear fiber stages use an SR-PCF. (a)
The SEM image shows a 55 $\mu$m diameter fiber with 8 capillaries. (b) Shows
the capillary diameter of 23 $\mu$m. (c) The core-wall thickness of the
capillary is 260 nm.  Figure 11: First fiber stage characterization. (a) The
first nonlinear fiber stage’s output spectrum (red), and the dashed line
represent the input laser spectrum. (b) The autocorrelation trace of the same
stage was measured at the input to the second stage, and the pulse duration at
the FWHM was found to be 25 fs. Figure 12: UFI IR7202 dispersion curve. Group
delay dispersion of a pair of double-angle chirp mirrors used for NIR pulse
compression. In total a pair of 3 double-angled chirp mirrors were used to
achieve the compression. Figure 13: EOS phase matching bandwidth. Calculated
the wavevector mismatch ($\Delta$k) for L = 10 $\mu$m BBO, which is related to
the EOS phase-matching bandwidth. The detection bandwidth of the EOS is
indicated between two dashed lines, which shows a nearly flat response. The
numerical calculations were performed using SISYFOS [77].

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