# Quantum trajectory theory and simulations of nonlinear spectra and multi-
photon effects in waveguide-QED systems with a time-delayed coherent feedback

Gavin Crowder<EMAIL_ADDRESS>Department of Physics, Queen’s University,
Kingston, Ontario, Canada, K7L 3N6  Department of Physics, University of
Ottawa, Ottawa, ON K1N 6N5, Canada  Lora Ramunno  Department of Physics,
University of Ottawa, Ottawa, ON K1N 6N5, Canada  Stephen Hughes  Department
of Physics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6

###### Abstract

We study the nonlinear spectra and multi-photon correlation functions for the
waveguide output of a two-level system (including realistic dissipation
channels) with a time-delayed coherent feedback. We compute these observables
by extending a recent quantum trajectory discretized-waveguide (QTDW) approach
which exploits quantum trajectory simulations and a collisional model for the
waveguide to tractably simulate the dynamics. Following a description of the
general technique, we show how to calculate the first and second order quantum
correlation functions, in the presence of a coherent pumping field. With a
short delay time, we show how feedback can be used to filter out the central
peak of the Mollow triplet or switch the output between bunched and anti-
bunched photons by proper choice of round trip phase. We further show how the
loop length and round trip phase effects the zero-time second order quantum
correlation function, an indicator of bunching or anti-bunching. New
resonances introduced through the feedback loop are also shown through their
appearance in the incoherent output spectrum from the waveguide. We explain
these results in the context of the waiting time distributions of the system
output and individual trajectories, uniquely stochastic observables that are
easily accessible with the QTDW model.

##  I Introduction

Feedback has been well used as a stabilizing and control mechanism both in
photonics and other areas of cutting-edge technology [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12]. Most commonly, measurement-based feedback has been employed
where the output of the system is used to act back on the system to achieve
better stability, state generation and error suppression [7, 13, 14, 8, 15, 9,
16, 17]. This feedback control is common practice in laser design and has also
been shown to improve quantum systems as well [10, 18]. However, for use in
many quantum technology applications, it is important to preserve the system
coherence. Thus, a time-delayed coherent feedback has been studied as an
avenue of increasing the coherent lifetime in such systems. In contrast to
measurement-based feedback, coherent feedback is integrated at the system
level and no measurements are taken to avoid introducing further decoherence
in the system [1, 19, 2, 3, 20, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 18, 37].

The regime of waveguide quantum electrodynamics (QED), where quantum systems
are coupled together via waveguide modes, is especially sensitive to loss of
coherence [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57]. These systems have many applications in quantum information
technology, where they can act as sources for single photons or pairs of
photons, photon frequency converters, and single photon detectors which can be
further integrated into circuit QED architectures [58]. Previous work has
shown that including coherent feedback in waveguide-QED systems can
significantly improve the coherent lifetime of the system or enhance the
system emission beyond the typical spontaneous emission rate [3, 20, 5, 27,
28, 32] as well as enable the generation of shaped single photon sources [30].
Not only has it been shown to improve these systems, but also to introduce new
system behaviour beyond well-known waveguide-QED results, such as new
resonances produced by the waveguide modes set up by the feedback [23, 34].

By including a time-delayed coherent feedback in the waveguide-QED system, a
non-Markovian dynamic is introduced in the evolution which adds an additional
complexity to the simulation of such systems. Typical models (e.g., Lindblad
master equations) make the Markovian approximation, that the evolution of the
system only depends on the state of the system at the present, which is no
longer valid when feedback is included. Instead, a “collisional model” of the
waveguide can be used where the waveguide is accounted for at the Hamiltonian
level as a series of interactions with a coupled quantum device [59, 33, 31,
60, 61]. By expanding the Hamiltonian in this way, Markovian numerical
solutions are again possible which are less restricting than the specialized
non-Markovian mathematical solutions which have also been used [3, 34]. Note
that we refer to the unique dynamics that arise from the introduction of the
feedback using the common nomenclature of non-Markovian, since the round trip
memory effects are still included in this model.

When the waveguide is included in the Hamiltonian, the Hilbert space can
quickly become very large and so specialized methods are used to model its
evolution. When limited to the linear regime, results can be computed
analytically but this significantly restricts the phenomena that can be
investigated [1, 3, 62, 63]. A popular method for modelling coherent feedback
is matrix product states (MPSs) [64, 23, 28, 65, 66, 37, 55, 67], a powerful
technique where tensor networks are used to limit the entanglement within the
Hilbert space. Quantum trajectory (QT) theory is a less popular technique
which has also recently been used to investigate the effects of a time-delayed
coherent feedback [31, 34, 37]. This technique uses stochastic individual
realizations of the system to obtain the ensemble average behaviour of the
system [68, 69, 70, 71, 72]. The advantage of this technique is that it can
give unique insights into the underlying stochastic phenomena of the system
behaviour and numerically, it scales linearly with the Hilbert space and is
completely parallelizable.

Previous QT approaches have been limited to the observables of the coupled
quantum system rather than the waveguide output, important for experimental
investigations of feedback. In this paper, we extend a previous QT discretized
waveguide (QTDW) model to investigate the quantum correlation functions and
waiting time distribution function, a uniquely accessible observable from QT
theory, of the waveguide output. These observables give intuitive and powerful
insight into the multi-quanta effects present when a time-delayed coherent
feedback is introduced to the system. Additionally, they are also readily
available to experimental realizations of these systems, and thus give fresh
insight into the experimental observables.

The rest of our paper is organized as follows: In Sec. II.1, we present the
waveguide-QED model of interest and introduce the QT formalism used in our
approach  which accounts for the full non-Markovian dynamics introduced by the
feedback loop. Subsequently, in Sec. II.2, we define the waveguide population
parameters of interest and in Sec. II.3, we explain how to compute the first
and second order quantum auto-correlation functions in the QT picture. In Sec.
III, we show our results, including how a time-delayed feedback can act to
increase the coherence of the system output and introduce multi-quanta
resonances. We also show how these results are affected by the inclusion of
Markovian output channels such as pure dephasing, which is important for
modelling realistic qubits. These results are explained through examples of
individual trajectories and the waiting time distribution of the waveguide
output, an experimentally accessible observable. Lastly, in Sec. V we
conclude.

##  II Theory

###  II.1 Model and Hamiltonian

We investigate a typical setup for including feedback in a waveguide-QED
system, which includes a two level system (TLS), such as a single atom or
quantum dot (QD) or flux qubit, coupled to a truncated waveguide depicted in
Fig. 1(a). The TLS has ground (excited) state represented by $\ket{g}$
($\ket{e}$) and couples to the waveguide to the left (into the feedback loop)
with radiative decay rate $\gamma_{\rm L}$, and to the right (out of the
system of interest) with rate $\gamma_{\rm R}$.  These rates need not be
equal, and indeed can be engineered to be zero in either direction.

The TLS is driven with a continuous wave (CW) laser of Rabi frequency $\Omega$
and the detuning between the laser and TLS is $\delta=\omega_{0}-\omega_{\rm
L}$, where $\omega_{\rm L}$ is the frequency of the laser and $\omega_{0}$ is
the frequency of the TLS. The location of the TLS is chosen as a reference to
be $x=0$, and the waveguide is truncated with a mirror (which we assume is
lossless and introduces a phase change of $\phi_{\rm M}$) at $x=-L_{0}/2$.
Then the round trip length of the feedback loop is $L_{0}$ which introduces a
delay time of $\tau=L_{0}/c(\omega_{0})$, where $c(\omega_{0})$ is the group
velocity of the waveguide mode of reference frequency $\omega_{0}$ in the
waveguide.

(a)  (b)

Figure 1: (a) Model of the waveguide-QED system of interest, which includes a
TLS with ground state $\ket{g}$ and excited state $\ket{e}$ embedded in a
terminated waveguide of length $L_{0}$. The TLS couples asymmetrically to the
waveguide with a decay rate to the right of $\gamma_{\rm R}$ and to the left
of $\gamma_{\rm L}$. The TLS is driven by a CW laser of strength $\Omega$ and
detuning $\delta=\omega_{0}-\omega_{L}$. The TLS also undergoes off-chip decay
with rate $\gamma_{0}$ and pure dephasing with rate $\gamma^{\prime}$. (b)
Representation in the QTDW picture where the modes of the waveguide have been
discretized into spatial bins. There are $N$ bins created by the
discretization and bin $n$ spans a spatial length from $-n\Delta t$ to
$-(n+1)\Delta t$ relative to the output bin. In the QTDW model, the TLS
couples to bins $N-1$ and $0$ with coupling constants $\lambda_{N-1}$ and
$\lambda_{0}$ respectively.

The main Hamiltonian for this system, $H=H_{0}+H_{\rm pump}+H_{\rm W}+H_{\rm
I}$, includes four important components: the non-interacting Hamiltonian for
the TLS, $H_{\rm S}$, the Hamiltonian of the CW-laser drive, $H_{\rm pump}$,
the non-interacting Hamiltonian for the waveguide, $H_{\rm W}$, and the
interaction Hamiltonian, $H_{\rm I}$, between the TLS and waveguide.

In the interaction picture of the laser drive frequency, $\omega_{\rm L}$, the
Hamiltonian of the TLS and the pump is

$H_{0}+H_{\rm
pump}=\delta\sigma^{+}\sigma^{-}+\frac{\Omega}{2}(\sigma^{+}+\sigma^{-}),$ (1)

where $\sigma^{+}$ ($\sigma^{-}$) is the Pauli raising (lowering) operator,
and we have applied a rotating wave approximation.  We have also adopted
natural units with $\hbar=1$. Next, in the (continuous) frequency domain, the
waveguide Hamiltonian is

$H_{\rm W}=\int_{-\infty}^{\infty}d\omega\omega b^{\dagger}(\omega)b(\omega),$
(2)

where we choose $b^{\dagger}(\omega)$ ($b(\omega)$) to be the raising
(lowering) operator for the propagating photon modes in the waveguide.  These
field operators have the usual commutator
$[b(\omega),b^{\dagger}(\omega^{\prime})]=\delta(\omega-\omega^{\prime})$.

Lastly, the influence of the feedback loop is seen in the interaction
Hamiltonian through a modification of the typical coupling rates:

$\displaystyle H_{\rm I}={}$
$\displaystyle\int_{-\infty}^{\infty}d\omega\left[\left(\sqrt{\frac{\gamma_{\rm
L}}{2\pi}}\sigma^{+}b(\omega)\right.\right.$ (3)
$\displaystyle{}+\left.\left.\sqrt{\frac{\gamma_{\rm
R}}{2\pi}}e^{i(\phi+\omega\tau)}\sigma^{+}b(\omega)\right)+\rm H.c.\right],$

where the coupling to the right propagating mode picks up the round trip phase
change, $\phi=\phi_{\rm M}-\omega_{\rm L}\tau$, composed of the phase change
from the mirror and the round trip phase of our reference frequency, in this
case the laser frequency.  The form of this interaction is derived in Appendix
A.

A full derivation of the QTDW model is given in Ref. [37],  including the
component Hamiltonians in the discrete frequency picture, thus we only
highlight the important points of the model below. In that paper the formalism
is also derived to allow for an open waveguide where the left and right moving
fields are treated independently, important for systems such as two QDs
spatially separated in the waveguide.  To represent the waveguide with a
collisional model, we transform Eq. (2) to the discrete time domain with
operators $B_{n}$, using a discrete Fourier transform on the waveguide mode
operators in the discrete frequency picture
($b(\omega)=\sqrt{2\pi/L_{0}}b_{k}$). This gives the explicit relationship

$\displaystyle B_{n}={}$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}b_{k}e^{i\omega_{k}n\Delta
t},$ (4) $\displaystyle b_{k}={}$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}B_{n}e^{{\color[rgb]{0,0,0}-}i\omega_{k}n\Delta
t},$

where $N$ is the total number of discrete bins used to model the waveguide,
$\Delta t$ is the corresponding time discretization, and $\omega_{k}=2\pi
k/L_{0}$, which assumes linear dispersion for the waveguide mode.  In this
domain, the delay time from the round trip can be expressed as $\tau=N\Delta
t$. Note, the commutator for the time domain operators is
$[B_{n},B^{\dagger}_{n^{\prime}}]=\delta_{n,n^{\prime}}$, and $B$ and
$B^{\dagger}$ have dimensionless units. We make the approximation that $N$ is
chosen sufficiently large (equivalently $\Delta t$ sufficiently small) that
the probability to have more than one photon in any individual bin is
negligible, which is valid when $\Delta t\ll 1/(\gamma_{\rm R}+\gamma_{\rm
L})$.  Formally, $B_{n}$ represents the slice of the waveguide field that
interacts with the TLS over a certain time interval, $[n\Delta t,(n+1)\Delta
t)$. We can equivalently represent these $B_{n}$ as a spatial section of the
waveguide through the intrinsic relationship between the space and time
domains related by the group velocity.

By transforming into this discrete spatial bin model, the waveguide is
effectively a sequence of bins which pass the feedback (and output) field
forward one bin each time step. Mathematically, this is seen through the
evolution of the time domain operator, $B_{n}$, under the waveguide
Hamiltonian:

$U_{\rm W}^{\dagger}(\Delta t)B_{n}U_{\rm W}(\Delta t)=B_{n-1},$ (5)

where $U_{\rm W}(\Delta t)=e^{-iH_{\rm W}\Delta t}$.  The derivation for this
relation is presented in Appendix B. Note that only the two spatial bins
located at the position of the TLS interact with the TLS at a single time.
This is shown schematically in Fig. 1(b).

The TLS and pump Hamiltonians, $H_{0}+H_{\rm pump}$, are unmodified by our
transformation into the discrete time domain. The interaction Hamiltonian
becomes

$H_{\rm I}=\left(\sqrt{\frac{\gamma_{L}}{\Delta
t}}\sigma^{+}B_{N-1}+e^{i\phi}\sqrt{\frac{\gamma_{R}}{\Delta
t}}\sigma^{+}B_{0}\right)+\rm{H.c.},$ (6)

so that the TLS first interacts with the $(N-1)^{\rm th}$ bin which then
passes the field from bin to bin around the loop and returns to the TLS after
$N$ time steps to interact with the TLS again in bin $0$. Thus, the TLS
couples to bins $N-1$ and $0$ with coupling constants
$\lambda_{N-1}=\sqrt{\gamma_{L}/\Delta t}$ and
$\lambda_{0}=e^{i\phi}\sqrt{\gamma_{R}/\Delta t}$.  The form of this
interaction Hamiltonian was fully derived in Sec. IV A (b) of Ref. [37] for
reference.

The ket vector,  in factorized form, for the complete TLS and waveguide system
is

$\ket{\psi(t)}=\ket{\psi_{\rm S}(t)}\ket{\psi_{\rm W}(t)},$ (7)

where $\ket{\psi_{\rm S}(t)}$ is the ket vector for the TLS and
$\ket{\psi_{\rm W}(t)}$ is the ket vector for the waveguide from $x=0$ to
$x=-L_{0}/2$, which we limit to a maximum of two photons in this section of
the waveguide.111These ket vectors are $\ket{\psi_{\rm
S}(t)}=a(t)\ket{g}+b(t)\ket{e}$ and $\ket{\psi_{\rm
W}(t)}=c^{(0)}(t)\ket{\\{0\\}}+\sum_{j=1}^{N}c^{(1)}_{j}(t)\ket{1_{j}}+\sum_{j=1}^{N-1}\sum_{k=j+1}^{N}c^{(2)}_{j,k}(t)\ket{1_{j}}\ket{1_{k}}$,
which can be multiplied together to get the form in Eq. (13). We justify this
approximation later in our results where we report the population for two
photons in the feedback loop.

The waveguide field in the $x>0$ region is modeled through simulated
measurements of the output field in the $0$’th bin. The ket vector can be
split as

$\ket{\psi(t)}=\ket{\psi_{0}(t)}\ket{0_{0}}+\ket{\psi_{1}(t)}\ket{1_{0}},$ (8)

where $\ket{0_{0}}$ is an empty bin 0 and $\ket{1_{0}}$ represents a photon in
bin $0$. Before moving the final bin forward and out of the waveguide section
of interest, its information is retrieved by projecting the full ket vector
into one of the two states $\ket{\psi_{0}(t)}$ or $\ket{\psi_{1}(t)}$ with
respective probabilities $\braket{\psi_{i}(t)}{\psi_{i}(t)}$. This process is
not norm conserving so renormalization must occur each time step.

Two dissipation channels are also included in our model as quantum jump
operators: off-chip decay from the TLS, $C_{0}=\sqrt{\gamma_{0}}\sigma^{-}$,
with rate $\gamma_{0}$ and pure dephasing in the TLS,
$C_{1}=\sqrt{\gamma^{\prime}}\sigma^{+}\sigma^{-}$, with rate
$\gamma^{\prime}$. Following standard QT theory, these are included in the
system evolution through stochastic application of the jump operator with
probabilities $\braket{C^{\dagger}_{0}C_{0}}$ and
$\braket{C^{\dagger}_{1}C_{1}}$ for off-chip decay and pure dephasing
respectively. The Hamiltonian is also modified to a non-Hermitian effective
Hamiltonian

$H_{\rm eff}=H_{\rm S}+H_{\rm pump}+H_{\rm
I}-\frac{i}{2}\sum_{j=0}^{1}C^{\dagger}_{j}C_{j},$ (9)

which is used to evolve the  combined TLS and waveguide ket vector whenever no
quantum jump occurs. The inclusion of the final term (which is non-Hermitian)
moves the model beyond that of the semi-classical weak excitation
approximation approaches. Both quantum jumps are Lindblad channels which have
the superoperator form

$\mathcal{L}(\rho)=-\frac{1}{2}\sum_{j=0}^{1}\left(C^{\dagger}_{j}C_{j}\rho+\rho
C^{\dagger}_{j}C_{j}\right)+\sum_{j=0}^{1}C_{j}\rho C_{j}^{\dagger},$ (10)

where $\rho$ is the density matrix for the TLS.  These dissipation channels
could be dealt with through additional collisional interactions, however, this
is not necessary as we take them to be Markovian, and doing so would add to
the size of the Hilbert space and increase the computational complexity of the
problem.

A single time step for a simulation with this model follows a four step
process:

  1. 1.

Evolve $\ket{\psi(t)}$ using a regular QT step under the effective
Hamiltonian, $H_{\rm eff}$, with the two quantum jump operators $C_{0}$ and
$C_{1}$.

  2. 2.

Take a simulated measurement on the final bin of the waveguide by calculating
the population of the bin, $N_{B_{0}}$ (defined in Eq. (11)), and comparing it
to a uniformly distributed random number, $x\in(0,1)$. The ket vector is then
projected according to whether a photon is detected or not.

  3. 3.

Evolve the waveguide bins under $H_{\rm W}$ which acts to step each bin
forward one space. Note that the final bin is emptied and contains no
information so it can be dropped while the incoming bin is empty for this
setup, although this is not a strict limitation in the model.

  4. 4.

Renormalize the complete system ket vector since the operation of measuring
and moving the bins along does not conserve the norm.

###  II.2 Waveguide Population Observables in the QTDW Approach

To investigate the photon population dynamics in the waveguide, we begin with
the probability of finding a photon in a single waveguide bin $j$ within the
feedback loop,

$N_{B_{j}}(t)=\braket{\psi(t)}{B^{\dagger}_{j}B_{j}}{\psi(t)},$ (11)

where bin $j$ is travelling towards the mirror if $j\geq N/2$, or travelling
away from the mirror if $j<N/2$. As the spatial length of the bins depends on
$\Delta t$, then so does $N_{B_{j}}$ so that the total population around the
whole loop does not. Thus, a more useful quantity to use is the flux through a
particular bin,

$n_{B_{j}}(t)=N_{B_{j}}(t)/\Delta t,$ (12)

which is in units of $1/{\rm s}$. The flux of the outgoing bin (at $j=0$) then
gives an observable for the photon flux leaving the system, independent of our
choice of $\Delta t$, which will prove very useful.

Another useful quantum quantity is the photon number distribution function in
the feedback loop, namely the probability to have zero, one, or two photons in
the feedback loop. With this metric, we can evaluate the applicability of our
“two-photons-in-the-loop” approximation. Since the QTDW model directly evolves
the ket vector for the system (Eq. (7)), this becomes a straight-forward
quantity to compute if desired. However, with other techniques such as MPS,
these quantities have been calculated via the correlation functions, which can
be computationally demanding and does not scale well with the total simulation
run time [73, 65].

Explicitly, the ket vector is

$\displaystyle\ket{\psi(t)}={}$
$\displaystyle\left[\alpha^{(0)}(t)\ket{g}+\beta^{(0)}(t)\ket{e}\right]\ket{\\{0\\}}$
(13)
$\displaystyle{}+\sum_{j=1}^{N}\left[\alpha_{j}^{(1)}(t)\ket{g}+\beta_{j}^{(1)}(t)\ket{e}\right]\ket{1_{j}}$
$\displaystyle{}+\sum_{j=1}^{N-1}\sum_{k=j+1}^{N}\left[\alpha_{j,k}^{(2)}(t)\ket{g}+\beta_{j,k}^{(2)}(t)\ket{e}\right]\ket{1_{j}}\ket{1_{k}},$

and thus the probabilities can be read off almost immediately as

$\displaystyle p(0,t)={}$
$\displaystyle|\alpha^{(0)}(t)|^{2}+|\beta^{(0)}(t)|^{2},$ (14a)
$\displaystyle p(1,t)={}$
$\displaystyle\sum_{j=1}^{N}|\alpha_{j}^{(1)}(t)|^{2}+|\beta_{j}^{(1)}(t)|^{2},$
(14b) $\displaystyle p(2,t)={}$
$\displaystyle\sum_{j=1}^{N-1}\sum_{k=j+1}^{N}|\alpha_{j,k}^{(2)}(t)|^{2}+|\beta_{j,k}^{(2)}(t)|^{2},$
(14c)

where $p(n,t)$ represents the probability of having $n$ photons in the loop at
time $t$.

Lastly, an important experimental observable is the waiting time distribution
(WTD) [74], $W(t^{\prime})$, for photons detected via the waveguide. This is
defined as the distribution of delay times, $t^{\prime}$, between detection
events via the waveguide output channel,

$W(t^{\prime})=\frac{N\left(\\{A\\}|t(A_{j})-t(A_{j-1})=t^{\prime}\right)}{N_{\rm
tot}},$ (15)

where $N_{\rm tot}$ is the total number of photons detected via the waveguide
and $\sum_{j}N\left(\\{A\\}|t(A_{j})-t(A_{j-1})=t^{\prime}\right)$ is the
number of detections in the detection record $\\{A\\}$ with delay time
$t^{\prime},$ between the $A_{j}$’th event and the previous event, $A_{j-1}$.
Since simulating these detection events are a direct part of the QTDW approach
to modelling the response, we can record a jump record for each trajectory and
then easily calculate the waiting times for each trajectory. This is uniquely
accessible by using a QT based approach as the WTD requires individual
realizations of the system rather than the ensemble average. This observable
can be very useful in explaining the phenomena seen in the ensemble average
response as we will show in Sec. III.

###  II.3 Output Quantum Correlation Functions

In previous work with QT based approaches to modelling feedback [34, 37], the
results were restricted to how the feedback affected the population of the
TLS. Although many useful insights can be gained from this population dynamic,
in this paper we focus our observables around the output photons leaving
through the waveguide, which is more practical and experimentally relevant.
Below, we also show how to calculate the first- and second-order quantum auto-
correlation functions for waveguide output photons where we use the first
order correlation function to calculate the incoherent spectra. This is
usually an extremely difficult theoretical problem for the usual master
equation approach because the quantum regression theorem cannot be used for a
non-Markovian dynamic [75]. By including the waveguide at the Hamiltonian
level, the QTDW model side steps this problem and we can directly calculate
the correlation functions at any photon bin, including the output bin.

We begin by describing how to obtain the second-order quantum auto-correlation
function as it is the more straightforward of the two. Since we are working
with a CW pump field, we are interested in the steady state behavior of the
following two-time correlation function:

$[g^{(2)}_{\rm out}(t_{2})]_{\rm
ss}=\frac{\braket{B_{0}^{\dagger}(0)B_{0}^{\dagger}(t_{2})B_{0}(t_{2})B_{0}(0)}_{\rm
ss}}{\braket{B_{0}^{\dagger}B_{0}}_{\rm ss}^{2}},$ (16)

where $t_{2}$ represents the time after steady state is reached (defined here
at $(0)$ in the correlation function). Due to the stochastic nature of the QT
technique, we calculate this quantity for individual trajectories and then
average over a large number of realizations to arrive at the ensemble average.
For good convergence, typically thousands of trajectories are required, which
in total take on the order of tens of minutes to run on a single computer.

In the numerical algorithm, an initial set of regular trajectories must first
be run to identify when steady state is reached, denoted by $t_{\rm ss}$ which
is defined as 0 in Eq. (16). Then, the trajectory, $\ket{\psi(t)}$, is run
following the prescribed algorithm until $t_{\rm ss}$ is reached. On the final
time step, the process is stopped before the simulated measurement is done on
the final bin of the system in step 2 of the algorithm (to avoid losing the
information in the $0^{\rm th}$ bin). At this point, the operator $B_{0}$ is
applied to the ket vector, $\ket{\psi^{\prime}(t_{\rm
ss})}=B_{0}\ket{\psi(t_{\rm ss})}$, i.e., we force a simulated detection of a
photon in the bin leaving the system. Then the final two steps of the
algorithm are finished by stepping all of the bins forward and renormalizing.
The new $\ket{\psi^{\prime}}$ is then evolved forward in time following the
four-step algorithm, with the observable

$G^{(2)}_{\rm
out}(t_{2})=\braket{\psi^{\prime}(t_{2})}{B_{0}^{\dagger}B_{0}}{\psi^{\prime}(t_{2})},$
(17)

being calculated at each time step. The final normalized second order
correlation function is

$[g^{(2)}_{\rm out}(t_{2})]_{\rm ss}=\frac{G^{(2)}_{\rm
out}(t_{2})}{\braket{B_{0}^{\dagger}B_{0}}_{\rm ss}},$ (18)

where, due to the renormalization that already occurs each time step when
evolving $\ket{\psi^{\prime}(t_{2})}$, $\braket{B_{0}^{\dagger}B_{0}}_{\rm
ss}$ is used rather than the square as appears in Eq. (16).

As a consequence of the approximation that there is only one photon in any
single bin222Note that we can still have two photons in different spatial
bins. (which is not a serious restriction as the bin populations are typically
very small), this formally sets

$[g^{(2)}_{\rm out}(0)]_{\rm ss}=0.$ (19)

However, the quantity $[g^{(2)}_{\rm out}(\Delta t)]_{\rm ss}$ can be used as
an analogous value to $[g^{(2)}_{\rm out}(0)]_{\rm ss}$ because it gives the
smallest time step between photon emissions in the QTDW model and can still
indicate whether sub- or super-Poissonian light is being emitted.

The incoherent spectrum from the waveguide output is also a good observable to
investigate multi-quanta effects, as it will contain resonances beyond those
from the TLS, and shows signatures of coherent bath control through feedback.
This spectrum is obtained from

$\displaystyle S_{\rm incoh}^{\rm out}(\omega)={}$
$\displaystyle\int_{0}^{\infty}dt_{2}e^{i(\omega-\omega_{\rm
L})t_{2}}\left[\braket{B_{0}^{\dagger}(t_{2})B_{0}(0)}_{\rm ss}\right.$ (20)
$\displaystyle\left.{}-\braket{B_{0}^{\dagger}(0)}_{\rm
ss}\braket{B_{0}(0)}_{\rm ss}\right],$

where the first term in the integrand is the unnormalized first order auto-
correlation function, $G_{\rm out}^{(1)}(t_{2})$, and the second term is
composed of steady-state expectation values that can be readily calculated
with the QTDW model.

Calculating $G_{\rm out}^{(1)}(t_{2})$ with the QTDW model is less intuitive
than the second order correlation function [68, 70, 71]. Similar to our
approach for the second order correlation function, the first step is to
identify when steady state is reached, $t_{\rm ss}$, from the ensemble average
system dynamics. Then the trajectory is evolved until $t_{\rm ss}$, but on the
final time step we stop before step 2 of the algorithm, as we did before when
calculating $[g^{(2)}_{\rm out}(t_{2})]_{\rm ss}$. Here, the process differs,
two copies of the system ket vector are made, a “lead trajectory” which is
unmodified, $\ket{\psi_{\rm lead}^{\prime}(0)}=\ket{\psi(t_{\rm ss})}$, and a
“follower trajectory” to which we apply $B_{0}$, $\ket{\psi_{\rm
fol}^{\prime}(0)}=B_{0}\ket{\psi(t_{\rm ss})}$ The time step is then completed
for the lead trajectory by completing steps 3 and 4 of the algorithm. For the
follower trajectory, the bins are stepped forward in step 3, but in step 4 the
trajectory is renormalized as

$\ket{\psi_{\rm fol}(0)}=\frac{\ket{\psi_{\rm
fol}^{\prime}(0)}}{\sqrt{\braket{\psi_{\rm lead}^{\prime}(0)}{\psi_{\rm
lead}^{\prime}(0)}}},$ (21)

so that it follows the normalization of the lead trajectory.

The lead trajectory then evolves following the four-step algorithm, during
which it continues to have stochastic jump events from $C_{0}$, $C_{1}$, and
(simulated) detections from the final bin. The follower trajectory evolves by
following the same sequence of jump events that the lead trajectory underwent
and at the end of each time step is then projected onto the length of the lead
trajectory following Eq. 21. To calculate the correlation function, we compute

$G_{\rm out}^{(1)}(t_{2})=\braket{\psi_{\rm
lead}(t_{2})}{B_{0}^{\dagger}}{\psi_{\rm fol}(t_{2})}.$ (22)

However, rather than calculating this at the end of each time step, it needs
to be calculated after step one of the algorithm is completed but before the
final bin is measured in step two.

If the normalized correlation function is desired, then one can calculate this
through

$[g_{\rm out}^{(1)}(t_{2})]_{\rm ss}=\frac{G_{\rm
out}^{(1)}(t_{2})}{\sqrt{\braket{B_{0}^{\dagger}(t_{2})B_{0}(t_{2})}_{\rm
ss}\braket{B_{0}^{\dagger}(0)B_{0}(0)}_{\rm ss}}},$ (23)

where the first expectation value in the denominator is the bin population of
$B_{0}$ at time $t_{2}$ from the lead trajectory and the second expectation
value is the population of $B_{0}$ from the lead trajectory when $t_{2}=0$.

##  III Results

In this section, we apply our extended QTDW approach to several systems. First
we show that it correctly produces feedback induced changes to the Mollow
triplet and explain these changes in the context of $[g^{(2)}_{\rm
out}(t_{2})]_{\rm ss}$ and $W(t^{\prime})$. Then we characterize the effect of
the delay time, $\tau$, and the round trip phase change, $\phi$, on the
waveguide output and the photon number distribution in the feedback loop.
Finally, we drive the system with a strong CW-pump to excite the additional
resonances from the feedback loop in the output spectrum.

A single TLS excited by an on-resonance CW laser, with Rabi frequency
$\Omega=2\pi\gamma$, will emit a spectrum with the distinctive Mollow triplet,
including a central resonance at the frequency of the laser $\omega_{\rm L}$
and two side peaks at $\omega-\omega_{\rm L}=\pm\Omega$. By introducing a
relatively short feedback loop, the central peak can disappear, such that with
the proper choice of the round trip phase, $\phi$, only the side peaks remain
[23]. In Fig. 2(b) the incoherent output spectra for the TLS without feedback
(with round trip time $\tau\rightarrow\infty$), and the TLS coupled to a short
feedback loop (with round trip time $\tau=0.1\gamma$) are shown, where in the
short loop case two values of round trip phase are considered, $\phi=0$ and
$\phi=\pi$. When $\phi=0$, the effect of feedback is such that the spectral
peaks have the same heights as the no feedback case, but are all broadened.
For $\phi=\pi$, the side peaks are significantly sharpened and the central
peak almost completely removed.

In the Heitler regime, when the Rabi frequency is small, the spectrum becomes
single peaked and the Mollow Triplet is no longer seen [76, 75, 77]. For a
Rabi frequency of $\Omega=0.2\pi\gamma$, Fig. 2(a) shows the output spectrum
both with and without feedback. With this weak pump and short delay time, the
field within the waveguide remains quite weak. A choice of $\phi=0$ again
leads to broadening of the central peak as the weak field constructively
interferes with itself. When $\phi=\pi$, the single photon in the loop field
destructively interferes with the TLS output and only a very small output flux
remains. This causes the central peak to be suppressed and only extremely
sharp sidepeaks remain. For a Mollow triplet, the side peaks are at
$\pm\Omega$, however the feedback modifies this as these two peaks are no
longer purely the dressed states of the TLS and laser. Instead, these states
are dressed by the additional field modes that are set up in the waveguide
causing the peaks to shift.

Note that we set $\gamma=\gamma_{\rm R}+\gamma_{\rm L}$ to be the total decay
rate into the waveguide and, for convenience, set $\gamma_{\rm R}=\gamma_{\rm
L}$, although this is not a restriction of the model and can easily be
generalized.  For the case of a single TLS with coherent feedback, the
phenomena we investigate is caused by the interference between the field
returning from the loop and the field emitted by the TLS. By introducing
asymmetric coupling, matching the magnitudes of the two fields becomes more
difficult and it typically reduces the effectiveness of the feedback. In
general, for multiple TLSs, with or without feedback, asymmetry can play an
important role when investigating interesting waveguide-QED regimes [78, 79,
80, 81, 82, 83, 84, 85, 86, 87, 55] so the generality is included in our
model.

Figure 2: Incoherent spectra of the waveguide output for (a)
$\Omega=0.2\pi\gamma$ and (b) $\Omega=2\pi\gamma$. For the spectrum in (b) the
output photon flux is in (c). For a TLS driven on resonance without feedback,
$\tau\rightarrow\infty$ (black) and with a short feedback loop,
$\tau=0.1\gamma^{-1}$, for $\phi=0$ (red) and $\phi=\pi$ (blue); 20,000
trajectories were used for $\tau\rightarrow\infty$ and $\phi=\pi$, and 50,000
trajectories for $\phi=0$.

To achieve maximal constructive and destructive interference, the condition
for the round trip phase change is $\Omega\tau/2-\phi=(2k-1)\pi$ for
destructive interference and $\Omega\tau/2-\phi=2k\pi$ for constructive
interference with $k\in\mathbb{Z}$. For the interference to be complete (i.e.
no output for the case of destructive interference) there is a further
requirement of $\Omega\tau=2n\pi$ for $n\in\mathbb{Z}$ to match the phase of
the Rabi oscillations. For a short loop and weak drive, this interference
occurs maximally when $\phi\approx 0$ for constructive interference and
$\phi\approx\pi$ for destructive interference.

This is readily seen by comparing the output flux from the loop,
$n_{B_{0}}(t)$, in Fig. 2(c), for the three setups. In the one-photon-in-the-
loop approximation, that the output from the waveguide at $\omega_{\rm L}$
will be suppressed when $\phi=\pi$ is consistent with the findings in Refs.
[21, 31, 34].  The approximation is good in the limit when
$\gamma\tau\rightarrow 0$, i.e. when the field in the feedback loop is weak
($\gamma\rightarrow 0$) or the dynamics can be treated in the Markovian regime
($\tau\rightarrow 0$), but fails when the non-Markovian dynamics are required.
Allowing for two photons in the loop, as we do here, means that the feedback
cannot match the phase requirements for destructive interference of the two
different photon frequencies simultaneously when $\phi=\pi$ [37]. Thus, the
loop acts to meet the phase requirements of the central $\omega=\omega_{\rm
L}$ peak (and thus suppress the peak) while the side peaks remain. When the
feedback interferes constructively when $\phi=0$, the waveguide emission from
the system is much faster and this acts to broaden the emission while
retaining the general Mollow triplet shape.  For intermediate choices of
$\phi$ between $0$ and $\pi$, the interference is no longer maximal and as the
phase is varied, the observables vary continuously between the behaviour of
the two maximal interference cases.

The photon counting statistics of the waveguide output are also strikingly
effected by the introduction of a time-delayed coherent feedback. By virtue of
having only two energy levels, a TLS in an infinite waveguide will nominally
emit sub-Poissonian light with $[g^{(2)}_{\rm out}(0)]_{\rm ss}=0$. The
response is similar when a short feedback loop, $\tau=0.1\gamma^{-1}$, is
included with $\phi=0$ and these are overlaid in Fig. 3(a). When feedback is
included, the response remains sub-Poissonian but loses the second-order
correlations in time much faster (the red line in Fig. 3(a) decaying more
quickly than the black) due to the increased output from the system. However,
when $\phi=\pi$, as in Fig. 3(b), the statistics of the outgoing light
switches to highly super-Poissonian over a period of time equal to the delay
time of the feedback. These two starkly different regimes can allow feedback
to be used to tune the characteristics of the outgoing light depending on the
intended use.

In Fig. 2, a relatively short feedback loop is used, $\tau=0.1\gamma^{-1}$, in
order to maintain coherence between the returning field in the feedback loop
and the emitted field from the TLS.  As the delay time increases, the feedback
becomes increasingly sensitive to meeting the exact phase matching conditions
for interference as the coherence between the two fields decreases. However,
other channels for decoherence, such as pure dephasing, also have an effect on
how well the feedback works.

In Fig. 4 we have introduced the Lindblad output channels (off chip decay and
pure dephasing), to see to what degree they disrupt the effect of feedback
when $\phi=\pi$ leading to destructive interference. Off chip decay from the
TLS is included in Fig. 4(a) at a rate of $\gamma_{0}=0.1\gamma$ which is
typical of state of the art TLS sources such as QDs [88, 89]. This output
channel does not greatly effect the response of the output flux because
whenever a quantum jump occurs, the TLS is emptied and so any returning
feedback is met with no field; thus, no interference occurs and the system
quickly returns to its steady state behaviour. However, when pure dephasing is
included in Fig. 4(b), we see a much more pronounced effect as the output flux
out of the system is increased approximately twentyfold when
$\gamma^{\prime}=\gamma$. In this case, when a jump occurs, the phase of the
TLS flips. The returning feedback is thus met with constructive interference
and the output from the system is enhanced leading to the increased flux out
of the system. Note though, that the steady state output flux when
$\gamma^{\prime}=\gamma$ is still much reduced from the case without feedback
which is shown as the black line in Fig. 2(c).  In the case of $\phi=0$,
similar physics is seen when including these output channels. Off-chip decay
continues to have little effect on the steady state output flux while
dephasing now works to spoil the constructive interference in the output
fields. This causes the steady state output flux to decrease as the dephasing
rate increases, from $[n_{B_{0}}/\gamma]_{\rm ss}\approx 0.45$ without any
output channels to $[n_{B_{0}}/\gamma]_{\rm ss}\approx 0.34$ when
$\gamma^{\prime}=\gamma$. To limit the amount of pure dephasing in QD systems,
usually one works at lower temperatures, and gated QDs can also reduce the
charge noise significantly [90, 91].

Pure dephasing can also act to qualitatively change the emitted spectrum.
Under off-resonant pumping, the Mollow triplet becomes asymmetrical when pure
dephasing is included [92, 93, 94]. We show how the inclusion of feedback
affects this result in Fig. 5. The detuning between the laser and TLS is
$\delta=5\gamma$ and the Rabi frequency if $\Omega=2\pi$. The rate of pure
dephasing is $\gamma^{\prime}=0.5\gamma$. When a short feedback loop is
included ($\tau=0.1\gamma^{-1}$) a similar effect is seen as when feedback is
included for the standard Mollow triplet in Fig. 2(b). By choice of round trip
phase the spectrum width broadens ($\phi=0$) or sharpens and filters out the
central peak ($\phi=\pi$). For the remaining results we will neglect the two
dissipation channels to present optimal results as we remain on resonance.

Figure 3: The second-order quantum correlation function for the three setups
described in Fig. 2. For both with and without feedback in (a) the output is
sub-Poissonian while in (b) it is super-Poissonian. Note that in (b) the
correlation function begins at 0 but then immediately jumps to $>$1 on the
next time step since there is a maximum of one photon in each bin. These
results are an average of 20,000 trajectories each. Figure 4: Effect on the
output photon flux from the inclusion of the Lindblad output channels; (a) off
chip decay and (b) pure dephasing. The TLS is driven on resonance with
$\Omega=0.4\pi\gamma$ with a feedback loop of delay time $\tau=0.1\gamma^{-1}$
and $\phi=\pi$. Each result is an average of 20,000 trajectories. Figure 5:
Asymmetric Mollow triplet from the TLS driven off resonance ($\delta=5\gamma$,
$\Omega=2\pi\gamma$) with pure dephasing ($\gamma^{\prime}=0.5\gamma$) with
and without feedback ($\tau=0.1\gamma^{-1}$). Each result is an average of
20,000 trajectories.

The WTDs of the waveguide output give us additional insight into how feedback
is changing the system dynamics. Without feedback, in Fig. 6(a), the system
goes through periodic peaks of photon emission which quickly die off as the
waiting time gets much longer than the lifetime of the TLS. As expected, there
is a very small probability of back to back jumps since the output light is
sub-Poissonian.

When feedback is introduced, as shown in Fig. 6(b), there is a sharp jump at
the beginning of the WTD, over the length of the delay time, indicating that
there is a significant population of back-to-back emissions occurring. After
this initial spike, the distribution is almost flat, out to very long waiting
times; indeed the graph in Fig. 6(b) is truncated for readability and the
distribution tapers out to $t^{\prime}=60\gamma^{-1}$. This indicates that if
a photon pair emission does not occur, the next emission is uncorrelated with
the previous emission, which agrees with the correlation function of Fig.
3(b). This dynamic also plays out in the output flux from the system after a
jump event occurs.

Figure 6(c) shows a snapshot of the output flux in a trajectory immediately
after a jump occurs for two separate jump events. The flux immediately
increases and becomes much larger over the timescale of a single round trip.
This sharp increase is due to the suppressed probability of a single photon in
the loop from the destructive interference of the output bin. If a photon is
detected, than it is more likely to have a second photon in the loop coming
around than it is to have an empty loop. If a second emission event does not
occur before the round trip ends then the flux settles back into the near
constant value before the jump occurred. There are slight fluctuations in the
between jump output flux, which is what determines the peak output flux after
a jump occurs.

Figure 6: The WTDs for a TLS driven on resonance with $\Omega=2\pi\gamma$ and
with (a) no feedback or (b) feedback from a short loop with
$\tau=0.1\gamma^{-1}$ and $\phi=\pi$. In (c) two jump events during a
trajectory are shown for the setup in (b) to illustrate the increase in output
flux after a jump occurs.

We now turn to investigating the effect of increasing the length of the
feedback loop (and thus the delay time). Figs. 7(a) and (b) show the effect of
increasing the delay time on the anti-bunching and bunching for both $\phi=0$
and $\phi=\pi$, respectively; here, the pump strength is decreased to
$\Omega=0.4\pi\gamma$ to avoid driving too many of the system resonances,
which increase the complexity of the photon counting statistics. As the loop
length increases, both setups show worse statistics as they become less anti-
bunched and less bunched. This is again due to the loss of coherence between
the feedback and TLS. When $\phi=0$, increasing the delay time acts to
increase probability of having two photons in the loop. Therefore, when a jump
occurs the remaining field in the feedback loop is non-zero. Conversely, when
$\phi=\pi$, the loss of coherence causes the destructive interference between
the one photon in the loop field and TLS to no longer be perfect. Somewhat
counter intuitively, this means when a jump occurs, it becomes less probable
that there are two photons in the loop as the loop increases in length.

It is interesting to note that the larger population in the loop actually
increases the likelihood of a photon pair emission when $\phi=\pi$. When
$\tau=2\gamma^{-1}$, the WTD for the waveguide output is shown in Fig. 7(c)
for $\phi=0$ and (d) for $\phi=\pi$. The lifetime of the WTD when $\phi=\pi$
is much longer showing the increased waiting time between individual jumps,
but also the initial peak is larger than the $\phi=0$ case showing an increase
in photon pair emission. Figure 7(e) and (f) show the probability for one and
two photons in the loop in the steady state as a function of delay time for
$\Omega=0.4\pi\gamma$ and $\Omega=2\pi\gamma$, respectively, when $\phi=0$
(note that the probabilities for one or two photons remain about same in the
steady state when $\phi=\pi$). The probability to have two photons in the loop
remains small past a delay time of $\tau=2\gamma^{-1}$. For example at
$\tau=2.5\gamma^{-1}$, the probability is $p(2,t_{\rm ss})\approx 0.03$ for
$\Omega=0.4\pi\gamma$ and the probability is $p(2,t_{\rm ss})\approx 0.11$ for
$\Omega=2\pi\gamma$, small enough for our approximation of a maximum of two
photons in the loop to be valid.

In Fig. 8, we compare the time dynamics of the $p(n,t)$ as we vary the round
trip phase change, delay time, pump strength and introduce off-chip decay and
pure dephasing. In Figs. 8(a) and (b) we plot $p(n,t)$ as a function of the
round trip phase change and the delay time, respectively, for $n=1$ and $2$.
As expected, we see that for $\phi=\pi$ the long-lived system Rabi
oscillations are passed on to the loop probabilities, while for $\phi=0$ we do
not see these oscillations. The increasing delay time also acts as expected to
increase the probabilities of having both one and two photons in the loop
since the feedback loop itself is longer and holds more of the emitted field
from the TLS. In Fig. 8(c) the pump strength is increased to
$\Omega=2\pi\gamma$. Although the probability to have one photon in the loop
remains approximately the same, the probability for two photons almost doubles
as the TLS is pumped harder. We also see the Rabi oscillations reflected in
the loop probabilities die out faster for the larger pump strength. Lastly, we
introduce both Lindblad output channels with $\gamma_{0}=0.1\gamma$ and
$\gamma^{\prime}=0.5\gamma$ in Fig. 8(d), which act to decrease the
probability to find either one photon or two photons in the loop and reduce
the Rabi oscillations.

Figure 7: Comparison of the near-equal time second-order correlation function
as a function of delay time for a TLS driven on resonance with
$\Omega=0.4\pi\gamma$ and round trip phase change of (a) $\phi=0$ and (b)
$\phi=\pi$. For the setup with the longest delay time in (b)
($\tau=2.0\gamma^{-1}$), we show the WTD in (c) for $\phi=0$ and in (d) for
$\phi=\pi$. We show the probability to have $n=1$ and $2$ photons in the loop
(the blue and red lines respectively) as a function of delay time when
$\phi=0$ for drive strengths of (e) $\Omega=0.4\pi\gamma$ and (f)
$\Omega=2\pi\gamma$. The results are an average of 10,000 trajectories for
each delay time. Figure 8: Time evolution of the probability of having $n=1$
and $2$ photons in the loop (the blue and red lines, respectively). In (a), we
set $\Omega=0.4\pi\gamma$ and $\tau=0.5\gamma^{-1}$ for $\phi=0$ (dashed) and
$\phi=\pi$ (solid). In (b), $\Omega=0.4\pi\gamma$ and $\phi=0$ for
$\tau=0.5\gamma^{-1}$ (dashed) and $\tau=2\gamma^{-1}$ (solid). In (c),
$\tau=2\gamma^{-1}$ and $\phi=\pi$ for $\Omega=0.4\pi\gamma$ (dashed) and
$\Omega=2\pi\gamma$ (solid). Lastly, in (d) when $\Omega=0.4\pi\gamma$,
$\tau=2\gamma^{-1}$, and $\phi=\pi$, we compare the case with no outputs
(solid, $\gamma_{0}=\gamma^{\prime}=0$) to that where both Lindblad output
channels are included (dashed, $\gamma_{0}=0.1\gamma$ and
$\gamma^{\prime}=0.5\gamma$). Each result is an average of 20,000
trajectories. Figure 9: Comparison of the near-equal time second-order
correlation function as a function of the round trip phase change, for three
delay times: (a) $\tau=0.5\gamma^{-1}$, (b) $\tau=1.0\gamma^{-1}$, and (c)
$\tau=2.0\gamma^{-1}$. The vertical lines denote the phase matching condition
$\Omega\tau/2-\phi=(2k-1)\pi,k\in\mathbb{Z}$. The TLS is driven on resonance
with $\Omega=0.4\pi\gamma$. The result for each round trip phase change is an
average of 10,000 trajectories. Figure 10: Four incoherent output spectra when
the system is driven on resonance with $\Omega=2\pi\gamma$ and the feedback is
introduced with a delay time of $\tau=2.0\gamma^{-1}$ for round trip phase
changes of $\phi=0$, $\phi=\pi/2$, $\phi=\pi$, and $\phi=3\pi/2$. Each spectra
is an average of 20,000 trajectories.

As the delay time from the feedback loop becomes non-negligible, the peak
bunching does not occur when $\phi=\pi$ as for the short loop but rather at
$\Omega\tau/2-\phi=(2k-1)\pi,k\in\mathbb{Z}$ as shown in Fig. 9 for three
different choices of loop length. This condition arises due to phase matching
of the Rabi oscillations in the TLS with the phase of the returning field.
Matching this condition would always give the peak bunching in the one photon
in the loop limit; however, when two photons are present this is not the case
as the peaks shift slightly and are not always the dominant phase choice for
bunching as shown in Fig. 9(c). Here, $\tau=2.0\gamma^{-1}$ and
$\Omega=0.4\pi\gamma$, so we would expect the peaks at $\phi=0.6\pi,1.4\pi$;
instead these are present but the central peak is at $\phi=\pi$. This likely
comes back to the fact that when there is a probability for two photons in the
loop, they both cannot be phase matched at the same time for complete
interference. Thus, at the longer loop lengths when the loop population
increases, a maximal $[g_{\rm out}^{(2)}(\Delta t)]_{\rm ss}$ occurs when
$\phi=\pi$ and the returning field interferes at the center frequency rather
than trying to match the one photon in the loop field.

Furthermore, as the population in the loop increases this gives rise to
additional resonances in the output spectrum arising from the feedback loop.
These additional resonances are at $\omega-\omega_{\rm L}=\left(2\pi
k+\phi\right)/2\pi\tau$ and we show example spectra in Fig. 10 for
$\Omega=2\pi\gamma$, $\tau=2.0\gamma^{-1}$, and  six choices of round trip
phase change. These resonances are a result of the multi-mode cavity
resonances set up via multiple round trips around the feedback loop, where the
TLS acts as a second mirror,  analogous to the Fabry-Pérot resonances set up
in a closed cavity coupling to the Mollow triplet. The cavity QED physics that
begins to be introduced is fully picked up by the QTDW model at the system
level where truncating to two photons in the loop is analogous to truncating
the cavity to a ground state and two excited state levels.

##  IV Discussion on Potential Experimental Systems to Exploit Coherent
Feedback

For optical frequencies, state of the art qubits in typical photonic
waveguides typically exhibit decay rates on the order of
$\gamma=1\,\,\rm{ns}^{-1}$ [89]. For example, if we embed such emitters into a
low loss $\rm{SiN}$ waveguide (assuming a group index of 2), then the required
waveguide length of $L_{0}/2=0.75$ cm for a “short” feedback loop, with delay
time $\tau=0.1\gamma^{{\color[rgb]{0,0,0}-1}}$. Such integration schemes have
now been demonstrated experimentally, using QD nanowires and SiN waveguides
[95]. For the more extreme delay time of $\tau=2\gamma^{-1}$, as used in Fig.
10, then $L_{0}/2=15$ cm. So low loss waveguides are required, though the
lengths could be reduced by designing slow light waveguides modes with a
larger group index. To control the tuning of the round trip phase, this would
require moving the QD a distance of half a wavelength to change the phase from
$\phi=0$ and $\phi=\pi$, or a means to tune the mirror phase or group
velocity.

In order to shorten the required feedback length, the QD can also be placed in
a cavity (such as a Fabry-Pérot cavity or ring resonator) to include a Purcell
factor, $P_{F}$, to the decay rate. This would decrease the required waveguide
length to be $L_{0}/2P_{F}$. For example, a modest Purcell factor of 10, would
decrease the range of the waveguide length to be $L_{0}/2\in[0.75,15]$ mm,
where waveguide loss is less of a problem.

At microwave frequencies, one can also exploit highly developed circuit QED
systems [96, 97], where multiple qubits can be controlled in position with
great precision [98] (even multiple qubits can be spatially controlled to
fractions of a wavelength). Here the physical length scales are of course much
larger, but all the results equally apply in scaled units.

##  V Conclusions

We have introduced an open-system quantum optics theory to study the nonlinear
behavior of an optically pumped TLS in a waveguide system, with a time-delayed
coherent feedback. To do this, we described an extension to a previous QT
model of coherent feedback [37] to explicitly characterize the waveguide
output from the system. We have shown how one can obtain the first- and
second-order correlation functions using the QTDW model and in turn the
incoherent output spectra. We then showed a number of results focusing on how
the multi quanta effects from the feedback loop effect the output observables.
Notably, with the proper phase matching conditions, the feedback can be used
to filter out the central laser peak, switch the system through anti-bunched
and bunched light output, and enhance the photon pair emission from the
system. We also show how the feedback loop can introduce new resonances into
the output spectra.

By focusing on incorporating practical experimental observables, along with
the unique stochastic insights from QT theory, this approach is useful to
connect to experimental realizations of time-delayed coherent feedback. The
inclusion of pure dephasing, an important experimental consideration, is a
natural addition to the QTDW model but can be added into the MPS formalism
with some added complexity [67]. To continue expanding on the practical
usefulness of this approach, the addition of incoming photon wavepackets from
the waveguide input rather than a CW-drive is necessary to investigate the
effect of feedback on the single photon source properties of the TLS. Also, an
expansion of the TLS to multi-exciton or biexciton systems will more
accurately represent the experimental reality of embedded QDs [99].

###### Acknowledgements.

This work was supported by the Natural Sciences and Engineering Research
Council of Canada, the Canadian Foundation for Innovation, the National
Research Council of Canada, Queen’s University and the University of Ottawa.
We thank Dan Dalacu and Robin Williams for support and useful discussions.

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##  Appendix A Deriving the Interaction Hamiltonian

The free field of the waveguide is described by

$\mathcal{E}(x)=\mathcal{E}_{-}(x)+\mathcal{E}_{+}(x),$ (24)

where $\mathcal{E}_{-}(x)$ and $\mathcal{E}_{+}(x)$ are the left and right
propagating fields respectively, with the explicit form

$\mathcal{E}_{\pm}(x)=\mp\frac{1}{\sqrt{2\pi}}\int_{\infty}^{\infty}e^{\pm
i\omega x/c}b(\omega)d\omega,$ (25)

where $c$ is the group velocity of the waveguide mode with frequency $\omega$.
Due to the reflection from the mirror, we can write $\mathcal{E}_{+}(0)$ in
terms of $\mathcal{E}_{-}(0)$ by including the phase change picked up from the
round trip,

$\mathcal{E}_{+}(0)=e^{i(\phi+\omega\tau)}\mathcal{E}_{-}(0),$ (26)

where $x=0$ is the location of the mirror. Then the interaction between the
TLS and this free field at $x=0$ is

$H_{\rm
I}=\sqrt{\gamma_{L}}\sigma^{+}\frac{\mathcal{E}_{-}(0)}{2}+\sqrt{\gamma_{R}}\sigma^{+}\frac{\mathcal{E}_{+}(0)}{2}+{\rm
H.c.}$ (27)

Next, substituting Eqs. (25) and (26) into Eq. (27), we get our interaction
Hamiltonian

$\displaystyle H_{\rm I}={}$
$\displaystyle\int_{-\infty}^{\infty}d\omega\left[\left(\sqrt{\frac{\gamma_{\rm
L}}{2\pi}}\sigma^{+}b(\omega)\right.\right.$ (28)
$\displaystyle{}+\left.\left.\sqrt{\frac{\gamma_{\rm
R}}{2\pi}}e^{i(\phi+\omega\tau)}\sigma^{+}b(\omega)\right)+\rm H.c.\right].$

##  Appendix B Deriving the Evolution of the Waveguide Operators

Generally, the free evolution of the waveguide is described by the unitary
operator $U_{\rm W}(t)=e^{-iH_{\rm W}t}$ and the evolution of each discrete
bin is $U_{\rm W}^{\dagger}(t)B_{n}U_{\rm W}(t)$. Over one time step these
exponentials can be expanded to first order in $\Delta t$ to be

$U_{\rm W}(\Delta t)=e^{-iH_{\rm W}\Delta t}=1-iH_{\rm W}\Delta t,$ (29)

which can be substituted into the evolution of $B_{n}$ to get

$U_{\rm W}^{\dagger}(\Delta t)B_{n}U_{\rm W}(\Delta t)=(1+iH_{\rm
W}^{\dagger}\Delta t)B_{n}(1-iH_{\rm W}\Delta t).$ (30)

Then expressing this in terms of the $b_{k}$ operators and expanding to first
order in $\Delta t$, we get

$\displaystyle U_{\rm W}^{\dagger}(\Delta t)B_{n}U_{\rm W}(\Delta t)$
$\displaystyle{}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}b_{k}e^{i\omega_{k}n\Delta
t}$ (31) $\displaystyle{}+\frac{i\Delta
t}{N}\sum_{k,k^{\prime}=0}^{N-1}\omega_{k}e^{i\omega_{k^{\prime}}n\Delta
t}\left(b_{k}^{\dagger}b_{k}b_{k^{\prime}}-b_{k^{\prime}}b_{k}^{\dagger}b_{k}\right).$

Noting that the commutation relation for $b_{k}$ is
$[b_{k},b_{k^{\prime}}]=\delta_{k,k^{\prime}}$ this reduces to

$\displaystyle U_{\rm W}^{\dagger}(\Delta t)B_{n}U_{\rm W}(\Delta t)$
$\displaystyle{}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}b_{k}e^{i\omega_{k}n\Delta
t}$ (32) $\displaystyle\hskip 7.11317pt{}-\frac{i\Delta
t}{N}\sum_{k,k^{\prime}=0}^{N-1}\omega_{k}\delta_{k,k^{\prime}}b_{k}e^{i\omega_{k^{\prime}}n\Delta
t},$
$\displaystyle{}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}b_{k}e^{i\omega_{k}n\Delta
t}\left(1-i\Delta t\omega_{k}\right),$
$\displaystyle{}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}b_{k}e^{i\omega_{k}(n-1)\Delta
t},$ $\displaystyle{}=B_{n-1},$

where we have used that $(1-i\Delta t\omega_{k})=e^{-i\Delta t\omega_{k}}$ to
first order in $\Delta t$.