# Non-classical excitation of a solid-state quantum emitter

L. M. Hansen<EMAIL_ADDRESS>University of Vienna, Faculty of
Physics, Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna,
Austria  Christian Doppler Laboratory for Photonic Quantum Computer, Faculty
of Physics, University of Vienna, Vienna, Austria  F. Giorgino  University of
Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology
(VCQ), 1090 Vienna, Austria  Christian Doppler Laboratory for Photonic Quantum
Computer, Faculty of Physics, University of Vienna, Vienna, Austria  L. Jehle
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science
and Technology (VCQ), 1090 Vienna, Austria  L. Carosini  University of Vienna,
Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ),
1090 Vienna, Austria  Christian Doppler Laboratory for Photonic Quantum
Computer, Faculty of Physics, University of Vienna, Vienna, Austria  J. C.
López Carreño  Institute of Theoretical Physics, University of Warsaw, ul.
Pasteura 5, 02-093 Warsaw, Poland  I. Arrazola  Instituto de Física Teórica,
UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040
Vienna, Austria  P. Walther<EMAIL_ADDRESS>University of Vienna,
Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ),
1090 Vienna, Austria  Christian Doppler Laboratory for Photonic Quantum
Computer, Faculty of Physics, University of Vienna, Vienna, Austria
University of Vienna, Research Network for Quantum Aspects of Space Time
(TURIS), 1090 Vienna, Austria  Institute for Quantum Optics and Quantum
Information (IQOQI) Vienna, Austrian Academy of Sciences, Vienna, Austria  J.
C. Loredo<EMAIL_ADDRESS>University of Vienna, Faculty of Physics,
Vienna Center for Quantum Science and Technology (VCQ), 1090 Vienna, Austria
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of
Physics, University of Vienna, Vienna, Austria

###### Abstract

The interaction between a single emitter and a single photon is a fundamental
aspect of quantum optics. This interaction allows for the study of various
quantum processes, such as emitter-mediated single-photon scattering and
effective photon-photon interactions. However, empirical observations of this
scenario and its dynamics are rare, and in most cases, only partial
approximations to the fully quantized case have been possible. Here, we
demonstrate the resonant excitation of a solid-state quantum emitter using
quantized input light. For this light-matter interaction, with both entities
quantized, we observe single-photon interference introduced by the emitter in
a coherent scattering process, photon-number-depended optical non-linearities,
and stimulated emission processes involving only two photons. We theoretically
reproduce our observations using a cascaded master equation model. Our
findings demonstrate that a single photon is sufficient to change the state of
a solid-state quantum emitter, and efficient emitter-mediated photon-photon
interactions are feasible. These results suggest future possibilities ranging
from enabling quantum information transfer in a quantum network to building
deterministic entangling gates for photonic quantum computing.

Studying light interacting with matter at the most fundamental level offers a
number of possibilities for the development of quantum technologies [1, 2]. On
the one hand, non-classical states of light are suitable as carriers and for
processing quantum information [3, 4] because they virtually do not interact
with their environment. However, the lack of direct photon-photon interactions
also prevents access to, e.g., deterministic entangling gates on quantized
states of light—an obstacle for building larger scalable quantum photonic
systems.

Light strongly interacts with resonant atomic systems. For instance, coherent
scattering involves the absorption of one photon subsequent re-emission of a
single photon; it has been theoretically shown that even one photon can
saturate an atomic transition [5, 6, 7, 8]. Moreover, an atomic system is
inherently quantum non-linear, mediating various phenomena of effective
photon-photon interactions, such as electromagnetically induced transparency
[9, 10], photon-blockade [11, 12, 13], and entangling quantum logic operations
[14, 15].

Figure 1: Single-photon interference mediated by a quantum emitter. a, A
single-photon wave packet is sent as input to a QD coupled to a cavity. The
light-matter interaction between the quantized light field and the quantum
emitter is revealed in the temporal profile of the output wave packet that
features single-photon interference. The process involves the quantum
interference of two scenarios: attenuated propagation of the input field and
re-emission after absorption. b, Experimentally, we investigate this concept
by generating a single photon from the emitter and sending it back again to
the emitter as quantized input field (blue dots). As predicted, the observed
output field (red dots) is governed by (1) the transmitted part of the photon
wave packet, (2) destructive interference, and (3) a second maxima indicating
the re-emission of a photon. Due to the high single-photon count rate the
Poissonian error bars are not visible in the plot. The observations are
confirmed by the results of theoretical simulations (solid lines) that follow
the cascaded master equation approach.

However, empirically observing an atomic system interacting with a single
photon has remained a formidable challenge, as one requires a sufficiently
controlled and strongly coupled quantum light-matter platform. Although
scarce, a few experiments have reported aspects of quantized input light
interacting with an atomic system, for instance, a quantum network of neutral
atoms interconnected via single photons [16], single photons scattered by a
single Rubidium atom [17], heralded single-photon absorption in a trapped ion
system [18], and electronic energy transfer in a photosynthetic complex
induced by a single photon [19]. Despite impressive demonstrations, these
previous works were limited by weak signals, which strongly hindered their
ability to investigate the rich photon dynamics and statistics inherent in the
fundamental process.

A solid-state platform offers a promising alternative for studying coherent
quantum light-matter interactions. Significant progress has been made in both
cavity and waveguide quantum electrodynamics [20, 21]. In particular,
semiconductor quantum dots (QDs) coupled to resonant cavities and photonic
crystal waveguides [22, 23] constitute a leading candidate for the generation
of various types of quantum light: highly-efficient sources of single and
entangled photons [24, 25, 26, 27], quantum superpositions and entangled
states of photon-numbers [28, 29], and deterministically generated squeezed
states [30, 31]. Moreover, these solid-state systems led to multiple
demonstrations of optical non-linearities involving a small number of photons
on average, with works including photon-number dependent spatial [32, 33, 34,
35, 36] and temporal [37, 38] ordering. However, these previous works employed
classical weak coherent input light, thus leaving non-classical light-matter
interactions in a solid-state platform without observation.

Here, we demonstrate a quantum light-matter interface in the solid-state. A
single QD coupled to an optical cavity is resonantly excited with non-
classical light. When the input light consists of a single photon, the output
field stems from the quantum interference between the re-emitted part of the
wave packet and the incoming part of the wave packet of the single photon. We
demonstrate that this process is coherent by performing first- and second-
order correlation measurements; indeed, the time-evolving output signal
exhibits a point of destructive interference, and consecutive output wave
packets display Hong-Ou-Mandel (HOM) two-photon interference [39]. Moreover,
we observe optical photon-number depended non-linearities occurring for
quantized input light, as revealed by changes in the photon-number statistics.
Further, we find signatures of stimulated emission processes between two
photons, verified by analyzing the temporal mode of the generated two-photon
Fock state.

### Atom-mediated single-photon interference

Figure 1a depicts the general concept of this work. Here, a quantum emitter,
an InAs/GaAs QD in a cavity, is excited with a non-classical wave packet. The
output signal resulting from the quantum light-matter interaction is then
collected for further analysis.

Figure 2: Measurements of single-photon purity and indistinguishability. a,
Comparing the photon statistics of the input field (top) and the output field
(bottom) reveals that single-photon components are suppressed in the output
field relative to multiphoton components. b, Our observations of the
indistinguishability of the quantized input fields (top) compared to the
scattered fields (bottom) indicate that indistinguishability remains in the
output field, validating the coherence of the process. c, Time-gated two-
photon probabilities $P_{2}$ estimated at the detector level for different
starting gate times indicate a different temporal structure of two-photon
components along the lifetime in the output field. The temporal histogram of
each signal is shown in gray as a reference.

In our experiment, the input non-classical light consists of resonance
fluorescence from the same QD. That is, we first prepare a highly efficient
single-photon source [40] (see Methods) and then use a setup with active path
control that allows us to choose between two options: either the signal is
back-reflected to the emitter or it is sent to the detection stage. With this
setup, we first send the resonance fluorescence signal back toward the QD and
later collect the resulting output signal; see the Supplementary Information
for details on the experimental setup.

We can tune the input light statistics by choosing the pulse area of the
excitation laser that generates the resonance fluorescence signal, from mostly
single photons—at $\pi$-pulse excitation—to states with one- and two-photon
terms—e.g., at $2\pi$-pulse excitation [28, 41]. We first study the resulting
time dynamics with an input signal generated by $\pi$-pulses. In this
scenario, the quantum emitter interacts with a resonant single-photon wave
packet. As the wave packet reaches the emitter, part of it is transmitted, and
the excitation probability of the emitter begins to increase. During the
interaction the emission probability keeps increasing, leading to quantum
interference between the incoming wave packet and the reemitted part. Once the
input wave packet duration ends, the emitter retains the probability of being
excited and spontaneously decays without interference. Hence, the observation
of destructive interference in the temporal output profile is due to coherent
single-photon scattering [8, 42].

The optimal input temporal shape for achieving atomic inversion is formed as
an exponentially increasing curve [5, 6]—the time-reversed profile of the
spontaneous emission decay. In our case, however, we see the opposite scenario
because we back-reflect resonance fluorescence: a single photon arrives at the
artificial atom with a field amplitude that first rises rapidly and then
exponentially decays. Figure 1b shows the measured input time profile with a
lifetime of $\tau\,{=}\,227$ ps, and the time profile of the output field
after the interaction with the emitter. We observe the predicted atom-mediated
single-photon interference phenomenon and identify three main regions: (1) an
earlier part governed by an undisturbed signal—i.e., the input photon is not
absorbed, it enters the optical cavity, and it is released back again into the
collection mode; (2) an intermediate part displaying a point of minimal
intensity given by destructive interference; and (3) a final part governed by
spontaneous emission of an atom subject to an exponentially-decaying input
field. Our measurements are confirmed by theoretical simulations that follow a
cascaded master equation approach in which two identical subsystems are
considered, with the output of the first system serving as input for the
second system (see Methods).

### Input-output photon statistics

We now describe the photon statistics of the input and output fields for the
scenario described above. We first measure the single-photon purity of the
input field by using our active path-controlled setup and choosing to directly
collect the resonance fluorescence at $\pi$-pulse excitation without sending
it back to the artificial atom. As shown in Fig. 2, a time-gated Hanbury Brown
and Twiss second-order autocorrelation measurement yields the value
$g^{(2)}_{\text{in}}(0)\,{=}\,2.540(2)\%$, and an HOM experiment on
consecutive emissions yields an indistinguishability value [43] of
$\mathcal{I}_{\text{in}}\,{=}\,91.90(5)\%$. These measurements demonstrate the
high fidelity of the single photon used as an input in the quantum light-
matter interface. We note that the multi-photon component in this state is
caused by re-excitation of the QD driven by ${\sim}25$ ps long pulses—shorter
pump pulses result in higher purities (see Supplementary Information).
However, we deliberately choose this pulse length to study scenarios with
stronger varying photon statistics along Rabi oscillations, which we describe
below. We now perform the same measurements on the output field resulting from
the atom-light interaction and find a higher value of
$g^{(2)}_{\text{out}}(0)\,{=}\,6.32(4)\%$, as well as a decrease in the output
field indistinguishability $\mathcal{I}_{\text{out}}\,{=}\,75.0(4)\%$.

Figure 3: Photon statistics. a, Characteristic Rabi oscillations of the non-
classical input field for resonant laser excitation are shown in blue, and the
output intensity resulting from non-classical excitation is plotted in red. b,
The zero-delay values of the second-order autocorrelation for the input (blue
dots) are measured for increasing pulse areas of the driving laser field, such
that the photon statistic of the input field is tuned. The non-linear
interaction between the input field and the QD causes a change in the measured
output photon statistics (red dots). The solid lines are theoretical
simulations of these measurements. Error bars are estimated following
Poissonian counting statistics and are not visible for the results of the
input field. c, The output field contains overall increased multiphoton
components compared to the input field, and the difference $\Delta
g^{(2)}(0)\,{=}\,g^{(2)}_{\text{out}}(0){-}g^{(2)}_{\text{in}}(0)$ is positive
except for points around $2\pi$-pulse excitation that generates the input.

The lower single-photon purity in the output field results from single-photon
interference and effective photon-photon interactions, leading to a higher
ratio of collected photon flux for multi-photon states [8, 35]. Our results
experimentally verify these effects for a non-classical input state. Our
findings presented in Fig. 2b are evidence of the coherence of this process,
showing for the first time that the output field maintains
indistinguishability, an important conclusion for network applications.

The origin of multi-photon components is fundamentally different for the input
and the output field. For the input field, these components are created from
re-excitation. That is, with a small probability, part of the driving
classical pulse area produces a first photon, leaving most of the pulse area
available to re-populate the excited state. Therefore, a second photon is
produced with the remaining pulse area, with this photon contributing
primarily to the high input single-photon purity. Note that in this scenario
the two photons occupy orthogonal time-modes along the input temporal profile.
However, the multi-photon components of the output field can exhibit a
different temporal structure depending on the interaction with the emitter.

This can be tested by time-gated photon statistics measurements and
calculating the resulting upper bound for two-photon contribution. For
simplicity, we consider multi-photon components up to two-photon terms since
they are dominant compared to higher multi-photon contributions. Figure 2c
(top) displays the upper-bound of the two-photon component of the input field
for increasing gate start times. The main feature of this analysis is the
rapid decrease in the amount of the two-photon component soon after gating
begins. That is, by neglecting the photon statistics within the signal rise-
time, the two-photon component of the remaining field is largely suppressed;
thus, the remaining state is much closer to one (and only one) photon. The
same gated measurement on the output field leads to a different conclusion, as
shown in Fig. 2c (bottom). Here, we do not observe a sharp decrease in the
relative two-photon component. In contrast, the amount of two-photon terms,
decreases more slowly, which is consistent with the decrease in intensity of
the entire wave packet. Thus, we observe two-photon components in the output
field at times when they were absent in the input field. In the input field, a
second photon is created only during the duration of the laser field due to
re-excitation. However, in the output field, two photons can be observed due
to temporal restructuring caused by the scattering dynamics from the emitter,
which we will investigate further in the following section.

### Effective photon-photon interaction

We now perform a scan of the classical pulse area responsible for generating
the input non-classical light. The resulting Rabi oscillations are shown in
Fig. 3a, together with the intensity of the output field stemming from the
respective atom-light interaction. We note an increase in the output intensity
for higher multi-photon contributions along the Rabi oscillations. This is
consistent with theoretical predictions [8, 44, 45], where higher photon
numbers lead to higher photon flux in the output field.

Figure 4: Evidence of effective photon-photon interactions. a, Temporal
histograms of the input signal for driving pulse areas around $2\pi$-pulse
laser excitation showing signatures of two-photon emission in separate
temporal modes. b, The resulting output signal has an increase in counts for
later time bins. c, Measured and theoretical time-resolved intensity
correlation maps of the input signal for $2\pi$-pulse laser excitation
evidence that two photons within the input field do not occur simultaneously.
The temporal axes of the measured time-resolved intensity correlation maps
have an experimental resolution of $50~{}\text{ps}$. d, The corresponding
measured and theoretical maps of the output signal, revealing simultaneous
correlations along the diagonal ($t_{1}\,{=}\,t_{2}$) as a signature of
stimulated emission. This is an observation of generating correlated two-
photon states from two separate single-photon states.

Moreover, second-order autocorrelation measurements along the Rabi
oscillations, as shown in Fig. 3b, exhibit an intriguing behavior: the output
$g_{\text{out}}^{(2)}(0)$ values are higher than the input
$g_{\text{in}}^{(2)}(0)$ values, except around $2{\pi}$-pulse excitation,
where the two-photon component of the input non-classical field is higher [28,
41]. Indeed, Fig. 3c displays the difference between these measurements,
clearly showing the change in the sign near the excitation using
$2{\pi}$-pulses. This is a consistent behavior that we observe also for other
values of the excitation pulse length that generate the input non-classical
field (see Supplementary Information). The observed feature is a result of the
non-linear interaction between the emitter and the input field. The incoming
temporally separated two-photon components are partially changed to one-photon
components in the detected output field, leading to lower
$g_{\text{out}}^{(2)}(0)$ values. We note that linear transformations, such as
unitary evolutions or optical losses, are not able to modify these photon
statistics within the light field. Therefore, these measurements demonstrate
the non-linear nature of the light-matter interface presented in this work.

A particularly interesting scenario occurs with the input state generated near
a $2\pi$-pulse excitation, where we can study instances involving two photons
arriving at the artificial atom. Now, the input field consists of one photon
that appears first and that is temporally shorter and a second photon with a
longer time-scale that follows immediately after, as shown in Fig. 4a. The
output field displays a similar two-photon structure except that the second
peak is relatively stronger in comparison to the case of the input field (see
Fig. 4b).

Time-resolved intensity correlation maps $G^{(2)}(t_{1},t_{2})$, as shown in
Figs. 4c,d provide insights into the processes occurring for this case of two
incident photons. For the input field, one finds two-photon correlations along
the horizontal and vertical axes of the map, which originate from temporally-
orthogonal events—one photon first, and a second photon after. However, the
output field also presents two-photon correlations along the diagonal, which
are events of detecting two photons simultaneously. This correlated two-photon
wave packet shows a measured decay time of $\tau^{\prime}\,{=}\,108(6)$ ps,
extracted from the exponential fit of the decay for $t_{1}\,{=}\,t_{2}$. This
corresponds to approximately half the spontaneous emission decay time
$\tau\,{=}\,227$ ps of the artificial atom (see Fig. 1b), a scaling factor
that we also predict theoretically (see Fig. 4d). This significant observation
explains the underlying mechanism: within all the complex quantum non-linear
processes at play, sometimes one photon arrives at the artificial atom and
excites it, and a subsequent photon arriving at the excited atom leads to the
generation of two photons sharing the same temporal mode. These observations
indicate stimulated emission processes [46] between only two quanta of light.
This represents a fundamental form of mediated photon-photon interaction in a
quantized field.

### Discussion

In this work, we reported the first observation of quantum light interacting
with an atomic system in a solid-state platform. We started by employing a QD-
cavity device that produces a high-fidelity single photon with high fiber-
efficiency. Indeed, device efficiency has been key in allowing the observation
of subtle changes in photon statistics within such processes, a task
previously out of experimental reach. We then used a setup with fast and
active path control that allows to send resonance fluorescence back to the
quantum emitter and later observe and analyze the results of their
interaction.

We studied different aspects of this process. First, we observed an artificial
atom being resonantly excited by one photon. Here, we demonstrated the
coherent nature of this process by observing single-photon interference
mediated by an atom. An interesting aspect of this scenario is the observation
that one photon is sufficient to excite the artificial atom, followed by the
re-emission of a later photon. Moreover, we performed HOM experiments and
showed that the output field produced at consecutive interaction instances
remains indistinguishable. We also observed non-linear aspects of the light-
matter interface acting on non-classical wave packets with one and two
photons. We find that photon statistics change as a result of the interaction,
with the output field containing more relative two-photon terms, which
represents the first observation of photon-number dependent sorting occurring
on non-classical input light.

Time-resolved intensity correlation maps also revealed a prime non-linear
aspect of a quantum light-matter interface: stimulated emission. We observe
that the output field contains simultaneous two-photon correlations, which are
not present in the input non-classical field. We argue that these events occur
when one resonant photon excites the atom, and a subsequent resonant photon
stimulates the emission of a photon in the same spatio-temporal mode. The
correlated output mode is a result of effective photon-photon interactions
introduced in the highly non-linear medium.

We believe that our observations open a novel path for exploring light-matter
interactions in a fully quantized system and will encourage a number of
follow-up studies. For instance, one can envision performing this protocol in
a sequential manner, where the output field resulting from the process is used
again as input to observe a field resulting from multiple interactions, thus
exploring the limits of the light-matter interface and answer questions such
as up to which point the process can remain coherent or what types of non-
classical light fields can be achieved asymptotically. Moreover, a QD emitting
a temporal stream of photons entangled, e.g., in their polarization [26] or
photon-number [29], could be used in a similar protocol, using either the same
or a different QD, to entangle spatio-temporally multiple quantum emitters
without requiring remote two-photon interference protocols or to create
complex states of multi-partite entangled quantum light-matter systems.

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### Methods

The InAs/GaAs QD investigated is coupled to a GaAs/AlAs distributive Bragg
reflector that form a micropillar. The device is connected to a surrounding
diode structure, allowing for electrical tuning through the Stark effect. For
the QD under study, the trion state is tuned into resonance with the cavity
mode by applying $0.75$ V. The Purcell enhancement results in a detected
radiation lifetime of $\tau\,{=}\,227~{}\text{ps}$ under $\pi$-pulse laser
excitation, see Supplementary Information, with the QD transition emitting
single photons at a wavelength of $922.1$ nm. All the measurements are
performed with the sample inside a closed-cycle cryostat (AttoDry800,
Attocube) at a temperature of 4 K. To focus the excitation onto the sample and
to collimate the output mode, a single aspheric lens with a numerical aperture
of $0.68$ is installed inside the chamber.

The initial excitation of the QD is performed using a pulsed Ti:Sapphire laser
(Tsunami, Spectra Physics) with a pulse repetition rate of $80$ MHz. For pump
preparation, we shape the laser pulse by spectral filtering with a 4$f$
system. The pulse length is set to ${\sim}25$ ps, assuming a transform-limited
Gaussian pulse duration. A cross-polarized configuration is used for
excitation and collection. In the collection path, linearly polarized laser
light is suppressed using a set of wave plates and a Glan-Taylor polarizer
with an extinction coefficient of more than seven orders of magnitude. The
measured single-photon collection efficiency in the single-mode fiber is
${\sim}28.7~{}\%$ [40], including all losses from excitation to detection.
Note that in the case of trion states, which have a circularly polarized
transition, the resonant excitation scheme inherently introduces $50~{}\%$
loss since linearly polarized light is collected.

For the preparation of non-classical excitation of a quantum emitter, the QD
is excited resonantly by the pulsed laser, generating a temporal sequence of
single photons that is then coupled to a single-mode fiber and sent to a
broadband electro-optic modulator (EOM, QUBIG GmbH). At this stage, single
photons, which serve as non-classical input, are deterministically sent back
to the QD. To ensure that only one type of excitation is present at a given
time, the delay of the returning single photons is adjusted accordingly such
that the input and output signal are separated by a delay of ${\sim}3$ ns. The
scattered Fock state then passes through the collection path and is actively
routed by the EOM to a Hanbury Brown and Twiss (HBT) setup consisting of a
balanced fiber beam splitter (BS) with two superconducting nanowire single-
photon detectors (SNSPDs, Single Quantum) connected to a correlation unit
(Time Tagger X, Swabian Instruments). The SNSPDs have a system detection
efficiency of $86\%$, a timing jitter of $30$ ps, and a dark count rate of
${<}\,10$ Hz. All devices are phase-locked to the pulsed laser excitation via
a field programmable gate array (FPGA), which operates the EOM at a 50% duty
cycle at a frequency of $8$ MHz and serves as a reference clock for the
measurement with a repetition rate of $8$ MHz.

For the data analysis, the raw time tags are recorded which allows the
histogram and correlation measurements to be carried out in post-processing.
Thus, the raw time tags can be analyzed in a time-gated manner for each
signal. For the time-resolved evaluation of the multi-photon probability
$P_{m}(t_{\text{start}})$ shown in Fig. 2, the gating is defined by a start
time $t_{\text{start}}$ and stop time $t_{\text{stop}}$ (in reference to the
$8$ MHz clock signal produced by the FPGA), and all events occurring outside
of the gating window $\Delta
t_{\text{gate}}\,{=}\,t_{\text{stop}}\,{-}\,t_{\text{start}}$ are disregarded.
Correlating the gated channels then gives access to a time-gated
$g^{(2)}(0,t_{\text{start}})$ value from which $P_{m}(t_{1})$ is inferred
[47]. Increasing $t_{\text{start}}$ step-wise (while $t_{\text{stop}}$ is
fixed) excludes the early fraction of the temporal profile and reveals a
significant decrease in $P^{(m)}$ when the rising edge is omitted. Notably,
the drop is much sharper for Fig. 2c (top) than for Fig. 2c (bottom).

Our theoretical simulations follow a cascaded master equation approach in
which two identical subsystems with lifetime
$\tau\,{=}\,\Gamma^{-1}\,{=}\,227$ ps are considered. The first system is
driven by a Gaussian pulse with a FWHM of $\tau_{p}=25$ ps, leading to the
output photon flux
$\propto\Gamma\langle\sigma^{1}_{+}\sigma_{-}^{1}\rangle(t)$, shown in Fig.
1b. Here, $\sigma_{\pm}^{1}$ and $\sigma_{\pm}^{2}$ are the usual atomic
ladder operators for the first and second system, respectively. The second
system’s output flux $\langle b^{\dagger}_{2}b_{2}\rangle(t)$, also shown in
Fig. 1b, results from the output field operator $b_{2}(t)$, which is a
weighted sum of the input field operator
$\propto\sqrt{\Gamma}\sigma_{-}^{1}(t)$ and a system’s response operator
$\propto\sqrt{\Gamma}\sigma_{-}^{2}(t)$. To account for experimental
imperfections, two different dephasing mechanisms were considered, and to
account for the detector time resolution, a Gaussian detector jitter with a
FWHM of $60$ ps was considered.

### Acknowledgments

The authors thank Michael Trupke, Peter Rabl, Yuri Minoguchi and Juan José
García-Ripoll for helpful discussions, and Virginia Oddi and Simone Piacentini
for help in initial steps of the project. This research was funded in whole,
or in part, from the European Union’s Horizon 2020 and Horizon Europe research
and innovation programme under grant agreement No 899368 (EPIQUS), No. 899354
(SuperQuLAN), and No 101135288 (EPIQUE), the Marie Sklodowska-Curie grant
agreement No 956071 (AppQInfo), and the QuantERA II Programme under grant
agreement No 101017733 (PhoMemtor); from the Austrian Science Fund (FWF)
through [10.55776/F71 $]$ (BeyondC), [10.55776/FG5 $]$ (Research Group 5), and
[10.55776/COE1 $]$ (Quantum Science Austria); by the Polish National Science
Center (NCN) “Sonatina” project CARAMEL with number 2021/40/C/ST2/00155. The
financial support by the Austrian Federal Ministry for Digital and Economic
Affairs, the National Foundation for Research, Technology and Development and
the Christian Doppler Research Association is gratefully acknowledged.

## Supplementary Information

###  .1 Experimental setup

The experimental setup for non-classical excitation of a QD is presented in
Fig.S1. The setup includes a cross-polarization excitation scheme, an active
stage for sending single photons back to the emitter, and an analysis setup.
The QD is first excited using resonant laser pulses. The sequence of single-
photon time bins is then collected in a single-mode fiber, and the
polarization state is set to horizontal polarization using fiber polarization
control.

Figure S1: Experimental setup. A pulsed laser is used to excite the QD
resonantly in a cross-polarized configuration. In the collection path, the
laser is suppressed, and single photons are coupled into the collection fiber.
In the following stage, an EOM is used to control the sequence in which single
photons are sent back to the QD and light after the interaction with the QD is
collected for analysis in a HBT setup.

During the first cycle, the EOM is in the off-state, which preserves the
polarization state of the single photons such that light is transmitted
through the PBS, is reflected by a mirror, and is coupled in reverse into the
collection fiber. Thus, during the off-phase of the EOM, single-photon time
bins return to the emitter. The scattered output signal after interaction with
the QD is coupled to the collection fiber in the forward direction and sent to
the EOM.

Following this, the EOM is set to the on-state, switching the polarization
state of the light to vertical polarization. The light reflects at the PBS and
is collected in the output fiber and sent to an HBT setup. This process allows
simultaneous observation of the initial single-photon stream and the output
signal, ensuring identical conditions for both excitation schemes. By
recording the raw time-tag stream of each detector and the reference clock,
both signals can be analyzed independently by time gating each signal in post
processing.

###  .2 Data set for different excitation parameters

For the measurements presented in the main text, we chose a laser pulse
duration of ${\sim}25$ ps such a duration enhances the two-photon probability
at the beginning of the lifetime because of reexcitation during the duration
of the laser pulse [41]. An additional set of measurements for a shorter laser
pump pulse is given in Fig. S2. For the additional data set, two-photon
contributions are suppressed by setting the laser pulse length of ${\sim}18$
ps, that enhances single-photon purity while maintaining a high count rate.

Figure S2: Additional data set. a, The time histograms shows single-photon
emission and Fock state scattering. b, Normalized second-order auto-
correlation function of the input field with a zero delay value of
$g_{\text{in}}^{(2)}(0)=1.73(2)\%$ close to $\pi$-pulse excitation. c, than
the normalized second-order auto-correlation function of the output field with
a zero delay value of $g_{\text{out}}^{(2)}(0)=3.4(2)\%$. d, Measured Rabi
oscillations for the input signal (blue) and the measured output count rate
(red), that shows an increased transmitted photon flux for higher multi-photon
contributions. e, Input photon statistic for increasing pulse area of the
classical driving field and resulting photon statistic of the output field. f,
Difference $\Delta
g^{(2)}(0)\,{=}\,g^{(2)}_{\text{out}}(0){-}g^{(2)}_{\text{in}}(0)$ in photon
statistics of input and output signal.

The time histograms in Fig. S2a indicate that the dynamics of th e output
signal, as discussed in the main text, are reproducible under similar
conditions. The interaction of the QD with the shorter laser pulse reduces the
probability of two-photon emission during pulse duration as it is shown in the
inset of the correlation measurement in Fig.S2b. Here, the central peak does
not exhibit the characteristic emission profile of re-excitation during the
duration of the laser pulse compared to the measurement presented in the main
text. This results in a higher single-photon purity of the first generation
with a second-order auto-correlation function value of
$g_{\text{in}}^{(2)}(0)\,{=}\,1.73(2)\%$. The correlation measurement of the
second generation is plotted in Fig.S2c. The measurement has a second-order
auto-correlation function value of $g_{\text{out}}^{(2)}(0){=3}.4(2)\%$,
indicating that the single-photon purity of the second generation is also
improved as compared to the measurement presented in the main text, while
remaining a lower single-photon purity relative to the first generation.

Further, we investigate Hong–Ou–Mandel (HOM) interference of each signal. The
measurement results are presented in Fig. S3. The raw HOM visibility of the
initial single-photon signal generated by laser excitation is
$V_{\text{in}}\,{=}\,90.0(2)\%$ (see Fig. S3a), yielding an
indistinguishability value [43] of $\mathcal{I}_{\text{in}}\,{=}\,93.3(2)\%$.
The raw HOM visibility of the signal after interaction with the QD is
$V_{\text{out}}\,{=}\,70.4(5)\%$ (see Fig. S3b), resulting in an
indistinguishability value of $\mathcal{I}_{\text{out}}\,{=}\,76.4(6)\%$. The
reported values do not account for unbalanced splitting ratios in the linear
network.

Figure S3: HOM interference. a, HOM measurement at $\pi$-pulse laser
excitation. b, HOM measurement for non-classical excitation.

The indistinguishability value is defined as
$\mathcal{I}\,{=}\,\frac{V+g^{(2)}(0)}{1-g^{(2)}(0)}$ [43]. The raw HOM
visibility is calculated as $V\,{=}\,1-A_{0}/A_{1}$, where $A_{0}$ is the
integrated area of the central peak and $A_{1}$ is the mean integrated area of
the peaks at a time delay of $12.5$ ns. The measurement uncertainties are
estimated following Poissonian counting statistics.

###  .3 Time histograms for classical and non-classical excitation

The time-resolved fluorescence measurements of the QD transition (trion) are
displayed in Fig. S4a. The data represent the detected raw measurements with a
resolution of $10$ ps (binwidth) and without corrections for a detector jitter
of $30$ ps. Figure S4b shows the detected time-resolved dynamics of the
interaction between a single-photon wave packet and the emitter. Note that the
dip and the attributed decay of the first peak originate from single-photon
interference and absorption, while the second decay indicates the decay rate
of the emission after the excitation of the emitter by a single photon [8].

Figure S4: Time histograms for single-photon emission and single-photon
scattering. a, The plot in a semilogarithmic scale shows the time-resolved
fluorescence measurement of the QD state tuned into resonance with the cavity
for $\pi$-pulse laser excitation. The system has a lifetime of $227$ ps, as
evaluated by fitting a monoexponential decay to the experimental data. b, The
plot shows the time-resolved histogram measurement of the scatter signal. The
fits reveal a monoexponential decay of 90 ps for the first peak and $422$ ps
for the second peak.

###  .4 Theoretical model

#### Cavity QED system coupled to a waveguide

We model the system as a two-level system (TLS) interacting with two
orthogonally polarized cavity modes (H and V) via the usual Jaynes-Cummings
interaction, whose Hamiltonian reads ($\hbar=1$) [48, 49]

$H_{\rm JC}=\frac{\omega_{0}}{2}\sigma_{z}+\omega_{\rm H}a^{\dagger}_{\rm
H}a_{\rm H}+\omega_{\rm V}a^{\dagger}_{\rm V}a_{\rm V}+ig_{\rm
H}(\sigma_{+}a_{\rm H}-\sigma_{-}a_{\rm H}^{\dagger})+g_{\rm
V}(\sigma_{+}a_{\rm V}+\sigma_{-}a_{\rm V}^{\dagger}).$ (1)

Here, $\omega_{0}$ and $\sigma_{z,+,-}$ are the frequency and Pauli operators
characterizing the TLS, $\omega_{\rm H,V}$ and $a^{\dagger}_{\rm H,V}$
($a_{\rm H,V}$) are the frequencies and creation (annihilation) operators
characterizing the cavity modes, and $g_{\rm H,V}$ are the light-matter
coupling strengths. The one-sided cavity is coupled to the continuum with a
total photon decay rate $\kappa$. In the “bad cavity” limit, $\kappa\gg g_{\rm
H,V}$, the cavity modes can be adiabatically eliminated, and the dynamics of
TLS is then governed by the Lindblad master equation

$\dot{\rho}=\mathcal{L}_{0}\rho=-i\delta_{L}[\sigma_{z},\rho]/2+\Gamma
D[\sigma_{-}]\rho+\gamma_{d}D[\sigma_{z}]\rho,$ (2)

where $\rho(t)$ is the density matrix, $\delta_{L}=\omega_{0}-\omega_{L}$ is
the detuning determined by the frequency of the rotating frame $\omega_{L}$,
and $D[X]\rho=X\rho X^{\dagger}-\frac{1}{2}(X^{\dagger}X\rho+\rho
X^{\dagger}X)$ is the Lindblad super-operator describing relaxation
($\sigma_{-}$) or dephasing ($\sigma_{z}$). On the one hand, the total decay
rate of the TLS $\Gamma=\Gamma_{\rm H}+\Gamma_{\rm V}+\gamma^{*}$ accounts for
losses via the cavity modes ($\Gamma_{\rm H,V}=4g_{\rm
H,V}^{2}/[\kappa(1+4\delta_{\rm H,V}^{2}/\kappa^{2})]$ with $\delta_{\rm
H,V}=\omega_{0}-\omega_{\rm H,V}$) or via other modes in the system
($\gamma^{*}\ll\Gamma$). On the other hand, $\gamma_{d}\ll\Gamma$ accounts for
a phenomenological dephasing rate.

For the sake of simplicity, in the following we consider $g_{\rm H}=g_{\rm
V}$, $\delta_{\rm H}=\delta_{\rm V}=0$ and $\gamma^{*}=0$. As a result,
$\Gamma_{H}=\Gamma_{V}=\Gamma/2$. Furthermore, we assume our cavity to be one
sided. Then, the following input-output relations hold

$b^{\rm H,V}_{out}=b_{in}^{\rm H,V}+\sqrt{\Gamma_{\rm
H,V}}\,\sigma_{-}=b_{in}^{\rm H,V}+\sqrt{\Gamma/2}\,\sigma_{-},$ (3)

where $b^{\rm H,V}_{in}$ and $b^{\rm H,V}_{out}$ are the input and output
field operators, respectively. Note that in the absence of a horizontally-
polarized input field, $\langle b_{in,\rm H}\rangle=0$, the measured output
field flux $N(t)=\langle b_{out,\rm H}^{\dagger}(t)b_{out,\rm H}(t)\rangle$ is
proportional to the probability that the TLS is in the excited state
$P_{e}(t)=\frac{1}{2}(1+\langle\sigma_{z}(t)\rangle)$, which can be calculated
via the density matrix $\rho(t)$ as $\langle X\rangle={\rm Tr}(\rho(t)X)$.

#### Laser driving and first-generation photon

In our system, one of the cavity modes (V) is used to drive the TLS, while the
other one (H) is used for measurement. We model the driving laser as a
Gaussian pulse leading to an input field [33, 41]

$\langle b^{\rm
V}_{in}\rangle(t)=\sqrt{n_{L}}\bigg{(}\frac{4\ln(2)}{\pi\tau_{p}^{2}}\bigg{)}^{1/4}\exp\Big{(}-\frac{2\ln(2)(t-t_{c})^{2}}{\tau_{p}^{2}}\Big{)}$
(4)

where $\tau_{p}$ is the pulse width, $t_{c}$ is the instant characterizing its
central point, and $n_{L}$ corresponds to the number of photons in the pulse,
proportional to the laser intensity. This translates into a driving term in
the effective TLS Hamiltonian, which in a frame rotating with the laser
frequency $\omega_{L}$ reads

$H(t)=\frac{\Omega(t)}{2}(\sigma_{+}+\sigma_{-}),$ (5)

where $\Omega(t)=4g_{\rm V}\langle b^{\rm V}_{in,+}\rangle(t)/\sqrt{\kappa}$
is the Rabi frequency. It is convenient to express the Rabi frequency
$\Omega(t)$ in terms of the pulse area $A$ as

$\Omega(t)=A\bigg{(}\frac{2\ln(2)}{\pi\tau_{p}^{2}}\bigg{)}^{1/2}\exp{(-2\ln(2)(t-t_{c})^{2}/\tau_{p}^{2})},$
(6)

ensuring $\int_{0}^{\infty}dt^{\prime}\Omega(t^{\prime})=A$ and removing
dependance on parameters $\kappa$, $g_{\rm V}$ or $n_{L}$. Furthermore, we
include a phonon-induced dephasing term as in Ref. [41]. Then, the Lindblad
master equation for the first generation photon looks

$\dot{\rho}=\mathcal{L}_{\Omega}\rho=-i[H(t),\rho]+\Gamma
D[\sigma_{-}]\rho+\gamma_{d}D[\sigma_{z}]\rho+B\Omega^{2}(t)D[\sigma_{z}]\rho,$
(7)

while the measured output flux is $N_{1}(t)\propto\langle b_{out,\rm
H}^{\dagger}(t)b_{out,\rm H}(t)\rangle=\Gamma P_{e}(t)/2$.

#### Single-photon driving and second-generation photon

In the experiment, the horizontally polarized light that comes out,
$b_{out,1}^{\rm H}$, is reflected back to the cavity, and becomes the input
field $b_{in,2}^{\rm H}$ at a later time, i.e. $b_{in,2}^{\rm
H}=\sqrt{\eta_{\rm loss}}\,b_{out,1}^{\rm H}$. Here, $\eta_{\rm loss}$
accounts for the fraction of photons that survive after traveling through the
optical setup back into the cavity. As the light entering the cavity is not
coherent, we cannot represent it as a Hamiltonian term. Instead, we make use
of a cascaded master equation approach [50], where the output of a first
system, driven by a laser pulse and with Pauli operators
$\sigma_{\alpha}^{(1)}$ (with $\alpha{=}\\{+,-,z\\}$), is used as the input
for a second system, with Pauli operators $\sigma_{\alpha}^{(2)}$. The full
master equation then reads

$\dot{\rho}=(\mathcal{L}^{(1)}_{\Omega}+\mathcal{L}^{(2)}_{0})\rho-\frac{\sqrt{\eta_{\rm
loss}}}{2}\Gamma\big{(}[\sigma_{+}^{(2)},\sigma_{-}^{(1)}\rho]+[\rho\sigma_{+}^{(1)},\sigma_{-}^{(2)}]\big{)}.$
(8)

The horizontally polarized output field operator for the second system now
reads

$b^{\rm H}_{out,2}=b_{in,2}^{\rm H}+\sqrt{\Gamma/2}\,\sigma_{-}^{(2)}.$ (9)

As in reality our cavity is not perfectly one-sided, only a fraction of the
light received by the atom will come out from the out mode $b^{\rm
H}_{out,2}$. We model this by assuming a percentage $\eta_{\rm re}$ of the
input light gets reflected right before entering the cavity. Then this
fraction destructively interferes with the output light field $b^{\rm
H}_{out,2}$. The modified measured field operator then reads

$b^{\rm H}_{out,2}=\sqrt{\Gamma/2}[(\sqrt{\eta_{\rm loss}}-\sqrt{\eta_{\rm
re}})\sigma_{-}^{(1)}+\sigma_{-}^{(2)}]$ (10)

where $\eta_{\rm loss}=(1-\eta_{\rm re})\eta^{\prime}_{\rm loss}$.

#### Two-photon correlation functions

The measured two-time correlation function $G^{(2)}(t_{1},t_{2})=\langle
b^{\dagger}_{out}(t_{1})b^{\dagger}_{out}(t_{2})b_{out}(t_{2})b_{out}(t_{1})\rangle$
is equivalent to $G^{(2)}(t,t+\tau)$, where $\tau=|t_{2}-t_{1}|$, thus, we
define $\overline{g^{(2)}(0)}$ as

$\overline{g^{(2)}(0)}=\frac{2\int_{0}^{\infty}dt\int_{0}^{\infty}d\tau
G^{(2)}(t,t+\tau)}{|\int_{0}^{\infty}dt\langle
b^{\dagger}_{out}(t)b_{out}(t)\rangle|^{2}},$ (11)

Under the previously considered approximations, the two-time correlation
function for the first generation signal is

$\displaystyle G^{(2)}(t,t+\tau)=\frac{\Gamma^{2}}{4}{\rm
Tr}(\sigma_{+}^{(1)}\sigma_{-}^{(1)}\,\mathcal{U}^{(\tau)}[\sigma_{-}^{(1)}\mathcal{U}^{(t)}[\rho_{0}]\sigma_{+}^{(1)}]),$
(12)

while, for the second generation, this is

$\displaystyle G^{(2)}(t,t+\tau)=\frac{\Gamma^{2}}{4}{\rm
Tr}(S_{+}S_{-}\,\mathcal{U}^{(\tau)}[S_{-}\mathcal{U}^{(t)}[\rho_{0}]S_{+}]),$
(13)

where $S_{\pm}=(\sqrt{\eta_{\rm loss}}-\sqrt{\eta_{\rm
re}})\sigma_{\pm}^{(1)}+\sigma_{\pm}^{(2)}$. Here, $\rho_{0}$ is the initial
state $\rho_{0}=|{\rm g}\rangle\langle{\rm g}|_{1}\otimes|{\rm
g}\rangle\langle{\rm g}|_{2}$, and $\mathcal{U}^{(\tau)}$ is the time-
evolution super-operator associated to the cascaded master evolution in Eq.
(8).

#### Numerical simulation and relevant parameters

The dynamics of our system is described by the cascaded master equation
introduced in Eq. (8). The free parameters are selected so as to maximise
agreement with the experimental results. In particular, the value of the decay
rate $\Gamma$ is chosen $\Gamma^{-1}=227$ ps, and the dephasing rates
$\gamma_{d}=0.035\Gamma$ and $B=1.3\times 10^{-4}\Gamma^{-1}$. Also, the
reflection parameter $\eta_{\rm re}$ is chosen to be $\eta_{\rm re}=0.05$,
while we find the studied observables independent of the loss parameter
$\eta^{\prime}_{\rm loss}$. Furthermore, the pulse parameters are chosen to be
$\tau_{p}=25$ ps and $t_{c}=200$ ps. Finally, all time-dependent observables,
e.g. $N(t)$, are assumed to have a finite time resolution limited by the
detector’s jitter, which is assume to be Gaussian with a full width at half
maximum of $60$ ps.