# On the nature of two-photon transitions for a Collection of Molecules in a
Fabry-Perot Cavity

Zeyu Zhou  Department of Chemistry, University of Pennsylvania, 231 South 34th
Street, Philadelphia, Pennsylvania 19104, United States  Hsing-Ta Chen
Department of Chemistry, University of Pennsylvania, 231 South 34th Street,
Philadelphia, Pennsylvania 19104, United States  Department of Chemistry and
Biochemistry, 251 Nieuwland Science Hall, Notre Dame, Indiana 46556, United
States  Maxim Sukharev  Department of Physics, Arizona State University,
Tempe, Arizona 85287, United States  College of Integrative Sciences and Arts,
Arizona State University, Mesa, Arizona 85212, United States  Joseph E.
Subotnik  Department of Chemistry, University of Pennsylvania, 231 South 34th
Street, Philadelphia, Pennsylvania 19104, United States  Abraham Nitzan
Department of Chemistry, University of Pennsylvania, 231 South 34th Street,
Philadelphia, Pennsylvania 19104, United States  Department of Physical
Chemistry, School of Chemistry, The Raymond and Beverly Sackler Faculty of
Exact Sciences and The Sackler Center for computational Molecular and
Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

We investigate the effect of a cavity on nonlinear two-photon transitions of a
molecular system and how such an effect depends on the cavity quality factor,
the field enhancement and the possibility of dephasing. We find that the
molecular response to strong light fields in a cavity with variable quality
factor can be understood as arising from a balance between (i) the ability of
the cavity to enhance the field of an external probe and promote multiphoton
transitions more easily and (ii) the fact that the strict selection rules on
multiphoton transitions in a cavity support only one resonant frequency within
the excitation range. Although our simulations use a classical-level
description of the radiation field (i.e. we solve Maxwell-Bloch or Maxwell-
Liouville equations within the Ehrenfest approximation for the field-molecule
interaction), based on experience with this level of approximation in past
studies of plasmonic and polaritonic systems, we believe that our results are
valid over a wide range of external probing.

##  I Introduction

Microcavities, including Fabry-Perot and plasmonic cavities, are well known
for changing the local density of radiative states within the cavity and
modifying the couplings between vibrational/electronic states to light,
leading to the Purcell effect in the weak coupling regime and the Rabi
splitting in the strong coupling regime at room temperature.[1, 2, 3, 4, 5, 6,
7, 8, 9] These alterations of the radiative couplings can result in drastic
changes to the energies of light-matter states and energy flow pathways; the
responses of molecules inside a cavity towards external light sources can be
very different from the response of bare molecules in solution.[10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]

Even though numerically exact simulations of quantum dynamics are
exponentially difficult, simulations of a classical electromagnetic field
(using finite-difference time-domain (FDTD) method) combined with a quantum,
mean-field description of the matter subsystem (usually comprising a set of
two-level emitters, representing quantum dots or Rydberg atoms) have shown
that a host of interesting cavity effects can in fact be captured by mixed
quantum-classical algorithms.[27, 28, 29, 30, 31, 32, 33, 34, 35, 36] As far
as one photon processes are concerned, the Purcell effect arises
semiclassically from constructive/destructive interference during multiple
reflections of the electromagnetic (EM) field by the cavity mirrors; the Rabi
splitting arises from the continuous oscillation of energy between the EM
field and the molecules inside the cavity; cavity leakage/finite polariton
lifetime can also be observed semiclassically if we simulate finite-size
cavity mirrors with a complex-valued dielectric constant. Nonlinear effects
can also be observed in semiclassical simulations, for example, the Rabi
splitting contraction results from ground state depletion can be simulated.
[37, 38]

Now, despite the power of FDTD approach, within the realm of polaritonics and
cavity effects, most research today relies on model Hamiltonians and/or
simulations with a single cavity mode. Such approaches can often capture a
great deal of physics, but one sacrifices the ability to capture quite a few
EM observables. For instance, an important example for this paper is the
recent CavityMD study [39, 40] demonstrating that, under illumination by a
strong EM field, if a lower vibrational polariton state is tuned to half the
frequency difference a $0\rightarrow 2$ vibrational excitation, exciting this
polariton state can enhance a nonlinear transition by facilitating the energy
transfer that arises from the intermolecular couplings between molecules
inside the cavity. This hypothetical numerical experiment suggests that
solvent polariton excitations in a cavity can efficiently pump solutes and
potentially promote chemical reactions. The results of ref. 40, however, were
limited to absorption and could not easily address transmission, which is the
most easily experimentally observed measurement. Here we present a mixed
quantum-classical FDTD calculation that directly addresses this observable.
[41, 42] We note that, in order to work with the largest signal possible, we
will simulate electronic transitions (instead of vibrational transitions).

To carry out this program, we consider a Fabry-Perot cavity formed by two gold
mirrors with different mirror thicknesses (and hence different quality
factors). Inside the cavity, there is a layer of 3-level molecules (which are
considered the solute) and the two-photon nonlinear transitions between the
ground and the second excited states are calibrated to be resonant with the
incoming laser field. The solvent is represented by two layers of 2-level
molecules. Their interaction with the cavity mode leads to the formation of
the lower/upper polariton states, thereby modifying the energy at which the EM
field can be transmitted efficiently through the cavity. Next, we pump the
system with external continuous wave (CW) light and compare the simulation
results inside and outside of the gold cavity. Our simulations show $(i)$ that
within a Maxwell-Bloch mean field treatment of the light-matter interaction,
there is a field enhancement in the classical cavity and this field
enhancement is the primary reason for altering two-photon nonlinear
transitions. Furthermore, $(ii)$ the cavity is able to suppress non-resonant
transitions between the second molecular excited state and the first excited
state (as well as between the first excited state and the ground state) as a
manifestation of the Purcell effect. Hence, the resonant 2-photon nonlinear
transitions are effectively shielded by the cavity mirrors, resulting in a
prolonged Rabi oscillation, in contrast to the scenarios outside the cavity.
$(iii)$ Increasing the cavity quality factor by making the gold mirrors
thicker, we observe a turnover in the rate of the 2-photon nonlinear
transitions. Finally, $(iv)$, we find that under constant illumination and in
the presence of pure dephasing, the Rabi oscillations in the cavity are damped
and the system reaches a time-independent steady state (which is often
observed[43, 44, 45]).

This manuscript is arranged as follows. In section II, we present four model
scenarios with layers of 2-level/3-level molecules and a cavity formed by two
parallel gold mirrors. In section III, we investigate population dynamics for
these scenarios with cavities displaying different mirror thickness (quality
factors) and we study how dephasing influences the two-photon transitions. In
section IV, we conclude and discuss the remaining questions.

##  II Model and Methods

###  II.1 Structure of the systems

Figure 1: Sketches of the four scenarios that are studied in this work. (a) A
single layer of 3-level molecules (a solute layer with states
$\left|0\right\rangle,\left|1\right\rangle,\left|2\right\rangle$); (b) a
single layer of 3-level molecules sandwiched by two layers of 2-level
molecules (two solvent layers with states
$\left|0^{\prime}\right\rangle,\left|1^{\prime}\right\rangle$); (c) Same as
(a) with two surrounding planar gold mirrors forming a Fabry-Perot cavity; (d)
Same as (b) but with two surrounding gold mirrors. All four scenarios are
pumped by monochromatic external laser with angular frequency
$\omega_{in}=1.5eV/\hbar$. In inset (e), we show the relevant transitions of
the solute molecular layer alone. An external monochromatic laser matches the
energy needed for 2-photon transition of the solute molecular layer. For
scenario (c), the distance between the gold mirrors is $354$ nm, which is
resonant with the incoming driving frequency ($\hbar\omega_{in}=E_{cav}=1.5$
eV). In inset (f), we draw the relevant lower and upper polariton states
formed by strong interaction between cavity EM field and solvent molecular
layers. For scenario (d), the cavity mode (with mirror distance $d_{2}=340$
nm) is on resonance with a bare solvent excitation
($E_{cav}=E_{0^{\prime}1^{\prime}}=1.55$ eV), while the frequency of the lower
polariton matches the external monochromatic light
($E_{LP}=\hbar\omega_{in}=1.50$ eV).

In this work, we consider four scenarios with continuous pumping by a laser
source of frequency $\omega_{in}=E_{in}/\hbar$. First, as shown in Fig 1(a),
we consider a single layer of 3-level (states
$\left|0\right\rangle,\left|1\right\rangle$ and $\left|2\right\rangle$) solute
molecules, with energy difference $E_{02}=2E_{in}$ between levels
$\left|0\right\rangle$ and $\left|2\right\rangle$. Level
$\left|1\right\rangle$ is placed so that the separations between levels
$\left|0\right\rangle$, $\left|2\right\rangle$ and level
$\left|1\right\rangle$ do not overlap with the incoming frequency (in the
simulations we take $E_{01}=2.0eV\neq\hbar\omega_{in}$,
$E_{12}=1.0eV\neq\hbar\omega_{in}$) as shown in inset (e). Thus, at early
times following excitation of the system, the population of
$\left|1\right\rangle$ is expected to be small. Second, as shown in Fig. 1
(b), we sandwich the 3-level molecules with two layers of 2-level
($\left|0^{\prime}\right\rangle$, $\left|1^{\prime}\right\rangle$) molecules
(blue, 4.5 nm thickness each). Here, the 3-level molecules are considered
solute molecules and the 2-level molecules are considered solvent molecules.
(We assume that the solvent and solute molecules are kept together by a non-
metallic structure that does not interact with the EM field.)

Next, as shown in Figs. 1 (c) and (d), we add gold mirrors (golden yellow, 50
nm thickness each) that encapsulate a microcavity. We use Drude model for the
bulk gold to characterize its linear response to the EM field. In scenario
(c), the distance between the two mirrors ($d_{1}$) is chosen such that the
cavity is on resonance with the incoming driving frequency
($E_{cav}=\hbar\omega_{in}$). In scenario (d), the distance between the two
mirrors ($d_{2}$) is taken such that the cavity is on resonance with solvent
molecules, i.e. $E_{cav}=E_{0^{\prime}1^{\prime}}$. The solvent-cavity
interaction now leads to the formation of upper/lower polariton states and we
adjust $E_{0^{\prime}1^{\prime}}$ and $d_{2}$ (which controls $E_{cav}$)
together to ensure $E_{LP}=\hbar\omega_{in}$; maintaining this condition
clearly depends on the transition dipole moment for solvent molecules between
$\left|0^{\prime}\right\rangle$ and $\left|1^{\prime}\right\rangle$) (i.e. the
light-matter coupling).

###  II.2 Maxwell-Bloch equations

The interaction between the external EM pumping field is modeled in one-
dimension. Thus, the one-dimensional Maxwell equations are

$\displaystyle\frac{\partial B_{y}}{\partial t}$
$\displaystyle=-\frac{\partial E_{x}}{\partial{z}}$ (1)
$\displaystyle\epsilon_{0}\frac{\partial E_{x}}{\partial t}$
$\displaystyle=-\frac{\partial B_{y}}{\mu_{0}\partial{z}}-J_{x}$ (2)

Here, $z$ is the longitudinal coordinate that is perpendicular to the
mirrors/slabs of nanolayers. $J_{x}$ is the polarization current

$\displaystyle
J_{x}=\frac{dP_{x}}{dt}=n_{0}\frac{d(\text{Tr}(\hat{\rho}\hat{\mu}_{x}))}{dt}$
(3)

Here, $n_{0}$ is the number density, $\mu_{x}$ is a matrix of transition
dipole moments between electronic states along $x$ direction. We simulate the
EM equations above (eqs 1 and 2) with an FDTD approach.

While the field is considered in the framework of the classical Maxwell
equations, the molecular layers are described by their density matrices
$\hat{\rho}$; $\hat{\rho}$ is $3\times 3$ for solute and $2\times 2$ for
solvent molecules under a mean-field approximation. In other words, there is
one density matrix for each grid point for each of the two molecular layers.
The Hamiltonian (and the corresponding Liouvillian) are time-dependent because
of the presence of the local time-dependent electric field.

$\displaystyle i\hbar\frac{d}{dt}\hat{\rho}$
$\displaystyle=[\hat{H}(t),\hat{\rho}]$ (4)

This field enters the Hamiltonian as the coupling between adjacent electronic
states, i.e. $\left\langle
0\right|\hat{H}\left|1\right\rangle=\mu_{x}^{01}E_{x}(t)$. In summary,

$\displaystyle\hat{H}=$ $\displaystyle E_{1}\left|1\right\rangle\left\langle
1\right|+E_{2}\left|2\right\rangle\left\langle
2\right|+\mu^{01}_{x}E_{x}(\left|0\right\rangle\left\langle
1\right|+\left|1\right\rangle\left\langle
0\right|)+\mu^{12}_{x}E_{x}(\left|1\right\rangle\left\langle
2\right|+\left|2\right\rangle\left\langle 1\right|)$
$\displaystyle\hat{H}^{\prime}=$ $\displaystyle
E_{1^{\prime}}\left|1^{\prime}\right\rangle\left\langle
1^{\prime}\right|+\mu^{0^{\prime}1^{\prime}}_{x}E_{x}(\left|0^{\prime}\right\rangle\left\langle
1^{\prime}\right|+\left|1^{\prime}\right\rangle\left\langle
0^{\prime}\right|)$ (5)

###  II.3 Initial condition

All molecular layers are initialized on the ground state
($\left|0\right\rangle$ and $\left|0^{\prime}\right\rangle$, respectively).
Near the edge of the simulation cell (e.g., far outside of any cavity), we
drive one grid point as a continuous wave source with frequency
$\omega_{in}=E_{in}/\hbar$ (which are chosen $1.5$ eV for all calculations
below).

$\displaystyle E_{x}(z=z_{0},t)=E_{0}\sin\omega_{in}t$ (6)

Here, $z_{0}$ is near the edge of the simulation grid. The EM field travels in
both directions. Along one direction, the EM field reaches the material
nanolayers; along the other direction, the EM field is absorbed by the
perfectly matched layer boundary conditions (PML) so that the molecular layers
are not affected by unpredictable reflections at the boundary of the
simulation system. [27]

##  III Results

In this section, we present numerical results from three perspectives. First,
we study the role of the cavity in enhancing the EM field inside the cavity
and in shielding the solute molecular nonlinear transitions. Second, we
investigate how the 2-photon nonlinear transitions between ground state
$\left|0\right\rangle$ and the second excited state $\left|2\right\rangle$ of
the solute molecules depend on the cavity quality factor (which is controlled
by the mirror thickness). Finally, we discuss how phenomenological pure
dephasing induced by nonradiative processes leads to damping of the 2-photon
Rabi oscillations.

###  III.1 Cavity-shielded 2-photon Rabi oscillation

We begin by studying the nature of 2-photon non-linear transitions outside vs
inside cavities where the $0\rightarrow 1$ and $1\rightarrow 2$ transitions
are not on resonance with $\omega_{in}$. In the context of Maxwell-Bloch
equations, because the EM field is purely classical and therefore there is no
simple conversion between the EM field amplitude and the cavity mode
population in the context of a quantum model, we will not be able to easily
measure the populations of the lower and upper polaritons. Therefore, instead,
we will calculate the population on molecular layers (the diagonal elements of
density matrices), keeping in mind that these molecular states are not
eigenstates of the system in the presence of interactions between the cavity
EM field and molecular excited states; some oscillations are expected (at
early times especially). The frequency of the Rabi oscillations between
$\left|0\right\rangle$ and $\left|2\right\rangle$ is dictated by the rate of
2-photon nonlinear transitions. To this end, we compare the dynamics for the
scenarios without a cavity ((a) and (b)) vs the scenarios with a cavity ((c)
and (d)) in Fig 1. Because (as is well known[46]) a cavity can trap the EM
field, for a fair comparison between outside vs inside cavity, we manually
increase the electric field of the external driving for scenarios (a) and (b)
by a factor of $3.4$ that was found empirically to be the field enhancement
inside the cavity compared with the incident field, i.e.
$E_{0}^{a,b}=3.4\times E_{0}^{c,d}$.

As a sanity check, in Fig 2 (e), we plot the population dynamics of
$\rho_{22}$ for scenario (b) with different EM field strengths. As one would
expect, stronger EM field strengths result in faster oscillations.

Figure 2: (a)-(d) The standard four scenarios under probing with monochromatic
light (see Fig. 1). (e) Population dynamics in time for state
$\left|2\right\rangle$ (corresponding to the solute molecular layer) for a
series of different intensities with geometry (b). (f)-(h) Population dynamics
of the excited states of solvent $\rho_{1^{\prime}1^{\prime}}$, solute
$\rho_{22}$ and $\rho_{11}$, respectively. In (f), the red and yellow lines
(showing results for scenarios (a) and (c)) overlap with the $X$ axis
($\rho_{1^{\prime}1^{\prime}}\equiv 0$); for the blue line [scenario (b)],
$\rho_{1^{\prime}1^{\prime}}$ reaches steady state very fast; for the green
line, corresponding to scenario (d), $\rho_{1^{\prime}1^{\prime}}$ oscillates
around the steady state value of (b). In (g), for the scenarios outside the
cavity [(a) and (b)], $\rho_{22}$ oscillates before reaching a steady state
with excitation balanced by emission; for scenarios inside the cavity (c) and
(d), $\rho_{22}$ oscillates for at least $100$ ps. Finally, in subfigure (h),
in scenarios (a) and (c), we find $\rho_{11}$ reaches a long-time, steady-
state limit while, by contrast, for scenarios (b) and (d), $\rho_{11}$ remains
$0$ within $100$ ps of our simulation. These effects can all be rationalized;
see the text.

In general, below, when discussing dynamics in a cavity, we will analyze first
the population of the solvent state $\left|1^{\prime}\right\rangle$, second
the population of state $\left|2\right\rangle$, and third the population of
state $\left|1\right\rangle$ in this order – because this ordering corresponds
to how the states are populated in time under an incoming cw em field. Let us
now analyze our observations about the Rabi oscillations and population
dynamics in Fig. 2.

  * •

Consider first the population dynamics of $\rho_{1^{\prime}1^{\prime}}(t)$. As
shown in Fig 2 (f), because there is no solvent molecular layer for (a) and
(c), obviously $\rho_{1^{\prime}1^{\prime}}\equiv 0$. More interestingly, for
scenario (b), the population reaches steady state very fast. Finally, for
scenario (d), there are persistent oscillations about the steady state of (b).
In other words, the coherent transitions last longer because of the cavity.

  * •

Next, consider the population dynamics of $\rho_{22}(t)$. As shown in Fig 2
(g), for scenarios (a) and (b), we observe damped Rabi oscillations; for
scenarios (c) and (d), we observe persistent Rabi oscillations. The Rabi
oscillations for scenario (d) match the oscillations in Fig 2(f); there is
clearly energy exchange between the solvent polariton and the solute
excitation. Note also that scenarios (c) and (d) yield identical oscillation
patterns. The incoming EM field enters the cavity with the same efficiency for
scenario (c) (where $\omega_{in}$ matches the cavity geometry) and scenario
(d) (where $\omega_{in}$ matches the lower polariton state of solvent and
solute); recall that, according to Fig. 1, the cavity lengths are slightly
different for scenarios (c) and (d).

  * •

Lastly, consider the population dynamics $\rho_{11}(t)$. As shown in Fig 2
(h), for the scenarios outside the cavity ((a) and (b)), although
$E_{01}\neq\hbar\omega_{in}$, $\rho_{11}$ becomes finite because of emission
from state $\left|2\right\rangle$ to state $\left|1\right\rangle$. In
contrast, for scenarios inside the cavity ((c) and (d)), $\rho_{11}$ remains
$0$. In other words, we find that inside a cavity, the
$\left|2\right\rangle\rightarrow\left|1\right\rangle$ transition is
effectively suppressed by the cavity as $E_{cav}\neq E_{12}$. The steady state
population $\rho_{11}(t\rightarrow\infty)$ is proportional to the incoming
electric field (in the weak field regime).

Now, our description above of the process whereby the $2\rightarrow 1$
transition is suppressed by the cavity was inferred from population dynamics.
To complement this point of view, and as mentioned in the introduction, we
would like to analyze the corresponding electric field as well using our FDTD
simulation. In Fig. 3, we plot the electric field on the solute molecular
layer in both time and frequency domains in logarithm scale. As shown in Fig 3
(e), for scenarios (c) and (d), the continuous Rabi oscillations in Fig 2
synchronize with the oscillation of the electric field envelope, which
demonstrates that energy is exchanged between the cavity EM field and the
solute molecular layer. In contrast, for scenarios (a) and (b), the envelope
is practically flat; there is no extra energy exchange after the absorption
and emission processes reach a steady state.

Next, we perform Fourier transformations on the signals in (e), and as shown
in Fig 3 (f), we observe two peaks at $1.0\text{ and }2.0$ eV for the scenario
with no cavity mirrors. The positions of the peaks correspond to
$\left|2\right\rangle\rightarrow\left|1\right\rangle$ and
$\left|1\right\rangle\rightarrow\left|0\right\rangle$ emissions from the
solute molecular layers respectively. That being said, note that Fig. 3 is
plotted on a log scale and these side peaks are very small. In the end, our
conclusion is clear: because these
$\left|2\right\rangle\rightarrow\left|1\right\rangle$ and
$\left|1\right\rangle\rightarrow\left|0\right\rangle$ transitions do not match
the cavity geometry, these transitions remain suppressed for scenarios inside
the cavity (c) and (d).

Figure 3: (a-d) The standard scenarios considered (see Fig. 1). We plot the
electric field envelope acting on the 3-level molecular layer in (e) time and
(f) the frequency domain (both in log scale). As shown in Fig. (e), the
envelope of the electric field for scenarios inside the cavity (c) and (d)
oscillates with an identical frequency as in Fig 2 (f) and (h), indicating
energy transfer between molecules and the EM field. For scenarios outside the
cavity (a) and (b), there is no such oscillation. In the frequency domain, the
major peak at $1.5$ eV corresponds to $\hbar\omega_{in}$ and appears in all
four scenarios. By contrast, the peaks at $E_{12}=1.0$ eV and $E_{01}=2.0$ eV
appear only for scenarios (a) and (b).

###  III.2 The role of mirror thickness and quality factor

In the previous subsection, we compared the population dynamics for the four
scenarios (a-d) in Fig. 1 where we implemented an adjustment to the magnitude
of the incoming field ($E_{0}^{a,b}=3.4\times E_{0}^{c,d}$) which was intended
to keep the local electric field roughly the same in and out of the cavity; in
other words, for scenarios (a-b) outside the gold cavity, we applied pumping
field with amplitude greater than that used to excite the systems for
scenarios (c-d) inside the cavity. In this subsection, we will ignore such a
difference and directly probe how the quality factor of the microcavity for
scenario (d) can increase/decrease the rate of nonlinear transitions under
identical external pumping (($E_{0}^{a,b}\equiv E_{0}^{c,d}$)). Note that the
quality factor is manifested by the thickness of the gold mirrors
($L=0-100$nm) in scenario (d) so that, when $L=0$, (d) reduces to (b) and
there is no cavity; again, we maintain a constant external pumping field
strength at all times. The mirror distance ($d_{2}=340$ nm) is calibrated such
that $E_{cav}\approx E_{0^{\prime}1^{\prime}}$ so that the lower polariton
state will form.

Figure 4: Population dynamics on (a) excited state
($\left|1^{\prime}\right\rangle$) of the solvent molecules and (b-c) excited
states $\left|2\right\rangle$ and $\left|1\right\rangle$ of the solute
molecules for scenario (d) in Fig. 1 with different choices of mirror
thickness $L=0-100$ nm. (d) Maximal circulating electric field amplitude
inside the cavity vs mirror thickness (log scale). Note that, in subfigures
(b) and (d), we find an optimal thickness $L\approx 50nm$ for the fastest
nonlinear transition corresponding to the maximal circulating electric field
amplitude (whose implications are discussed in the text). The distance between
the mirrors is kept constant ($d_{2}\equiv 340$ nm) and in resonance with the
solvent molecules ($E_{cav}=E_{0^{\prime}1^{\prime}}$) for all choices of
mirror thickness.

In Fig 4, we scan over different thicknesses for the mirror ($L$) and compare
the resulting population dynamics ($\rho_{1^{\prime}1^{\prime}}(t)$,
$\rho_{11}(t)$ and $\rho_{22}(t)$).

  * •

In Fig 4(a), we plot the time dynamics for the excited state population of the
2-level solvent molecular layers $\rho_{1^{\prime}1^{\prime}}(t)$ for a
varying mirror thickness ($L=0-100$ nm, $Y$ axis). We find that
$\rho_{1^{\prime}1^{\prime}}(t)$ reaches a steady state because of the balance
between the external pumping and the decay induced by cavity leakage/mirror
absorption. Moreover, as the mirror thickness increases, the steady state
value reaches a maximum near a $40-60$ nm mirror thickness and becomes smaller
for thicker mirrors. As shown in Fig 4 (d), this maximum corresponds to the
greatest circulating electric field amplitude inside the cavity at steady
state.

  * •

In Fig 4(b), we plot the population of the second excited state of the 3-level
solute molecular layers $\rho_{22}(t)$. We find that $\rho_{22}(t)$ undergoes
Rabi oscillations for all choices of the mirror thicknesses (note that similar
oscillations were also found in Fig 2(g)). In agreement with subfigure (a), we
observe that the frequency of the Rabi oscillation shows a non-monotonic
behavior. Starting from a cavity with a very low quality factor ($L=10$nm),
the Rabi frequency increases when the mirror thickness increases. 111Note
that, the cavity mode energy is never strictly $1.5$ eV as we vary mirror
thicknesses. Here we choose to keep a constant distance between two mirrors
for consistency. However, when the quality factor becomes very large
($L=100$nm), the external EM field cannot enter the cavity efficiently such
that the effective amplitude of the pumping field inside the cavity near the
molecular layers is reduced (and the Rabi frequency decreases). Note that the
Rabi oscillations do not last forever. For example, for mirror thickness
$L=50$nm, the Rabi oscillations stop after $200$ ps. In other words, as the
mirrors thickness $L$ increases, the Rabi oscillations last longer until
$\rho_{22}$ decays to $\rho_{11}$ as induced by the cavity leakage (discussed
in the following).

  * •

In Fig 4(c), we plot the population of the first excited state of the solute
molecular layers, $\rho_{11}(t)$. For low-quality factors ($L<40$ nm), even
though $E_{01}\neq E_{in}$, $\rho_{11}$ is not zero for the same reason as in
Fig. 2 (h) (corresponding to scenario (d)), we also observe that the
population $\rho_{11}$ becomes non-zero only after at least one cycle of the
Rabi oscillation of $\rho_{22}$ in (b). This observation suggests that the
population $\rho_{11}$ arises from the transition from $\left|2\right\rangle$
to $\left|1\right\rangle$. As the thickness of the mirrors becomes larger
($40<L<60$ nm), it takes more Rabi oscillation cycles of $\rho_{22}$ to make
such a transition to populate $\rho_{11}$ and, after the transition, the
maximal population of $\rho_{22}$ drop while maintaining the Rabi oscillation.
When $L>60$ nm, we find that $\rho_{11}(t\leq 100\text{ps})=0$.

The bottom line is that these plots show that increasing the quality factors
(i.e. making the mirror thicker) can lead to non-monotonic behavior: there is
a trade-off insofar as collective response being enhanced when a photon
bounces back and forth in a cavity versus collective behavior being reduced
when a photon cannot enter the cavity to begin with.

###  III.3 Dynamics with phenomenological pure dephasing

Figure 5: Population dynamics (a) $\rho_{1^{\prime}1^{\prime}}(t)$, (b)
$\rho_{11}(t)$ and (c) $\rho_{22}(t)$ when we dephase the solvent molecular
layers for scenario (d) ($\tau_{d}=1-10$ ps). For slow dephasing, the Rabi
splitting remains unperturbed. When the dephasing rate is increased
(corresponding to a smaller dephasing timescale in the lower part of the
figures), the excited state population $\rho_{1^{\prime}1^{\prime}}$ of the
solvent molecules is trapped, leading to ground state depletion and Rabi
splitting contraction. Hence, given the identical incoming pumping frequency
$\hbar\omega_{in}=1.5$ eV, the external pumping cannot enter the cavity
efficiently and $\rho_{11}$ is not populated within the first 100 ps. Figure
6: Population dynamics (a) $\rho_{1^{\prime}1^{\prime}}(t)$, (b)
$\rho_{11}(t)$ and (c) $\rho_{22}(t)$ when we dephase the 3-level solute
molecular layer for scenario (d) ($\tau_{d}=1-10$ ps). Difference arises
versus Fig 5 because now the dephasing is on the solute molecules. We find
that the Rabi oscillation pattern in (b) is damped in the strong dephasing
limit and the system reaches steady state at $\rho_{22}=0.5$. The time
dynamics of the solvent molecules remains mostly the same as for scenario (d)
in Fig 2 (f), except for the fact that the persistent small oscillation is
damped in the strong dephasing limit. As in Fig. 5(c), we do not see any
substantial population in $\rho_{11}$.

In practice, the 2-photon Rabi oscillation pattern does not last forever as
shown in 4. The Rabi oscillations can be damped by other non-radiative
processes and dephasing. In this subsection, we investigate how pure dephasing
as induced by the environment changes the time dynamics of scenario (d). The
pure dephasing is implemented by manually damping the coherence $\rho_{01}$,
$\rho_{02}$, $\rho_{0^{\prime}1^{\prime}}$ and $\rho_{12}$ by a factor of
$\exp(-dt/\tau_{d})$ for each timestep $dt$. For a typical experiment, the
pure dephasing timescale ($\tau_{d}$) is between $1$ ps to $100$ ps. To best
isolate the effects of pure dephasing, we perform two parallel sets of
calculations with dephasing either (i) exclusively on the solvent molecular
layers, as shown in Fig 5, or (ii) exclusively on the solute molecular layer,
as shown in Fig 6. In practice, pure dephasing should apply to both molecular
layers with similar rates, but the following gedanken experiments will make
clear different effects.

First, for pure dephasing on the solvent molecular layers, the direct effect
is that the radiative decay from $\left|1^{\prime}\right\rangle$ to the ground
state $\left|0^{\prime}\right\rangle$ is suppressed, which leads to ground
state depletion and Rabi splitting contraction. Hence, the resonant frequency
of lower polariton is no longer the same as the incoming driving
($E_{LP}>E_{in}$). As a result, the maximal electric field amplitude decreases
inside the cavity and the Rabi oscillations of the solute molecular layer
between $\left|0\right\rangle$ and $\left|2\right\rangle$ are slowed down, as
shown in Fig 5 (b).

Second, in Fig 6 we investigate dynamics with pure dephasing on the solute
layer. As shown in (a), as the dephasing rate increases, the small
oscillations of $\rho_{1^{\prime}1^{\prime}}$ (see Fig 2(f)) are damped, yet
the population remains around the same steady state value. Similarly, as far
as $\rho_{22}$ is concerned, the Rabi oscillation pattern is damped (but not
slowed down). Dephasing the solute is not as interesting as dephasing the
solvent.

##  IV Conclusion and Outlook

In conclusion, we have performed finite difference time domain (FDTD)
calculations within a mean field approximation to simulate 2-photon nonlinear
transitions between a continuous wave light and 3-level solute molecular
layers. The population dynamics of the molecular layers are obtained outside
and inside a one-dimensional cavity made of two gold mirrors. We find that the
presence of a cavity changes the dynamics in two aspects: (i) the cavity
amplifies the EM field such that the transitions occur faster when the quality
factor is moderately large; and (ii) the cavity inhibits transitions that are
not on resonance, which effectively protects the coherent Rabi oscillation
dynamics from relaxation. However, if we further increase the quality factor
of the cavity, the external electromagnetic field cannot enter the cavity
efficiently, leading to slower Rabi oscillations. We also investigate the role
of dephasing on the solute and the solvent layers and, in short, both types
demolish the two-photon transition.

Looking forward, one can envision several improvements to the current
calculations. First, it is well-known that under a mean field assumption
(where the EM field is classical and modulates all interactions between
molecules), some elements of static intermolecular coupling are not included.
A proper configuration interaction Hamiltonian that incorporates
intermolecular coupling may well reveal new physics – but this approach would
certainly be more expensive. Second, the current calculations lack any (even
phenomenological) nonradiative decay between excited states and ground states.
In the future, one might include such an effect by introducing a Lindblad term
into the equation of motion; such a term would certainly damp the Rabi
oscillations. Third and finally, we note that the calculations presented here
are only one-dimensional: the cavity has multiple modes but only along one
polarization. Practically speaking, our setup corresponds to concentric
cavities, i.e. there is effectively a non-zero curvature on the mirrors. In
the future, one would like to perform two/three-dimensional FDTD calculations
to include more polarization directions.

Beyond improving the present calculations, ideally the next step of this
research is to go beyond two-level systems and model realistic systems with
realistic interactions between EM fields molecular layers inside a multi-mode
cavity. For example, in the future, one would like to generalize the present
set of simulations to study the behavior of matter in a cavity under very
strong photonic excitation where, for example, BEC phases appear[48, 49, 50].
The intersection of electronic dynamics and strong coupling to light fields
form a very rich area for future exploration.

## Acknowledgements

This work has been supported by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, under Award No. DE-SC0019397
(J.E.S.); the U.S. National Science Foundation under Grant No. CHE1953701
(A.N.). MS acknowledges support by the Air Force Office of Scientific Research
under Grant No. FA9550-22-1-0175.

##  Appendix A Parameters for FDTD simulation

In this appendix, we list all parameters in our simulation.

Table 1: Parameters for Maxwell-Bloch simulation. Name | Value$\dagger$
---|---
Grid resolution ($dx$) |  $0.1$ nm
Time step (dt) | $dx/2/c_{0}$
Thickness of solute |  $1$ nm
Thickness of solvent |  $4.5$ nm $\times 2$
Thickness of gold |  $50$ nm $\times 2$
Distance between gold mirrors in Fig 1(c) |  $354$ nm
Distance between gold mirrors in Fig 1(d) |  $340$ nm
Drude model for gold |  $\omega_{d}=7.039$eV
|  $\Gamma_{d}=0.1809$ eV
Transition dipole moment |  $\mu_{x}^{01}=\mu_{x}^{12}=\mu_{x}^{0^{\prime}1^{\prime}}=10$ Debye

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# Element resolved evidence of superdiffusive spin current arising from
ultrafast demagnetization process

R. Gupta<EMAIL_ADDRESS>Department of Materials Science and
Engineering, Uppsala University, Box 35, SE-751 03, Uppsala, Sweden
Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20,
Uppsala, Sweden  F. Cosco  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20, Uppsala, Sweden  R. S. Malik  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden
X. Chen  Department of Physics and Astronomy, Uppsala University, Box 516,
SE-751 20, Uppsala, Sweden  Department of Physics and Shenzhen Institute for
Quantum Science & Engineering, Southern University of Science and Technology,
Shenzhen, Guangdong 518055, China  S. Saha  Department of Physics and
Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden  Department
of Physics, Ashoka University, 131029 Hariyana, India  A. Ghosh  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden
MAX IV Laboratory, Lund University, SE-22 100, Lund, Sweden  T. Pohlmann
Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg,
Germany  J. R. L. Mardegan  Deutsches Elektronen-Synchrotron (DESY),
Notkestrasse 85, D-22607 Hamburg, Germany  S. Francoual  Deutsches Elektronen-
Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany  R. Stefanuik
Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20,
Uppsala, Sweden  J. Söderström  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20, Uppsala, Sweden  B. Sanyal  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden
O. Karis  Department of Physics and Astronomy, Uppsala University, Box 516,
SE-751 20, Uppsala, Sweden  P. Svedlindh  Department of Materials Science and
Engineering, Uppsala University, Box 35, SE-751 03, Uppsala, Sweden  P. M.
Oppeneer  Department of Physics and Astronomy, Uppsala University, Box 516,
SE-751 20, Uppsala, Sweden  R. Knut<EMAIL_ADDRESS>Department of
Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden

###### Abstract

Using element-specific measurements of the ultrafast demagnetization of
Ru/Fe65Co35 hetero-structures, we show that Ru can exhibit a significant
magnetic contrast (3% asymmetry) resulting from ultrafast spin currents
emanating from the demagnetization process of the FeCo layer. We use this
magnetic contrast to investigate how superdiffusive spin currents are affected
by the doping of heavy elements in the FeCo layer. We find that the spin
currents are strongly suppressed, and that the recovery process in Ru slows
down, by Re doping. This is in accordance with a change in interface
reflectivity of spin currents as found by the superdiffusive spin transport
model.

††preprint: APS/123-QED

In 1996 it was found by Beaurepaire et al. Beaurepaire  _et al._ (1996) that a
ferromagnetic Ni film can be demagnetized on a subpicosecond time scale by
femtosecond (fs) laser pulse excitations. The possibility to optically
manipulate the magnetization on such fast time scales paves the way for
terahertz (THz) operation of spintronic devices Zhang  _et al._ (2020).
Moreover, ultrafast demagnetization of Ni results in THz emission being
proportional to the second time-derivative of magnetization Beaurepaire  _et
al._ (2004). Besides technological relevance, research in this area has led to
discoveries of novel phenomena like all-optical magnetization switching
Stanciu  _et al._ (2007); Gorchon  _et al._ (2017), transfer of THz spin
currents Rudolf  _et al._ (2012); Kampfrath  _et al._ (2013); Schellekens  _et
al._ (2014) and the optical inter-site spin transfer effect Dewhurst  _et al._
(2018); Willems  _et al._ (2020); Tengdin  _et al._ (2020). For the last two
decades, ultrafast magnetization dynamics of magnetic materials has been
probed with fs laser pulses using time-resolved techniques such as high
harmonic generation (HHG) Mathias  _et al._ (2012); Turgut  _et al._ (2013),
magnetic circular dichroism Stamm  _et al._ (2007), and spin-polarized two-
photon photoemission Schmidt  _et al._ (2010). Several microscopic mechanisms
have been proposed to explain the ultrafast dynamics including electron-photon
mediated spin-flip scattering Koopmans  _et al._ (2005, 2010); Steiauf and
Fähnle (2009), electron-electron scattering Krauß  _et al._ (2009), electron-
magnon scattering Carpene  _et al._ (2008); Jana  _et al._ (2020a), and direct
transfer of angular momentum from photon to electron mediated by spin-orbit
coupling Dalla Longa  _et al._ (2007); Zhang and Hübner (2000). Malinowski et
al. Malinowski  _et al._ (2008) first showed that laser-induced spin transport
can enhance the demagnetization in metallic heterostructures. Superdiffusive
spin transport Battiato  _et al._ (2010) was proposed to play an important
role in the laser-induced demagnetization process of metallic magnetic films
Rudolf  _et al._ (2012); Eschenlohr  _et al._ (2013); Bergeard  _et al._
(2016); Hofherr  _et al._ (2017). However, other mechanisms to generate fast
spin currents such as spin pumping were also proposed Choi  _et al._ (2015);
Remy  _et al._ (2020); Beens  _et al._ (2020) and the origin of optically
induced spin currents is still being debated Lichtenberg  _et al._ (2022).
Spin-current injection through spin pumping Choi  _et al._ (2015); Beens  _et
al._ (2020) should be favored by a stronger damping, but, although the latter
can be engineered e.g. by doping of the ferromagnetic layer, such
investigations are scarce.

Kampfrath et al. investigated the THz emission caused by the ultrafast laser-
induced demagnetization of an Fe layer adjacent to nonmagnetic Ru and Au
capping layers Kampfrath  _et al._ (2013). The measured temporal behavior of
the THz spin currents was very different between the Ru and Au layers, which
was explained by the different absorption of superdiffusive spin currents in
the Ru and Au layers. In the case of Au, there are no empty $d$-states and the
hot electrons injected into the $sp$ states with high electron-band velocity
will therefore be quickly reflected back into the magnetic layer. In the case
of Ru, the $d$-band states with lower band velocities will transiently trap
the injected spins and thus delay the spin current. However, as only the THz
emission was measured, there was no de facto evidence of a superdiffusive spin
current in the nonmagnetic Ru layer.

In this work, we investigate the relation between ultrafast demagnetization
and spin current generation as well as effects of doping and choice of capping
layer by using the element-specific transverse magneto-optic Kerr effect
(T-MOKE) and superdiffusive spin current simulations. Ru/Fe65Co35/Ru and
Cu/Fe65Co35/Cu heterostructures are utilized as model systems, where rhenium
(Re) is used as dopant in Fe65Co35 to modify the spin dynamics (cf. Fig. S1 in
Supplemental Materials (SM) SM ). The ultrafast spin dynamics is shown to be
strongly dependent on the Re doping concentration. A transient spin
accumulation, that is significantly reduced with increasing Re doping, is
evident in the Ru layer. This decreased spin injection with Re doping can
partly be assigned to a decreasing sample magnetization, and potentially a
decreased spin transport due to scattering at Re sites. However, our
simulations show that an increased electron reflectivity at the Ru interface,
due to Re doping, results in a good qualitative correspondence to the temporal
behaviour of the Ru spin accumulation found experimentally.

The Re doped (0, 3, 6, and 12.6 at.%) Fe65Co35 thin films were grown on
SiO2/Si substrate at room temperature using DC sputtering. Layers of Ru(3 nm)
and Cu(5 nm) were used as seed and capping layers. The nominal thickness of
Fe65Co35 was 20 nm, which was verified by x-ray reflectometry. All
characterization details can be found in section I of the SM SM  and in Refs.
Akansel  _et al._ (2018); Gupta  _et al._ (2020); Akansel  _et al._ (2019).

The element-specific ultrafast magnetization dynamics of Fe65Co35 interfaced
with Ru and Cu films as function of Re doping was studied using a table-top
HHG setup Mathias  _et al._ (2012); Plogmaker  _et al._ (2015); Jana  _et al._
(2017); Stefanuik  _et al._ (2018). The table-top HHG setup uses the T-MOKE to
measure the magnetic asymmetry of the signal (cf. section I, SM SM ). In the
T-MOKE spectrometer, the laser is a chirped pulse amplification system running
at a central wavelength of 800 nm with a maximum energy of 2 mJ per pulse and
a pulse length of 35 fs. The sample is magnetized, using a magnetic field
strength of $\pm$ 80 mT, perpendicularly to the plane of incidence of the
incoming $p$-polarized light Jana  _et al._ (2017). The reflected light from
the sample is recorded with a CCD camera. In order to obtain a magnetic
contrast, the change in reflected intensity at the absorption edges of the
different elements in the heterostructure was recorded for opposite directions
of the magnetization. The resulting photon-energy dependent magnetic asymmetry
$A(E)$ is the normalized difference of the reflected intensities for the two
magnetization directions Mathias  _et al._ (2012); Jana  _et al._ (2020b) and
is defined as follows,

$A(E)=\frac{I_{\mathrm{+}}(E)-I_{\mathrm{-}}(E)}{I_{\mathrm{+}}(E)+I_{\mathrm{-}}(E)},$
(1)

where $I_{+}(E)$ and $I_{-}(E)$ are the reflected intensities measured for the
two different magnetization directions. This asymmetry is proportional to the
magnetization of the sample as discussed in previous studies Mathias  _et al._
(2012, 2013); La-O-Vorakiat  _et al._ (2012); Jana  _et al._ (2020b).

Figure 1:  Static T-MOKE asymmetries of Ru/Fe65Co35/Ru heterostructures with
different Re doping (left axis). The different shaded regions represent the
absorption edges of Ru (purple), Fe (red), and Co (blue). The black solid line
corresponds to the calculated asymmetry for the undoped sample. The right axis
shows the asymmetry of an Cu/Fe65Co35/Cu heterostructure, both experimentally
(red squares) and calculated (red dashed line).

Figure 1 shows the measured static Kerr asymmetry of Ru/Fe65Co35/Ru for
different Re concentrations. The asymmetry is $\sim$10% and $\sim$4% at the
M2,3-absorption edge of Fe ($\sim$54 eV) and Co ($\sim$60 eV), respectively.
The asymmetry is reduced with increasing Re doping following the same trend as
the saturation magnetization, which has already been reported Akansel  _et
al._ (2019); Gupta  _et al._ (2020). An additional large asymmetry is observed
around 44 eV (purple region), which corresponds to the Ru N2,3-absorption
edge. The magnitude of the Ru asymmetry is comparable to the Fe asymmetry at
54 eV. To verify that the increase in asymmetry at 44 eV is related to Ru, a
Cu/Fe65Co35/Cu heterostructure was studied for which no significant increase
in magnetic asymmetry near to 44 eV was observed (cf. lower part of Fig. 1).
Using the formalism given by Zak et al. Zak  _et al._ (1991), we have
calculated the magnetic asymmetry for the undoped heterostructures and present
them as black solid and red dashed lines in Fig. 1, for the Ru and Cu capped
samples, respectively. The magneto-optical (MO) parameters for Fe and Co were
obtained from Willems et al. Willems  _et al._ (2020) and Valencia et al.
Valencia  _et al._ (2006). MO parameters are only available above 45 eV and
was hence interpolated down to 44 eV (cf. Fig. S2, section I in SM SM ). Both
Ru and Cu are non-magnetic in these simulations. From these simulations it is
clear that the weak MO component from Fe around 44 eV will result in a large
asymmetry at the Ru N2,3 absorption edge, while for the Cu capped sample the
asymmetry at 44 eV remains low. Characterization of the Ru interface utilizing
element-specific synchrotron based x-ray techniques and first-principles
calculations can be found in sections II and III of the SM SM .

Figure 2:  The time-dependent magnetic asymmetry measured at the Ru edge shown
with blue circles for the 6% Re doped sample. The green squares correspond to
the Ru asymmetry after subtracting the Fe contribution (red solid line). The
asymmetry at 44 eV for the Cu capped sample, after subtracting the Fe
contribution is shown with red triangles. Inset shows the zoom view of Ru
asymmetry.

Figure 2 shows time- and element-resolved measurements for the Ru capped 6% Re
doped sample. The blue circles shows the Ru asymmetry during the
demagnetization. A large increase of the asymmetry ($\sim$ 21%, as indicated
in Figure 2) is observed after 100 fs, which has decreased to below its
initial value after 500 fs. The initial rise of the asymmetry is attributed to
spin currents that spin polarize Ru and the subsequent decreased asymmetry is
related to the decreased asymmetry of Fe, which has a notable contribution at
the Ru edge. The Fe contribution, represented by the fitted red solid line,
can easily be subtracted from the Ru asymmetry at 44 eV. Note that the Fe
asymmetry, measured at 54 eV, has been normalized to the Ru value before time
zero. The remaining asymmetry then corresponds to the pure spin polarization
of Ru, shown as green squares. It is clear that the peak of the Ru spin
polarization occurs before the maximum demagnetization of Fe, and that this
spin polarization is short lived ($\lesssim$ 500) fs. For comparison, we also
show the asymmetry for the Cu capped sample at 44 eV after subtracting the Fe
contribution (red triangles). It is clear that in this case there is no sign
of any spin accumulation and that the asymmetry follows the Fe asymmetry (cf.
Fig. S5, section I in SM SM ).

Figure 3: (a) The time resolved reflectivity at the Ru edge (44 eV) for both
magnetization directions ($M+$ and $M-$). At time zero there is a sharp
increase in the reflectivity that is strongly asymmetric between $M+$ and $M-$
(cf. arrow 1). After about 1 ps there is still a significant difference in
reflectivity due to the demagnetization of the FeCo layer (cf. arrow 2). After
about 4 ps there has been a slow increase in the reflectivity for both $M+$
and $M-$ (cf. arrow 3), suggesting that the index of refraction has changed.
(b) Calculated dependence of the reflectivity (left axis, blue and red lines)
and asymmetry (right axis, black line) on the index of refraction (IOR) of Ru.
The transparent blue box indicates the experimentally found range of change in
the reflectivity obtained from (a).

It is possible that index of refraction of Ru at the absorption edge will be
sensitive to absorption of the pump light and potentially create artefacts in
the asymmetry signal Koopmans  _et al._ (2000); Carva  _et al._ (2009). To
investigate this, we show the reflectivity at the Ru edge (44 eV) for the two
different magnetization directions in Fig. 3(a). The data has been normalized
to the value of $M+$ (blue circles) before time zero, after which the $M-$
data (red squares) has been shifted up by $0.13$ to facilitate the comparison
between $M-$ and $M+$ data. In Fig. 3(b) we show the calculated reflectivity
for the two magnetization directions (blue and red lines, left scale) and the
corresponding asymmetry (black line, right scale) as a function of the index
of refraction (IOR) of Ru. The horizontal scale corresponds to a
multiplicative factor in the IOR and $1.00$ hence corresponds to the unpumped
case. The blue shaded box indicates the range that results in the same
reflectivity changes as observed experimentally in Fig. 3(a). Since changing
the IOR has a similar effect on both $M+$ and $M-$, we find that the change in
the observed asymmetry due changes in the IOR of Ru is fairly small. The
asymmetry goes from 5.22% to 5.11% as one moves from 1.00 to 1.005 on the
multiplication factor of IOR, i.e. left and right edges of the blue shaded
box, and it should be noted that the asymmetry decreases rather than increases
when the reflectivity increases. This would suggest that the spin current
asymmetry in Fig. 2 (green squares) is slightly underestimated, instead of
peaking at about 2.5% it should peak at 2.6% if compensated for changes in the
IOR of Ru. In Fig. 3(a) we observe an increase of both $M+$ and $M-$
intensities at time zero, suggesting a change in the IOR, however, there is a
large difference in the amplitude of this increase (indicated by the vertical
arrow denoted 1) which highlights the magnetic origin of the asymmetry change.
The difference between $M+$ and $M-$ after 1 ps (vertical arrow denoted 2), is
mainly due to the demagnetization of the FeCo layer. There is a slow increase
in the reflectivity which saturates after 4 ps (vertical arrow denoted 3).
This would suggest that the initial increase in reflectivity, at time zero, is
due to a non-thermalized electron distribution. As the electron distribution
thermalizes the reflectivity decreases, and finally the heat is transfered
from the electron system to the phonon system, again resulting in an increased
reflectivity.

Figure 4: (a) Ultrafast magnetization dynamics of Ru (after subtracting the Fe
contribution) in Fe65Co35 films with different Re doping. An increase in the
asymmetry signal is observed after time zero. The time-dependent change in
asymmetry is strongest for the undoped sample and decreases with increasing Re
doping. The vertical black dashed line provides a guide for the position of
the Re 6% sample peak position at 130 fs. (b) The extracted mean time of the
experimentally observed asymmetry versus Re doping. The mean time of the
asymmetry provides information about temporal shifts in the amplitude.

The ultrafast dynamics at the Ru absorption edge for different Re
concentrations are shown in Fig. 4(a). In all measurements, the pump fluence
was adjusted so that all samples showed a similar amount of demagnetization
($\sim$20%) at the Fe edge. All curves have been smoothed by taking an average
of each point by its nearest neighbouring points, cf. Fig. S6 in SM for non-
smoothed data SM . For the undoped sample, the peak asymmetry at the Ru edge
is about $\sim$3% (black curve). For the samples with 3 at% and 6 at% Re
doping, the peak asymmetry signal decreases to $\sim$2%. Notably, Re doping
attenuates the time dependent Ru asymmetry significantly, and for the 12.6
at.% Re doped sample the asymmetry is close to the detection limit. A
reasonable explanation for this effect is that Re doping results in enhanced
spin scattering, which effectively blocks the spin current from reaching the
Ru layer as illustrated in Fig. S1(c) of the SM SM . Also, the saturation
magnetization decreases by 25% from undoped to 12.6 % Re doped samples Gupta
_et al._ (2020), which certainly should have a detrimental effect on the spin
current. In Fig. 4(b) we show the mean time of the asymmetry amplitude (i.e.
mean time $=\sum_{t=0}^{t=1.5ps}A(t)\cdot t/\sum_{t=0}^{t=1.5ps}A(t)$, where
$A(t)$ is the time dependent asymmetry). Since the asymmetry at shorter time
scales $\sim$ 130 fs appears to be affected more by the Re doping than the
asymmetry after 0.5 ps, the mean time shifts to higher values with increasing
Re doping.

Figure 5:  (a) Computed time-dependent magnetization in Ru for diﬀerent
reﬂection coeﬃcients at the Ru/FeCo interface. (b) Calculated spin current in
Ru for different reﬂection coeﬃcients at the Ru/FeCo interface. (c) The
extracted mean time of the simulated asymmetry versus reflection coefficient.
The mean time of the asymmetry amplitude moves to higher values with an
increasing reflection coefficient.

In Figure 5 we show the calculated superdiffusive spin current in the Ru layer
using the superdiffusive spin transport model. In this model the laser-induced
demagnetization of FeCo is the source of the spin current Battiato  _et al._
(2010). The details of the model can be found in section IV, SM SM . Figure
5(a) displays the computed laser-induced transient magnetization of the Ru
layer as a function of the spin current reflectivity at the FeCo/Ru interface,
assuming an Fe0.5Co0.5 composition and using ab initio computed hot electron
velocities and spin lifetimes from Ref. Nechaev and Chulkov (2009). The
magnetization of the Ru layer consists of two contributions to properly treat
the interface, $\Delta M_{\mathrm{Ru}}(3\,\mathrm{nm})$ and a small fraction
of the magnetization of the ferromagnetic layer, $\Delta
M_{\mathrm{FeCo}}(0.2\,\mathrm{nm})$. This is done because in the
superdiffusive model material properties change abruptly from one layer to the
other without taking into account hybridization and other interface effects.
We notice that the Ru layer reaches its maximum magnetization at $t\sim 60$
fs, where $t=0$ is taken at the peak of the gaussian laser pulse. After this
fast magnetization, the layer is demagnetized within 0.6 ps. Note that the
magnetization becomes slightly negative due to the demagnetization of the
$\Delta M_{\mathrm{FeCo}}(0.2\,\mathrm{nm})$ layer that was included at the
interface. In Figure 5(b) we show the calculated spin current in Ru. Around
the peak Ru magnetization the spin current changes its direction and flows
back into the FeCo layer. Except for the trivial reduction in spin current
amplitude when increasing the reflection coefficient, since less spins will
transmit through the interface, we also see a significant reduction in the
spin current that flows back into the FeCo layer. Effectively the increased
reflection coefficient traps the spins more effectively within the Ru layer,
and the reduction in Ru magnetization will hence be increasingly more driven
by spin-flip processes rather than by the back flow. In Figure 5(c) we have
calculated the mean time of the magnetization in the range between 0 and 0.6
ps, similarly as the mean time was calculated for the asymmetry in Fig. 4(b).
We find that an increased reflectivity at the interface results in a similar
behaviour as we find experimentally when increasing the Re doping.

The present measurements are consistent with superdiffusive spin-current
injection into Ru, but other models based on spin injection through magnon
spin pumping have also been proposed Choi  _et al._ (2015); Beens  _et al._
(2020); Lichtenberg  _et al._ (2022). In the former, the spin current is
generated through the difference in transport efficiency between majority and
minority spins. In the latter, mainly majority electrons are generated during
demagnetization via electron-magnon interactions, which creates a spin
imbalance and a resulting spin-current. Interestingly, the spin-mixing
conductance Tserkovnyak  _et al._ (2002) of the FeCo/Ru interface has been
found to increase with Re doping Gupta  _et al._ (2020), suggesting that the
strong electron energy dependence of the reflectivity Lu  _et al._ (2020) can
result in an enhanced spin-mixing conductance (low energy electrons) at the
same time as the superdiffusive spin current (high energy electrons) decreases
due to higher reflectivity. The Gilbert damping parameter ($\alpha_{G}$) is
known to increase when transition-metal films are doped with 4d, 5d or rare-
earth elements Rantschler  _et al._ (2007); Woltersdorf  _et al._ (2009);
Rebei and Hohlfeld (2006); Akansel  _et al._ (2018). For our samples,
$\alpha_{G}$ increases more than four-fold for 12.6% Re doping in FeCo Gupta
_et al._ (2020). A magnon dominated demagnetization process would likely be
sensitive to the Gilbert damping Koopmans  _et al._ (2005), however, we find
no change of the demagnetization time for the different samples (cf. Fig.
S4(a), SM SM ). Furthermore, in models based on spin pumping through magnon
damping Choi  _et al._ (2015); Beens  _et al._ (2020); Lichtenberg  _et al._
(2022) more Gilbert damping would imply an increased loss of spin angular
momentum and a spin voltage imbalance building up at the FeCo/Ru interface,
leading to a larger spin current pumped into Ru. However, this behavior is not
seen in our measurements.

We expect that other non-magnetic (NM) transition metals should exhibit a
similar contrast at their respective M or L-edges, if they are used as capping
layers. For buried NM layers it is unlikely that the signal can be separated
from the much stronger signal resulting from the ferromagnetic layer.
Previously, observation of laser-excited spin currents were made with visible
MOKE Hofherr  _et al._ (2017); Alekhin  _et al._ (2019) and magnetic second-
harmonic generation Melnikov  _et al._ (2011). Recently, injection of
nonequilibrium electrons from Fe into Au were measured with time-resolved
photoemission Kühne  _et al._ (2022). However, direct element-selective
measurements, as presented here, of spin injection in NM metals due to
superdiffusive spin currents have not been reported before. These results are
of particular use for studying devices that emit THz radiation generated from
spin currents Kampfrath  _et al._ (2013), a phenomenon that has received
attention recently Nenno  _et al._ (2019); Dang  _et al._ (2020); Gupta  _et
al._ (2021a); Papaioannou and Beigang (2021); Bull  _et al._ (2021); Gupta
_et al._ (2021b); Mottamchetty  _et al._ (2023). Figure S10 of the SM depicts
the Fourier transform of the superdiffussive THz spin current in the Ru layer;
the bandwidth is found to be $\sim 18$ THz for the undoped sample, which
decreases with increasing Re doping.

This work is supported by the Swedish Research Council (VR, contracts
2017-03799, 2021-04658 and 2021-5395), the Olle Engkvists Stiftelse (Grant No.
182-0365), Carl Tryggers Stiftelse (Grant No. 17:241), and by the Swedish
National Infrastructure for Computing (SNIC), through grant agreement No.
2018-05973. This work is furthermore funded by the European Union’s
Horizon2020 Research and Innovation Programme under FET-OPEN Grant agreement
No. 863155 (s-Nebula). P.M.O. acknowledges support by the Deutsche
Forschungsgemeinschaft through CRC/TRR 227 “Ultrafast Spin Dynamics”, project
MF. R.G. acknowledges DESY (Hamburg, Germany) for the provision of
experimental facilities. Parts of this research were carried out at beamline
P09 at PETRA III under proposal I-20200894 EC. We are grateful to M.Sc.
Dominik Graulich and Dr. Timo Kuschel (Bielefeld University) for developing
within the long-term project LTP I-20170016 the vacuum chamber for the vector
electromagnet used in this work and Prof. Günter Reiss (Bielefeld University)
for contributing within LTP II-20190009 to the in-vacuum silicon drift
detector used in this work.

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