# The SUPER excitation of quantum emitters is a multi-photon process

Luca Vannucci* and Niels Gregersen  DTU Electro, Department of Electrical and
Photonics Engineering, 2800 Kongens Lyngby, Denmark<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

The swing-up of quantum emitter population (SUPER) scheme allows to populate
the excited state of a quantum emitter with near-unity fidelity using two red-
detuned laser pulses. Its off-resonant, yet fully coherent nature has
attracted significant interest in quantum photonics as a valuable tool for
preparing single-photon sources in their excited state on demand, while
simultaneously ensuring straightforward spectral filtering of the laser.
However, the physical understanding of this mechanism in terms of energy
exchange between the electromagnetic field and the emitter is still lacking.
Here, we present a fully quantized model of the swing-up excitation and
demonstrate that it is in fact a multi-photon process, where one of the modes
loses two or more photons while the other gains at least one. Our findings
provide an unexpected physical interpretation of the SUPER scheme and unveil a
new non-linear interaction between single emitters and multiple field modes.

##  1 Introduction

Efficient sources of indistinguishable single photons are essential components
of optical quantum computers and simulators [1, 2]. Whereas spontaneous
parameteric downconversion allows for straightforward generation of highly
indistinguishable heralded photons, its probabilistic nature leads to low
single-photon purity and collection efficiency, which can only be improved
with complex multiplexing schemes [3, 4]. On the other hand, quantum emitters
based on semiconductor quantum dots yield nowadays highly indistinguishable
single photons with high purity and collection efficiency of the order of
$\approx 0.6$ [5, 6, 2]. Activating such sources on demand calls for a fast
and coherent optical excitation protocol that prepares the emitter in the
desired excited state.

In the so-called swing-up of quantum emitter population (SUPER) scheme [7],
two red-detuned laser pulses interacting simultaneously with the emitter cause
fast oscillations of the excited state population. For a proper choice of the
laser parameters (frequency, amplitude, and duration of each pulse), the
emitter is finally left in the excited state with fidelity close to 100%. The
technological advantage of this technique, as compared for example with
resonant fluorescence, is that laser rejection is straightforwardly achieved
via spectral filtering due to the red-detuned nature of the pump [8, 9].
Moreover, it populates the excited state directly with a fast and coherent
process, as opposed for example to pumping into higher energy levels (which
requires incoherent internal relaxation to create the exciton). This
attractive scenario had stimulated intense theoretical and experimental
research, including the prospect of applying this technique to cavity-coupled
quantum dots [10, 11], the analysis of the important role played by phonon
scattering [12, 13, 14], and the possibility of achieving a similar effect
with hybrid acousto-optical methods [15]. While the SUPER scheme was
originally devised for semiconductor quantum dots, it relies in fact on
general properties of quantum emitters coupled to the electromagnetic field
and has now been investigated in different material platforms, such as tin
vacancies in diamond [16] and emitters in the layered transition metal
dichalcogenide WSe2 [13].

In spite of this active interest, the physical origin of this effect is not
yet fully understood. Bracht et al. [17] have explained the swing-up effect
with a dressed state picture of the emitter-field system. While one of the
laser pulses is dressing the bare emitter states and effectively modify their
energy splitting, the other one drives transition between the dressed states.
However, a fundamental interpretation in terms of energy exchange between the
emitter and the electromagnetic field is still missing. For example, whereas
resonance fluorescence is easily interpreted in terms of absorption of one
single photon from the laser pump, such a simple picture is currently
unavailable for the SUPER scheme. This is further complicated by the off-
resonant nature of this protocol and by the presence of two simultaneous laser
fields, which makes the physical interpretation less intuitive.

In this work, we surprisingly show that the SUPER scheme excitation involves
multiple photon exchanges between the emitter and the laser. We achieve this
by developing a quantum-mechanical model for an emitter coupled to two
quantized field modes, which allows to count the number of photons in each
mode before and after interaction with the emitter. Specifically, we find that
two photons are typically absorbed from one of the modes during the swing-up
process. Of these two excitations, one is stored in the emitter by raising its
population from the ground to the excited state, while the remaining one is
absorbed into the second field mode. Interestingly, we find that population
inversion may still occur even when one of the field modes is in the vacuum
state, provided that the other one is populated with at least two photons.

This article is organized as follows. First, we describe the model and the
methodology in Section 2. Adopting a Fock state assumption for the
electromagnetic field, we analyze the dynamics of the SUPER and the red-and-
blue dichromatic scheme [18, 12] in the large photon number regime in Sections
3 and 4, respectively. We then focus on the few-photon regime in Section 5,
where we consider the case where one of the field modes is in the vacuum
state. In Section 6, we describe the field in terms of coherent states, and
compare the results with the ones obtained under a Fock state assumption. We
finally discuss our findings in Section 7, before drawing our conclusions in
Section 8.

##  2 Model

We model the quantum emitter as a two-level system with ground state $\ket{G}$
and excited state $\ket{X}$. Its Hamiltonian is
$H_{0}=\hbar\omega_{0}\sigma^{\dagger}\sigma$, with
$\sigma^{\dagger}=\outerproduct{X}{G}$ the raising operator. The emitter is
coupled to two quantized electromagnetic field modes with Hamiltonian
$H_{j}=\hbar\omega_{j}a^{\dagger}_{j}a_{j}$, $j\in\quantity{1,2}$, where
$a_{j}$ is the annihilation operator for mode $j$. Using the rotating wave
approximation, the emitter-field coupling is modeled as a two-mode Jaynes-
Cummings Hamiltonian,

$H_{\rm int}(t)=\sum_{j=1,2}\hbar
g_{j}\exp\quantity(-\frac{t^{2}}{t_{p}^{2}})\quantity(\sigma^{\dagger}a_{j}+\sigma
a_{j}^{\dagger}).$ (1)

The coupling constants $g_{j}$ in Eq. (1) are modulated by a time-dependent
Gaussian envelope $\exp\quantity(-\frac{t^{2}}{t_{p}^{2}})$ to simulate pulsed
excitation.

In the single-mode Jaynes-Cummings model, the excitation number
$\mathcal{N}=\sigma^{\dagger}\sigma+a^{\dagger}a$ is a conserved quantity. It
follows that the subspace spanned by the pair
$\quantity{\ket{X,n},\ket{G,n+1}}$, satisfying
$\expectationvalue{\mathcal{N}}=n+1$, is decoupled from the rest of the
Hilbert space. This makes it possible to solve the problem analytically, even
in the presence of time-dependent coupling [19, 20, 21]. Similarly, the two-
mode Jaynes-Cummings model has a conserved excitation number
$\mathcal{N}=\sigma^{\dagger}\sigma+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}$.
However, in contrast with the single-mode model, the subspace with constant
excitation number $\expectationvalue{\mathcal{N}}=n+1$ is spanned by the set
of states
$\quantity{\ket{G,n+1,0},\ket{X,n,0},\ket{G,n,1},\ket{X,n-1,1},\ldots,\ket{G,0,n+1}}$
and has dimension $2(n+1)+1$, thereby making the analytical solution
challenging.

We thus calculate the dynamics numerically by solving the Schrodinger equation
$i\hbar\partial_{t}\ket{\psi(t)}=\widehat{H}_{\rm int}(t)\ket{\psi(t)}$ in the
interaction picture with respect to $H_{0}+H_{1}+H_{2}$, with

$\displaystyle\widehat{H}_{\rm int}(t)$
$\displaystyle=\exp\quantity[\frac{i}{\hbar}(H_{0}+H_{1}+H_{2})t]H_{\rm
int}(t)\exp\quantity[-\frac{i}{\hbar}(H_{0}+H_{1}+H_{2})t]$
$\displaystyle=\sum_{j=1,2}\hbar
g_{j}\exp\quantity(-\frac{t^{2}}{t_{p}^{2}})\quantity(e^{-i\delta_{j}t}\sigma^{\dagger}a_{j}+e^{+i\delta_{j}t}\sigma
a_{j}^{\dagger}),$ (2)

and $\delta_{j}=\omega_{j}-\omega_{0}$ the detuning of each mode with respect
to the emitter frequency. Specifically, noting that
$\exp\quantity(-3^{2})\approx 10^{-4}$, we initialize the system in the chosen
initial state $\ket{\psi(t_{\rm i})}$ at time $t_{\rm i}=-3t_{p}$ and we
calculate the evolution of $\ket{\psi(t)}$ with a 4th order Runge-Kutta
algorithm until the final state $\ket{\psi(t_{\rm f})}$ at time $t_{\rm
f}=+3t_{p}$, where the population of the emitter and each field mode is
inspected. Here, the use of the interaction picture is beneficial to avoid
fast oscillating terms, especially when dealing with field states with a large
number of photons.

It is worth noticing that the numerical algorithm does not preserve the state
normalization. This is easily verified with a 1st order Euler method by
noticing that the evolved state
$\ket{\psi(t+\differential{t})}=\ket{\psi(t)}-\frac{i}{\hbar}\differential{t}\widehat{H}_{\rm
int}\ket{\psi(t)}$ satisfies
$\innerproduct{\psi(t+\differential{t})}{\psi(t+\differential{t})}=\innerproduct{\psi(t)}{\psi(t)}+\frac{(\differential
t)^{2}}{\hbar^{2}}\innerproduct{\widehat{H}_{\rm int}\psi(t)}{\widehat{H}_{\rm
int}\psi(t)}>1$, even for $\innerproduct{\psi(t)}{\psi(t)}=1$. Therefore, we
take care of normalizing the quantum state to 1 after each step in the
calculation.

For the numerical implementation, the state of each field mode $j$ is expanded
on a truncated number state basis $\quantity{\ket{n_{j}}}$, with
$n_{j}\in(N_{j}^{\rm min},\ldots,N_{j}^{\rm max})$, and operators acting on
mode $j$ are represented as $M_{j}\times M_{j}$ matrices, with
$M_{j}=N_{j}^{\rm max}-N_{j}^{\rm min}+1$. We make sure that results are
converged with respect to the choice of $N_{j}^{\rm max}$ and $N_{j}^{\rm
min}$. This can be checked, for instance, by ensuring that the expectation
value of the excitation number
$\mathcal{N}=\sigma^{\dagger}\sigma+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}$
is conserved throughout the simulation, as required by the fact that
$\commutator{\widehat{H}_{\rm int}(t)}{\mathcal{N}}=0$.

##  3 Dynamics of the SUPER scheme

Figure 1:  (a–f): Quantum state of the coupled emitter-field system after a
SUPER pulse, as a function of the detuning $\Delta_{2}=t_{p}\delta_{2}$ and
the initial number of photons $n_{2}^{\rm i}$ in mode 2. The system is
initialized at time $t_{\rm i}$ in the state $\ket{G,n_{1}^{\rm i},n_{2}^{\rm
i}}$, with $\ket{n_{j}^{\rm i}}$ representing Fock states. The quantities
plotted are the final excited state population
$\expectationvalue{\sigma^{\dagger}\sigma}$ (panels a, b) and the variation
$\expectationvalue{\Delta n_{j}}=\expectationvalue{a_{j}^{\dagger}a_{j}(t_{\rm
f})}-n_{j}^{\rm i}$ of the average photon number in mode 1 (c, d) and 2 (e,
f). (g–m): Evolution of $\expectationvalue{\sigma^{\dagger}\sigma}$,
$\expectationvalue{\Delta n_{1}}$, and $\expectationvalue{\Delta n_{2}}$ for
different configurations corresponding to local maxima in panels a and b, see
red arrows. Parameters for the first mode are fixed at $\Delta_{1}=6$
(everywhere), $\sqrt{n_{1}^{\rm i}}=62.83$ (panels a, c, e), and
$\sqrt{n_{1}^{\rm i}}=157.08$ (panels b, d, f). The dimensionless interaction
strength is $G_{1}=G_{2}=0.1$ (everywhere).

We begin by considering the system dynamics when the field modes are both red-
detuned with respect to the emitter and initialized in large-number Fock
states $\ket{n_{j}^{\rm i}}$. This is motivated by the fact that optical
excitation of a quantum emitter involves laser pulses with a large number of
photons. Although Fock states are generally inaccurate to describe the quantum
field of a laser, their use allows for a substantial numerical simplification
of the problem in terms of the number $M_{j}$ of states needed. The case of
coherent states, which represent a more accurate description of the laser
field, will be examined later in Section 6, where we find good qualitative
agreement between the two pictures. We note that the dynamics is governed by
the dimensionless quantities $\Delta_{j}=t_{p}\delta_{j}$, and
$G_{j}=t_{p}g_{j}$, and we will discuss the results in terms of these
parameters.

In Fig. 1, we explore the population
$P_{X}=\expectationvalue{\sigma^{\dagger}\sigma}$ of the $\ket{X}$ state and
of both field modes at time $t_{\rm f}=3t_{p}$, i.e. after interaction has
occurred. Following the literature [8, 9, 12, 13], we fix the parameters for
the first mode (dimensionless detuning $\Delta_{1}=t_{p}\delta_{1}$ and
initial photon number $n_{j}^{\rm i}$), and scan the corresponding parameters
for the second mode. Note that the terms “first” and “second” identify the
mode with shorter and larger detuning in absolute value, respectively. In the
SUPER scheme literature based on semiclassical models of light-matter
interaction, results are typically shown as a function of the electric field
amplitude [7, 12], which is proportional to the square root of the photon
number. Therefore, we choose to plot the results as a function of
$\sqrt{n_{2}^{\rm i}}$. The dimensionless coupling is fixed to
$G_{1}=G_{2}=0.1$.

In Fig. 1a, which is taken at fixed $\sqrt{n_{1}^{\rm i}}=62.83$ ($n_{1}^{\rm
i}=3947$), we identify a region of near-unity exciton preparation fidelity,
with a maximum of $P_{X}=0.99$. This is accompanied by additional resonances
at shorter detuning with incomplete population inversion of the order of
$\approx 0.4$ at maximum. In contrast, by increasing the initial photon number
in mode 1 to $\sqrt{n_{1}^{\rm i}}=157.08$ ($n_{1}^{\rm i}=24673$), we observe
multiple resonances with $P_{X}\geq 0.99$ (see Fig. 1b). Similar features have
been observed in the literature using a classical description of the field and
represent the signature of the SUPER scheme. We conclude that our model of a
two-level emitter interacting with two quantized field modes captures the
physics of the SUPER swing-up effect satisfactorily.

Differently from the semiclassical description, a quantized model give access
to the exact number of photons in each mode. We thus calculate the difference
$\expectationvalue{\Delta n_{j}}=\expectationvalue{a_{j}^{\dagger}a_{j}(t_{\rm
f})}-\expectationvalue{a_{j}^{\dagger}a_{j}(t_{\rm i})}$ of the photon number
in each mode before and after interaction, which is plotted in Fig. 1c–f. In
correspondence with each resonance of $P_{X}$, we observe variations of the
photon number in both modes. Remarkably, mode 1 shows always a decrease in
photon number, whereas mode 2 shows an unexpected increase. Exploring the
dynamics in correspondence of $P_{X}\approx 1$ (see Fig. 1g–l), we find that
exactly two photons are subtracted from mode 1, while mode 2 gains one photon
in the process. This demonstrates surprisingly that SUPER scheme is a multi-
photon process involving a redistribution of photons between the field modes.
We also identify a higher-order process leading to large but incomplete
population inversion $P_{X}>0.85$ (see Fig. 1m) where almost three photons
($\expectationvalue{\Delta n_{1}}=-2.76$) are subtracted from mode 1, while
mode 2 gains almost two photons ($\expectationvalue{\Delta n_{2}}=1.90$). We
hypothesize that a process leading to $\expectationvalue{\Delta n_{1}}=-3$ and
$\expectationvalue{\Delta n_{2}}=2$ exactly might be found with a thorough
investigation of the parameter space, which is not performed here. Despite
these variations in the photon number, we observe that the total excitation
number
$\mathcal{N}=\sigma^{\dagger}\sigma+a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}$
is indeed conserved in the process, i.e. $P_{X}+\expectationvalue{\Delta
n_{1}}+\expectationvalue{\Delta n_{2}}=0$.

In contrast with the semiclassical description of the SUPER scheme dynamics,
which shows fast and ample oscillations in time of the $\ket{X}$ state
population within one pulse (of the order of 10–30), the first leftmost
resonance in Fig. 1a–b is characterized by a smooth and monotonic dynamics.
Higher order resonances at shorter detuning and larger photon number display
few oscillations in the emitter population and photon number of the field
modes (of the order of 2–3).

##  4 Dynamics of the red-and-blue dichromatic scheme

In the red-and-blue dichromatic scheme, two laser pulses are symmetrically
detuned to the red and blue side of the spectrum with respect to the emitter.
It has been demonstrated that specific configurations involving pulses of
different amplitude generate complete population inversion, whereas symmetric
configurations with identical pulse amplitude lead necessarily to $P_{X}=0$
[18]. Exciton preparation with the red-and-blue scheme occurs through a
similar swing-up effect of the population [18, 12].

Figure 2:  (a, b, c): Quantum state of the coupled emitter-field system after
a red-and-blue dichromatic pulse, as a function of the initial number of
photons $n_{j}^{\rm i}$ in each mode $j$. The system is initialized in the
state $\ket{G,n_{1}^{\rm i},n_{2}^{\rm i}}$, with $\ket{n_{j}^{\rm i}}$
representing Fock states. The quantities plotted are the final $\ket{X}$ state
population $\expectationvalue{\sigma^{\dagger}\sigma}$ (panel a) and the
variations $\expectationvalue{\Delta
n_{j}}=\expectationvalue{a_{j}^{\dagger}a_{j}(t_{\rm f})}-n_{j}^{\rm i}$ of
the average photon numbers in modes 1 (b) and 2 (c). (d–g): Evolution of
$\expectationvalue{\sigma^{\dagger}\sigma}$, $\expectationvalue{\Delta
n_{1}}$, and $\expectationvalue{\Delta n_{2}}$ in time for different
configurations corresponding red arrows in panel a. In all panels, the mode
detunings are fixed at $\Delta_{1}=-\Delta_{2}=6$, and the dimensionless
interaction strength is $G_{1}=G_{2}=0.1$.

To model this scenario, we constrain the detuning to
$\Delta_{1}=-\Delta_{2}=6$ and explore the dependence of the final exciton
population as a function of the initial photon number in each mode. Once
again, we initialize the field modes in Fock states $\ket{n_{j}^{\rm i}}$. As
shown in Fig. 2a, we obtain several resonances featuring $P_{X}\approx 1$ in
quantitative agreement with results based on semiclassical models [12]. No
resonance is indeed observed along the diagonal $n_{1}^{\rm i}=n_{2}^{\rm i}$,
in agreement with the requirement of different pulse amplitudes.

In contrast with the SUPER scheme, we observe that this process involves
photon exchange with a single mode, see Figs. 2b and Fig. 2c. For example, at
$\quantity(\sqrt{n_{1}^{\rm i}},\sqrt{n_{2}^{\rm i}})$ = (16.00, 61.25) and
(29.21, 89.84), exactly one photon is subtracted from mode 1 (the one at
positive detuning) to populate the excited state, while no photon exchange
occurs with mode 2 (negative detuning). The opposite holds true by exchanging
$n_{1}^{\rm i}$ with $n_{2}^{\rm i}$. There, one photon is subtracted from
mode 2 with no gain/loss in mode 1. We notice, however, that in both cases the
mode with $\expectationvalue{\Delta n_{j}}=0$ participates actively in the
dynamics, as demonstrated by oscillations of the average photon number in time
reported in Fig. 2d–g.

Similarly to the SUPER scheme, we notice that the fundamental resonance
(namely, the one occurring at smaller photon number) shows a monotonic
increase in $P_{X}$ with no oscillations (Fig. 2d–g). Higher order resonances
display few small oscillations.

##  5 Strong emitter-field coupling: “vacuum” swing-up

So far, the emitter dynamics under a two-color excitation protocol involving a
large number of photons is consistent with a semiclassical approach where the
electromagnetic field is treated as a classical object. We now explore a truly
quantum regime where the field modes are populated by a few photons only.

The strength of the emitter-field coupling is governed by the coupling
constants $G_{j}$ and the electric field amplitude
$E_{j}\propto\sqrt{n_{j}^{\rm i}}$, as one can readily verify by calculating
the matrix elements of $H_{\rm int}$. Therefore, to maintain the same
effective light-matter coupling while scaling down the field occupation to the
few photon regime, we simultaneously increase the value of the dimensionless
coupling constants $G_{j}=t_{p}g_{j}$. In experiments, this can be done by
increasing the emitter-field coupling $g_{j}$ via optical engineering of the
electromagnetic environment, or by increasing the pulse duration $t_{p}$.

Figure 3:  (a–d): Final $\ket{X}$ state population
$\expectationvalue{\sigma^{\dagger}\sigma}$ after a SUPER pulse in the low
photon number regime. The system is initialized in the state
$\ket{G,n_{1}^{\rm i},n_{2}^{\rm i}}$, with $\ket{n_{j}^{\rm i}}$ representing
Fock states. In (a) and (b), the detuning $\Delta_{1}$ and initial number of
photons $n_{1}^{\rm i}$ in mode 1 are fixed as reported in the Figure,
$\Delta_{2}$ is varied continuously along the $x$-axis, and $n_{2}^{\rm i}$ is
varied in integer steps along the $y$-axis. In (c) and (d), the initial photon
numbers $n_{j}^{\rm i}$ are fixed and the detunings $\Delta_{j}$ are varied
continuously. (e, f): Evolution of
$\expectationvalue{\sigma^{\dagger}\sigma}$, $\expectationvalue{\Delta
n_{1}}$, and $\expectationvalue{\Delta n_{2}}$ in time for different
configurations corresponding red arrows in panels c) and d). (g): Evolution of
$\expectationvalue{\sigma^{\dagger}\sigma}$, $\expectationvalue{n_{1}}$, and
$\expectationvalue{n_{2}}$ in time for a red-and-blue dichromatic
configuration with $\Delta_{1}=-\Delta_{2}=3.12$. (h): Evolution of
$\expectationvalue{\sigma^{\dagger}\sigma}$, $\expectationvalue{n_{1}}$, and
$\expectationvalue{n_{2}}$ in time for a SUPER configuration with
$\Delta_{1}=-15.68$, $\Delta_{2}=-3.78$, and where the emiter is initially in
the excited state $\ket{X}$. Note that panels g) and h) report the absolute
photon numbers $\expectationvalue{n_{j}}$ rather than the variation
$\expectationvalue{\Delta n_{j}}$. In all panels, the dimensionless
interaction strength is $G_{1}=G_{2}=5$.

Starting with the SUPER scheme, we explore this scenario in Fig. 3 by
increasing the coupling to $G_{1}=G_{2}=5$ and scaling the initial number of
photons in mode 1 down to $n_{1}^{\rm i}=5$ and $n_{1}^{\rm i}=2$ (panels a
and b, respectively). Interestingly, we observe that the typical resonances of
the SUPER scheme are still clearly visible, and only few photons are needed to
achieve full population inversion. Quite surprisingly, exciton preparation is
still possible even when the second field mode is prepared in the vacuum state
$n_{2}^{\rm i}=0$. We observe $P_{X}=0.97$ for $(n_{1}^{\rm i},n_{2}^{\rm
i})=(5,0)$ and $P_{X}=0.88$ for $(n_{1}^{\rm i},n_{2}^{\rm i})=(2,0)$, where
incomplete population inversion $P_{X}<1$ is explained by the fact that we
have chosen an arbitrary value of $\Delta_{1}=-6$. By fixing the initial
number of photons in each mode and continuously varying both detunings
$\Delta_{j}$, we obtain indeed full population inversion with $P_{X}=1.00$ at
$(n_{1}^{\rm i},n_{2}^{\rm i},\Delta_{1},\Delta_{2})=(5,0,-7.56,-28.56)$, and
$P_{X}=0.99$ at $(n_{1}^{\rm i},n_{2}^{\rm
i},\Delta_{1},\Delta_{2})=(2,0,-4.06,-15.96)$, see Figs. 3c and d. The
corresponding time-dependent dynamics is plotted in Figs. 3e and f,
demonstrating that full population inversion with exchange of multiple photons
in indeed possible even when one of the field modes is in the vacuum state.
For the case of Fig. 3f, where the field is initialized in $(n_{1}^{\rm
i},n_{2}^{\rm i})=(2,0)$, we notice that mode 1 is left in the vacuum state,
while mode 2 is in the single-photon Fock state $\ket{n_{2}^{\rm f}}=\ket{1}$
after interaction.

We observe no population inversion when $(n_{1}^{\rm i},n_{2}^{\rm i})=(1,0)$,
for any value of the detunings. This demonstrates that the minimum excitation
number to activate the SUPER mechanism is $\mathcal{N}=2$, consistent with the
fact that the SUPER scheme cannot be explained with the exchange of one single
photon between the emitter and the field.

Applying similar considerations, we find signatures of the red-and-blue
dichromatic scheme in the few-photon regime as well. Despite the fact that
dichromatic excitation in the large photon number regime is explained with the
subtraction of a single photon from one of the modes, we do not observe
population inversion for initial state $(n_{1}^{\rm i},n_{2}^{\rm i})=(1,0)$
or $(0,1)$, indicating that the requirement on the excitation number is still
$\mathcal{N}\geq 2$. On the other hand, we find that preparing the field modes
in two single-photon states $\quantity(n_{1}^{\rm i},n_{2}^{\rm i})=(1,1)$ at
detuning $\Delta_{1}=-\Delta_{2}=3.12$ leads to population of the exciton
state with fidelity $P_{X}=0.96$, see Fig. 3g. Here, the system is
approximately left in the entangled state $\ket{\psi(t_{\rm
f})}=\frac{1}{\sqrt{2}}\quantity(\ket{X,1,0}+\ket{X,0,1})$ after interaction,
meaning that half a photon has been subtracted from both modes on average.

Finally, it is worth noting that the multi-photon resonances of the SUPER
scheme work in the reverse fashion as well. In other words, it is possible to
completely depopulate an initially excited emitter by subtracting one
excitation from mode 1, while increasing the population of the other field
mode by two excitations. This may be used to engineer non-linear interaction
between field modes at different frequency. For example, as shown in Fig. 3h,
a single-photon state at detuning $\Delta_{1}=-15.68$ is converted into a two-
photon state at $\Delta_{2}=-3.78$ accompanied by relaxation of the emitter to
the ground state.

##  6 Coherent states

Figure 4:  Evolution of $\expectationvalue{\sigma^{\dagger}\sigma}$,
$\expectationvalue{n_{1}}$, and $\expectationvalue{n_{2}}$ in time for
different initial states of the electromagnetic field. The field modes are
initialized either in Fock states containing $n_{1}^{\rm i}$ and $n_{2}^{\rm
i}$ photons respectively (dashed lines), or in coherent states with the same
average number of photons (solid lines). Detunings $\Delta_{j}$ and initial
photon numbers $n_{j}^{\rm i}$ are indicated in the top panel title. The
dimensionless interaction strength is $G_{1}=G_{2}=1$ (a–c) and
$G_{1}=G_{2}=5$ (d–f).

So far we have considered Fock states of the field modes for simplicity.
However, the quantum state of a laser pulse is best described by a coherent
state $\ket{\alpha}=e^{-|\alpha^{2}|/2}\sum_{n}\alpha^{n}/(\sqrt{n!})\ket{n}$.
In Fig. 4, we compare the dynamics for a case where the field is initialized
in the multi-mode Fock state $\ket{n_{1}^{\rm i},n_{2}^{\rm i}}$ with the case
where the initial field state is $\ket{\alpha_{1}^{\rm i},\alpha_{2}^{\rm
i}}$, with $\ket{\alpha_{j}^{\rm i}}$ representing coherent states containing
the same number of photons $\matrixelement{\alpha_{j}^{\rm
i}}{a_{j}^{{\dagger}}a_{j}}{\alpha_{j}^{\rm i}}=|\alpha_{j}^{\rm
i}|^{2}=n_{j}^{\rm i}$ on average. For sufficiently high excitation number
(see Fig. 4a–c), we observe that the two situations are qualitatively similar.
Indeed, the description based on coherent states confirms that the SUPER
scheme excitation takes place by removing approximately two photons from mode
1 while simultaneously increasing the population of mode 2 by one photon.
However, whereas the Fock state description yields a monotonic dynamics, the
coherent state model reproduces the typical swing-up pattern that
characterizes the SUPER scheme, resulting in fast oscillations of the exciton
population. Despite this remarkable qualitative difference, the attained value
of $P_{X}$ using coherent states is quantitatively similar to the one obtained
using Fock states.

However, the agreement breaks down in the deep quantum regime of low
excitation number. As shown in Figs. 4d–f, a model using coherent states
predicts $P_{X}=0.32$ for $(|\alpha_{1}^{\rm i}|^{2},|\alpha_{2}^{\rm
i}|^{2})=(2,1)$ and $(\Delta_{1},\Delta_{2})=(-6,-19.81)$, while the value
predicted with a Fock state assumption is $P_{X}=0.96$. We conclude that SUPER
scheme excitation in the low photon number regime necessitates of few-photon
Fock states as compared with coherent states of small amplitude. Whereas the
latter can be realized, for example, with attenuated laser pulses, the former
are fundamentally different and more challenging to realize [22, 23, 24].

##  7 Discussion

We have investigated the physics of two-color excitation schemes in terms of
energy exchanges between the emitter and the laser pump. To do so, we have
developed a model for a two-level emitter interacting with two quantized field
modes, where interaction occurs during short pulses of duration $t_{p}$ in
time. Both the SUPER and the red-and-blue dichromatic schemes have been
considered.

We have observed that exciton population under the SUPER scheme, where both
modes are red-detuned with respect to the emitter, is accompanied by
variations in the photon number of both modes, unveiling the multi-photon
nature of this process. We find clear signatures of events with
$(\expectationvalue{\sigma^{\dagger}\sigma},\expectationvalue{\Delta
n_{1}},\expectationvalue{\Delta n_{2}})=(+1,-2,+1)$ and $(+1,-3,+2)$, where
multiple excitations are subtracted from the 1st field mode (the one with
smaller detuning, in absolute value). While one excitation is used to raise
the emitter population to the excited state, the remaining are absorbed by the
2nd field mode at larger detuning. Interestingly, this holds true in the
regime where the field modes are populated with only few photons, and even
when one of the field is in the vacuum state. We speculate that events
involving exchange of higher number of photons, i.e.
$(\expectationvalue{\sigma^{\dagger}\sigma},\expectationvalue{\Delta
n_{1}},\expectationvalue{\Delta n_{2}})=(1,-m,m-1)$ with $m>3$, might be found
at larger total power, although we have not observed such a case in this work.

This finding is in stark contrast with the paradigm of resonant excitation,
where the emitter exchanges one energy quantum with a single field mode. We
have indeed verified that our model reproduces this scenario by removing the
2nd field mode (i.e. setting $G_{2}=0$) and moving the 1st mode into resonance
with the emitter ($\Delta_{1}=0$). By scanning the initial number of photons
$n_{1}^{\rm i}$ in mode 1, we have found clear signatures of Rabi
oscillations. For configurations resulting in $P_{X}=1$, we have observed that
one single photon is removed from the field. A similar scenario is expected to
take place in phonon-assisted excitation [25, 26, 27], although we have not
included phonon coupling in this work.

Reversing the working principle of the SUPER scheme, our model shows that it
is possible to stimulate emission from an excited emitter with a simultaneous
redistribution of photons among the field modes. For instance, a system
initialized in the state $\ket{X,n_{1},n_{2}}$ is found in the state
$\ket{G,n_{1}-1,n_{2}+2}$ after interaction with two red-detuned modes. Once
again, this is valid in the case $n_{2}=0$ where mode 2 is in the vacuum
state. Quite interestingly, this observation challenges the typical paradigm
of spontaneous emission stating that the emitted photon must have the same
frequency as the photons in the incident wave. These findings may be used to
engineer a non-linear interaction where one single photon stimulates the
emission of two identical photons at a different frequency.

Considering the red-and-blue dichromatic excitation, where one of the field
modes is moved to the blue side of the spectrum, we have observed that it
involves energy exchange with only one of the field modes. However, our
results demonstrate that the presence of a second field mode is necessary to
engineer full population inversion, even when no variation in its the photon
number is recorded. An exception to this paradigm is the case of dichromatic
excitation for an initial state $\ket{G,1,1}$, where we find that half a
photon is subtracted on average from both modes.

It is worth noting that we have obtained a good qualitative agreement with the
description of the SUPER and red-and-blue dichromatic scheme under a
semiclassical model for light-matter interaction, where the field is not
quantized. However, the advantage of the quantum model presented here is that
it makes it possible to calculate the variation in photon number for each
mode. We have furthermore observed that a description of the laser field in
terms of coherent states is needed to reproduce the typical swing-up behavior
of the excited state population under SUPER scheme excitation, which is
characterized by fast oscillations. However, we have surprisingly found that a
description in term of Fock states yields qualitatively similar results for
the final population inversion, and demands less computational resources. For
example, the Fock state dynamics shown in Fig. 4a is fully converged when
using $M_{1}=M_{2}=11$ states in the field Hilbert space, whereas
$M_{1}=M_{2}=91$ states are required for full convergence of the corresponding
coherent state simulation. The qualitative agreement between the two
descriptions is however limited to the large photon number regime. For small
photon numbers of the order of few units, fundamental differences in the
behavior of Fock and coherent states emerge.

Finally, we have used values in the range 0.1–5 for the dimensionless coupling
constants $G_{j}=g_{j}t_{p}$. Considering short pulses of the order of
$t_{p}=$ 2–3 ps, a value $G_{j}=0.1$ is obtained with a light-matter coupling
strength of the order of $g_{j}=$ 0.03–0.05 THz. The latter corresponds to
typical values attained in high-Q micropillars [28, 12] or open tunable
microcavities [6], and similar values are found elsewhere in the literature
[26]. Two strategies may be pursued to attain stronger dimensionless coupling
$G_{j}=5$. On one hand, the pulse duration may be increased by one order of
magnitude to $t_{p}\approx 100$ ps. We notice, however, that the system in
this regime is more prone to environment induced decoherence, such as charge
and spin noise, and phonon scattering. It has been shown that the latter is
especially detrimental for the SUPER scheme, leading to reduced population
inversion [12, 13]. As alternative, a light-matter coupling strength
$g_{j}\approx 2$ THz is needed to obtain $G_{j}=5$. Whereas such a strong
coupling maybe challenging to realize, we notice that nanocavities with
extreme photon confinement well below the diffraction limit in dielectrics
have been designed and fabricated in silicon [29]. Here, ultra-strong light-
matter coupling is expected due to the extremely low mode volume.

As mentioned, phonon scattering is generally detrimental to the SUPER scheme.
Here, we have considered an ideal case of unitary dynamics with no
decoherence, where the Schrödinger equation is solved exactly with a numerical
algorithm to obtain the quantum state $\ket{\psi(t)}$ of the emitter-field
system. To include phonon coupling, a master equation description in terms of
the density operator $\rho$ is needed. However, the quantum state $\ket{\psi}$
is implemented numerically as a vector with dimension $2M_{1}M_{2}$. Using the
same basis, the density operator $\rho$ becomes a $2M_{1}M_{2}\times
2M_{1}M_{2}$ matrix, which poses serious numerical challenges due to the
substantial increase in computational resources.

During the writing of this manuscript, we became aware of a related work which
also demonstrates the multi-photon nature of the SUPER scheme [30].

##  8 Conclusions

We have studied the exchange of energy between a two-level emitter and two
quantized field modes under the SUPER scheme excitation. Surprisingly, we have
found that one mode loses two or more photons while the other gains at least
one, proving that the SUPER is a multi-photon process involving a
redistribution of photons between the field modes. Our results unveil a novel
and unexpected off-resonant interaction of light and matter, and open new
possibilities for manipulating the quantum state of atoms using only few off-
resonant photons.

Funding This work received supported from the European Research Council (ERC-
CoG “Unity”, grant no.865230) and the European Union’s Horizon 2020 Research
and Innovation Programme under the Marie Skłodowska-Curie Grant (Agreement No.
861097).

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper may be
obtained from the authors upon reasonable request.

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