# Dynamics of Photoexcited Cs Atoms Attached to Helium Nanodroplets

Nicolas Rendler  Audrey Scognamiglio  Institute of Physics, University of
Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany  Manuel Barranco
Departament FQA, Facultat de Física, Universitat de Barcelona, Diagonal 645,
08028 Barcelona, Spain  Institute of Nanoscience and Nanotechnology (IN2UB),
Universitat de Barcelona, Barcelona, Spain  Marti Pí  Departament FQA,
Facultat de Física, Universitat de Barcelona, Diagonal 645, 08028 Barcelona,
Spain  Institute of Nanoscience and Nanotechnology (IN2UB), Universitat de
Barcelona, Barcelona, Spain  Nadine Halberstadt  Laboratoire des Collisions,
Agrégats, Réactivité, IRSAMC, Université de Toulouse, CNRS UMR 5589, 31062
Toulouse Cedex 09, France  Katrin Dulitz  Frank Stienkemeier  Institute of
Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg,
Germany<EMAIL_ADDRESS>
###### Abstract

We present an experimental study of the dynamics following the photoexcitation
and subsequent photoionization of single Cs atoms on the surface of helium
nanodroplets. The dynamics of excited Cs atom desorption and readsorption as
well as CsHe exciplex formation are measured using femtosecond pump-probe
velocity map imaging spectroscopy and ion-time-of-flight spectrometry. The
time scales for the desorption of excited Cs atoms off helium nanodroplets as
well as the time scales for CsHe exciplex formation are experimentally
determined for the 6p states of Cs. For the 6p ${}^{2}\Pi_{1/2}$ state, our
results confirm that the excited Cs atoms only desorb from the nanodroplet
when the excitation wavenumber is blue-shifted from the
$6p\,^{2}\Pi_{1/2}\leftarrow 6s\,^{2}\Sigma_{1/2}$ resonance. Our results
suggest that the dynamics following excitation to the 6p ${}^{2}\Pi_{3/2}$
state can be described by an evaporation-like desorption mechanism, whereas
the dynamics arising from excitation to the 6p ${}^{2}\Sigma_{1/2}$ state is
indicative for a more impulsive desorption process. Furthermore, our results
suggest a helium-induced spin-orbit relaxation from the 6p
${}^{2}\Sigma_{1/2}$ state to the 6p ${}^{2}\Pi_{1/2}$ state. Our findings
largely agree with the results of time-dependent 4He-density-functional theory
(DFT) simulations published earlier [Eur. Phys. J. D 2019, 73, 94].

helium nanodroplet, femtosecond pump-probe spectroscopy, short-time
dynamics,desorption dynamics, rare gas clusters

##  I Introduction

Due to the inertness and the exceptional characteristics of superfluid helium
(He), He nanodroplets are widely used as a weakly perturbing matrices for
atoms, molecules, aggregates and clusters. The applications of He nanodroplets
to experimental investigations are extensive and include, for instance, high-
resolution frequency-resolved spectroscopy [1, 2, 3], time-resolved
spectroscopy [4, 5, 6], chemical reaction [7] and nanoplasma formation studies
[8, 5, 9], and the study of nanostructure formation and deposition [10, 11,
12]. Despite the inertness of the He nanodroplets, dopant-host dynamics can be
triggered by the electronic excitation and subsequent ionization of dopants
which can involve, for example, the desorption of a dopant from the droplet,
the broadening and shift of spectral features, the formation of exciplexes and
the ignition of a nanoplasma [9]. This motivates further studies aimed at
understanding these dynamics in detail. The dynamics of photoexcited alkali
metal atom dopants on the surface of He nanodroplets have already been
intensively investigated in both theoretical and experimental studies. For the
theoretical studies, a pseudo-diatomic model was used, in which the He
nanodroplet, HeN (where $N$ is the number of He atoms contained in the
droplet), is treated like a heavy constituent atom of the alkali atom-HeN
complex. This model, in which the potential energy curves of the alkali atom
are calculated as a function of the atom’s distance $R$ from the center of the
He nanodroplet, has proven to be suitable for predicting the dynamics of such
systems to a good extent [13, 14, 15, 16]. Due to the weak attraction between
the alkali metal atom and the He atoms of the nanodroplet, alkali atoms reside
in a dimple at the surface of the droplet [17, 18]. After electronic
excitation of the alkali atom, Pauli repulsion between the electronic
distributions of the excited alkali atom and the surrounding ground-state He
atoms usually leads to the desorption of the excited alkali atom, with
exceptions for some of the lowest excited states of Rb and Cs [19, 20, 21].
The desorption and solvation dynamics of excited alkali atoms from He
nanodroplets depend on the time delay between the excitation and ionization
steps, on the specific excited state and on the excitation wavenumber. The
involved time scales can thus vary significantly in between the different
alkali-atom species. Therefore, separate studies for all alkali atom He-
nanodroplet systems are needed to understand the underlying dynamics.

As a response to photoexcitation, alkali-He exciplexes can be formed on the
surface of He nanodroplets due to local minima in the pseudo-diatomic
potentials of the dopant-droplet system [22]. Exciplex formation has already
been observed for Li [23], Na [22], K [24, 23, 25, 26, 22], Rb [27], Cs [28,
29], Cr [30] and Ag [31]. Due to the higher number of degrees of freedom, the
dynamics of exciplexes attached to He nanodroplets are more complicated as
compared to atom-droplet dynamics. Despite the inertness of the superfluid He,
it is well known that the He environment can induce shifts and strong
broadenings of atomic transitions. Transitions that are electric-dipole
forbidden in free atoms can become allowed in the droplet-bound atom due to
the reduced symmetry of the dopant-He-nanodroplet system. In previous
experiments of Cs-doped He nanodroplets, the desorption and solvation dynamics
of photoexcited Cs atoms have been investigated using laser-induced
fluorescence spectroscopy, beam depletion and time-of-flight mass spectrometry
[19]. The experimental results presented here were obtained using femtosecond-
pump-probe spectroscopy and thus provide additional insights into the time
scales of the involved dynamics.

Figure 1: (a) Schematic pseudo-diatomic potential energy diagram of a Cs-HeN
nanodroplet complex and illustration of the femtosecond pump-probe
photoionization scheme used in the experiments. (b) Correlation diagram
showing the lowest atomic states of Cs (right) and the corresponding molecular
states of a Cs-HeN complex in the approximation of a pseudo-diatomic model
(left). The pump laser wavenumbers $E_{\mathrm{pump}}/(hc)$ used to excite the
6p ${}^{2}\Pi_{1/2}$, 6p ${}^{2}\Pi_{3/2}$, and 6p ${}^{2}\Sigma_{1/2}$ states
are indicated by blue, orange, and green arrows, respectively. The probe-laser
wavenumber $E_{\mathrm{probe}}/(hc)$ (red color) is depicted for the
ionization of the 6p ${}^{2}\Pi_{3/2}$ state only. The wavenumber axis of the
diagram is not to scale.

The interaction between an electronically excited Cs atom and a He nanodroplet
is often repulsive because of the high delocalization of the valence electron
in excited-state Cs [32, 26]. The 6p ${}^{2}\Pi_{1/2}$, 6p ${}^{2}\Pi_{3/2}$
and 6p ${}^{2}\Sigma_{1/2}$ molecular states, which are excited near the 6p
$\leftarrow$ 6s transitions in atomic Cs, have bound-state character, but
resonant excitation near the equilibrium position of the Cs-HeN complex is
predicted to produce excited-state Cs atoms in molecular states far above the
dissociation limit, so that a desorption of the excited Cs atom from the
droplet is expected [33]. Using time-dependent 4He-DFT simulations, the same
authors have found that excited-state Cs atoms desorb after excitation to the
6p ${}^{2}\Sigma_{1/2}$ state, while for the 6p ${}^{2}\Pi_{1/2}$ state, the
atoms are predicted to desorb from the droplet when excitation is close to the
maximum of the ${}^{2}\Pi_{1/2}$ band, at $E_{\mathrm{pump}}/(hc)$ = 11260
cm-1, and remain attached when excitation is done at lower wavenumbers. The
results of the theoretical studies also imply that CsHe exciplex formation is
supported for excitation to the 6p ${}^{2}\Pi_{3/2}$ and 6p ${}^{2}\Pi_{1/2}$
states only, and that the desorption of the exciplex in the 6p
${}^{2}\Pi_{3/2}$ state is due to nonradiative relaxation to the 6p
${}^{2}\Pi_{1/2}$ state [33].

Figure 1a shows a schematic illustration of the dynamics following the
photoexcitation and subsequent photo-ionization of a Cs atom on the surface of
a He nanodroplet. Schematic pseudo-diatomic potential curves are shown as a
function of distance between the He nanodroplet and the Cs atom. The
excitation (pump) step, indicated by the blue arrow, leads to the excitation
of a wavepacket in the excited molecular state. In the case depicted here, the
excited state is repulsive, so that the excited-state Cs atom desorbs from the
droplet. In the ionization (probe) step (red arrow), a Cs+-HeN molecular ion
is formed. This potential is attractive because of the attractive force
between the positive charge of the Cs+ ion and the He nanodroplet [34]. During
the time delay between the excitation and ionization steps, the excited
wavepacket evolves along the excited state of the potential energy curve.
Depending on this time delay, two possible scenarios can occur. The first
possibility is that the excited Cs atom has gained enough kinetic energy and
has reached a distance at which the droplet attraction is weaker, so that the
ion formed after photo-ionization can overcome the attractive force in the
ionic potential. In this case, the Cs+ ion is liberated from the droplet and
detected. The second possibility is that the kinetic energy of the produced
ion is not sufficient for the ion to escape from the He nanodroplet. In this
case, the ion falls back into the droplet and is not detected. Its kinetic
energy is dissipated as a result of its interaction with the droplet during
the solvation process. The critical value of the time delay between the
excitation and ionization steps, which separates the desorption from the fall
back and solvation of Cs+, is usually termed the fall-back time.

In the following, experimental results on the dynamics of Cs atoms attached to
He nanodroplets and photoexcited to the three molecular states depicted in
Figure 1b will be presented. Two different detection techniques, velocity map
imaging (VMI) and ion-time-of-flight (ion-TOF) mass spectrometry, were used.
The ion-TOF measurements have allowed us to assign the masses of the detected
ions. Electron- and ion-VMI detection have allowed us to extract the relative
state populations and the kinetic energies of the detected ions. Furthermore,
VMI detection provides information about possible anisotropies in the ion
emission.

##  II Experimental Section

Figure 2 shows a schematic drawing of the setup used for the experiments
presented here. In brief, a beam of He nanodroplets with a mean size ranging
from 1000 to 10000 atoms per droplet is generated by a continuous supersonic
expansion of He through a nozzle with a 5 $\mu$m opening diameter. The central
part of the beam passes through a 0.4 mm diameter skimmer placed 13 mm
downstream from the nozzle, which reduces the transverse velocity spread of
the beam and allows for a downstream propagation of the supersonic beam under
high-vacuum conditions. Thereafter, the beam enters a vacuum chamber which
contains a radiatively-heated pick-up cell held at 353 K. These temperature
conditions ensure that single Cs atom doping of the He nanodroplets is
predominant. The Cs-doped He nanodroplets then intersect with the pump and
probe laser pulses in vacuum chambers which contain a VMI detector and an ion-
TOF mass spectrometer, respectively. The VMI detector used here, which is
oriented in a direction perpendicular with respect to the He nanodroplet beam,
is described in detail in ref. [35]. To record the velocity-map images, a fast
CMOS camera (acA1920-155um - Basler ace) and a centroiding computer algorithm
are used. The electron- and ion-velocity-map distributions are reconstructed
by using the maximum entropy velocity map reconstruction method [36]. The ion-
TOF spectrometer is mounted in-line with the He nanodroplet beam. It includes
two high-voltage extraction electrodes, a field-free drift region and a Daly-
type detector [37], which is composed of a Faraday cup, a scintillator plate,
an optical bandpass filter to block stray laser light and a photomultiplier
tube (PMT). The ion-TOF signals are recorded using a fast digitizer card
(Acqiris DP105/U1067A-001).

Figure 2: Schematic drawing of the experimental setup. Abbreviations: Ti:Sa =
titanium:sapphire laser system, BS = nonpolarizing beamsplitter, SHG =
$\beta$-barium borate crystal for second harmonic generation, NOPA = nonlinear
optical parametric amplifier, PC = prism compressor, DM = dichroic mirror, FM
= flip mirror, L1 and L2 = lenses with a focal length of 75 cm each, VMI =
velocity map imaging spectrometer, iTOF = ion time-of-flight mass
spectrometer.

The femtosecond excitation (pump) pulses at center wavenumbers
$E_{\mathrm{pump}}/(hc)$ between 11120 and 11990 cm-1 (where $h$ is Planck’s
constant and $c$ is the speed of light; see Figure 1) are produced by feeding
a part of the output of a regenerative Titanium:Sapphire (Ti:Sa) amplifier
($E/(hc)$ = 12500 cm-1) into a nonlinear optical parametric amplifier (NOPA).
For the ionization (probe) step, we produce frequency-doubled femtosecond
laser pulses at $E_{\mathrm{probe}}/(hc)=25000$ cm-1 by sending a portion of
the laser light of the regenerative amplifier through a $\beta$-barium borate
(BBO) crystal. The 5 kHz repetition rate of the regenerative amplifier assures
a sufficiently large period between subsequent pulse pairs to prevent multiple
photoexcitations of the alkali atoms. The time delay between the excitation
pulse and the ionization pulse is scanned using a mechanical delay stage
before the pulses are recombined collinearly with a dichroic mirror. The
duration of the laser pulses is in between $100\leq t_{\mathrm{p}}\leq 200$ fs
depending on the used center wavenumber. The pulses are focused into the
interaction region by using a lens (75 cm focal length) resulting in
intensities of $\approx\,10^{11}$ W/cm2 for the pump pulses and
$\approx\,10^{13}$ W/cm2 for the probe pulses. The beam paths FM-L1 and FM-L2
shown in Figure 2 are used for the VMI and the iTOF measurements,
respectively. Signal contributions produced by the pump or the probe pulses
only are subtracted from the pump-probe-correlated signal. The polarizations
of the pump and probe laser beams are set perpendicular to the VMI
spectrometer axis.

##  III Results and Discussion

Figure 3: Results obtained using laser excitation of the 6p ${}^{2}\Pi_{1/2}$,
6p ${}^{2}\Pi_{3/2}$ and ${}^{2}\Sigma_{1/2}$ states at the wavenumbers
$E_{\mathrm{pump}}/(hc)$ given in the figure, followed by laser ionization at
$E_{\mathrm{probe}}/(hc)=25000$ cm-1 and subsequent ion-TOF detection. (a, c,
e) Exemplary ion-TOF mass spectra obtained at pump-probe delays of 200 ps, 200
ps and 150 ps, respectively. (b, d, f) Integrated Cs+ and CsHe+ yields
obtained by integration of the ion-TOF traces in the intervals marked by the
red and green shadings in (a, c, e) and subsequent smoothing as a function of
pump-probe delay, respectively. The experimental data obtained from one
measurement are shown as black and blue markers, respectively. The solid red
lines are obtained from fits to the experimental data (see the main text). The
ion yield data and fit curves are normalized to the values of the respective
fit curves at the largest pump-probe delay.

As can be seen from Figures 3a,c,e, most ion-TOF spectra contain contributions
from Cs+, CsHe+, Cs${}_{2}^{+}$ and Cs${}_{3}^{+}$. For Cs${}_{2}^{+}$
(Cs${}_{3}^{+}$) resulting from the desorption of Cs2 (Cs3) and subsequent
photo-ionization of the molecule, we did (did not) observe delay-dependent
dynamics at all excitation wavenumbers studied here. Because there is
currently no theory work available which would allow us to describe the
desorption dynamics of Csx (where $x=2,3,...$), only the results obtained for
the desorption of photoexcited Cs atoms and CsHe exciplexes are discussed
below. For all ion-VMI measurements presented in this work, no pronounced
anisotropies have been found.

We have also conducted experiments at different doping conditions by varying
the temperature of the pick-up cell in the range of 328–348 K. This also leads
to a shrinking of droplet sizes as a result of evaporative cooling which
arises from the energy deposition upon pick-up. At the different doping
conditions studied here, we did not observe a change in the desorption
dynamics. This behavior can be attributed to the local character of the
interaction of the Cs atom with the He nanodroplet. This finding is in
agreement with previous experimental and theory work on other alkali-doped He
nanodroplets, such as Na-HeN [38]. In the following, the subscripts “V” and
“T” are used to distinguish in between results obtained by using VMI and ion-
TOF detection, respectively. The superscript “e” (no superscript) is used to
label results obtained by using electron-VMI (ion-VMI and ion-TOF) detection.
The subscript “c” denotes calculated results from theory work. Because two
different functional forms are used to fit the experimental results, the
subscript “R” is used to label the results obtained from a growth function
(see below). An overview of all the experimental results obtained in this work
and the corresponding theory values is given in Tab. 2.

###  III.1 Desorption and Solvation of Cs Atoms Excited to the 6p 21/2 State

Figure 4: (a, d, g, j) Normalized ion velocity distributions, (b, e, h, k)
normalized ion yields, and (c, f, i, l) most probable ion velocities as a
function of pump-probe delay obtained by using ion-VMI detection. Each row
corresponds to a selected wavenumber $E_{\mathrm{pump}}/(hc)$ (given in the
figure). The wavenumber for the probe pulse was
$E_{\mathrm{probe}}/(hc)=25000\,cm^{-1}$. The experimental data are shown as
filled contour lines and as black and blue markers. The solid red lines are
the results of fits to the experimental data (see the main text). In (g–l),
two features A and B are observed which exhibit different delay-dependent
characteristics. The fit curve shown in (i) is for feature A only. The data
used for the dashed black lines in (c) and (i) are extracted from ref. [33].
The intensity scales in (a, d, g, j) range from 0 to 1. The ion yield data and
fit curves are normalized to the values of the respective fit curves at the
largest pump-probe delay.

Figures 4a–c show the ion yields and ion velocities as a function of pump-
probe delay which were obtained by resonant excitation of the 6p
${}^{2}\Pi_{1/2}$ state at $E_{\mathrm{pump}}/(hc)=11260$ cm-1 and subsequent
ionization at $E_{\mathrm{probe}}/(hc)=25000$ cm-1 using VMI detection. In all
of the ion-VMI measurements presented in this article, the arrival times of
the ions were not gated due to the low overall ion signal intensity. From the
results of the ion-TOF measurements discussed in more detail below, we can
infer that for excitation to the 6p 21/2 state the ion signal intensity in the
VMI traces is mainly from Cs+. Therefore, the time dependence shown in Figures
4a–c can be ascribed mainly to Cs+. Only a very weak signal intensity arising
from CsHe exciplexes is recorded in the ion-TOF measurements, which does not
allow for a measurement of its dynamics. This observation is in agreement with
the dynamics expected from the calculated diatomic potential energy curve for
CsHe exciplexes in the 6p ${}^{2}\Pi_{1/2}$ state, which has a barrier that
leads to a strong suppression of exciplex formation [33]. Exciplex formation
was also not reported for Cs-doped He nanodroplets after excitation to the 6p
${}^{2}\Pi_{1/2}$ state [19] and for Rb-doped He nanodroplets after excitation
to the 5p ${}^{2}\Pi_{1/2}$ state [39, 21].

The time-dependent ion yield shown in Figure 4b was fitted to an error
function of the form

$\emph{I}(\emph{t})=A_{0}\\{\,\mathrm{erf}[\left(t-\tau_{\mathrm{V}}\right)/\sigma_{\mathrm{V}}]+1\\},$
(1)

with variable amplitude $A_{0}$, fall-back time constant $\tau_{\mathrm{V}}$
and width $\sigma_{\mathrm{V}}$. This functional form, which has also
previously been used to describe the desorption dynamics of alkali atoms from
He nanodroplets [13], contains an exponential term which fits to the
characteristics of a desorption process. The fit yields a fall-back time of
$\tau_{\mathrm{V}}$ = 72 ps and a width of $\sigma_{\mathrm{V}}$ = 44 ps. In
Figure 4 c, the most probable ion velocity is plotted as a function of pump-
probe delay. This velocity is determined by fitting the velocity distribution
at each delay step (shown in Figure 4a) with a Gaussian function. The most
probable ion velocity at each pump-probe delay is then given by the peak value
of the Gaussian fit to the ion velocity distribution. At pump-probe delays,
where the ion signal intensity is very low, and the corresponding fits yield
very noisy or unphysical values for the extracted velocities, the results are
not used for the further analysis, and they are thus not shown in Figure 4.
The asymptotic ion velocity, $v_{\mathrm{V}}$, which is the maximum ion
velocity reached at an infinitely long pump-probe delay, is obtained by
fitting the delay-dependent most probable ion velocities in Figure 4c to a
function of the form

$\gamma(t)=v_{\mathrm{V}}\\{1-\exp[{-(t-t_{0})/s_{\mathrm{V}}}]\\},$ (2)

with the asymptotic ion velocity $v_{\mathrm{V}}$, the time offset $t_{0}$,
and width $s_{\mathrm{V}}$. At this excitation wavenumber, we obtain an
asymptotic ion velocity $v_{\mathrm{V}}$ = 35 m/s and a width $s_{\mathrm{V}}$
$<$ 1 m/s.

The obtained fall-back times and asymptotic ion velocities are close to the
theory values of $\tau_{\mathrm{c}}$ =77.8 ps and $v_{\mathrm{c}}$ = 23.8 m/s
given by Coppens et al. [33] which were derived by using time-dependent 4He-
DFT simulations and attributed to to impulsive dissociation, which is expected
from the corresponding repulsive He-Cs(6p) interaction potential. The
calculated time-dependent velocities, extracted from ref.[33] , are included
as dashed lines in Figures 4c,i. In comparison, the most probable velocities
obtained from the measurements miss the steep increase starting at the fall-
back time which is apparent from the theory curves, and they display high
velocity values close to or even below the nominal fall-back times. We
attribute this behavior to the broad distribution of excitation energies in
combination with the steep repulsive potential. The calculations do not
include the distribution of excitation energies. In the experiment, around the
fall-back time the average ion intensity is low; on the other hand, the blue
tail provides already contributions with high ion velocity.

The measurements were repeated at a slightly red-detuned excitation wavenumber
of $E_{\mathrm{pump}}/(hc)=11120$ cm-1. Even though the recorded number of
photoelectrons is comparable to the number of photoelectrons obtained for the
resonant excitation of this state at $E_{\mathrm{pump}}/(hc)=11260$ cm-1, we
were unable to detect ions under these excitation conditions in the VMI
measurements. These observations are consistent with previous results by
Theisen et al. [19] who explained these findings by the nondesorptive nature
of this state when excitation occurs at the lower energy tail of its dipole
absorption spectrum. The nondesorptive character is due to the less strong
repulsion of the excited Cs atom from the droplet at this excitation
wavenumber, which does not allow the excited Cs atom to gain enough kinetic
energy to leave the Cs-HeN potential before the interaction is switched to
attraction in the ionization step.

However, we were able to detect a small amount of Cs+ in the corresponding
ion-TOF measurements at an excitation wavenumber of
$E_{\mathrm{pump}}/(hc)=11120$ cm-1. This can be explained by some overlap of
the broad laser excitation spectrum (full width at half-maximum
$\mathrm{(FWHM)}\approx 149\,\mathrm{cm}^{-1}$) with the high-energy tail of
the absorption profile for this state. Because of the different sensitivities
of the ion-VMI and ion-TOF detection schemes, weak ion signal contributions
may not show up in the VMI measurements.

Table 1: Summary of fall-back times $\tau_{\mathrm{V}}$ and $\tau_{\mathrm{T}}$ inferred from the ion-VMI and from the ion-TOF measurements, respectively, at different central excitation wavenumbers $E_{\mathrm{pump}}/(hc)$. The full width at half-maximum (FWHM) bandwidth of each laser excitation spectrum is provided in brackets behind the respective excitation wavenumber. The widths $\sigma_{\mathrm{V}}$ and $\sigma_{\mathrm{T}}$ obtained from the corresponding fits of the ion yields to error functions are given in parantheses. Likewise, the obtained asymptotic ion velocities $v_{\mathrm{V}}$ as well as the 50 % rise times $\tau_{\mathrm{T,R}}$ and $\tau^{\mathrm{e}}_{\mathrm{V,R}}$ from the ion-TOF and from the electron-VMI measurements are given, respectively. The uncertainty of each asymptotic ion velocity $s_{\mathrm{V}}$ (given in parantheses) is quoted either as the error of the corresponding fit to Eq. 2 or as the standard deviation of the most probable ion velocities (see the main text).222In several VMI traces, two features (labeled as “A” ,“B” for ions and “I”, “II” for electrons, respectively) with different time constants were found to be indicative for desorption processes. These time constants are thus given separately. In the ion-TOF measurements, the Cs+ and CsHe+ ion masses were found to exhibit different delay dependencies. Therefore, $\tau_{\mathrm{T}}$ and $\tau_{\mathrm{T,R}}$ are given for both ion masses separately. The calculated fall-back times $\tau_{\mathrm{c}}$ and the asymptotic ion velocities $v_{\mathrm{c}}$ for Cs+ obtained by Coppens et al. [33] are shown for comparison. For the measurement at $E_{\mathrm{pump}}/(hc)=11120$ cm-1 ($E_{\mathrm{pump}}/(hc)=11640$ cm-1), no fall-back times and asymptotic ion velocities are given, since desorption dynamics was not observed (the signal intensity was not sufficient). state |  $E_{\mathrm{pump}}/(hc)$ (FWHM) [cm-1] | label |  $\tau_{\mathrm{V}}$($\sigma_{\mathrm{V}}$) [ps] |  $\tau_{\mathrm{T}}$($\sigma_{\mathrm{T}}$) [ps] |  $\tau_{\mathrm{T,R}}$ [ps] |  $\tau^{\mathrm{e}}_{\mathrm{V,R}}$ [ps] |  $\tau_{\mathrm{c}}$ [ps] |  $v_{\mathrm{V}}$($s_{\mathrm{V}}$) [m/s] |  $v_{\mathrm{c}}$ [m/s]
---|---|---|---|---|---|---|---|---|---
6p 21/2 | 11260 (152) |  | 72(44) |  Cs+: 82(73) |  |  | 77.8 |  35($<$1) | 23.8
6p 21/2 | 11120 (149) |  |  |  |  |  |  |  |
6p 23/2 | 11750 (166) |  | 55(51) |  |  Cs+: 29 |  |  | 22(4) |
|  |  |  |  |  CsHe+: 152 |  |  |  |
6p 23/2 | 11640 (163) |  |  |  |  |  |  |  |
6p 21/2 | 11900 (170) | A, I | 36(19) |  |  | 21 | 20.2 |  36($<$1) | 43.6
|  | B, II | 16(26) |  |  |  |  | 169(1) |
6p 21/2 | 11990 (173) | A, I | 35(26) |  Cs+: 30(27) |  | 23 |  | 36(3) |
|  |  |  |  CsHe+: 49(41) |  |  |  |  |
|  | B, II | 20(17) |  |  |  |  | 173(3) |

The delay-dependent Cs+ ion yields from complementary ion-TOF measurements,
following excitation to the 6p ${}^{2}\Pi_{1/2}$ state, were obtained by using
the same experimental parameters as in the VMI measurements. We only observe
electrons and no bare Cs+ ions and assign the missing positive charge to He
droplet attached Cs+ ions, which we cannot detect because of the very large
mass. In this way the results confirm the solvation of Cs+ inside the He
droplet upon excitation at $E_{\mathrm{pump}}/(hc)<11120$ cm-1.

At an excitation wavenumber of $E_{\mathrm{pump}}/(hc)=11260$ cm-1, the Cs+
ions were found to desorb with a fall-back time of $\tau_{\mathrm{T}}=$ 82 ps.
This time constant was obtained from a fit of the experimental Cs+ ion data
(see Figure 3b) to the error function given by eq. 1. The corresponding width
of the error function was found to be $\sigma_{\mathrm{T}}=$ 73 ps. These
values are fairly consistent with the results obtained from the ion-VMI
measurements.

###  III.2 Dynamics of Cs Atoms Excited to the 6p 23/2 State

Figure 5: (black, solid): Calculated dipole absorption spectrum of a Cs-HeN
complex containing $N=1000$ 4He atoms (black curve) and two measured NOPA
excitation spectra at different center wavenumbers (blue and red curves, see
legend). The NOPA spectra were normalized to their corresponding maxima. The
data for the dipole absorption spectrum were obtained from ref. [33]

.

The results of the time-dependent 4He-DFT simulations for the excitation of
the 6p ${}^{2}\Pi_{3/2}$ state predict that the excited Cs atom quickly forms
a linear exciplex molecule which does not desorb from the He nanodroplet but
stays at the droplet surface [33]. Figure 4d–f show the results of the ion-VMI
measurements obtained upon resonant excitation of this state at
$E_{\mathrm{pump}}/(hc)=11750$ cm-1. In Figure 5, the measured spectra of the
broad-band NOPA laser are plotted together with the dipole absorption spectrum
of a CsHeN complex. Because of the large width of the laser excitation
spectrum ($\mathrm{FWHM}\approx 166\,$cm-1), it is expected that the 6p
${}^{2}\Sigma_{1/2}$ state is also populated during the excitation step.
However, the results from the ion-VMI measurements imply that the excited Cs
atoms do desorb from the He nanodroplet. The fall-back time of the ions and
the corresponding width (obtained by fitting the experimental ion traces to
eq. 1) are $\tau_{\mathrm{V}}=$ 55 ps and $\sigma_{\mathrm{V}}=$ 51 ps,
respectively. The asymptotic ion velocity, which was determined to be the mean
value of the most probable ion velocities shown in Figure 4d, is
$v_{\mathrm{V}}$ = 22 m/s (standard deviation of $s_{\mathrm{V}}$ = 4 m/s).

From the delay-dependent ion-TOF yields in Figure 3d, it can be inferred that
both excited-state Cs atoms and CsHe exciplexes appear to desorb from the He
nanodroplets. At long pump-probe delays, the detected number of Cs+ ions seems
to have reached a constant value which suggests that all excited Cs atoms have
desorbed. In contrast to that, the CsHe+ yield is still increasing at the
longest measured delays.

Because fits to Eq. 1 did not yield satisfactory results, the Cs+ and CsHe+
ion yields (see Figure 3d) are fit to the following functional form

$\tilde{I}(t)=\tilde{A}_{0}\\{1-\exp[{-\mathrm{\ln\left(2\right)}(t-t_{0})/\tau}]\\},$
(3)

with variable amplitude $\tilde{A}_{0}$, offset $t_{0}$ and width $\tau$. The
fits yield 50 $\%$-rise times of $\tau_{\mathrm{T,R}}=t_{0}+\tau=$ 29 ps and
$\tau_{\mathrm{T,R}}=t_{0}+\tau=$ 152 ps for Cs+ and CsHe+, respectively. The
fit to the CsHe+ yield was started at a nonzero pump-probe delay of $t_{0}=$
30 ps to account for the delayed appearance of the CsHe+ signal in the ion-TOF
measurements. This value is fairly consistent with the predicted CsHe exciplex
formation time of 25 ps [33]. Even though desorption is predicted not to occur
for bare Cs atoms after resonant excitation of the 6p ${}^{2}\Pi_{3/2}$ state
[33], we do observe a considerable amount of Cs+ ions (see Figure 3c). A small
fraction of this Cs+ yield is expected to be due to a parasitic population of
the 6p ${}^{2}\Sigma_{1/2}$ state due to the overlap of the laser excitation
spectrum with the lower energy tail of the dipole absorption peak
corresponding to the 6p ${}^{2}\Sigma_{1/2}\leftarrow$ 6s ${}^{2}\Sigma_{1/2}$
transition (see Figure 5). However, the main signal contribution is expected
to arise from the excitation of the narrow and much stronger 6p
${}^{2}\Pi_{3/2}\leftarrow$ 6s ${}^{2}\Sigma_{1/2}$ transition. Furthermore, a
comparison of the Cs+ yields from ion-TOF spectrometry in Figure 3 e (obtained
by resonant excitation of the 6p ${}^{2}\Sigma_{1/2}$ state) with those shown
in Figure 3c (obtained by resonant excitation of the 6p ${}^{2}\Pi_{3/2}$
state) suggests that the involved solvation dynamics of the excited Cs atoms
are different. In contrast to the other excitation schemes, for which error
functions were used for fitting the experimental data, the delay-dependent
increase of the Cs+ ion yield in Figure 3d is described best by using the
growth function given by eq. 3. In addition to that, the ion velocity
distribution shown in Figure 4d appears to be broader compared to the
distributions observed for the excitation of the other states in this work.
These findings indicate that the desorption could be driven by a more complex,
evaporative-like process which deviates from the impulsive, dissociation-like
desorption of Cs atoms in the 6p ${}^{2}\Pi_{1/2}$ and 6p ${}^{2}\Sigma_{1/2}$
states. Such evaporative-like desorption dynamics were suggested for the 5p
states of Rb which were observed to exhibit much slower desorption dynamics
compared to other states [14, 27].

As proposed by Coppens et al. [33], CsHe exciplexes can also form on the
surface of the droplet after excitation of the 6p ${}^{2}\Pi_{3/2}$ state,
since there is no barrier in the calculated diatomic potential of the CsHe
exciplex in this case. After exciplex formation, spin-orbit relaxation to the
6p ${}^{2}\Pi_{1/2}$ state may occur which could lead to the desorption of
CsHe exciplexes or of bare Cs atoms. The contribution of spin-orbit relaxation
to desorption depends on how the energy, which is released by this relaxation
process, is shared in between the He nanodroplet and the CsHe exciplex.
Because we do not detect photoelectrons with an energy corresponding to the 6p
${}^{2}\Pi_{1/2}$ state, this relaxation process does not seem to occur.
Vibrational predissociation could also lead to the desorption of CsHe
exciplexes, since the vibrational energy of the CsHe exciplexes is sufficient
to release CsHe from the He nanodroplet. This process would be similar to the
vibrational predissociation of rare gas-diatomic molecule van der Waals
complexes reported in the literature [40, 41, 42, 43]. The results of a
theoretical study by Leino et al., which addresses the formation and possible
desorption of RbHe exciplexes in the 5p ${}^{2}\Pi_{3/2}$ state, suggest that
the internal binding energy of the RbHe exciplex is approximately 10 times
higher than the binding energy of the RbHe exciplex to the He nanodroplet
[44].

To evaluate whether vibrational predissociation could be a possible process
leading to the desorption of CsHe exciplexes after excitation of the 6p
${}^{2}\Pi_{3/2}$ state, we have fitted the pseudo-diatomic potential energy
curve of the 6p ${}^{2}\Pi_{3/2}$ state, given in ref. [33], to a Morse
potential function, from which the vibrational energy eigenvalues of the
Cs(6p)-HeN complex can be calculated analytically. Using this procedure, we
obtain four bound vibrational levels with principal quantum numbers $n=0,1,2$
and $3$ and the corresponding binding energies $E_{n}$ = $-88.0$
cm${}^{-1}\cdot hc$, $-46.6$ cm${}^{-1}\cdot hc$, $-18.2$ cm${}^{-1}\cdot hc$,
$-2.9$ cm${}^{-1}\cdot hc$, respectively. It is likely that the binding energy
of CsHe exciplexes to the He nanodroplet is close to the estimated binding
energy of $\approx$ 10 cm${}^{-1}\cdot hc$ for RbHe exciplexes [44].
Vibrational relaxation can therefore easily provide enough energy to break the
bond between the CsHe exciplex and the He nanodroplet which supports the
above-mentioned explanation for the desorption of CsHe exciplexes.
Furthermore, the delay-dependent ion-TOF yield for CsHe+ obtained from this
measurement follows a different time dependence as compared to the other CsHe+
ion-yield curves shown in this work. This points to a desorption process which
is different from impulsive desorption, for which the delay-dependent ion
yield usually follows an error function. Even though our results provide
strong evidence for CsHe exciplex desorption as a result of vibrational
predissociation, more experimental and theoretical efforts are required to
allow for a confident conclusion on the process that causes the desorption of
CsHe exciplexes after resonant excitation to the 6p ${}^{2}\Pi_{3/2}$ state.

Because VMI detection could not be used for mass-selected ion spectrometry,
the velocity at which the CsHe exciplexes desorb from the droplet is not
known. Judging from the relative ion yields obtained from the ion-TOF
measurements, it is likely that the delay-dependent dynamics observed in the
ion-VMI data arise from both excited Cs atoms and CsHe exciplexes. This could
also explain the different fall-back times obtained using ion-VMI and ion-TOF
detection (see Tab. 2). Since the ion-TOF data were recorded in a mass-
selected manner, the fall-back times for Cs+ obtained from the ion-TOF
measurements are more reliable than the fall-back times obtained from the ion-
VMI measurements.

###  III.3 Dynamics of Cs Atoms Excited to the 6p 21/2 State

Because of the broad excitation spectrum of the NOPA (width of
$\mathrm{FWHM}\approx 166$ cm-1) and due to the closely-spaced absorption
bands for excitation to the 6p ${}^{2}\Sigma_{1/2}$ and 6p ${}^{2}\Pi_{3/2}$
states (see Figure 5), VMI measurements were done at two different laser
excitation wavenumbers to determine the dynamics of the 6p
${}^{2}\Sigma_{1/2}$ state. In the first measurement, the wavenumber of the
excitation laser is centered near the maximum absorption cross section for the
6p ${}^{2}\Sigma_{1/2}$ state ($E_{\mathrm{pump}}/(hc)=11900$ cm-1). In the
second measurement, the wavenumber of the excitation laser is blue-detuned
from the resonance ($E_{\mathrm{pump}}/(hc)=11990$ cm-1) to ensure that the
contribution of the 6p ${}^{2}\Pi_{3/2}$ state to the signal intensity is
negligible.

Figure 6: Photoelectron kinetic energy distributions resulting from electron-
VMI detection following laser excitation of the 6p ${}^{2}\Sigma_{1/2}$ state
at (a) $E_{\mathrm{pump}}/(hc)=11900$ cm-1 and (c)
$E_{\mathrm{pump}}/(hc)=11990$ cm-1 and subsequent laser ionization at
$E_{\mathrm{probe}}/(hc)=25000$ cm-1 at different pump-probe delays. (b), (d)
Delay-dependent photoelectron yields for features I (blue markers), II (black
markers) and III (green markers) obtained by integrating the photoelectron
kinetic energy distributions shown in (a) and (b) in the corresponding feature
ranges, respectively. The red curves were obtained from fits to the delay-
dependent photoelectron yields (see the main text).

Figure 6a shows the delay-dependent photoelectron kinetic energy distributions
which were obtained by using electron-VMI detection at an excitation
wavenumber of $E_{\mathrm{pump}}/(hc)=11900$ cm-1. As can be seen from the
figure, three different features are detected which are labeled as I, II and
III. The photoelectrons for feature I have an energy of 0.60 eV, which can be
assigned to ionization out of the 6p ${}^{2}\Pi_{1/2}$ state. The
photoelectrons obtained for feature II have a mean kinetic energy of 0.66 eV
which corresponds to the electron energy resulting from the ionization out of
the 6p ${}^{2}\Sigma_{1/2}$ and 6p ${}^{2}\Pi_{3/2}$ states at
$E_{\mathrm{probe}}/(hc)=25000$ cm-1. At this excitation wavenumber, the
number of detected electrons for feature I is lower than for feature II and
the electrons for feature I appear to occur at a time delay compared to those
for feature II. The $50\,\%$ rise time for feature I, which is obtained by
fitting the corresponding photoelectron yield to Eq. 3, was determined to be
$\tau^{\mathrm{e}}_{\mathrm{V,R}}=$ 21 ps.

The corresponding laser excitation spectrum is well separated from the
absorption band corresponding to the excitation of the 6p
${}^{2}\Pi_{1/2}\leftarrow$ 6s ${}^{2}\Sigma_{1/2}$ transition. In addition to
that, the natural lifetime of the 6p state of the free Cs atom (30.41 ns [45])
is much longer than the maximum pump-probe delay. This suggests that the
delayed population of the 6p ${}^{2}\Pi_{1/2}$ state arises from the
perturbation of the 6p ${}^{2}\Sigma_{1/2}$ state by the He droplet. This
perturbation leads to spin-orbit relaxation. Spin-orbit relaxation has been
observed previously for other dopants attached to He droplets [46, 47, 48, 49,
50, 30, 51, 52, 53]. For Cs-doped He nanodroplets, spin-orbit relaxation was
proposed to occur upon excitation to the 6p ${}^{2}\Pi_{3/2}$ state. At this
stage, it is not clear to us why spin-orbit relaxation occurs upon excitation
of the 6p ${}^{2}\Sigma_{1/2}$ state, but not for excitation of the 6p
${}^{2}\Pi_{3/2}$ state. Because the atomic states are identical, spin-orbit
relaxation would be expected. However, it should be less likely for the 6p
${}^{2}\Sigma_{1/2}$ state than for the 6p ${}^{2}\Pi_{3/2}$ state considering
the stronger droplet repulsion in the 6p ${}^{2}\Sigma_{1/2}$ state which is
evident from the pseudo-diatomic potentials in the article by Coppens et al.
[33].

The results of the ion-VMI measurement shown in Figures 4g–i were obtained at
the same excitation wavenumber of $E_{\mathrm{pump}}/(hc)=11900$ cm-1 as the
data shown in Figures 6a,b. Two ion features A and B are observed in Figure4g
which appear at different pump-probe delays. The ion velocity distributions
are broad, but they are clearly separated from each other. The ion yield for
feature B is $\approx 30\%$ of the ion yield for feature A. The corresponding
pump-probe-delay dependency of the ion yield is shown in Figure 4h. The fall-
back times $\tau_{\mathrm{V}}=$ 36 ps and $\tau_{\mathrm{V}}=$ 16 ps for
features A and B, respectively, are obtained from fits to the error function
given by eq. 1. The kinetic energies of the two ion features are significantly
different. The most probable ion velocity for feature A increases from 32 m/s
at a pump-probe delay of 40 ps to 35 m/s at a delay of 62 ps (Figure 4i).
Because of the low signal intensity for feature B, the most probable ion
velocities (inferred from the fits to the ion velocity distribution in Figure
4g in between pump-probe delays of 38 ps and 60 ps) vary in between 166 m/s
and 171 m/s, respectively. A delay dependency could not be inferred from the
most probable ion velocities for feature B in Figure 4i. Asymptotic ion
velocities for features A and B of $v_{\mathrm{V}}$ = 36 m/s and
$v_{\mathrm{V}}$ = 169 m/s, respectively, were obtained from fits of the
delay-dependent most probable ion velocities for features A and B in Figure 4i
with a function of the same form as given by Eq. 2. Here, the uncertainties of
the most probable ion velocities for features A and B, $s_{\mathrm{V}}<$ 1 m/s
and $s_{\mathrm{V}}$ = 1 m/s, respectively, are given as the standard
deviation of the most probable ion velocity over the ten largest pump-probe
delays.

The results obtained by using electron- and ion-VMI detection at an excitation
wavenumber of $E_{\mathrm{pump}}/(hc)=11990$ cm-1 are illustrated in Figures
4j–l and 6c,d, respectively. Owing to the little overlap of the laser
excitation spectrum with the absorption band of the close-lying 6p
${}^{2}\Pi_{3/2}\leftarrow$ 6s ${}^{2}\Sigma_{1/2}$ transition, the signal
contributions arising from excitation of the 6p ${}^{2}\Sigma_{1/2}$ state are
expected to be dominant.

At $E_{\mathrm{pump}}/(hc)=11990$ cm-1, we observe that the relative signal
intensities of the ion and electron features are different compared to the
results obtained at $E_{\mathrm{pump}}/(hc)=11900$ cm-1. The electron signals
which presumably arise from spin-orbit relaxation from the 6p
${}^{2}\Sigma_{1/2}$ state to the 6p ${}^{2}\Pi_{1/2}$ state (feature I) are
increased compared to the measurement at the other excitation wavenumber. At
long pump-probe delays, feature I reaches almost the same yield as feature II
which in turn is due to excitation via the 6p ${}^{2}\Sigma_{1/2}$ state. The
$50\,\%$ rise time for feature I, which is obtained by fitting the
corresponding photoelectron yield to Eq. 3, was determined to be
$\tau^{\mathrm{e}}_{\mathrm{V,R}}=$ 23 ps. The ion yield for feature B even
exceeds the ion yield for feature A. This observation indicates a clear
correlation between the ions for feature B and the electrons arising from
spin-orbit relaxation.

In contrast to that, the fall-back times ($\tau_{\mathrm{V}}=$ 35 ps and
$\tau_{\mathrm{V}}=$ 20 ps) and asymptotic ion velocities ($v_{\mathrm{V}}$ =
36 m/s and $v_{\mathrm{V}}$ = 173 m/s) for features A and B, respectively,
obtained at $E_{\mathrm{pump}}/(hc)=11990$ cm-1 are similar to the values
obtained at $E_{\mathrm{pump}}/(hc)=11900$ cm-1. At
$E_{\mathrm{pump}}/(hc)=11990$ cm-1, the most probable ion velocity for
feature B increases from 168 m/s at a delay of 20 ps to 173 m/s at a delay of
65 ps. For feature A, the most probable ion velocity is also observed to
increase, from 33 m/s at a delay of 31 ps to 36 m/s at a delay of 65 ps. The
most probable ion velocities and asymptotic ion velocities for features A and
B are slightly increased when the excitation wavenumber is changed from
$E_{\mathrm{pump}}/(hc)=11900$ cm-1 to $E_{\mathrm{pump}}/(hc)=11990$ cm-1.
The observed increase of the asymptotic ion velocity at the higher excitation
wavenumber is in agreement with the results reported for the desorption of
other excited-state alkali atoms, e.g. Li, Na and Rb [52, 49, 13, 14]. From
the simulations by Coppens et al. [33], ions with a velocity of 170 m/s are
not expected to arise from a desorption mechanism caused by a repulsive
pseudo-diatomic potential. Because of the obvious correlation of the signal
intensities attributed to the spin-orbit relaxation from the 6p
${}^{2}\Sigma_{1/2}$ state to the 6p ${}^{2}\Pi_{1/2}$ state and the ions for
feature B, we have estimated the energy release from spin-orbit relaxation in
order to check whether this process can provide enough energy to explain the
observed ion velocities. The energy released by this relaxation process, which
was estimated from the energy difference of the potential energy curves given
in ref. [33], amounts to 550 cm${}^{-1}\cdot hc$. The energy corresponding to
the most probable ion velocity for feature B is about 200 cm${}^{-1}\cdot hc$.
Following this consideration, 36 $\%$ of the released energy would be
converted into the kinetic energy of the ions, and the remaining energy would
be transferred to the droplet and dissipated by the evaporation of He atoms.
For RbHe exciplexes, which were found to be desorbed after the relaxation from
the 5p $\Pi_{3/2}$ state to the 5p $\Pi_{1/2}$ state, it is reported that 21
$\%$ of the energy is released as kinetic energy of the ions [54]. Comparably
high kinetic ion energies are reported for Ba+ [55]. However, the authors of
this latter study do not provide a final explanation for a possible process
which leads to the production of such fast ions.

The black and blue markers in Figure 3f show data sets for the delay
dependence of the Cs+ and CsHe+ ion yields obtained from the ion-TOF
measurements at an excitation wavenumber of
$E_{\mathrm{pump}}/(hc)=11990\,$cm-1. The corresponding fall-back times,
calculated via eq. 1, are $\tau_{\mathrm{T}}=$ 30 ps (with a width of
$\sigma_{\mathrm{T}}=$ 27 ps) for Cs+ and $\tau_{\mathrm{T}}=$ 49 ps (with a
width of $\sigma_{\mathrm{T}}=$ 41 ps) for CsHe+. The fall-back time for Cs+
is again in fair agreement with the results obtained from the ion-VMI
measurements and with the simulation results by Coppens et al. [33]. For the
excitation to the 6p ${}^{2}\Sigma_{1/2}$ state, the formation of CsHe
exciplexes is not expected, since the corresponding Cs-He potential energy
curve for this state is purely repulsive. For this reason, Coppens et al. [33]
do not provide fall-back times for CsHe+ ions arising from the excitation of
the 6p ${}^{2}\Sigma_{1/2}$ state. We attribute the observation of CsHe
exciplexes in these measurements to the overlap of the laser excitation
spectrum with the absorption band corresponding to the 6p
${}^{2}\Pi_{3/2}\leftarrow$ 6s ${}^{2}\Sigma_{1/2}$ transition. The effective
overlap area between the absorption band and the laser excitation spectrum
provides an estimate of the expected parasitic population of the 6p
${}^{2}\Pi_{3/2}$ state. To check whether the parasitic population of the 6p
${}^{2}\Pi_{3/2}$ state can explain the observed CsHe+ yield in the
measurements for the 6p ${}^{2}\Sigma_{1/2}$ state shown in Figure 3f, we
compared the effective overlap area between the absorption band and the laser
excitation spectrum for two measurements. For the first measurement the
excitation spectrum was centered at $E_{\mathrm{pump}}/(hc)=11990\,$cm-1,
whereas for the second measurement the excitation spectrum was slightly red-
detuned to increase the overlap with the high-energy tail of the 6p
${}^{2}\Pi_{3/2}$ dipole absorption band. Both the effective overlap and the
CsHe+ yield increased by the same amount for the red-detuned condition.
Therefore, it is justified to attribute the observation of CsHe exciplexes at
$E_{\mathrm{pump}}/(hc)=11990$ cm-1 to the parasitic population of the 6p
${}^{2}\Pi_{3/2}$ state.

For excitation via the 6p ${}^{2}\Sigma_{1/2}$ state, photoelectrons with a
mean kinetic energy of 1.3 eV are also observed (feature III in Figure 6). At
large pump-probe delays, this photoelectron yield is delay-independent.
Compared to that, the photoelectron yield is increased by $\approx 50\,\%$ at
short pump-probe delays. The photoelectron yield from feature III is
significantly increased when the excitation wavenumber is increased from
$E_{\mathrm{pump}}/(hc)=11900$ cm-1 to $E_{\mathrm{pump}}/(hc)=11990$ cm-1.
Unfortunately, we were not able to assign this measured photoelectron feature.
Further investigations are necessary to disclose the underlying process which
leads to the emission of photoelectrons at this specific kinetic energy.
Because the overall signal intensity for feature III is small compared to the
signal intensities for features I and II and the time constant for the decay
of feature III (3 ps for excitation at $E_{\mathrm{pump}}/(hc)=11900$ cm-1 and
7 ps for excitation at $E_{\mathrm{pump}}/(hc)=11990$ cm-1) is quite different
from the time constant obtained for the increase of feature I
($\tau^{\mathrm{e}}_{\mathrm{V,R}}=21$ ps for excitation at
$E_{\mathrm{pump}}/(hc)=11900$ cm-1 and $\tau^{\mathrm{e}}_{\mathrm{V,R}}=23$
ps for excitation at $E_{\mathrm{pump}}/(hc)=11990$ cm-1), we believe that the
contribution of this unknown reaction channel to the other dynamics is small.

##  IV Summary

Using pulsed femtosecond pump-probe spectroscopy in combination with
electron-/ion-VMI and ion-TOF detection, we have resolved the picosecond
desorption dynamics of excited Cs atoms and CsHe exciplexes attached to the
surface of He nanodroplets. The dynamics were studied following the pulsed
laser excitation of Cs-HeN states close to the atomic 6p $\leftarrow$ 6s
transitions in Cs and the subsequent detection of ions and/or electrons
produced by a second, time-delayed ionization laser pulse. The results of the
ion-TOF measurements imply pump-probe-delay-dependent dynamics for both
excited-state Cs atoms and CsHe exciplexes.

Our results indicate an impulsive-like desorption of Cs atoms after excitation
of the 6p ${}^{2}\Pi_{1/2}$ state at an excitation wavenumber of
$E_{\mathrm{pump}}/(hc)$ = 11260 cm-1. Compared to that, we have observed a
significantly reduced ion yield and a high photoelectron yield at an
excitation wavenumber of $E_{\mathrm{pump}}/(hc)$ = 11120 cm-1. The
photoelectron yield resulting from the excitation at $E_{\mathrm{pump}}/(hc)$
= 11120 cm-1 is a factor of $\approx\,4.5$ higher than for excitation at
$E_{\mathrm{pump}}/(hc)$ = 11260 cm-1. These findings confirm the results of
previous experimental studies in which Cs atoms, excited at a wavenumber
corresponding to the 6p ${}^{2}\Pi_{1/2}\leftarrow$ 6s ${}^{2}\Sigma_{1/2}$
transition in Cs-HeN, were found not to desorb from the He nanodroplet at
excitation wavenumbers $E_{\mathrm{pump}}/(hc)<11200$ cm-1. Following
photoexcitation of the 6p ${}^{2}\Pi_{1/2}$ state and subsequent photo-
ionization, we have measured only a very small number of CsHe+ ions,
corresponding to the desorption of CsHe exciplexes. This confirms the absence
of exciplex formation in this state, due to a barrier which prevents access to
the corresponding well in the CsHe potential [33]. For the excitation of the
6p ${}^{2}\Pi_{1/2}$ state, the determined fall-back times are in qualitative
agreement with the results from time-dependent 4He-DFT simulations by Coppens
et al. [33]. The experimentally obtained asymptotic ion velocity of the
desorbing ions in the ${}^{2}\Pi_{1/2}$ state is higher than the theory value.
This may be explained by signal contributions of the excitation at the blue
edge of the dipole absorption spectrum, whereas the theory value is obtained
at the maximum intensity of the dipole absorption spectrum.

After excitation of the 6p ${}^{2}\Sigma_{1/2}$ state, we report an impulsive
desorption of excited-state Cs atoms which is also supported by the results of
the DFT simulations by Coppens et al. [33]. Additionally, we have observed
exceptionally high ion kinetic energies which cannot be explained by using the
calculated pseudo-diatomic potential energy curve for this state. Our results
strongly suggest that spin-orbit relaxation from the 6p ${}^{2}\Sigma_{1/2}$
state to the 6p ${}^{2}\Pi_{1/2}$ state leads to the desorption of the
excited-state Cs atoms. We estimate that 36 $\%$ of the energy released by
spin-orbit relaxation is converted to the kinetic energy of the ions.

We have also observed the desorption of excited-state Cs atoms and CsHe
exciplexes after excitation to the 6p ${}^{2}\Pi_{3/2}$ state, which is not
predicted by theory. Because the observed pump-probe dependence for the
excitation of this state is very different compared to the other states, we
believe that a mechanism is at play which is distinctively different from
impulsive desorption. For this state, we suggest an evaporative-like
desorption mechanism. Furthermore, we show that the vibrational relaxation of
CsHe exciplexes in the 6p ${}^{2}\Pi_{3/2}$ state can release enough energy to
cause the desorption of CsHe exciplexes from the droplet surface. The same
process was proposed before for RbHe exciplexes [44] and NaHe exciplexes [56].
CsHe exciplexes formed after the excitation of the 6p ${}^{2}\Sigma_{1/2}$
state show impulsive dynamics which are also not predicted by theory. We
explain this observation by a parasitic population of the 6p ${}^{2}\Pi_{3/2}$
state as a result of the broad laser excitation spectrum.

The desorption dynamics of Cs2, which are observed as an increase of the
Cs${}_{2}^{+}$ ion yield as a function of pump-probe delay, are in the same
time range as those observed for excited-state Cs atoms and CsHe exciplexes.
Additional theoretical studies are necessary to understand the dynamics
observed for Cs2 as well as their possible role in the CsHe exciplex dynamics
observed after excitation of the 6p ${}^{2}\Sigma_{1/2}$ state.

##  V Acknowledgements

This work was financially supported by the German Research Foundation (DFG;
GRK 2079 ‘Cold Controlled Ensembles in Physics and Chemistry’) and by the
European Research Council (ERC) Advanced Grant COCONIS (694965). K.D.
acknowledges financial support by the Fonds der Chemischen Industrie (Liebig
Fellowship). The authors declare no competing financial interest.

## References

  * Toennies and Vilesov [2004] J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004).
  * Stienkemeier and Vilesov [2001] F. Stienkemeier and A. F. Vilesov, J. Chem. Phys. 115, 10119 (2001).
  * Choi  _et al._ [2006] M. Y. Choi, G. E. Douberly, T. M. Falconer, W. K. Lewis, C. M. Lindsay, J. M. Merritt, P. L. Stiles, and R. E. Miller, Int. Rev. Phys. Chem. 25, 15 (2006).
  * Stienkemeier and Lehmann [2006] F. Stienkemeier and K. K. Lehmann, J. Phys. B: At. Mol. Opt. Phys. 39, R127 (2006).
  * Mudrich and Stienkemeier [2014] M. Mudrich and F. Stienkemeier, Int. Rev. Phys. Chem. 33, 301 (2014).
  * Ziemkiewicz  _et al._ [2015] M. P. Ziemkiewicz, D. M. Neumark, and O. Gessner, Int. Rev. Phys. Chem. 34, 239 (2015).
  * Mauracher  _et al._ [2018] A. Mauracher, O. Echt, A. M. Ellis, S. Yang, D. K. Bohme, J. Postler, A. Kaiser, S. Denifl, and P. Scheier, Phys. Rep. 751, 1 (2018).
  * Krishnan  _et al._ [2011] S. R. Krishnan, L. Fechner, M. Kremer, V. Sharma, B. Fischer, N. Camus, J. Jha, M. Krishnamurthy, T. Pfeifer, R. Moshammer, J. Ullrich, F. Stienkemeier, M. Mudrich, A. Mikaberidze, U. Saalmann, and J.-M. Rost, Phys. Rev. Lett. 107, 173402 (2011).
  * Heidenreich  _et al._ [2016] A. Heidenreich, B. Grüner, M. Rometsch, S. R. Krishnan, F. Stienkemeier, and M. Mudrich, New J. Phys. 18, 073046 (2016).
  * Mozhayskiy  _et al._ [2007] V. Mozhayskiy, M. N. Slipchenko, V. K. Adamchuk, and A. F. Vilesov, J. Chem. Phys. 127, 094701 (2007).
  * Ernst and Hauser [2021] W. E. Ernst and A. W. Hauser, Phys. Chem. Chem. Phys. 23, 7553 (2021).
  * Emery  _et al._ [2013] S. B. Emery, K. B. Rider, B. K. Little, and C. M. Lindsay, J. Phys. Chem. C 117, 2358 (2013).
  * von Vangerow  _et al._ [2015] J. von Vangerow, O. John, F. Stienkemeier, and M. Mudrich, J. Chem. Phys. 143, 034302 (2015).
  * von Vangerow  _et al._ [2017] J. von Vangerow, F. Coppens, A. Leal, M. Pi, M. Barranco, N. Halberstadt, F. Stienkemeier, and M. Mudrich, J. Phys. Chem. Lett. 8, 307 (2017).
  * Stienkemeier  _et al._ [1996] F. Stienkemeier, J. Higgins, C. Callegari, S. I. Kanorsky, W. E. Ernst, and G. Scoles, Z. Phys. D: At., Mol. Clusters 38, 253 (1996).
  * Bünermann  _et al._ [2007] O. Bünermann, G. Droppelmann, A. Hernando, R. Mayol, and F. Stienkemeier, J. Phys. Chem. A 111, 12684 (2007).
  * Ancilotto  _et al._ [1995] F. Ancilotto, E. Cheng, M. W. Cole, and F. Toigo, Z. Phys. B: Condens. Matter 98, 323 (1995).
  * Dalfovo [1994] F. Dalfovo, Z. Phys. D 29, 61 (1994).
  * Theisen  _et al._ [2011a] M. Theisen, F. Lackner, and W. E. Ernst, J. Chem. Phys. 135, 074306 (2011a).
  * Theisen  _et al._ [2011b] M. Theisen, F. Lackner, and W. E. Ernst, J. Phys. Chem. A 115, 7005 (2011b).
  * Auböck  _et al._ [2008] G. Auböck, J. Nagl, C. Callegari, and W. E. Ernst, Phys. Rev. Lett. 101, 035301 (2008).
  * Reho  _et al._ [1997] J. Reho, C. Callegari, J. Higgins, W. E. Ernst, K. K. Lehmann, and G. Scoles, Faraday Discuss. 108, 161 (1997).
  * Enomoto  _et al._ [2004] K. Enomoto, K. Hirano, M. Kumakura, Y. Takahashi, and T. Yabuzaki, Phys. Rev. A 69, 012501 (2004).
  * Schulz  _et al._ [2001] C. P. Schulz, P. Claas, and F. Stienkemeier, Phys. Rev. Lett. 87, 153401 (2001).
  * Reho  _et al._ [2000a] J. Reho, J. Higgins, K. K. Lehmann, and G. Scoles, J. Chem. Phys. 113, 9694 (2000a).
  * Reho  _et al._ [2000b] J. Reho, J. Higgins, C. Callegari, K. K. Lehmann, and G. Scoles, J. Chem. Phys. 113, 9686 (2000b).
  * Coppens  _et al._ [2018] F. Coppens, J. von Vangerow, M. Barranco, N. Halberstadt, F. Stienkemeier, M. Pi, and M. Mudrich, Phys. Chem. Chem. Phys. 20, 9309 (2018).
  * Nettels  _et al._ [2005] D. Nettels, A. Hofer, P. Moroshkin, R. Müller-Siebert, S. Ulzega, and A. Weis, Phys. Rev. Lett. 94, 063001 (2005).
  * Moroshkin  _et al._ [2006] P. Moroshkin, A. Hofer, D. Nettels, S. Ulzega, and A. Weis, J. Chem. Phys. 124, 024511 (2006).
  * Koch  _et al._ [2014] M. Koch, A. Kautsch, F. Lackner, and W. E. Ernst, J. Phys Chem. A 118, 8373 (2014).
  * Persson  _et al._ [1996] J. L. Persson, Q. Hui, Z. J. Jakubek, M. Nakamura, and M. Takami, Phys. Rev. Lett. 76, 1501 (1996).
  * Callegari and Ancilotto [2011] C. Callegari and F. Ancilotto, J. Phys. Chem. A 115, 6789 (2011).
  * Coppens  _et al._ [2019] F. Coppens, J. von Vangerow, A. Leal, M. Barranco, N. Halberstadt, M. Mudrich, M. Pi, and F. Stienkemeier, Eur. Phys. J. D 73, 94 (2019).
  * Toennies and Vilesov [1998] J. P. Toennies and A. F. Vilesov, Annu. Rev. Phys. Chem. 49, 1 (1998).
  * Fechner  _et al._ [2012] L. Fechner, B. Grüner, A. Sieg, C. Callegari, F. Ancilotto, F. Stienkemeier, and M. Mudrich, Phys. Chem. Chem. Phys. 14, 3843 (2012).
  * Dick [2014] B. Dick, Phys. Chem. Chem. Phys. 16, 570 (2014).
  * Daly [1960] N. R. Daly, Rev. Sci. Instrum. 31, 264 (1960).
  * Hernando  _et al._ [2012] A. Hernando, M. Barranco, M. Pi, E. Loginov, M. Langlet, and M. Drabbels, Phys. Chem. Chem. Phys. 14, 3996 (2012).
  * Theisen  _et al._ [2011c] M. Theisen, F. Lackner, F. Ancilotto, C. Callegari, and W. E. Ernst, Eur. Phys. J. D 61, 403 (2011c).
  * Beswick and Jortner [1977] J. Beswick and J. Jortner, Chem. Phys. Lett. 49, 13 (1977).
  * Beswick and Jortner [1981] J. A. Beswick and J. Jortner, Intramolecular dynamics of van der waals molecules, in  _Advances in Chemical Physics_ (John Wiley & Sons, Ltd.:, 1981) pp. 363–506.
  * Rohrbacher  _et al._ [2000] A. Rohrbacher, N. Halberstadt, and K. C. Janda, Annu. Rev. Phys. Chem. 51, 405 (2000).
  * Beswick  _et al._ [2012] J. A. Beswick, N. Halberstadt, and K. C. Janda, Chem. Phys. 399, 4 (2012).
  * Leino  _et al._ [2011] M. Leino, A. Viel, and R. E. Zillich, J. Chem. Phys. 134, 024316 (2011).
  * Young  _et al._ [1994] L. Young, W. T. Hill, S. J. Sibener, S. D. Price, C. E. Tanner, C. E. Wieman, and S. R. Leone, Phys. Rev. A: At., Mol., Opt. Phys. 50, 2174 (1994).
  * Loginov and Drabbels [2007] E. Loginov and M. Drabbels, J. Phys. Chem. A 111, 7504 (2007).
  * Loginov  _et al._ [2011] E. Loginov, C. Callegari, F. Ancilotto, and M. Drabbels, J. Phys. Chem. A 115, 6779 (2011).
  * Loginov  _et al._ [2015] E. Loginov, A. Hernando, J. A. Beswick, N. Halberstadt, and M. Drabbels, J. Phys. Chem. A 119, 6033 (2015).
  * Loginov and Drabbels [2014] E. Loginov and M. Drabbels, J. Phys. Chem. A 118, 2738 (2014).
  * Brühl  _et al._ [2001] F. R. Brühl, R. A. Trasca, and W. E. Ernst, J. Chem. Phys. 115, 10220 (2001).
  * Lindebner  _et al._ [2014] F. Lindebner, A. Kautsch, M. Koch, and W. E. Ernst, Int. J. Mass Spectrom. 365-366, 255 (2014).
  * Kautsch  _et al._ [2013] A. Kautsch, M. Koch, and W. E. Ernst, J. Phys. Chem. A 117, 9621 (2013).
  * von Vangerow  _et al._ [2014] J. von Vangerow, A. Sieg, F. Stienkemeier, M. Mudrich, A. Leal, D. Mateo, A. Hernando, M. Barranco, and M. Pi, J. Phys. Chem. A 118, 6604 (2014).
  * von Vangerow [2017] J. von Vangerow,  _Femtosecond time-resolved photoion and femtosecond time-resolved photoion and photoelectron imaging spectroscopy of doped helium nanodroplets of doped helium nanodroplets_ , Ph.d. dissertation, University of Freiburg, Freiburg (2017).
  * Leal  _et al._ [2016] A. Leal, X. Zhang, M. Barranco, F. Cargnoni, A. Hernando, D. Mateo, M. Mella, M. Drabbels, and M. Pi, J. Chem. Phys. 144, 094302 (2016).
  * Loginov [2008] E. Loginov,  _Photoexcitation and photoionization dynamics of doped liquid helium-4 nanodroplets_ , Ph.d. dissertation, École polytechnique fédérale de Lausanne, Lausanne (2008).


# Neural networks for on-the-fly single-shot state classification

Rohit Navarathna1,2<EMAIL_ADDRESS>Tyler Jones1,2,3 Tina Moghaddam1,2
Anatoly Kulikov1,2,4 Rohit Beriwal1,2 Markus Jerger1,2,5 Prasanna Pakkiam1,2
Arkady Fedorov1,2 1 ARC Centre of Excellence for Engineered Quantum Systems,
St Lucia, Queensland 4072, Australia
2 School of Mathematics and Physics, University of Queensland, St Lucia,
Queensland 4072, Australia
3 Max Kelsen, Spring Hill, Queensland 4000, Australia
4 Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
5 JARA-FIT Institute for Quantum Information, Forschungszentrum Jülich, 52425
Jülich, Germany

###### Abstract

Neural networks have proven to be efficient for a number of practical
applications ranging from image recognition to identifying phase transitions
in quantum physics models. In this paper we investigate the application of
neural networks to state classification in a single-shot quantum measurement.
We use dispersive readout of a superconducting transmon circuit to demonstrate
an increase in assignment fidelity for both two and three state
classification. More importantly, our method is ready for on-the-fly data
processing without overhead or need for large data transfer to a hard drive.
In addition we demonstrate the capacity of neural networks to be trained
against experimental imperfections, such as phase drift of a local oscillator
in a heterodyne detection scheme.

††preprint: APS/123-QED

##  I Introduction

Machine learning (ML) is ubiquitous in modern computer science, with
applications ranging from image and speech recognition to self-driving vehicle
and automated medical diagnostic systems. By virtue of its archetypal problem
classes – regression and classification – ML algorithms and neural networks in
particular have recently found a number of applications in quantum computing,
helping researchers to tackle such tasks as optimizing gates and pulse
sequences [Zahedinejad2016, August2017, Ding2021, Baum2021], identifying phase
transitions [Rem2019, Dong2019], correcting imperfections of measurement
apparatus [Palmieri2020, Zwolak2020, Durrer2020], classifying states
[Hueaav2761, Cimini2020] or evolution [Stenberg2016, Flurin2020, Gentile2021]
of a quantum system with little or no a priori knowledge, and even optimizing
the fabrication process [Mei2021a].

A proposal to use machine learning to discriminate measurement trajectories
and outcomes was one of the first and most natural applications of ML in the
field, and has led to improvements in readout assignment fidelity
[Magesan2015]. The technique is now being regularly implemented across the
community [Dickel2018, Kono2018, Martinez2020]. As these methods continue to
see success, neural networks have become a promising technique for
incorporation into the readout procedure, due to their generalisability and
capacity to extract useful features from dense data. A recent advancement uses
neural networks to compensate for system dependent errors due to processes
such as cross-talk in multiplexed qubit readout [Lienhard2021]. In this work
we also apply neural networks to the readout of a superconducting transmon
system. However, our approach works on-the-fly with no data processing
overhead and can be trained against experimental parameter drifts, in addition
to increasing readout fidelity in two and three state discrimination
scenarios.

To deploy our neural-network-based state classification, we use an open source
machine learning PyTorch library [NEURIPS2019_9015]. Geared towards computer
vision and natural language processing, it includes the capability to realise
deep neural networks and contains built-in functionality for data processing
on a graphics processing unit (GPU). GPU integration enables our pipeline to
be fast enough to perform on-the-fly data classification without the need to
transfer raw measured signal to a hard drive. Amongst other advantages, it
allows monitoring the readout assignment fidelity in real time.

With the initial training of the neural network taking on the order of
minutes, consequent retraining of the network weights requires several seconds
and allows the readout assignment fidelity to return to the optimal value.
More importantly, the convolutional neural network used in the present work
may be designed and trained in a way resilient to certain experimental
parameter drifts. Specifically, we present a strategy to eliminate the effect
of local relative phase drifts induced by generating microwave equipment on
the readout assignment fidelity.

The paper is organized as follows. Section II contains the description of our
measurement setup. In Section III we detail the optimal neural network
architecture as well as compare the performance of competing methods. In
Section IV, we present the real-time classification using GPU processing and
investigate robustness of our methods against phase drift. In the final
section we summarise the results, report conclusions and outline possible
future extensions of our proposed methodology.

##  II Measurement setup

The measurement setup caters to the repeated preparation and measurement of
the transmon. We prepare the transmon in one of the two or three basis states,
followed by a measurement. The assignment fidelity, as defined in Equation 1
is used to compare the efficacy of different classification methods.

Figure 1: Experimental setup. The cavity drive is generated by a pulsed
microwave signal generator. The transmon drive uses an upconversion scheme
with two AWG channels and a microwave signal generator leading to the charge
line of the transmon. The transmission through the cavity is amplified using a
JPA, HEMT and room temperature amplifiers. The signal is then downconverted to
25 MHz using a mixer and digitized. The digitized traces are sent to the GPU
in batches, then downconverted to DC and filtered using an FIR low pass
filter. This is the data which is processed and classified.

###  II.1 Samples, transmon control and detection chain

We performed the experiments with two different samples, each comprising a
primitive of the circuit quantum electrodynamics platform: a transmon coupled
to a readout cavity. The first run of experiments was used as a test-bed to
compare the quality of various ML methods. It was conducted using a single
transmon embedded in a three-dimensional copper cavity (Sample A). The second
set of experiments demonstrated on-the-fly data processing using a GPU and the
methods’ stability against phase drifts; here one of the transmons on a multi-
transmon chip (Sample B) was used.

Both samples utilised frequency-tunable transmons, where the frequency of the
primary transition between the ground (first excited) state and the first
(second) excited state $\omega_{ge}$ ($\omega_{ef}$) is controllable by
external magnetic coils. The interaction between the transmons and their
respective resonators leads to the corresponding state-dependent dispersive
shifts of the resonator frequency $\omega_{r}$ specified as $\chi_{ge}$ and
$\chi_{ef}$. The dispersive shifts allow performing readout of the qutrit
states by sending a probe tone to the resonator at frequency $\omega_{d}$ and
subsequently detecting the transmitted signal with a standard heterodyne
detection scheme. These and other relevant sample parameters are shown in
Table 1 for both samples.

After an initialization period which thermalizes the qubit to the ground
state, we used Gaussian pulses of length $20$ ns to prepare the transmon in
the three basis states; applying no pulse to keep the transmon in its ground
state $\left|g\right>$, applying one $\pi$-pulse at $\omega_{ge}$ to prepare
the transmon in its first excited state $\left|e\right>$, and applying two
consecutive $\pi$-pulses at $\omega_{ge}$ and $\omega_{ef}$ respectively to
prepare a transmon in the second excited state $\left|f\right>$. These
protocols are illustrated in Fig. 2a.

Immediately after execution of the control pulses, a rectangular measurement
pulse is applied to the input of the readout cavity. This measurement pulse
populates the resonator with photons which leak out through an output port at
a rate characterised by the photon decay rate $\kappa$. To achieve a
sufficiently high SNR for transmon readout in a single-shot regime, we use a
Josephson parametric amplifier (JPA) similar to one described in Ref.
[Eichler2014]. Following amplification through the JPA, the readout signal is
further amplified using a High Electron Mobility Transistor (HEMT) amplifier
and multiple room temperature amplifiers. The signal is then downconverted to
25 MHz and acquired by a digitizer.

###  II.2 GPU processing

For the on-the-fly experiments with Sample B we acquired 512 time points per
measurement, recorded to the buffer of a 500 MSa/s digitizer Spectrum M4i.
After populating the buffer with 2048 time traces, we transferred the digital
waveforms to PC memory (RAM) and then to the GPU memory for batch processing.
While the data is being processed, the digitizer buffer is populated with new
acquired waveforms. This parallel workflow enables our acquisition system to
circumvent any overhead due to the data processing.

Within our data processing workflow, each acquired waveform undergoes digital
downconversion (DDC) by multiplying the acquired signal with $\cos(\omega_{\rm
DDC}t$) ($\sin[\omega_{\rm DDC}t]$) where $\omega_{\rm DDC}/2\pi=25~{}$MHz, to
obtain the in-phase quadrature $I(t)$ (out-of-phase quadrature $Q(t)$). A
Finite Impulse Response (FIR) Filter with a window of $40$ samples ($20$ ns)
and a cutoff frequency of $20$ MHz is applied to the signal to eliminate the
signal image at $50$ MHz along with $25$ MHz noise (originally DC offset).
After obtaining $I(t)$ and $Q(t)$, the signal undergoes further post-
processing. This may include time integration, channel correlation, or even
being fed through trained PyTorch neural networks (see more details in Section
III).

Due to the large number of cores in the GPU the data can be processed in
parallel, which allowed us to perform real time data acquisition for the on-
the-fly experiment described in Section IV. Although the results of this paper
were obtained with a repetition time of $40~{}\mu$s, our GPU data processing
can be run without overhead as fast as $3.2~{}\mu$s per repetition, obtaining
$19$ million traces ($19$ trillion samples) in $1$ minute.

Parameter | Transmon A | Transmon B
---|---|---
$\omega_{cav}/2\pi$ | 7.08 GHz | $7.63~{}\rm{GHz}$
$\omega_{ge}/2\pi$ | 6.27 GHz | $5.49~{}\rm{GHz}$
$\omega_{ef}/2\pi$ | 5.95 GHz | $5.16~{}\rm{GHz}$
$2\chi_{ge}/2\pi$ | 8.00 MHz | $8.50~{}\rm{MHz}$
$2\chi_{ef}/2\pi$ | 5.35 MHz | $15.57~{}\rm{MHz}$
$\kappa/2\pi$ | 1.31 MHz | $1.56~{}\textrm{MHz}$
$T_{1}$ | 11.75 $\mu$s | $4.07~{}\textrm{\mu s}$
$T_{2}$ | 3.17 $\mu$s | $4.29~{}\textrm{\mu s}$
Table 1: Device parameters

###  II.3 State classification by signal integration

In order to determine the baseline readout fidelity, we first employed the
conventional method of state classification. Following this method, we prepare
the relevant basis states of the transmon followed by a measurement pulse. The
heterodyne measurement signal which resulted was integrated in time, giving us
one complex number for each acquisition with integrated in-phase and out-of-
phase quadratures $I$ and $Q$. By repeating the measurements, we can populate
measurement outcome histograms for every prepared state on the I-Q plane, as
shown in Fig. 2. The mean values of the state responses are then calculated
and stored as calibration data. A particular measurement response can be
classified by selecting the state whose mean response is closest to this
recorded point on the I-Q plane. The assignment fidelity can be evaluated as

$\mathcal{F}_{a}=\frac{1}{N}\sum_{i=1}^{N}\mathbb{P}(i|i)$ (1)

where $\mathbb{P}(i|j)$ is the probability of obtaining outcome “$i$” given
the system was prepared in $j^{\textrm{th}}$ state. Here, we use $N=2(3)$ and
states ${\left|g\right>,\left|e\right>}$
(${\left|g\right>,\left|e\right>,\left|f\right>}$) to calculate qubit (qutrit)
assignment fidelity.

Figure 2: a) Transmon control pulses used for preparing the three basis states
of the transmon. b) Histogram of integrated cavity responses $I+iQ$ for
different prepared states. The blue, red and green points correspond to the
transmon being prepared in the $\left|g\right>$, $\left|e\right>$ and
$\left|f\right>$ states respectively. The relaxation of the f-state to the
excited state and excited state to ground state is visible. Some of the points
that are decaying from the f-state to the excited state will be classified as
ground state due to their proximity to the ground state cluster.

For the coherence time of Sample B we found the optimal integration time of
$1~{}\mu s$, which provides a qutrit assignment fidelity of $83\%$. Given the
rather short $T_{1}$ times of the transmon in Sample B, this fidelity is
strongly affected by relaxation during measurement.

Alternatively, one can apply a matched filter to the heterodyne measurement
signal prior to integration in the conventional method. Matched filters are
calibrated by taking the means of all acquired signals corresponding to each
basis states, and storing the complex conjugates of these expected responses.
The complex conjugate is used to inflate the weight of signal samples with
highest amplitude while enforcing constant phase, leading to a more meaningful
final value for discrimination purposes than simple integration. An incoming
signal for classification is passed through each of these three filters, and
the differences between filtered signals are evaluated. The filter which
returns maximum average amplitude upon application to the readout signal
determines the classified basis state.

##  III Machine Learning

Integrated cavity response provides a simple way to perform state
classification, but neglects many properties of the signal. Fig. 3 shows the
mean readout signal trajectories of the three transmon states on the I-Q
plane, and corresponding examples of their single shots. Certain features of
these responses (such as the fact that the response for the ground state
curves clockwise while the responses for the first and second states curve
anticlockwise) are not recoverable in the signal after time integration. While
the matched filter processing may help to compensate for this deficiency, it
is proposed that neural network-based machine learning may present a superior
method for classifying trajectories according to these intrinsic features.

Figure 3: Trajectories for the three states. The solid lines are the mean of
2048 traces acquired for each prepared state. The dashed lines are examples of
single shots, which are the inputs to the machine learning model. The arrows
indicate the direction of the single shot trajectories. The mean trajectories
start from $(\rm{I},\rm{Q})=(0,0)$. a) Both the conventional and CNN correctly
classify the shots. b) The CNN correctly classifies all shots, but the
conventional method wrongly classifies the $\left|f\right>$ shot as
$\left|e\right>$. This is likely because the transmon decayed from
$\left|f\right>$ to $\left|e\right>$ early in its trajectory.

To identify the best ML learning method we first collected data from Sample A.
The transmon was prepared in each of the three basis states ($\left|g\right>$,
$\left|e\right>$ and $\left|f\right>$) followed by the 2 $\mu$s measurement
pulse, resulting in 50 time samples for each trace at 100 MHz. In total, we
collected 16384 traces corresponding to each state for analysis. $90\%$ of
this data was used for training, and the rest was used to test and obtain the
classification fidelity.

###  III.1 Developing the model

Model | 2 state fidelity | 3 state fidelity
---|---|---
Conventional | 0.841 | $0.711$
Matched Filter | 0.913 | $0.747$
K Nearest Neighbours | 0.902 | $0.845$
Support Vector Machine | 0.917 | $0.851$
Random Forest Classifier | 0.917 | $0.874$
Vanilla Neural Network | 0.912 | $0.925$
LSTM | 0.909 | $0.904$
CNN | 0.919 | $0.928$
Table 2: Assignment fidelities for different machine learning models evaluated
on an identical test data set generated with Sample A. For all methods, there
are significant improvements over the conventional method (time integration of
readout signal and setting classification thresholds). CNN is the best model
for classifying between three levels. This method was selected to be used in
Section IV of the paper for on-the-fly data analysis. Note that the same data
was used to extract both the qubit and qutrit fidelities. Since the readout
parameters were optimised for the qutrit case the CNN model returns higher
value for the 3 state fidelity.

Before using neural networks, we evaluated the assignment fidelity of other
common machine learning methods including the support vector machine, a random
forest classifier, and a k-nearest-neighbors algorithm. After this, we
deployed several neural network architectures, including a single hidden layer
(‘vanilla’) feedforward neural network, several versions of convolutional
neural networks (CNN), and long short term memory networks (LSTM). The
assignment fidelities obtained using each of these algorithms is shown in
Table 2.

The k-nearest neighbour algorithm bears the most similarity to the baseline
assignment fidelity method, using the mean trajectory instead of just the
integrated point from a trajectory. The LSTM algorithm is popular in language
processing, and was chosen because they are designed to deal with sequences of
data, and can therefore process long time correlations between data points. We
hypothesised that this property would help to detect relaxation events, just
as one might detect a syllabic transition while processing language.

A CNN is most popularly used for pattern recognition and image classification.
It is a neural network where the hidden layer is a convolution of the input
with a kernel (or filter). We feed the time-domain signal data to the CNN with
the I(t) and Q(t) traces as two channel inputs, analogous to the red, blue and
green channels of a color image. The CNN model parameters which resulted in
the highest fidelity are described in more detail in Sec. IV.

###  III.2 Global phase robustness

Single-shot readout trajectories are well separated in the I-Q plane by their
relative differences in amplitude and phase. While the signal amplitude is
generally stable the global phase of the readout signal can sometimes vary
throughout the course of an experiment schedule. This is particularly relevant
for our experimental setup, where the readout signal itself and the local
oscillator for the signal downconversion were produced by separate microwave
signal generators. In addition, time jitter of the acquisition trigger pulse
for the digitizer can also introduce a global phase shift to the resultant
readout signal. Drifts in phase will manifest themselves as overall rotation
in the I-Q plane and will affect the classification fidelities for the
conventional threshold methods.

The neural networks used for state classification can in principle learn to
discriminate on the basis of relevant features (general behaviours of single-
shot readout trajectories associated with each state, ‘ground states curve
clockwise’) rather than features which are localised to individual datasets
(global phase rotation, ‘ground states project rightward’). To enforce
robustness to such input parameter variations in a neural network, the
training dataset should contain samples which are broadly distributed across
the domain of that input parameter. This effectively removes the parameter
(global phase) from the possible set of features upon which the model can
learn.

To endow the network with global phase robustness, a dataset was created by
obtaining 256 traces for each of the three basis states ($\left|g\right>$,
$\left|e\right>$ and $\left|f\right>$) at each of 500 different global phases
(between 0 and 2$\pi$). Global phase was applied by imposing an arbitrary wait
time at the beginning of each experiment repetition. The readout signals
obtained within each experiment are downconverted to 25 MHz using a local
oscillator which is phase-locked to the generator producing the readout signal
itself. This phase-locking ensures that if measurements occur at integer
multiples of 40 ns within the pulse sequence, phase coherence is conserved. By
generating data with uniformly distributed initial wait times between 0 and 40
ns, a predictable global phase was no longer a feature of the training
dataset, forcing the model to learn a phase-robust mapping from readout
trajectory to qutrit state. We fed this data into a CNN network in the same
manner as described in Sec. III.1.

##  IV On-the-fly acquisition

After selecting CNN as the method with the highest assignment fidelity, we
apply this model to state classification on-the-fly. This on-the-fly protocol
minimizes the size of data stored on the hard disk, reducing temporal overhead
for data processing and allowing for real-time monitoring of the assignment
fidelity.

###  IV.1 FIR bandwidth optimization

We used a 5 MHz FIR low pass filter for Sample A. However, we found it was
beneficial to increase the cutoff frequency of the FIR filter to 20 MHz when
using CNN data processing for Sample B. This exposes the CNN to more noise,
but potentially allows access to more information about the prepared states.
Incidentally, this operation also requires fewer coefficients on the FIR
filter, reducing computation complexity and significantly lowering the time
required by the CNN to reach the highest fidelity.

###  IV.2 The model

Here we describe the specific architecture of the network used in our CNN
model. The network consists of the following layers:

  1. 1.

1D convolution: A convolution layer with 2 input channels (corresponding to I
and Q data channels), 16 output channels, and a kernel size of 128. The
relatively large kernel size is hypothesised (but rigorously confirmed) to
filter out higher frequency noise more effectively. The initial kernel weights
(or filter coefficients) are manually set using the He initialisation function
[he2015delving], with experimentation demonstrating that CNN performance is
very sensitive to this weight initialisation.

  2. 2.

ReLu activation: After passing through the first layer the data undergoes
mapping $f(x)=\rm{max}(0,x)$ in order to expedite the learning process
[Nair2010].

  3. 3.

1D convolution: A convolution layer with 16 input channels (corresponding to
the output of the previous convolution layer), 32 output channels, and a
kernel size of 5. This expands the previous 16 features to 32, by taking
various linear combinations of the prior layer outputs.

  4. 4.

ReLu activation: See above for details.

  5. 5.

Max pooling: The maximums of every three neighbouring output values of the
previous layer are evaluated, to reduce the size of the data representation
and therefore computational time in the remainder of the network.

  6. 6.

Flattening: This step reshapes data to a one-dimensional array removing input
channel and time indices. This is necessary for data processing in upcoming
layers. No data is fundamentally altered here.

  7. 7.

Dropout: 50% of the data points are randomly selected to be set to zero
(called ‘neuron deactivation’). This step was initially introduced to prevent
the model from being overtrained. Overtraining or overfitting of a ML model is
known behaviour of the ML method when it learns ‘too much’ from the training
data, showing an artificially high accuracy of classification while training,
but failing to maintain performance when tested on new data. This may occur if
there are certain features of the signal present in the training dataset due
to noise or other conditions relevant to the data-taking process, which are
not present in the testing data set (see more discussion on overfitting in
Subsection IV.3). The dropout layer safeguards against a heavy reliance on any
single weight or feature.

  8. 8.

Linear: This layer applies a linear transformation to the incoming data:
$y=Ax+b$, where weights (denoted as a matrix $A$) and biases (denoted as a
vector $b$) are optimized. Here, the size of the output $y$ is half of the
size of the input $x$.

  9. 9.

ReLu activation: See above for details.

  10. 10.

Linear: In this last linear layer the size of the output data is designed to
be two for the qubit and three for the qutrit classification, corresponding to
the possible preparation states.

All optimization in the neural network was performed with the Adam optimizer
[kingma2017adam] and a mean squared error (MSE) loss function. The output was
classified using a softmax function; for the qutrit case, this results in the
mapping of three outputs to one state, $\left|g\right>$, $\left|e\right>$ or
$\left|f\right>$.

###  IV.3 Training

Generation of the labelled data required for model training involved preparing
the transmon in each of the three states, probing with a measurement signal
and recording the output. For each training cycle, we record 2048 traces of
each state, and pass these through the model. The loss was calculated and
gradient descent was undertaken at a learning rate of $10^{-3}$. This cycle
takes $\sim 3$ seconds.

We trained the model on new data at every training cycle, acquired in real-
time from the sample in the dilution refrigerator. This framework provides an
intrinsic protection from overfitting; since there is a new dataset each time
the loss is calculated, the model cannot learn on any spurious signal features
localised to a single dataset. The low learning rate assists in helping the
model to learn patterns that are common to data across training cycles,
thereby increasing model stability.

###  IV.4 Testing

After each update of the model weights, a further 2048 acquisitions were made
to test the model. The loss and assignment fidelity on test data were stored
for monitoring. Fig. 4 shows the model learning over 100 training cycles.

Figure 4: Training over 100 training cycles and re-training over 20 cycles.
The left plot shows the CNN learning to differentiate between the three (or
two) states. The model provides better fidelity than the baseline fidelity
within 30 training cycles over approximately one minute. The plot on the right
shows the model being re-trained after running the experiment for a few hours.
Retraining takes only 20 training cycles.

##  V Results

Figure 5: Fidelity of different classification methods. Each point represents
a fidelity evaluation using 2048 traces for each of the three states gathered
on the fly, while each plot represents a different training regime. The black
dashed lines indicate when the model was retrained. The retrain CNN model
consistently does better than the cal-baseline measurement, even though it is
only trained at discrete intervals. The no retraining model generally has a
better fidelity than the baseline, but not always, due to parameter drifts.
The 5 retrains and 10 retrains models are the same models as the retrain
model, frozen to prevent further training at their respective number of
retrains. The black arrows show which frozen model is used.

To investigate robustness of the CNN classification model against system
parameter drifts, we performed continuous measurements over 24 hours and
monitored the fidelity values as shown in Fig. 5. Each plot shows three
fidelity values obtained from the same data but using different data
processing techniques. First, we evaluated the fidelities obtained from the
integrated responses as described in Sec. II.3, where the mean responses for
$|g\rangle$, $|e\rangle$ and $|f\rangle$ states are calibrated only once prior
to the experiment “Baseline”). Second, we obtained the fidelity using the mean
responses re-calibrated every 2048 repeated measurements (“Cal-Baseline”).
Finally, we plot the fidelity obtained by the on-fly-processing with CNN model
(“CNN”) as described in Sec. IV.

To acquire additional insight into behaviours of our CNN model we show two
separate plots corresponding to different training regimes. Each of these
scenarios are described in more detail below:

  * •

Retrain model The model is trained for an initial 100 training cycles. The
fidelity is then repeatedly tested for $\sim 1$ to $3.5$ hours, before
training the model again for another $20$ training cycles. This retraining
process is akin to recalibration in the “Cal-Baseline” measurement. This model
performs better than both the “Baseline” and “Cal-Baseline” measurements,
despite the frequency of recalibration being significantly lower than its
“Cal-Baseline” counterpart.

  * •

No Retraining This model is never retrained after the initial 100 training
cycles. It also performs better than the Baseline throughout the 24 hours, but
loses its enhanced fidelity after a few hours.

###  V.1 Global phase robustness

In contrast to retraining a CNN model at set time intervals to combat system
parameter drifts, we also demonstrate the ability to generate a static CNN
model robust to parameter drift by training and evaluating in an experimental
configuration with induced phase shift as described in Sec. III.2. The design
of the neural network remains identical for this experiment, aside from using
a kernel size of 10 in the first convolutional layer, exhibiting its
versatility. Assignment fidelities are evaluated at 500 intervals with phase
shifts ranging from 0 to 2$\pi$ over the course of 9 minutes in Fig. 6, using
both the baseline mean integration and CNN processing methods
(trained/calibrated using data gathered immediately prior).

The CNN model classifies readout signals at a qutrit fidelity of
(77.9$\pm$2.5)%, with uncertainty representing the standard deviation over
global phase. This is comparable to the baseline state classification with no
phase drift calibrated prior to the experiment (79.4$\pm$1.7%), but any non-
trivial phase drift will degrade the conventional approach markedly while the
robust model maintains accuracy without any recalibration required.

Figure 6: Assignment fidelities of the baseline mean calibration method and a
phase-robust CNN model classifying data with an induced phase drift. Each
point represents a fidelity evaluation using 2048 traces for each of the three
states. The phase-robust CNN model displays considerably better performance
than the baseline method in general, and comparable performance at zero phase
shift (where the baseline was calibrated). Neither method undergoes any
retraining or recalibration over the course of the experiment.

##  VI Conclusion

In this work, we have investigated a set of machine learning methods for the
two and three state classification of a transmon with the dispersive
measurement. All ML methods trialled achieved higher assignment fidelities
than the conventional method of readout signal time-integration and subsequent
threshold calibration. Convolutional neural network methods demonstrated the
highest performance consistent with the results from [Lienhard2021]. In
particular, CNN methods were not only performant in the detection of
relaxation events during measurement, but could also be trained against
experimental imperfections such as the local oscillator phase drift.

The open-source library PyTorch facilitated the implementation of on-the-fly
CNN data processing on a GPU. Due to batch processing this method is not
readily suitable for quantum feedback but reduces data transfer to a hard
drive and allowed us to investigate the performance of the CNN method in real
time.

We found that after sufficient training, the assignment fidelity is able to
maintain its value for a few hours. After this time, retraining is required to
prevent a drop in assignment fidelity. We also found that retraining takes
substantially less time than the initial training, and training on new
measurement data taken during these few hours reduces overfitting of the
model.

Improved assignment fidelity and ability to directly train the model against
system imperfections are the key advantages for performing this readout signal
processing using neural networks. The open-source, GPU-friendly, and easily
implementable nature of PyTorch makes these neural networks an attractive tool
for state classification.

###### Acknowledgements.

We thank Andreas Wallraff, Markus Oppliger, Anton Potočnik, and Mintu Mondal
for fabricating JPA used in the measurements. The authors were supported by
the Australian Research Council Centre of Excellence for Engineered Quantum
Systems (EQUS, CE170100009) and by Lockheed Martin Corporation via research
contract S19 004.