# Coherent scattering from coupled two level systems

Thomas Nutz  Controlled Quantum Dynamics Theory Group, Imperial College
London, London SW7 2AZ, United Kingdom  Quantum Engineering Technology Labs,
H. H. Wills Physics Laboratory and Department of Electrical and Electronic
Engineering, University of Bristol, BS8 1FD, UK  Samuel T. Mister
<EMAIL_ADDRESS>These authors contributed equally to this work
Quantum Engineering Technology Labs and Quantum Engineering Centre for
Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical
and Electronic Engineering, University of Bristol, BS8 1FD, UK  Petros
Androvitsaneas  Andrew Young  E. Harbord  J. G. Rarity  Ruth Oulton  Dara P.
S. McCutcheon  Quantum Engineering Technology Labs, H. H. Wills Physics
Laboratory and Department of Electrical and Electronic Engineering, University
of Bristol, BS8 1FD, UK

###### Abstract

We study the resonance fluorescence properties of an optically active spin
$1/2$ system, elucidating the effects of a magnetic field on the coherence of
the scattered light. We derive a master equation model for this system that
reproduces the results of a two level system (TLS) while also being applicable
to a spin system with ground state coupling. This model is then solved
analytically in the weak excitation regime. The inclusion of spin dynamics in
our model alters the properties of the coherently scattered light at a
fundamental level. For a TLS the coherence properties are known to be
determined by the input laser. We show that spin scattered light inherits the
coherence properties of the spin. This mapping allows us to measure spin
dynamics and coherence time through direct measurement of the scattered
fields. Furthermore, we show the ability to resolve sub-natural linewidth
zeeman splittings. Along with representing an invaluable tool for spin
spectroscopy understanding the coherence properties of the spin-scattered
field will be vital for spin-photon based quantum technologies.

##  I Introduction

Spin–photon interactions give rise to effects and applications beyond those
possible with a simple two level system, as output photons become entangled
with the internal spin states. This has a myriad of applications involving
entanglement distributon Humphreys-2018 , and the production of multi-photon
entangled states lindner_proposal_2009 ; Gershoni-2023 . In particular,
semiconductor quantum dot (QDs) and defect centre single photon sources have
been shown theoretically lindner_proposal_2009 ; economou_optically_2010 ;
lee_quantum_2019 ; denning_protocol_2017-1  and now experimentally
schwartz_deterministic_2016 ; vasconcelos_scalable_2019 ; scerri_frequency-
encoded_2018 ; delteil_generation_2016 ; stockill_phase-tuned_2017  to emit
entangled photons. While some methods of conventional quantum optics have been
successfully applied to these solid-state systems, new approaches are needed
to realize the potential of this hybrid domain of quantum physics.

A crucial step in the theoretical description of these entanglement schemes is
the assumption that the four-level system formed by the two states of a spin
$1/2$ particle and their corresponding optically accessible excited states can
be treated as two independent two-level schemes (TLS), as far as the
light–matter interaction is concerned. This assumption makes it possible to
use central techniques and results of quantum optics such as single-photon
wavepacket emission economou_unified_2005 ; lindner_proposal_2009 ;
nguyen_ultra-coherent_2011 , the giant phase shift hu_giant_2008 ;
androvitsaneas_charged_2016 ; hofmann_optimized_2003 ; nutz_stabilization_2019
, or the spectrum of resonance fluorescence mollow_power_1969 ;
cresser1982resonance ; matthiesen_subnatural_2012 ; proux_measuring_2015 ;
bennett_cavity-enhanced_2016 . Such a treatment is valid in the regime of
negligible coupling between the two TLS, however some amount of coupling is
required in most entanglement schemes. We address this challenge by developing
the theory of a simple and yet important and versatile system, namely a single
spin in a magnetic field interacting with laser light. This theory allows us
to elucidate several effects arising due to the replacement of the
conventionally considered ground-excited state system with a spin system as a
quantum emitter. Most importantly the presence of this spin gives rise to a
new type of coherent scattering with a characteristic spectrum that we refer
to as spin scattering. The system we consider consists of two coupled two-
level systems (TLS). This system is naturally found in charged III-V
semiconductor QDs warburton_single_2013 , where the two ground states can be
identified as electron spin states and the coupling is provided by a magnetic
field orthogonal to the optical axis. We investigate the coherence properties
of the light scattered from this system, and find it to differ qualitatively
from light scattered by a single or two independent TLS emitters. Most
notably, in the conventional coherent scattering regime of a single TLS,
characterised by low excitation probability and weak laser driving, the
scattered light inherits the coherence properties of the laser, such that a
single-frequency excitation laser would lead to scattered light with a Dirac
$\delta$-distribution-like emission spectrum mollow_power_1969 ;
carmichael1999statistical ; GIBBS197687 ; volz_atom_2007 . This central
quantum optical result, however, does not hold true for our spin system. We
show the scattered light does not inherit the coherence properties of the
laser, but those of the spin. The mapping of the spin coherence on to the spin
scattered field manifests as the spectrum being split into two narrow spectral
lines. The separation of these lines is related to the coupling between the
two ground states. Additionally the bandwidth of the individual spectral
features is related to dephasing processes. We further show that even if this
spin lives in a completely decoherence-free environment, the excitation laser
itself acts as a source of dephasing, such that some amount of spectral
broadening of the scattered light is unavoidable.

##  II Modeling the dynamics of the combined spin-photon system

We frame our analysis in terms of the optically active states in a charged
III-V quantum dot (QD). The ground states constitute an effective spin $1/2$
system with basis states obeying $\sigma_{z}\ket{\uparrow/\downarrow}=\pm
1/2\ket{\uparrow/\downarrow}$, where the quantisation axis $z$ with Pauli
operator $\sigma_{z}$ is chosen along the optical and growth axis of the QD
(Faraday basis). This spin resides in a magnetic field and is exposed to laser
light via an optically driven cavity. The two spin ground states couple to two
orthogonally polarised optical modes of the cavity which in turn couple to two
orthogonally polarised continua of free-space modes. For concreteness we
assume that the ground state $\ket{\uparrow}$ couples to a right-hand
circularly polarised cavity mode through an excited state $\ket{\Uparrow}$,
while $\ket{\downarrow}$ couples to a left-hand circularly polarised cavity
mode via $\ket{\Downarrow}$. In a III-V QD these excited states correspond to
charged excitons (trions) of angular momentum $\pm 3/2$. A schematic of this
system is shown in Fig. 1, and our aim is describe the effect of the laser
excitation on the system, and in turn the emitted field characteristics.

The full Hamiltonian containing spin, cavity and port mode degrees of freedom
is

$H=H^{0}_{s}+H^{0}_{c}+H^{0}_{p}+H_{cp}^{I}+H_{sc}^{I}$ (1)

where (set $\hbar=1$)

$\begin{split}H^{0}_{s}&=\omega_{0}P_{e}+\omega_{B}\sigma_{x}\\\
H^{0}_{c}&=\omega_{c}\left(a_{R}^{\dagger}a_{R}+a_{L}^{\dagger}a_{L}\right)\\\
H^{0}_{p}&=\sum_{k}\omega_{k}\left(r_{k,R}^{\dagger}r_{k,R}+r_{k,L}^{\dagger}r_{k,L}\right)\end{split}$
(2)

with excited state energy $\omega_{0}$ and excited state projector
$P_{e}=\outerproduct{\Uparrow}{\Uparrow}+\outerproduct{\Uparrow}{\Uparrow}$.
The magnetic field is assumed perpendicular to the optical axis, and is
described by the Zeeman term
$\omega_{B}\sigma_{x}=\omega_{B}\left(\outerproduct{\uparrow}{\downarrow}+\outerproduct{\downarrow}{\uparrow}\right)$.
The right- and left-handed cavity modes with annihilation operators $a_{R}$
and $a_{L}$ have photon energy $\omega_{c}$, and each couples to a quasi-
continuum of port modes with annihilation operators $r_{k,R}$ and $r_{k,L}$,
with energies $\omega_{k}$. The interaction part of the Hamiltonian is given
by

$\begin{split}H_{cp}^{I}&=\sum_{k}\kappa^{*}_{k}\left(r_{k,R}a_{R}^{\dagger}+r_{k,L}a_{L}^{\dagger}\right)+\mathrm{H.c.}\\\
H_{sc}^{I}&=g\left(S_{-}^{R}a_{R}^{\dagger}+S_{-}^{L}a_{L}^{\dagger}+\mathrm{H.c.}\right),\end{split}$
(3)

where $S_{-}^{R}=\outerproduct{\uparrow}{\Uparrow}$ and
$S_{-}^{L}=\outerproduct{\downarrow}{\Downarrow}$, while $\kappa_{k}$ and $g$
denote coupling constants.

Figure 1: Schematic of a spin-photon system realized by a charged quantum dot
inside an optical cavity. The round inset shows a level scheme with couplings
as described by the Master equation of Eq. 8, where $\Omega_{R/L}$ are Rabi
frequencies and $\Gamma$ denotes the Purcell-enhanced decay rate.

###  II.1 Output field

The derivation of the output field in terms of the spin system proceeds
analogously for both polarizations, and as such we only present the essential
steps for right-handed ($R$) polarization. In a standing-wave basis the
electric field can be written as
$\bm{E}_{R}=\bm{\hat{e}}_{R}E_{R}^{(+)}(z,t)+\bm{\hat{e}}^{*}_{R}E_{R}^{(-)}(z,t)$
with positive frequency component

$E_{R}^{(+)}(z,t)=\sum_{k}\sqrt{\frac{\omega_{k}}{4\epsilon_{0}AL}}\Big{(}\mathrm{e}^{i\phi_{R}}\mathrm{e}^{i\omega_{k}z/c}+\mathrm{e}^{-i\omega_{k}z/c}\Big{)}r_{k,R}(t),$
(4)

for $z>0$. We write the Jones vectors for right/left handed circularly
polarized light as $\bm{\hat{e}}_{R/L}=1/\sqrt{2}(1,\pm i)$, respectively, and
will also use $\bm{\hat{e}}_{H}=1/\sqrt{2}(\bm{\hat{e}}_{R}+\bm{\hat{e}}_{L})$
and $\bm{\hat{e}}_{V}=-i/\sqrt{2}(\bm{\hat{e}}_{R}-\bm{\hat{e}}_{L})$ for
horizontally and vertically polarized light in the following. $AL$ is the mode
volume of the imagined port mode cavity, and $\phi_{R}$ denotes the phase
incurred upon reflection from the top mirror.

Following a procedure detailed in Appendix A we can decompose the field
operator $E_{R}^{(+)}(z,t)$ into the free field $E_{R,f}^{(+)}(z,t)\sim
r_{k,R}(0)\exp(-i\omega_{k}t)\equiv r_{k,R}^{f}(t)$ reflected by the top
mirror of the cavity, and the source field $E_{R,s}^{(+)}(z,t)\sim a_{R}(t)$,
i.e. the field inside the cavity leaking through the top mirror into to port
modes. This cavity field depends on the field of the dipole $\propto
S_{-}^{R}$ due to the coupling Hamiltonian $H_{sc}^{I}$. We then assume that
the dipole $S_{-}^{R}(t)$ undergoes dynamics with a Fourier transform centered
at some frequency $\omega_{d}$ and a bandwidth much smaller than the cavity
bandwidth $\kappa=2\pi g(\omega_{c})|\kappa(\omega_{c})|^{2}$, where
$g(\omega_{c})$ is the density of port modes at the cavity resonance frequency
$\omega_{c}$. This assumption will be confirmed self-consistently in the
Master equation treatment presented in Subsec. II.2, where we find
$\omega_{d}$ to correspond to the input laser frequency, while the bandwidth
of dipole dynamics is limited by the decay rate $\Gamma\sim g^{2}/\kappa$.
Working in the Purcell regime given by $\Gamma\ll\kappa$ we can then decompose
the outgoing field evolving with retarded time $t_{o}=t-z/c$ as

$E^{(+)}_{R,\mathrm{out}}(z,t)=\sum_{k}\sqrt{\frac{\omega_{k}}{4\epsilon_{0}AL}}R_{k}r_{k,R}^{f}(t_{o})+\xi
S_{-}^{R}(t_{o}),$ (5)

with
$\xi=-2g\frac{\kappa}{\kappa_{c}^{*}}\sqrt{\omega_{c}/(4\epsilon_{0}AL)}(i(\omega_{c}-\omega_{d})+\kappa/2)^{-1}$
and where the reflection coefficient of the empty cavity is given by

$R_{k}=\frac{i(\omega_{c}-\omega_{k})-\kappa/2}{i(\omega_{c}-\omega_{k})+\kappa/2}.$
(6)

The first term in Eq. (5) can be interpreted as the light reflected by the
cavity without having interacted with the electronic system, while the second
term $\propto S_{-}^{R}(t_{o})$ describes the light that is radiated by the
dipole into the cavity mode and then transmitted through the top mirror into
the output field. Note that solving the Heisenberg equation $\dot{S}_{-}(t)$
by Fourier transform in the weak-driving limit where $P_{e}\approx 0$ yields
the phase shift obtained using input-output theory for a two-level system
coupling to a cavity auffeves-garnier_giant_2007 , as shown in Appendix A.

While the selection rules single out the circular polarizations $R$ and $L$ as
good bases for calculations involving the coupling Hamiltonian $H_{sc}^{I}$,
an interesting experimental configuration is given by a horizontally polarized
excitation laser and measurement of the scattered light in the orthogonal
linear polarization. The corresponing fields are readily calculated from Eq.
(5), for instance the positive frequency component of the vertically polarized
($V$) field is given by

$\begin{split}E_{V}^{(+)}(t)&=\frac{i}{\sqrt{2}}\left(E^{(+)}_{R,\mathrm{out}}(t)-E^{(+)}_{L,\mathrm{out}}(t)\right)\\\
&=\sum_{k}\sqrt{\frac{\omega_{k}}{4\epsilon_{0}AL}}R_{k}r_{k,V}^{f}(t_{o})+\xi
S_{-}^{V}(t_{o})\end{split}$ (7)

The operators corresponding to vertical polarization are
$r_{k,V}=i/\sqrt{2}\left(r_{k,R}-r_{k,L}\right)$ and
$S_{-}^{V}(t)=i/\sqrt{2}\left(S_{-}^{R}(t)-S_{-}^{L}(t)\right)$. We can
already see from this expression that if the input field is horizontally
polarized, i.e. $\langle r_{k,R}(0)\rangle=\langle r_{k,L}(0)\rangle$, then
$\langle r_{k,V}^{f}(t_{o})\rangle=0$ and therefore the vertically polarized
output field (referred to as cross-polarized field) is solely dependent on the
dipole field $\propto S_{-}^{V}$. In this way the excitation laser can be
filtered out and we only need to consider the dipole field in the description
of the cross-polarized field. We furthermore find that for a resonant and weak
horizontally polarized input field all the laser power is scattered into the
vertical polarization, in accordance with the input-output calculation used to
explain the giant phase shift measured in androvitsaneas_efficient_2019 .

###  II.2 Master equation for the spin system

To calculate field expectation values using Eq. (5) we need to find the time
evolution of the dipole operator $S_{-}^{R}(t)$, and in order to do so we
derive a master equation for the spin state $\rho$ in the Hilbert space
spanned by
$\\{\ket{\uparrow},\ket{\Uparrow},\ket{\downarrow},\ket{\Downarrow}\\}$. The
derivation proceeds in two steps, the first tracing over the port modes to
yield a master equation for the joint cavity mode–spin system, and the second
step eliminates the cavity mode in the weak coupling Purcell regime. We assume
the initial state $\ket{\psi_{0}}$ of the port mode system to be the vacuum
except for two modes $r_{d,R}$ and $r_{d,L}$ satisfying
$r_{d,R}\ket{\psi_{0}}=\alpha_{R/L}\ket{\psi_{0}}$, i.e. a coherent state
describing a laser field oscillating at frequency $\omega_{k}=\omega_{d}$.
Once again, the derivation for the two polarisation proceeds analogously. We
refer to Appendix C for the detailed derivation and state the resulting master
equation for the spin system with adiabatically eliminated cavity mode in a
frame rotating at $\omega_{d}$ as

$\dot{\rho}(t)=-i[H_{s},\rho(t)]+\sum_{k=R,L}L_{k}\rho(t)L_{k}^{\dagger}-\frac{1}{2}\left\\{L_{k}^{\dagger}L_{k},\rho(t)\right\\},$
(8)

with

$H_{s}=\Delta
P_{e}+\omega_{B}\sigma_{X}+\left(\Omega_{R}S_{-}^{R}+\Omega_{L}S_{-}^{L}+\mathrm{H.c.}\right)$
(9)

and Lindblad operator $L_{R/L}=\sqrt{\Gamma}S_{-}^{R/L}$, where the Purcell
enhanced decay rate and renormalised detuning are

$\Gamma=\frac{g^{2}\kappa}{\Delta_{c}^{2}+\kappa^{2}/4}\ \mathrm{and}\
\Delta=\Delta_{0}+\frac{\Delta_{c}g^{2}}{\Delta_{c}^{2}+\kappa^{2}/4},$ (10)

where $\Delta_{0/c}=\omega_{0/c}-\omega_{d}$. The driving parameters are given
by

$\Omega_{R/L}=i\frac{g\alpha_{R/L}}{\kappa/2-i(\omega_{c}-\omega_{d})}.$ (11)

The spin system therefore obeys a master equation describing classical driving
with Rabi frequencies $\Omega_{R/L}$ and a cavity-mediated decay rate
$\Gamma$. Note that the polarization of the laser is encoded in the relative
magnitude and phase of $\Omega_{R}$ and $\Omega_{L}$.

While the spin system basis spanned by
$\left\\{\ket{\uparrow},\ket{\Uparrow},\ket{\downarrow},\ket{\Downarrow}\right\\}$
allowed us to treat polarizations $R$ and $L$ separately in the previous
derivations, the so-called Voigt basis can be advantageous in the analysis of
the dynamics of this spin system. The Voigt basis is
$\\{\ket{+},\ket{P},\ket{-},\ket{M}\\}$ with basis states

$\begin{split}\ket{\pm}&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}\pm\ket{\downarrow}\right),\\\
\ket{P/M}&=\frac{1}{\sqrt{2}}\left(\ket{\Uparrow}\pm\ket{\Downarrow}\right).\end{split}$
(12)

In this basis the system Hamiltonian for a horizontally polarized laser
($\Omega_{R}=\Omega_{L}\equiv\Omega$) is given by (c.f. Eq. (9))

$H_{s}=\begin{pmatrix}\Delta&\Omega&0&0\\\ \Omega&\omega_{B}&0&0\\\
0&0&-\omega_{B}&\Omega\\\ 0&0&\Omega&\Delta\end{pmatrix},$ (13)

while the spontaneous decay of the excited states is described by two Lindblad
operators

$\begin{split}L_{H}&=\sqrt{\frac{\Gamma}{2}}\left(\outerproduct{+}{P}+\outerproduct{-}{M}\right)\equiv\sqrt{\Gamma}S^{H}_{-}\\\
L_{V}&=\sqrt{\frac{\Gamma}{2}}\left(\outerproduct{+}{M}+\outerproduct{-}{P}\right)\equiv\sqrt{\Gamma}S^{V}_{-}.\end{split}$
(14)

Note that unlike in the Faraday basis, the Hamiltonian is block diagonal in
the Voigt basis, while the decay operators are not. Which means that in this
basis the the ground states are static within the magnetic field and
population exchange is mediated through the optical transition which are
excited by the drive laser. The energy level scheme and its couplings in this
basis are shown in Fig. 2.

In this basis it is straight forward to apply the methods described in Ref.
reiter_effective_2012  to adiabatically eliminate the excited states (weak
excitation approximation) which yields an effective master equation for the
approximate state $\rho_{\mathrm{eff}}$ which exists in the ground state
subspace spanned by $\\{\ket{+},\ket{-}\\}$ alone.

$\dot{\rho}_{\mathrm{eff}}=-i[H_{\mathrm{eff}},\rho_{\mathrm{eff}}]\\!+\\!\sum
L_{\mathrm{eff}}^{k}\rho_{\mathrm{eff}}L_{\mathrm{eff}}^{k\dagger}-\frac{1}{2}\\{L_{\mathrm{eff}}^{k\dagger}L_{\mathrm{eff}}^{k},\rho_{\mathrm{eff}}\\}$
(15)

where we have defined

$\begin{split}H_{\mathrm{eff}}&=\tilde{\omega}_{B}\big{(}\outerproduct{+}{+}-\outerproduct{-}{-}\big{)},\\\
L_{\mathrm{eff}}^{H}&=\gamma_{-}\outerproduct{+}{+}+\gamma_{+}\outerproduct{-}{-},\\\
L_{\mathrm{eff}}^{V}&=\gamma_{-}\outerproduct{-}{+}+\gamma_{+}\outerproduct{+}{-},\end{split}$
(16)

with renormalised Zeeman energy

$\tilde{\omega}_{B}=\omega_{B}+\frac{1}{2}\left(\frac{\Omega^{2}(\omega_{B}-\Delta)}{(\omega_{B}-\Delta)^{2}+\left(\frac{\Gamma}{2}\right)^{2}}+\frac{\Omega^{2}(\omega_{B}+\Delta)}{(\omega_{B}+\Delta)^{2}+\left(\frac{\Gamma}{2}\right)^{2}}\right)$
(17)

and complex rates

$\gamma_{-}=\frac{\Omega\sqrt{\frac{\Gamma}{2}}}{\Delta-\omega_{B}-i\frac{\Gamma}{2}},\qquad\gamma_{+}=\frac{\Omega\sqrt{\frac{\Gamma}{2}}}{\Delta+\omega_{B}-i\frac{\Gamma}{2}}.$
(18)

Inspection of this effective master equation reveals that optical driving
directly affects the ground state spin dynamics. There is a threefold effect
of the optical driving, namely an energy shift described by
$H_{\mathrm{eff}}$, a dephasing process described by $L_{\mathrm{eff}}^{H}$,
and a spin flip process via virtual optical excitation corresponding to
vertical decay processes described by $L_{\mathrm{eff}}^{V}$.

To understand the dynamics described by this effective master equation is it
convenient to recast it in terms of a Bloch vector $\bm{r}$ which describes
the quantum state of the spin $1/2$ particle in the ground state manifold
spanned by $\\{\ket{\pm}\\}$. That is, we write
$\rho_{\mathrm{eff}}=\frac{\mathbbm{1}}{2}+r_{x}\sigma_{x}+r_{y}\sigma_{y}+r_{z}\sigma_{z}$
where we define the Pauli matrices in the Voigt basis as
$\sigma_{z}=\frac{1}{2}\left(\outerproduct{+}{+}-\outerproduct{-}{-}\right)$,
$\sigma_{x}=\frac{1}{2}\left(\outerproduct{+}{-}+\outerproduct{-}{+}\right)$,
$\sigma_{y}=i[\sigma_{x},\sigma_{z}]$ and Bloch vector components
$r_{i}=\mathrm{Tr}\\{\sigma_{i}\rho_{\mathrm{eff}}\\}$. We then find the
components $r_{x}$, $r_{y}$ to obey

$\frac{d}{dt}\begin{pmatrix}r_{x}\\\
r_{y}\end{pmatrix}=\begin{pmatrix}R-\gamma_{\Sigma}&-(2\tilde{\omega}_{B}-I)\\\
2\tilde{\omega}_{B}&-\gamma_{\Sigma}\end{pmatrix}\begin{pmatrix}r_{x}\\\
r_{y}\end{pmatrix},$ (19)

Where
$\gamma_{\Sigma}=\absolutevalue{\gamma_{+}}^{2}+\absolutevalue{\gamma_{-}}^{2}$
and further $R$ and $I$ are defined through the following decomposition of
$\gamma_{+}\gamma_{-}^{*}=\frac{R}{2}+i\frac{I}{2}$. This is a system of
linear differential equations and can be solved by standard methods. We find
the following solution.

$\begin{pmatrix}r_{x}\\\ r_{y}\end{pmatrix}=e^{-\gamma
t}(c_{+}\textbf{v}_{+}e^{2i\omega_{e}t}+c_{-}\textbf{v}_{-}e^{-2i\omega_{e}t})$
(20)

with total dephasing rate $\gamma=\gamma_{\Sigma}-R/2$. $\omega_{e}$ is given
by

$\omega_{e}=\sqrt{\tilde{\omega}_{B}(\tilde{\omega}_{B}-I/2)-(R/4)^{2}}.$ (21)

This effective precession frequency depends on the renormalised Zeeman energy
and the parameters R and I, this term characterises the competition between
Larmor precession and dephasing processes within the ground state manifold.
Finally, the vectors $\textbf{v}_{\pm}$ are given by

$\textbf{v}_{\pm}=\begin{pmatrix}\frac{2(\tilde{\omega}_{B}-I/2)}{R/2\mp
2i\omega_{e}}\\\ 1\end{pmatrix}=\begin{pmatrix}\mathrm{v}_{\pm}\\\
1\end{pmatrix},$ (22)

and the coefficients $c_{\pm}$ are determined through initial conditions. Eq.
20 shows the spin dynamics are described by damped harmonic motion, with
frequency $2\omega_{e}$.

###  II.3 Derivation of optical coherence properties

With our master equations in hand we can calculate the coherence properties of
the scattered field. The properties of interest are captured by the steady
state first order correlation function $g^{(1)}(\tau)=\langle
E^{(-)}(\tau)E^{(+)}(0)\rangle$ which, using Eq.7, can be show to be
proportional to $\langle S^{V}_{+}(\tau)S^{V}_{-}(0)\rangle$. Making use of
the quantum regression formula carmichael1999statistical ;
mccutcheon2015optical  this is given by

$\langle
S^{V}_{+}(\tau)S^{V}_{-}(0)\rangle=\mathrm{e}^{-i\omega_{d}\tau}\mathrm{Tr}\left\\{\bar{S}_{+}^{V}\exp(\mathcal{L}\tau)[\bar{S}_{-}^{V}\rho_{\mathrm{std}}]\right\\},$
(23)

where $\bar{S}^{V}_{-}=\exp(i\omega_{d}\tau)S^{V}_{-}$ is the dipole operator
in the rotating frame, $\mathcal{L}$ denotes the Liovillian superoperator
defined by the right hand side of Eq. (8), and $\rho_{\mathrm{std}}$ is the
steady-state density operator of the same equation.

Figure 2: Level scheme in the Voigt basis for horizontally polarized driving.
While the Hamiltonian does not couple the subsystems corresponding to
right/left handed polarization any more, there is a decay term $\propto
S^{V}_{-}$ that does introduce such transitions.

We now calculate the steady state first order correlation function in the
weak-excitation limit where $\Omega/\Gamma\ll 1$. Our procedure is to define
an auxiliary density operator $\rho^{\prime}(\tau)$ which obeys
$\dot{\rho}^{\prime}(\tau)=\mathcal{L}\rho^{\prime}(\tau)$ with
$\rho^{\prime}(0)=\bar{S}_{-}^{V}\rho_{\mathrm{std}}$, from which the relevant
correlation function can be found from
$\mathrm{Tr}\left\\{\bar{S}_{+}^{V}\rho^{\prime}(\tau)\right\\}=\langle-|\rho^{\prime}(\tau)|P\rangle+\langle+|\rho^{\prime}(\tau)|M\rangle\equiv\rho^{\prime}_{mP}+\rho^{\prime}_{pM}$.
The relevant matrix elements of $\dot{\rho}^{\prime}(\tau)$ are

$\begin{split}\frac{d}{d\tau}\rho^{\prime}_{mP}&=\left(i(\Delta+\omega_{B})-\frac{\Gamma}{2}\right)\rho^{\prime}_{mP}-i\Omega\left(\rho^{\prime}_{MP}-\rho^{\prime}_{mp}\right)\\\
\frac{d}{d\tau}\rho^{\prime}_{pM}&=\left(i(\Delta-\omega_{B})-\frac{\Gamma}{2}\right)\rho^{\prime}_{pM}-i\Omega\left(\rho^{\prime}_{PM}-\rho^{\prime}_{pm}\right),\end{split}$
(24)

where $\rho^{\prime}_{PM}\equiv\langle P|\rho^{\prime}(\tau)|M\rangle$,
$\rho^{\prime}_{MP}\equiv\langle M|\rho^{\prime}(\tau)|P\rangle$. We can
distinguish three types of matrix elements in these differential equations.
First there are elements $\rho^{\prime}_{mp/pm}$ living in the ground state
subspace. These matrix elements obey the effective master equation of Eq. 15
solved by Eq. 20 and decay on a long timescale
$1/\gamma\sim\Gamma/\Omega^{2}$. Secondly there are the matrix elements
$\rho^{\prime}_{mP/pM}$ that obey Eq. 24 and behave like dipoles, i.e. they
are suppressed by a decay term $\sim\Gamma$ but rise up due to a source term
$\sim\Omega\rho^{\prime}_{mp/pm}$, which yields a term
$\propto\Omega/\Gamma\rho^{\prime}_{mp/pm}$ upon integration. The third type
of matrix elements is given by terms $\rho_{PM/MP}$ living entirely in the
excited state subspace. These terms behave like excitation probabilities upon
integration, i.e. they are suppressed by another factor $\Omega/\Gamma$
relative to dipole-like terms. In the weak-excitation approximation we neglect
the terms $\rho_{PM/MP}$ in the excited state subspace, such that the
differential equations of Eq. 24 simplify and can be integrated using the
effective master equation solutions of Eq. 20 to yield

$\begin{split}\langle\bar{S}^{V}_{+}(\tau)\bar{S}^{V}_{-}(0)\rangle=\
&s_{+}\mathrm{e}^{(2i\omega_{e}-\gamma)\tau}+s_{-}\mathrm{e}^{(-2i\omega_{e}-\gamma)\tau},\end{split}$
(25)

where we have furthermore neglected fast-decaying terms
$\propto\exp(-\Gamma\tau)$. The coefficients $s_{\pm}$, can be determined
using the initial condition
$\rho^{\prime}(0)=\bar{S}_{V}^{-}\rho_{\mathrm{std}}$ and are found to be

$s_{\pm}=\frac{2\Omega^{2}}{N}\frac{(\mathrm{v}_{\mp}\omega_{B}+i\Delta-\frac{\Gamma}{2})(\mathrm{v}_{\pm}\alpha_{\pm}-\omega_{B})}{(\mathrm{v}_{\mp}-\mathrm{v}_{\pm})(\alpha_{\pm}^{2}+\omega_{B}^{2})},$
(26)

In this expression we have defined
$\alpha_{\pm}=\left(\frac{\Gamma}{2}-i\Delta-\gamma\right)\pm 2i\omega_{e}$
and $N=\Gamma^{2}+4\Delta^{2}+4\omega_{B}^{2}+8\Omega^{2}$.

This correlation function captures the coherence of cross-polarized light
scattered by two coupled two-level systems in the weak-driving regime and is a
generalisation of the seminal analysis of resonance fluorescence of a two-
level system (TLS) mollow_power_1969 .

It is easier to visualise the properties of this correlation function in the
frequency domain hence we calculate the emission spectrum using the Wiener-
Khinchin theorem. It states that in the steady-state the emission spectrum is
proportional to the fourier transform of the two time electric field
correlation function.

$S(\omega)=2\mathrm{Re}\left[\int_{0}^{\infty}d\tau\exp[-i\omega\tau]\langle
S^{V}_{+}(\tau)S^{V}_{-}(0)\rangle\right].$ (27)

Hence, for general $\omega_{B}$ but weak excitation we find from Eq. (25)

$S(\omega-\omega_{d})=2\mathrm{Re}\Bigg{[}\frac{s_{+}}{i\left(\omega-2\omega_{e}\right)+\gamma}+\frac{s_{-}}{i\left(\omega+2\omega_{e}\right)+\gamma}\Bigg{]}.$
(28)

When the coupling parameter $\omega_{B}=0$ our theory reproduces the TLS
analysis. This is the well-known coherent scattering (equivalently Heitler)
limit in which the correlation function becomes $\langle
S^{V}_{+}(\tau)S^{V}_{-}(0)\rangle=2\Omega^{2}/N$ and the corresponding
emission spectrum takes the form $S(\omega)\propto\delta(\omega-\omega_{d})$.
It should be noted that in any real system environmental noise, such as that
caused by nuclear spins, will induce additional spin dephasing effects that
will not map onto the TLS analysis. The form of this spectrum constitutes one
of our core findings, and is markedly different from the conventional
resonance fluorescence spectrum in the coherent scattering regime. We will
explore its implications in the following section. Furthermore, it should be
noted that the resulting spectrum of the cross-polarized field is independent
of the polarisation of the input field and it does not rely on the alignment
along the field.

##  III Coherence and Spectral properties of the cross-polarized field

Figure 3: a) The blue curve shows the cross-polarised emission spectrum for
the over damped regime, where depahsing induced by the drive field suppresses
oscillations generated by the Larmor precession. The orange curve shows the
cross-polarised emission spectrum for the under damped regime where the Larmor
precession dominates. b) These plots show the absolute value of the first
order correlation function for the same regimes (blue curve) over damped and
(orange curve) under damped. Figure 4: A selection of emission spectrum plots
for different parameter regimes. a)The cross polarised emission spectrum for
fixed driving power and varying magnetic field strength. This shows how the
ground state coupling splits the spectrum by $4\omega_{B}$. b) The cross
polarised emission spectrum for fixed magnetic field strength,
$\omega_{B}/\Gamma=0.25$, and varying Rabi frequency. These plots show that
increased drive power leads to increase in the spectral broadening. This
increase in damping also slightly modifies the natural precession frequency
leading to a shift in the spectral positions. c) The cross polarised emission
spectrum beyond the weak excitation approximation. Here we can see four peaks.
The two outer peaks correspond to spin scattered light while the two inner
peaks, centered at $\pm\omega_{B}$, are spontaneous emission from real
excitation. d) The cross polarised emission spectrum with $\Omega/\Gamma=1.5$
and varying spin dephasing rates. Showing that the outer spin scattered peaks
are broadened by the ground state dynamics while the real excitation peaks are
not.

In the previous section we showed that the spin dynamics are described by a
damped harmonic system (Eq. 20), which will have regimes of qualitatively
different behaviour, i.e., under/over damped. The novel observation here is
that the source of damping derives from the optical driving of the system and
scales with the strength of the drive field. The competition between this
process and the Larmor precession induced by the applied magnetic field now
define the regimes of harmonic motion for the spin, which is controlled by the
parameter $\omega_{e}$.

When $\omega_{e}$ is a real number and $\gg 0$, i.e. large field strengths,
the spin is under damped and Eq. (20) describes Larmor precession with a
frequency defined by the applied magnetic field ($2\omega_{B}$) and a damping
rate $\gamma\sim{\Omega}^{2}/\Gamma$, related to the strength of the driving
field. This behaviour is mapped to the output field and in the time domain
gives rise to an oscillating first order correlation function (Eq. 25) with an
exponentially decaying envelope, this has been observed experimentally
androvitsaneas2022quantum . Transforming in to the frequency domain we find a
split spectrum consisting of two lines at $\omega=\omega_{d}\pm 2\omega_{e}$,
both broadened by the optically induced spin dephasing rate $\gamma$, the
orange curve in Fig. 3(a). At the point where optical dephasing dominates the
spin becomes over damped and $\omega_{e}$ become an imaginary number. The spin
now dephases faster than the Larmor precession and Eq. (20) describes
biexponential decay with rates $\gamma\pm 2|\omega_{e}|$. In the time domain
of the output field the first order correlation function (Eq. 25) no longer
oscillates and simply decays, i.e., two overlapping spectral features both
centred at $\omega=\omega_{d}$ of widths $\gamma\pm 2|\omega_{e}|$, the blue
curve in Fig. 3(a).

These spectral features constitute a novel and interesting regime of optical
scattering. The position of the spectral lines is determined by the applied
magnetic field and their width by the optical driving power. We see that the
spectrum is split by approximately $4\omega_{B}$ (Fig. 4(a)), i.e., twice the
Zeeman energy. This is due to energy conservation, where production of
vertically polarised light must coincide with a spin flip transition
($\ket{+}\Leftrightarrow\ket{-}$). This energy loss(gain) of $2\omega_{B}$ is
compensated via a corresponding shift in the emitted photon to a higher(lower)
energy. It is also clear from Fig. 4(a) that increasing the magnetic field
reduces the amount of light scattered into the vertically polarised output
mode as the optical transitions detune from the input laser. Furthermore, in
Fig. 4(b) one observes that the spectral broadening of the individual peaks is
controlled by the driving power. This gives rise to two sub natural linewidth
peaks whose bandwidth is limited by the optically induced dephasing. This
effect constitutes a new self-broadening mechanism of laser scattering that
needs to be taken into account in optical studies of spin dephasing in the
weak excitation regime. Taken together these unique features are not be
captured via conventional treatment of resonance florescence from a TLS.

As the driving power is increased beyond the weak excitation limit our
analytic solution for the spectrum begins to breakdown (see Appendix D). To
provide insight into the system’s behavior beyond the weak excitation limit we
can numerically solve Eq. 8. This leads to Fig. 4(c) showing that the outer
spin scattered peaks are broadened but persist beyond the weak excitation
regime. Additionally, the emergence of two other peaks centered at
$\pm\omega_{B}$ is observed. These are due to real excitation giving rise to
two spontaneous emission lines with bandwidth $\Gamma$. To verify that the
outer peaks are indeed spin scattered light we add additional dynamics to the
ground state subspace that model non optical environmental dephasing effects.
To this end a term $L_{pd}=\sqrt{\frac{\gamma_{pd}}{2}}\sigma_{x}$ is added to
the Master equation Eq. 8. The emission spectra for different amounts of spin
dephasing are shown in Fig. 4(d). Clearly only the outer peaks are broadened
by this dissipation term, further confirming the mapping of the spin dynamics
to the output spectrum.

##  IV Conclusion

We study an optically active spin $1/2$ system and find it to be an
insightful, solvable model of coupled quantum emitters, thus generalizing the
analysis of coherent scattering that has long served as a paradigm in quantum
optics. Our analysis shows that in the weak driving limit the spectrum of spin
scattered light is determined by the spin dynamics rather than the coherence
of the excitation laser. This mapping could have useful implications for the
measurement of quantum systems, particularly for the nuclear spin environments
found in quantum dots, as numerous studies have shown that strong driving can
lead to a complicated feedback mechanism between the electron and nuclear spin
systems Economou-2014 ; Evers-2014 . Furthermore, the properties of the spin
scattered field are important for understanding the application of these
systems to quantum information tasks. For example, the efficiency/fidelity of
protocols for generating entangled photons denning_protocol_2017-1 ;
hu_giant_2008 , or implementing quantum memories or repeaters WaksDrop  can be
sensitive to these spectral and coherence properties. The methods used in this
analysis are also applicable to different configurations of coupled emitters,
such as neutral or multiply charged quantum dots, defect centres or atoms.

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##  Appendix A Derivation of the output field

Measured quantities are calculated from the upwards-propagating electric field
$\bm{E}(z,t)=\bm{E}_{R}(z,t)+\bm{E}_{L}(z,t)$, which we decompose into
positive and negative frequency components as
$\bm{E}_{R/L}=\bm{E}_{R/L}^{(+)}(z,t)+\bm{E}_{R/L}^{(-)}(z,t)$. In terms of
mode operators we write

$\bm{E}_{R}^{(+)}(z,t)=i\bm{\hat{e}}_{R}\sum_{k}\sqrt{\frac{\omega_{k}}{\epsilon_{0}AL^{\prime}}}\frac{1}{2i}\Big{(}\mathrm{e}^{i\phi_{R}}\mathrm{e}^{i\omega_{k}z/c}+\mathrm{e}^{-i\omega_{k}z/c}\Big{)}r_{k,R}(t)$
(29)

for $z>0$ (similarly for $\bm{E}^{(+)}_{L}(z,t)$). In this expression
$\bm{\hat{e}}_{R}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ i\end{pmatrix}$ is the
Jones vector for right-handed circularly polarized light propagating along
$z$, $AL^{\prime}$ is the mode volume of the imagined port mode cavity, and
$\phi_{R}$ denotes the optical phase incurred upon reflection from the top
mirror.

Solving the Heisenberg equation
$\dot{r}_{k,R}(t)=-i\omega_{k}r_{k,R}(t)-i\kappa_{k}a_{R}(t)$ obtained from
Eq. 1 yields

$r_{k,R}(t)=\mathrm{e}^{-i\omega_{k}t}r_{k,R}(0)-i\kappa_{k}\int_{0}^{t}dt^{\prime}\mathrm{e}^{-i\omega_{k}(t-t^{\prime})}a_{R}(t^{\prime}).$
(30)

Substituting this formal solution into Eq. 29, converting
$\sum_{k}\rightarrow\int_{0}^{\infty}d\omega g(\omega)$ with a mode density
$g(\omega)$, and approximating
$g(\omega)\sqrt{\frac{\omega}{\epsilon_{0}AL^{\prime}}}\kappa(\omega)$
constant over the bandwidth of the slowly varying cavity mode
$\tilde{a}_{R}(t)=\exp(i\omega_{c}t)a_{R}(t)$ (Markov approximation) yields
$\bm{E}^{(+)}_{R}(z,t)=\bm{E}^{(+)}_{R,f}(z,t)+\bm{E}^{(+)}_{R,s}(z,t)$, where

$\bm{E}^{(+)}_{R,f}(z,t)=i\bm{\hat{e}}_{R}\sum_{k}\sqrt{\frac{\omega_{k}}{\epsilon_{0}AL^{\prime}}}\frac{1}{2i}\Big{(}\mathrm{e}^{i\phi_{R}}\mathrm{e}^{-i\omega_{k}(t-\frac{z}{c})}+\mathrm{e}^{-i\omega_{k}(t+\frac{z}{c})}\Big{)}r_{k,R}(0)$
(31)

and

$\bm{E}^{(+)}_{R,s}(z,t)=-i\bm{\hat{e}}_{R}\mathrm{e}^{i\phi_{R}}\frac{\kappa}{2\kappa(\omega_{c})^{*}}\sqrt{\frac{\omega_{c}}{\epsilon_{0}AL^{\prime}}}a_{R}(t-z/c),$
(32)

and where we have defined $\kappa\equiv 2\pi
g(\omega_{c})|\kappa(\omega_{c})|^{2}$. In the next step we decompose the
source field $\propto a_{R}(t-z/c)$ into the reflected free field and the
field emitted by the dipole $S_{-}^{R}$ in the adiabatic approximation, which
assumes that the cavity bandwidth is much greater than the bandwidth of the
dipole dynamics. The validity of this approximation can be confirmed self-
consistently later on when we derive a master equation for the electron
dynamics.

The Heisenberg equation for mode operator $a_{R}(t)$ reads

$\dot{a}_{R}(t)=-i\omega_{c}a_{R}(t)-i\sum_{k}\kappa_{k}^{*}r_{k,R}(t)-igS_{-}^{R}(t).$
(33)

substituting the formal solution $r_{k,R}(t)$ from Eq. 30 yields

$\dot{a}_{R}(t)=-\left(i\omega_{c}+\frac{\kappa}{2}\right)a_{R}(t)-i\sum_{k}\kappa_{k}^{*}r_{k,R}(0)\mathrm{e}^{-i\omega_{k}t}-igS_{-}^{R}(t)$
(34)

where again the Markov approximation for the cavity mode has been made.
Integrating $\dot{a}_{R}(t)$ yields

$a_{R}(t)=-i\sum_{k}\kappa_{k}^{*}r_{k,R}(0)\frac{\mathrm{e}^{-i\omega_{k}t}}{i(\omega_{c}-\omega_{k})+\kappa/2}-i\frac{g}{i(\omega_{c}-\omega_{d})+\kappa/2}S_{-}^{R}(t)$
(35)

for $t\gg 1/\kappa$, where $\omega_{d}$ gives the centre frequency of the
oscillating dipole $S_{-}^{R}(t)$ (shown to equal the frequency of a driving
field later on). A measured signal is given by the upwards propagating
component of $\bm{E}(z,t)$, which is given by the components of
$\bm{E}_{R/L}(z,t)$ that evolve with the retarded time $t-z/c$. Collecting
these terms from Eq. 31, 32, and substituting from Eq. 35, yields a signal
field

$\bm{E}^{(+)}_{R,out}(z,t)=\frac{\bm{\hat{e}}_{R}\mathrm{e}^{i\phi_{R}}}{\sqrt{\epsilon_{0}AL^{\prime}}}\left[\frac{1}{2}\sum_{k}\mathrm{e}^{-i\omega_{k}(t-z/c)}r_{k,R}(0)\sqrt{\omega_{k}}\left[1-\sqrt{\frac{\omega_{c}}{\omega_{k}}}\frac{\kappa_{k}^{*}}{\kappa_{c}^{*}}\frac{\kappa}{i(\omega_{c}-\omega_{k})+\kappa/2}\right]-\frac{\sqrt{\omega_{c}}}{2\kappa_{c}^{*}}\frac{\kappa
gS_{-}^{R}(t-z/c)}{i(\omega_{c}-\omega_{d})+\kappa/2}\right].$ (36)

For a near-resonant input field, i.e.
$\innerproduct{r_{k,R}^{\dagger}(0)r_{k,R}(0)}{r_{k,R}^{\dagger}(0)r_{k,R}(0)}\approx
0$ for $|\omega_{c}-\omega_{k}|\gg\kappa$, we can approximate
$\sqrt{\frac{\omega_{c}}{\omega_{k}}}\approx\frac{\kappa_{k}^{*}}{\kappa_{c}^{*}}\approx
1$ and obtain

$\bm{E}^{(+)}_{R,out}(z,t)=\frac{\bm{\hat{e}}_{R}\mathrm{e}^{i\phi_{R}}}{2}\left[\sum_{k}\sqrt{\frac{\omega_{k}}{\epsilon_{0}AL^{\prime}}}\mathrm{e}^{-i\omega_{k}(t-z/c)}r_{k,R}(0)\frac{i(\omega_{c}-\omega_{k})-\kappa/2}{i(\omega_{c}-\omega_{k})+\kappa/2}-\sqrt{\frac{\omega_{c}}{\epsilon_{0}AL^{\prime}}}\frac{\kappa
g}{\kappa_{c}^{*}}\frac{S_{-}^{R}(t-z/c)}{i(\omega_{c}-\omega_{d})+\kappa/2}\right].$
(37)

The first term in the brackets of Eq. 5 can be interpreted as the light
reflected by the cavity without having interacted with the electronic system
($\frac{i(\omega_{c}-\omega_{k})-\kappa/2}{i(\omega_{c}-\omega_{k})+\kappa/2}$
is the corresponding phase shift). The second term $\propto gS_{-}^{R}(t-z/c)$
then gives the light that is radiated by the dipole into the cavity mode and
then transmitted through the top mirror into the output field.

Figure 5: Components of the model. Showing the FLS coupled to the two cavity
modes which in turn are coupled to the port modes.

##  Appendix B Master equation for the cavity-spin system

We eliminate the port modes $r_{k}$ from the full $\\{r_{k},a,S_{-}\\}$ system
to obtain a Master equation for the driven, lossy cavity-spin subsystem
described by $\\{a,S_{-}\\}$ (we omit the labels $R$, $L$ in this section, as
the derivation is the same for both polarizations). This derivation makes the
assumption of a Markovian cavity, i.e. a cavity with a coupling parameter
$\kappa_{k}$ that is essentially constant over its bandwidth. We begin by
splitting the Hamiltonian given in Eq. 2 and Eq. 3 into system, reservoir
(i.e. port modes), and interaction components as
$H=H_{R}^{R}+H_{SR}^{R}+H_{R}^{L}+H_{SR}^{L}+H_{S}$, where

$\begin{split}H_{R}^{R/L}&=\sum_{k}\omega_{k}r_{R/L}^{\dagger}r_{R/L}\\\
H_{SR}^{R/L}&=\sum_{k}\left(\kappa_{k}^{*}r_{k,R/L}a^{\dagger}+\kappa_{k}r_{k,R/L}^{\dagger}a\right),\end{split}$
(38)

and where terms not containing port mode operators $r_{k}$ are contained in
$H_{S}$. The following derivation applies to both circular polarizations $R$
and $L$ and we can omit these labels to yield a Hamiltonian
$H=H_{R}+H_{SR}+H_{S}$, with

$H_{S}=\omega_{c}a^{\dagger}a+\omega_{0}P_{e}+g(S_{+}a+S_{-}a^{\dagger})+\omega_{B}\sigma_{x},$
(39)

where $P_{e}$ projects onto the excited state of a two-level system (TLS)
(either $\\{\ket{\uparrow},\ket{\Uparrow}\\}$ or
$\\{\ket{\downarrow},\ket{\Downarrow}\\}$) and $S_{+}$ and $S_{-}$ excite and
de-excite this TLS, respectively.

Starting with a state $\chi(t)$ of the combined reservoir-cavity-TLS system
evolving under the above Hamiltonian, we define an interaction picture state
$\tilde{\chi}(t)=\mathrm{e}^{iH_{0}t}\chi(t)\mathrm{e}^{-iH_{0}t}$ where
$H_{0}=H_{R}+\omega_{d}(a^{\dagger}a+P_{e})$. This state evolves as
$\dot{\tilde{\chi}}(t)=-i[\tilde{H}(t),\tilde{\chi}(t)]$, with

$\tilde{H}(t)=\sum_{k}\left(\kappa_{k}^{*}r_{k}a^{\dagger}\mathrm{e}^{-i(\omega_{k}-\omega_{d})t}+\kappa_{k}r_{k}^{\dagger}a\mathrm{e}^{i(\omega_{k}-\omega_{d})t}\right)+H_{S}-\omega_{d}(a^{\dagger}a+P_{e}).$
(40)

We are interested in the reduced state of the cavity-TLS system
$\tilde{\rho}(t)=\mathrm{Tr}\left\\{\tilde{\chi}(t)\right\\}$, where the
reservoir modes with operators $r_{k}$ are assumed to initially be in the
vacuum state except for one mode $r_{d}$, which is in a coherent state
$D_{d}\ket{0}$ due to the narrowband laser driving. To take this initial state
into account we consider a state $\tilde{\chi}_{D}(t)\equiv
D^{\dagger}_{d}\tilde{\chi}(t)D_{d}$, where $D_{d}$ is the displacement
operator $D_{d}=\exp(\alpha r_{d}^{\dagger}-\alpha^{*}r_{d})$ with dispacement
amplitude $\alpha$. This displaced state $\tilde{\chi}_{D}(t)$ evolves as
$\dot{\tilde{\chi}}_{D}(t)=-i[\tilde{H}_{D}(t),\tilde{\chi}_{D}(t)]$, where
$\tilde{H}_{D}(t)=D^{\dagger}_{p}\tilde{H}(t)D_{d}=\tilde{H}(t)+\kappa_{d}^{*}\alpha
a^{\dagger}+\kappa_{d}\alpha^{*}a$, and importantly
$\tilde{\rho}(t)=\mathrm{Tr}\left\\{\tilde{\chi}(t)\right\\}=\mathrm{Tr}\left\\{\tilde{\chi}_{D}(t)\right\\}$.
We have therefore related a cavity interacting with a reservoir initially in a
single-mode coherent state to a classically driven cavity interacting with a
reservoir initially in the vacuum state. To remove the remaining system part
of $\tilde{H}_{D}(t)$ we define another interaction picture by
$\bar{\chi}_{D}(t)=\mathrm{e}^{iH_{c}t}\tilde{\chi}_{D}(t)\mathrm{e}^{-iH_{c}t}$
with $H_{c}=H_{S}-\omega_{d}(a^{\dagger}a+P_{e})+\kappa_{d}^{*}\alpha
a^{\dagger}+\kappa_{d}\alpha^{*}a$ to find
$\dot{\bar{\chi}}_{D}(t)=-i[\bar{H}_{SR}(t),\bar{\chi}_{D}(t)]$, with
$\bar{H}_{SR}(t)=\bar{a}(t)\bar{\Gamma}^{\dagger}(t)+\bar{a}^{\dagger}(t)\bar{\Gamma}(t)$
and

$\begin{split}\bar{a}(t)&=\mathrm{e}^{iH_{c}t}a\mathrm{e}^{-iH_{c}t}\mathrm{e}^{i\omega_{d}t}\\\
\bar{\Gamma}(t)&=\sum_{k}\kappa_{k}^{*}r_{k}\mathrm{e}^{-i\omega_{k}t}.\end{split}$
(41)

Given this interaction Hamiltonian we can proceed with the derivation of a
Master equation in the Born-Markov approximation following the steps given in
detail in, for example carmichael1999statistical . Assuming an initially
uncorrelated system-reservoir state $\bar{\chi}(0)$ we write

$\dot{\bar{\rho}}(t)\approx-\int_{0}^{t}dt^{\prime}\mathrm{Tr}\left\\{\left[\bar{H}_{SR}(t),\left[\bar{H}_{SR}(t^{\prime}),\bar{\rho}(t^{\prime})\otimes
R_{0}\right]\right]\right\\},$ (42)

where the Born approximation consists of the replacement
$\bar{\chi}_{D}(t^{\prime})\rightarrow\bar{\rho}(t^{\prime})\otimes R_{0}$
where $R_{0}$ denotes the vacuum state of the reservoir. This approximation
therefore assumes that the effect of changes to the reservoir on the system
dynamcics due to its interaction with the system are negligible, which is
justified in the case of the large reservoir considered here. More explicitly
we write

$\begin{split}\dot{\bar{\rho}}(t)=-\int_{0}^{t}dt^{\prime}\Big{(}&\bar{a}^{\dagger}(t)\bar{a}(t^{\prime})\bar{\rho}(t^{\prime})\mathrm{Tr}\left\\{\bar{\Gamma}(t)\bar{\Gamma}^{\dagger}(t^{\prime})R_{0}\right\\}+\bar{a}(t)\bar{\rho}(t^{\prime})\bar{a}^{\dagger}(t^{\prime})\mathrm{Tr}\left\\{\bar{\Gamma}^{\dagger}(t)R_{0}\bar{\Gamma}(t^{\prime})\right\\}+\\\
&\bar{a}(t^{\prime})\bar{\rho}(t^{\prime})\bar{a}^{\dagger}(t)\mathrm{Tr}\left\\{\bar{\Gamma}^{\dagger}(t^{\prime})R_{0}\bar{\Gamma}(t)\right\\}+\bar{\rho}(t^{\prime})\bar{a}^{\dagger}(t^{\prime})\bar{a}(t)\mathrm{Tr}\left\\{\bar{\Gamma}(t^{\prime})\bar{\Gamma}^{\dagger}(t)R_{0}\right\\}\Big{)},\end{split}$
(43)

Evaluating

$\mathrm{Tr}\left\\{\bar{\Gamma}(t)\bar{\Gamma}^{\dagger}(t^{\prime})R_{0}\right\\}=\int_{0}^{\infty}d\omega
g(\omega)|\kappa(\omega)|^{2}\mathrm{e}^{i\omega(t^{\prime}-t)}$ (44)

with $g(\omega)$ denoting the density of states of the port modes, we arrive
at four terms of the form

$\int_{0}^{t}dt^{\prime}\bar{a}^{\dagger}(t)\bar{a}(t^{\prime})\bar{\rho}(t^{\prime})\mathrm{Tr}\left\\{\bar{\Gamma}(t)\bar{\Gamma}^{\dagger}(t^{\prime})R_{0}\right\\}=\int_{0}^{\infty}d\omega
g(\omega)|\kappa(\omega)|^{2}\int_{0}^{t}dt^{\prime}\bar{a}^{\dagger}(t)\bar{a}(t^{\prime})\bar{\rho}(t^{\prime})\mathrm{e}^{-i\omega(t-t^{\prime})}.$
(45)

The term $\bar{a}^{\dagger}(t)\bar{a}(t^{\prime})\bar{\rho}(t^{\prime})$
varies slowly with $t^{\prime}$, as its evolution is generated by
$\tilde{H}_{D}(t)$, which only features terms much smaller than the optical
frequency $\omega_{d}$. The integral therefore averages to zero for
frequencies $\omega$ that significantly differ from
$\omega_{c}\approx\omega_{0}\approx\omega_{d}$. Thus we can approximate
$g(\omega)|\kappa(\omega)|^{2}\approx g(\omega_{c})|\kappa(\omega_{c})|^{2}$
under the integral $\int_{0}^{\infty}d\omega$, extend the lower limit
$0\rightarrow-\infty$, evaluate
$\int_{-\infty}^{\infty}d\omega\exp(-i\omega(t-t^{\prime}))=2\pi\delta(t-t^{\prime})$,
and arrive at

$\dot{\bar{\rho}}(t)=\kappa\left[\bar{a}(t)\bar{\rho}(t)\bar{a}^{\dagger}(t)-\frac{1}{2}\left(\bar{a}^{\dagger}(t)\bar{a}(t)\bar{\rho}(t)+\bar{\rho}(t)\bar{a}^{\dagger}(t)\bar{a}(t)\right)\right],$
(46)

where $\kappa\equiv 2\pi g(\omega_{c})|\kappa(\omega_{c})|^{2}$. Reversing the
transformations $\bar{\rho}(t)\rightarrow\tilde{\rho}(t)\rightarrow\rho(t)$
and including the mode polarization labels yields the master equation for the
cavity-TLS system in the Born-Markov approximation:

$\begin{split}\dot{\rho}(t)&=-i[H,\rho(t)]+\sum_{k=R,L}\left(L_{k}\rho(t)L_{k}^{\dagger}-\frac{1}{2}\left(L_{k}^{\dagger}L_{k}\rho(t)+\rho(t)L_{k}^{\dagger}L_{k}\right)\right)\\\
&\mathrm{where}\\\
H&=\omega_{c}(a_{R}^{\dagger}a_{R}+a_{L}^{\dagger}a_{L})+\epsilon_{R}\mathrm{e}^{i\omega_{d}t}a_{R}+\epsilon_{R}^{*}\mathrm{e}^{-i\omega_{d}t}a^{\dagger}_{R}+\epsilon_{L}\mathrm{e}^{i\omega_{d}t}a_{L}+\epsilon_{L}^{*}\mathrm{e}^{-i\omega_{d}t}a^{\dagger}_{L}\\\
&\ \ \
+\omega_{0}(P_{\Uparrow}+P_{\Downarrow})+\omega_{B}\sigma_{X}+g\left(S_{-}^{R}a_{R}^{\dagger}+S_{+}^{R}a_{R}+S_{-}^{L}a_{L}^{\dagger}+S_{+}^{L}a_{L}\right),\\\
L_{R}&=\sqrt{\kappa}a_{R},\\\ L_{L}&=\sqrt{\kappa}a_{L}.\end{split}$ (47)

##  Appendix C Adiabatic elimination of the cavity mode

The procedure presented in Appendix B has eliminated the cavity modes
$r_{k,R/L}$ from the system description to yield a model in terms of two
cavity modes $a_{R}$ and $a_{L}$ coupling to the electronic four-level system
(FLS). In order to evaluate the output field given in Eq. 32, however, we only
need the FLS dynamics, such that we can further simplify the analysis by
eliminating the cavity modes from the description to arrive at a master
equation of the FLS alone. This elimination is possible in the adiabatic
limit, which assumes that the cavity linewidth $\sim\kappa$ is much greater
than the bandwidth of the FLS dynamics. In physical terms this adiabatic
approximation assumes that the cavity modes couple much stronger to the port
modes than to the FLS dipoles. As far as the FLS dynamics are concerned these
cavity modes can therefore be assumed in a state governed by the driving of
the cavity rather than photons emitted by the FLS, which escape from the
cavity so fast that they do not interact significantly with the FLS.
Mathematically the derivation of the FLS Master equation is somewhat different
from the previous calculation, as the starting point is the Master equation
Eq. 47 rather than unitary evolution under the Hamiltonian defined in Eq. 2
and 3. We make use of a superoperator formalism and closely follow
carmichael2007statistical .

Again the derivation proceeds independently and in the same way for the two
polarizations $R$ and $L$, such that we can consider a simplified version of
Eq. 47, i.e. a cavity-TLS state $\rho(t)$ evolving as
$\dot{\rho}(t)=-i[H,\rho(t)]+L\rho(t)L^{\dagger}-\frac{1}{2}\left(L^{\dagger}L\rho(t)+\rho(t)L^{\dagger}L\right)$
with

$\begin{split}H&=\omega_{c}a^{\dagger}a+\omega_{0}P_{e}+\epsilon\mathrm{e}^{i\omega_{d}t}a+\epsilon^{*}\mathrm{e}^{-i\omega_{d}t}a^{\dagger}+g(S_{-}a^{\dagger}+S_{+}a)\\\
L&=\sqrt{\kappa}a.\end{split}$ (48)

In a rotating frame
$\tilde{\rho}(t)=\exp(i\omega_{d}(a^{\dagger}a+P_{e}))\rho(t)\exp(-i\omega_{d}(a^{\dagger}a+P_{e}))$
the Hamiltonian becomes

$H=\Delta_{c}a^{\dagger}a+\Delta_{0}P_{e}+\epsilon
a+\epsilon^{*}a^{\dagger}+g(S_{-}a^{\dagger}+S_{+}a)$ (49)

with $\Delta_{c/0}=\omega_{c/0}-\omega_{d}$. We can again introduce a
displacement operator in order to have a more convenient form of the Born
approximation later on. Consider
$\tilde{\rho}_{D}(t)=D(\beta)^{\dagger}\tilde{\rho}(t)D(\beta)$ with
displacement operator $D(\beta)=\exp(\beta a^{\dagger}+\beta^{*}a)$. This
displaced state evolves as

$\dot{\tilde{\rho}}_{D}(t)=-i[H_{D},\tilde{\rho}_{D}(t)]+\kappa\left(a_{D}\tilde{\rho}_{D}(t)a_{D}^{\dagger}-\frac{1}{2}\left(a_{D}^{\dagger}a_{D}\tilde{\rho}_{D}(t)+\tilde{\rho}_{D}(t)a_{D}^{\dagger}a_{D}\right)\right)$
(50)

where $a_{D}\equiv D(\beta)^{\dagger}aD(\beta)=a+\beta$. Choosing
$\beta=-i\frac{\epsilon^{*}}{\kappa/2+i\Delta_{c}}$ we find

$\dot{\tilde{\rho}}_{D}(t)=-i[H_{D},\tilde{\rho}_{D}(t)]+\kappa\left(a\tilde{\rho}_{D}(t)a^{\dagger}-\frac{1}{2}\left(a^{\dagger}a\tilde{\rho}_{D}(t)+\tilde{\rho}_{D}(t)a^{\dagger}a\right)\right),$
(51)

where
$H_{D}=\Delta_{c}a^{\dagger}a+\Delta_{0}P_{e}+g(a^{\dagger}S_{-}+aS_{+})+\Omega
S_{-}+\Omega^{*}S_{+}$ and where we define the Rabi frequency
$\Omega=g\beta^{*}$. As in the derivation of the cavity master equation we
have arrived at a classical driving Hamiltonian, where in this case it is the
TLS rather than the cavity that is classically driven.

The following steps are conveniently expressed in a superoperator formalism
following carmichael2007statistical . We write $\mathcal{L}=(\ \bullet\ A)$ to
denote an operation acting on an arbitrary operator $O$ as $\mathcal{L}A=OA$.
The bullet $\bullet$ in the definition denotes whether the operator to be
acted on is to be right-: $(\bullet\ A)$ or left-multiplied: $(A\ \bullet\ )$,
or sandwiched in between: $(A\ \bullet\ B)$. We can then write Eq. 51 as
$\dot{\tilde{\rho}}_{D}(t)=(\mathcal{L}_{S}+\mathcal{L}_{R}+\mathcal{L}_{SR})\tilde{\rho}_{D}(t)$,
where

$\begin{split}\mathcal{L}_{S}&=-i\Big{(}(H_{S}\ \bullet\ )-(\ \bullet\
H_{S})\Big{)}\\\ \mathcal{L}_{R}&=-i\Delta_{c}\Big{(}(a^{\dagger}a\ \bullet\
)-(\ \bullet\ a^{\dagger}a)\Big{)}+\kappa\Big{(}(a\ \bullet\
a^{\dagger})-\frac{1}{2}(a^{\dagger}a\ \bullet\ )-\frac{1}{2}(\ \bullet\
a^{\dagger}a)\Big{)}\\\ \mathcal{L}_{SR}&=g\Big{(}(a^{\dagger}\ \bullet\
)(S_{-}\ \bullet\ )-(a\ \bullet\ )(S_{+}\ \bullet\ )+(\ \bullet\ a)(\ \bullet\
S_{+})-(\ \bullet\ a^{\dagger})(\ \bullet\ S_{-})\Big{)}.\end{split}$ (52)

All TLS terms have been subsumed in $H_{S}$, i.e.
$H_{D}=\Delta_{c}a^{\dagger}a+H_{S}+g(a^{\dagger}S_{-}+aS_{+})$.

Our goal is to derive a Master equation for the TLS state
$\dot{\tilde{\sigma}}(t)=\mathrm{Tr}_{R}\left\\{\dot{\tilde{\rho}}(t)\right\\}=\mathrm{Tr}_{R}\left\\{\dot{\tilde{\rho}}_{D}(t)\right\\}$.
We begin by defining an interaction picture as
$\bar{\rho}(t)=\exp(-(\mathcal{L}_{P}+\mathcal{L}_{S})t)\tilde{\rho}(t)$,
where $\dot{\bar{\rho}}(t)=\bar{\mathcal{L}}_{SR}\bar{\rho}(t)$,
$\bar{\mathcal{L}}_{SR}=\exp(-(\mathcal{L}_{P}+\mathcal{L}_{S})t)\mathcal{L}_{SR}\exp((\mathcal{L}_{P}+\mathcal{L}_{S})t)$,
and $\bar{\sigma}(t)=\exp(-\mathcal{L}_{S})\tilde{\sigma}(t)$. Then we find

$\dot{\bar{\sigma}}(t)=\int_{0}^{t}dt^{\prime}\mathrm{Tr}_{R}\left\\{\bar{\mathcal{L}}_{SR}(t)\bar{\mathcal{L}}_{SR}(t^{\prime})\bar{\sigma}(t^{\prime})\otimes
A_{0}\right\\}$ (53)

given an uncorrelated initial state $\bar{\rho}(0)$ and making the Born
approximation
$\bar{\rho}(t^{\prime})\rightarrow\bar{\sigma}(t^{\prime})\otimes A_{0}$ under
the integral $\int_{0}^{t}dt^{\prime}$, where $A_{0}$ denotes the vacuum state
of the cavity mode obeying $aA_{0}=A_{0}a^{\dagger}=0$. This Born
approximation constitutes the first part of the adiabatic approximation, which
assumes that the effect of the cavity mode on the TLS is well described by the
interaction with a coherent state. We can then write explicitly

$\mathcal{\bar{L}}_{SR}=-ig\Big{(}(\overline{a^{\dagger}\ \bullet\
})(\overline{S_{-}\ \bullet\ })+(\overline{a\ \bullet\
\vphantom{+1}})(\overline{S_{+}\ \bullet\ })-(\overline{\ \bullet\
a\vphantom{+1}})(\overline{\ \bullet\ S_{+}})-(\overline{\ \bullet\
a^{\dagger}})(\overline{\ \bullet\ S_{-}})\Big{)}$ (54)

where $\\{(\overline{S_{-}\ \bullet\ }),(\overline{S_{+}\ \bullet\
}),(\overline{\ \bullet\ S_{+}}),(\overline{\ \bullet\
S_{-}})\\}=\exp(-\mathcal{L}_{S}t)\\{(S_{-}\ \bullet\ ),(S_{+}\ \bullet\ ),(\
\bullet\ S_{+}),(\ \bullet\ S_{-})\\}\exp(\mathcal{L}_{S}t)$ and similarly for
reservoir superoperators such as $(\overline{\ \bullet\
a\vphantom{+1}})=\exp(-\mathcal{L}_{R}t)(\ \bullet\ a)\exp(\mathcal{L}_{R}t)$.
Following carmichael2007statistical  we evaluate

$\begin{split}(\overline{a\ \bullet\
\vphantom{+2}})&=\exp\left(-\left(i\Delta_{c}+\frac{\kappa}{2}\right)t\right)(a\
\bullet\ )\\\ (\overline{a^{\dagger}\ \bullet\
\vphantom{+2}})&=\exp\left(\left(i\Delta_{c}+\frac{\kappa}{2}\right)t\right)(a^{\dagger}\
\bullet\
)+\left[\exp\left(\left(i\Delta_{c}+\frac{\kappa}{2}\right)t\right)-\exp\left(-\left(i\Delta_{c}+\frac{\kappa}{2}\right)t\right)\right](\
\bullet\ a^{\dagger}).\end{split}$ (55)

Using these expressions as well as their conjugates, the integrand of Eq. 53
becomes

$\begin{split}\mathrm{Tr}_{R}&\left\\{\bar{\mathcal{L}}_{SP}(t)\bar{\mathcal{L}}_{SP}(t^{\prime})\bar{\sigma}(t^{\prime})\otimes
A_{0}\right\\}=\\\
g^{2}\Big{[}&\exp\left(-\left(i\Delta_{c}+\frac{\kappa}{2}\right)(t-t^{\prime})\right)\left[(\overline{\
\bullet\ S_{+}})(t)(\overline{S_{-}\ \bullet\ })(t^{\prime})-(\overline{S_{+}\
\bullet\ })(t)(\overline{S_{-}\ \bullet\ })(t^{\prime})\right]+\\\
&\exp\left(-\left(-i\Delta_{c}+\frac{\kappa}{2}\right)(t-t^{\prime})\right)\left[(\overline{S_{-}\
\bullet\ })(t)(\overline{\ \bullet\ S_{+}})(t^{\prime})-(\overline{\ \bullet\
S_{-}})(t)(\overline{\ \bullet\
S_{+}})(t^{\prime})\right]\Big{]}\bar{\sigma}(t^{\prime}).\end{split}$ (56)

The second part of the adiabatic approximation assumes that the cavity
bandwidth $\kappa$ is much greater than the bandwidth of the system dynamics,
which means for the above expression that $(\overline{\ \bullet\
S_{+}})(t)(\overline{S_{-}\ \bullet\ })(t^{\prime})\approx(\overline{\
\bullet\ S_{+}})(t)(\overline{S_{-}\ \bullet\ })(t)$ for
$t-t^{\prime}\in[0,1/\kappa]$. We can therefore evaluate the integral
$\int_{0}^{t}dt^{\prime}$ in Eq. 53 to obtain

$\begin{split}\dot{\bar{\sigma}}(t)=\frac{\kappa
g^{2}}{2\Delta_{c}^{2}+\kappa^{2}/2}\left[2(\overline{S_{-}\bullet
S_{+}})(t)-(\overline{S_{+}S_{-}\bullet})(t)-(\overline{\bullet
S_{+}S_{-}})(t)\right]\\\
-\frac{i\Delta_{c}}{\Delta_{c}^{2}+\kappa^{2}/4}\left[(\overline{S_{+}S_{-}\bullet})(t)-(\overline{\bullet
S_{+}S_{-}})(t)\right].\end{split}$ (57)

The first line of this expression describes the cavity-mediated decay rate of
the TLS (Purcell effect), while the second line gives a Lamb shift of the TLS
resonance energy. Undoing the interaction picture (but maintaining the frame
rotating at $\omega_{d}$) and including both polarizations we therefore obtain
the Master equation for the FLS with adiabatically eliminated cavity modes:

$\begin{split}\dot{\tilde{\sigma}}(t)&=-i[H,\tilde{\sigma}(t)]+\sum_{k=R,L}\left(L_{k}\tilde{\sigma}(t)L_{k}^{\dagger}-\frac{1}{2}\left(\tilde{\sigma}(t)L_{k}^{\dagger}L_{k}+L_{k}^{\dagger}L_{k}\tilde{\sigma}(t)\right)\right)\\\
\mathrm{where}\
H&=\Delta^{\prime}(P_{\Uparrow}+P_{\Downarrow})+\omega_{B}\sigma_{X}+\Omega_{R}S_{-}^{R}+\Omega_{R}^{*}S_{+}^{R}+\Omega_{L}S_{-}^{L}+\Omega_{L}^{*}S_{+}^{L},\\\
L_{R/L}&=\sqrt{\Gamma}S_{-}^{R/L},\\\ \mathrm{with}\
\Gamma&=\frac{g^{2}\kappa}{\Delta_{c}^{2}+\kappa^{2}/4}\ \ \mathrm{and}\ \
\Delta=\left(\Delta_{0}+\frac{\Delta_{c}g^{2}}{\Delta_{c}^{2}+\kappa^{2}/4}\right).\end{split}$
(58)

##  Appendix D Breakdown of Approximation

The analytical expression Eq. 28 is only valid in the weak excitation regime.
For comparison to arbitrary driving powers we use a numerical solution to the
full system master equation. Examples are shown in Fig. 6. Here, we have good
agreement between the numerical and analytic methods when $\Omega$ is two
orders of magnitude less than $\Gamma$ and the approximation breaks down as
this difference approaches a single order of magnitude. This is when the
lifetime-broadened component of the scattered field become significant and can
no longer be neglected. This incoherent component can be recovered by
including the matrix elements $\rho^{\prime}_{MP/PM}$ in Eq. (24).

Figure 6: Plots of analytic emission spectrum (Eq. 28, solid lines) and
numerically calculated emission spectrum, dashed lines, derived from solving
Eq. 8 directly. a) Here $\Omega/\Gamma=0.05$ and $\Delta/\Gamma=0$. This plot
shows that our analytic expression is in excellent agreement with the
numerical solution for this parameter regime. This plot additionally
highlights the distinct spectrum types with the spectrum appearing split for
large magnetic fields but reduces to a single peak for small but non zero
magnetic field strengths. b) Here $\Omega/\Gamma=0.1$ and $\Delta/\Gamma=0$.
In this regime the weak excitation approximation begins to break down but the
overall features of the spectrum are still captured. The discrepancy here is
due to the lifetime broadened component of the scattered field that is
neglected by the weak excitation approximation. c) Here $\Omega/\Gamma=0.15$
and $\Delta/\Gamma=0$. In this plot the lifetime broadened component is fairly
significant as seen in the inset showing significant discrepancy between the
two spectra.


# Relation of the thermodynamic parameter of disordering with the width of
structure factor and defect concentration in a metallic glass

G.V. Afonin  A.S. Makarov  R.A. Konchakov  J.C. Qiao  A.N. Vasiliev  N.P.
Kobelev  V.A. Khonik<EMAIL_ADDRESS>Department of General Physics,
Voronezh State Pedagogical University, Lenin St. 86, Voronezh 394043, Russia
School of Mechanics, Civil Engineering and Architecture, Northwestern
Polytechnical University, Xi’an 710072, China  National University of Science
and Technology MISiS, Moscow 119049, Russia  Institute of Solid State Physics
RAS, Chernogolovka, 142432, Russia

###### Abstract

In this work, we show that above the glass transition there exists a strong
unique interrelationship between the thermodynamic parameter of disorder of a
metallic glass derived using its excess entropy, diffraction measure of
disorder given by the width of the X-ray structure factor and defect
concentration derived from shear modulus measurements. Below the glass
transition, this relationship is more complicated and depends on both
temperature and thermal prehistory.

###### keywords:

metallic glasses, thermodynamic and diffraction disorder parameters, defects

Understanding of general key features of relaxation behavior of metallic
glasses (MGs) is of vital importance for the adequate description of their
structural state and atomic rearrangements [1, 2]. In particular, current
literature gives quite a few examples for the interpretation of relaxation
properties in terms of different types of defects assumed to exist in real MGs
[3, 4, 5, 6, 7, 8, 2]. However, any relation of the defect structure with
macroscopic structure parameters remains mostly out of view. An attempt to
advance in this direction was recently reported in our work [9], in which it
was found that the width of the structure factor given by the full width at
half maximum (FWHM) of the first peak of structure factor is related to the
concentration $c_{def}$ of interstitial-type defects derived from shear
modulus measurements of a Pt-based metallic glass. It was shown, in
particular, that above the glass transition temperature $T_{g}$, the FWHM
linearly increases with $c_{def}$. This indicates that heat absorption and
related increase of $c_{def}$ provide a significant defect-induced disruption
of the dominant short-range order increasing thus the integral structural
disordering. Below $T_{g}$, the relationship between the FWHM and $c_{def}$ is
more complicated but defect-induced ordering is observed upon approaching
$T_{g}$. In any case, it is clear that the diffraction measure of disorder is
related to the defect concentration.

On the other hand, within the view of statistical physics [11], any
ordering/disordering should be related to a change of glass entropy. This
approach was developed in our recent work [12], in which a dimensionless
thermodymanic parameter of structural order was introduced as $\xi(T)=1-\Delta
S(T)/\Delta S_{melt}$, where $\Delta S(T)$ is temperature-dependent excess
entropy of glass with respect to the maternal crystal and $\Delta S_{melt}$ is
the entropy of melting. The above order parameter varies in the range
$0<\xi<1$ upon changing the structure from the fully disordered (liquid-like)
state with $\Delta S\rightarrow\Delta S_{melt}$ and $\xi\rightarrow 0$ towards
the fully ordered state (crystal-like) characterized by $\Delta S\rightarrow
0$ and $\xi\rightarrow 1$. An application of this approach to 13 metallic and
2 tellurite glasses showed that the order parameter $\xi$ provides very useful
qualitative and quantitative information on structural ordering of various
glasses and its evolution upon relaxation. This order parameter is quite
sensitive to the glass state and its chemical composition. In particular,
structural relaxation below $T_{g}$ results in a significant increase of
$\xi$\- parameter while heating above $T_{g}$ leads to its rapid decrease. It
was also found that order parameter $\xi_{sql}$ calculated for the supercooled
liquid state is related to the glass forming ability (GFA). An increase of
$\xi_{sql}$ strongly worsens the GFA.

While the parameter $\xi$ describes the order in glass, the quantity

$\alpha(T)=\frac{\Delta S(T)}{\Delta S_{melt}}$ (1)

describes structural disordering. In this work, we determined the parameter
$\alpha$ for the aforementioned Pt-based glass and found that it is strongly
related to the FWHM and $c_{def}$ indicating thus that the thermodynamic
entropy-based and diffraction characteristics of disordering are i)
interrelated and ii) linked to the defect concentration.

We studied the same glass Pt42.5Cu27Ni9.5P21 (at.%) produced in the bulk form
by melt jet quenching [9] while its FWHM and defect concentration $c_{def}$ as
functions of temperature were also taken from this work. The excess entropy of
glass with respect to the maternal crystal was calculated from calorimetric
measurements as [12, 13]

$\Delta S(T)=\frac{1}{\dot{T}}\int_{T}^{T_{cr}}\frac{\Delta W(T)}{T}dT,$ (2)

where $\Delta W(T)=W_{gl}(T)-W_{cr}(T)$ is the differential heat flow,
$W_{gl}(T)$ and $W_{cr}(T)$ are the heat flows coming from glass and its
maternal crystal, respectively, $\dot{T}$ is the heating rate and $T_{cr}$ is
the temperature of the complete crystallization. Specific details of
calorimetric measurements are given in Ref.[12]. These measurements were
performed at a heating rate of 3 K/min on both initial and relaxed samples,
where those latter were prepared by heating to 559 K (deep in the supercooled
liquid state) and cooling back to room temperature at the same rate. Note that
if current temperature $T=T_{cr}$ then $\Delta S=0$ and, therefore, the
function $\Delta S(T)$ represents temperature dependence the excess entropy of
glass with respect to the maternal crystal. The melting entropy $\Delta
S_{melt}$ was accepted to be 11.3 J/(K$\times$mol) [15]. This allows
calculating the parameter of disordering $\alpha$, which changes from
$\alpha\rightarrow 1$ for the fully disordered (liquid-like) state to
$\alpha\rightarrow 0$ for fully ordered (crystal-like) state.

Figure 1:  Differential heat flow $\Delta W$ and disorder parameter
$\alpha=\Delta S/\Delta S_{melt}$ for the glass under investigation in the
initial and relaxed states, where the excess entropy of glass with respect to
maternal crystal $\Delta S$ is calculated using Eq.(2).

Figure 1 gives temperature dependences of the differential heat flow $\Delta
W$ for the glass under investigation in the initial and relaxed states. These
curves were used to calculate the excess entropy $\Delta S$ with Eq.(2) and
next determine the disorder parameter $\alpha=\Delta S/\Delta S_{melt}$, which
is also given in Fig.1. It is seen that exothermal heat flow of the initial
sample in the range 400 K$<K<$510 K (below $T_{g}$) leads to a decrease of
disorder parameter $\alpha$ in the same range, just as one would expect.
Continued heating leads to a strong heat absorption near and above $T_{g}$
(i.e. in the supercooled liquid state), which is accompanied by a marked
increase of disorder (as anticipated) determined by $\alpha$-increase from
0.52 to 0.71 in the same range. Relaxed sample instead of exothermal flow
displays notable endothermal effect, which is coupled to moderate disordering
described by an increase of $\alpha$ from 0.42 to 0.52 in the aforementioned
temperature range. Disordering of the relaxed sample above $T_{g}$ is
described by the same rise of $\alpha$ as in the case of initial sample. The
latter behavior is expected as well since MGs’ structural order in the
supercooled liquid state is independent of thermal prehistory (see e.g.
Ref.[14]). Thus, the evolution of the disorder parameter qualitatively agrees
with the character of structure evolution upon heating of initial and relaxed
samples.

One can now combine these results with the information on the FWHM and defect
concentration data reported earlier in Ref.[9]. In this work, the FWHM was
taken as the width $\Gamma$ of the 1st peak of the structure factor $S(q)$
normalized by the scattering vector $q_{0}$ corresponding to this peak, i.e.
$\gamma=\Gamma/q_{0}$. The defect concentration was calculated withing the
framework of the Interstitialcy theory [10] using high-frequency shear modulus
data as $c_{def}(T)=\beta^{-1}ln\left[\mu(T)/G(T)\right]$, where $G(T)$ and
$\mu(T)$ are temperature dependences of the shear modulus of the glass under
investigation in the initial state and after complete crystallization,
respectively, and dimensionless shear susceptibility $\beta\approx 18$ [9].
Temperature dependences of the FWHM $\gamma$, defect concentration $c_{def}$
and disorder parameter $\alpha=\Delta S/\Delta S_{melt}$ given above (see
Fig.1) are shown in Fig.2.

It is seen that the relationship between $\gamma$, $c_{def}$ and $\alpha$
above $T_{g}$ is quite clear and straightforward: the disorder parameter
$\alpha$, FWHM $\gamma$ and defect concentration $c_{def}$ sharply increase
with temperature. The interpretation of this behavior is rather obvious.
Heating above $T_{g}$ leads to rapid defect multiplication (increase of
$c_{def}$), which is accompanied by strong heat absorption (see the
endothermal effect demonstrated by the heat flow $\Delta W$ in Fig.1 above
$T_{g}$). On the other hand, an increase of the defect concentration rises the
amount of regions with damaged short-range order increasing thus the FWHM, as
discussed earlier [9]. Basing on the results of the present work, one can now
argue that an increase of the defect concentration also increases the
thermodynamic parameter of disordering $\alpha$ determined by the glass excess
entropy. This is the main finding of the present work. It is also worth
emphasizing that all quantities $\gamma$, $c_{def}$ and $\alpha$ almost do not
depend on thermal prehistory, as one would expect. Overall, a quantitative
analysis of essentially different phenomena, the shear elasticity, X-ray
diffraction and calorimetric behavior of the glass under investigation gives
unequivocally consistent results.

Figure 2:  Temperature dependences of the FWHM $\gamma$ (original data are
taken from the work by Neuber et al. [15]) and defect concentration $c_{def}$
reproduced from Ref.[9] together with the disorder parameter $\alpha=\Delta
S/S_{mel}$ (right axes) shown in Fig.1. Calorimetric glass transition
temperature $T_{g}$ (see Fig.1) is indicated by the arrow.

The situation below the glass transition is more complicated but nonetheless
quite understandable. A decrease of $\alpha$ (in other words, structural
ordering) of initial samples with temperature well below $T_{g}$ is
conditioned by exothermal structural relaxation (see exothermal $\Delta
W$-rise for the initial state in the range 400 K$<T<500$ K in Fig.1), which
is, however, poorly seen in temperature dependences of the FWHM $\gamma$ and
$c_{def}$. Preliminary relaxation of samples by heating into the supercooled
liquid state and cooling back to room temperature results in endothermal heat
flow even well below $T_{g}$ (see the same range the range 400 K$<T<500$ K in
Fig.1 for relaxed samples) and related disordering (increase of $\alpha$) even
below $T_{g}$, which is quite well manifested in an increase of the defect
concentration but almost invisible in the FWHM (see $c_{def}(T)$ and
$\gamma(T)$ dependences for relaxed samples). Structural disordering with
temperature above $T_{g}$ becomes much stronger.

In conclusion, we performed calorimetric measurements on bulk glassy
Pt42.5Cu27Ni9.5P21, which was earlier used to uncover the relationship between
the width of X-ray structure factor FWHM $\gamma$ and concentration of defects
$c_{def}$ derived using shear modulus data as a function of temperature and
thermal prehistory [9]. On this basis, we determined the excess entropy of
glass with respect to the maternal crystal using Eq.(2) and calculated the
thermodynamic parameter of structural disordering $\alpha$ defined by Eq.(1).
It was found that the parameter $\alpha$ above the glass transition
temperature $T_{g}$ rapidly increases with the FWHM $\gamma$ and $c_{def}$
proving an interconnection between the thermodynamic excess entropy-based and
diffraction measures of disorder and their relation to the defect
concentration. Overall, this investigation confirms earlier idea [9] that
macroscopic structural disordering in metallic glasses above $T_{g}$ is
intrinsically related to their structural defects. Below $T_{g}$, the
interrelation between $\alpha$, $\gamma$ and $c_{def}$ is more complicated but
nonetheless follows reasonable expectations.

## Acknowledgments

The work was supported by the Russian Science Foundation under the project
23-12-00162.

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