# Experimental analysis of energy transfers between a quantum emitter and
light fields

I. Maillette de Buy Wenniger1, S. E. Thomas1, M. Maffei2, S. C. Wein2,3,
M. Pont1, N. Belabas1, S. Prasad2, A. Harouri1, A. Lemaître1, I. Sagnes1, N.
Somaschi4,
A. Auffèves∗5,6, P. Senellart∗ Centre for Nanosciences and Nanotechnology,
CNRS, Université Paris-Saclay, UMR 9001, 10 Boulevard Thomas Gobert, 91120
Palaiseau, France
2Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble,
France
3Institute for Quantum Science and Technology and Department of Physics and
Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4
4Quandela SAS, 10 Boulevard Thomas Gobert, 91120 Palaiseau, France
5 MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Laboratory
6Centre for Quantum Technologies, National University of Singapore, 117543
Singapore, Singapore

###### Abstract

Energy transfer between quantum systems can either be achieved through an
effective unitary interaction or through the generation of entanglement. This
observation defines two types of energy exchange: unitary and correlation
energy. Here we propose and implement experimental protocols to access these
energy transfers in interactions between a quantum emitter and light fields.
Upon spontaneous emission, we measure the unitary energy transfer from the
emitter to the optical field and show that it never exceeds half of the total
energy and is reduced when introducing decoherence. We then study the
interference of the emitted field and a laser field at a beam splitter and
show that the energy transfers quantitatively depend on the quantum purity of
the emitted field.

The study of energy exchanges between closed, isolated quantum systems is
highly relevant to quantum technologies, be it to study quantum batteries
Quantum_battery ; Quantacell ; Polini2019  or the fundamental energetic cost
of quantum information processing Stevens2022 ; GB ; Ozawa . These studies are
important to understand the energetic costs not only at the fundamental level,
but also at the macroscopic level Alexia . In this context, understanding and
probing the nature of energy exchanges is essential. In the canonical case of
a bipartite system, two different types of energy transfers can be identified.
We consider two quantum systems $A$ and $B$ with the energy initially in $A$
and subsequently transferred to $B$ such that at the end of the interaction we
have $-\Delta{\cal E}_{A}=\Delta{\cal E}_{B}$. After the interaction, we can
define the unitary energy ${E}^{A,B}_{\mathrm{unit}}$ as the energy
transferred from $A$ to $B$ through an effective unitary interaction from the
point of view of $B$. This unitary energy is limited by quantum correlations
that build up during the interaction between the two systems. The remaining
energy, dubbed correlation energy ${E}^{A,B}_{\mathrm{corr}}=\Delta{\cal
E}_{B}-{E}^{A,B}_{\mathrm{unit}}$, signals the presence of correlations or
entanglement during the interaction. In the limit where $A$ becomes classical,
correlations disappear and $B$ only receives unitary energy. From a quantum
thermodynamics perspective, this corresponds to the unambiguous case where a
closed quantum system $B$ is classically driven, such that it only receives
work Binder2018 . As such, unitary energy is naturally associated with a
notion of work for closed quantum systems Alipour2016 ; Hossein-Nejad2015 ;
Weimer2008 ; Schroder2018 .

Theoretically, this framework has been applied to the the case of a qubit
coupled to a reservoir of electromagnetic modes Ciccarello2021 ; fan2010 ;
landi2019 , a situation relevant for quantum light generation and qubit
manipulation Andolina2018 ; Ferraro2018 ; Monsel2020 ; Maffei2021 . As an
example, when a coherent and intense classical field drives a qubit, both
systems remain non-entangled and only exchange unitary energy: the qubit
purity is unaltered, an ideal regime to implement single qubit gates Mikko2017
; Monsel2020 . The case of spontaneous emission, where a quantum emitter
releases its energy into a mode of the electromagnetic field has theoretically
been studied. It was predicted that the fraction of unitary energy transferred
is proportional to the initial quantum coherence between the ground and
excited state of the qubit and represents at most 50 % of the total energy
transferred Monsel2020 .

Figure 1:  Measuring energetic transfers. (a) Sketch of the first scenario
studied: energetic transfers in spontaneous emission, from a quantum emitter
“q” to the vacuum of the electromagnetic field “f”. (b) The second scenario
studied: energetic transfers from the spontaneously emitted field to a
classical field “c” at a beam splitter. (c) Experimental implementation: a
resonant pulsed laser excites the QD-cavity device. The emitted photonic field
is filtered from the laser drive in cross-polarization at a polarizing beam
splitter (PBS). In configuration (1) with an additional flip mirror (FM), the
photonic field path is split into two paths at a beam splitter (BS1) and one
arm is delayed such that two subsequently emitted photonic fields interfere at
a 50:50 beam splitter (BS2). In configuration (2) the classical field is
derived from the same drive laser, reflected off a mirror (M) in the same
cryostation as the QD device “q” to minimize the impact of mechanical
vibrations and shaped using a Fabry-Pérot etalon (FP). Removing the flip
mirror, the quantum and classical fields interfere at BS2. For both steps, the
output intensities are recorded using two superconducting nanowire single-
photon detectors D1 and D2.

Experimentally exploring these situations is very challenging: it not only
requires the ability to accurately access correlated (i.e. respecting global
energy conservation) energy changes within two coupled quantum systems, but
also to trace back how these systems exchanged energy. So far experiments have
focused on energy changes experienced by only one (open) quantum system, e.g.
polarized photons cimini2020 , ions VonLindenfels2019 , spins in MNR
Peterson2019 , or on average energy exchanges between two systems, like a
superconducting qubit and a field Cottet2017 ; Stevens2022 . Here, we take a
leap forward, by proposing and implementing experimental protocols to access
the unitary and correlation energies exchanged in a closed bipartite system in
two scenarios (Fig. 1(a,b)): the spontaneous emission of a quantum emitter,
and interference between the emitted field and a classical field. The emitter
is a single semiconductor quantum dot inserted in an optical microcavity. We
experimentally access the fraction of the unitary energy emitted during
spontaneous emission by performing self-homodyne measurements with the emitted
light field. At low temperature, we measure a fraction very close to the
$50\,\%$ theoretical limit. By increasing the emitter temperature, we study
the impact of pure dephasing and observe a reduction in the fraction of
unitary energy transfer evidencing loss of coherence induced by the
environment of the emitter. In the second scenario, we probe energetic
exchanges between the spontaneously emitted field and a coherent field by
interfering them on a beam splitter. The energy transferred is shown to be
limited by the purity of the quantum field and the relative coherence of both
fields.

The energy exchanges between a qubit and a field during spontaneous emission
are theoretically described in Ref. Monsel2020 . A qubit (q) resonantly
excited by a laser drive is brought into the pure quantum superposition state
$\ket{\Psi_{\mathrm{q}}}=\cos(\theta/2)\ket{g}+\sin(\theta/2)\mathrm{e}^{i\alpha}\ket{e}$,
with $\ket{g}$ and $\ket{e}$ ground and excited states separated by an energy
$\hbar\omega_{0}$, and $\theta,\alpha\in[0,\pi]$ are the pulse area and the
classical phase of the driving laser, respectively. The total initial energy
brought to the emitter is
$\mathcal{E}_{\mathrm{q}}=\hbar\omega_{0}\sin^{2}(\theta/2)$. At the end of
the spontaneous emission process, this energy is entirely transferred to the
emitted field (f),
$\Delta\mathcal{E}_{\mathrm{q,f}}=\mathcal{E}_{\mathrm{q}}$. In the absence of
decoherence, the emitted field state is pure and reads
$\ket{\Psi_{\mathrm{f}}}=\cos(\theta/2)\ket{0}+\sin(\theta/2)\mathrm{e}^{i\alpha}\ket{1}$,
where $\ket{0}$ and $\ket{1}$ are the photon-number states in the mode of the
electromagnetic field with populations $p_{0}=\cos^{2}(\theta/2)$ and
$p_{1}=\sin^{2}(\theta/2)$, see Fig. 1(a). The unitary energy provided by the
qubit to the field corresponds to the coherent part of the emitted field
energy Monsel2020 ; Maffei2021 ; Cottet2017  and reads
$E^{\mathrm{q,f}}_{\mathrm{unit}}=\hbar\omega_{0}s^{2}$, where $s$ is the
coherence of the qubit before spontaneous emission:
$s=\cos(\theta/2)\sin(\theta/2)$. The fraction of unitary energy transferred
is thus maximal at $\theta=\pi/2$ and amounts to $50\,\%$ of the initial
energy. Conversely, the correlation energy corresponds to the incoherent
component of the emitted field and corresponds to
$E^{\mathrm{q,f}}_{\mathrm{corr}}=\Delta\mathcal{E}_{\mathrm{q,f}}-E^{\mathrm{q,f}}_{\mathrm{unit}}$.
It is maximal for $\theta=\pi$ where all the energy is transferred in the form
of correlation energy.

We experimentally probe these energy processes with an InGaAs quantum dot (QD)
coupled to a micropillar cavity giesz2016 ; Somaschi2016 . At a temperature of
5 K, these artificial atoms have been shown to be close to the text book atom-
cavity system with negligible influence of the solid-state environment. This
is attested by the generation of single photons with near-unity
indistinguishability with similar devices Somaschi2016  and the unique
demonstration that they can emit light wave packets in a quantum superposition
of 0 and 1 photon Loredo2019 . The QD in a cryostation is resonantly driven by
7 ps laser pulses at 925 nm derived from a Ti:Sapphire laser operating at a 81
MHz repetition rate (see Fig. 1(c)). The emitted photonic field is separated
from the laser drive using a cross-polarization configuration. Black symbols
in Fig. 2(a) correspond to the normalized intensity $\mu_{\text{f}}$ of the
emitted field as a function of the pulse area, $\theta$, of the driving laser.
We observe the onset of Rabi oscillations attesting the coherent control over
the qubit, i.e. the ability to generate arbitrary quantum superpositions of
the qubit ground and excited state. We assume a near-unity occupation of the
qubit excited state at the power $P_{\pi}$ corresponding to the highest
intensity so that the pulse area is $\theta=2\arcsin(\sqrt{P/P_{\pi}})$. This
hypothesis is validated by our measurement that allows to actually measure the
excited state probability as we show in Supplementary Fig. S.1. This
experimental curve then corresponds to the normalized energy of the emitted
field $\Delta{\cal
E}_{\mathrm{q,f}}/(\hbar\omega_{0})={\mu_{\text{f}}}=\sin^{2}(\theta/2)$
(black line).

To experimentally measure the unitary energy transferred during spontaneous
emission, we perform self-homodyne measurements with the emitted field. In the
absence of decoherence, we show in the supplementary information that the
resulting visibility of interference, $v$, directly corresponds to the
fraction of unitary energy – i.e. efficiency of unitary energy transfer –
$v=E^{\mathrm{q,f}}_{\mathrm{unit}}/\Delta\mathcal{E}_{\mathrm{q,f}}=\cos^{2}(\theta/2)$.
As shown in configuration 1 of Fig. 1(c), we temporally overlap two
successively emitted fields separated in time by 12.5 ns at beam splitter BS2.
With two single-photon detectors, we record the single counts that show
anticorrelated intensities as a function of the relative phase between the two
interfering fields Loredo2019 . This allows us to deduce the interference
visibility $v$.

Figure 2: Energy transfer during spontaneous emission. (a) Total energy
transferred from the qubit “q” to the vacuum of the electromagnetic field “f”
$\Delta\mathcal{E}_{\mathrm{q,f}}$ (diamonds), unitary energy
$E^{\mathrm{q,f}}_{\mathrm{unit}}$ (open circles) and correlation energy
$E^{\mathrm{q,f}}_{\mathrm{corr}}$ (filled circles). (b) Measured visibility
$v$ of the photonic field self-homodyne interference as a function of the
pulse area $\theta$. For both plots, blue (red) symbols corresponds to
measurements at 5 K (20 K) respectively. The purple dashed lines correspond to
theoretical expectations without any decoherence. The red and blue lines
correspond to the theoretical expectations considering the introduction of
pure dephasing, and with small corrections accounting for imperfect laser
rejection.

Fig. 2(b) shows the measured visibility of the interference fringes $v$ as a
function of $\theta$ (see Supplementary). The experimentally measured
visibility decreases continuously as $\theta$ increases, as theoretically
expected. Combining this visibility measurement with the measurement of the
total energy given to the field $\Delta\mathcal{E}_{\mathrm{q,f}}$, we deduce
the amount of energy stemming from unitary interaction between the qubit and
the field:
$E^{\mathrm{q,f}}_{\mathrm{unit}}=\Delta\mathcal{E}_{\mathrm{q,f}}\times v$.
Fig. 2(a) shows the experimental normalized values of
$E^{\mathrm{q,f}}_{\mathrm{unit}}$ as a function of $\theta$ (open symbols) as
well as the corresponding correlation energy
$E^{\mathrm{q,f}}_{\mathrm{corr}}=\Delta\mathcal{E}_{\mathrm{q,f}}\times(1-v)$
(filled symbols) and the corresponding theoretical predictions without
decoherence (purple dashed lines). At low $\theta$, our observations at 5 K
are very close to the theoretical predictions in the absence of decoherence
that most of the energy transferred from the qubit to the electromagnetic
field corresponds to unitary energy Monsel2020 . Such behavior can be
understood considering that, in the low excitation regime, the radiated field
stems from the qubit dipole: no light-matter entanglement takes place, and the
emitted field is remarkably close to a coherent field truncated for photon-
number components above 1. Conversely, light-matter entanglement emerges when
qubit population is created for increasing $\theta$. Although at the end of
the spontaneous emission process the qubit and field are in a separable state,
quantum correlations taking place during spontaneous emission reduce the
amount of unitary energy eventually transferred to the electromagnetic field,
reaching the situation where all the energy is transferred in the form of
correlation energy for $\theta=\pi$. In this situation, the emitted field
contains no coherent component and gets completely squeezed. Remarkably, this
is when the qubit acts as a deterministic single-photon source – a key device
for discrete variable optical quantum technologies. The unitary energy
transfer is maximal for $\theta=\pi/2$, with an equipartition of unitary and
correlation energy. This corresponds to a maximal initial coherence in the
qubit state.

We now explore the effect of decoherence by increasing the QD temperature. At
20 K, the Rabi oscillations are mostly unchanged compared to 5 K (see Suppl.)
showing that although the QD is coupled to the phonon bath, it does not
receive energy from it. It is mostly subject to phonon-induced pure dephasing
with the phonon sideband strongly suppressed by the Purcell effect of the
micropillar cavity in which the QD is embedded Grange2017 . The same
theoretical framework still holds (see Suppl.): the unitary energy corresponds
to the coherent part of the emitted light field that is now described by the
density matrix
$\hat{\rho}_{\text{f}}=\cos^{2}(\theta/2)\hat{\rho}_{0}+\sin^{2}(\theta/2)\hat{\rho}_{1}+\cos(\theta/2)\sin(\theta/2)(\hat{\rho}_{01}+\hat{\rho}_{10})$
where the subscripts 0 and 1 stand for the vacuum or one-photon part of the
field, respectively. $M_{\mathrm{s}}=\mathrm{Tr}[\hat{\rho}_{1}^{2}]$ is the
single-photon indistinguishability or purity of the single-photon component in
the temporal domain santori2002 . The reduction of quantum coherence between
the vacuum and the one-photon component is captured by ${\cal
C}=\mathrm{Tr}[\hat{\rho}_{01}\hat{\rho}_{10}]$. The unitary energy provided
by the qubit to the field now reads:
$E^{\mathrm{q,f}}_{\mathrm{unit}}=\hbar\omega_{0}\mathcal{C}\cos^{2}{(\theta/2)}\sin^{2}{(\theta/2)}$.
In the case of pure dephasing, we further show that ${\cal
C}={M_{\mathrm{s}}}$ (see Suppl.).

Figure 3:  Energy exchanges in a quantum interference. (a) The theoretical
total energy, unitary and correlation energy transfers from the photonic field
“f” to the classical field “c” in the ideal case. (b) Measured visibility $v$
as a function of $\theta$ when interfering the photonic field with the
classical field (blue 5 K, red 20 K ). The solid lines are a theoretical fit
to the data using $v=\cos(\theta/2)\mathcal{C}_{\text{f,c}}$, and the shaded
regions represent $2\sigma$ uncertainty in the fit. (c) Measured total energy
transferred from the quantum field to the classical field,
$\Delta\mathcal{E}_{\mathrm{f,c}}$ as a function of $\theta$, the correlation
energy $E^{\mathrm{f,c}}_{\mathrm{corr}}$ deduced from Fig. 2(a), and the
unitary energy transferred $E^{\mathrm{f,c}}_{\mathrm{unit}}$ (blue 5 K, red
20 K). Lines correspond to experimental data fit with deduced parameters
$\mathcal{C}_{\text{f,c}}$ and $\mathcal{C}$ at each temperature.

Fig. 2(b) shows the visibility of the self-homodyne measurement at 20 K
evidencing a unitary energy transfer between the QD and the optical field.
However, the efficiency is reduced compared to 5 K. We fit both data sets and
find $\mathcal{C}(5~{}\mathrm{K})=0.975\pm 0.007$ and
$\mathcal{C}(20~{}\mathrm{K})=0.594\pm 0.007$. This is in reasonable agreement
with the independently measured single-photon indistinguishability
$M_{\mathrm{s}}=(92.6\pm 0.1)\%$ at 5 K, and $M_{\mathrm{s}}=(58.0\pm 1.0)\%$
at 20 K. The unitary energy is still maximum for $\theta=\pi/2$ but its
maximum value has reduced from
$E^{\mathrm{q,f}}_{\mathrm{unit}}/\hbar\omega_{0}=(27.9\pm 1.2)\%$ at 5 K to
$E^{\mathrm{q,f}}_{\mathrm{unit}}/\hbar\omega_{0}=(15.8\pm 0.6)\%$ at 20 K.
These observations attest for the phonon bath-induced loss of coherence of the
quantum emitter.

In the second step of our protocol, we couple the emitted photonic field with
a coherent (classical) field on a 50:50 beam splitter (see Fig. 1(b)). This
standard configuration to obtain effective light-light interaction in optical
quantum technologies leads to the same splitting in the energy transfers
between light modes: unitary versus correlation (see Suppl.). A transfer of
unitary energy corresponds to a change in the coherent energy of the receiving
field. We set the input energies of both fields to be equal to maximize the
interference visibility $v$ i.e. ${\cal E}_{\mathrm{f}}^{\mathrm{in}}={\cal
E}_{\mathrm{c}}^{\mathrm{in}}=\hbar\omega_{0}\sin^{2}(\theta/2)$, and consider
one of the outputs as hosting the resulting field. Contrary to the previous
scenario where the interference visibility gave direct access to the unitary
energy, our calculations show that, in the case of a classical field
interfering with a quantum field, the interference visibility gives now access
to the total energy exchanged (see Suppl.). The energy transferred to the
classical field now reads $\Delta{\cal
E}_{\mathrm{f,c}}=\mathcal{E}^{\mathrm{in}}_{\mathrm{f}}\times v$ with $v=1$
signaling a complete transfer. The visibility is given by
$v=\mathcal{C}_{\mathrm{f,c}}\cos(\theta/2)$ where
$\mathcal{C}_{\mathrm{f,c}}$ quantifies the overall classical and quantum
coherence between both fields. In the case where the QD is initially subject
to pure dephasing, we show that $\mathcal{C}_{\mathrm{f,c}}={M_{\text{f,c}}}$,
where $M_{\text{f,c}}$ is the mean wave packet overlap between the quantum and
the classical field.

We find that the correlation energy is dictated by the quantum field (see
Suppl.): $E^{\mathrm{f,c}}_{\mathrm{corr}}=E^{\mathrm{q,f}}_{\mathrm{corr}}/2$
which is consistent with the fact that the quantum field is the only field
carrying an incoherent component. As a result, we can deduce the unitary
energy component according to
$E^{\mathrm{f,c}}_{\mathrm{unit}}=\mathcal{E}^{\mathrm{in}}_{\mathrm{f}}\times
v-E^{\mathrm{q,f}}_{\mathrm{corr}}/2$, which is upper-bounded by the unitary
energy initially given to the quantum field:
$E^{\mathrm{q,f}}_{\mathrm{unit}}\geq E^{\mathrm{f,c}}_{\mathrm{unit}}$ (see
Suppl.). Fig. 3(a) presents the theory curves for the unitary, correlation and
total energy transfers in the ideal situation of a pure quantum state. Most of
the energy is transferred in the form of unitary energy in the limit where
$\theta\rightarrow 0$, i.e. when the state of the quantum field is the closest
to a coherent field and the energy transfer is complete.
$\Delta\mathcal{E}_{\mathrm{f,c}}$ is maximum for $\pi/2<\theta<\pi$, a
behavior that results from a trade-off between the maximization of the quantum
field coherence reached at $\theta=\pi/2$ while the coherence of the classical
field continuously increases with $\theta$. For high values of $\theta$, the
unitary energy becomes negative, signaling flow of unitary energy in the
opposite direction: from the classical field to the quantum field. Finally,
when $\theta=\pi$ the quantum field consists of a single-photon pulse, giving
rise to a maximum correlation energy input of half a photon into the classical
field. No interference takes place, hence, no energy is transferred to the
classical field. In this situation, correlation and unitary energy transfers
perfectly cancel out.

We experimentally study this interference both at 5 K and 20 K using the
experimental configuration 2 in Fig. 1(c). To minimize the effect of
vibrations in our closed-cycle cryostation, the classical field is sent into
the same cryostation and focused by an objective lens (L) onto a mirror. To
further increase the signal-to-noise ratio, the classical field is spectrally
shaped using Fabry-Pérot etalons (FP) to increase its temporal overlap with
the quantum field (see Fig. S.3). We observe phase-dependent oscillations in
the counts at the two detectors at the output of the beam splitter from which
we extract an interference visibility plotted as a function of pulse area
$\theta$ in Fig. 3(b). As theoretically predicted, an increasing visibility is
observed when reducing $\theta$, both at 5 K and 20 K, evidencing energy
transfer towards the classical field of increasing efficiency. Our
observations are well reproduced by $v=\Delta\mathcal{E}_{\text{f,c}}/{\cal
E}_{\text{f}}^{\mathrm{in}}=\cos(\theta/2)\mathcal{C}_{\text{f,c}}$ for
$\mathcal{C}_{\text{f,c}}(5~{}\mathrm{K})=(36.3\pm 0.4)\%$ and
$\mathcal{C}_{\text{f,c}}(20~{}\mathrm{K})=(27.2\pm 0.4)\%$.

Fig. 3(c) shows the measured unitary, correlation and total energy transfers
deduced from the visibility measured in Fig. 3(b) and the measurement of the
correlation energy received by the quantum field in Fig. 2(b). While the
energy transfer during spontaneous emission at 5 K was close to the ideal
case, this second step deviates from theoretical maximum with significantly
reduced maximum energy exchanged, and positive unitary energy flows for
smaller $\theta$ range. The situation is even worse at 20 K, where the unitary
energy flows the opposite direction for most of the $\theta$ range.

To better quantitatively analyze these observations, we measured the mean wave
packet overlaps between the quantum field and the classical field. We find
$M_{\text{f,c}}=(48.9\pm 0.3)\%$ at 5 K and $M_{\text{f,c}}=(32.3\pm 0.7)\%$
at 20 K. These values are lower than expected considering the single-photon
mean wave packet overlap and the good temporal overlap of the envelop
functions of the both fields (see Fig. S.3). This could indicate some spectral
diffusion of the qubit resonance over long time scales that is not properly
captured by the $M_{s}$ extracted from interfering two quantum fields
separated by $12{.5}$ ns only. Overall, the expected upper bound of
$\mathcal{C}_{\mathrm{f,c}}={M_{\text{f,c}}}$ is not reached here, an
observation that could be due to some residual blinking of the QD transition
that affect differently our measurements of $\mathcal{C}_{\mathrm{f,c}}$ and
${M_{\text{f,c}}}$.

In conclusion, we have proposed and implemented experimental protocols to
measure the unitary and correlation energy exchanged between a two-level
system and the electromagnetic field as well as between two light fields. The
unitary energy bears similarities with the still investigated concept of work
in quantum thermodynamics Binder2018 . Thus, our experiment is an important
step to measure work exchanges between systems, and not only the work received
by a quantum open system. This capacity holds the promise to monitor directly
how a battery provides or stores energy, and to explore how quantum features
impacts these exchanges RMP_Coherence . This feature has remained elusive so
far in experimental studies. Remarkably, it is achieved here without the need
to resort to time-resolved reconstruction of quantum trajectories Murch2020 .
Our study reveals that the maximum unitary energy transfer efficiency is
obtained during spontaneous emission at the onset of the qubit population
inversion. It also demonstrates how both classical and quantum coherence
impact the nature of energetic transfers and can be accessed through homodyne
measurements. The two experimental situations studied here constitute key
building blocks for a multitude of quantum technologies from atom-based
quantum memories, linear optical gates to Bell state measurements among
others. The present work thus carries the seeds of an energetic investigation
of realistic processes at the core of optical quantum technologies Alexia .

∗Correspondence and requests for data should be addressed to A.A.
<EMAIL_ADDRESS>and P.S. (pascale.senellart-mardon@c2n.upsaclay.fr).

We acknowledge financial support by the Agence Nationale de la Recherche
(QuDICE project), the H2020-FET OPEN project number 899544 - PHOQUSING, the
French RENATECH network, the Paris Ile-de-France Region in the framework of
DIM SIRTEQ, the Foundational Questions Institute Fund (Grant number FQXi-
IAF19-05), the Templeton World Charity Foundation, Inc (Grant No. TWCF0338).

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Supplementary Material

## I. Description of the photonic field

In this section we provide a description of the photonic field state in
realistic conditions where decoherence perturbs the spontaneous emission
mechanism. The photonic density operator in the pulse-mode formalism
ozdemir2002pulse  of a single propagating mode $\hat{a}_{\mathrm{f}}(t)$
containing at most one photon reads:

$\displaystyle\hat{{\rho}}_{\mathrm{f}}$
$\displaystyle=p_{0}\ket{0}\bra{0}+p_{1}\iint
dtdt^{\prime}\xi(t,t^{\prime})\hat{a}_{\mathrm{f}}^{\dagger}(t)\ket{0}\bra{0}\hat{a}_{\mathrm{f}}(t^{\prime})+\sqrt{p_{0}p_{1}}\int
dt\zeta(t)\hat{a}_{\mathrm{f}}^{\dagger}(t)\ket{0}\bra{0}+\text{h.c.}$ (S.1)
$\displaystyle=p_{0}\hat{{\rho}}_{0}+p_{1}\hat{{\rho}}_{1}+\sqrt{p_{0}p_{1}}\left(\hat{{\rho}}_{01}+\hat{{\rho}}_{10}\right),$

where $\xi(t,t^{\prime})=\xi^{*}(t^{\prime},t)$ is a Hermitian function
describing the temporal shape and coherence of the single photon and
$\zeta(t)$ is the complex amplitude describing the time dynamics of the
photon-number coherence. The photon-number probabilities $p_{0}$ and $p_{1}$
satisfy $p_{0}+p_{1}=1$ and so $\text{Tr}[\hat{{\rho}}_{\mathrm{f}}]=1$
implies that $\int\xi(t,t)dt=1$. The total purity of this photonic state is

$\displaystyle\mathcal{P}$
$\displaystyle=\text{Tr}\left[\hat{{\rho}}_{\mathrm{f}}^{2}\right]=p_{0}^{2}+p_{1}^{2}M_{\mathrm{s}}+2p_{0}p_{1}\mathcal{C},$
(S.2)

where

$M_{\mathrm{s}}=\text{Tr}\left[\hat{{\rho}}_{1}^{2}\right]=\iint
dtdt^{\prime}\left|\xi(t,t^{\prime})\right|^{2}$ (S.3)

is the single-photon indistinguishability, or purity in the temporal domain,
and

$\mathcal{C}=\text{Tr}\left[\hat{{\rho}}_{01}\hat{{\rho}}_{10}\right]=\int
dt\left|\zeta(t)\right|^{2}$ (S.4)

is the number purity of the coherence between the single photon and the
vacuum.

We can compute the expected values for $M_{\text{s}}$ and $\mathcal{C}$ for a
two-level emitter that is ideally prepared in an initial state, but that emits
into the waveguide under the influence of pure dephasing and spectral
diffusion. The temporal wavefunction of a single photon produced by a two-
level emitter in the absence of pure dephasing is given by
$f(t,\omega_{0})=\sqrt{\gamma}e^{-\gamma t/2-i\omega_{0}t}$, where $\gamma$ is
the spontaneous emission rate of the two-level emitter and $\omega_{0}$ is the
transition frequency. Pure dephasing, by definition, causes an exponentially
decaying temporal coherence between the photonic state at two instantaneous
times $t$ and $t^{\prime}$. This is captured by the temporal density function
and coherence function of the forms

$\xi(t,t^{\prime},\omega_{0})=f(t,\omega_{0})f^{*}(t^{\prime},\omega_{0})e^{-\gamma^{\star}|t-t^{\prime}|}\hskip
28.45274pt\zeta(t,\omega_{0})=f(t,\omega_{0})e^{-\gamma^{\star}t}$ (S.5)

One can see that the absolute value of the difference in times $t$ and
$t^{\prime}$ prevents this temporal density function $\xi$ from being factored
using some new wavefunction $f^{\prime}$, hence the single-photon component of
the photonic state is no longer temporally pure.

When measuring $M_{\text{s}}$ or $\mathcal{C}$, two photonic states arising
from successive excitations of the emitter are interfered in a path-unbalanced
Mach-Zehnder interferometer (see section IV.2). The time delay between
excitations allows the emitter to shift emission frequency by some small
amount $\delta\omega$, although we still consider the frequency of the
transition for each individual emission event to be fixed on the timescale of
$1/\gamma$. This spectral diffusion of the emission implies a reduced mean
wave packet overlap, and hence a reduced single-photon indistinguishability of

$M_{\text{s}}(\delta\omega)=\iint
dtdt^{\prime}\text{Re}\left[\xi(t,t^{\prime},\omega_{0})\xi^{*}(t,t^{\prime},\omega_{0}+\delta\omega)\right].$
(S.6)

Evaluating this expression for the photonic state of the form given by Eq.
(S.1) using the function $\xi$ from Eq. (S.5) gives

$M_{\text{s}}(\delta\omega)=\frac{\gamma(\gamma+2\gamma^{\star})}{(\gamma+2\gamma^{\star})^{2}+\delta\omega^{2}}.$
(S.7)

In this expression, we can identify $\gamma+2\gamma^{\star}=2/T_{2}$ which is
related to the total decoherence rate and gives the homogeneously-broadened
line width of the emitter for a single emission. The line is further broadened
by a classical averaging over $\delta\omega$ following a distribution
describing the spectral diffusion of the emitter.

Similarly, we can find that the number purity of the coherence between the
single photon and the vacuum measured in the presence of spectral diffusion is
susceptible to the decoherence caused by pure dephasing, but also by the
random phase fluctuations caused by the spectral shift of the emitter:

$\mathcal{C}(\delta\omega)=\int
dt\text{Re}\left[\zeta(t,\omega_{0})\zeta^{*}(t,\omega_{0}+\delta\omega)\right].$
(S.8)

Using our temporal wavefunction solutions for an ideally-prepared two-level
emitter, we obtain exactly the same result as for the mean wave packet overlap

$\mathcal{C}(\delta\omega)=\frac{\gamma(\gamma+2\gamma^{\star})}{(\gamma+2\gamma^{\star})^{2}+\delta\omega^{2}}=M_{\text{s}}(\delta\omega).$
(S.9)

Thus, regardless of the distribution of spectral fluctuations, both pure
dephasing and spectral diffusion degrade $M_{\text{s}}$ and $\mathcal{C}$
equivalently.

Note that the result $\mathcal{C}=M_{\text{s}}$ is not necessarily true for
all dephasing effects. In general, the fact that the photonic density matrix
must remain positive semi-definite implies that
$\mathcal{C}\leq\sqrt{M_{\text{s}}}$, which arises from the Cauchy-Schwarz
inequality and implies that $\mathcal{C}$ may be larger than $M_{\text{s}}$.
This upper bound scaling was observed when varying the polarization of
interfering wave packets in Ref. Loredo2019 , and it held because modifying
the polarization in a controlled way does not induce additional phase
fluctuations between the vacuum and single-photon states that impact
$\mathcal{C}$. Hence, in that work, $\mathcal{C}$ was forced to follow the
trend set by the physical upper bound, which is set by the single-photon
indistinguishability.

### II. Effect of the pure dephasing on the energetic analysis

The analysis of the energetics during spontaneous emission builds on the
general two-body quantum energetic analysis of Alipour2016 ; Hossein-Nejad2015
; Weimer2008  applied to the specific case where the closed two-body system
comprises a qubit and a traveling electromagnetic field Maffei2021 . Here we
show this framework remains valid even when the qubit, while interacting with
the electromagnetic field, is coupled to a thermal bath of phonons causing its
pure dephasing. In order to show this result, let us first summarize the main
results of Refs. Alipour2016 ; Hossein-Nejad2015 ; Weimer2008  that represent
the starting point of our energetic analysis. For any pair of systems (A and
B) coupled by an Hamiltonian interaction $V_{A,B}$, the joint state at time t
can be written as $\rho(t)=\rho^{A}(t)\otimes\rho^{B}(t)+\chi_{A,B}(t)$, where
$\rho^{A(B)}(t)=\text{Tr}_{B(A)}\\{\rho(t)\\}$ and $\chi_{A,B}(t)$ is the
correlation matrix.

The definitions of unitary energy and correlation energy follow naturally from
this separation of the dynamics:

$\dot{{E}}^{A(B)}_{\mathrm{unit}}(t)=-\frac{i}{\hbar}\text{Tr}_{A,B}\left\\{\left[H_{A(B)},V_{A,B}\right]\rho^{A}\left(t\right)\otimes\rho^{B}\left(t\right)\right\\},$
(S.10)
$\dot{{E}}^{A(B)}_{\mathrm{corr}}(t)=-\frac{i}{\hbar}\text{Tr}_{A,B}\left\\{\left[H_{A(B)},V_{A,B}\right]\chi_{A,B}(t)\right\\},$
(S.11)

with $H_{A(B)}$ being the bare Hamiltonian of system A(B). For each system the
change of the internal energy is given by the sum of unitary and correlation
energy, i.e.
$\Delta\mathcal{U}_{A(B)}={{E}}^{A(B)}_{\mathrm{unit}}+{{E}}^{A(B)}_{\mathrm{corr}}$.
The two fluxes in unitary energy $\dot{{E}}^{A}_{\mathrm{unit}}$ and
$\dot{{E}}^{B}_{\mathrm{unit}}$ are connected by the balance relation:

$\displaystyle\dot{{E}}^{A}_{\mathrm{unit}}+\dot{{E}}^{B}_{\mathrm{unit}}-\frac{i}{\hbar}\text{Tr}\\{[V_{A,B},H_{A}+H_{B}]\rho^{A}(t)\otimes\rho^{B}(t)\\}=0.$
(S.12)

When A is the qubit and B is its spontaneously emitted field (battery),
$\text{Tr}\\{[V_{A,B},H_{A}+H_{B}]\rho^{A}(t)\otimes\rho^{B}(t)\\}=0$ at any
time. This can be shown by replacing the operators in the commutator with
their explicit expressions: $H_{A}=\hbar\omega_{0}\sigma^{\dagger}\sigma$,
$H_{B}=\hbar\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}$, $V_{A,B}=\hbar
g_{0}\sum_{k}(b_{k}\sigma^{\dagger}+b^{\dagger}_{k}\sigma)$, where
$\sigma=\ket{g}\bra{e}$. Injecting these expressions in the commutator we get:
$\text{Tr}\\{[V_{A,B},H_{A}+H_{B}]\rho^{A}(t)\otimes\rho^{B}(t)\\}=\hbar
g_{0}\sum_{k}(\omega_{k}-\omega_{0})\text{Re}\\{\langle
b_{k}\rangle\langle\sigma\rangle^{*}\\}$. When the field is the qubit’s
spontaneous emission, $\text{Re}\\{\langle
b_{k}\rangle\langle\sigma\rangle^{*}\\}\propto[\gamma^{2}/4+(\omega_{k}-\omega_{0})^{2}]^{-1}$,
the spectrum is symmetric around $\omega_{0}$, and hence summing over $k$,
positive and negative contributions cancel each other out. This hypothesis is
fulfilled here with the cavity suppressing phonon sideband emission by several
orders of magnitude Grange2017 . This implies
$\dot{{E}}^{B}_{\mathrm{unit}}=-\dot{{E}}^{A}_{\mathrm{unit}}=\hbar\omega_{0}|\langle
a_{f}(t)\rangle|^{2}$, where the last equality has been derived in Ref.
Maffei2021 .

In order to include pure dephasing in the analysis we have to write down the
full Hamiltonian of the three-body system comprising qubit, electromagnetic
field, phonons. As there is no direct coupling between the electromagnetic
field and the phonons the Hamiltonian will have the form:
$H_{A,B,C}=H_{A}+H_{B}+H_{C}+V_{A,B}+V_{A,C}$, where now A=qubit,
B=electromagnetic field, and C=phonons. The joint system state can be written
as $\rho(t)=\rho^{A}(t)\otimes\rho^{B}(t)\otimes\rho^{C}(t)+\chi_{A,B,C}(t)$.
Now the unitary and correlation energy received by A read:

$\dot{{E}}^{A}_{\mathrm{unit}}(t)=-\frac{i}{\hbar}\text{Tr}_{A,B}\left\\{\left[H_{A},V_{A,B}\right]\rho^{A}\left(t\right)\otimes\rho^{B}\left(t\right)\right\\}-\frac{i}{\hbar}\text{Tr}_{A,C}\left\\{\left[H_{A},V_{A,C}\right]\rho^{A}\left(t\right)\otimes\rho^{C}\left(t\right)\right\\},$
(S.13)
$\dot{{E}}^{A}_{\mathrm{corr}}(t)=-\frac{i}{\hbar}\text{Tr}_{A,B}\left\\{\left[H_{A},V_{A,B}\right]\chi_{A,B}(t)\right\\}-\frac{i}{\hbar}\text{Tr}_{A,C}\left\\{\left[H_{A},V_{A,C}\right]\chi_{A,C}(t)\right\\},$
(S.14)

while for the systems B and C we find:

$\dot{{E}}^{B(C)}_{\mathrm{unit}}(t)=-\frac{i}{\hbar}\text{Tr}_{A,B(C)}\left\\{\left[H_{B(C)},V_{A,B(C)}\right]\rho^{A}\left(t\right)\otimes\rho^{B(C)}\left(t\right)\right\\},$
(S.15)
$\dot{{E}}^{B(C)}_{\mathrm{corr}}(t)=-\frac{i}{\hbar}\text{Tr}_{A,B(C)}\left\\{\left[H_{B(C)},V_{A,B(C)}\right]\chi_{A,B(C)}(t)\right\\},$
(S.16)

with $\chi_{A,C}(t)=\text{Tr}_{B}\\{\chi(t)\\}$ and
$\chi_{A,B}(t)=\text{Tr}_{C}\\{\chi(t)\\}$. For each system the change of the
internal energy is still given by the sum of unitary and correlation energy,
i.e.
$\Delta\mathcal{U}_{A(B)}={{E}}^{A(B)}_{\mathrm{unit}}+{{E}}^{A(B)}_{\mathrm{corr}}$.

Let us now write explicitly the Hamiltonian $V_{A,C}$ coupling qubit and
phonons’ bath; as we are considering pure dephasing, this interaction reads
$V_{A,C}=\sum_{k}(\xi_{k}c_{k}+\xi^{*}_{k}c^{\dagger}_{k})\otimes\sigma_{z}$.
This Hamiltonian commutes with the qubit’s bare Hamiltonian, hence, in both
Eqs. (S.13) and (S.14), the last terms vanish and the unitary and correlation
energy on the qubit take the same expression as in the zero-temperature case,
i.e. $\dot{{E}}^{A}_{\mathrm{unit}}(t)=-\hbar\omega_{0}|\langle
a_{f}(t)\rangle|^{2}$. The relation linking the unitary energy on the photonic
field with the unitary energy on the qubit is still given by Eq. (S.12). The
term $\text{Tr}\\{[V_{A,B},H_{A}+H_{B}]\rho^{A}(t)\otimes\rho^{B}(t)\\}$ is
still identical to zero at any time as the spectrum of the emission is still
symmetric around $\omega_{0}$. In particular we are assuming that the qubit’s
pure dephasing only implies a spectral broadening of the spontaneous emission,
leaving the center in $\omega_{0}$, mathematically this means: $\langle
a_{f}(t)\rangle\propto e^{-(\gamma/2+\gamma^{*})t-i\omega_{0}t}$. Hence we
obtain again
$\dot{{E}}^{B}_{\mathrm{unit}}(t)=-\dot{{E}}^{A}_{\mathrm{unit}}(t)=\hbar\omega_{0}|\langle
a_{f}(t)\rangle|^{2}$ as in the zero-temperature case.

## III. Relating energetic and optical quantities

In this section, we propose an experimental protocol to measure the energy
exchanges between the qubit and the electromagnetic field as well as between
the photonic field and the classical field. We demonstrate that homodyne
measurements, where two optical fields interfere on a balanced beam splitter
(see Fig. 1), give direct access to the unitary energy exchanged between two
fields. For two input fields impinging a balanced beam splitter, the
transformation is given by:

$\begin{pmatrix}\hat{a}_{3}(t)\\\
\hat{a}_{4}(t)\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&e^{i\phi}\\\
-e^{i\phi}&1\end{pmatrix}\begin{pmatrix}\hat{a}_{1}(t)\\\
\hat{a}_{2}(t)\end{pmatrix}$ (S.17)

The visibility of interference is defined as:

$v=\frac{\mu_{3}-\mu_{4}}{\mu_{3}+\mu_{4}},$ (S.18)

where $\mu_{j}=\int\langle\hat{a}^{\dagger}_{j}(t)\hat{a}_{j}(t)\rangle dt$ is
the average photon number detected in mode $j$ at the output of the beam
splitter.

In what follows, we successively study the case of a self-homodyne experiment,
where the input fields are copies of the photonic field of Eq. (S.1), i.e.
$a_{1}(t)=a_{\mathrm{f1}}(t)$, and $a_{2}(t)=a_{\mathrm{f2}}(t)$, and the case
of a standard homodyne experiment where the input fields are the photonic
field (f) and the classical field (c), i.e.
$a_{1}(t)=a_{\text{c}}^{\mathrm{in}}(t)$,
$a_{2}(t)=a_{\mathrm{f}}^{\mathrm{in}}(t)$,
$a_{3}(t)=a_{\text{c}}^{\mathrm{out}}(t)$,
$a_{4}(t)=a_{\mathrm{f}}^{\mathrm{out}}(t)$.

### III.1. Measuring the efficiency of energetic transfers in spontaneous
emission with self-homodyne interference

In Refs. Monsel2020 ; Maffei2021 , it has been demonstrated that when the
qubit-field system is isolated, the total unitary energy provided by the qubit
to the field’s mode can be extracted from the final field’s state:

$\displaystyle E_{\mathrm{unit}}^{\mathrm{q,f}}=\hbar\omega_{0}\int
dt|\langle\hat{a}_{\mathrm{f}}(t)\rangle|^{2}$ (S.19)

We extended the definition above to the experimental situation where the joint
qubit-field state may not be perfectly pure. By evaluating Eq. (S.19) on the
general state in Eq. (S.1) with $p_{0}=\cos^{2}{\left(\theta/2\right)}$, and
$p_{1}=\sin^{2}{\left(\theta/2\right)}$, we find
$E_{\mathrm{unit}}^{\mathrm{q,f}}=\hbar\omega_{0}\cos^{2}{\left(\theta/2\right)}\sin^{2}{\left(\theta/2\right)}\int
dt|\zeta(t)|^{2}$.

Now using Eq. (S.4), we find the expression for the unitary energy presented
in the main text,
$E_{\mathrm{unit}}^{\mathrm{q,f}}=\hbar\omega_{0}\cos^{2}{\left(\theta/2\right)}\sin^{2}{\left(\theta/2\right)}\mathcal{C}$.

$\mathcal{C}$ can be experimentally accessed from the visibility of the self-
homodyne interference where two copies of the emitted photonic field are
incident on a balanced beam splitter. The two input states are identical
except for a relative phase $\phi$ on the number coherence between them which
is due to the difference in path length.

The numerator of Eq. (S.18) is given by:

$\displaystyle 2\mathrm{Re}\left[\int dt\langle
e^{i\phi}\hat{a}^{\dagger}_{\mathrm{f1}}(t)\hat{a}_{\mathrm{f2}}(t)\rangle\right]=2\mathrm{Re}\left[\int
dte^{i\phi}\langle\hat{a}^{\dagger}_{\mathrm{f1}}(t)\rangle\langle\hat{a}_{\mathrm{f2}}(t)\rangle\right]=2\mathrm{cos}(\phi)\int
dt|\langle\hat{a}_{\mathrm{f}}(t)\rangle|^{2}$ (S.20)

where the first equality comes from the fact that the states of modes
$\mathrm{f1}$ and $\mathrm{f2}$ are uncorrelated, the second from the fact
that the states are identical, i.e.
$\langle\hat{a}_{\mathrm{f1}}\rangle=\langle\hat{a}_{\mathrm{f2}}\rangle=\langle\hat{a}_{\mathrm{f}}\rangle$.

Since the two fields carry the same number of photons, the denominator of Eq.
(S.18) is simply
$2\mu_{\mathrm{f}}=2\mathcal{E}_{\mathrm{f}}/(\hbar\omega_{0})$. Putting all
together we get:

$v=\mathrm{cos}(\phi)\frac{\int
dt|\langle\hat{a}_{\mathrm{f}}(t)\rangle|^{2}}{\mu_{\mathrm{f}}}=\mathrm{cos}(\phi)\frac{E_{\mathrm{unit}}^{\mathrm{q,f}}}{\mathcal{E}_{\mathrm{f}}}$
(S.21)

The visibility of the interference varies with the relative phase between the
two inputs, $\phi$, and reaches maximum constructive (destructive)
interference when $\phi=0\,(\pi)$.

### III.2. Transfer of energy from the photonic field to the classical field
through homodyne interference

We transfer energy from the photonic field via interference to a coherent
field. The visibility of this classical-homodyne measurement allows us to
quantify the energy and the unitary energy transfer. The photonic field (f)
enters the balanced beam splitter in channel 2, with
$\langle\hat{a}^{\mathrm{in}}_{\mathrm{f}}(t)\rangle=\sin{\left(\theta/2\right)}\cos{\left(\theta/2\right)}\zeta(t)$.
The classical field (c) field, $\ket{\beta}$, enters the beam splitter in
channel 1, with $\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle=\beta(t)$.

The final energy of the classical receiver field, reads:

$\displaystyle\mathcal{E}^{\mathrm{out}}_{\text{c}}$
$\displaystyle=\frac{1}{2}\left[\mathcal{E}^{\mathrm{in}}_{\text{c}}+\mathcal{E}^{\mathrm{in}}_{\mathrm{f}}\right]+\hbar\omega_{0}\mathrm{Re}\left[\int
dt\langle\hat{a}^{\mathrm{in}}_{\mathrm{f}}(t)\rangle\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle^{*}\right]$
(S.22) $\displaystyle=\frac{\hbar\omega_{0}}{2}\left[\int
dt|\beta(t)|^{2}+\sin^{2}{(\theta/2)}\right]+\hbar\omega_{0}\cos{(\theta/2)}\sin{(\theta/2)}\mathrm{Re}\left[\int
dt\zeta(t)\beta(t)^{*}\right]$

The efficiency of the energy transfer reads
$G=\mathcal{E}^{\mathrm{out}}_{\text{c}}/\left[\hbar\omega_{0}\int
dt|\beta(t)|^{2}+\hbar\omega_{0}\sin^{2}{\left(\theta/2\right)}\right]$,
replacing the numerator with the expression above, we find that $G$ is maximal
when $\int dt|\beta(t)|^{2}=\sin^{2}{\left(\theta/2\right)}$, i.e.
$\beta(t)=\sin{(\theta/2)}\sqrt{\xi(t,t)}e^{-i\phi(t)}$, namely the two fields
are matched in intensity. This condition is met in the experiment presented in
the main and it corresponds to an interference visibility reading:

$v=\frac{1}{\mu_{\mathrm{f}}}\mathrm{Re}\left[\int
dt\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle\langle\hat{a}^{\mathrm{in}}_{\mathrm{f}}(t)\rangle^{*}\right]=\cos{\left(\theta/2\right)}\mathrm{Re}\left[\int
dt\zeta(t)\sqrt{\xi(t,t)}e^{i\phi(t)}\right]=\cos{\left(\theta/2\right)}\mathcal{C}_{\text{f,c}}$
(S.23)

where, in the last equality, we defined the quantity
$\mathcal{C}_{\text{f,c}}=\text{Re}\left[\int
dt\zeta(t)\sqrt{\xi(t,t)}e^{i\phi(t)}\right]$ accounting for both the
classical and the quantum coherence of the process. It is clear that the
visibility and hence the relative efficiency $G=(1+v)/2$ are maximal when
$\mathcal{C}_{\text{f,c}}=1$. The energy change of the two fields hence reads:

$\displaystyle\Delta\mathcal{E}_{\text{f,c}}=\hbar\omega_{0}\cos{(\theta/2)}\sin^{2}{(\theta/2)}\mathcal{C}_{\text{f,c}}=-\Delta\mathcal{E}_{\mathrm{f}}$
(S.24)

We can split this energy change into unitary and correlation energy terms:

$\displaystyle\Delta\mathcal{E}_{\mathrm{f,c}}=E^{\text{f,c}}_{\mathrm{unit}}+E^{\text{f,c}}_{\mathrm{corr}}$
(S.25)

Using the analysis introduced in Refs. Alipour2016 ; Hossein-Nejad2015 ;
Weimer2008 ; Schroder2018 ; Maffei2021 , we find the unitary energy:

$\displaystyle E^{\text{f,c}}_{\mathrm{unit}}$
$\displaystyle=\hbar\omega_{0}\left(\int
dt|\langle\hat{a}^{\mathrm{out}}_{\text{c}}(t)\rangle|^{2}-\int
dt|\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle|^{2}\right)$ (S.26)
$\displaystyle=\hbar\omega_{0}\mathrm{Re}\left[\int
dt\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle\langle\hat{a}^{\mathrm{in}}_{\mathrm{f}}(t)\rangle^{*}\right]+\frac{\hbar\omega_{0}}{2}\left(\int
dt|\langle\hat{a}^{\mathrm{in}}_{\mathrm{f}}(t)\rangle|^{2}-\int
dt|\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle|^{2}\right)$
$\displaystyle=\hbar\omega_{0}\sin^{2}{(\theta/2)}\left[\cos{(\theta/2)}\mathcal{C}_{\text{f,c}}+\left(\cos^{2}{(\theta/2)}\mathcal{C}-1\right)/2\right]$

It can be easily verified that for any choice of
$\langle\hat{a}^{\mathrm{in}}_{\text{c}}(t)\rangle$ we find
$E^{\text{f,c}}_{\mathrm{unit}}\leq E^{\text{q,f}}_{\mathrm{unit}}$.

Subtracting the unitary energy from the total energy change we find the
correlation energy:

$\displaystyle
E^{\text{f,c}}_{\mathrm{corr}}=\hbar\omega_{0}\sin^{2}{(\theta/2)}\left(1-\cos^{2}{(\theta/2)}\mathcal{C}\right)/2=\left[\mathcal{E}^{\mathrm{in}}_{\text{f}}-\hbar\omega_{0}\int
dt|\langle\hat{a}^{\mathrm{in}}_{\mathrm{f}}(t)\rangle|^{2}\right]/2=E^{\text{q,f}}_{\mathrm{corr}}/2$
(S.27)

Similar to the self-homodyne interference, we can derive the explicit results
for $\mathcal{C}_{\mathrm{f,c}}$ and $M_{\mathrm{f,c}}$ under the effects of
pure dephasing and spectral diffusion, assuming a perfectly-prepared two-level
emitter. In this case, we must interfere the photonic state from the quantum
dot described by $\xi$ and $\zeta$ with the classical field described by the
amplitude $\beta(t)=\sin(\theta/2)\sqrt{\xi(t,t)}e^{-i\phi(t)}$. Thus,

$M_{\mathrm{f,c}}(\delta\omega)=\iint
dtdt^{\prime}\text{Re}\left[\sqrt{\xi(t,t,\omega_{0})\xi(t^{\prime},t^{\prime},\omega_{0})}e^{-i\omega_{0}(t-t^{\prime})}\xi^{*}(t,t^{\prime},\omega_{0}+\delta\omega)\right]=\frac{\gamma(\gamma+\gamma^{\star})}{(\gamma+\gamma^{\star})^{2}+\delta\omega^{2}}.$
(S.28)

In this case, the factor $\sin(\theta/2)$ from $\beta(t)$ is canceled in the
normalization and we take $\phi(t)=\omega_{0}t$. First, we can notice the
similarity between this expression and $M_{\text{s}}(\delta\omega)$. The only
difference is that the effect of pure dephasing via $\gamma^{\star}$ is halved
compared to the self-homodyne scenario due to the fact that the classical
field does not experience pure dephasing. Second, we can also solve for

$\mathcal{C}_{\mathrm{f,c}}(\delta\omega)=\int
dt\text{Re}\left[\sqrt{\xi(t,t,\omega_{0})}e^{-i\omega_{0}t}\zeta^{*}(t,t,\omega_{0}+\delta\omega)\right]=\frac{\gamma(\gamma+\gamma^{\star})}{(\gamma+\gamma^{\star})^{2}+\delta\omega^{2}}=M_{\mathrm{f,c}}(\delta\omega).$
(S.29)

Thus, just like in the self-homodyne case, we find that the values
$M_{\mathrm{f,c}}$ and $\mathcal{C}_{\mathrm{f,c}}$ extracted from homodyne
interference are equally degraded by pure dephasing and spectral diffusion.
Also similar to the self-homodyne case, we can again note that
$\mathcal{C}_{\mathrm{f,c}}=M_{\mathrm{f,c}}$ is not a general result for
every model since a Cauchy-Schwarz inequality will imply a general bound of
$\mathcal{C}_{\mathrm{f,c}}\leq\sqrt{M_{\mathrm{f,c}}}$. This more general
upper bound may be observed, for example, by rotating the polarization of one
of the two inputs.

## IV. Experimental methods

### IV.1. Photonic field emission

In the main text, we assumed near unity population inversion both at 5 K and
at 20 K at $\pi$ pulse for the sake of simplicity in our data analysis. We
show here that this hypothesis is verified a posteriori. Indeed, the self-
homodyne measurement also allows us to measure the excited state population
probability $p_{1}$ from the visibility of interference. We can deduce the
excited state probability at $\theta=\pi$ according to

$p_{1}(\pi)=1-\frac{v(\pi)}{\mathcal{C}}$ (S.30)

where we extract $\mathcal{C}$ from the fit in Fig. 2(a):
$v(\theta)=\mathcal{C}\cos^{2}{(\theta/2)}$. From this we deduce the values
$p_{1}(\pi)=0.95\pm 0.19$ at 5K and $p_{1}(\pi)=0.92\pm 0.02$ at 20K.

Figure S.1: Population excited state (a) The qubit emission intensity as a
function of pulse area $\theta$ at 5 K (blue) and 20 K (red). (b) The
corresponding excited state population as a function of pulse area for the
qubit performing at 5 K (blue) and at 20 K (red).

In Fig. S.1(a) we show the qubit raw emission intensity measured both at 5 K
and 20 K which shows very similar trend - evidencing a very limited effect on
the coherence imprinted on the qubit. Fig. S.1(b) shows the corresponding
occupation of the qubit excited state $p_{1}(\theta)$ using the above measured
values $p_{1}(\pi)$.

### IV.2. Photonic field indistinguishability

To measure $\mathcal{C}$ and $M_{\mathrm{s}}$ for the emitted photonic field,
two sequentially generated copies of the state interfere at a 50:50 beam
splitter. We monitor the single-photon intensity at the two outputs of the
beam splitter to observe interference fringes, and also simultaneously monitor
the two-photon coincidences where we observe bunching due to Hong-Ou-Mandel
interference Loredo2019 ; ollivier2021hong ; Wein2021PhotonnumberEG . To
obtain the single-photon indistinguishability $M_{\mathrm{s}}$, we first
measure the mean wave packet overlap between two subsequently emitted photonic
fields, $M$ ollivier2021hong . The mean wave packet overlap is defined by
$M=(1/\mu_{\mathrm{f}}^{2})\iint
dtdt^{\prime}\left|G^{(1)}(t,t^{\prime})\right|^{2}$, where
$\mu_{\mathrm{f}}=\int dtI_{\mathrm{f}}(t)=\sum_{n}np_{n}$ is the average
photon number,
$I_{\mathrm{f}}(t)=\braket{\hat{a}_{\mathrm{f}}^{\dagger}(t)\hat{a}_{\mathrm{f}}(t)}$
is the wave packet temporal envelop, and
$G^{(1)}(t,t^{\prime})=\braket{\hat{a}_{\mathrm{f}}^{\dagger}(t^{\prime})\hat{a}_{\mathrm{f}}(t)}$
is the first-order (amplitude) correlation function. Experimentally, the
integrated coincident counts after HOM interference binned with respect to the
detection delay $\tau$ produce histograms proportional to $\int
dtG^{(2)}_{\text{HOM}}(t,t+\tau)$, where
$2G^{(2)}_{\text{HOM}}(t,t^{\prime})=I_{\mathrm{f}}(t)I_{\mathrm{f}}(t^{\prime})+G^{(2)}(t,t^{\prime})-\left|G^{(1)}(t,t^{\prime})\right|^{2}$
is the phase-averaged intensity correlation after HOM interference and
$G^{(2)}(t,t^{\prime})=\braket{\hat{a}_{\mathrm{f}}^{\dagger}(t)\hat{a}_{\mathrm{f}}^{\dagger}(t^{\prime})\hat{a}_{\mathrm{f}}(t^{\prime})\hat{a}_{\mathrm{f}}(t)}$
is the intensity autocorrelation measured using a Hanbury Brown–Twiss setup.

The required normalization $\mu_{\mathrm{f}}^{2}$ for $M$ is obtained by
comparing the coincident counts for when the inputs are co-polarized to when
they are cross-polarized. In this latter case, we have $G^{(1)}=0$. Then, if
the fully integrated and normalized intensity correlation
$g^{(2)}=(1/\mu_{\mathrm{b}}^{2})\iint dtdt^{\prime}G^{(2)}(t,t^{\prime})$ is
very small, the HOM histograms allow us to quantify $M$ via the HOM visibility
defined as:

${V_{\text{HOM}}=}\frac{g^{(2)}_{\mathrm{HOM},\perp}-g^{(2)}_{\mathrm{HOM},\parallel}}{g^{(2)}_{\mathrm{HOM},\perp}},$
(S.31)

where
$g^{(2)}_{\mathrm{HOM},\parallel}=A_{\tau_{0},\parallel}/\overline{A}_{\tau_{>1},\parallel}$
and
$g^{(2)}_{\mathrm{HOM},\perp}=A_{\tau_{0},\perp}/\overline{A}_{\tau_{>0},\perp}$
are the central peak areas ($A_{\tau_{0}}$) in co- ($\parallel$) and cross-
($\perp$) input polarization configurations normalized by the average peak
area of the histogram peaks arising from uncorrelated counts
($\overline{A}_{\tau>1,\parallel}$ or $\overline{A}_{\tau>0,\perp}$). Note
that, due to the interferometer delay needed to interfere subsequently
generated states, the first side peak of the $\parallel$ case is partially
suppressed due to anti-bunching and hence is excluded from the normalization.
In addition, Eq. (S.31) is an accurate measurement of $M$ only when there is
not much first-order coherence in the number basis so that
$\overline{A}_{\tau_{>0},\parallel}$ is not suppressed by possible
interference fringes. Hence, this measurement approach is accurate for pulse
areas $\theta\simeq\pi$; however, $M$ is expected to take similar values for
all $\theta$.

If the mode contains no more than one photon, then
$G^{(1)}(t,t^{\prime})=\mu_{\mathrm{f}}^{2}\xi(t,t^{\prime})$ and so
$V_{\mathrm{HOM}}=M=M_{\mathrm{s}}$. For small nonzero $g^{(2)}$, both
$A_{\tau_{0},\parallel}$ and $A_{\tau_{0},\perp}$ are increased equally, which
increases the denominator in Eq. (S.31) and causes $V_{\text{HOM}}$ to
underestimate $M$. This small underestimate of $M$ can be corrected by taking
$M=V_{\text{HOM}}(1+g^{(2)})$. Note that this measurement approach and
subsequent $g^{(2)}$ correction are different than those used in Ref.
ollivier2021hong , where the visibility was defined as
$V_{\text{HOM}}=1-2g^{(2)}_{\text{HOM},\parallel}$, which implies
$M=V_{\text{HOM}}+g^{(2)}$. Both approaches should predict identical values of
$M$. Furthermore, if $g^{(2)}$ is small but nonzero and one of the two photons
is approximately distinguishable from the emitted single photon state
ollivier2021hong , then $M$ is related to $M_{\mathrm{s}}$ via $M\simeq
M_{\mathrm{s}}(1-g^{(2)})$. Hence, in our experiments, $M_{\mathrm{s}}$ is
accurately estimated by

$M_{\mathrm{s}}\simeq V_{\text{HOM}}\frac{1+g^{(2)}}{1-g^{(2)}}.$ (S.32)

Fig. S.2 shows the results of these second-order intensity correlation
measurements with Fig. S.2(a) and Fig. S.2(b) (Fig. S.2(c) and Fig. S.2(d))
corresponding to the auto-correlation $g^{(2)}$, and the time integrated HOM
measurement $g^{(2)}_{\mathrm{HOM}}$ at 5 K (20 K), all taken at $\theta=\pi$.
We extract similar auto-correlation values at 5 K and 20 K: $g^{(2)}$ of
$(2.84\pm 0.08)\%$ and $(2.28\pm 0.08)\%$, respectively. However, the
indistinguishability between two subsequently emitted single-photon wave
packets is reduced at the higher temperature. From the indistinguishability
measurements we deduce a single-photon indistinguishability of
$M_{\mathrm{s}}=(92.6\pm 0.1)\%$ at 5 K and $M_{\mathrm{s}}=(58.0\pm 1.0)\%$
at 20 K.

Figure S.2:  Performance of Single-Photon Source (a) Auto-correlation
measurement of the single-photon wave packets generated by the single-photon
source with $\theta=\pi$ at 5 K. The resulting coincidence histogram allows
for extraction of the single-photon purity $g^{(2)}$. (b) The coincidence
histogram obtained from Hong-Ou-Mandel interference measurements
$g^{(2)}_{\mathrm{HOM}}$ for $\theta=\pi$ at 5 K. From Eq. 24 and the single-
photon purity extracted from (a) we can deduce the mean wave packet overlap,
or single-photon indistinguishability, $M_{\mathrm{s}}$. (c) and (d) the same
as (a) and (b) but at 20 K. Note that the small peaks observed in the
coincidence histograms at non-integer multiples of the pulse separation time
of 12.5 ns are due to electronic reflections in our measurement setup and can
be neglected.

### IV.3. Photonic and classical fields overlap

To generate temporal mode overlap between the emitted photonic field and the
classical field we shape the latter with a Fabry-Pérot etalon. By time
resolving the emission dynamics of both fields (see Fig. S.3), we
qualitatively ensure temporal overlap between the classical field (black
dashed) and the photonic field (blue 5 K, red 20 K). To then quantify the mean
wave packet overlap between the two fields, $M_{\text{f,c}}$, we perform a HOM
experiment with the two fields for $\theta=\pi$. Here it is important to
account for the non-negligible classical intensity correlation
$g^{(2)}_{\mathrm{c}}=1$ of the coherent state input. The coincidence counts
can arise in three ways. (1) A photon from each input do not bunch when
leaving the beam splitter, leading to a contribution of
$2\mu_{\text{f}}\mu_{\mathrm{c}}(1-M_{\text{f,c}})$. The factor of two in this
term arises from the two ways to obtain a coincidence count (both photons
reflected or both photons transmitted). (2) Two photons arrive from the
coherent state when the photonic field is vacuum (or lost), leading to an
additional contribution
$\mu_{\mathrm{c}}^{2}g^{(2)}_{\mathrm{c}}=\mu_{\mathrm{c}}^{2}$ to the
coincidence counts. (3) Two photons arrive from the photonic field input,
leading to a small contribution $\mu_{\text{f}}^{2}g^{(2)}$ to the coincidence
counts. If we now assume that $\theta$ is near $\pi$-pulse, the average
uncorrelated histogram peak area $\overline{A}_{\tau>0,\parallel}$ is given by
$2\mu_{\mathrm{c}}\mu_{\text{f}}$. Thus, for the co-polarized case we have
$g^{(2)}_{\mathrm{HOM},\parallel}=1-M_{\text{f,c}}+\overline{g}^{(2)}$, where
$\overline{g}^{(2)}=(\mu_{\text{c}}/\mu_{\mathrm{f}})g^{(2)}_{\text{c}}/2+(\mu_{\mathrm{f}}/\mu_{\text{c}})g^{(2)}_{\mathrm{f}}/2$
is the weighted average input intensity correlation. The intensity correlation
after interference of inputs in cross-polarization is then simply obtained by
$g^{(2)}_{\mathrm{HOM},\perp}=1+\overline{g}^{(2)}$. Hence, $M_{\text{f,c}}$
is measured by

$M_{\text{f,c}}=\frac{g^{(2)}_{\mathrm{HOM},\perp}-g^{(2)}_{\mathrm{HOM},\parallel}}{g^{(2)}_{\mathrm{HOM},\perp}}\left(1+\overline{g}^{(2)}\right).$
(S.33)

When the inputs are balanced ($\mu_{\mathrm{c}}=\mu_{\text{f}}$), we have
$1+\overline{g}^{(2)}=1+(g^{(2)}+1)/2$. If the photonic field’s $g^{(2)}\ll
1$, the correction factor becomes 3/2.

Fig. S.4 shows the coincidence histograms of HOM measurements with the quantum
photonic field and the classical field at 5 K and 20 K, Fig. S.4(a) and Fig.
S.4(b), respectively. We perform the measurements with the two fields cross-
polarized ($g^{(2)}_{\mathrm{HOM},\perp}$, grey), and co-polarized
($g^{(2)}_{\mathrm{HOM},\parallel}$, blue and red). From the coincidence
histograms we extract a mean wave packet overlap of $M_{\text{f,c}}=(48.9\pm
0.3)\%$ at 5 K and $M_{\text{f,c}}=(32.3\pm 0.7)\%$ at 20 K.

Figure S.3: Temporal profiles photonic field and classical field. The
classical field (dashed black), derived from the same coherent laser field
used to drive the quantum dot, is temporally shaped to overlap with the
temporal profile of the emitted photonic field at 5K and 20K (blue and red,
solid). Figure S.4:  Mean wave packet overlap between classical coherent field
and single-photon field. (a) The coincidence histograms of time integrated
Hong-Ou-Mandel interference measurements between the single-photon field
(generated with pulse area $\theta=\pi$) and the classical coherent field at 5
K, measured in co- ($g^{(2)}_{\mathrm{HOM}}\parallel$) and cross-
($g^{(2)}_{\mathrm{HOM}}\perp$) polarization configuration. (b) Same as (a)
but at 20 K. Using Eq. S.33 we obtain mean wave packet overlaps of
$M_{\text{f,c}}=(48.9\pm 0.3)\%$ at 5 K and $M_{\text{f,c}}=(32.3\pm 0.7)\%$
at 20 K.

### IV.4. Measuring $\mathcal{C}$ and $\mathcal{C}_{\mathrm{f,c}}$

Figure S.5:  Extracting visibility of interference $v$. (a) Intensities
measured by detector 1 (red) and 2 (blue) at the output of the interference
beam splitter (BS1 in Figure 1b) as a function of time, where each point at
$t_{i}$ is normalized by the sum of the two intensities at $t_{i}$. The inset
shows a zoom of the signal, demonstrating anti-correlated intensities. Over
the full measurement time we let the relative phase between the two signals
freely evolve. (b) The visibility $V$ as a function of time obtained from Eq.
S.18 corresponding to the data shown in (a). (c) Histogram of visibility time
trace presented in (b). Dashed vertical lines indicate the extracted maximum
visibility $v$. This data set was taken at $\theta=0.55\pi$.

To determine both $\mathcal{C}$ and $\mathcal{C}_{\mathrm{f,c}}$, homodyne
measurements are performed where we measure the difference in count rate at
each detector as the interferometer phase evolves. We detail here the protocol
for the self-homodyne measurement (step 1). The same protocol applies for the
energetic transfers in spontaneous emission with some small adjustments as
indicated later on. The maximum visibility of these interference fringes $v$
give the integrated first-order coherence $c^{(1)}$ Loredo2019 ;
Wein2021PhotonnumberEG  of the photonic field, where $v\equiv
c^{(1)}=(1/\mu_{\mathrm{f}})\int
dt\left|\braket{\hat{a}_{\mathrm{f}}(t)}\right|^{2}$. If the mode contains no
more than one photon, then
$\braket{\hat{a}_{\mathrm{f}}(t)}=\sqrt{p_{0}p_{1}}\zeta(t)$ and
$\mu_{\mathrm{f}}=p_{1}$. Thus, $\mathcal{C}$ can be determined from
$c^{(1)}=p_{0}\mathcal{C}$.

Experimentally, we interfere two fields which are matched in intensity,
polarization, and time-of-arrival at a 50:50 beam splitter. We monitor the
intensity at the two outputs of the beam splitter using two superconducting
nanowire single photon detectors (SNSPDs, Single Quantum Eos), which register
the single-photon counts as a function of time, with 100 ms resolution. When
interfering two fields that show photon-number coherence, the count rates at
the two outputs of the beam splitter are found be to anti-correlated. If the
two fields are in (out of) phase then we see maximum constructive
(destructive) interference in output 1, corresponding to a maximum (minimum)
count rate in detector 1, respectively. In order to determine the maximum
interference visibility we record the intensities measured by the two
detectors for approximately 20 minutes whilst we let the phase between the two
input fields, $\phi$, freely evolve, to ensure that the full phase space has
been explored. An example of the intensities measured by two detectors as a
function of time for $\theta=0.55\pi$ is given in Fig. S.5(a) with red (blue)
being the intensity measured by detector 1 (2). The inset of Fig. S.5(a) shows
a zoom of the signal, clearly displaying anti-correlated intensity signals.
The measured intensities vary over time due to the freely evolving phase.

Fig. S.5(b) shows the extracted visibility (`Eq. 9`) corresponding to the raw
data shown in Fig. S.5(a). We build a histogram of the measured visibility
values over time, as shown in Fig. S.5(c). We extract the maximum visibility
of interference, $v$, by taking the average of the absolute value of the $N$
highest, and $N$ lowest visibility data points (with $N=100$). The error is
calculated through error propagation and standard counting error. For the
example data set shown in Fig. S.5, we have indicated in Fig. S.5(c) the
corresponding $v$ with two vertical dashed lines and label.

If the wave packet has a small probability of containing more than one photon,
then $v\equiv c^{(1)}$ can overestimate $\mathcal{C}$. To correct for this
effect, we use the approach detailed in Ref. ollivier2021hong  and consider
that a small amount of noise is added to the ideal state in Eq. S.1 by a beam
splitter interaction. We then decompose the photonic field amplitude
$\braket{\hat{a}_{\mathrm{f}}(t)}$ into a contribution from the desired
quantum state (subscript $\mathrm{s}$) and the additional noise (subscript
$\mathrm{n}$)
$\braket{\hat{a}_{\mathrm{f}}(t)}=\cos(\vartheta)\braket{\hat{a}_{\text{s}}(t)}+\sin(\vartheta)\braket{\hat{a}_{\text{n}}(t)}$,
where $\vartheta$ is a noise parameter governing the amount of added noise.
Then, the integrated first-order coherence becomes

$\mu_{\mathrm{f}}c^{(1)}=p_{1}p_{0}\mathcal{C}\cos^{2}(\vartheta)+2\sqrt{p_{1}\mu_{\text{n}}}c^{(1)}_{\text{s,n}}\cos(\vartheta)\sin(\vartheta)+\mu_{\text{n}}c^{(1)}_{\text{n}}\sin^{2}(\vartheta),$
(S.34)

where
$\mu_{\mathrm{f}}=\mu_{\mathrm{s}}+\mu_{\mathrm{n}}=p_{1}+\mu_{\mathrm{n}}$,
$c^{(1)}_{\text{s,n}}=(1/\sqrt{\mu_{\mathrm{s}}\mu_{\mathrm{n}}})\int
dt\text{Re}\left[\braket{\hat{a}_{\mathrm{s}}(t)}\braket{\hat{a}_{\mathrm{n}}(t)}\right]$
quantifies the first-order coherence between the noise and the quantum state,
and $c^{(1)}_{\text{n}}=(1/\mu_{\mathrm{n}})\int
dt\left|\braket{\hat{a}_{\mathrm{n}}(t)}\right|^{2}$ quantifies the first-
order coherence of the classical noise itself. In our experiments, the noise
arises from reflected unfiltered laser from the fast qubit excitation pulse,
which is temporally separate from the light emitted into the mode of the
electromagnetic field by the qubit. Hence, the noise is not coherent with the
quantum state $c^{(1)}_{\mathrm{s,n}}=0$ but is itself coherent by definition
$c^{(1)}_{\mathrm{n}}=1$. In this notation, $g^{(2)}\ll 1$ can be similarly
written as in the supplementary of Ref. ollivier2021hong

$\displaystyle\mu_{\mathrm{f}}^{2}g^{(2)}=2(1+M_{\text{s,n}})p_{1}\mu_{\text{n}}\cos^{2}(\vartheta)\sin^{2}(\vartheta)+\mu_{\mathrm{n}}^{2}\sin^{4}(\vartheta)$
(S.35)

where $M_{\text{s,n}}=(1/\mu_{\mathrm{s}}\mu_{\mathrm{n}})\iint
dtdt^{\prime}\text{Re}\left[G^{(1)}_{\mathrm{s}}(t,t^{\prime})G^{(1)}_{\mathrm{n}}(t^{\prime},t)\right]\simeq
0$ is the mean wave packet overlap between the emitted quantum state and the
temporally separate classical noise photons. Note that here we used
$g^{(2)}_{\mathrm{s}}=0$ and $g^{(2)}_{\mathrm{n}}=1$. Defining
$\cos^{2}(\eta)=p_{1}\cos^{2}(\vartheta)/\mu_{\mathrm{f}}$ and
$\sin^{2}(\eta)=\mu_{\text{n}}\sin^{2}(\vartheta)/\mu_{\mathrm{f}}$, we can
re-write our expressions in terms of a single parameter:

$\displaystyle c^{(1)}(\eta)$
$\displaystyle=p_{0}\mathcal{C}\cos^{2}(\eta)+\sin^{2}(\eta)$ (S.36)
$\displaystyle g^{(2)}(\eta)$
$\displaystyle=2\cos^{2}(\eta)\sin^{2}(\eta)+\sin^{4}(\eta)$

Clearly, we have $c^{(1)}=p_{0}\mathcal{C}$ when $\eta\rightarrow 0$ as
expected for the ideal case. If nonzero, the lowest-order correction is then
given by $\lim_{\eta\rightarrow
0}(dc^{(1)}(\eta)/dg^{(2)}(\eta))=(1-p_{0}\mathcal{C})/2$. Hence we have

$p_{0}\mathcal{C}\simeq\frac{c^{(1)}-g^{(2)}/2}{1-g^{(2)}/2}.$ (S.37)

Note that this formalism allows to account for the residual $g^{(2)}$ in the
visibility measurement (blue and red curves in Fig. 2(a) and Fig. 2(b)).
Indeed, a small fraction of laser signal in the measured field leads to
classical interference that artificially increases the amount of unitary work
transferred. This effect contributes all the more as we increase $\theta$ as
attested by an increased second-order intensity correlation of the emitted
photonic field.

## V. Glossary

Table S.1: List of terms, symbols, and definitions for quantities related to energetics. Name | Symbol | Definition or constraint
---|---|---
Qubit initial state | $\ket{\Psi_{\mathrm{q}}}$ | $\cos(\theta/2)\ket{g}+\sin(\theta/2)e^{i\phi}\ket{e}$
Qubit initial energy | $\mathcal{E}_{\mathrm{q}}$ | $\hbar\omega_{0}\sin^{2}(\theta/2)$
Qubit initial coherence | $s$ | $\cos(\theta/2)\sin(\theta/2)$
Photonic field mode | $\hat{a}_{\mathrm{f}}$ | -
Photonic field envelope (intensity profile) | $I_{\mathrm{f}}(t)$ | $\braket{\hat{a}^{\dagger}_{\mathrm{f}}(t)\hat{a}_{\mathrm{f}}(t)}$
Photonic field initial energy | $\mathcal{E}_{\mathrm{f}}$ | $\hbar\omega_{0}\mu_{\mathrm{f}}=\mathcal{E}_{\mathrm{q}}(0)$
Photonic field ideal state | $\ket{i_{\mathrm{f}}}$ | $\cos(\theta/2)\ket{0}+\sin(\theta/2)e^{i\phi}\ket{1}$
Photonic field ideal unitary energy | $E^{\mathrm{q,f}}_{\mathrm{unit}}$ | $\hbar\omega_{0}s^{2}$
Qubit-Photonic field energy transfer efficiency | $\eta_{\mathrm{q,f}}$ | $E^{\mathrm{q,f}}_{\mathrm{unit}}/\mathcal{E}_{\mathrm{f}}$
Classical field mode (coherent state) | $\hat{a}_{\mathrm{c}}$ | -
Classical field amplitude | $\beta(t)$ | $\braket{\hat{a}_{\mathrm{c}}(t)}$
Classical field envelope (intensity profile) | $I_{\mathrm{c}}(t)$ | $\braket{\hat{a}_{\mathrm{c}}^{\dagger}(t)\hat{a}_{\mathrm{c}}(t)}=|\beta(t)|^{2}$
Classical field initial energy | $\mathcal{E}_{\mathrm{c}}$ | $\hbar\omega_{0}\mu_{\mathrm{c}}$
Table S.2: List of terms, symbols, and definitions for quantities for the
analysis of photonic states.

Name | Symbol | Definition or constraint
---|---|---
Photon creation operator (time basis) | $\hat{a}^{\dagger}(t)$ | $[\hat{a}(t),\hat{a}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$
Photonic density operator (number basis) | $\hat{{\rho}}$ | $p_{0}\hat{{\rho}}_{0}+p_{1}\hat{{\rho}}_{1}+\sqrt{p_{0}p_{1}}(\hat{{\rho}}_{01}+\hat{{\rho}}_{10})+\cdots$
Vacuum state | $\hat{{\rho}}_{0}$ | $\ket{0}\\!\bra{0}$
Single-photon state | $\hat{{\rho}}_{1}$ | $\iint dtdt^{\prime}\xi(t,t^{\prime})\hat{a}^{\dagger}(t)\ket{0}\\!\bra{0}\hat{a}(t^{\prime})$
Single-photon temporal density function | $\xi(t,t^{\prime})$ | $\text{Tr}[\hat{a}^{\dagger}(t^{\prime})\hat{a}(t)\hat{\rho}_{1}]$
Number coherence (between $\ket{0}$ and $\ket{1}$) | $\hat{{\rho}}_{01}$ | $\int dt\zeta(t)\hat{a}^{\dagger}(t)\ket{0}\\!\bra{0}$
Temporal number coherence amplitude | $\zeta(t)$ | $\text{Tr}\left[\hat{a}(t)\hat{\rho}_{01}\right]$
Temporal wave packet (first-order correlation)∗ | $G^{(1)}(t,t^{\prime})$ | $\braket{\hat{a}^{\dagger}(t^{\prime})\hat{a}(t)}$
Temporal envelope (intensity profile) | $I(t)$ | $\braket{\hat{a}^{\dagger}(t)\hat{a}(t)}$
Average photon number | $\mu$ | $\int dtI(t)=\sum_{n}np_{n}$
Mean wave packet overlap | $M_{i,j}$ | $\frac{1}{\mu_{i}\mu_{j}}\iint dtdt^{\prime}\text{Re}\left[G^{(1)}_{i}(t,t^{\prime})G^{(1)}_{j}(t^{\prime},t)\right]$
Indistinguishability | $M$ | $\frac{1}{\mu^{2}}\iint dtdt^{\prime}\left|G^{(1)}(t,t^{\prime})\right|^{2}$
Single-photon indistinguishability (trace purity) | $M_{\mathrm{s}}$ | $\iint dtdt^{\prime}\left|\xi(t,t^{\prime})\right|^{2}=\text{Tr}[\hat{{\rho}}_{1}^{2}]$
Normalized† first-order coherence amplitude overlap | $c^{(1)}_{i,j}$ | $\frac{2}{\mu_{i}+\mu_{j}}\int dt\text{Re}\left[\braket{\hat{a}_{i}(t)}\braket{\hat{a}_{j}^{\dagger}(t)}\right]$
Number purity | $\mathcal{C}$ |  $\int dt\left|\zeta(t)\right|^{2}=\text{Tr}\left[\hat{{\rho}}_{01}\hat{{\rho}}_{10}\right]$, $0\leq\mathcal{C}\leq\sqrt{M_{\mathrm{s}}}$
Number purity parameter | $\lambda^{2}$ |  $\mathcal{C}/\sqrt{M_{\mathrm{s}}}$, $0\leq\lambda\leq 1$
Second-order (intensity) correlation | $G^{(2)}(t,t^{\prime})$ | $\braket{\hat{a}^{\dagger}(t)\hat{a}^{\dagger}(t^{\prime})\hat{a}(t^{\prime})\hat{a}(t)}$
Normalized time-integrated second-order correlation‡ | $g^{(2)}(\tau)$ | $\frac{1}{\mu}\int dtG^{(2)}(t,t+\tau)$
Normalized fully integrated second-order correlation‡ | $g^{(2)}$ | $\frac{1}{\mu^{2}}\iint dtdt^{\prime}G^{(2)}(t,t^{\prime})=\frac{2}{\mu^{2}}\sum_{n}\binom{n}{2}p_{n}$

∗ Upper-case $G$ is used for unnormalized correlations, lower-case $g$ is used
for normalized correlations.
† An alternative convenient normalization is $1/\sqrt{\mu_{i}\mu_{j}}$, which
is equivalent when $\mu_{i}=\mu_{j}$.
‡ Note that $g^{(2)}\neq g^{(2)}(0)$ in this notation.