††thanks: Present adress: Faculty of Physics, University of Warsaw, 02-093
Warsaw, Poland

# A chiral one-dimensional atom using a quantum dot in an open microcavity

Nadia O. Antoniadis<EMAIL_ADDRESS>Department of Physics,
University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
Natasha Tomm  Department of Physics, University of Basel, Klingelbergstrasse
82, CH-4056 Basel, Switzerland  Tomasz Jakubczyk  Department of Physics,
University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
Rüdiger Schott  Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität
Bochum, D-44780 Bochum, Germany  Sascha R. Valentin  Lehrstuhl für Angewandte
Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany  Andreas D.
Wieck  Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum,
D-44780 Bochum, Germany  Arne Ludwig  Lehrstuhl für Angewandte
Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany  Richard J.
Warburton  Department of Physics, University of Basel, Klingelbergstrasse 82,
CH-4056 Basel, Switzerland  Alisa Javadi<EMAIL_ADDRESS>Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland

###### Abstract

In nanostructures, the light-matter interaction can be engineered to be
chiral. In the fully quantum regime, a chiral one-dimensional atom, a photon
propagating in one direction interacts with the atom; a photon propagating in
the other direction does not. Chiral quantum optics has applications in
creating nanoscopic single-photon routers, circulators, phase-shifters and
two-photon gates. Furthermore, the directional photon-exchange between many
emitters in a chiral system may enable the creation of highly exotic quantum
states. Here, we present a new way of implementing chiral quantum optics – we
use a low-noise quantum dot in an open microcavity. Specifically, we
demonstrate the non-reciprocal absorption of single photons, a single-photon
diode. The non-reciprocity, the ratio of the transmission in the forward-
direction to the transmission in the reverse direction, is as high as 10.7 dB,
and is optimised in situ by tuning the photon-emitter coupling to the optimal
operating condition ($\beta=0.5$). Proof that the non-reciprocity arises from
a single quantum emitter lies in the nonlinearity with increasing input laser
power, and in the photon statistics – ultralow-power laser light propagating
in the diode’s reverse direction results in a highly bunched output
($g^{(2)}(0)=101$), showing that the single-photon component is largely
removed. The results pave the way to a single-photon phase shifter, and, by
exploiting a quantum dot spin, to two-photon gates and quantum non-demolition
single-photon detectors.

In a non-chiral one-dimensional atom, an atom is coupled equally to a right-
propagating and to a left-propagating mode in a waveguide. There are two
input/output ports, one on the left (port 1) and one on the right (port 2). In
the ideal limit (perfect atom with $\beta=1$, where $\beta$ is the probability
that the atom emits a photon into the waveguide, a single photon at the input
in resonance with the atom), the atom acts as a perfect mirror: the
reflectivity is $R=1$; the transmission $T=0$ [1, 2]. This changes completely
in a chiral one-dimensional atom: $R$ and $T$ depend on the propagation
direction, left-to-right ($1\rightarrow 2$) or right-to-left ($2\rightarrow
1$), i.e. the system exhibits non-reciprocity ($T_{1\rightarrow 2}\neq
T_{2\rightarrow 1}$). There are two simple cases [3]. First, for $\beta=1$,
the atom now becomes perfectly transparent ($T_{1\rightarrow 2}=1$,
$T_{2\rightarrow 1}=1$, $R_{1\rightarrow 1}=0$, $R_{2\rightarrow 2}=0$). In
one direction, the $2\rightarrow 1$-direction, say, the photon is phase-
shifted by $\pi$ via the interaction with the atom; in the other direction,
$1\rightarrow 2$, the photon phase-shift is zero. Second, for
$\beta=\frac{1}{2}$: in the $2\rightarrow 1$-direction, the photon is
scattered by the atom into non-waveguide modes – the photon is absorbed – such
that $T_{2\rightarrow 1}=0$ and $R_{2\rightarrow 2}=0$, whereas in the
$1\rightarrow 2$-direction, the photon does not interact with the atom,
$T_{1\rightarrow 2}=1$ and $R_{1\rightarrow 1}=0$.

Chiral quantum optics has been implemented by using a single emitter in a
nano-engineered waveguide, for instance a Rb atom in the evanescent field of a
dielectric nanofibre [4, 5], or a semiconductor quantum dot in a waveguide [6,
7, 8]. In the semiconductor case, $\beta$-factors can be high in nano-beam
structures and particularly high in photonic-crystal waveguides. The system
becomes chiral provided the quantum dot is located off-centre in a nano-beam
[8]; and at the centre of an inversion-asymmetric photonic-crystal waveguide
[6].

We report here a different approach to engineering a chiral one-dimensional
atom. A single-mode optical fibre constitutes the waveguide on the “left” of
the quantum dot; another single-mode optical fibre constitutes the waveguide
on the “right” of the quantum dot; the atom itself is a quantum dot in a low-
volume one-sided microcavity, where the microcavity is coupled with high
efficiency to the single-mode fibres. Chirality is induced by applying a
magnetic field to a neutral quantum dot: the quantum dot’s $\sigma^{+}$\-
transition couples to the microcavity and is addressed with
$\sigma^{+}$-polarised photons. The advantage of this approach is that the
resonant microcavity boosts the light-matter interaction in a controllable
way: the $\beta$-factor can be tuned from small to extremely high values
(99.7% has been achieved [9]). Also, the good mode-matching [10] implies that
a high-efficiency, fibre-coupled platform for chiral quantum optics can be
constructed.

Here, we implement the chiral scheme with $\beta=\frac{1}{2}$. In one
direction, a single photon is transmitted; in the other direction, the photon
is absorbed, Fig. 1a. We call this device a “single-photon diode” in analogy
to its electronic counterpart. The challenge is twofold: to achieve exactly
the right $\beta$; and to achieve a close-to-perfect (transform-limited)
quantum dot. These challenges were met: we achieve an isolation of 10.7 dB,
the highest non-reciprocal response recorded with a single quantum emitter. In
addition, the high overall efficiency [10] enables us to observe optical
nonlinearities already at an input power of just 100 pW. The quantum nature of
this nonlinearity is validated by observation of photon bunching by a factor
of 101 compared to that of a laser field.

Figure 1: Schematic and operating principle of the chiral one-dimensional atom
(a) The optical system showing the two input/output ports. (b) Illustration of
the open microcavity. The heterostructure consists of a GaAs/AlAs distributed
Bragg reflector (the “bottom” mirror) and self-assembled InAs quantum dots
embedded in an n-i-p diode. Nano-positioners allow precise tuning of both
microcavity frequency (via Z) and also the $\beta$-factor by positioning the
quantum dots with respect to the anti-node of the microcavity (via XY). An
external magnetic field of 2.0 T splits the neutral quantum dot into two
circularly-polarised transitions. The polarisation of the light is controlled
by a polarising beam-splitter (PBS) and a quarter-wave plate ($\lambda/4$) in
the microscope. The $\sigma^{+}$-polarised exciton creates a photon in the
microcavity with probability $\beta$. (c) Transmission from port 1 to port 2
versus quantum dot detuning for three different lateral positions of the
quantum dot: orange triangles ($\beta\approx 0.72$), purple circles
($\beta\approx 0.50$) and blue rectangles ($\beta\approx 0.12$). The solid
lines are theory curves. The off-resonant transmission is limited to 0.7 by
the residual non-degeneracy of the microcavity. The transmission is defined
without taking into account the losses in the optical setup. The overall end-
to-end throughput of the setup, taking into account all the optical elements
and the fibre-coupling, is 56%: this corresponding to $T=1$. (d) Contrast in
the transmission as a function of the lateral position. Coloured data-points
indicate the positions used in (b). The black solid line is the theory.
Figure 2: Non-reciprocal response of the diode. Transmission through the
system versus quantum dot and microcavity detuning for the backward (a) (port
1 to 2) and forward (b) (port 2 to 1) directions. Experimental data (left) and
theory (right) show an excellent match. The black lines show the Lamb shift of
the quantum dot resonance induced by the vacuum field of the microcavity.
(c)/(d) Transmission through the diode on resonance with the
microcavity/quantum dot in the forward and the backward direction versus
quantum dot/cavity detuning. The purple and turquoise data are cut-throughs of
(a) (2$\rightarrow$1) and (b) (1$\rightarrow$2). The black solid lines are the
theory

Figure 1b shows a schematic of the setup and the operation principle of the
diode. The optical setup consists of a polarising beam-splitter and a quarter-
wave plate set at 45∘ with respect to the polarising beam-splitter axes.
Consequently, light propagating in the forward direction is mapped to a left-
handed field at the input of the microcavity, while light propagating in the
backward direction is mapped to a right-handed field at the microcavity input,
thereby creating the spin-momentum locking. The microcavity (Fig. 1b)
comprises a highly reflective “bottom” mirror and a much less reflective “top”
mirror [10]. The top mirror is a dielectric distributed-Bragg-reflector
deposited on a crater in a silica substrate with a radius-of-curvature of 11
$\upmu$m. The bottom mirror is a semiconductor distributed-Bragg-reflector on
top of which InAs quantum dots are embedded in an n-i-p diode in the
heterostructure. The open nature of the microcavity allows the lateral
position of the quantum dots to be controlled precisely with respect to the
anti-node of the microcavity. This allows in situ tuning of the
$\beta$-factor. Furthermore, adjusting the distance between the bottom mirror
and the top mirror provides precise control over the microcavity’s frequency,
enabling the quantum dot’s $\sigma^{+}$-polarised exciton and the microcavity
to be brought into spectral resonance.

We define the transmission as the propagation through the entire diode, i.e.
from port 1 to 2 (2 to 1) in the forward (backward) direction. Ideally, the
transmission in the forward direction $T_{1\rightarrow 2}$ is unity as the
left-handed optical field is orthogonal to the dipole-moment of the quantum
dot, i.e. there is no interaction. In the backward direction, the transmission
amplitude is given by $t_{2\rightarrow 1}=1-2\beta$ where $\beta$ is the
$\beta$-factor describing the interaction of the $\sigma^{+}$-polarised
exciton with an empty microcavity. The full transmission in the backward
direction is $T_{2\rightarrow 1}=\left|t_{2\rightarrow
1}\right|^{2}=\left|1-2\beta\right|^{2}$. In the over-coupled regime,
$\beta\approx 1$, the backwards-propagating photons receive a $\pi$-phase
shift as they transit through the coupled system. At the critical coupling,
$\beta=0.5$, the light reflected by the quantum dot should interfere
destructively with the light directly reflected from the microcavity and the
transmission through the system vanishes; instead, the photons are scattered
into non-microcavity modes.

For this scheme to operate, the microcavity itself should not rotate the
polarisation. However, the fundamental microcavity mode is typically split
into two modes with orthogonal linear polarisations, a consequence of a
birefringence in the semiconductor heterostructure [11]. This mode-splitting
decreases with increasing wavelength while the microcavity linewidth,
$\kappa/(2\pi)$, increases (Supplementary Information Sec. II). We operate at
a wavelength of 945 nm where the mode-splitting is on the order of 29 GHz and
$\kappa/(2\pi)=102$ GHz such that the mode-splitting is much smaller than the
linewidth of the microcavity, and the microcavity mode is nearly degenerate.
The microcavity losses are dominated by the transmission through the top
mirror, i.e. $\kappa_{\rm top}\gg\kappa_{\rm bottom}$ (Supplementary
Information Sec. II). Note that in this regime, the residual mode-splitting
acts as an overall loss channel and does not limit the non-reciprocity of the
system; it mostly affects the insertion loss of the system (here 1.5 dB).

The inset in Fig. 1b shows the level structure of a neutral quantum dot under
a magnetic field along the growth direction (Faraday geometry). An out-of-
plane magnetic field of 2.0 T splits the right-handed and left-handed
transitions by 63 GHz, enough that only one transition interacts with the
laser; here, the right-handed dipole $\sigma^{+}$ (see Supplementary
Information Sec. II for characterisation of the quantum dot). The microcavity
system resides in a cryostat at a temperature of 4.2 K.

Figure 3: Nonlinear response of the diode. (a) Transmission on resonance with
the microcavity versus quantum dot detuning and optical input power. The
transmission saturates at 0.7 on increasing the power. (b) Power dependence of
the transmission on resonance with the quantum dot. The black solid line
corresponds to the theory. The insets illustrate absorption by the quantum dot
at small powers, and transmission at higher powers due to the increasing
transparency of the emitter. The critical power is indicated with a vertical
line. (c) Autocorrelation function of the backwards propagating light for
three different input optical powers. At the lowest optical power (5 pW) a
bunching of 101 is observed. As the power increases the bunching decreases.

The transmission of the diode is probed with coherent laser light at very low
powers, a regime dominated by single-photon components. We exploit the
$\beta$-factor’s dependence on the lateral position of the quantum dot with
respect to the centre of the microcavity to tune the system to the critical-
coupling condition. Figure 1c shows the transmission in the backward
direction, $T_{2\rightarrow 1}$, as a function of the quantum dot frequency
for three different positions, i.e. three different $\beta$-factors. The
orange data points show the transmission for the quantum dot centred in the
microcavity. For this position we extract $\beta=0.72$, matching well the
theoretical expectations for the maximum achievable $\beta$ at this wavelength
(Supplementary Information Sec. III). The $\beta$-factor is evaluated via the
linewidth and the contrast of the dip. When the quantum dot is laterally
displaced such that $\beta=0.50$ (purple data points), the $T_{2\rightarrow
1}$ is lowest: $T_{2\rightarrow 1}=0.07$. Also, the linewidth of the quantum
dot transition decreases with decreasing $\beta$ due to a reduced Purcell-
enhancement. We define the transmission contrast as one minus the ratio
between transmission on resonance and the transmission far off-resonance with
the quantum dot ($1-\frac{T_{2\rightarrow 1}^{0}}{T_{2\rightarrow
1}^{\infty}}$). The transmission contrast is measured while scanning the
lateral position of the quantum dot relative to the microcavity’s optical axis
(Fig. 1d). For a well-centred position, the contrast in the transmission is
around 0.67, increasing to a value of 0.9 as $\beta$ approaches 0.5, and
decreasing as $\beta$ is further reduced. The solid black lines in Fig. 1c,d
are the theoretically expected behaviour. (A comprehensive model for
transmission through a two-level system coupled to a non-degenerate one-sided
microcavity is derived and discussed in detail in Supplementary Information
Sec. I.) An average spectral fluctuation of the quantum dot of 40 MHz was
found by comparing the theoretical model and the measurements [12].

For the operation of the single-photon diode, we focus at the position with
the largest transmission contrast, $\beta=0.50$. The non-reciprocal nature of
the diode is demonstrated by measuring the transmission in both directions at
the critical-coupling condition. Figure 2a shows the transmission through the
diode in the backward direction as a function of the microcavity detuning and
the quantum dot detuning. The transmission contrast shown in these maps
decreases with microcavity detuning on account of a reduced $\beta$-factor. On
the contrary, the transmission in the forward direction (Fig. 2b) presents an
almost flat behaviour independent of the detuning from the quantum dot’s
resonance. The panels on the right side of Fig. 2a,b show the theoretically
predicted behaviour from our model (Supplementary Information Sec. I). A
comparison of the transmission in the forward and backward directions as a
function of the quantum dot (microcavity) detuning is shown in Fig. 2c(d). At
resonance, the transmission in the forward direction is around 0.82. The
slight increase in the transmission signal in the forward direction is
attributed to the mode-splitting. A figure of merit for a diode is the
isolation. It is defined as $T_{1\rightarrow 2}/T_{2\rightarrow 1}$ and found
to be a factor 11.9 (corresponding to 10.7 dB).

To prove that the non-reciprocity arises from a single emitter, we probe both
the power dependence and the photon statistics of the output. In the power
dependence, we find a striking nonlinearity of the transmission in the
backward direction. Figure. 3a shows the transmission in the backward
direction as a function of the optical power and detuning from the quantum
dot’s resonance. On resonance with the quantum dot (Fig. 3b), the backward
transmission increases with a power-law dependence with a slope of one and is
described by $T_{2\rightarrow
1}^{0}=\frac{P/P_{\textrm{C}}}{1+P/P_{\textrm{C}}}$, where $P$ is the input
optical power and $P_{\textrm{C}}$ the critical power. The experimental data
in Fig. 3b match this behaviour very well for $P_{\textrm{C}}\,=\,213$ pW.
This behaviour is characteristic of the saturation of a two-level system –
while the interaction between the input field and the quantum dot is linear in
power at very low powers, the quantum dot saturates at higher input powers,
which leads to a strong power-dependent transmission. The critical power is
very close to the theoretically expected value of 198 pW (Eq. 9 in
Supplementary Information Sec I). This power level corresponds to an average
photon flux of $\innerproduct{n}{n}=0.27$ at the input of the microcavity per
lifetime of the quantum dot ($\tau_{\textrm{QD}}\,=\,0.26$ ns).

The very low onset of the nonlinearity implies that the quantum statistics of
the output field are affected by interaction with the quantum dot [13]. We
verify this by measuring the second-order auto-correlation function,
$g^{(2)}(\tau)$, of the backward transmitted light [14, 15, 16, 17]. The
$g^{(2)}(\tau)$ was measured for three different powers (Fig. 3c). At the
lowest power (5 pW), a very strong bunching of 101 is observed, proving that
the single-photon components of the laser have been largely removed by the
quantum dot. With increasing power, the bunching decreases rapidly and
eventually vanishes, corresponding exactly to the expected behaviour on
saturating the quantum dot: at high powers, most of the laser light is
transmitted without interaction, resulting in $g^{(2)}(\tau)=1$, the auto-
correlation function of the laser light. Additionally, the auto-correlation
function of the transmission in the forward direction is constant and unitary
– this confirms the non-reciprocal transmission in the system (see
Supplementary Information Sec. IV). The measurements were modelled (see
Supplementary Information Sec. IV) and the results are depicted as a solid
black line in Fig. 3c. We emphasise that all the data in Fig. 2 and Fig. 3 are
modelled with the same set of parameters, in particular $\beta=0.50$ and a
free-space decay rate $\gamma_{0}/(2\pi)=300$ MHz.

The reported experiments reveal a strong non-reciprocal and highly nonlinear
transport of optical photons through a quantum dot-microcavity system. Model
calculations based on the canonical chiral one-dimensional atom describe the
system extremely well. The single-photon diode is realised with a modest
emitter-microcavity coupling, $\beta=0.5$. We can foresee a range of
applications. For example, the non-reciprocal behaviour can be dynamically
controlled by driving the $\sigma^{-}$ transition, by using the spin-state of
a charge carrier in the quantum dot, or by fast Stark tuning of the quantum
dot, opening possibilities for optical switches and transistors [18, 19, 20,
21, 22]. Theory predicts that the strong bunching of the photons in the
transmission of the system presages the formation of a two-photon bound state
– it is a first step in creating exotic photonic states and simulating many-
body dynamics using photons [23, 24, 25, 26]. The performance of the system
can be further improved by eliminating the mode-splitting of the microcavity.
The mode-splitting can be minimised, perhaps eliminated, by exploiting the
electro-optic effect [27] or by applying uni-axial stress to the semiconductor
heterostructure [11]. This would not only reduce the insertion losses but also
bring the regime $\beta=1$ within range. Such a device would be ideal for
achieving a single-photon phase-shifter and has a strong potential for
deterministic two-photon quantum gates, either by using spin-state of the
quantum dot or by exploiting photonic bound states [28, 29].

## Acknowledgements

We thank Sahand Mahmoodian and Daniel Najer for stimulating discussions. The
work was supported by NCCR QSIT and SNF Projects No.s 200020$\\_$175748 and
200020$\\_$204069\. AJ received funding from the European Unions Horizon 2020
Research and Innovation Programme under the Marie Skłodowska-Curie grant
agreement No. 840453 (HiFig). TJ acknowledges support from the European Unions
Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-
Curie grant agreement No. 792853 (Hi-FrED). RS, ADW and AL acknowledge
financial support from the grants DFH/UFA CDFA05-06, DFG TRR160, DFG project
383065199, and BMBF Q.Link.X 16KIS0867.

## Supplementary Information:
A chiral one-dimensional atom using a quantum dot in an open microcavity

##  I Transmission of a two-level emitter coupled to a one-sided cavity

The full model describing the transmission of a two-level system coupled to a
one-sided cavity is derived based on the theory in Ref. [1]. We consider the
situation in Fig. 4a. The cavity has two orthogonally polarised modes (H and
V). Both modes are coupled to a one-dimensional waveguide. The Hamiltonian for
the system is:

$\displaystyle\begin{split}\textrm{H}=~{}&\hbar\omega_{0}\sigma_{z}+\hbar(\omega_{0}+\delta_{\textrm{H}})a_{\textrm{H}}^{\dagger}a_{\textrm{H}}+\hbar(\omega_{0}+\delta_{\textrm{V}})a_{\textrm{V}}^{\dagger}a_{\textrm{V}}\\\
&+i\hbar
g_{\textrm{H}}(\sigma_{+}a_{\textrm{H}}-\sigma_{-}a_{\textrm{H}}^{\dagger})+i\hbar(g_{\textrm{V}}^{*}\sigma_{+}a_{\textrm{V}}-g_{\textrm{V}}\sigma_{-}a_{\textrm{V}}^{\dagger})\\\
&+\sum_{k}\hbar\omega
b_{\textrm{kH}}^{\dagger}b_{\textrm{kH}}+\sum_{k}\hbar\omega
b_{\textrm{kV}}^{\dagger}b_{\textrm{kV}}\\\
&+\sum_{k}\hbar\kappa_{-}(a_{\textrm{V}}b_{\textrm{kH}}^{\dagger}+a_{\textrm{V}}^{\dagger}b_{\textrm{kH}})+\sum_{k}\hbar\kappa_{+}(a_{\textrm{V}}b_{\textrm{kV}}^{\dagger}+a_{\textrm{V}}^{\dagger}b_{\textrm{kV}})\end{split}$
(1)

The cavity contains a single two-level system (TLS) with angular resonance
frequency $\omega_{0}$. $a_{\textrm{H/V}}$ ($b_{\textrm{kH/kV}}$) describes
the annihilation operator for the cavity mode (waveguide fields).
$\omega_{0}+\delta_{\textrm{H/V}}$ is the cavity angular frequency, $\omega$
is the laser angular frequency and $\Delta\omega=\omega_{0}-\omega$. The
atomic operators are given by
$\sigma_{z}=\frac{1}{2}(\outerproduct{e}{e}-\outerproduct{g}{g})$ and
$\sigma_{-}=\outerproduct{g}{e}$. The coupling strengths between the cavity
modes and the TLS are given by $g_{\textrm{H,V}}$ where $g_{\textrm{V}}$ can
be complex. Applying the same procedure as Ref. [1], we end up with the
following equations of motion for the cavity modes and the atomic operators in
the laser frame:

$\displaystyle\begin{split}\dot{a}_{\textrm{H/V}}&=-i\left(\Delta\omega+\delta_{\textrm{H/V}}\right)\cdot
a_{\textrm{H/V}}-\frac{\kappa}{2}a_{\textrm{H/V}}-g_{\textrm{H/V}}\cdot\sigma_{-}+i\sqrt{\kappa}\cdot
b_{\textrm{in,H/V}},\\\
\dot{\sigma}_{-}&=-i\Delta\omega\sigma_{-}-2g_{\textrm{H}}\cdot\sigma_{z}\cdot
a_{H}-2g^{*}_{\textrm{V}}\cdot\sigma_{z}\cdot
a_{V}-\gamma/2\cdot\sigma_{-},\\\
\dot{\sigma}_{z}&=g_{\textrm{H}}\cdot\left(\sigma_{+}a_{H}+a_{H}^{\dagger}\sigma_{-}\right)+\left(g^{*}_{\textrm{V}}\sigma_{+}a_{V}+g_{\textrm{V}}a_{V}^{\dagger}\sigma_{-}\right)-\gamma\cdot\left(\sigma_{z}+\frac{1}{2}\right),\\\
b_{\textrm{out,H/V}}&=b_{\textrm{in,H/V}}+i\sqrt{\kappa}\cdot
a_{\textrm{H,V}},\end{split}$ (2)

where $\gamma$ is the decay rate of the two-level system into non-cavity
“leaky” modes and is introduced using a Lindblad operator, and $\kappa$ is the
photon decay rate through the top mirror. $b_{\textrm{in,H/V}}$ describe the
horizontally/vertically polarised components of the input field, and
$b_{\textrm{out,H/V}}$ describe the corresponding output fields.

Equations 2 are the quantum coupled mode equations for the evolution of a TLS
and a cavity driven by $b_{\textrm{in}}$. In the “bad cavity” regime
($\kappa\gg g$), the cavity mode can adiabatically be eliminated from the
equations by setting $\dot{a}=0$. This implies:

$\displaystyle\begin{split}a_{\textrm{H,V}}&=\frac{i\sqrt{\kappa}\cdot
b_{\textrm{in,H,V}}-g_{\textrm{H,V}}\cdot\sigma_{-}}{\frac{\kappa}{2}+i(\Delta\omega+\delta_{\textrm{H,V}})}.\\\
\end{split}$ (3)

The linearly-polarised field at the input of the diode (the diode consists of
the cavity, the TLS and the optical components, as shown in Fig. 4),
$b_{\textrm{in}}$, is converted to a circularly polarised field at the input
of the cavity, and $b_{\textrm{in,H}}=1/\sqrt{2}b_{\textrm{in}}$ and
$b_{\textrm{in,V}}=i/\sqrt{2}b_{\textrm{in}}$. Moreover, the TLS has a
circularly-polarised optical dipole, hence, $g_{\textrm{V}}=ilg_{\textrm{H}}$,
where $l=\pm 1$ sets the handedness of the optical dipole. In the following,
we drop the subscript H, setting $g_{\textrm{H}}=g$. Equation LABEL:eq:cavity
may be written in a more compact form by using the definition
$t_{\textrm{H,V}}=\left(1+{2i\left(\Delta\omega+\delta_{\textrm{H,V}}\right)}/{\kappa}\right)^{-1}$:

$\displaystyle\begin{split}a_{\textrm{H}}&=\sqrt{\frac{2}{\kappa}}\left(ib_{\textrm{in}}-\sqrt{\frac{2g^{2}}{\kappa}}\sigma_{-}\right)t_{\textrm{H}},\\\
a_{\textrm{V}}&=i\sqrt{\frac{2}{\kappa}}\left(ib_{\textrm{in}}-l\sqrt{\frac{2g^{2}}{\kappa}}\sigma_{-}\right)t_{\textrm{V}}.\\\
\end{split}$ (4)

The output of the cavity passes a quarter wave-plate and is converted to a
vertical field at the output of the diode, and hence,
$b_{\textrm{out}}=\frac{1}{\sqrt{2}}(b_{\textrm{out,H}}-ib_{\textrm{out,V}})$.
Combining this equation with the last term in Eq. 2, the field amplitude at
the output of the diode can be related to the input fields and the cavity
operators as:

$\displaystyle\begin{split}b_{\textrm{out}}&=b_{\textrm{in}}+\sqrt{\frac{\kappa}{2}}(ia_{\textrm{H}}+a_{\textrm{V}}),\\\
&=b_{\textrm{in}}-b_{\textrm{in}}(t_{\textrm{H}}+t_{\textrm{V}})-i\sqrt{\frac{\Gamma_{1}}{2}}(t_{\textrm{H}}+lt_{\textrm{V}})\cdot\sigma_{-},\end{split}$
(5)

and the equations motion become:

$\displaystyle\begin{split}\dot{\sigma}_{-}&=-i\Delta\omega\sigma_{-}-\frac{\Gamma_{1}}{2}\left(t_{H}+t_{V}+\frac{1}{F_{p1}}\right)~{}\sigma_{-}+ib_{\textrm{in}}\sqrt{\frac{\Gamma_{1}}{2}}(-2\sigma_{z})(t_{H}+lt_{V}),\\\
\dot{\sigma}_{z}&=-{\Gamma_{1}}\left(\textrm{Re}(t_{H}+lt_{V})+\frac{1}{F_{p1}}\right)\left(\sigma_{z}+\frac{1}{2}\right)+b_{\textrm{in}}\sqrt{\frac{\Gamma_{1}}{2}}\left(i(t_{H}+t_{V})\sigma_{+}+h.c.\right),\end{split}$
(6)

where $\Gamma_{1}=\frac{4g^{2}}{\kappa}$,
$F_{p1}=\frac{4g^{2}}{\kappa\gamma}$. Note that the decay rate of the TLS is
$\Gamma=2\Gamma_{1}$ when the cavity modes are degenerate. In this limit, the
Purcell-factor is $F_{p}=2F_{p1}$ as both cavity modes contribute to the total
Purcell-factor.

Next, we solve the steady-state equations
$\dot{\sigma}_{-}=\dot{\sigma}_{z}=0$ for $\sigma_{z}$ and $\sigma_{-}$.
Following the Ehrenfest theorem, we can replace the operators in the steady
state by their expectation values: $\expectationvalue{\sigma_{-}}$ = $S_{-}$,
$\expectationvalue{\sigma_{z}}$ = $S_{z}$ and
$\expectationvalue{b_{\textrm{in}}}$ = $b_{\textrm{in}}$. We end up with:

$\displaystyle\begin{split}S_{-}&=\frac{ib_{\textrm{in}}\sqrt{\frac{\Gamma}{4}}(-2S_{z})(t_{\textrm{H}}+lt_{\textrm{V}})}{-i\Delta\omega+\Gamma/4\left(t_{H}+t_{V}+\frac{2}{F_{p}}\right)~{}},\\\
S_{z}&=\frac{-1/2\left(\textrm{Re}(t_{H}+lt_{V})+\frac{2}{F_{p}}\right)}{\textrm{Re}(t_{H}+lt_{V})+\frac{2}{F_{p}}+2b^{2}_{\textrm{in}}\textrm{Re}[\frac{(t_{\textrm{H}}+t_{\textrm{V}})(t^{*}_{\textrm{H}}+lt^{*}_{\textrm{V}})}{i\Delta\omega+\Gamma/4(t^{*}_{\textrm{H}}+t^{*}_{\textrm{V}}+2/F_{p})}]}.\end{split}$
(7)

For the simplified case of resonant excitation ($\Delta\omega=0$) and
degenerate cavity modes ($\delta_{H}=\delta_{V}$), $S_{z}$ reduces to:

$S_{z}=-\frac{1}{2}\frac{1}{1+|b_{\textrm{in}}|^{2}/P_{c}}=-\frac{1}{2}\Pi,$
(8)

with critical power:

$P_{c}=\frac{\Gamma}{8\beta^{2}}\cdot\hbar\omega.$ (9)

$\beta$ is the coupling efficiency of the emitter to the cavity and can be
described by: $\beta=\frac{\Gamma}{\Gamma+\gamma}=\frac{F_{p}}{F_{p}+1}$.
Plugging Eq. 7 into Eq. 5, we determine the full transmission coefficient of
the cavity-emitter system via $b_{\textrm{out}}=t\cdot b_{\textrm{in}}$ to be:

$t=~{}1~{}-~{}t_{H}~{}-t_{V}~{}+~{}\frac{\Pi}{\frac{2\Delta\omega}{\frac{\Gamma}{2}(t_{H}+t_{V})}+1+\frac{2}{F_{p}(t_{H}+t_{V})}}\cdot\frac{(t_{H}+lt_{V})^{2}}{(t_{H}+t_{V})}.$
(10)

When $l=1$, the optical dipole and the incoming light have the same
handedness, while $l=-1$ for a non-interacting optical dipole. At low input
powers, with the same simplified condition as in Eq. 8 ($\Delta\omega=0$,
$\delta_{H}=\delta_{V}$) and $l=1$, the reflection coefficient can be reduced
to the simple form:

$t(0)~{}=-1+2\beta.$ (11)

In this case, $t(0)=0$ at the critical coupling condition $\beta=1/2$.

So far, $t$ includes only the coherent part of the transmitted field. At
higher excitation powers, the transmitted field is a sum of coherent and
incoherent components. The incoherent component arises from the spontaneous
emission of photons from the TLS. The incoherent part of the scattering
($P_{\textrm{incoh}}$) can be extracted by subtracting the coherent scattering
of the TLS (both scattered into the cavity and into non-cavity modes) from the
total power [30]:

$\frac{P_{\textrm{incoh}}}{|b_{\textrm{in}}|^{2}}=1-|t|^{2}-\gamma\cdot\frac{|\expectationvalue{S_{-}}|^{2}}{|b_{\textrm{in}}|^{2}}.$
(12)

The intensity transmission from port 1 to port 2 (or vice versa) in Fig. 1b of
the main text is given by

$T=|t|^{2}+\beta\cdot\frac{P_{\textrm{incoh}}}{2\cdot|b_{\textrm{in}}|^{2}}.$
(13)

In the simplest ideal case (resonance condition as in Eq. 8 and $\beta=0.5$),
this leads to a power-dependent transmission of the form:

$T=\frac{{|b_{\textrm{in}}|^{2}}/{P_{c}}}{(1+{|b_{\textrm{in}}|^{2}}/{P_{c}})}.$
(14)

The average photon flux per lifetime of the emitter is related to the input
power via
$\hbar\omega\expectationvalue{n}=|b_{\textrm{in}}|^{2}/(\Gamma+\gamma)$.

Figure 4: (a) Illustration of the concept and definition of the parameters of
a one-sided cavity with cavity modes $a_{\textrm{H}}$ and $a_{\textrm{V}}$,
and incoming (outgoing) field $b_{\textrm{in}}(b_{\textrm{out}})$. A two-level
emitter with ground state $\ket{g}$ and excited state $\ket{e}$ is coupled to
the cavity modes. (b) Transmission through the full system in (a) as a
function of the laser frequency detuning, $\Delta f=\Delta\omega/(2\pi)$, for
different $\beta$-factors in the simplified, ideal case:
$\delta_{\textrm{H}}/(2\pi)=\delta_{\textrm{V}}/(2\pi)=0$, without spectral
fluctuation, $\kappa/(2\pi)=$ 102 GHz and $\gamma/(2\pi)=$ 0.30 GHz. (c)
Transmission through the full system in (a) as a function of the laser
frequency detuning, $\Delta f=\Delta\omega/(2\pi)$, for different
$\beta$-factors under experimental conditions:
$\delta_{\textrm{H}}/(2\pi)-\delta_{\textrm{V}}/(2\pi)=$ 29 GHz,
$\delta_{\textrm{sf}}/(2\pi)=40$ MHz, $\kappa/(2\pi)=$ 102 GHz and
$\gamma/(2\pi)=$ 0.30 GHz. The inset shows how the depth of the dip and the
linewidth depend on the $\beta$-factor.

In order to match the conditions of the experiment, we include a spectral
fluctuation. This is done by convoluting a Lorentzian distribution with the
quantum dot transmission [12]:

$T_{\textrm{end}}=\int T~{}(\Delta\omega_{\textrm{QD}}+\sigma)\cdot
L(\sigma,\delta_{\textrm{sf}})\cdot d\sigma.$ (15)

$L(\sigma,\delta_{\textrm{sf}})$ describes a Lorentzian distribution and
$\delta_{\textrm{sf}}$ is the FWHM of this distribution, i.e. the spectral
fluctuation.

Figures 4b and c show the transmission through the system as a function of the
laser frequency for different values of the $\beta$-factor. We use
$\kappa/(2\pi)=102$ GHz and $\gamma/(2\pi)=0.30$ GHz for both figures. In Fig.
4b we show the simplified, ideal case without mode-splitting and without
spectral fluctuation
($\delta_{\textrm{H}}/(2\pi)=\delta_{\textrm{V}}/(2\pi)=$0,
$\delta_{\textrm{sf}}/(2\pi)=0$). The transmission is unity in the case of
zero coupling between TLS and cavity ($\beta=0$). With increasing $\beta$, a
dip appears on resonance with the TLS. This dip reaches zero at the critical
coupling condition $\beta=0.5$. Increasing $\beta$ further reduces the depth
of the dip again. In the strongly over-coupled regime ($\beta\approx 1$), the
photons that interact with the TLS acquire a $\pi$-phase shift according to
Eq. 11. Figure 4c shows the same plot after adapting the theory to the
experimental conditions, $\delta_{\textrm{sf}}/(2\pi)=40$ MHz and
$\delta_{\textrm{H}}/(2\pi)-\delta_{\textrm{V}}/(2\pi)=$ 29 GHz. With no TLS-
cavity coupling ($\beta=0$), the transmission through the system is governed
by the two cavity modes and a dip of 30% arises due to the finite mode-
splitting of the cavity. In the under-coupled regime ($\beta<0.5$), a small
dip is visible on resonance with the quantum dot, similar to the ideal case.
This dip reaches 90% for the critical-coupling condition ($\beta=0.5$). At the
critical-coupling condition, most of the photons impinging on the system are
dissipated by the TLS to the non-cavity modes. Increasing $\beta$ further
results in a reduced dip size.

We model the transmission plots in the main text using Eq. 15 with the same
parameters as in Fig. 4. For the results in Fig. 2 and Fig. 3b of the main
text, we set $F_{p}=1$. The saturation power extracted by fitting Fig. 3b in
the main text with Eq. 14 is $P_{c}=213$ pW, which is in agreement within the
errorbar with the theoretical saturation power $P_{c}=198$ pW that is expected
from Eq. 9.

##  II Microcavity and quantum dot characterisation

###  II.1 Characterisation of the cavity parameters

By measuring the cross-polarised [31] transmission through the bare cavity
setup (from port 2 to port 1 in Fig. 4a) and fitting a double-Lorentzian to
the data, we can characterise the decay rate, $\kappa$, and the mode-
splitting, $\Lambda=\delta_{H}-\delta_{V}$, of the cavity; see the top panel
in Fig. 5b for an example of the transmission spectrum. The top panel in Fig.
5a shows $\kappa$ and $\Lambda$ as a function of wavelength; $\Lambda$
decreases for higher wavelength while $\kappa$ increases [11]. By comparing
$\kappa$ to the cavity losses reported in Ref. [32], which were measured with
the same bottom mirror but with a highly reflective top mirror, we conclude
that $\kappa\approx\kappa_{\textrm{top}}$ ($\kappa_{\textrm{top}}$ describes
the cavity losses through the top mirror), i.e. close to all losses arise via
transmission through top mirror for all wavelengths, assuring that the cavity
is indeed one-sided in the full range of wavelengths presented.

###  II.2 Characterisation of the insertion loss of the diode and the optimal
wavelength for its operation

Neglecting interaction terms in Eq. 10, we can calculate the transmission
through both cavity modes in the co-polarised configuration as:

$T=\left|1-\frac{1}{1+{2i(\Delta\omega+{\Lambda}/{2})}/{\kappa}}-\frac{1}{1+{2i(\Delta\omega-{\Lambda}/{2})}/{\kappa}}\right|^{2}.$
(16)

Using Eq. 16 and $\kappa$ and $\Lambda$ extracted in Sec. IIA (shown in the
top panel of Fig. 5a), we calculate the transmission from port 2 to port 1 for
circularly polarised light at different wavelengths for $\Delta\omega=0$, i.e.
the laser frequency lies exactly between the two cavity modes (bottom panel of
Fig. 5a). We observe that the mode-splitting induces an overall loss in the
transmission, which decreases with increasing wavelength. Eventually, the
transmission reaches unity at high wavelengths.

In order to reduce the insertion loss, working at high wavelengths, i.e. above
950 nm, seems obvious. However, as is explained in more detail in Sec. III, we
are limited by the maximum achievable $\beta$. Given these constraints, we
determined 945 nm to be an ideal wavelength to carry out the experiments.
Figure 5b shows the cross-(top) and co-polarised (bottom) cavity transmission
for this optimised operation-wavelength. The co-polarised data could be
reconstructed using the extracted cavity parameters of Fig. 5a and Eq. 16.

Figure 5: Characterisation of the cavity parameters without a QD (a) Measured
cavity decay rate ($\kappa/(2\pi)$, red circles) and mode-splitting
($\Lambda/(2\pi)$, blue triangles) versus wavelength. The transmission through
the one-sided cavity on resonance with the cavity mode (black stars) is
calculated with these two parameters and Eq. 16. The transmission increases,
i.e. the loss decreases, with increasing wavelength. (b) The measured
transmission through the cavity mode in the cross- (top) and co-polarised
(bottom) configuration at a wavelength of 945 nm versus the detuning of the
cavity frequency. The peak in cross-polarisation is fitted with the response
of the two linear cavity modes (H and V) that are plotted in purple and
turquoise. The cavity dip was modelled using the parameters from (a).

###  II.3 Characterisation of the quantum dot

The Purcell factor, $F_{p}$, is a critical parameter characterising the
operation of our diode. It is defined through
$\Gamma_{\textrm{tot}}=(1+F_{p})\gamma$, where $\Gamma$ is the overall
linewidth of the quantum dot, and $\gamma$ is the linewidth of the quantum dot
outside the cavity. The linewidth of a bare quantum dot is typically
$\gamma/(2\pi)\approx 300$ MHz, as measured in [10]. We estimate the Purcell-
enhanced linewidth of the quantum dot by measuring the resonance fluorescence
spectrum from the quantum dot. Figure 6a shows a sample spectrum from a
quantum dot placed in the centre of the cavity. We extract
$\Gamma_{\textrm{tot}}/(2\pi)=1.2$ GHz by fitting two Lorentzians to the data.
The observed Purcell factor is $F_{p}=3$, matching predictions from COMSOL
simulations shown in Fig. 8.

The two peaks in Fig 6a show the fine-structure splitting of the optical
transition. This splitting between the two transitions increases on applying a
magnetic field (B), as seen in Fig. 6b. At B=2.00 T, the transitions are split
by 63 GHz (0.189 nm), large enough that only one of them couples to the laser.
The two curves in Fig. 6b show the photoluminescence (PL) signal of the
quantum dot for different positions of the quarter-wave plate (QWP), hence,
for different detection polarisations. The purple (turquoise) line represents
the QWP setting where the incoming light matches the lower (higher)-wavelength
transition. Therefore, only signal from the corresponding transition is
collected. The data in Fig. 6 were taken on a different quantum dot to that
used in the rest of the paper, but they show the principle of operation.

Figure 6: Characterisation of the QD (a) Resonance fluorescence from an over-
coupled quantum dot in the microcavity in the cross-polarised configuration.
The Purcell-enhanced linewidth is around 1.20 GHz, corresponding to
$\beta=0.72$. (b) Photoluminescence spectrum of the quantum dot in a magnetic
field of 2.00 T. A Zeeman splitting of 0.189 nm is observed. The lower
wavelength (purple), i.e. higher energy, transition was used for all
measurements. The two curves were measured by changing the angle of the
quarter-wave plate between the two circular configurations.

###  II.4 Overcoming the dragging effect from the nuclei

While operating in a magnetic field, we observed a nuclear-spin related effect
in the transmission data, see Fig. 7. We found that the transmission depends
on the scanning direction of the gate voltage and is asymmetric around the
resonance of the quantum dot. In order to overcome these so-called dragging
and anti-dragging effects, the gate voltage was switched rapidly (100 Hz)
between the resonance voltage and the co-tunnelling voltage using a square
function from an arbitrary waveform generator. In the co-tunnelling regime, an
electron is exchanged frequently with another electron, randomising the
electron’s spin state and eliminating the spin polarisation of the nuclei
[33]. Transmission data were acquired only when the gate voltage was at the
resonance voltage. The procedure to overcome nuclear spin effects was applied
for all transmission data shown in the main text and in the remainder of the
supplementary information.

Figure 7: Overcoming nuclear spin induced effects (a)/(b) Transmission through
the diode as a function of quantum dot detuning for the high/low energy
transition. The so-called “dragging” and “anti-dragging” effects
(purple/turquoise lines) are overcome (black lines) by jumping back and forth
between the measurement voltage and a voltage in the co-tunnelling regime.

##  III Dependence of the $\beta$-factor on the wavelength and the position of
the quantum dot

###  III.1 The wavelength dependence

The coupling efficiency between the quantum dot and the cavity mode is given
by the $\beta$-factor which has a wavelength dependence arising from the
dispersion of the top and the bottom mirrors. We simulate this dependence by
calculating the mode volume and the quality factor of the cavity in COMSOL.
The results are shown in Fig. 8. The $\beta$-factor drops with increasing
wavelength and falls below 0.5 at a wavelength of 955 nm. This effect arises
from the fact that the centre of the stop-band for the top and the bottom
mirrors are 920 nm. Away from 920 nm the reflection of the top mirror drops
and results in a reduced quality factor and, hence, a reduction in the
$\beta$-factor.

###  III.2 The wavelength for the diode operation

Using the model derived in Sec. I, $\Lambda$ and $\kappa$ extracted in Sec.
II, and leaving $\beta$ a free parameter, we estimate the highest observable
contrast ($1-T^{0}/T^{\infty}$) for each wavelength (bottom part of Fig. 8)
and extract the $\beta$ at which it is maximised (top panel in Fig. 8). The
achievable contrast is nearly constant between 945 and 955 nm, giving a broad
range of quantum dots to work with. This also shows that the insertion losses
discussed in Sec. II do not influence the isolation, i.e. non-reciprocity, of
the system, but only describe overall losses. We chose an emitter at 945 nm,
as the maximum $\beta$ is above the critical value of 0.5, and the position
dependence of $\beta$ can be exploited to tune the system into the critical
coupling conditions.

Figure 8: Wavelength dependence of the $\beta$-factor. COMSOL simulations (red
circles) show the maximum achievable $\beta$ for a quantum dot positioned in
the centre of the cavity. At wavelengths above 955 nm, the critical $\beta$ of
0.5 is not reachable. Calculations using the model in Sec. I enable us to
predict the maximal contrast achievable for different wavelengths (black
stars) and the optimal $\beta$-factor to observe the strongest contrast (blue
triangles). The range between 945 and 955 has the highest contrast at
$\beta=0.5$ and is therefore ideal. The experiments in the main text were
carried out at 945 nm (vertical line).

###  III.3 The position dependence

The cavity mode has a Gaussian intensity profile centred around the crater.
The coupling between the cavity and the quantum dot is maximum when the
quantum dot is placed at the centre of the cavity mode. Figure. 9a shows the
resonance fluorescence of a quantum dot as a function of its lateral position.
The origin of the coordinate system corresponds to the centre of the cavity
mode. The fluorescence from the quantum dot drops as it is displaced from the
origin, which shows that the coupling efficiency between the cavity and the
quantum dot is reduced. This position dependence can be used to achieve
$\beta=0.5$ with high precision as shown in the main text. The position with
$\beta=0.5$, used in most measurements in the main text, is indicated by the
blue circle in Fig. 9a.

###  III.4 Over-coupled cavity-quantum dot system

Figure 9b shows the transmission as a function of cavity and quantum dot
detuning, i.e. the same as Fig. 2a,b in the main text, for an over-coupled
quantum dot (the same quantum dot but now positioned in the centre of the
cavity). For this situation, the strongest contrast appears when the quantum
dot is spectrally detuned from the cavity resonance (contrast 0.89, red
circles); the Purcell factor, $F_{p}$, hence also $\beta$, decrease as a
function of cavity detuning. In this case, the isolation resulting from the
detuned quantum dot-cavity system is 11.2 (10.5 dB), which is similar to the
value reported in the main text (10.7 dB). The panels on the right side of
Fig. 9 are reproduced using the model in Sec. I, with $\beta=0.72$,
$\kappa/(2\pi)=102$ GHz, $\Lambda/(2\pi)=29$ GHz, and a spectral fluctuation
of $\delta_{\textrm{sf}}/(2\pi)=35$ MHz. The theoretical plots in Fig. 2 of
the main text are reproduced with the same parameters but with $\beta=0.5$.

Figure 9: (a) Intensity of resonance fluorescence as a function of the
displacement of the quantum dot from the cavity mode centre. The position is
defined as the distance from the anti-node with respect to the top mirror of
the cavity, i.e. $r=\sqrt{(X^{2}+Y^{2})}$. We apply a bias to the X and Y
nanopositioners to move the semiconductor sample. The applied voltages are
translated to distance using the specifications from the manufacturer
(attocube systems AG). The position of the quantum dot for most of the
measurements in the main text (highest transmission contrast) is indicated
with a blue circle. (b) Transmission as a function of cavity and quantum dot
detuning for the same quantum dot as in the main text, but in the centre of
the cavity ($r=0$) where $\beta=0.72$. The strongest contrast is 0.89 which
appears at a spectral detuning of 64 GHz from the cavity resonance (red
circles). The model explains the data very well.

##  IV Second-order correlation function

We measured the intensity auto-correlation function $g^{(2)}(\tau)$ as a
function of time delay $\tau$ for the photons propagating in the forward (port
1 to port 2) and in the backward (port 2 to port 1) directions. The recorded
data for both directions at the lowest power is shown in Fig. 10a.
$g^{(2)}(\tau)$ in the forward direction is a flat line, corresponding to that
of coherent light, as there is no interaction with the emitter in the forward
direction and the light’s photon statistics are not affected. In the backward
direction, a $g^{(2)}(\tau=0)=101$ is observed at zero delay and low power of
5 pW. This bunching decays exponentially at higher power as can be observed in
Fig. 10b.

In order to model $g^{(2)}(\tau)$ of the backwards-propagating photons, we
base ourselves on the derivation of Ref. [13]. A comparison of the equations
of motion in Ref. [13] to the ones in Eq. 10 (in resonance condition:
$\Delta\omega=0$, $\delta_{\textrm{H/V}}=0$ $\rightarrow$ $t_{H/V}=1$) reveals
that the equations of motion for a two-sided (2S) cavity are the same as the
equations of motion for a one-sided (1S) cavity under the transformation:

$\frac{1}{1/F_{\textrm{p,2S}}+1}=\frac{2}{1/F_{\textrm{p,1S}}+1},$ (17)

where $F_{\textrm{p,2S}}$ is the Purcell factor as defined in Ref. [13] and
$F_{\textrm{p,1S}}$ is the Purcell factor in Eq. 10 (labelled $F_{p}$ in Sec.
I-III). This transformation is equivalent to transforming the $\beta-$factor
as $\beta_{\textrm{2S}}=2\beta_{\textrm{1S}}$. Note that this substitution can
also be seen from the difference in the simplified transmission equations,
which for a two-sided cavity is $T=|1-\beta|^{2}$ and $T=|1-2\beta|^{2}$ for a
one-sided cavity.

We use the result for $g^{(2)}(\tau)$ in Ref. [13] to model $g^{(2)}(\tau)$ in
our experiments. The full formula for $g^{(2)}(\tau)$ is given by

$g^{(2)}(\tau)=1+\left(\frac{\sqrt{2}F_{\textrm{p,2S}}}{1+Y^{2}}\right)^{2}\cdot~{}e^{(3/4)\tau^{\prime}}\cdot\Bigg{\\{}\left(Y^{2}+\frac{F_{\textrm{p,2S}}^{2}}{2}-1\right)\cdot\cosh(\Omega\tau^{\prime})+\frac{1}{4\Omega}\cdot\left[Y^{2}\cdot\frac{5-F_{\textrm{p,2S}}}{1+F_{\textrm{p,2S}}}-1-\frac{F_{\textrm{p,2S}}^{2}}{2}\right]\cdot\sinh(\Omega\tau^{\prime})\Bigg{\\}},$
(18)

with $\Omega=1/4\cdot\sqrt{1-(8Y)^{2}/((1+F_{\textrm{p,2S}})^{2}}$; $Y$ is the
Rabi angular frequency in units of $\gamma$
($\textrm{Y}=\sqrt{\Gamma}/\sqrt{\gamma}\cdot b_{\textrm{in}}$); and
$\tau^{\prime}$ is a dimensionless delay,
$\tau^{\prime}\,=\,\gamma(1~{}+~{}F_{\textrm{p,2S}})\cdot~{}\tau$. In the low
power limit ($Y\ll 1$), this reduces to:

$g^{(2)}(\tau)=(1-F_{\textrm{p,2S}}^{2}\cdot e^{-\tau^{\prime}/2})^{2},$ (19)

and

$g^{(2)}(0)=(1-F^{2}_{\textrm{p,2S}})^{2}.$ (20)

The power dependence of the bunching is modelled by setting the time delay
$\tau=0$, resulting in:

$g^{(2)}(0)=1+\left(\frac{\sqrt{2}F_{p,2S}}{1+Y^{2}}\right)^{2}\cdot\left(Y^{2}+\frac{F_{p,2S}^{2}}{2}-1\right).$
(21)

These equations describe our measurements very well. Note that all the
modelling for the correlation function was carried out for the case of a
degenerate cavity for simplicity. We use $F_{\textrm{p,1S}}=0.8$ to model our
results in the backward direction. The discrepancy with the expected value of
$F_{\textrm{p,1S}}=1$ likely results from the residual spectral fluctuation
(see last part of Sec. I and Eq. 15) as $g^{(2)}(\tau)$ has a strong
dependence on dephasing at low powers [34].

Figure 10: (a) Auto-correlation function $g^{(2)}(\tau)$ for a critically-
coupled quantum dot-cavity system for the two propagation directions. Top:
Backward direction showing a high bunching of 101. Bottom: Forward direction
showing a flat curve indicating that in the forward direction there is no
interaction between the photons and the quantum dot such that the light
remains coherent after passing through the cavity. The input power is around 5
pW. The black, solid lines are based on the model in Eq. 19. (b) $g^{(2)}(0)$
as a function of power in the backward direction. The observed bunching
behaviour is a very strong function of power. The solid line is based on the
model in Eq. 21.

## References

  * [1] A. Auffèves-Garnier, C. Simon, J.-M. Gérard, and J.-P. Poizat, _Giant optical nonlinearity induced by a single two-level system interacting with a cavity in the Purcell regime_ , Physical Review A 75, 053823 (2007).
  * [2] J.-T. Shen and S. Fan, _Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom_ , Physical Review A 79, 023837 (2009).
  * [3] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, _Chiral quantum optics_ , Nature 541, 473 (2017).
  * [4] C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, _Nanophotonic Optical Isolator Controlled by the Internal State of Cold Atoms_ , Physical Review X 5, 041036 (2015).
  * [5] M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, _Quantum optical circulator controlled by a single chirally coupled atom_ , Science 354, 1577 (2016).
  * [6] I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, _Deterministic photon–emitter coupling in chiral photonic circuits_ , Nature Nanotechnology 10, 775 (2015).
  * [7] R. J. Coles, D. M. Price, J. E. Dixon, B. Royall, E. Clarke, P. Kok, M. S. Skolnick, A. M. Fox, and M. N. Makhonin, _Chirality of nanophotonic waveguide with embedded quantum emitter for unidirectional spin transfer_ , Nature Communications 7, 11183 (2016).
  * [8] D. L. Hurst, D. M. Price, C. Bentham, M. N. Makhonin, B. Royall, E. Clarke, P. Kok, L. R. Wilson, M. S. Skolnick, and A. M. Fox, _Nonreciprocal Transmission and Reflection of a Chirally Coupled Quantum Dot_ , Nano Letters 18, 5475 (2018).
  * [9] D. Najer, I. Söllner, P. Sekatski, V. Dolique, M. C. Löbl, D. Riedel, R. Schott, S. Starosielec, S. R. Valentin, A. D. Wieck, N. Sangouard, A. Ludwig, and R. J. Warburton, _A gated quantum dot strongly coupled to an optical microcavity_ , Nature 575, 622 (2019).
  * [10] N. Tomm, A. Javadi, N. O. Antoniadis, D. Najer, M. C. Löbl, A. R. Korsch, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig, and R. J. Warburton, _A bright and fast source of coherent single photons_ , Nature Nanotechnology 16, 399 (2021).
  * [11] N. Tomm, A. R. Korsch, A. Javadi, D. Najer, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig, and R. J. Warburton, _Tuning the Mode Splitting of a Semiconductor Microcavity with Uniaxial Stress_ , Physical Review Applied 15, 054061 (2021).
  * [12] A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, and R. J. Warburton, _Charge noise and spin noise in a semiconductor quantum device_ , Nature Physics 9, 570 (2013).
  * [13] P. R. Rice and H. J. Carmichael, _Single-atom cavity-enhanced absorption. I. Photon statistics in the bad-cavity limit_ , IEEE Journal of Quantum Electronics 24, 1351 (1988).
  * [14] D. Wang, H. Kelkar, D. Martin-Cano, D. Rattenbacher, A. Shkarin, T. Utikal, S. Götzinger, and V. Sandoghdar, _Turning a molecule into a coherent two-level quantum system_ , Nature Physics 15, 483 (2019).
  * [15] A. Pscherer, M. Meierhofer, D. Wang, H. Kelkar, D. Martín-Cano, T. Utikal, S. Götzinger, and V. Sandoghdar, _Single-molecule vacuum Rabi splitting: four-wave mixing and optical switching at the single-photon level_ , arXiv:2105.02560 (2021).
  * [16] T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, _Nanophotonic quantum phase switch with a single atom_ , Nature 508, 241 (2014).
  * [17] H. Snijders, J. A. Frey, J. Norman, M. P. Bakker, E. C. Langman, A. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, _Purification of a single-photon nonlinearity_ , Nature Communications 7, 12578 (2016).
  * [18] D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, _A single-photon transistor using nanoscale surface plasmons_ , Nature Physics 3, 807 (2007).
  * [19] D. Witthaut, M. D. Lukin, and A. S. Sørensen, _Photon sorters and QND detectors using single photon emitters_ , Europhysics Letters 97, 50007 (2012).
  * [20] A. Javadi, D. Ding, M. H. Appel, S. Mahmoodian, M. C. Löbl, I. Söllner, R. Schott, C. Papon, T. Pregnolato, S. Stobbe, L. Midolo, T. Schröder, A. D. Wieck, A. Ludwig, R. J. Warburton, and P. Lodahl, _Spin–photon interface and spin-controlled photon switching in a nanobeam waveguide_ , Nature Nanotechnology 13, 398 (2018).
  * [21] I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan, _All-optical routing of single photons by a one-atom switch controlled by a single photon_ , Science 345, 903 (2014).
  * [22] D. O’Shea, C. Junge, J. Volz, and A. Rauschenbeutel, _Fiber-Optical Switch Controlled by a Single Atom_ , Physical Review Letters 111, 193601 (2013).
  * [23] C. Noh and D. G. Angelakis, _Quantum simulations and many-body physics with light_ , Reports on Progress in Physics 80, 016401 (2016).
  * [24] M. Hafezi, M. D. Lukin, and J. M. Taylor, _Non-equilibrium fractional quantum Hall state of light_ , New Journal of Physics 15, 063001 (2013).
  * [25] M. Gullans, S. Diehl, S. Rittenhouse, B. Ruzic, J. D’Incao, P. Julienne, A. Gorshkov, and J. Taylor, _Efimov States of Strongly Interacting Photons_ , Physical Review Letters 119, 233601 (2017).
  * [26] M. Maghrebi, M. Gullans, P. Bienias, S. Choi, I. Martin, O. Firstenberg, M. Lukin, H. Büchler, and A. Gorshkov, _Coulomb Bound States of Strongly Interacting Photons_ , Physical Review Letters 115, 123601 (2015).
  * [27] J. A. Frey, H. J. Snijders, J. Norman, A. C. Gossard, J. E. Bowers, W. Löffler, and D. Bouwmeester, _Electro-optic polarization tuning of microcavities with a single quantum dot_ , Optics Letters 43, 4280 (2018).
  * [28] T. Ralph, I. Söllner, S. Mahmoodian, A. White, and P. Lodahl, _Photon Sorting, Efficient Bell Measurements, and a Deterministic Controlled-Z Gate Using a Passive Two-Level Nonlinearity_ , Physical Review Letters 114, 173603 (2015).
  * [29] S. Mahmoodian, G. Calajó, D. E. Chang, K. Hammerer, and A. S. Sørensen, _Dynamics of Many-Body Photon Bound States in Chiral Waveguide QED_ , Physical Review X 10, 031011 (2020).
  * [30] A. Javadi, I. Söllner, M. Arcari, S. L. Hansen, L. Midolo, S. Mahmoodian, G. Kiršanskė, T. Pregnolato, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, _Single-photon non-linear optics with a quantum dot in a waveguide_ , Nature Communications 6, 8655 (2015).
  * [31] A. V. Kuhlmann, J. Houel, D. Brunner, A. Ludwig, D. Reuter, A. D. Wieck, and R. J. Warburton, _A dark-field microscope for background-free detection of resonance fluorescence from single semiconductor quantum dots operating in a set-and-forget mode_ , Review of Scientific Instruments 84, 073905 (2013).
  * [32] D. Najer, N. Tomm, A. Javadi, A. R. Korsch, B. Petrak, D. Riedel, V. Dolique, S. R. Valentin, R. Schott, A. D. Wieck, A. Ludwig, and R. J. Warburton, _Suppression of Surface-Related Loss in a Gated Semiconductor Microcavity_ , Physical Review Applied 15, 044004 (2021).
  * [33] C. Latta, A. Högele, Y. Zhao, A. N. Vamivakas, P. Maletinsky, M. Kroner, J. Dreiser, I. Carusotto, A. Badolato, D. Schuh, W. Wegscheider, M. Atature, and A. Imamoglu, _Confluence of resonant laser excitation and bidirectional quantum-dot nuclear-spin polarization_ , Nature Physics 5, 758 (2009).
  * [34] P. Lodahl, S. Mahmoodian, and S. Stobbe, _Interfacing single photons and single quantum dots with photonic nanostructures_ , Reviews of Modern Physics 87, 347 (2015).