# Mitigating Landside Congestion at Airports through
Predictive Control of Diversionary Messages
††thanks: *Pacific Northwest National Laboratory is operated by Battelle
Memorial Institute for the U.S. Department of Energy under Contract No. DE-
AC05-76RL01830. This work was supported by the U.S. Department of Energy
Vehicle Technologies Office.

Nawaf Nazir  Pacific Northwest National Lab
Richland, USA
<EMAIL_ADDRESS>Soumya Vasisht  Pacific Northwest National Lab
Richland, USA
<EMAIL_ADDRESS>Shushman Choudhury  Stephen Zoepf  Lacuna
Technologies
Palo Alto, USA
<EMAIL_ADDRESS>Lacuna Technologies
Palo Alto, USA
<EMAIL_ADDRESS>Chase Dowling  Pacific Northwest National Lab
Richland, USA
<EMAIL_ADDRESS>
###### Abstract

We present a data-driven control framework for adaptively managing landside
congestion at airports. Ground traffic significantly impacts airport
operations and critical efficiency, environmental, and safety metrics. Our
framework models a real-world traffic intervention currently deployed at
Seattle-Tacoma International Airport (SEA), where a digital signboard
recommends drivers to divert to Departures or Arrivals depending on current
congestion. We use measured vehicle flow/speed and passenger volume data, as
well as time-stamped records of diversionary messages, to build a macroscopic
system dynamics model. We then design a model predictive controller that uses
our estimated dynamics to recommend diversions that optimize for overall
congestion. Finally, we evaluate our approach on $50$ real-world historical
scenarios at SEA where no diversions were deployed despite significant
congestion. Our results suggest that our framework would have improved speed
in the congested roadway by up to three times and saved between $20$ and $80$
vehicle-hours of cumulative travel time in every hour of deployment. Overall,
our work emphasizes the opportunity of algorithmic decision-making to augment
operator judgment and intuition and yield better real-world outcomes.

###### Index Terms:

Airports, Traffic congestion, Data-driven control, Predictive control,
Diversionary messages

##  I Introduction

The problem of ground traffic congestion at airports has persisted for decades
and major airports have invested considerable time and money to alleviate it
[1]. The policies enacted range from hiring ground staff that resolve specific
incidents to multi-year planning and infrastructure projects [2]. Targeted and
effective strategies for managing landside congestion can greatly influence
overall quality of service by reducing travel times, idle vehicle emissions,
and adverse events, and improving traveler satisfaction [3, 4].

Our paper focuses on a specific adaptive intervention: messages on a digital
signboard that recommend incoming vehicles to divert from Departures to
Arrivals when the former is congested (and vice versa). We do so because such
signboard systems are already in use in certain airports, where the diversions
are deployed based on operator intuition and judgment rather than algorithmic
decision-making (e.g., Seattle-Tacoma International Airport, or SEA, whose
real-world data we use to build a macroscopic traffic model and to evaluate
our approach). But our methodology could apply to other adaptive congestion
management tactics at airports, and even more widely to other ground traffic
problems for which such data are available.

The most relevant prior research on this problem used so-called microscopic,
i.e., high-fidelity simulations that modeled the behaviour of individual
vehicles [5, 6, 7]. Such approaches can represent detailed scenarios and
evaluate them comprehensively, but are typically data and time intensive to
create and computationally expensive to query. They are far more suitable for
assessing the long-term effects of operational changes than for producing
real-time interventions. A mesoscopic airport curbside model addressed some
but not all of the challenges of microscopic simulations, and focused on
analysing various curbside policies rather than optimizing them [8]. In
contrast, to make real-time interventions feasible, we use a macroscopic model
with only aggregate traffic information.

Figure 1: An overhead view of the physical layout of Seattle-Tacoma
International Airport, whose real-world data we use. We illustrate the
variable messaging sign, traffic bottlenecks, Arrival and Departure drives,
and the locations of respective speed/flow sensors, labeled A and D. The
figure is used from our previous work [9].

Our key idea is to formulate the problem of adaptive congestion management
using diversionary messages as one of model predictive control or MPC, a
widely successful framework for adaptively controlling complex systems [10].
Using real-world historical data on vehicle flow and speed and the displayed
messages, we estimate a system dynamics model. This model captures both the
natural evolution of the macroscopic traffic state as well as its response to
interventions. We then design a discrete-time MPC formulation where the
control variables are the two kinds of interventions (Departures to Arrivals,
and vice versa) and the predictive model is our estimated system dynamics.

To evaluate our methodology, we use $50$ historical scenarios from Seattle-
Tacoma airport, where one of Departures/Arrivals was congested but no
diversionary messages were deployed. Our simulations suggest that intervening
adaptively with our framework would have mitigated congestion in all cases.
The speed in the congested roadway would have improved significantly (by up to
three times), and between $20$ to $80$ vehicle-hours of cumulative travel time
would have been saved in every hour of intervention.

##  II Related Work

This section reviews three areas of work that we build upon. The first is the
general idea of using model predictive control and related ideas from optimal
control theory to manage ground traffic. The second and third are related but
distinct approaches to understanding and improving airport curbside
management: modeling and analysis of existing operations and detailed
simulations of various potential long-term policies to guide future decision-
making.

###  II-A Model Predictive Control for Traffic Management

The idea of using MPC to solve variants of ground traffic management has been
studied in the literature [11]. A comprehensive summary of early approaches
can be found in the PhD thesis [12]. The majority of existing work differs
from ours in the specific real-world control strategy that they encode: the
timing cycle of traffic signals [13]; so-called ramp-metering to limit inflow
of vehicles into a freeway at rush hour [14]; dynamic curb allocation in urban
areas [15]; and perimeter control that applies the general idea of ramp-
metering to a large-scale urban network [16, 17]. None of these strategies
apply to balancing and diverting inflow between two facilities, each with
their unique traffic characteristics. Some recent research exist on the
traffic control community considers our flavor of problem, e.g., a theoretical
model of flow exchanges in multi-reservoir systems [18] and dynamically
reorganizing and rearranging inflow on congested lanes and roadways [19, 20].
But they do not consider the complex time-varying nature of airport traffic
and do not develop and evaluate their approach on real traffic and
intervention data.

###  II-B Understanding Airport Landside Operations

Since as far back as the late 1960s, travelers to airports have reported
facing delays and frustration due to ground traffic congestion and bottlenecks
[1]. Over the decades, the literature on evaluating and analysing landside
facilities has covered a number of areas: general terminal operations [21,
22], access management [2, 23], level of service [3, 24], and customer
experience for both quantitative metrics like dwell time [25] and subjective
ones like satisfaction [4]. Two papers from this line of work are more
directly related to our specific problem. One studied how advance signs
(albeit for airlines, not diversions) guide and improve vehicle search times
[26] and the other compared the throughput characteristics of pickup to
dropoff facilities (of which Arrivals and Departures are canonical examples)
[27]. This body of work is useful for developing context and understanding the
general problems and opportunities to improve, but they have no notion of
optimization or control that we can use in a computational framework.

###  II-C Landside Management Policies for Airports

In addition to understanding the problem of landside operations at airports,
the research community has been trying to address it as well. Numerical
simulations have emerged as a natural and effective tool of choice; an early
example is of Las Vegas McCarran International Airport trying to redesign its
ground transportation access system [5]. A related team of authors then used
discrete event simulation to model curbside vehicular traffic at airports and
applied it to Austin-Bergstrom International Airport [28]. Most early efforts
relied on simple bespoke models, e.g., dual roadways for loading/unloading and
circulation [6] and a mathematical queueing framework [29]; such approaches do
not benefit and improve from real-time data about the airports. More recent
works have harnessed vastly improved computational resources to develop high
fidelity microsimulations that are carefully calibrated to facility data; two
notable examples are a study on improving taxi drop-off throughput at a
terminal [30] (they modeled a railway station in Nanjing but the principles
hold), and a comprehensive modeling framework of Dallas Fort-Worth
International Airport [7]. Such detailed simulations can be quite realistic
(though they require their fair share of assumptions) and are well-suited for
long-term strategic planning. But their data and computation intensive nature
is what makes them unsuitable to be used in the loop of a real-time control
algorithm.

##  III Methodology

In this section, we summarize our problem formulation and approach, to
emphasize the various interacting components. We operate a variable message
signboard (VMS) that incoming drivers observe before they decide whether to go
to the Departures or Arrivals roadways (see Section I). If we infer that
either roadway is significantly more congested than the other, we can
recommend that drivers heading to the congested roadway divert to the other
one. Our problem is to develop a controller that computes a good sequence of
diversions for the current and expected traffic conditions. The controller’s
high-level objective is to mitigate congestion, i.e., improve travel speeds in
the congested roadway.

The specifics of our problem formulation depend on the real-world data from
SEA that we use to demonstrate and evaluate our approach. We have measurements
of vehicle flow (volume per unit time) and average speed over 4 months of the
year 2022, binned to 15-minute intervals. The hourly departure and arrival
average speeds over a 24-hour horizon are depicted in Figure 2(a) and Figure
2(b) respectively. Here, each dot represents the average speed of all vehicles
at the terminal within the 15-minute interval. The speed data highlights the
difference in congestion times between arrivals and departures and hence the
opportunity to divert traffic from one terminal to another when necessary. We
also have time-stamped records of messages deployed by SEA, recommending
traffic to divert from one terminal to the other. In recent work, we inferred
the response rates of incoming traffic to the suggestions [9]. Here, we will
build upon our understanding of the system and design a controller that
mitigates congestion.

(a)

(b)

(c)

Figure 2: Average vehicle speeds at (a) Departures and (b) Arrivals and (c)
normalized cumulative passenger volumes.

Our approach consists of two key components. First, we estimate a model of the
traffic dynamics from the available data, both the natural variation and the
response to interventions. We use least-squares regression to fit a linear
discrete-time dynamical model. Second, we use the estimated dynamics model in
the loop of a model predictive controller. The controller optimizes over
sequence of binary control inputs that encode the two kinds of diversions.

##  IV Estimating Macroscopic Traffic Dynamics

Consider a linear discrete-time dynamic model to represent the evolution of
airport traffic dynamics:

$\displaystyle x_{k+1}=Ax_{k}+Bu_{k}$ (1)

where $x_{k}\in\mathbb{R}^{4\times 1}$, $u_{k}\in\mathbb{R}^{4\times 1}$,
$A\in\mathbb{R}^{4\times 4}$ is the auto-regressive co-efficient matrix and
$B\in\mathbb{R}^{4\times 4}$ is the control co-efficient matrix. The state
$x_{k}$ describes Departures flow and speed (DF/DS) and Arrivals flow and
speed (AF/AS) at time $k$ and can be concisely denoted as
$x_{k}=\begin{bmatrix}\textrm{DF, DS, AF, AS}\end{bmatrix}^{\top}$. The
control input $u_{k}$ represents the treatment applied at time step $k$ for
diverting traffic from Departures to Arrivals (TD) or vice versa (TA). We also
consider the Departures and Arrivals passenger volume data (DV and AV,
respectively), also obtained from SEA [31], as additional exogenous inputs in
our predictive model at each time step. AV and DV for a given time step are
determined from the preceding two hours of aggregated incoming volume and the
subsequent two hours of aggregated departing volume data, respectively. Figure
2(c) depicts the normalized cumulative passenger volume. The resulting input
vector $u_{k}$ is given as, $u_{k}=\begin{bmatrix}\textrm{TD, TA, DV,
AV}\end{bmatrix}^{\top}$. To determine the macroscopic traffic dynamics model
in (1), we train a linear regression model that estimates matrices $A$ and
$B$.

Let $X$ represent a collection of states, $x_{k},\ k\in\\{1,\ldots,N-1\\}$,
where $N$ is the size of our dataset. Let $Y$ be the set of states at the
following time step, $x_{k+1}$ and $U$ the set of inputs, whose elements
correspond to each entry in $X$. We randomly select 80% of the data points as
the training set, $X_{\mathrm{Tr}}\subset X$ (and $U_{\mathrm{Tr}}\subset
U,Y_{\mathrm{Tr}}\subset Y$) to represent the training dataset and reserve the
rest for validation. The state and input matrices may be stacked to obtain
$X^{\prime}=[X_{\mathrm{Tr}};U_{\mathrm{Tr}}]$. The resulting system matrix is
$A^{\prime}=[A\ B]\in\mathbb{R}^{4\times 8}$. The optimization problem to
estimate the best fit model in (1) is given by:

$\displaystyle\operatorname*{arg\,min}_{A^{\prime}}||Y-A^{\prime}X^{\prime}||_{2}+\rho||A^{\prime}||_{2,1},$
(2)

where $||M||_{2,1}=\sum_{i=1}^{n}\sqrt{\sum_{j=1}^{m}m_{ij}^{2}}$ is the
$l_{2,1}$-norm of a matrix first introduced in [32] as a rotational invariant
$l_{1}$ norm shown to be less sensitive to outliers [33].

We can augment the objective function above with constraints that encode
domain knowledge about traffic flow and other system characteristics. For
example, Departures and Arrivals flow are measured on separate and independent
roadways and are likely to be uncorrelated. Similarly, treating Arrivals is
expected to positively impact the speed at Arrivals (and similarly for
Departures). To include this knowledge, we change Equation 2 to:

$\displaystyle\operatorname*{arg\,min}_{A^{\prime}}$
$\displaystyle||Y-A^{\prime}X^{\prime}||_{2}+\rho||A^{\prime}||_{2,1}$ (3a)
s.t. $\displaystyle CA^{\prime}=0$ (3b) $\displaystyle DA^{\prime}\leq 0$ (3c)

where $C\in\mathbb{R}^{4\times 4}$ encodes the independence of Departures and
Arrivals flow, and $D\in\mathbb{R}^{4\times 4}$ encodes the positive impacts
of treating roadways on their corresponding speeds. Finally, $\rho$ is a
regularization term. The values of these parameters can be found in the code
on the Github page.

Figure 3: Error histograms for arrival and departure flows and speed using the
model obtained from solving Equation 3.

The optimization problem in (3) is convex and can be solved with a generic
optimization solver. We evaluate the prediction accuracy of the resulting $A$
and $B$ matrices against a held-out portion of our ground truth dataset. Each
held-out dataset point is a tuple of the previous and next system state. We
apply the trained linear dynamics model (consisting of $A$ and $B$) on the
previous state to predict the next state, and compute the error between the
predicted and true system state at the next time interval. This model abets
our primary goal of control design for automated decision-making. More
accurate models may be built to estimate macroscopic traffic dynamics, but it
is beyond the scope of our current research.

Variable | MAE | RMSE
---|---|---
Departure Flow (count) | 22.87 | 31.73
Departure Speed (kph) | 2.70 | 4.99
Arrival flow (count) | 28.12 | 36.85
Arrival speed (kph) | 2.87 | 4.89
TABLE I: Error statistics on our regression model.

Figure 3 shows a histogram of the errors for each state variable (DF, DS, AF,
AS), while Table I shows the mean absolute error (MAE) and root mean square
error (RMSE) for the same variables. These results indicate that the model is
good enough to predict a congested or uncongested scenario and inform the
design of TA or TD. In the following section, we formulate an MPC framework
that predicts traffic congestion and designs the timing of the diversionary
treatment signals to shorten the time spent in a congested state. We use the
distribution of the errors in Figure 3 to augment the trained model
predictions to closely mimic that of the true system and demonstrate the
improved performance.

##  V Model Predictive Control for Diversions

We now integrate the macroscopic landside airport traffic model from Section
IV with the following MPC formulation:

$\displaystyle\min_{u_{k}[1:2]}$
$\displaystyle\sum_{k=0}^{T}((\mathcal{D}_{k}-1)^{2}+(\mathcal{A}_{k}-1)^{2}+\gamma\sum
u_{k}[1:2])$ (4a) s.t $\displaystyle\mathcal{D}_{k}\leq 1$ (4b)
$\displaystyle\mathcal{D}_{k}\leq\frac{x_{k}[2]}{\widehat{DS}}$ (4c)
$\displaystyle\mathcal{A}_{k}\leq 1$ (4d)
$\displaystyle\mathcal{A}_{k}\leq\frac{x_{k}[4]}{\widehat{AS}}$ (4e)
$\displaystyle x_{k+1}=Ax_{k}+Bu_{k}$ (4f) $\displaystyle
x_{0}=x_{\text{init}}$ (4g) $\displaystyle x_{k}\geq 0$ (4h)
$\displaystyle\sum u_{k}[1:2]\leq 1$ (4i) $\displaystyle
u_{k}[1:2]\in\\{0,1\\}$ (4j)

where $\widehat{DS}=35$ km per hour and $\widehat{AS}=45$ km per hour are the
critical speeds for Departures and Arrivals respectively, determined from the
speed flow fundamental diagrams [9]. $u_{k}[1:2]$ denotes the values of TD and
TA at time $k$. $x_{\text{init}}$ is the initial state of the system at time
$t=0$ and $\gamma$ is the regularization factor for the control signal.
$\mathcal{D}_{k}$ and $\mathcal{A}_{k}$ are Departures and Arrivals critical
speed ratios, defined as the ratio of the average speed to the critical speed.
A critical ratio value well below $1$ indicates significant congestion.

The objective (4a) is to minimize the deviation of $\mathcal{D}_{k}$ and
$\mathcal{A}_{k}$ from $1$. The constraints in (4b)-(4e) penalize Departures
and Arrivals speeds only when the respective critical ratios fall below $1$,
but not when they are above $1$, which is consistent with our goal of
increasing average speeds (and hence reducing congestion). Additional
constraints, such as system dynamics, initial state and restricting the state
variables to be positive are described in (4f) - (4h). Finally, the
constraints in (4i)-(4j) limit the control inputs (TD and TA) to be binary
variables and that at most only one diversionary signal is active at each time
step (since the VMS can only display one message at a time).

We solve the problem in Equation 4 in a receding horizon manner, i.e., at each
time step, the above objective is minimized over an appropriate time horizon
to obtain the optimal sequence of TD or TA signals. The first elements of this
sequence are applied as input to the system and the dynamics is propagated
forward in time. This process is repeated by resetting $x_{0}$ with $x_{k+1}$
until the final time step. Since we do not have access to real-time
measurements from the actual system or a simulator to represent it, we use the
distributions of the model errors shown in Figure 3 to generate system outputs
as a response to control (or no control). The optimal control signals obtained
from solving Equation 4 are applied to the model:

$\displaystyle x_{k+1}=Ax_{k}+Bu_{k}+\mathcal{U}(a,b)$ (5)

where $\mathcal{U}(a,b)$ is a uniform distribution between $a$ and $b$, based
on the 90th percentile values of the errors shown in Figure 3. The values of
$a$ and $b$ for the four states are provided in Table II.

Variable | a | b
---|---|---
Departure Flow (DF) error | -34.6 | 38.5
Departure Speed (DS) error | -2.9 | 3.4
Arrival flow (AF) error | -43.3 | 46.5
Arrival speed (AS) error | -3.5 | 3.9
TABLE II: Parameters $a$ and $b$ describing the uniform distribution
$\mathcal{U}(a,b)$ from Equation 5.

We demonstrate the effectiveness of our formulation by performing several runs
of the optimization loop with the randomized system output from Equation 5 and
recording the mean and standard deviation of the resulting speeds and flows.
We now discuss how we use these statistics to compute metrics such as travel
time savings and speed improvement.

##  VI Evaluation

Figure 4: An untreated congestion scenario from our dataset. On May 1, 2022,
the Departures roadway (blue line) at SEA was congested for nearly three hours
from 0800-1100, while Arrivals was operating normally.

This section describes how we evaluate the effectiveness of our approach in
recommending diversions to ease congestion. We selected $50$ historical
scenarios from our SEA dataset that represent _untreated congestion_ , i.e.,
where either Arrivals or Departures was congested but no diversion was
deployed (the scenarios were evenly split between those with Arrivals
congested and those with Departures congested). To extract such scenarios, we
looked for periods of time, typically 2-3 hours, where the critical speed
ratio of either facility dropped significantly while that of the other
facility was normal, e.g., Figure 4. These $50$ scenarios serve as the
grounding examples on which we run our controller and measure how much
congestion would have been eased.

We run our controller on each untreated congestion scenario to yield a
sequence of control actions that encode either diversionary messages or no
intervention. For any MPC approach, two key parameters are the planning
horizon (inner loop) and the number of execution steps (outer loop). Our
planning horizon is $12$ time-steps, or $3$ hours, as congestion typically
builds up and dissipates within that time-frame. However, we set the number of
execution steps to $4$, or $1$ hour. For each scenario, we diverge from
history after the first execution step, since we use our one-step dynamics
model to simulate the system state for the next execution step. Therefore, we
limit execution steps to $4$ to allow enough time-steps to evaluate our
controller meaningfully, but not so much as to diverge significantly.

(a) Departures Treated

(b) Arrivals Treated

Figure 5: Our MPC controller computes diversions that, if followed, would have
improved speed in the treated roadway up to three times the actual speed in
the historical scenario. The speed improvement for treating Departures is much
higher because our dynamics model estimates a higher impact of treating
Departures on its speed, than for treating Arrivals.

In addition to a sequence of $4$ control actions, the controller also outputs
a sequence of $4$ counterfactual states (flow, speed etc.), for each scenario.
These values represent the expected future states of the system, up to $1$
hour, had our controller’s diversions been deployed. Therefore, we use the
actual and counterfactual states of each scenario to obtain two key
performance metrics.

The first performance metric uses the actual and counterfactual vehicle
speeds. Recall from Equation 4a that our optimization objective is to improve
the speed of the treated roadway without significantly worsening the other
roadway. Therefore, we compute the ratios of the counterfactual and actual
speeds for each roadway, up to $4$ execution steps, averaged over the $50$
scenarios. Figure 5 shows that our diversions would have improved speed in the
treated roadway by up to three times without affecting the other roadway.

The speed improvement ratios for treating Departures (Figure 5(a)) are
considerably higher than for treating Arrivals (Figure 5(b)). The primary
reason for this difference is our dynamics model from Section IV, which we use
to simulate the counterfactual states. The model estimates a higher impact of
treating Departures (TD) on improving Departures speed, than the impact of
treating Arrivals (TA) on improving Arrivals speed. The higher impact of TD is
in turn due to the higher response rates of drivers to the corresponding
diversionary message; our prior work on inferring the response rates explores
their differences in detail [9].

Note also from Figure 5(b) that when Arrivals is treated, the standard error
around the average counterfactual/actual speed ratio reduces with increasing
15-minute intervals. This reduction contradicts both our intuition that the
counterfactual outcomes should become noisier as we simulate further into the
future and the fact that this intuition is reflected by the Departures line in
Figure 5(a). The reason is that congestion in Arrivals typically dissipates
more quickly than for Departures, usually within an hour from when congestion
is highest. Therefore, in the untreated congestion scenarios used for Figure
5(b), the actual speed goes back to hovering around or above the critical
speed within the hour. Our controller’s diversions seek to bring the
counterfactual speed to the critical speed, thus explaining why the ratio of
counterfactual to actual speeds is consistently close to 1 over all such
scenarios.

Figure 6: The distributions of the vehicle hours saved due to our controller’s
diversions, over the $50$ evaluation scenarios. The upper bound for each
scenario is $4$ times the corresponding lower bound, given the range of
lengths of plausible traffic bottlenecks at Seattle-Tacoma airport.

The second performance metric uses both speed and flow; it computes the
counterfactual cumulative vehicle hours saved by the vehicles that go to the
treated roadway. At SEA, the traffic bottleneck can be anywhere from $0.5$ to
$2$ kilometres in length, yielding corresponding lower and upper bounds of
travel time improvement respectively. We compute the dot product of the travel
time improvement in each execution step with the number of incoming vehicles
in the corresponding $15$ minute interval, to get the metric of cumulative
vehicle hours saved (per hour).

Figure 6 plots the means and inter-quartile ranges of the lower and upper
bounds of the cumulative vehicle hours saved, over the $50$ scenarios. It
shows that our diversions could have saved roughly $20$ to $80$ vehicle hours
per hour of deployment on average. Reducing vehicle hours also reduces idle
time fuel wastage and emissions, both important downstream operational
metrics. The specific savings depend on vehicle make and model, but we can
derive coarse estimates for a mid-sized sedan [34]. The average idle time fuel
expense is $0.35$ gallons per hour per vehicle, which translates to a savings
of $7$ to $28$ gallons of fuel per hour. The average idle time emissions is
$2100$ grams of carbon dioxide per vehicle per hour, which translates to a
total savings of $90$ to $360$ kilograms of $\text{CO}_{2}$ per hour. Given
that the airport experiences around $6$ to $8$ hours of congestion every
business day, all these savings could accumulate significantly over business
months and years.

##  VII Conclusion

We presented a comprehensive data-driven framework for controlling variable
message signs to reduce ground traffic congestion and thus travel time at
airports. Using available real-world speed and flow data from SEA, we obtain a
linear dynamical model of the system that we then use in a model predictive
controller to optimize for our congestion objective. Our simulation results on
several historical congested scenarios at SEA showcase the efficacy of our
approach and how algorithmic decision-making can augment human judgment in
real-world transportation problems. Our specific application requires only one
control decision per time-step, but our methodology is general and capable of
solving more complex and multi-dimensional control problems.

Future efforts will work towards a pilot deployment of this controller at SEA,
to automate the traffic diversion messages and quantify the benefits to
travelers in terms of travel time saved. Future work will also explore the
human response to such variable message signs and how that can be included
within the predictive control framework.

## Acknowledgements

We thank the landside operations team at Seattle-Tacoma International Airport
and Port of Seattle for sharing data and insights.

## References

  * [1] Marjorie Brink and Donald Maddison  “Identification and measurement of capacity and levels of service of landside elements of the airport”  In _Transportation Research Board Special Report_ , 1975
  * [2] Heather Wishart Shriner and Lester A Hoel  “Evaluating improvements in landside access for airports”  In _Transportation research record_ 1662.1 SAGE Publications Sage CA: Los Angeles, CA, 1999, pp. 32–40
  * [3] Carlos Müller and Geoffrey D Gosling  “A framework for evaluating level of service for airport terminals”  In _Transportation Planning and Technology_ 16.1 Taylor & Francis, 1991, pp. 45–61
  * [4] Salvatore Failla, Enzo Bivona and Nazareno Ventola  “Exploring Airports’ Landside Congestion Impacts on the dynamic of Passengers Satisfaction.”  In _International Conference of the 14 System Dynamics Society_ , 2014
  * [5] Gloria G Bender and Kuo Y Chang  “Simulating roadway and curbside traffic at las vegas mccarran international airport”  In _IIE Solutions_ 29.11 Institute of IndustrialSystems Engineers (IISE), 1997, pp. 26–32
  * [6] Gavin Duncan and Hugh Johnson  “Development and application of a dynamic simulation model for airport curbsides”  In _The 2020 Vision of Air Transportation: Emerging Issues and Innovative Solutions_ , 2000, pp. 153–164
  * [7] Juliette Ugirumurera et al.  “A modeling framework for designing and evaluating curbside traffic management policies at Dallas-Fort Worth International Airport”  In _Transportation Research Part A: Policy and Practice_ 153 Elsevier, 2021, pp. 130–150
  * [8] Tyler M Harris, Mehdi Nourinejad and Matthew J Roorda  “A mesoscopic simulation model for airport curbside management”  In _Journal of Advanced Transportation_ 2017 Hindawi, 2017
  * [9] Soumya Vasisht et al.  “Estimating Driver Response Rates to Variable Message Signage at Seattle-Tacoma International Airport”  In _Findings_ Findings Press, 2022
  * [10] Eduardo F Camacho and Carlos Bordons Alba  “Model predictive control”  Springer science & business media, 2013
  * [11] Nathan H Gartner  “Development of demand-responsive strategies for urban traffic control”  In _System Modelling and Optimization_ Springer, 1984, pp. 166–174
  * [12] András Hegyi  “Model predictive control for integrating traffic control measures”  Netherlands TRAIL Research School, 2004
  * [13] Bao-Lin Ye et al.  “A survey of model predictive control methods for traffic signal control”  In _IEEE/CAA Journal of Automatica Sinica_ 6.3 IEEE, 2019, pp. 623–640
  * [14] Tom Bellemans, B De Schutter and Bart De Moor  “Model predictive control for ramp metering of motorway traffic: A case study”  In _Control Engineering Practice_ 14.7 Elsevier, 2006, pp. 757–767
  * [15] Nawaf Nazir et al.  “Optimal, centralized dynamic curbside parking space zoning”  In _2022 IEEE 25th International Conference on Intelligent Transportation Systems (ITSC)_ , 2022, pp. 91–98  IEEE
  * [16] Isik Ilber Sirmatel and Nikolas Geroliminis  “Economic model predictive control of large-scale urban road networks via perimeter control and regional route guidance”  In _IEEE Transactions on Intelligent Transportation Systems_ 19.4 IEEE, 2017, pp. 1112–1121
  * [17] Isik Ilber Sirmatel and Nikolas Geroliminis  “Stabilization of city-scale road traffic networks via macroscopic fundamental diagram-based model predictive perimeter control”  In _Control Engineering Practice_ 109 Elsevier, 2021, pp. 104750
  * [18] Guilhem Mariotte and Ludovic Leclercq  “Flow exchanges in multi-reservoir systems with spillbacks”  In _Transportation Research Part B: Methodological_ 122 Elsevier, 2019, pp. 327–349
  * [19] Paul Anderson and Nikolas Geroliminis  “Dynamic lane restrictions on congested arterials”  In _Transportation research part A: policy and practice_ 135 Elsevier, 2020, pp. 224–243
  * [20] Yan Huang, Bei Zhou, Zongzhi Li and Lei Zhang  “Evaluation of Travel Lane Narrow Configurations and Inflow Restrictions on Congested Arterials”  In _CICTP 2021_ , 2021, pp. 1808–1815
  * [21] Vojin Tošić  “A review of airport passenger terminal operations analysis and modelling”  In _Transportation Research Part A: Policy and Practice_ 26.1 Elsevier, 1992, pp. 3–26
  * [22] Scott A Parr, Nikola Mitrovic and Aleksandar Stevanovic  “Evaluating landside transportation operations for general aviation airports: estimating vehicle efficiency and pedestrian safety”  In _World Review of Intermodal Transportation Research_ 4.1 Inderscience Publishers Ltd, 2013, pp. 1–22
  * [23] Thomas Budd, Stephen Ison and Tim Ryley  “Airport surface access management: Issues and policies”  In _Journal of Airport Management_ 6.1 Henry Stewart Publications, 2011, pp. 80–97
  * [24] Anderson Correia and SC Wirasinghe  “Evaluating level of service at airport passenger terminals: Review of research approaches”  In _Transportation research record_ 1888.1 SAGE Publications Sage CA: Los Angeles, CA, 2004, pp. 1–6
  * [25] Wonkyu Kim, Yonghwa Park and Byung Jong Kim  “Estimating hourly variations in passenger volume at airports using dwelling time distributions”  In _Journal of Air Transport Management_ 10.6 Elsevier, 2004, pp. 395–400
  * [26] Reddy Kichhanagari, Rosa DeRamus Motley, Susan A Duffy and Donald L Fisher  “Airport terminal signs: Use of advance guide signs to speed search times”  In _Transportation research record_ 1788.1 SAGE Publications Sage CA: Los Angeles, CA, 2002, pp. 26–32
  * [27] Anton J Kleywegt and Xinyu Liu  “Throughput Capacity Comparison for Airport Pickup and Dropoff Facilities”  In _Transportation Research Record_ 2676.2 SAGE Publications Sage CA: Los Angeles, CA, 2022, pp. 148–164
  * [28] Cenk Tunasar, Gloria Bender and Holland Young  “Modeling curbside vehicular traffic at airports”  In _1998 Winter Simulation Conference. Proceedings (Cat. No. 98CH36274)_ 2, 1998, pp. 1113–1117  IEEE
  * [29] Leigh Fisher  “Acrp report 40: Airport curbside and terminal area roadway operations”  In _Transportation Research Board, Washington, DC, USA_ , 2010
  * [30] Fangyi Yang et al.  “Achieving higher taxi outflows from a drop-off lane: A simulation-based study”  In _Transportation Research Part C: Emerging Technologies_ 115 Elsevier, 2020, pp. 102623
  * [31] SeaTac Airport Seattle  “Passenger Volume Data”, https://www.portseattle.org/page/airport-statistics, 2022 (accessed January 31st 2023)
  * [32] Chris Ding, Ding Zhou, Xiaofeng He and Hongyuan Zha  “R 1-pca: rotational invariant $l_{1}$-norm principal component analysis for robust subspace factorization”  In _Proceedings of the 23rd international conference on Machine learning_ , 2006, pp. 281–288
  * [33] Feiping Nie, Heng Huang, Xiao Cai and Chris Ding  “Efficient and robust feature selection via joint $\ell_{2,1}$-norms minimization”  In _Advances in neural information processing systems_ 23, 2010
  * [34] Linda Gaines, Eric Rask and Glenn Keller  “Which is greener: idle, or stop and restart? Comparing fuel use and emissions for short passenger-car stops”, 2013


††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Direct detection of quantum non-Gaussian light from a dispersively coupled
single atom

Jitendra K. Verma  Lukáš Lachman  Radim Filip  Department of Optics, Faculty
of Science, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech
Republic

###### Abstract

Many applications in quantum communication, sensing and computation need
provably quantum non-Gaussian light. Recently such light, witnessed by a
negative Wigner function, has been estimated using homodyne tomography from a
single atom dispersively coupled to a high-finesse cavity [17]. This opens an
investigation of quantum non-Gaussian light for many experiments with atoms
and solid-state emitters. However, at their early stage, an atom or emitter in
a cavity system with different channels to the environment and additional
noise are insufficient to produce negative Wigner functions. Moreover,
homodyne detection is frequently challenging for such experiments. We analyse
these issues and prove that such cavities can be used to emit quantum non-
Gaussian light employing single-photon detection in the Hanbury Brown and
Twiss configuration and quantum non-Gaussianity criteria suitable for this
measurement. We investigate in detail cases of considerable cavity leakage
when the negativity of the Wigner function disappears completely.
Advantageously, quantum non-Gaussian light can be still conclusively proven
for a large set of the cavity parameters at the cost of overall measurement
time, even if noise is present.

##  1 Introduction

Discrete electronic level structures in the individual atoms and solid-state
systems embedded in a cavity are excellent candidates for future deterministic
and scalable light sources for quantum technology [31, 1]. Such emitted light
is frequently nonclassical, however, quantum communication [42], quantum
sensing [43], quantum simulations [4] and quantum computing [9] require light
beyond the Gaussian squeezed light produced in nonlinear optics. The
literature primary focuses on high quality and scalable generation of single-
photon states [27, 3, 37, 29, 44, 21].

However, current quantum technology already requires a broader class of
sources than ones repeatedly producing single-photon states [26, 33, 6, 8] or
photon-pairs [34, 13] on demand. Encoding information through the large
dimensional space of continuous variables needs some form of coherent-state
superposition instead [11]. At optical frequencies, only conditional
generation of such states has been available so far [40]. However, that method
is not scalable, and it seriously limits optical continuous variable
experiments. Atomic and solid-state cavity experiments have the potential to
be deterministic and scalable sources of such quantum non-Gaussian states [35,
39]. Therefore, the generation of a coherent-state superposition from a single
atom dispersively coupled to an optical cavity was a pioneering step [17].
This superposition exhibited highly nonclassical negative values of the Wigner
function, indirectly estimated from homodyne tomography [28]. This opens new
territory for atomic and solid-state experiments that could finally cover
demands for hybrid quantum optical technologies and their applications [2].
However, initially, the cavity development and overall loss are still limiting
factors for many experiments to observe such negative Wigner functions [19].
This then limits the transition from routinely detected nonclassical features
of such light to more challenging quantum non-Gaussian aspects. While Wigner
function negativity provides a conclusive proof that light is beyond any
mixture of Gaussian states it is unnecessarily high threshold [20]. In
addition, homodyne tomography is too demanding for many experiments due to the
requirement of mode matching with the local oscillator.

In this paper, we propose direct detection of quantum non-Gaussian light from
a single atom in the cavity, which employs single-photon avalanche photo-
diodes (SPADs) in Hanbury Brown and Twiss configuration instead of homodyne
tomography. Importantly, this method allows us to detect quantum non-Gaussian
light from a single atom in the leaky cavity limit when the negative Wigner
function is not observable. This enables investigation of quantum non-Gaussian
features produced by discrete levels in a broad class of current atomic and
solid-state experiments. Such experiments will further open a general
theoretical and experimental investigation of such light sources and stimulate
further steps in this part of photonics and quantum technology.

Figure 1: Quantum non-Gaussian light from a single atom in a leaky cavity.
(_a_) A scheme for preparation and detection of quantum non-Gaussian light
from a cavity. A pulsed laser beam (orange) goes through a BS with very low
reflectivity and a small portion of the beam is reflected towards a cavity
where it interacts dispersively with an atom (see (_b_)). The mirrors of the
cavity exhibit losses $\kappa_{l}$ and $\kappa_{r}$. The state of the atom is
simultaneously manipulated by a microwave field during the interaction. The
light leaving the cavity is measured in the Hanbury Brown and Twiss detection
set-up. It comprises a beam-splitter (BS), which divides the impinging state
of light between two single-photon avalanche photo-diodes SPAD1 and SPAD2.
(_b_) A scheme of the employed energy transitions forming the
$\Lambda$-system. The laser pulse (orange wave) entering the cavity drives the
transition between states $|e\rangle$ and $|u\rangle$ resonantly and
$|g\rangle$ and $|u\rangle$ off-resonantly through the virtual level
$|v\rangle$. The state $|u\rangle$ can decay spontaneously (light orange wave)
with the rate $\gamma$. The initial atomic state
$(|g\rangle+|e\rangle)/\sqrt{2}$ causes entanglement between the light
emerging from the cavity and the atom. Simultaneous interaction of the atom
with a $\pi/2$ microwave pulse (yellow wave) addressing the transition between
$|g\rangle$ and $|e\rangle$ yields the state (6) that can exhibit heralded
quantum non-Gaussian light by measuring the atomic state.

##  2 Quantum non-Gaussianity

Quantum non-Gaussianity refers to overcoming any limitation given by the
mixtures of states $|\psi_{G}\rangle=S(\xi)D(\alpha)|0\rangle$, where the
squeezing operator
$S(\xi)=\exp\left[\xi\left(a^{\dagger}\right)^{2}-\xi^{*}a^{2}\right]$ and the
displacement operator $D(\alpha)=\exp(\alpha a^{\dagger}-\alpha^{*}a)$ acts on
the vacuum. Therefore, the density matrix $\rho$ exhibits quantum non-
Gaussianity formally when

$\rho\neq\int\mathrm{d}^{2}\xi\mathrm{d}^{2}\alpha
P(\xi,\alpha)S(\xi)D(\alpha)|0\rangle\langle
0|D^{\dagger}(\alpha)S^{\dagger}(\xi),$ (1)

where $P(\xi,\alpha)$ stands for a probability density function of the
parameters $\xi$ and $\alpha$. Quantum non-Gaussianity can be recognized under
realistic conditions of current experiments employing the Hanbury Brown and
Twiss detection scheme [5], which is depicted in Fig. 1 (_a_). It consists of
a beam-splitter (BS) and two single-photon avalanche photo-diodes SPAD1 and
SPAD2. A criterion [10, 22] for the quantum non-Gaussianity compares their
response with a threshold defined in terms of success probability $P_{s}$ of a
click on SPAD1 and error probability $P_{e}$ of clicks on both detectors SPAD1
and SPAD2. The quantum non-Gaussianity manifests itself when a state exhibits
the probabilities $P_{s}$ and $P_{e}$ that surpass a threshold given by the
parametric expression

$\displaystyle P_{s}(t)$
$\displaystyle=1-4e^{-\frac{1-t^{2}}{2t(1+3t)}}\sqrt{\frac{t}{3t^{2}+10t+3}}$
(2) $\displaystyle P_{e}(t)$
$\displaystyle=1-8e^{-\frac{1-t^{2}}{2t(1+3t)}}\sqrt{\frac{t}{3t^{2}+10t+3}}$
$\displaystyle+e^{-\frac{(t-1)(t+3)}{2t(1+3t)}}\frac{\sqrt{t}}{1+t},$

where $t\in\left(0,1\right]$ corresponds to a parameter identifying the tuple
$(P_{s},P_{e})$ giving the threshold values. Surpassing the threshold (2)
means a measured state achieves success probability $P_{s}$ greater for a
given error probability $P_{e}$ than any mixture of Gaussian states allows.
Alternatively, modern experimental platforms exploit homodyne tomography to
perform the detection. It relies on the difficult matching of a local
oscillator to the output mode of a cavity with an atom [17] and can be used to
estimate the density matrix in the Fock state basis or the Wigner function in
phase space. With the knowledge of the density matrix, the quantum non-
Gaussianity manifests itself already in two probabilities $p_{1}=\langle
1|\rho|1\rangle$ and $p_{2+}=1-\langle 0|\rho|0\rangle-\langle
1|\rho|1\rangle$ by exceeding the threshold

$p_{1}=2e^{({\frac{t-1}{2t}})}\frac{1-t}{\sqrt{t}(1+t)^{2}};p_{2+}=1-2e^{({\frac{t-1}{2t}})}\frac{1+t^{2}}{\sqrt{t}(1+t)^{2}},$
(3)

where $t\in\left(0,1\right]$ parmetrises the threshold again. This criterion
is also suitable for partial photon-resolving detectors faithfully
distinguishing zero, single and more photons.

Using homodyne tomography, quantum non-Gaussianity is certified by the
negativity of the Wigner function [28]. However, the negativity always
vanishes for $3$ dB of loss, which is too challenging for many experimental
platforms in optics [1]. For large losses, certification of the quantum non-
Gaussianity employing the Wigner function becomes less demanding on the losses
if a constraint on the mean number of photons is involved in the criterion
[32]. However, this requires either quantum tomography using homodyne
detection, challenging for many atomic and solid-state experiments or,
ideally, direct detection resolving all the photon numbers to determine the
mean number of photons without systematic errors. on the contrary, criterion
(2) uses only the most common method of photon auto-correlation measurement
and criterion (3) employs only detectors distinguishing no photons, single
photon and more than one photon. Compared to [32], this largely relaxes the
conditions for conclusive and faithful photon number resolution.
Simultaneously, they are more demanding on the quality of prepared states than
the nonclassicality criterion [16, 22], rejecting all classical waves. In
spite of that, it was demonstrated both theoretically [23] and experimentally
that criteria (2) and (3) are feasible for solid-state sources [38] and atomic
sources [18] when the negativity of the Wigner function disappears completely
due to optical losses. The quantum non-Gaussianity of such states was
recognized as a useful indicator for safety in quantum communication [25].
However, no test has been done yet for the atomic or solid-state emitters in
the cavity.

##  3 Single atom dispersively coupled to a cavity

A prospective platform that enables diverse generation and detection of the
quantum non-Gaussian light from a single atom coupled dispersively to a cavity
is depicted in Fig. 1 (_a_). The BS with very small reflectivity splits a weak
coherent state from the pumping beam and the coherent state enters the cavity
mediating the interaction between light in the cavity mode and an atom. Fig. 1
(_b_) presents relevant atomic energy levels forming the $\Lambda$-scheme.
Whereas the transition between states $|e\rangle$ and $|u\rangle$ is coupled
to the cavity mode, the transition between states $|g\rangle$ and $|u\rangle$
is strongly detuned from the cavity mode. A cavity operating in the strong
coupling regime [35] induces dispersive coupling between the atom and light
inside the cavity, which allows the atom to control the phase of the coherent
state leaving the cavity [9]. If the atom is in the state $|g\rangle$, the
cavity acts as a mirror due to off-resonant coupling with cavity mode and, in
this case, the coherent state is reflected with a $\pi$ phase shift. When the
atom occupies the state $|e\rangle$, the strong coupling with the cavity mode
induces significant detuning of the cavity dressed states from the frequency
of the incoming coherent state. Thus, the reflected state remains a coherent
state, which experiences no phase shift [41, 14]. Preparing the atom initially
in the superposition $(|g\rangle+|e\rangle)/\sqrt{2}$ gives rise to
entanglement between light leaving the cavity and the atomic state [17].
Employing simultaneously a microwave pulse that causes $\pi/2$ rotation
between the states $|g\rangle$ and $|e\rangle$ allows us to generate the state

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{N(\alpha)}}\left[(|\alpha\rangle-|-\alpha\rangle)|e\rangle\right.$
(4) $\displaystyle\left.+(|\alpha\rangle+|-\alpha\rangle)|g\rangle\right],$

where $\alpha$ is the amplitude of the coherent state entering the cavity and
$N(\alpha)$ is such that $\langle\psi|\psi\rangle=1$ [7]. Thus, projection to
the atomic state $|e\rangle$ heralds the state of light

$|\psi_{-}\rangle=\frac{1}{\sqrt{2-2\exp(-|\alpha|^{2})}}(|\alpha\rangle-|-\alpha\rangle),$
(5)

where the superposition of coherent states with different phase has the
capacity to exhibit quantum non-Gaussianity due to negative Wigner function,
in principle, for any $\alpha$.

Figure 2: Quantum non-Gaussianity of light emitted by a single atom in a
cavity in terms of the parameters of the dispersive interaction. The quantum
non-Gaussianity manifests itself up to some threshold intensity $|\alpha|^{2}$
of the incident coherent light that is depicted against the cumulative
parameter $\eta$ (_a_) and cavity output decay rate $\kappa_{r}$ normalized to
atom-cavity coupling rate $g$ (_b_). In both figures, the solid and dashed
lines represent the thresholds $|\alpha|^{2}$ varying with $\kappa_{r}$ for
fixed spontaneous emission rate $\gamma=0.32g$ and cavity damping rate either
$\kappa_{l}=0.05g$ or $\kappa_{l}=0.5g$. The colors differentiate values of
$\kappa_{l}$. Whereas the red and green solid lines present the threshold
imposed by criterion (2), employing responses from SPADs, the same lines in
dashed show the threshold using criterion (3) derived for homodyne detection.
The red and green dashed vertical lines in (_a_) represent the physical
boundary for the parameter $\eta$ when $\kappa_{l}=0.5g$ and
$\kappa_{l}=0.05g$ respectively. In both figures, the thresholds for the
cavity with damping rate $\kappa_{l}=0.05g$ (green lines) are compared with
the boundary for the negativity in the Wigner function (the light blue
region). The cavity with damping rate $\kappa_{l}=0.5g$ does not enable
observation of the negativity. A condition imposed by the criterion of
nonclassicality $P_{s}^{2}-P_{e}>0$ employing SPADs [22] is given by the
yellow and pink solid lines. The inset plot in (_a_) zooms the region of small
$|\alpha|^{2}$ and $\eta$ in logarithmic scale. Briefly, (_a_) demonstrates
that the quantum non-Gaussianity can be detected for much smaller $\eta$ than
the negativity of the Wigner function if the pumping power is low. In
contrast, (b) shows that varying $\kappa_{r}$ strongly affects the maximal
intensity $|\alpha|^{2}$ allowing the quantum non-Gaussianity to be exposed.

A realistic platform suffers from optical leakage from the cavity and
dissipating emission from the atom. Simultaneously, the temporal shape of the
reflected optical pulse gets distorted from the entering pulse when the
amplitude $\alpha$ becomes large [41]. Both these imperfections limit
recognition of the quantum non-Gaussianity not only for atoms but also for
solid-state systems. Further, we consider realistic states induced by the
incident optical pulse with small amplitude $\alpha$. Then, the pulse
distortion can be neglected and the quantum non-Gaussianity gets lost mainly
due to the leakage from the cavity or dissipating emission from the atom. Let
$g$ denote the atom-cavity coupling rate between the transition
$|e\rangle$-$|u\rangle$ and the cavity mode. The excitation to the atomic
state $|u\rangle$ can cause a photon to be spontaneously emitted with rate
$\gamma$. Also, the mirrors effects leakage of the light from the cavity
towards the detector and towards the environment, which happens with rates
$\kappa_{r}$ and $\kappa_{l}$, respectively. Considering these processes in
the dynamics of the light coupled to the atom allows us to establish the
approximate density matrix of the heralded state emerging from the cavity as
[41]

$\displaystyle\rho_{-}=\frac{1}{2(1-e^{-2\eta|\alpha|^{2}})}\left[|\alpha_{e}\rangle\langle\alpha_{e}|+|\alpha_{g}\rangle\langle\alpha_{g}|\right.$
(6) $\displaystyle\
\left.-e^{-2(1-\eta)\eta|\alpha|^{2}}\left(|\alpha_{e}\rangle\langle\alpha_{g}|+|\alpha_{g}\rangle\langle\alpha_{e}|\right)\right],$

where the amplitudes of the coherent states $|\alpha_{e}\rangle$ and
$|\alpha_{g}\rangle$ obey

$\alpha_{g}=\frac{\kappa_{l}-\kappa_{r}}{\kappa_{l}+\kappa_{r}}\alpha,\
\alpha_{e}=\frac{g^{2}+(\kappa_{l}-\kappa_{r})\gamma}{g^{2}+(\kappa_{l}+\kappa_{r})\gamma}\alpha.$
(7)

The cumulative parameter $\eta$ in (6) reads

$\eta=\frac{\kappa_{r}}{\kappa_{l}+\kappa_{r}}\frac{g^{2}}{g^{2}+\gamma(\kappa_{l}+\kappa_{r})}$
(8)

and it determines the purity of the state (6). The strong-coupling regime [35]
signifies $1-\eta\ll 1$, for which $\rho_{-}$ approaches (5) asymptotically.
According to (8), the cavity has inherent imperfections quantified by
$\kappa_{l}$ and $\gamma$, which reduce $\eta$ beyond the strong coupling
regime. The parameter $\kappa_{r}$ has non-trivial impacts on $\eta$. Very
small $\kappa_{r}$ causes the emission of the light towards the SPADs to be
weak since the light is kept inside the cavity. On the other hand, high
$\kappa_{r}$ increases the intensity of the light impinging on the SPADs but
reduces the parameter $\eta$. With fixed $\gamma>0$ and $\kappa_{l}>0$, the
maximal $\eta$ occurs for [15]

$\kappa_{r}=\sqrt{\frac{\kappa_{l}}{\gamma}}\sqrt{g^{2}+\gamma\kappa_{l}},$
(9)

which expresses how the cavity can be engineered to maximize purity of the
state $\rho_{-}$.

The state (5) exhibits a Wigner function with fringes of negative regions,
which proves the quantum non-Gaussianity. However, the negativity is sensitive
to optical leakage from the cavity and dissipative emission from the atom.
Thus, the state $\rho_{-}$ manifests negative Wigner function only for
$\eta>1/2$ (see Appendix for details concerning derivation). Considering
$\kappa_{r}$ in (9), the other parameters allow us to detect the negative
Wigner function only when

$\kappa_{l}\gamma<\frac{g^{2}}{8}.$ (10)

This shows that the product $\kappa_{l}\gamma$ is a key parameter for the
exhibition of the negativity in the Wigner function. It implies that the
negativity requires sufficient suppression of either $\gamma$ or $\kappa_{l}$
(ideally both) with respect to small $g$ and $\kappa_{r}$ approaching the
optimal one given by (9).

The threshold (2) employing a response from SPADs can be surpassed even when
condition (10), which is necessary for the negativity of the Wigner function,
fails. We evaluated the manifestation of the quantum non-Gaussianity by using
the criterion that the threshold (2) yields. It imposes a condition on the
parameter $\eta$ that varies with the intensity $|\alpha|^{2}$ of the incident
coherent state. Fig. 2 presents the results of this numerical analysis for two
different states (6). One of the states is given by $\gamma=0.32g$ and
$\kappa_{l}=0.05g$, which represent the parameters of the performed experiment
in [17]. These parameters fulfill condition (10) allowing us to observe the
negativity of the Wigner function for a proper choice of $\kappa_{r}$. The
other state emerges from the cavity with $\kappa_{l}=0.5g$ and $\gamma=0.32g$.
Due to (10), its Wigner function is positive for any $\kappa_{r}$ and
$|\alpha|^{2}>0$. Fig. 2 (a) presents a condition the quantum non-Gaussianity
imposes on the cumulative parameter $\eta$ with respect to the intensity
$|\alpha|^{2}$ for the fixed parameters $\kappa_{l}$ and $\gamma$. In both
considered states, the criterion (2) reveals the quantum non-Gaussianity even
for situations when the Wigner function is completely positive as the figure
demonstrates. Although the parameter $\eta$ is convenient for identifying the
negativity of the Wigner function, its quantification of the cavity for the
quantum non-Gaussianity is not so intelligible. Thus, the same results are
presented in Fig. 2 (b) in terms of $\kappa_{r}$ (normalized to fixed $g$)
instead of $\eta$. It clearly demonstrates $\kappa_{r}$ reaching the value in
(9) determines the maximal $|\alpha|^{2}$ that allows the state (6) to fulfill
the criterion. The size of $|\alpha|^{2}$ becomes crucial in an analysis of
the experimental error bars as described further. Also, both Figs. 2 (a) and 2
(b) present the difference in the quantum non-Gaussianity manifestation
employing the detection scheme with SPADs and homodyne tomography according to
the criteria (2) and (3) respectively. Both requirements become almost
identical for states with $|\alpha|^{2}\ll 1$. Beyond that regime, the
conditions diversify themselves substantially only for the cavity obeying
(10), in which case the criterion (2) employing SPADs tolerates higher
$|\alpha|^{2}$. Finally, these two figures compare the conditions for the
quantum non-Gaussianity with the nonclassical condition that is less demanding
on $\eta$ for a given $|\alpha|^{2}$ than quantum non-Gaussianity.

Figure 3: The maximal probability $P_{h}\times P_{e}$ that allows recognition
of the quantum non-Gaussianity against the product of parameters
$\kappa_{l}\times\gamma/g^{2}$ and $k_{r}$ achieving the value $\kappa_{r,0}$
from (9) (blue), $75\%$ of $\kappa_{r,0}$ (red) and $50\%$ of $\kappa_{r,0}$
(green). The vertical axis quantifies the probability $P_{h}\times P_{e}$
exhibited by the states where the intensity $|\alpha|^{2}$ achieves its
maximal value that allows certification of the quantum non-Gaussianity.
Further growth of $P_{h}\times P_{e}$ effected by increasing the intensity
$|\alpha|^{2}$ of the pumping beam causes the failure in the recognition. The
light blue stripe depicts cases with $\kappa_{l}\gamma/g^{2}<1/8$, which
enables detection of the negativity in the Wigner function when $\kappa_{r}$
obeys (9).

Criterion (2) imposes no fundamental limit on the parameter $\eta$, i.e. any
$\eta>0$ allows us to observe the quantum non-Gaussianity if $|\alpha|^{2}$ is
sufficiently small. However, the intensity $|\alpha|^{2}$ influences the
experimental time in recognition of the quantum non-Gaussianity since
uncertainty in experimental estimation of the probabilities $P_{s}$ and
$P_{e}$ depends on a number of click events, which grows with $|\alpha|^{2}$.
Simultaneously, $|\alpha|^{2}$ affects the probability $P_{h}$ of successfully
heralding the state $\rho_{-}$, which is given by projection on the atomic
state $|e\rangle$, according to $P_{h}=1-e^{-2\eta|\alpha|^{2}}$. This
suggests the appropriate intensity $|\alpha|^{2}$ is given by the trade-off
between the overall measurement time and requirement imposed by the
suppression of the threshold (2). Fig. 3 presents the rate $P_{h}\times P_{e}$
for states achieved from the maximal intensity $|\alpha|^{2}$ of the pumping
beam that the quantum non-Gaussianity tolerates. Note, the rate is always
lower in real detection of the quantum non-Gaussianity since it has to be
arranged by $|\alpha|^{2}$ such that the measured probabilities $P_{s}$ and
$P_{e}$ surpass the threshold even within their error bars.

##  4 Realistic cavity with noise contributions

Realistic systems can be further affected by small noise contributions from
emitter, stray light or measurement dark counts. We test our prediction under
noise impacts by considering multiphoton effects of the displacement operator
$D(\nu)=\exp(\nu a^{\dagger}-\nu^{*}a)$ with random amplitude $\nu$ on the
emerging light. Thus, the density matrix deteriorated by the noise obtains

$\rho=\int\mathrm{d}^{2}\nu P(\nu)D(\nu)\rho_{-}D^{\dagger}(\nu)$ (11)

with $P(\nu)$ being a probability density function identifying the stochastic
processes affecting the amplitude $\nu$. We can analyse two typically relevant
stochastic processes. In the first case only the phase of $\nu$ undergoes the
fluctuation, i.e. $P_{P}(\nu)=\delta(|\nu|^{2}-\bar{n})/(2\pi)$. In the second
case, we allow for
$P_{BE}(\nu)=\exp\left\\{-\left[(\nu+\nu^{*})^{2}+(\nu-\nu^{*})^{2}\right]/(4\bar{n})\right\\}/(\pi\bar{n})$,
and therefore the amplitude $\nu$ randomly changes both the phase and its
size. The model of the noise in (11) with the probability density function
$P_{P}(\nu)$ corresponds to the Poissonian noise and the model induced by the
probability density function $P_{BE}(\nu)$ describes the Bose-Einstein noise.
Appendix C presents how the Poissonian noise and the Bose-Einstein noise
influence the manifestation of quantum non-Gaussianity for specific values of
$\bar{n}$. In the region of very weak pumping power $|\alpha|^{2}\ll 1$ and
small mean number of noisy photons $\bar{n}\ll 1$, the probabilities obey the
approximation

$\displaystyle P_{s}$ $\displaystyle\approx\frac{\eta}{2}$ (12) $\displaystyle
P_{e}$
$\displaystyle\approx\frac{\eta\left[(\eta-1)g^{2}+2\eta\gamma\kappa\right]^{2}}{g^{4}}|\alpha|^{2}+\eta\bar{n}$

for both the Poissonian noise and the thermal noise. Since the criterion
becomes approximated by $P_{s}^{3}>P_{e}/4$ for the surpassed probability
$P_{e}$, the quantum non-Gaussianity imposes an approximate condition

$|\alpha|^{2}+\bar{n}<\frac{\eta^{2}}{2},$ (13)

which expresses the limitation on $|\alpha|^{2}$ involving the impacts of the
noise. If $\bar{n}\geq\eta^{2}/2$ quantum non-Gaussianity is recognized for no
values of $|\alpha|^{2}$. Let us recall that (11) introduces a model of
narrow-band noise, which contributes to the mode occupied by the light from
the cavity. A model of broad-band (incoherent) Poissonian noise, which
contributes independently to many modes, gives rise to the less demanding
approximate condition $2|\alpha|^{2}+\bar{n}<\eta^{2}$ with $\bar{n}$ being
the mean number of noisy photons.

##  5 Conclusion and Outlook

To conclude, we proved that light emitted from a dispersively coupled atom or
solid state emitter inside a cavity manifests quantum non-Gaussianity much
more broadly under realistic conditions of current experiments. The only
limitation for the recognition is a background noise deteriorating the
measured light. High intensity of the pumping coherent beam, which prevents
the recognition as well, can always be reduced with the cost of decreasing the
overall rates. Simultaneously, this detection can be employed to expose the
quantum non-Gaussianity of light from a quantum dot inside a micropillar
cavity [30] or a photonic crystal [27]. Using more atoms in the cavity, future
higher Fock state preparation can be evaluated using the existing hierarchy of
quantum non-Gaussianity [24].

R. F. and L. L. acknowledge project 21-13265X of the Czech Science Foundation.
J. K. V. acknowledges the MSMT of the Czech Republic and Horizon 2020
Framework Programme (731473, project 8C20002 ShoQC). We acknolwedge the
support of H2020 Spreading Excellence and Widening Participation (951737,
NONGAUSS).

##  Appendix A Appendix: Criterion of quantum non-Gaussianity

The direct experimental recognition of quantum non-Gaussianity exploits the
Hanbury Brown and Twiss configuration as depicted in Fig. 1 of the main text.
It allows us to determine the probability $P_{s}$ of a click event on the
detector SPAD1 and the probability $P_{e}$ of double-click happening
simultaneously on both the detectors SPAD1 and SPAD2. The criterion rejects
directly the detectors’ response on all the mixtures of Gaussian states in
this specific scheme. It is represented by establishing a witness

$F_{a}(\rho)=P_{s}+aP_{e},$ (14)

which exceeds a threshold value given for some freely chosen parameter $a$.
The threshold value $F(a)$ corresponds to the maximum

$F(a)=\max_{\alpha,\xi}F_{a}(|\psi_{G}\rangle)$ (15)

over the parameters $\alpha$ and $\xi$ identifying any pure Gaussian state
$|\psi_{G}\rangle=S(\xi)D(\alpha)|0\rangle$ with $S(\xi)$ and $D(\alpha)$
being the squeezing and displacement operator. Importantly, the function
$F(a)$ covers all mixtures of Gaussian states as well, i.e.

$F_{a}(\rho_{G})\leq F(a)$ (16)

with $\rho_{G}$ being an incoherent mixture of pure states $|\psi_{G}\rangle$,
and therefore violation of the inequality in (16) certifies the quantum non-
Gaussianity.

To express formulas for probabilities $P_{s}$ and $P_{e}$ that the state
$|\psi_{G}\rangle$ exhibits, we introduce auxiliary probabilities $P_{0}$ and
$P_{00}$ referring to the probability of no-click on SPAD1 and to the
probability of no-click on neither detector SPAD1 and SPAD2, respectively.
These introduced probabilities obey the identities $P_{s}=1-P_{0}$ and
$P_{e}=1-2P_{0}+P_{00}$. Then, the state $|\psi_{G}\rangle$ yields

$\displaystyle P_{0}$
$\displaystyle=4\sqrt{\frac{V}{3V^{2}+10V+3}}e^{-\frac{|\alpha|^{2}\left[1+6V+V^{2}+(1-V^{2})\cos
2\phi\right]}{2(3+V)(1+3V)}}$ $\displaystyle P_{00}$
$\displaystyle=2\frac{\sqrt{V}}{1+V}e^{-\frac{|\alpha|^{2}\left[\cos^{2}\phi+V\sin^{2}\phi\right]}{1+V}},$

where $V=\exp(-2|\xi|)$ and $\phi$ is difference in phases of the squeezing
$\xi$ and displacement $\alpha$. All these steps allow us to express
$F_{a}(|\psi_{G}\rangle)$ as a function of the parameters $|\alpha|^{2}$, $V$
and $\phi$. Its maximum occurs when the parameters fulfill

$\displaystyle\partial_{|\alpha|^{2}}P_{s}\partial_{\phi}P_{e}$
$\displaystyle=\partial_{|\alpha|^{2}}P_{e}\partial_{\phi}P_{s}$
$\displaystyle\partial_{|\alpha|^{2}}P_{s}\partial_{V}P_{e}$
$\displaystyle=\partial_{|\alpha|^{2}}P_{e}\partial_{V}P_{s}.$

These equations provide a solution $\phi=0$ and
$|\alpha|^{2}=(V+3)(1-V^{2})/\left[2V(1+3V)\right]$ with
$V\in\left(0,1\right]$. Thus, all mixtures of Gaussian states $\rho_{G}$
satisfy

$\displaystyle F_{a}(\rho_{G})$ $\displaystyle\
\leq\max_{V}\left\\{1-P_{0}(V)\right.$
$\displaystyle+\left.a\left[1-2P_{0}(V)+P_{00}(V)\right]\right\\}$

with

$\displaystyle P_{0}(V)$
$\displaystyle=4e^{\frac{-1+V^{2}}{2V(1+3V)}}\sqrt{\frac{V}{3+10V+3V^{2}}}$
(17) $\displaystyle P_{00}(V)$
$\displaystyle=2e^{\frac{(-1+V)(1+3V)}{2V(1+3V)}}\frac{\sqrt{V}}{1+V}.$

Recall, the quantum non-Gaussianity is witnessed by such a parameter $a$ that
implies surpassing the right hand side of (A) by $F_{a}$ of a measured state.
This $a$ can be always found if the measured state yields probabilities above
a threshold in the space of probabilities $P_{s}$ and $P_{e}$. Its parametric
form gains

$\displaystyle P_{s}$ $\displaystyle=1-P_{0}(V)$ (18) $\displaystyle P_{e}$
$\displaystyle=1-2P_{0}(V)+P_{00}(V),$

where $V\in\left(0,1\right]$ plays the role of a parameter.

##  Appendix B Appendix: A single atom dispersively coupled to a cavity

Let pulsed laser light enter a cavity where an atom with energy levels forming
the $\Lambda$-system is coupled to the cavity mode. Under assumptions
mentioned in [41], the heralded light leaving the cavity approaches the
density matrix [17]

$\displaystyle\
\rho_{-}=\frac{1}{2(1-e^{-2\eta|\alpha|^{2}})}\left[|\alpha_{e}\rangle\langle\alpha_{e}|+|\alpha_{g}\rangle\langle\alpha_{g}|\right.$
(19) $\displaystyle\
\left.-e^{-2(1-\eta)\eta|\alpha|^{2}}\left(|\alpha_{e}\rangle\langle\alpha_{g}|+|\alpha_{g}\rangle\langle\alpha_{e}|\right)\right],$

where $|\alpha_{e}\rangle$ and $|\alpha_{g}\rangle$ are coherent states with
the amplitudes

$\alpha_{g}=\frac{\kappa_{l}-\kappa_{r}}{\kappa_{l}+\kappa_{r}}\alpha,\
\alpha_{e}=\frac{g^{2}+(\kappa_{l}-\kappa_{r})\gamma}{g^{2}+(\kappa_{l}+\kappa_{r})\gamma}\alpha.$
(20)

with $\alpha$ being the amplitude of the incoming coherent state. The
parameter $\eta$ in (19) is defined by the identity

$\eta|\alpha|=\frac{1}{2}|\alpha_{e}-\alpha_{g}|,$ (21)

which gives rise to

$\eta=\frac{\kappa_{r}}{\kappa_{l}+\kappa_{r}}\frac{g^{2}}{g^{2}+\gamma(\kappa_{l}+\kappa_{r})}.$
(22)

It shows how the cavity parameters reduce the difference (21). A cavity
operating in the strong coupling regime when $g\gg\gamma$ and
$g\gg\kappa_{r}\gg\kappa_{l}$ allows us to approach $\eta=1$. Beyond that
regime, the cumulative parameter $\eta$ is always reduced below one, which
affects the purity of the state $\rho_{-}$.

###  B.1 Wigner function

The Wigner function $W(\alpha,\alpha^{*})$ enables a recognition of the
quantum non-Gaussianity when it exhibits negativity. To determine its value
for the states $\rho_{-}$, we consider the relation

$W(\beta,\beta^{*})=\frac{1}{2\pi}\langle
D^{\dagger}(\beta)(-1)^{a^{\dagger}a}D(\beta)\rangle,$ (23)

where $a^{\dagger}$ and $a$ are creation and annihilation operator and
$D(\beta)=\exp\left(\beta a^{\dagger}-\beta^{*}a\right)$ is the displacement
operator [36]. Let us introduce the state

$\widetilde{\rho}_{-}=D\left[-(\alpha_{e}+\alpha_{g})/2\right]\rho_{-}D^{\dagger}\left[-(\alpha_{e}+\alpha_{g})/2\right].$
(24)

The displacement operator in (24) shifts the state $\rho_{-}$ such that the
negative region (if there is any) is at the origin. Thus, employing the
identities (6) and (20) yields

$\displaystyle\ \widetilde{\rho}_{-}=$ (25) $\displaystyle\
\frac{1}{2(1-e^{-2\eta|\alpha|^{2}})}\left[|\eta\alpha\rangle\langle\eta\alpha|+|-\eta\alpha\rangle\langle\eta\alpha|\right.$
$\displaystyle-\left.e^{-2(1-\eta)\eta|\alpha|^{2}}\left(|\eta\alpha\rangle\langle-\eta\alpha|+|-\eta\alpha\rangle\langle-\eta\alpha|\right)\right].$

The state $\widetilde{\rho}_{-}$ is formally identical to the state
$|\psi\rangle\propto(|\alpha\rangle-|-\alpha\rangle)$ that undergoes losses
quantified by the transmission $\eta$. Due to that, the negativity remains
only for $\eta>1/2$ [20].

###  B.2 Criterion from Ref.[12]

The certification of the quantum non-Gaussianity for states having completely
positive Wigner function is also possible employing the criterion combining
the value of the Wigner function at origin
$W(0,0)=\mbox{Tr}\left[\rho(-1)^{a^{\dagger}a}\right]$ with the mean number of
photons $\bar{N}=\langle a^{\dagger}a\rangle$. Explicitly, the criterion
recognizes the quantum non-Gaussianity when

$W(0,0)<\frac{1}{2\pi}e^{-2\bar{N}(1+\bar{N})}.$ (26)

The condition converges to the requirement on the negativity for large
$\bar{N}$. However, the quantum non-Gaussianity can manifest itself even for
$W(0,0)$ arbitrarily close to $\frac{1}{2\pi}$ (a value achieved by the
vacuum) when $\bar{N}$ is sufficiently suppressed. The criterion (26) applied
on the state (19) reveals that the quantum non-Gaussianity can manifests
itself for arbitrarily small $\eta>0$. Fig. 4 compares the criterion (3) in
the main text with the criterion (26) with respect to their capability to
recognize the quantum non-Gaussianity of the state (19). Both criteria impose
similar demands on the state (19). Let us emphasise, however, that the
criteria are very different concerning the requirement for detection.
Criterion (3) requires only faithful resolution of zero, single and more
photons, whereas (26) needs reliable and faithful photon-number resolving
detection.

Figure 4: Comparing thresholds in terms of intensity of pumping $|\alpha|^{2}$
and the parameter $\eta$ that enable certifying the quantum non-Gaussianity of
the state (19). The thresholds imply from obeying the criterion (3) in the
main text (yellow) or from satisfying the criterion (26) (blue). The
spontaneous emission rate $\gamma$ obtains $\gamma=0.32$ and the values of the
parameter $\kappa_{l}$ are fixed to $\kappa_{l}=0.05$ and $\kappa_{l}=0.5$.

##  Appendix C Appendix: Effects of the noise

The density matrix (6) represents an emitted light, which is not affected by
noise. To include the noise contributions in the description, we allow for a
scenario when the light is influenced by the action of a displacement operator
with a fluctuating amplitude. Thus, the noise affects the density matrix
according to

$\rho=\int\mathrm{d}^{2}\beta P(\beta)D(\beta)\rho_{-}D^{\dagger}(\beta),$
(27)

where $D(\beta)=\exp(\beta a^{\dagger}-\beta^{*}a)$ is the displacement
operator. The amplitude $\beta$ undergoes random fluctuations that the
probability density function $P(\beta)$ determines. Further, we assume two
fundamental stochastic processes for which the probability density functions
yield

$\displaystyle P_{P}(\beta)$
$\displaystyle=\frac{1}{2\pi}\delta\left(|\beta|-\sqrt{\bar{n}}\right)$ (28)
$\displaystyle P_{BE}(\beta)$
$\displaystyle=\frac{1}{\pi\bar{n}}e^{-\frac{|\beta|^{2}}{\bar{n}}},$

where $\bar{n}$ is mean number of the noisy photons. When $P_{P}(\beta)$ is
taken into account, the amplitude $\beta$ experiences random fluctuation in
phase only. On the contrary, $P_{BE}(\beta)$ represents fluctuation in both
the phase and the intensity $|\beta|^{2}$. Let us call the noise identified by
the probability density functions $P_{P}(\beta)$ Poissonian noise and the
noise given by $P_{BE}(\beta)$ Bose-Einstein noise according to the
distribution of noisy photons that are produced when the stochastic
displacement affects the vacuum states.

Figure 5: Thresholds of the quantum non-Gaussianity and nonclassicality of
noisy states defined by (27) and (28). The thresholds are shown in terms of
pumping intensity $|\alpha|^{2}$ and either cumulative parameter $\eta$
(_a_)-(_d_) or the parameter $\kappa_{r}$ normalized to $g$ (_e_)-(_h_). The
quantum non-Gaussianity and nonclassicality are certified for $|\alpha|^{2}$
below a respective threshold. Each sub-figure allows for different mean number
of noisy photons $\bar{n}$. Whereas the blue and green solid lines represent
the Poissonian noise deteriorating the light, the red and golden solid lines
correspond to the Bose-Einstein noise. Also, the colors differentiate the
parameter $\kappa_{l}$, which obtain either $\kappa_{l}=0.05g$ (red and blue)
or $\kappa_{l}=0.5g$ (green and golden) as labeled above the plots. The
spontaneous emission rate $\gamma$ is set to $\gamma=0.32g$ for each sub-
figure. Finally, a line of a given color is plotted as solid and dashed, which
allows comparison of the thresholds for the quantum non-Gaussianity (solid)
with thresholds for the nonclassicality (dashed). Sub-figures (_c_), (_d_) and
(_g_), (_h_) do not present the solid golden and green lines since they depict
thresholds of states where the noise contributions effects lost of the quantum
non-Gaussianity when $\kappa_{l}=0.5$. In sub-figures (_a_)-(_d_), the
horizontal red and golden dashed lines represent the physical boundary for
$\eta$ when $\kappa_{l}=0.05g$ and $\kappa_{l}=0.5g$, respectively.

To obtain formulas for the click probabilities, let us deal with the state
$\rho_{-,\beta}=D(\beta)\rho_{-}D^{\dagger}(\beta)$ before considering the
stochastic processes. Its density matrix works out to

$\displaystyle\
\rho_{-,\beta}=\frac{1}{2(1-e^{-2\eta|\alpha|^{2}})}\left\\{|\bar{\alpha}_{e,\beta}\rangle\langle\bar{\alpha}_{e,\beta}|+|\bar{\alpha}_{g,\beta}\rangle\langle\bar{\alpha}_{g,\beta}|\right.$
(29) $\displaystyle\
-e^{-2(1-\eta)\eta|\alpha|^{2}}\times\left[e^{\eta(\beta\alpha^{*}-\beta^{*}\alpha)}|\bar{\alpha}_{e,\beta}\rangle\langle\bar{\alpha}_{g,\beta}|\right.$
$\displaystyle\
\left.\left.+e^{\eta(\beta^{*}\alpha-\beta\alpha^{*})}|\bar{\alpha}_{g,\beta}\rangle\langle\bar{\alpha}_{e,\beta}|\right]\right\\},$

where $\bar{\alpha}_{e,\beta}=\alpha_{e}+\beta$ and
$\bar{\alpha}_{g,\beta}=\alpha_{g}+\beta$. Let the state $\rho_{-,\beta}$ go
through a balanced BS. Outgoing state $\bar{\rho}_{-,\beta}$ in the
transmitted mode is given by

$\displaystyle\ \bar{\rho}_{-,\beta}=$ (30) $\displaystyle\
\frac{1}{2(1-e^{-2\eta|\alpha|^{2}})}\left\\{|\bar{\alpha}_{e,\beta}/\sqrt{2}\rangle\langle\bar{\alpha}_{e,\beta}/\sqrt{2}|\right.$
$\displaystyle\
+|\bar{\alpha}_{g,\beta}/\sqrt{2}\rangle\langle\bar{\alpha}_{g,\beta}/\sqrt{2}|$
$\displaystyle\
-e^{-2(1-\eta)\eta|\alpha|^{2}-\eta^{2}|\alpha|^{2}+\left[\beta^{*}(\alpha_{g}+\alpha_{e})+\beta(\alpha_{g}^{*}+\alpha_{e}^{*})\right]/4}$
$\displaystyle\
\left.\times\left[e^{(\alpha^{*}_{e}\beta+\alpha_{g}\beta^{*})/2+\eta(\beta\alpha^{*}-\beta^{*}\alpha)}|\bar{\alpha}_{e,\beta}/\sqrt{2}\rangle\langle\bar{\alpha}_{g,\beta}/\sqrt{2}|\right.\right.$
$\displaystyle\
+e^{(\alpha_{e}\beta^{*}+\alpha_{g}^{*}\beta)/2+\eta(\beta^{*}\alpha-\beta\alpha^{*})}$
$\displaystyle\
\left.\left.\times|\bar{\alpha}_{g,\beta}/\sqrt{2}\rangle\langle\bar{\alpha}_{e,\beta}/\sqrt{2}|\right]\right\\}.$

The density matrices of the incident state (29) and transmitted state (30)
allows us to determine how $\beta$ influences the click probability
$P_{s}(\beta)=1-\langle 0|\bar{\rho}_{-,\beta}|0\rangle$ and double click
$P_{e}(\beta)=1-2\langle 0|\bar{\rho}_{-,\beta}|0\rangle+\langle
0|\rho_{-,\beta}|0\rangle$. Thus, the click probability $P_{s}$ and double
click probability $P_{c}$ of the noisy state (27) read

$\displaystyle P_{s}=\int\mathrm{d}^{2}\beta P(\beta)P_{s}(\beta)$ (31)
$\displaystyle P_{e}=\int\mathrm{d}^{2}\beta P(\beta)P_{e}(\beta)$

with $P(\beta)$ being some probability density function from (28). We provide
only the approximate formulas for $P_{s}$ and $P_{e}$ appropriate for states
with $|\alpha|^{2}\ll 1$ and $\bar{n}\ll 1$. Under these assumptions, they get
simplified to

$\displaystyle P_{s}$ $\displaystyle=\frac{\eta}{2}$ (32) $\displaystyle
P_{e}$
$\displaystyle=\frac{\eta\left[(\eta-1)g^{2}+2\eta\gamma\kappa\right]^{2}}{g^{4}}|\alpha|^{2}+\eta\bar{n},$

for both the considered noise. Fig. 5 depicts when the noise contribution
enables recognition of the quantum non-Gaussianity for some specific settings
of the cavity parameters.

The model of noisy states (27) can be modified to a model in which the noise
contributes as independent background noise, i.e. the density matrix reads
$\rho=\rho_{-}\otimes\rho_{\bar{n}}$, where $\rho_{\bar{n}}$ obeys the
Poissonian distribution of photons with mean number of photons $\bar{n}$. In
this case, the click probabilities in the limit of $|\alpha|^{2}\ll 1$ and
$\bar{n}\ll 1$ works out to be

$\displaystyle P_{s}$ $\displaystyle=\frac{\eta}{2}$ (33) $\displaystyle
P_{c}$
$\displaystyle=\frac{\eta\left[(\eta-1)g^{2}+2\eta\gamma\kappa\right]^{2}}{g^{4}}|\alpha|^{2}+\frac{\eta\bar{n}}{2}.$

Comparing (32) with (33) suggests that the background noise has less impacts
on the probability $P_{c}$. Consequently, the quantum non-Gaussianity is more
sensitive to contributions of noise in (27) than to the contributions of the
background noise.

## References

  * Aharonovich et al. [2016] I. Aharonovich, D. Englund, and M. Toth.  Solid-state single-photon emitters.  _Nature Photonics_ , 10(10):631–641, 2016.  doi: 10.1038/nphoton.2016.186.
  * Andersen et al. [2015] U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa.  Hybrid discrete- and continuous-variable quantum information.  _Nature Physics_ , 11(9):713–719, 2015.  doi: 10.1038/nphys3410.
  * Arcari et al. [2014] M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. Lee, J. Song, S. Stobbe, and P. Lodahl.  Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide.  _Physical Review Letters_ , 113(9):093603, 2014\.  doi: 10.1103/physrevlett.113.093603.
  * Aspuru-Guzik and Walther [2012] A. Aspuru-Guzik and P. Walther.  Photonic quantum simulators.  _Nature Physics_ , 8(4):285–291, 2012.  doi: 10.1038/nphys2253.
  * Brown and Twiss [1954] R. H. Brown and R. Twiss.  LXXIV. a new type of interferometer for use in radio astronomy.  _The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science_ , 45(366):663–682, 1954.  doi: 10.1080/14786440708520475.
  * Chu et al. [2016] X.-L. Chu, S. Götzinger, and V. Sandoghdar.  A single molecule as a high-fidelity photon gun for producing intensity-squeezed light.  _Nature Photonics_ , 11(1):58–62, 2016.  doi: 10.1038/nphoton.2016.236.
  * Daiss et al. [2019] S. Daiss, S. Welte, B. Hacker, L. Li, and G. Rempe.  Single-photon distillation via a photonic parity measurement using cavity QED.  _Physical Review Letters_ , 122(13):133603, 2019\.  doi: 10.1103/physrevlett.122.133603.
  * Ding et al. [2016] X. Ding, Y. He, Z.-C. Duan, N. Gregersen, M.-C. Chen, S. Unsleber, S. Maier, C. Schneider, M. Kamp, S. Höfling, C.-Y. Lu, and J.-W. Pan.  On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar.  _Physical Review Letters_ , 116(2):020401, 2016\.  doi: 10.1103/physrevlett.116.020401.
  * Duan and Kimble [2004] L.-M. Duan and H. J. Kimble.  Scalable photonic quantum computation through cavity-assisted interactions.  _Physical Review Letters_ , 92(12):127902, 2004\.  doi: 10.1103/physrevlett.92.127902.
  * Filip and Mišta [2011] R. Filip and L. Mišta.  Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states.  _Physical Review Letters_ , 106(20):200401, 2011\.  doi: 10.1103/physrevlett.106.200401.
  * Flühmann et al. [2019] C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home.  Encoding a qubit in a trapped-ion mechanical oscillator.  _Nature_ , 566(7745):513–517, 2019.  doi: 10.1038/s41586-019-0960-6.
  * Genoni et al. [2013] M. G. Genoni, M. L. Palma, T. Tufarelli, S. Olivares, M. S. Kim, and M. G. A. Paris.  Detecting quantum non-Gaussianity via the Wigner function.  _Physical Review A_ , 87(6), jun 2013.  doi: 10.1103/physreva.87.062104.
  * Ginés et al. [2021] L. Ginés, C. Pepe, J. Gonzales, N. Gregersen, S. Höfling, C. Schneider, and A. Predojević.  Time-bin entangled photon pairs from quantum dots embedded in a self-aligned cavity.  _Optics Express_ , 29(3):4174, 2021.  doi: 10.1364/oe.411021.
  * Goto and Ichimura [2005] H. Goto and K. Ichimura.  Quantum trajectory simulation of controlled phase-flip gates using the vacuum Rabi splitting.  _Physical Review A_ , 72(5):054301, 2005.  doi: 10.1103/physreva.72.054301.
  * Goto et al. [2019] H. Goto, S. Mizukami, Y. Tokunaga, and T. Aoki.  Figure of merit for single-photon generation based on cavity quantum electrodynamics.  _Physical Review A_ , 99(5):053843, 2019.  doi: 10.1103/physreva.99.053843.
  * Grangier et al. [1986] P. Grangier, G. Roger, and A. Aspect.  Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences.  _Europhysics Letters (EPL)_ , 1(4):173–179, 1986.  doi: 10.1209/0295-5075/1/4/004.
  * Hacker et al. [2019] B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li, and G. Rempe.  Deterministic creation of entangled atom–light Schrödinger-cat states.  _Nature Photonics_ , 13(2):110–115, 2019.  doi: 10.1038/s41566-018-0339-5.
  * Higginbottom et al. [2016] D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt.  Pure single photons from a trapped atom source.  _New Journal of Physics_ , 18(9):093038, 2016\.  doi: 10.1088/1367-2630/18/9/093038.
  * Huber et al. [2017] D. Huber, M. Reindl, Y. Huo, H. Huang, J. S. Wildmann, O. G. Schmidt, A. Rastelli, and R. Trotta.  Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots.  _Nature Communications_ , 8(1), 2017.  doi: 10.1038/ncomms15506.
  * Hudson [1974] R. Hudson.  When is the wigner quasi-probability density non-negative?  _Reports on Mathematical Physics_ , 6(2):249–252, 1974.  doi: 10.1016/0034-4877(74)90007-x.
  * Huh et al. [2015] J. Huh, G. G. Guerreschi, B. Peropadre, J. R. McClean, and A. Aspuru-Guzik.  Boson sampling for molecular vibronic spectra.  _Nature Photonics_ , 9(9):615–620, 2015.  doi: 10.1038/nphoton.2015.153.
  * Lachman and Filip [2013] L. Lachman and R. Filip.  Robustness of quantum nonclassicality and non-Gaussianity of single-photon states in attenuating channels.  _Physical Review A_ , 88(6):063841, 2013.  doi: 10.1103/physreva.88.063841.
  * Lachman and Filip [2016] L. Lachman and R. Filip.  Quantum non-Gaussianity from a large ensemble of single photon emitters.  _Optics Express_ , 24(24):27352, 2016.  doi: 10.1364/oe.24.027352.
  * Lachman et al. [2019] L. Lachman, I. Straka, J. Hloušek, M. Ježek, and R. Filip.  Faithful hierarchy of genuine n -photon quantum non-Gaussian light.  _Physical Review Letters_ , 123(4), 2019.  doi: 10.1103/physrevlett.123.043601.
  * Lasota et al. [2017] M. Lasota, R. Filip, and V. C. Usenko.  Sufficiency of quantum non-Gaussianity for discrete-variable quantum key distribution over noisy channels.  _Physical Review A_ , 96(1):012301, 2017.  doi: 10.1103/physreva.96.012301.
  * Lodahl et al. [2004] P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos.  Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals.  _Nature_ , 430(7000):654–657, 2004.  doi: 10.1038/nature02772.
  * Lodahl et al. [2015] P. Lodahl, S. Mahmoodian, and S. Stobbe.  Interfacing single photons and single quantum dots with photonic nanostructures.  _Reviews of Modern Physics_ , 87(2):347–400, 2015\.  doi: 10.1103/revmodphys.87.347.
  * Lvovsky and Raymer [2009] A. I. Lvovsky and M. G. Raymer.  Continuous-variable optical quantum-state tomography.  _Reviews of Modern Physics_ , 81(1):299–332, 2009\.  doi: 10.1103/revmodphys.81.299.
  * Morin et al. [2019] O. Morin, M. Körber, S. Langenfeld, and G. Rempe.  Deterministic shaping and reshaping of single-photon temporal wave functions.  _Physical Review Letters_ , 123(13):133602, 2019\.  doi: 10.1103/physrevlett.123.133602.
  * Najer et al. [2019] 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(7784):622–627, 2019.  doi: 10.1038/s41586-019-1709-y.
  * O'Brien et al. [2009] J. L. O'Brien, A. Furusawa, and J. Vučković.  Photonic quantum technologies.  _Nature Photonics_ , 3(12):687–695, 2009.  doi: 10.1038/nphoton.2009.229.
  * Palma et al. [2014] M. L. Palma, J. Stammers, M. G. Genoni, T. Tufarelli, S. Olivares, M. S. Kim, and M. G. A. Paris.  Detecting quantum non-Gaussianity of noisy Schrödinger cat states.  _Physica Scripta_ , T160:014035, apr 2014.  doi: 10.1088/0031-8949/2014/t160/014035.
  * Peter et al. [2005] E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch.  Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity.  _Physical Review Letters_ , 95(6):067401, 2005\.  doi: 10.1103/physrevlett.95.067401.
  * Prilmüller et al. [2018] M. Prilmüller, T. Huber, M. Müller, P. Michler, G. Weihs, and A. Predojević.  Hyperentanglement of photons emitted by a quantum dot.  _Physical Review Letters_ , 121(11):110503, 2018\.  doi: 10.1103/physrevlett.121.110503.
  * Reiserer and Rempe [2015] A. Reiserer and G. Rempe.  Cavity-based quantum networks with single atoms and optical photons.  _Reviews of Modern Physics_ , 87(4):1379–1418, 2015.  doi: 10.1103/revmodphys.87.1379.
  * Royer [1977] A. Royer.  Wigner function as the expectation value of a parity operator.  _Physical Review A_ , 15(2):449–450, feb 1977\.  doi: 10.1103/physreva.15.449.
  * Somaschi et al. [2016] N. Somaschi, V. Giesz, L. D. Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaítre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart.  Near-optimal single-photon sources in the solid state.  _Nature Photonics_ , 10(5):340–345, 2016.  doi: 10.1038/nphoton.2016.23.
  * Straka et al. [2014] I. Straka, A. Predojević, T. Huber, L. Lachman, L. Butschek, M. Miková, M. Mičuda, G. S. Solomon, G. Weihs, M. Ježek, and R. Filip.  Quantum non-Gaussian depth of single-photon states.  _Physical Review Letters_ , 113(22):223603, 2014\.  doi: 10.1103/physrevlett.113.223603.
  * Thomas and Senellart [2021] S. Thomas and P. Senellart.  The race for the ideal single-photon source is on.  _Nature Nanotechnology_ , 16(4):367–368, jan 2021\.  doi: 10.1038/s41565-021-00851-1.
  * Wakui et al. [2007] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki.  Photon subtracted squeezed states generated with periodically poled KTiOPO_4.  _Optics Express_ , 15(6):3568, 2007.  doi: 10.1364/oe.15.003568.
  * Wang and Duan [2005] B. Wang and L.-M. Duan.  Engineering superpositions of coherent states in coherent optical pulses through cavity-assisted interaction.  _Physical Review A_ , 72(2):022320, 2005.  doi: 10.1103/physreva.72.022320.
  * Wang et al. [2021] X.-J. Wang, S.-J. Yang, P.-F. Sun, B. Jing, J. Li, M.-T. Zhou, X.-H. Bao, and J.-W. Pan.  Cavity-enhanced atom-photon entanglement with subsecond lifetime.  _Physical Review Letters_ , 126(9):090501, 2021\.  doi: 10.1103/physrevlett.126.090501.
  * Wolf et al. [2019] F. Wolf, C. Shi, J. C. Heip, M. Gessner, L. Pezzè, A. Smerzi, M. Schulte, K. Hammerer, and P. O. Schmidt.  Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions.  _Nature Communications_ , 10(1), 2019.  doi: 10.1038/s41467-019-10576-4.
  * Zapletal and Filip [2017] P. Zapletal and R. Filip.  Multi-copy quantifiers for single-photon states.  _Scientific Reports_ , 7(1), 2017.  doi: 10.1038/s41598-017-01333-y.