WGMwgm QEqe EPep PMSpms BECbec DEde

[orcid=0000-0002-1488-5829] [1]

[orcid=0000-0003-0365-4731] [1] [orcid=0000-0003-2596-8264] [1] [1]
[orcid=0000-0001-6138-8633] [2] [orcid=0000-0001-9854-8100] [2]
[orcid=0000-0003-2813-8469] [2] [orcid=0000-0001-7520-4364] [2]
[orcid=0000-0002-1622-9761] [2] [orcid=0000-0002-4534-7484] [2]

1]organization=University of Groningen, Groningen, Netherlands
2]organization=The Australian National University, Canberra, Australia
3]organization=Insight Edge Inc., Tokyo, Japan 4]organization=Cavendish
Laboratory, University of Cambridge, Cambridge, United Kingdom
5]organization=Georgia Institute of Technology, Atlanta, USA
6]organization=ETH Zürich, Zürich, Switzerland 7]organization=University of
Oxford, UK 8]organization=George Mason University, Fairfax, USA
9]organization=The SETI Institute Carl Sagan Center, California, USA
10]organization=Santa Fe Institute, New Mexico, USA 11]organization=Google
Inc., Mountain View, California, USA

[cor1]NASA Frontier Development Lab 2018 Participant [cor2]NASA Frontier
Development Lab 2018 Mentor

# PyATMOS: A Scalable Grid of Hypothetical Planetary Atmospheres

A Chopra<EMAIL_ADDRESS>A. Bell  W. Fawcett  R. Talebi  D.
Angerhausen  A.G. Baydin  A. Berea  N.A. Cabrol  C. Kempes  M. Mascaro  [  [
[  [  [  [  [  [  [  [  [

###### Abstract

Cloud computing offers an opportunity to run compute-resource intensive
climate models at scale by parallelising model runs such that datasets useful
to the exoplanet community can be produced efficiently. To better understand
the statistical distributions and properties of potentially habitable
planetary atmospheres we implemented a parallelised climate modelling tool to
scan a range of hypothetical atmospheres.Starting with a modern day Earth
atmosphere, we iteratively and incrementally simulated a range of atmospheres
to infer the landscape of the multi-parameter space, such as the abundances of
biological mediated gases (O2, CO2, H2O, CH4, H2, and N2) that would yield
‘steady state’ planetary atmospheres on Earth-like planets around solar-type
stars. Our current datasets comprises of 124,314 simulated models of exoplanet
atmospheres and is available publicly on the NASA Exoplanet Archive. Our
scalable approach of analysing atmospheres could also help interpret future
observations of planetary atmospheres by providing estimates of atmospheric
gas fluxes and temperatures as a function of altitude. Such data could enable
high-throughput first-order assessment of the potential habitability of
exoplanetary surfaces and sepcan be a learning dataset for machine learning
applications in the atmospheric and exoplanet science domain.

###### keywords:

atmospheres terrestrial planets Earth astrobiology cloud computing

##  1 Introduction

Understanding the nature and distribution of habitable environments in the
universe, and the life forms that may inhabit them, is increasingly part of
the primary science goals of remote and in situ planetary exploration
missions. Strategies (NASEM, 2018, 2019) and roadmaps (Des Marais et al.,
2008; Achenbach et al., 2015; Horneck et al., 2016) all suggest that
identifying, exploring, and characterising extraterrestrial environments for
habitability and biosignatures will be a key focus of planetary science
research endeavors in the coming decade. Remote spectroscopy allows us to
infer the atmospheric composition of exoplanets, but it will be challenging
for even the latest generation of ground- and space-based telescopes to
characterise the surface habitability of an exoplanet (Robinson, 2018).
Additionally, while visible and infrared spectroscopy can help quantify the
abundances of atmospheric gases such as O2, H2O, CO2, CH4 and CO, other gases
such as H2 and N2 have no permanent dipole moment and are difficult to
quantify through spectroscopic observations (Kaltenegger, 2017; Woitke et al.,
2021).

All these gases collectively have significant control on the oxidation state
of the atmosphere and the extent of disequilibrium in an atmosphere which is
available to any potential surface biochemistry. Until ground-based Extremely
Large Telescopes and or space-based mission concepts like HabEx, LUVOIR, or
LIFE come to fruition, the expected SNRs associated with observations in the
JWST-era are unlikely to be able to place strong constraints on the
atmospheric compositions of exoplanets in circumstellar habitable zones
(Seager, 2017; Fujii et al., 2018; Kaltenegger et al., 2020). Modelling of
explanatory atmospheres with limited observational constraints will remain
modi operandi of planetary habitologists for the foreseeable future.

In an effort to more holistically understand the nature of potentially
habitable atmospheres, we designed a modelling framework that allows
concurrent simulation of hundreds of thousands of planetary atmospheres so
that it would become possible to undertake ‘parameter sweeps’ in a high-
dimensional parameter space. Here we present the PyATMOS dataset, and the
associated scalable modelling framework produced as part of the 2018 NASA
Frontier Development Lab to explore the parameter space of planetary
atmospheres that are conducive to habitable conditions on the planetary
surface.

The universe is filled with stars similar to our Sun (Robles et al., 2008) and
exoplanet statistics suggest that rocky planets similar to our Earth are
common (Burke et al., 2015; Petigura et al., 2013; Bovaird et al., 2015; Hsu
et al., 2019; Bryson et al., 2020). Water, heat, chemical disequilibria, and
energy sources would have been present on early wet rocky planets because of
the universal nature of the processes that produced them (Chopra and
Lineweaver, 2018; Lineweaver and Chopra, 2012a).

Since all life on Earth needs liquid water during some part of its life cycle,
and the surface of the Earth is covered with it, the presence of liquid water
on a planet’s surface is taken as a necessary (but not sufficient) condition
for life (Lineweaver and Chopra, 2012b; McKay, 2014). Even if water is a
constituent of the initial inventory of volatiles on rocky planets in the
circumstellar habitable zones of their host stars, surface liquid water can
exist only within the relatively narrow range of pressures and temperatures
and thus may be only a transient feature of most habitable planets (Lineweaver
et al., 2018; Chopra and Lineweaver, 2016). Thus, the search for extra-
terrestrial life on exoplanets is in a large part a search for extra-
terrestrial surface pressures and temperatures that are conducive to liquid
water. The pressure and temperature on a planetary surface is in large part a
function of the properties of the atmosphere above the surface.

The exoplanetary science community has been studying factors that can
influence surface habitability of exoplanets such as surface temperatures,
densities, compositions, tectonic regimes, atmospheric chemistry, and albedos
(Kasting and Catling, 2003; Gaidos et al., 2005; Nisbet et al., 2007; Zahnle
et al., 2007; Lammer et al., 2009; Kopparapu et al., 2013; Seager, 2013;
Cockell, 2016; Godolt et al., 2016; Kaltenegger, 2017; Boutle et al., 2017;
Meadows and Barnes, 2018; Kite and Ford, 2018; Keles et al., 2018). When it
comes to remote detection in the near future, our search for life on
potentially habitable planets will almost exclusively depend on our ability to
spectrally characterise and understand the abiotic and potentially biotic
contributions to atmospheric chemical disequilibria (Kasting et al., 2009;
Krissansen-Totton et al., 2018; Seager and Deming, 2010; Vázquez et al.,
2010). If we are to find an unambiguous biosignature that can be remotely
detected, and design instruments to detect them, we need to identify the range
of atmospheres that should be priority targets for future observations
(Lovelock, 1965; Seager, 2017; Meadows et al., 2018; Schwieterman et al.,
2018; Catling et al., 2018; Kiang et al., 2018; Walker et al., 2018). We will
also need to understand about what type and extent of biology could support,
or at least be compatible with, the different atmospheres that could exist on
exoplanets.

Planets within our solar system have strikingly different surface conditions,
in large part because of the composition of the atmospheres they host. The
next generation of telescopes will have the sensitivity required to determine
the composition of exoplanetary atmospheres (Fujii et al., 2018; Wang et al.,
2018; Venot et al., 2018). Remotely assessing the potential for life on the
surface of a planet will require us to estimate the surface pressure and
temperature to assess the likelihood of surface liquid water. The parameter
space of possible atmospheres on exoplanets is large and exploring it is
computationally intensive.

In order to investigate such a large parameter space, we created a ‘set &
forget’ workflow to run planetary atmosphere models within a scalable
framework. To test the framework, we simulated a wide distribution of
atmospheric compositions by varying six input parameters of the model. The six
parameters varied were the concentrations gases: O2, CO2, H2O, CH4, H2, and
N2. The gases were chosen because they are the most abundant gases in Earth’s
atmosphere and thus likely to be the gases whose concentrations will be of
interest to future observations of potentially habitable exoplanets.
Additionally, the surface fluxes of these gases have been biologically
mediated by life on Earth through different metabolisms ever since the
emergence of life on Earth (Nealson and Conrad, 1999; Nisbet and Sleep, 2001).
Thus, studying atmospheres with different concentrations and fluxes of these
gases can not only better enable us to evaluate the surface habitability of
potentially inhabited exoplanets but also inform estimates of the likelihood
and type of life being present on a remotely characterised exoplanet. Our
approach will help transition from a zero-dimensional model of a circumstellar
habitable zone to a more nuanced Gaussian distribution which can parametrise
the extent of habitability (Lineweaver et al., 2018).

##  2 Method

###  2.1 Simulation of atmospheres with ATMOS

To scan the parameter space of atmospheres, we employed the ATMOS software
package (Arney et al., 2016; Meadows et al., 2016) on a massively parallelised
cloud-based process to create a database of exoplanet atmospheres.

The ATMOS package, a coupled photochemistry-climate model111
https://github.com/VirtualPlanetaryLaboratory/atmos, considers a 1-D column of
gas through the atmosphere. It is configurable with input parameters such as
the concentration or surface fluxes of different species of gases, the stellar
type of the planet’s host star, the gravitational field strength of the
planet, and the distance between the planet and the host star. The output of
ATMOS is a 1-D column of the resultant atmosphere’s temperature, pressure, gas
concentrations and gas fluxes as a function of altitude.

ATMOS uses a photochemical model to calculate the effect of UV radiation on
the different gas species, and a climate model to calculate the temperature
and pressure profile, as a function of altitude, of the different gases.

The photochemical model includes particle microphysics and is run first to
generate an initial atmospheric state based on user-specified boundary
conditions (gas mixing ratios and fluxes, the temperature-pressure profile and
the incident stellar spectrum). For our analyses, we started with planetary
boundary conditions set to the present-day Earth and stellar parameters set to
the present-day Sun. Output files from the photochemical model for altitude,
pressure and gas mixing ratios are then passed into the climate model as its
initial conditions and the climate model runs until it reaches a converged
state. The climate model then feeds updated temperature and gas profiles back
into the photochemical model. The models iterate back and forth in this manner
until convergence is reached222For the development history and details on the
coupling and convergence of the two models, see Arney et al. (2016); Meadows
et al. (2016). ATMOS can thus be described as a coupled set of differential
equations, and the software works to find a local ‘steady state’ solution for
a given set of gas concentrations and fluxes as a function of altitude. A
consequence of this is a strong dependence on the initial ‘seeded’ state of
atmospheric concentrations. The software can only solve the set of
differential equations provided that the next set of initial conditions is not
too far from that of the previous set of initial conditions, and therefore one
must take small steps in parameter space to get from one set of gas
concentrations to another. The increments were determined empirically in past
usage of this code by Arney et al. (2016).

Gas | Scan range | Increment | Modern Earth
---|---|---|---
O2 | 0.0–0.3 | 0.02 | 0.21
0.3–1.0 | 0.05
CO2 | 0.0–0.1 | 0.01 | 4.00 $\times$ 10$-$4
0.1–1.0 | 0.05
H2O | 0.0–0.9 | 0.05 | 1.23 $\times$ 10$-$ 2
CH4 | 0.0–0.1 | 0.005 | 1.63 $\times$ 10$-$ 6
H2 | 0.0–10$-$7 | 10$-$9 | 8.13 $\times$ 10$-$8
N2+trace gases | — | — | 0.78
Table 1: Fractional scan range and increments of gases varied in order to
explore the parameter space of atmospheres. We note that N2 was not varied in
a step-wise manner as was done with the other gasses but was instead used to
‘fill’ the remainder of the atmosphere if the combination of other gas
concentrations did not add to 100%. Trace gases were not varied and included
as a fixed portion of the atmosphere.

Table 1 contains the list of scan ranges and increments which correspond to
the step sizes, and the initial conditions corresponding to Modern Earth. The
gas concentrations chosen to vary were O2, H2O, CH4, CO2, H2 and N2. Other
trace gases (including O3) important to the composition of Earth’s current
atmosphere were incorporated into the models at Modern Earth concentrations,
and not varied between the scans. Starting with a present-day Earth
atmosphere, we iteratively and incrementally sampled atmospheres with
different gas mixtures.

In order to explore the parameter space of atmospheric concentrations in a
systematic manner, PyATMOS was configured to iteratively use previous
atmosphere solutions that were within the defined increments of the ‘target’
conditions as ‘seeds’. A finished run would then go on to seed the initial
state for a subsequent run, which would solve the state for some small
permutation in each gas relative to the previous state. The ‘origin’ state for
the whole search strategy was defined by a Modern Earth template (a complete
set of parameters corresponding to the present-day atmosphere of the Earth)
and subsequent runs computed the atmospheric profiles in a parameter space
‘similar’ to Modern Earth. The process would repeat until either the model run
timed-out or the defined parameter space was scanned.

###  2.2 Software and compute environment

The ATMOS software exhibits platform dependencies, in part attributable to its
legacy piece-wise development in Fortran. To streamline the ATMOS runs and
maintain cross-platform consistency, we created a Docker image of ATMOS based
on the Ubuntu Linux distribution. This image guaranteed consistent performance
on all host platforms. To automate the process of configuring ATMOS for
individual runs, we wrote a package called PyATMOS in Python 3 (chosen for its
flexibility, extensive community-driven resources and potential for further
development by end-users). PyATMOS allows one to easily configure ATMOS, run
it, and extract the relevant results.

A Docker image loaded with PyATMOS, which inherited the original ATMOS image,
was created to instantiate thousands of individual cloud instances, all of
which worked in parallel to search the atmospheric parameter space. Additional
Python scripts were written to supervise a work-queue and designed to manage
the run-constraints of ATMOS outlined in Section 2.1. The work-queue is
visualised in Fig. 1.

Figure 1: Cloud-computing work-queue for exploring a large parameter space of
atmospheres.

The cloud instances spawned off thousands of identical virtual environments to
compute the individual atmospheric concentrations with PyATMOS. Google Cloud
Storage hosted all the data output by each run, and a SQL server stored a
parsed log of all completed and queued runs. A Redis server tracked the
completion of runs and allocated new work to each virtual machine.

Listing 1 shows how a series of gas concentrations is input to PyATMOS, the
code is then run, and the results are stored in a specified output directory.
Since ATMOS requires a previously found stable atmosphere to ‘step’ from in
order to perform the new calculation, we set the previous_photochem_solution
and previous_clima_solution parameters of the atmos.run function to strings
containing the path to the relevant previous solution.

⬇

import pyatmos

atmos = pyatmos.Simulation(

docker_image =
"registry.gitlab.com/frontierdevelopmentlab/astrobiology/pyatmos")

# setup the docker container

atmos.start()

# Configuration for ATMOS

concentrations = {’H2O’: 0.2, ’CO2’: 0.0004, ’CH4’: 1.63e-06, ’O2’: 0.2, ’H2’:
8.13e-08}

args = {

’species_concentrations’ : concentrations,

’output_directory’ : "/home/results/"}

# Run the code

atmos.run(**args)

# Close the docker container

atmos.close()

Listing 1: Simple example of running PyATMOS. A set of gas concentrations as
inputs are run in a single iteration of ATMOS with PyATMOS. When executed via
a batch scheduling script, the same code enables parameter sweeps across a
range of atmospheres.

##  3 Results

Figure 2: Histograms of input concentrations for CH4, CO2, H2, H2O and O2.  Figure 3: Histograms of output surface fluxes for CH4, CO2, H2, H2O and O2.  Column Name | Table Label | Units | Description
---|---|---|---
input_CH4 | Input CH4 concentration | fractional |  CH4 concentration at planet surface input to model
input_CO2 | Input CO2 concentration | fractional |  CO2 concentration at planet surface input to model
input_H2 | Input H2 concentration | fractional |  H2 concentration at planet surface input to model
input_H2O | Input H2O concentration | fractional |  H2O concentration at planet surface input to model
input_O2 | Input O2 concentration | fractional |  O2 concentration at planet surface input to model
concentration_CH4 | CH4 Concentration | fractional |  CH4 concentration at planet surface*
concentration_CO2 | CO2 Concentration | fractional |  CO2 concentration at planet surface*
concentration_H2 | H2 Concentration | fractional |  H2 concentration at planet surface*
concentration_H2O | H2O Concentration | fractional |  H2O concentration at planet surface*
concentration_O2 | O2 Concentration | fractional |  O2 concentration at planet surface*
flux_CH4 | CH4 Flux | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$ |  CH4 flux required to maintain concentration at planet surface*
flux_CO2 | CO2 Flux | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$ |  CO2 flux required to maintain concentration at planet surface*
flux_H2 | H2 Flux | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$ |  H2 flux required to maintain concentration at planet surface*
flux_H2O | H2O Flux | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$ |  H2O flux required to maintain concentration at planet surface*
flux_O2 | O2 Flux | $\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$ |  O2 flux required to maintain concentration at planet surface*
pressure_bar | Pressure | $\mathrm{bar}$ | Pressure at planet surface*
temperature_kelvin | Temperature | $\mathrm{K}$ | Temperature at planet surface*
Table 2: Column definitions of the summary table which contains 124,314 rows,
where the data in each row summarises the varied input parameters, and the
resulting output parameters (indicated by * in the description) for each
atmosphere model.

A total of 124,314 atmospheres were simulated and Fig. 2 shows the
distribution of concentrations sampled in this study. Although this represents
only a small fraction of the 6-D parameter space possible within the scan
constraints (limited by our compute-resource limits), the resulting dataset
could be expanded for future studies. For each atmosphere, the temperature,
pressure, gas concentration and gas fluxes were calculated as a function of
altitude from 0–80 km in 100 steps (a mean step-size of approximately 800 m
normally distributed with a standard deviation of 118 m). The concentrations
and fluxes for each of the gases listed in Table 1 were calculated, along with
several other trace gases with concentrations less than 1 %. Figure 3 shows
the distribution of surface fluxes for the five gases which were varied as
input parameters in the atmosphere simulations.

The data from the simulated atmospheres are available to the community on the
NASA Exoplanet Archive333https://exoplanetarchive.ipac.caltech.edu/cgi-
bin/FDL/nph-fdl?atmos. The interactive portal enables users to filter,
preview, and download one or more models of interest. Table 2 describes the
summary table which shows the input concentrations and the output parameters.

Figure 4: Temperature (dashed lines) and pressure (solid lines) profiles for
three of the simulated atmospheres. The black lines are for the ‘Modern Earth’
atmosphere, the orange and blue lines correspond to the atmospheres simulated
in this study with the largest concentrations of CH4 (13%) and CO2 (40%)
respectively.  Figure 5: Distribution of atmospheres in the temperature versus
pressure plane. The graphs above the upper and right axes are the 1-D density
profiles of temperature and pressure, respectively. Although the distribution
is sensitive to the priors applied to the scan and the total number of
atmospheres scanned, similar analyses with larger datasets could help infer
the frequency of different classes of habitable exoplanetary atmospheres and
enable interpretation of biosignatures.

Figure 4 shows an example of three temperature- and pressure-profiles: the
present-day Earth, the atmosphere with the largest concentration of CO2 (40%),
and the atmosphere with the largest concentration of CH4 (13%) at the surface
of the planet. The pressure variation between the modelled planets shown in
the figure is not significant at lower altitudes but grows as the altitude
increases.

Unsurprisingly, planets with the large concentrations of CO2 and CH4 have
hotter surface temperatures than on Modern Earth. However, the thermal
inversion observed in the Modern Earth’s stratosphere due to absorption of
ultraviolet radiation by ozone (at around 50km altitude) is significantly
affected by higher concentrations of CH4 and CO2. In the case of higher CO2
concentration, although the surface is warmer due to the increased greenhouse
effect, CO2 is also better able to cool in the infrared at altitudes above
30kms.

To gain a holistic view of the space of atmospheres simulated, the
temperatures and pressures at the planetary surfaces were extracted, and the
distribution of these atmospheres is shown in Fig. 5. Since the distribution
is sensitive to the scan parameters employed in the search, only limited
conclusions are possible with the data collected here. Among the simulated
atmospheres, there are three “islands” of atmospheres that can be identified
in Fig. 5. The bottom left-most of these contained the atmosphere
corresponding to present-day Earth (average surface temperature of 15$\circ$C
and pressure of 1.02 atm). These islands are probably more a facet of the
parameter space exploration strategy than indicative of planetary regimes.

Figure 6: 2-D histograms (heatmaps) of the density of scanned atmospheres in
the surface temperature versus gas mixing ratio plane, overlain with the
profile histogram of atmosphere temperatures as a function of the gas mixing
ratio (O2-a, CO2-b, CH4-b). Each bin of gas mixing ratio contains many
atmospheres, with all the combinations of other gasses that were simulated.
Red points show the mean of the temperatures of the atmospheres in each bin,
and the error bars show the standard deviation. As the heatmaps and the
profile histograms depend significantly on the priors applied to the
atmosphere scan and the number of scanned atmospheres, limited interpretation
is possible with the current dataset and plots here only demonstrate the
concept.

Figure 6 shows 2-D heatmaps and profile histograms for the O2 (6a) and CO2
(6b) concentrations versus temperature. The heatmaps are binned, and each bin
shows the number of atmospheres generated as a function of temperature and the
gas mixing ratio, with darker regions indicating a greater proportion of
atmospheres in a given histogram bin. The profile histograms (red bars) show
the average temperature for all the atmospheres in that particular range of
gas mixing ratio; for example, the first red point on Fig. 6a corresponds to
the average of all the atmospheres with O2 mixing ratio between 0.00–0.05
(regardless of the concentrations of other gases). The red point shows the
mean of the temperatures of the atmospheres, and the error bars indicate the
standard deviation.

Plots such as Fig. 6 offer a simple way of determining the surface temperature
of a planet to first-order. Such an approximation would be particularly
valuable where remote characterisation has only been able to constraint the
abundances of some gases. For example, based on our current dataset, if we
were to find that a planet had an O2 mixing ratio between 0.35–0.40, then
there would be a 68% chance that the surface temperature of that planet is in
the range 30–50 ∘C – a potentially useful result given that liquid water on
the surface of a planet may be an indicator for life (Chopra and Lineweaver,
2016; Lineweaver et al., 2018). Further constraints on the temperatures could
be provided by a concordance of results from other gases. However, to infer
realistic surface temperatures, we would need to simulate a representative set
of all possible exoplanetary atmospheres and expand our current dataset.

##  4 Future Directions

  1. 1.

This work set the stellar parameters to that of the Sun and an Earth-sized
planet at 1AU from the host star. The work could be expanded to include M and
K stars which are of particular interest to exoplanet habitability studies,
and model a range of planetary sizes, insolation and obliquities. While we
have used a relatively simple 1-D model in our study, the batch-processing
framework developed to utilise cloud computing is sufficiently flexible to
enable more recently developed and complex 3D-GCM models such as ROCKE-3D (Way
et al., 2017), ExoCAM (Wolf et al., 2018), LMD-Generic (Wordsworth et al.,
2011; Turbet et al., 2018), and the MET Office Unified Model (Boutle et al.,
2017) to be run at scale and conduct parameter sweeps. Such a grid of models,
potentially validated with data from future exoplanet observations, could help
estimate the statistical distributions of habitable zones.

  2. 2.

Large collections of atmosphere models are valuable as synthetic training
datasets for machine learning applications in the exoplanet science domains.
For example, a neural network model trained on various stellar types,
planetary radii, planet-star distances, and atmospheric compositions would
reduce the need to run resource-intensive models. Similarly, ML-based
atmospheric retrieval frameworks (Soboczenski et al., 2018; Cobb et al., 2019)
used to determine an exoplanetary atmosphere’s temperature structure and
composition from an observed spectrum, can benefit from the large repository
to atmospheric models to efficiently generate training spectra libraries.

  3. 3.

In this study, we do not attempt to infer distribution of bio-masses and/or
metabolisms capable of sustaining and co-existing with the surface gases
fluxes of the modelled atmospheres. However, future simulations that couple
planetary atmosphere models such as ATMOS to biogeochemical models in a
similar manner to Kharecha et al. (2005) and Gebauer et al. (2017), could
enable characterisation of the potential role of biology in regulating
planetary atmospheres (Harding and Margulis, 2010; Lenton et al., 2018; Lyons
et al., 2015).

Such efforts would lead to more nuanced context-dependent interpretations of
habitability parameters such as surface temperatures, photon and redox free
energy availability for different classes of planetary systems (Lineweaver et
al., 2018; Lenardic and Seales, 2021) and assessments of potential
biosignatures in exoplanetary atmospheres.

##  5 Conclusions

A set of 124,314 explanatory atmospheres have been simulated using a new
framework, PyATMOS. For each simulated atmosphere, the temperature, pressure,
gas concentration and gas flux were calculated as a function of altitude,
ranging from 0–80 km. The resulting dataset is much larger than was available
before to the exoplanet habitability community. Using the computational
framework, future work could facilitate statistical inferences on quantities
such as an exoplanet’s surface temperature and pressure based on more readily
measurable quantities such as gas concentrations in a planetary atmosphere.

## References

  * Achenbach et al. (2015) Achenbach, L., Bailey, J., Barnes, R.K., Baross, J.A., Bertka, C., Boston, P., Boyd, E., Cable, M., Chen, I., Ciesla, F.J., Des Marais, D.J., Domagal-Goldman, S.D., Cook, J.E., Goldman, A., Hud, N., Laine, P., Lloyd, K., Lyons, T., Meadows, V.S., Mix, L., Mojzsis, S.J., Muller, U., Pasek, M., Powell, M., Robinson, T.D., Rosenzweig, F., Schmidt, B., Seelig, B., Springsteen, G., Vance, S., Welander, P., Williams, L., Wordsworth, R.D., 2015.  NASA Astrobiology Strategy 2015.  NASA Astrobiology , 1–236URL: https://nai.nasa.gov/media/medialibrary/2016/04/NASA_Astrobiology_Strategy_2015_FINAL_041216.pdf, doi:10.1017/CBO9781107415324.004, arXiv:arXiv:1011.1669v3.
  * Arney et al. (2016) Arney, G.N., Domagal-Goldman, S.D., Meadows, V.S., Wolf, E.T., Schwieterman, E.W., Charnay, B., Claire, M.W., Hébrard, E., Trainer, M.G., 2016.  The Pale Orange Dot: The Spectrum and Habitability of Hazy Archean Earth.  Astrobiology 16, 873--899.  URL: http://online.liebertpub.com/doi/10.1089/ast.2015.1422, doi:10.1089/ast.2015.1422, arXiv:1610.04515.
  * Boutle et al. (2017) Boutle, I.A., Mayne, N.J., Drummond, B., Manners, J., Goyal, J., Lambert, F.H., Acreman, D.M., Earnshaw, P.D., 2017\.  Exploring the climate of Proxima B with the Met Office Unified Model.  Astronomy & Astrophysics 120, 1--13.  doi:10.1051/0004-6361/201630020, arXiv:1702.08463.
  * Bovaird et al. (2015) Bovaird, T., Lineweaver, C.H., Jacobsen, S.K., 2015.  Using the inclinations of Kepler systems to prioritize new Titius-Bode-based exoplanet predictions.  Monthly Notices of the Royal Astronomical Society 448, 3608--3627.  URL: http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stv221, doi:10.1093/mnras/stv221.
  * Bryson et al. (2020) Bryson, S., Kunimoto, M., Kopparapu, R.K., Coughlin, J.L., Borucki, W.J., Koch, D., Aguirre, V.S., Allen, C., Barentsen, G., Batalha, N.M., Berger, T., Boss, A., Buchhave, L.A., Burke, C.J., Caldwell, D.A., Campbell, J.R., Catanzarite, J., Chandrasekaran, H., Chaplin, W.J., Christiansen, J.L., Christensen-Dalsgaard, J., Ciardi, D.R., Clarke, B.D., Cochran, W.D., Dotson, J.L., Doyle, L.R., Duarte, E.S., Dunham, E.W., Dupree, A.K., Endl, M., Fanson, J.L., Ford, E.B., Fujieh, M., Gautier III, T.N., Geary, J.C., Gilliland, R.L., Girouard, F.R., Gould, A., Haas, M.R., Henze, C.E., Holman, M.J., Howard, A.W., Howell, S.B., Huber, D., Hunter, R.C., Jenkins, J.M., Kjeldsen, H., Kolodziejczak, J., Larson, K., Latham, D.W., Li, J., Mathur, S., Meibom, S., Middour, C., Morris, R.L., Morton, T.D., Mullally, F., Mullally, S.E., Pletcher, D., Prsa, A., Quinn, S.N., Quintana, E.V., Ragozzine, D., Ramirez, S.V., Sanderfer, D.T., Sasselov, D., Seader, S.E., Shabram, M., Shporer, A., Smith, J.C., Steffen, J.H., Still, M., Torres, G., Troeltzsch, J., Twicken, J.D., Uddin, A.K., Van Cleve, J.E., Voss, J., Weiss, L.M., Welsh, W.F., Wohler, B., Zamudio, K.A., 2020\.  The Occurrence of Rocky Habitable-zone Planets around Solar-like Stars from Kepler Data.  The Astronomical Journal 161, 36\.  URL: http://dx.doi.org/10.3847/1538-3881/abc418, doi:10.3847/1538-3881/abc418, arXiv:2010.14812.
  * Burke et al. (2015) Burke, C.J., Christiansen, J.L., Mullally, F., Seader, S., Huber, D., Rowe, J.F., Coughlin, J.L., Thompson, S.E., Catanzarite, J., Clarke, B.D., Morton, T.D., Caldwell, D.A., Bryson, S.T., Haas, M.R., Batalha, N.M., Jenkins, J.M., Tenenbaum, P., Twicken, J.D., Li, J., Quintana, E.V., Barclay, T., Henze, C.E., Borucki, W.J., Howell, S.B., Still, M., 2015.  Terrestrial Planet Occurrence Rates for the Kepler GK Dwarf Sample.  The Astrophysical Journal 809, 19\.  URL: http://iopscience.iop.org/article/10.1088/0004-637X/809/1/8/meta, doi:10.1088/0004-637X/809/1/8, arXiv:1506.04175.
  * Catling et al. (2018) Catling, D.C., Krissansen-Totton, J., Kiang, N.Y., Crisp, D., Robinson, T.D., DasSarma, S., Rushby, A.J., Del Genio, A., Bains, W., Domagal-Goldman, S., 2018\.  Exoplanet Biosignatures: A Framework for Their Assessment.  Astrobiology 18, 709--738.  URL: http://www.liebertpub.com/doi/10.1089/ast.2017.1737, doi:10.1089/ast.2017.1737, arXiv:1705.06381.
  * Chopra and Lineweaver (2016) Chopra, A., Lineweaver, C.H., 2016\.  The Case for a Gaian Bottleneck: the Biology of Habitability.  Astrobiology 16, 7--22.  URL: http://online.liebertpub.com/doi/abs/10.1089/ast.2015.1387, doi:10.1089/ast.2015.1387.
  * Chopra and Lineweaver (2018) Chopra, A., Lineweaver, C.H., 2018\.  The Cosmic Evolution of Biochemistry, in: Habitability of the Universe Before Earth. Elsevier. volume 1, pp. 75--87.  URL: http://linkinghub.elsevier.com/retrieve/pii/B9780128119402000046, doi:10.1016/B978-0-12-811940-2.00004-6.
  * Cobb et al. (2019) Cobb, A.D., Himes, M.D., Soboczenski, F., Zorzan, S., O’Beirne, M.D., Güneş Baydin, A., Gal, Y., Domagal-Goldman, S.D., Arney, G.N., Angerhausen, D., 2019\.  An Ensemble of Bayesian Neural Networks for Exoplanetary Atmospheric Retrieval.  The Astronomical Journal 158, 33\.  doi:10.3847/1538-3881/ab2390, arXiv:1905.10659.
  * Cockell (2016) Cockell, C.S., 2016.  The similarity of life across the universe.  Molecular Biology of the Cell 27, 1553--1555.  URL: http://www.molbiolcell.org/cgi/doi/10.1091/mbc.E15-11-0809, doi:10.1091/mbc.E15-11-0809.
  * Des Marais et al. (2008) Des Marais, D.J., Nuth, J.a., Allamandola, L.J., Boss, A.P., Farmer, J.D., Hoehler, T.M., Jakosky, B.M., Meadows, V.S., Pohorille, A., Runnegar, B., Spormann, A.M., 2008.  The NASA Astrobiology Roadmap.  Astrobiology 8, 715--730.  URL: http://www.ncbi.nlm.nih.gov/pubmed/18793098, doi:10.1089/ast.2008.0819.
  * Fujii et al. (2018) Fujii, Y., Angerhausen, D., Deitrick, R., Domagal-Goldman, S.D., Grenfell, J.L., Hori, Y., Kane, S.R., Palle, E., Rauer, H., Siegler, N., Stapelfeldt, K.R., Stevenson, K.B., 2018\.  Exoplanet Biosignatures: Observational Prospects.  Astrobiology 18, 739--778.  doi:10.1089/ast.2017.1733, arXiv:1705.07098.
  * Gaidos et al. (2005) Gaidos, E., Deschenes, B., Dundon, L., Fagan, K., McNaughton, C., Menviel-Hessler, L., Moskovitz, N., Workman, M., 2005\.  Beyond the principle of plentitude: A review of terrestrial planet habitability.  URL: https://www.liebertpub.com/doi/abs/10.1089/ast.2005.5.100, doi:10.1089/ast.2005.5.100.
  * Gebauer et al. (2017) Gebauer, S., Grenfell, J.L., Stock, J., Lehmann, R., Godolt, M., von Paris, P., Rauer, H., 2017.  Evolution of Earth-like Extrasolar Planetary Atmospheres: Assessing the Atmospheres and Biospheres of Early Earth Analog Planets with a Coupled Atmosphere Biogeochemical Model.  Astrobiology 17, 27--54.  URL: http://online.liebertpub.com/doi/10.1089/ast.2015.1384, doi:10.1089/ast.2015.1384, arXiv:1807.06844.
  * Godolt et al. (2016) Godolt, M., Grenfell, J.L., Kitzmann, D., Kunze, M., Langematz, U., Patzer, A.B., Rauer, H., Stracke, B., 2016\.  Assessing the habitability of planets with Earth-like atmospheres with 1D and 3D climate modeling.  Astronomy and Astrophysics 592, 1--12.  doi:10.1051/0004-6361/201628413, arXiv:1605.08231.
  * Harding and Margulis (2010) Harding, S., Margulis, L., 2010\.  Water Gaia: 3.5 Thousand Million Years of Wetness on Planet Earth, in: Crist, E., Rinker, H.B. (Eds.), Gaia in Turmoil: Climate Change, Biodepletion, and Earth Ethics in an Age of Crisis. MIT Press, p. 371.
  * Horneck et al. (2016) Horneck, G., Walter, N., Westall, F., Grenfell, J.L., Martin, W.F., Gomez, F., Leuko, S., Lee, N., Onofri, S., Tsiganis, K., Saladino, R., Pilat-Lohinger, E., Palomba, E., Harrison, J., Rull, F., Muller, C., Strazzulla, G., Brucato, J.R., Rettberg, P., Capria, M.T., 2016\.  AstRoMap European Astrobiology Roadmap.  Astrobiology 16, 201--243.  doi:10.1089/ast.2015.1441.
  * Hsu et al. (2019) Hsu, D.C., Ford, E.B., Ragozzine, D., Ashby, K., 2019\.  Occurrence Rates of Planets orbiting FGK Stars: Combining Kepler DR25, Gaia DR2 and Bayesian Inference.  The Astronomical Journal 158, 109\.  URL: http://arxiv.org/abs/1902.01417, doi:10.3847/1538-3881/ab31ab, arXiv:1902.01417.
  * Kaltenegger (2017) Kaltenegger, L., 2017.  How to Characterize Habitable Worlds and Signs of Life.  Annual Review of Astronomy and Astrophysics 55, 433--485.  URL: http://www.annualreviews.org/doi/10.1146/annurev-astro-082214-122238, doi:10.1146/annurev-astro-082214-122238.
  * Kaltenegger et al. (2020) Kaltenegger, L., Lin, Z., Rugheimer, S., 2020.  Finding Signs of Life on Transiting Earthlike Planets: High-resolution Transmission Spectra of Earth through Time around FGKM Host Stars.  The Astrophysical Journal 904, 10\.  doi:10.3847/1538-4357/abb9b2.
  * Kasting and Catling (2003) Kasting, J.F., Catling, D.C., 2003\.  Evolution of a habitable planet.  Annual Review of Astronomy and Astrophysics 41, 429--463.  URL: http://www.annualreviews.org/doi/abs/10.1146/annurev.astro.41.071601.170049, doi:10.1146/annurev.astro.41.071601.170049.
  * Kasting et al. (2009) Kasting, J.F., Traub, W.A., Roberge, A., Léger, A., Schwartz, A., Wootten, A., Vosteen, A., Lo, A., Brack, A., Tanner, A., Coustenis, A., Lane, B., Oppenheimer, B.R., Mennesson, B., Lopez, B., Grillmair, C., Beichman, C., Cockell, C.S., Hanot, C., McCarthy, C., Stark, C., Marois, C., Aime, C., Angerhausen, D., Montes, D., Wilner, D., Defrere, D., Mourard, D., Lin, D., Kite, E., Chassefiere, E., Malbet, F., Tian, F., Westall, F., Illingworth, G., Vasisht, G., Serabyn, G., Marcy, G.W., Bryden, G., White, G., Laughlin, G., Torres, G., Hammel, H., Ferguson, H., Shibai, H., Rottgering, H., Surdej, J., Wiseman, J., Ge, J., Bally, J., Krist, J., Monnier, J., Trauger, J., Horner, J., Catanzarite, J., Harrington, J., Nishikawa, J., Stapelfeldt, K., von Braun, K., Biazzo, K., Carpenter, K., Balasubramanian, K., Kaltenegger, L., Postman, M., Spaans, M., Turnbull, M.C., Levine, M., Burchell, M., Ealey, M., Kuchner, M., Marley, M., Dominik, M., Mountain, M., Kenworthy, M., Muterspaugh, M., Shao, M., Zhao, M., Tamura, M., Kasdin, N., Haghighipour, N., Kiang, N.Y., Elias, N., Woolf, N., Mason, N., Absil, O., Guyon, O., Lay, O., Borde, P., Fouqué, P., Kalas, P., Lowrance, P., Plavchan, P., Hinz, P., Kervella, P., Chen, P., Akeson, R., Soummer, R., Waters, R., Barry, R., Kendrick, R., Brown, R., Vanderbei, R., Woodruff, R., Danner, R., Allen, R., Polidan, R., Seager, S., MacPhee, S., Hosseini, S., Metchev, S., Kafka, S., Ridgway, S., Rinehart, S., Unwin, S., Shaklan, S., ten Brummelaar, T., Mazeh, T., Meadows, V.S., Weiss, W., Danchi, W., Ip, W., Rabbia, Y., 2009.  Exoplanet Characterization and the Search for Life, in: Astro2010: The Astronomy and Astrophysics Decadal Survey, p. 151.  URL: https://arxiv.org/abs/0911.2936.
  * Keles et al. (2018) Keles, E., Grenfell, J.L., Godolt, M., Stracke, B., Rauer, H., 2018.  The effect of varying atmospheric pressure upon habitability and biosignatures of earth-like planets.  Astrobiology 18, 116--132.  doi:10.1089/ast.2016.1632.
  * Kharecha et al. (2005) Kharecha, P., Kasting, J.F., Siefert, J., 2005.  A coupled atmosphere -- ecosystem model of the early.  Geobiology 3, 53--76.
  * Kiang et al. (2018) Kiang, N.Y., Domagal-Goldman, S.D., Parenteau, M.N., Catling, D.C., Fujii, Y., Meadows, V.S., Schwieterman, E.W., 2018.  Exoplanet Biosignatures: At the Dawn of a New Era of Planetary Observations.  Astrobiology 18, 619--629.  doi:10.1089/ast.2018.1862.
  * Kite and Ford (2018) Kite, E.S., Ford, E.B., 2018\.  Habitability of exoplanet waterworlds.  The Astrophysical Journal 864, 75\.  URL: http://iopscience.iop.org/article/10.3847/1538-4357/aad6e0/meta, doi:10.3847/1538-4357/aad6e0.
  * Kopparapu et al. (2013) Kopparapu, R.K., Ramirez, R.M., Kasting, J.F., Eymet, V., Robinson, T.D., Mahadevan, S., Terrien, R.C., Domagal-Goldman, S.D., Meadows, V.S., Deshpande, R., 2013\.  Habitable zones around main-sequence stars: New estimates.  The Astrophysical Journal 765, 16\.  doi:10.1088/0004-637X/765/2/131, arXiv:1301.6674.
  * Krissansen-Totton et al. (2018) Krissansen-Totton, J., Olson, S., Catling, D.C., 2018.  Disequilibrium biosignatures over Earth history and implications for detecting exoplanet life.  Science Advances 4.  URL: https://www.science.org/doi/10.1126/sciadv.aao5747, doi:10.1126/sciadv.aao5747.
  * Lammer et al. (2009) Lammer, H., Bredehöft, J.H., Coustenis, A., Khodachenko, M.L., Kaltenegger, L., Grasset, O., Prieur, D., Raulin, F., Ehrenfreund, P., Yamauchi, M., Wahlund, J.E., Grießmeier, J.M., Stangl, G., Cockell, C.S., Kulikov, Y.N., Grenfell, J.L., Rauer, H., 2009.  What makes a planet habitable?  The Astronomy and Astrophysics Review 17, 181--249.  URL: http://www.springerlink.com/index/10.1007/s00159-009-0019-z, doi:10.1007/s00159-009-0019-z.
  * Lenardic and Seales (2021) Lenardic, A., Seales, J., 2021\.  Habitability: A process versus a state variable framework with observational tests and theoretical implications.  International Journal of Astrobiology 20, 125--132.  doi:10.1017/S1473550420000415.
  * Lenton et al. (2018) Lenton, T.M., Daines, S.J., Dyke, J.G., Nicholson, A.E., Wilkinson, D.M., Williams, H.T.P., 2018\.  Selection for Gaia across Multiple Scales.  Trends in Ecology & Evolution URL: http://www.sciencedirect.com/science/article/pii/S0169534718301186, doi:10.1016/j.tree.2018.05.006.
  * Lineweaver and Chopra (2012a) Lineweaver, C.H., Chopra, A., 2012a.  The Habitability of Our Earth and Other Earths: Astrophysical, Geochemical, Geophysical, and Biological Limits on Planet Habitability.  Annual Review of Earth and Planetary Sciences 40, 597--623.  URL: http://www.annualreviews.org/doi/abs/10.1146/annurev-earth-042711-105531, doi:10.1146/annurev-earth-042711-105531.
  * Lineweaver and Chopra (2012b) Lineweaver, C.H., Chopra, A., 2012b.  What can Life on Earth Tell Us about Life in the Universe?, in: Seckbach, J. (Ed.), Genesis - In The Beginning: Precursors of Life, Chemical Models and Early Biological Evolution. Springer, pp. 799--815.  URL: http://www.springer.com/gp/book/9789400729407.
  * Lineweaver et al. (2018) Lineweaver, C.H., Chopra, A., McIntyre, S.R.N., 2018.  The evolution of habitability: Characteristics of habitable planets, in: Kolb, V.M. (Ed.), Handbook of Astrobiology. 1 ed.. CRC Press, Boca Raton. chapter 10.1, pp. 685--698.  URL: https://www.crcpress.com/Handbook-of-Astrobiology/Kolb/p/book/9781138065123.
  * Lovelock (1965) Lovelock, J.E., 1965.  A physical basis for life detection experiments.  Nature 207, 568--570.  doi:10.1038/207568a0.
  * Lyons et al. (2015) Lyons, T.W., Fike, D.A., Zerkle, A.L., 2015.  Emerging biogeochemical views of Earth’s ancient microbial worlds.  Elements 11, 415--421.  doi:10.2113/gselements.11.6.415.
  * McKay (2014) McKay, C.P., 2014.  Requirements and limits for life in the context of exoplanets.  Proceedings of the National Academy of Sciences 111, 12628--12633.  URL: http://www.pnas.org/cgi/doi/10.1073/pnas.1304212111, doi:10.1073/pnas.1304212111.
  * Meadows et al. (2016) Meadows, V.S., Arney, G.N., Schwieterman, E.W., Lustig-Yaeger, J., Lincowski, A.P., Robinson, T.D., Domagal-Goldman, S.D., Barnes, R.K., Fleming, D.P., Deitrick, R., Luger, R., Driscoll, P.E., Quinn, T.R., Crisp, D., 2016\.  The Habitability of Proxima Centauri b: Environmental States and Observational Discriminants.  Astrobiology 18, 133--189.  URL: http://arxiv.org/abs/1608.08620, doi:10.1089/ast.2016.1589, arXiv:1608.08620.
  * Meadows and Barnes (2018) Meadows, V.S., Barnes, R.K., 2018\.  Factors affecting exoplanet habitability, in: Deeg, H.J., Belmonte, J.A. (Eds.), Handbook of Exoplanets. Springer Nature, pp. 2771--2794.  doi:10.1007/978-3-319-55333-7_57.
  * Meadows et al. (2018) Meadows, V.S., Reinhard, C.T., Arney, G.N., Parenteau, M.N., Schwieterman, E.W., Domagal-Goldman, S.D., Lincowski, A.P., Stapelfeldt, K.R., 2018.  Exoplanet Biosignatures: Understanding Oxygen as a Biosignature in the Context of Its Environment.  Astrobiology 18, 630----662.  doi:10.1089/ast.2017.1727, arXiv:1705.07560.
  * NASEM (2018) NASEM, 2018.  Exoplanet Science Strategy.  The National Academies Press, Washington, DC.  URL: https://doi.org/10.17226/25187, doi:10.17226/25187.
  * NASEM (2019) NASEM, 2019.  An Astrobiology Strategy for the Search for Life in the Universe.  The National Academies Press, Washington, DC.  URL: https://www.nap.edu/catalog/25252/an-astrobiology-strategy-for-the-search-for-life-in-the-universe, doi:10.17226/25252.
  * Nealson and Conrad (1999) Nealson, K.H., Conrad, P.G., 1999\.  Life: past, present and future.  Philosophical transactions of the Royal Society of London. Series B, Biological sciences 354, 1923--39.  URL: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1692713&tool=pmcentrez&rendertype=abstract, doi:10.1098/rstb.1999.0532.
  * Nisbet and Sleep (2001) Nisbet, E.G., Sleep, N.H., 2001\.  The habitat and nature of early life.  Nature 409, 1083--91.  URL: http://www.ncbi.nlm.nih.gov/pubmed/11234022, doi:10.1038/35059210.
  * Nisbet et al. (2007) Nisbet, E.G., Zahnle, K.J., Gerasimov, M.V., Helbert, J., Jaumann, R., Hofmann, B.A., Benzerara, K., Westall, F., 2007\.  Creating Habitable Zones, at all Scales, from Planets to Mud Micro-Habitats, on Earth and on Mars.  Space Science Reviews 129, 79--121.  URL: http://www.springerlink.com/index/10.1007/s11214-007-9175-5, doi:10.1007/s11214-007-9175-5.
  * Petigura et al. (2013) Petigura, E.A., Howard, A.W., Marcy, G.W., 2013.  Prevalence of Earth-size planets orbiting Sun-like stars.  Proceedings of the National Academy of Sciences 110, 19273--8.  URL: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3845182&tool=pmcentrez&rendertype=abstract, doi:10.1073/pnas.1319909110.
  * Robinson (2018) Robinson, T.D., 2018.  Characterizing Exoplanet Habitability, in: Deeg, H.J., Belmonte, J.A. (Eds.), Handbook of Exoplanets. Springer International Publishing, Cham, pp. 3137--3157.  URL: http://link.springer.com/10.1007/978-3-319-55333-7_67, doi:10.1007/978-3-319-55333-7_67, arXiv:1911.04441.
  * Robles et al. (2008) Robles, J., Lineweaver, C.H., Grether, D., Flynn, C., Egan, C.A., Pracy, M.B., Holmberg, J., Gardner, E., 2008\.  A Comprehensive Comparison of the Sun to Other Stars: Searching for Self-Selection Effects.  The Astrophysical Journal 684, 691--706.  URL: http://stacks.iop.org/0004-637X/684/i=1/a=691, doi:10.1086/589985.
  * Schwieterman et al. (2018) Schwieterman, E.W., Kiang, N.Y., Parenteau, M.N., Harman, C.E., DasSarma, S., Fisher, T.M., Arney, G.N., Hartnett, H.E., Reinhard, C.T., Olson, S.L., Meadows, V.S., Cockell, C.S., Walker, S.I., Grenfell, J.L., Hegde, S., Rugheimer, S., Hu, R., Lyons, T.W., 2018\.  Exoplanet Biosignatures: A Review of Remotely Detectable Signs of Life.  Astrobiology 18, 663--708.  URL: http://www.liebertpub.com/doi/10.1089/ast.2017.1729, doi:10.1089/ast.2017.1729, arXiv:1705.05791.
  * Seager (2013) Seager, S., 2013.  Exoplanet Habitability.  Science 340, 577--581.  URL: http://science.sciencemag.org/content/340/6132/577.abstract, doi:10.1126/science.1232226.
  * Seager (2017) Seager, S., 2017.  The search for habitable planets with biosignature gases framed by a ‘Biosignature Drake Equation’.  International Journal of Astrobiology , 1--9doi:10.1017/S1473550417000052.
  * Seager and Deming (2010) Seager, S., Deming, D., 2010\.  Exoplanet Atmospheres.  Annual Review of Astronomy and Astrophysics 48, 631--672.  URL: http://www.annualreviews.org/doi/abs/10.1146/annurev-astro-081309-130837, doi:10.1146/annurev-astro-081309-130837.
  * Soboczenski et al. (2018) Soboczenski, F., Himes, M.D., O’Beirne, M.D., Zorzan, S., Baydin, A.G., Cobb, A.D., Gal, Y., Angerhausen, D., Mascaro, M., Arney, G.N., Domagal-Goldman, S.D., 2018.  Bayesian Deep Learning for Exoplanet Atmospheric Retrieval, in: NeurIPS - Third workshop on Bayesian Deep Learning, Montreal, Canada.  URL: http://arxiv.org/abs/1811.03390, arXiv:1811.03390.
  * Turbet et al. (2018) Turbet, M., Bolmont, E., Leconte, J., Forget, F., Selsis, F., Tobie, G., Caldas, A., Naar, J., Gillon, M., 2018.  Modeling climate diversity, tidal dynamics and the fate of volatiles on TRAPPIST-1 planets.  Astronomy and Astrophysics 612, 1--22.  doi:10.1051/0004-6361/201731620, arXiv:1707.06927.
  * Vázquez et al. (2010) Vázquez, M., Pallé, E., Rodríguez, P.M., 2010.  Biosignatures and the Search for Life on Earth, in: The Earth as a Distant Planet. Springer New York, NY. Astronomy and Astrophysics Library, pp. 197--249.  URL: http://link.springer.com/10.1007/978-1-4419-1684-6_5.
  * Venot et al. (2018) Venot, O., Drummond, B., Miguel, Y., Waldmann, I.P., Pascale, E., Zingales, T., 2018\.  A better characterization of the chemical composition of exoplanets atmospheres with ARIEL.  Experimental Astronomy 46, 101--134.  URL: https://doi.org/10.1007/s10686-018-9597-y, doi:10.1007/s10686-018-9597-y, arXiv:1711.08433.
  * Walker et al. (2018) Walker, S.I., Bains, W., Cronin, L., Kiang, N.Y., Schwieterman, E.W., Shkolnik, E.L., Smith, H.B., 2018.  Exoplanet Biosignatures: Future Directions.  Astrobiology 18, 779--824.  URL: http://doi.org/10.1089/ast.2018.1862, doi:10.1089/ast.2017.1738.
  * Wang et al. (2018) Wang, J., Mawet, D., Hu, R., Ruane, G., Delorme, J.r., Klimovich, N., 2018.  Baseline requirements for detecting biosignatures with the HabEx and LUVOIR mission concepts.  Journal of Astronomical Telescopes, Instruments, and Systems 4, 1.  URL: http://arxiv.org/abs/1806.04324, doi:10.1117/1.jatis.4.3.035001, arXiv:1806.04324.
  * Way et al. (2017) Way, M.J., Aleinov, I., Amundsen, D.S., Chandler, M., Clune, T., Del Genio, A.D., Fujii, Y., Kelley, M., Kiang, N.Y., Sohl, L., Tsigaridis, K., 2017.  Resolving Orbital and Climate Keys of Earth and Extraterrestrial Environments with Dynamics 1.0: A General Circulation Model for Simulating the Climates of Rocky Planets.  The Astrophysical Journal Supplement Series 231, 12.  URL: http://arxiv.org/abs/1701.02360%0Ahttp://dx.doi.org/10.3847/1538-4365/aa7a06, doi:10.3847/1538-4365/aa7a06, arXiv:1701.02360.
  * Woitke et al. (2021) Woitke, P., Herbort, O., Helling, C., Stüeken, E., Dominik, M., Barth, P., Samra, D., 2021.  Coexistence of CH4, CO2, and H2O in exoplanet atmospheres.  Astronomy and Astrophysics 646.  doi:10.1051/0004-6361/202038870.
  * Wolf et al. (2018) Wolf, E.T., Haqq-Misra, J., Toon, O.B., 2018.  Evaluating Climate Sensitivity to CO2 Across Earth’s History.  Journal of Geophysical Research: Atmospheres 123, 11,861--11,874.  doi:10.1029/2018JD029262.
  * Wordsworth et al. (2011) Wordsworth, R.D., Forget, F., Selsis, F., Millour, E., Charnay, B., Madeleine, J., 2011\.  Gliese 581d is the first discovered terrestrial-mass exoplanet in the habitable zone.  The Astrophysical Journal Letters 733, L48.  URL: http://iopscience.iop.org/2041-8205/733/2/L48, doi:10.1088/2041-8205/733/2/L48.
  * Zahnle et al. (2007) Zahnle, K.J., Armstrong, R., Cockell, C.S., Halliday, A.N., Nisbet, E.G., Selsis, F., Sleep, N.H., 2007.  Emergence of a Habitable Planet.  Space Science Reviews 129, 35--78.  URL: http://www.springerlink.com/index/10.1007/s11214-007-9225-z, doi:10.1007/s11214-007-9225-z.


††thanks: MGA and FDC contributed equally to this work.

# Large-scale spin-orbit photonic circuits in two dimensions

Maria Gorizia Ammendola  Dipartimento di Fisica, Università degli Studi di
Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, Via Cintia,
80126 Napoli, Italy  Scuola Superiore Meridionale, Via Mezzocannone, 4, 80138
Napoli, Italy  Francesco Di Colandrea<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di Napoli Federico II,
Complesso Universitario di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy
Nexus for Quantum Technologies, University of Ottawa, K1N 5N6, Ottawa, ON,
Canada  Lorenzo Marrucci  Dipartimento di Fisica, Università degli Studi di
Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, Via Cintia,
80126 Napoli, Italy  CNR-ISASI, Institute of Applied Science and Intelligent
Systems, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy  Filippo Cardano
<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di
Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, Via Cintia,
80126 Napoli, Italy

###### Abstract

Photonic circuits, optical platforms that connect input and output modes
according to a specific map, serve as effective optical processors for both
classical and quantum states of light. The number of optical elements
typically scales with that of processed modes, leading to a direct correlation
between system size, circuit complexity, and optical losses. Here we present a
photonic circuit technology implementing large-scale unitary maps, linking a
single input to hundreds of output modes in a two-dimensional compact layout.
The map corresponds to the outcome of a quantum walk of structured photons,
realized experimentally through light propagation in three liquid-crystal
metasurfaces, having the local orientation of optic axes artificially
engineered in a complex pattern. Theoretically, the walk length and the number
of connected modes can be arbitrary, keeping optical losses constant. The
patterns can be designed to accurately replicate multiple unitary maps. We
also discuss limited reconfigurability by adjusting the overall birefringence
and the relative displacement of the three optical elements. These results lay
the basis for the design of low-loss photonic circuits that target a broader
range of unitary maps, primarily for manipulating multi-photon states in
genuinely quantum regimes.

## Introduction

Optical degrees of freedom, such as those associated with spatial, spectro-
temporal, or polarization features of the optical field, serve as a convenient
resource for encoding information. The abundance of tools for their accurate
manipulation established photonics as a versatile platform for both classical
and quantum information processing tasks.

Currently, a variety of platforms have been demonstrated that can realize
different operations on optical modes [1], including vector-matrix
multiplication [2], nonlinear maps [3], and unitary transformations [4].
Optical processors based on linear circuits provide key applications in
optical computing [5, 6] and are emerging as building blocks of future optical
neural networks and AI systems [7].

When used as optical simulators, the processed optical modes encode the
degrees of freedom of a target system (typically, lattice models describing
electronic systems in condensed matter), and the overall transformation maps
to a unitary temporal evolution operator. By monitoring the system output one
can observe directly optical analogues of classical or quantum dynamics [8, 9,
10].

Quantum light may be injected at the input ports of these systems, yielding
output states strongly affected by the quantum interference of two (or more)
photons [11, 12, 13, 14, 15]. The complexity associated with multi-particle
interference phenomena underlies Boson sampling problems, extremely popular in
the recent past as they provided the playground for the first instances of
quantum advantage [16, 17].

Optical platforms like those mentioned above, performing a variety of tasks,
are often referred to as photonic circuits. This classification considers
their analogy with other circuits where the information carriers, like
electric signals, are routed to distinct channels and processed. In integrated
systems, this analogy is straightforward, as optical signals (both as
macroscopic wave-packets or single photons) are spatially localized (like
electrical currents), travel along distinct waveguides, and are manipulated
through integrated beam splitters and phase shifters [18].

Optical modes building a photonic circuit may not correspond to separated
paths for traveling light, but may correspond to co-propagating modes that are
orthogonal because of alternative degrees of freedom like spectro-temporal
ones or those associated with transverse spatial modes, like those carrying
orbital angular momentum [19, 20].

In the first case, trains of pulses associated with non-overlapping time-bins
are conveniently manipulated via propagation into fibers or paths of different
lengths, with a variety of applications such as quantum information [21],
quantum computing [17] and quantum communication [22].

In the second case, the manipulation of co-propagating transverse modes of
structured light via propagation through multi-mode fibers, complex
diffractive elements, or multi-plane light converters [23, 24, 25, 26] has
been successfully demonstrated in the recent years. While these alternative
circuits have not yet reached the technological maturity of integrated
solutions, they offer advantages in terms of the number of addressable modes,
reconfigurability, and the alternative detection schemes of quantum light by
using camera-like sensors [14].

Within this context, photonic circuits based on liquid-crystal metasurfaces
(LCMSs) have been recently introduced for the realization of quantum walks
(QWs). A LCMS is an ultra-thin, transmissive plate made of a micrometric layer
of LC molecules, with their local orientation being artificially patterned at
the micrometric scale. Essentially, they act as standard waveplates for
polarization manipulation, but with a spatially varying optic-axis orientation
[27]. When exhibiting periodic patterns, LCMSs couple circularly polarized
modes of light that carry quantized transverse momentum [28].

The original scheme for the realization of QWs with this platform required a
long sequence of periodic LCMSs, coupling modes arranged both in 1D [29] and
2D [28] grids. In the 1D case, a technique has been recently demonstrated that
allows compressing the entire transformation into only three metasurfaces
[30]. This result is independent of the walk length and the number of involved
modes, which is strictly related to the size of the implemented unitary
matrix, thus dramatically reducing optical losses.

To fully exploit the two-dimensional nature of transverse modes, it would be
highly desirable to implement this concept with modes arranged in a 2D grid.
However, this presents crucial difficulties related to the requirement of
continuity in the LC patterns in a 2D plane. Here we propose and validate a
scheme achieving this goal by tolerating the presence of isolated vortices of
LCs, each carrying an elementary charge. We report instances of unitary
transformations that are equivalent to 2D QWs up to $20$ time steps, mapping
localized inputs to superpositions of up to $800$ modes arranged in a square
lattice. The same amount of modes would have required hundreds of time steps
in the 1D case, leading to beams with much higher transverse momentum, which
inherently suffer faster diffraction and need larger camera sensors to be
detected.

Figure 1: QWs in the space of light transverse momentum. (a) Photonic modes
implementing the position states on the lattice. For each mode carrying
$m_{x}$ and $m_{y}$ units of transverse momentum $\Delta k_{\perp}$ in the $x$
and $y$ directions, respectively, we plot the linear phase profile in the
transverse $xy$ plane. (b) LC pattern of a $g$-plate. The local molecular
director forms an angle $\theta$ with the $x$ axis. In a $g$-plate, we have
$\theta(x)=\pi x/\Lambda$, with $\Lambda$ being the spatial period. The
birefringence is uniform and electrically tunable by applying a voltage to the
cell [31].

## Results

### QWs in the light transverse momentum space via LCMSs

Figure 2: Large-scale mode mixing via LCMSs. (a) Three LCMSs
($Q_{1},Q_{2},Q_{3}$) implement the optical transformation corresponding to
the desired multi-mode mixing $\mathcal{U}$. The inset illustrates a LCMS with
its LC optic-axis pattern. Off-diagonal elements of the LCMS Jones matrix flip
the polarization handedness and add a space-dependent conjugate phase
modulation on orthogonal circular polarization components $\ket{L}$ and
$\ket{R}$. (b) Top to bottom. Values of $\theta(x,y)$ obtained by
straightforward resolution of Eq. (7) are typically discontinuous. A numerical
routine is devised to match different solutions at each transverse position to
enforce continuity, which is necessary for a real device to be fabricated. The
resulting pattern is often characterized by the emergence of vortices,
isolated points where the LC orientation is undefined. A microscope image of a
LCMS taken between crossed polarizers reveals its optic-axis pattern. (c) The
mode sorting is realized in the focal plane of a lens (F), where modes appear
as a 2D array of Gaussian beams separated by $\Delta k_{\perp}$. Each spot is
a superposition of the polarization (coin) states $\\{\ket{L},\ket{R}\\}$.

QWs represent a convenient framework for building unitary transformations to
be directly implemented in a photonic circuit. These are prototypical
discrete-time dynamics of a particle (walker) on a lattice, whose motion is
conditioned by a spin-like internal degree of freedom (coin). In a 2D
configuration, position states $\ket{m_{x},m_{y}}$ ($m_{x},m_{y}$ are
integers), associated with lattice sites and spanning the Hilbert space
$\mathcal{H}_{\ell}$, are combined with coin states $(\ket{0},\ket{1})$
spanning the space $\mathcal{H}_{s}$ to form the circuit modes. We consider
two-level coin systems and assume that the system is prepared in a single
input mode, that is a localized walker
${\ket{\psi_{0}}=\ket{m_{x}=0,m_{y}=0}\otimes\ket{\phi_{0}}}$, where
$\ket{\phi_{0}}$ is the input coin state. After $t$ steps, the system is
mapped into a linear superposition of multiple modes, whose number scales
linearly with step number:

$\displaystyle\begin{split}\ket{\psi_{t}}&=U_{0}^{t}\ket{\psi_{0}}=\\\
&=\sum_{m_{x},m_{y}}\sum_{j\in\\{0,1\\}}c_{m_{x},m_{y},j}\ket{m_{x},m_{y}}\otimes\ket{j}.\end{split}$
(1)

Here, $U_{0}$ is the single-step evolution operator. We assume this operator
to be identical at each step, though this condition can be relaxed to obtain
more general transformations associated with time-dependent QWs.

In this paper, we focus on the QWs introduced in Ref. [28], where the single-
step evolution operator is

$U_{0}(\alpha)=T_{y}(\alpha)T_{x}(\alpha)W.$ (2)

Here, $W$ is the coin rotation operator, reading

$W=\frac{1}{\sqrt{2}}\begin{pmatrix}1&&i\\\ i&&1\end{pmatrix},$ (3)

and

$T_{x}(\alpha)=\begin{pmatrix}\cos(\alpha/2)&&i\sin(\alpha/2)\hat{t}_{x}\\\
i\sin(\alpha/2)\hat{t}^{\dagger}_{x}&&\cos(\alpha/2)\end{pmatrix}$ (4)

is the translation operator along $m_{x}$, with
${\hat{t}_{x}\ket{m_{x},m_{y}}=\ket{m_{x}-1,m_{y}}}$. A similar expression
holds for $T_{y}(\alpha)$. The parameter $\alpha$ tunes the hopping amplitudes
between neighboring sites. We specifically set $\alpha=\pi/2$.

Our photonic implementation of the QW states defined in Eq. (1) employs
optical modes having the following expression:

$\ket{m_{x},m_{y},j}=A(x,y,z)e^{ik_{z}z}e^{i(m_{x}x+m_{y}y)\Delta
k_{\perp}}\ket{j},$ (5)

where $A(x,y,z)$ is a Gaussian envelope with a beam waist $w_{0}$, $k_{z}$ is
the wavevector $z$ component, $\Delta k_{\perp}$ is a unit of transverse
momentum, and $\ket{j}$ is a left/right circular polarization state
$\ket{L}$/$\ket{R}$, respectively (see Fig. 1(a)). To have a negligible cross-
talk between these modes, their waist radius must be greater than $2\pi/\Delta
k_{\perp}$ [28]. The most straightforward way to engineer the QW dynamics with
these modes is by cascading a sequence of polarization gratings having
${\Lambda=2\pi/\Delta k_{\perp}}$ as their spatial period. Intuitively, these
give photons a transverse momentum kick equal to $\pm\Delta k_{\perp}$,
depending on the polarization being left or right circular, respectively, thus
implementing the QW shift operator. This is the key idea at the basis of the
first experiment demonstrating QWs with such transverse modes, with
polarization gratings realized in terms of LCMSs termed _g_ -plates [28] (see
Fig. 1(b)).

As anticipated, LCMSs consist of a micrometric nematic LC layer sandwiched
between two glass plates, whose internal sides are coated with a transparent
conductive material to enable the application of electric fields. Such devices
can be modeled as standard waveplates with an inhomogeneous optic-axis
orientation. In the circular polarization basis, their Jones matrix reads

$Q_{\delta}(\theta)=\begin{pmatrix}\cos(\delta/2)&&i\sin(\delta/2)e^{-2i\theta(x,y)}\\\
i\sin(\delta/2)e^{2i\theta(x,y)}&&\cos(\delta/2)\end{pmatrix}.$ (6)

Here, $\delta$ is the optical birefringence parameter determined by the out-
of-plane tilt angle of LC molecules, controlled by the electric field
amplitude, and $\theta(x,y)$ is the optic-axis orientation with respect to the
reference $x$ axis. Patterns of LC orientation are imprinted via a
photoalignment technique [31, 27]. Diagonal elements of the LCMS matrix
(proportional to $\cos{(\delta/2)}$) leave part of the beam unaltered. Off-
diagonal elements (proportional to $\sin{(\delta/2)}$) flip the polarization
handedness and add a space-dependent geometric phase (equal to $2\theta$ and
opposite for orthogonal circular polarizations), as pictorially shown in Fig.
2(a). The action of a _g_ -plate, where ${\theta(x)=\pi x/\Lambda}$, is
equivalent to the translation operator of Eq. (4), with $\alpha=\delta$. Using
classical light, 1D QWs up to 14 steps [29, 32] and 2D QWs up to 5 steps [28]
have been realized via the action of several cascaded _g_ -plates. Using a
two-photon input state, 3 steps of a 2D QW have been reported, with the walk
length limited by the number of available single-photon detectors and optical
losses [33]. The latter indeed represents a key limiting factor in multi-
photon experiments. The number of devices (or the circuit depth in integrated
architectures) scales linearly with the number of steps, therefore losses
increase exponentially and severely limit the possibility of implementing
large-scale evolutions in a genuinely quantum regime.

### Large-scale mode mixing via three LCMSs

#### Minimal LCMSs scheme

In typical experiments using LCMSs to realize photonic circuits for QWs,
diffraction between consecutive devices is avoided. As such, the action of a
long sequence of LCMSs is captured by the product of the Jones matrices of
individual LCMSs, each featuring the form of Eq. (6). The resulting matrix
$\mathcal{L}$ is thus the Jones matrix associated with the entire system,
having spatial frequencies that increase with the number of steps to be
realized.

The entire sequence can be replaced by a shorter chain of LCMSs. It is well
known that an arbitrary polarization transformation can be realized via a
minimal sequence of three waveplates [34, 35]. A possible choice is a half-
wave plate sandwiched between two quarter-wave plates, that is
$Q_{\pi/2}(\theta_{3})Q_{\pi}(\theta_{2})Q_{\pi/2}(\theta_{1})$ (see Eq. (6)).

The possibility of patterning the optic axis of LCMSs allows us to implement
an arbitrary transformation at each transverse position, thus decomposing the
target unitary $\mathcal{L}(x,y)$ into the action of three plates (see Fig.
2(a)):

$\mathcal{L}(x,y)=Q_{\pi/2}(\theta_{3}(x,y))Q_{\pi}(\theta_{2}(x,y))Q_{\pi/2}(\theta_{1}(x,y)).$
(7)

To achieve this goal, we first compute the overall Jones matrix $\mathcal{L}$
associated with the entire walk, and then solve Eq. (7) in terms of
$\theta_{1},\theta_{2},\theta_{3}$ at each transverse position [30]. These
equations admit multiple analytical solutions, and their straightforward use
typically leads to discontinuous LC patterns featuring several disclinations
along extended lines. As illustrated in the inset of Fig. 2(b), here we
develop an optimization routine (detailed in the Methods) enabling us to
enforce continuous patterns for LC angles. This procedure tolerates
singularities for the LC orientation, appearing as vortices with elementary
charge, that are clearly visible in the example in Fig. 2 and in the other
patterns presented in Fig. 3. The vortex charge quantifies the rotation of LC
molecules (modulo $\pi$) when following a closed trajectory around the
singular point. The elementary charges are $\pm 1/2$.

In Fig. 3(a), we plot the optic-axis modulation of the first LCMS
($\theta_{1}(x,y)$) employed for the simulation of $3$, $5$, $10$, and $20$
steps of the QW protocol described above. The minimal transformation naturally
preserves the spatial periodicity $\Lambda$ characterizing the original
cascaded scheme. The plotted modulations are relative to a $3\Lambda\times
3\Lambda$ square, with $\Lambda=5$ mm in the experiment.

#### Reading out the power spectrum of output states

Figure 3: 2D QWs via spin-orbit photonics. (a) Optic-axis modulation of the
first metasurface ($\theta_{1}(x,y)$) employed for the simulation of the 2D
QW. (b) Experimental images obtained for a $\ket{R}$-polarized input state,
from which the walker probability distribution $P_{\text{exp}}(m_{x},m_{y})$
is extracted (c), and compared with the theoretical prediction
$P_{\text{th}}(m_{x},m_{y})$ (d). For each realization, we report the value of
the similarity, computed as the average of four independent measurements. Rows
refer to $3$, $5$, $10$, and $20$ time steps ($t$), respectively.

The final stage of a mode-mixing experiment consists of the mode sorting and
detection stage. The modes of Eq. (5) can be spatially resolved on a CCD
camera placed in a focal plane of a lens, implementing an optical Fourier
Transform (see Fig. 2(c)). As discussed above, these modes have negligible
overlap as long as $w_{0}\geq\Lambda$ [28], where $w_{0}$ is the beam waist. A
complete description of the experimental setup is provided in the Methods.

Representative experimental images for a $\ket{R}$-polarized localized input
after ${3,5,10,}$ and 20 steps are shown in Fig. 3(b), from which the QW
probability distributions can be extracted (see Fig. 3(c)). Each light spot is
associated with a walker site, with probability given by the normalized light
intensity within that spot. The output modes distribution is directly related
to the unitary map, in this case, our specific QW protocol. The specific
orientation of the walker distribution reflects the structure of the QW
protocol, that misses a coin rotation between consecutive translations along
the $x$ and $y$ directions (see Ref. [28]). When adding such additional
operation, the walker symmetrically spreads across the entire lattice [33].
The procedure to extract the walker probability distribution is outlined in
the Methods.

Figure 3(d) shows the corresponding theoretical probability distributions. The
agreement between the theoretical predictions and the experimental
observations is quantified in terms of the similarity

$S=\left(\sum_{m_{x},m_{y}}\sqrt{P_{\text{{exp}}}(m_{x},m_{y})P_{\text{{th}}}(m_{x},m_{y})}\right)^{2},$
(8)

where $P_{\text{{exp}}}$ and $P_{\text{{th}}}$ are the normalized experimental
and theoretical probability distributions, respectively. A good agreement with
the theory is observed in all our experiments, with similarity always above
$87\%$. The uncertainties are computed as the standard deviation over $4$
independent measurements. The decrease in similarity observed with the
increase in the number of modes is ascribed to the increasing complexity of
the LCMS patterns when targeting longer evolutions. Experimental results
obtained with different input coin states are reported in the Supplementary
Material.

Simple propagation through a lens does not allow accessing all output modes.
At each spot we are not discriminating light polarization, and intensities are
given by the incoherent sum of both left and right states. However, we can use
a $g$-plate [28], that is a LCMS with a linear variation of optic-axis angle,
having a spatial period ${\Lambda_{g}=250\,\mu\text{m}\ll\Lambda}$ before the
lens. In this way, left (right) polarized modes get a large momentum kick
$\Delta k_{g}\gg\Delta k_{\perp}$ in the positive (negative) direction, so
that they are imaged at different positions on the camera sensor. Figure
4(a-c) shows the projections on $\ket{L}$ and $\ket{R}$ of the probability
distribution for $5$ steps and a localized $\ket{R}\text{-}\text{polarized}$
input. Since the $g$-plate is partially tuned, a fraction of the beam does not
get any polarization-dependent kick, as such in the central part of the camera
we obtain the total intensity distribution.

Contrary to the classical case, the output mode distribution depends on the
input coin state. This is a consequence of the interference among different
paths, which intrinsically distinguishes the QW from its classic counterpart.
Nevertheless, the quantum process always presents ballistic features [36].
Figure 4(d) shows the variance over time of the output probability
distributions for our QW protocol, both in the $x$ and $y$ direction. We
report the measured $\sigma_{x}^{2}$ and $\sigma_{y}^{2}$ for $3$, $5$, $10$,
and $20$ time steps for a localized $\ket{R}$-polarized input. The ballistic
trend ${\sigma^{2}\propto t^{2}}$ is well captured in our experiments.
Deviations at 20 time steps are probably due to a larger fraction of the field
that remains close to the central mode (see Fig. 3(c-d), bottom row). Variance
plots relative to different input polarizations are provided in the
Supplementary Material.

Figure 4: Resolving the totality of the modes. A $g$-plate with a smaller
spatial period $\Lambda_{g}\ll\Lambda$ placed before the Fourier lens allows
us to resolve separately light with orthogonal circular polarizations. (a)
Experimental images, (b) experimental reconstructions
$P_{\text{exp}}(m_{x},m_{y})$ and (c) theoretical predictions
$P_{\text{th}}(m_{x},m_{y})$ of the output distribution and its projections on
$\ket{L}$ and on $\ket{R}$. A localized $\ket{R}$-polarized input after 5
steps is considered. (d) Variance of the output distribution along $x$ and
$y$. The experimental points correctly reproduce the expected ballistic
behavior. Figure 5:  Unitary maps obtained by reconfiguring a sequence of
three plates. (a) LCMSs’ optical birefringence parameters
${\delta_{i}\in[0,2\pi)}$ (here represented as the tilt of the LC molecules
with respect to the propagation axis $z$) can be electrically tuned. Moreover,
their lateral relative position can also be adjusted, both in the $x$ and $y$
direction (red arrows). (b) When shifting the plates, the overall
transformation is still a unitary circuit coupling transverse wavevector
modes. The three panels show the output intensity distribution computed
numerically when the LCMSs designed to implement the 5-step QW are not
shifted, when the second and the third are laterally shifted in opposite
directions along both $x$ and $y$ of $\pm 1\text{ mm}$ and $\pm 2\text{ mm}$,
respectively. (c) Histogram of the number of active modes for $500$ unitary
maps, numerically realized by randomly varying the birefringence parameters,
with ${\delta_{i}\in[0,2\pi)}$ (red), and $500$ maps realized by randomly
varying the relative position in a range $\leq 2.5$ mm (blue) of the three
LCMSs implementing $20$ QW steps.

#### Reconfigurability

In the experiments described above, LCMS patterns have been computed to yield
the transformation associated with a target QW. To reproduce the correct map,
they must be stacked carefully matching their transverse modulations to make
Eq. (7) valid in each point. Moreover, the applied voltages must be adjusted
so that they work as half-wave and quarter-wave plates. In its current
implementation, the platform cannot be reprogrammed: if the target QW changes,
a new set of three plates should be fabricated with the correct pattern of
optic axes. However, when changing the plates’ birefringence and relative
positions (see Fig. 5(a)), the overall transformation remains a unitary mode
coupler for the transverse modes defined in Eq. (5). This result is not
trivial if one considers that these modes are a discrete subset in a continuum
of modes associated with the transverse wavevector, which is a 2D continuous
variable. In Fig. 5(b), we compare the output intensity distributions computed
numerically when adding lateral shifts to the LCMSs designed for a 5-step QW.
Importantly, the output field corresponds to a well-defined grid of Gaussian
modes in all cases.

To provide a quantitative analysis of the properties of achievable
transformations, we computed the number of times a lattice mode
$\ket{m_{x},m_{y}}$ is activated when varying some of the adjustable
parameters. In particular, an output state $\ket{m_{x},m_{y}}$ is considered
to be active when its intensity is above the threshold value $1/d^{2}$, where
${d=2t+1}$ and $t$ is the number of QW steps. This value corresponds to the
intensity of a flat probability distribution of $d^{2}$ lattice modes. The
histogram distribution of the number of active modes in the two configurations
for the LCMSs designed for a 20-step QW is plotted in Fig. 5(c). The latter
shows that changing the plates’ birefringence can significantly alter the
connectivity of the circuit, which eventually approaches the identity
transformation when all values of $\delta$ are close to $2\pi$. On the other
hand, adjusting the plates’ relative displacements (while keeping the
retardations fixed) have a much less pronounced impact on this aspect. The set
of unitary processes explored so far are measurably diverse from the initial
$20$-step QW process $U_{20}$. To provide a quantitative estimate of this
diversity, we computed their average fidelity with respect to $U_{20}$, which
reads $\bar{F}=(45\pm 10)\%$, where
$F(U)=\frac{1}{2}\absolutevalue{\text{Tr}(U^{\dagger}U_{20})}$ and the
uncertainty is the standard deviation. A more detailed analysis of the
properties of the achievable transformations goes beyond the scope of the
present work and will be investigated in the near future.

## Discussion and Conclusions

We realized a compact photonic circuit that implements unitary transformations
associated with 2D QWs on transverse modes of structured light. Compressing
multiple-step QW dynamics into a limited number of spin-orbit devices leads to
greater complexity in their optic-axis patterns while keeping the size of the
setup the same.

The complexity of the explorable evolutions is currently limited by our
fabrication routine, but this can be overcome in the future by optimizing
specific stages of the procedure, or by choosing a different type of spin-
orbit devices, like dielectric metasurfaces featuring sub-wavelength
resolution [37].

Our platform is versatile, scalable, and couples optical modes of free space
propagation, with partial reconfigurability given by the tunable birefringence
parameter and the relative displacements of the plates. This reconfigurability
might be further amplified by replacing our metasurfaces with LC displays with
locally tunable birefringence. However, these typically operate in reflection
mode, while our platform works in transmission, with more than 80$\%$ of the
input light transmitted by each device.

The unitary transformations we have presented are not arbitrary and are
inherently characterized by translation invariance. The roadmap to leverage
this approach to realize more general transformations, possibly universal,
necessarily requires considering diffraction between consecutive LCMSs to
break the translation symmetry, using for instance concepts already
demonstrated for multi-plane light converters [38, 39].

The limited amount of losses will allow employing these circuits to explore
multi-photon evolutions, by leveraging also novel detection systems like SPAD
arrays [14], single photon cameras with high temporal resolution [40, 41], or
ultra-sensitive cameras based on superconducting nanowires detectors [42].

## Acknowledgements

We acknowledge Alexandre Dauphin and Alioscia Hamma for fruitful discussions.
MGA, FDC, LM, and FC acknowledge support from the PNRR MUR project
PE0000023-NQSTI.

## Methods

Figure 6: Experimental implementation. (a) Experimental setup to engineer QW
dynamics. The entire evolution is compressed within only three LCMSs. (b)
Reconstruction of the probability distribution $P_{\text{exp}}(m_{x},m_{y})$
from the experimental image. After the central ${\ket{m_{x},m_{y}}=\ket{0,0}}$
spot has been determined, the probability of each site is computed as the
normalized integrated intensity within the corresponding light spot. Figure 7:
Numerical routine to retrieve 2D continuous LCMS optic-axis modulations. (a)
Different scenarios (i)-(ii)-(iii) are illustrated, depending on the specific
current position on the plate ${\mathbf{r}_{ij}}$ (yellow square). The violet
path contains discrete positions where the optimization algorithm has already
been executed. The green crosses mark neighboring elements ${\mathbf{r}_{n}}$
where a continuous modulation has already been found, and are therefore
involved in the optimization of the metric $d$ (see text). Neighboring
elements where the algorithm has not been executed yet are marked by red
crosses. (b) Full pattern of one of the LCMSs designed to implement 10-step QW
($3\Lambda\times 3\Lambda$ square, with $\Lambda=5$ mm), imaged when the plate
is positioned between crossed polarizers revealing the LC’s in-plane
orientation.

###  .1 Experimental Setup

Our experiments are realized with the setup sketched in Fig. 6(a). A He–Ne
laser beam (wavelength ${\lambda=633}$ nm) passes through a telescope system,
consisting of two aspheric lenses $L_{1}$ and $L_{2}$ (with focal lengths
${f_{1}=5}$ cm and ${f_{2}=30}$ cm) and a $25$ $\mu$m-pinhole (Ph). The latter
operates as a spatial filter. As discussed in the text, a convenient choice
for the beam waist is ${w_{0}\simeq\Lambda=5}$ mm. A combination of a half-
wave plate (HWP) and a quarter-wave plate (QWP) sets the desired coin-
polarization input state. The beam then propagates through the three LCMSs
implementing the full dynamics. These are mounted in practical mounts allowing
us to adjust their transverse displacement with micrometric precision, both in
the $x$ and $y$ direction. This is needed for an accurate alignment of the
plates, which makes Eq. (7) valid at each point. At the exit of the last
metasurface, we set a lens $L_{3}$ (with focal length $f_{3}=50$ cm), Fourier-
transforming light momenta into positions on the CCD camera placed in the
focal plane.

###  .2 Search for continuous optic-axis modulations

To solve Eq. (7), we decompose the optical sequence $\mathcal{L}$ and the
target operator $\mathcal{U}$ in terms of the generators of SU(2):

$\sum_{i=0}^{3}\ell_{i}(x,y)\sigma_{i}=\sum_{i=0}^{3}c_{i}(x,y)\sigma_{i},$
(9)

where $\sigma_{0}$ is the identity matrix and $\sigma_{1}$, $\sigma_{2}$, and
$\sigma_{3}$ are the three Pauli matrices. By equalizing one by one the
corresponding terms of the two sums, one can determine the optic-axis
modulations for the three LCMSs: $\theta_{\alpha}(x,y)$,
$\alpha\in\\{1,2,3\\}$. However, multiple solutions exhibiting singular points
and sudden jumps are possible. To avoid artificial jumps, a dedicated
algorithm is devised to pick among all possible solutions the one that
minimizes the following metric at each transverse position:

$d_{ij}=\sum_{n=1}^{N_{ij}}\bigg{[}\sum_{\alpha=1}^{3}\left(\theta_{\alpha}({\mathbf{r}_{ij}})-\theta_{\alpha}({\mathbf{r}_{n}})\right)^{2}\bigg{]}.$
(10)

The latter provides a measure of the distance between the orientation of the
optic axis of the three LCMSs at the current transverse position
${\mathbf{r}_{ij}}$ and its $N_{ij}$ neighboring elements ${\mathbf{r}_{n}}$
(chosen within a tunable range), where a possible modulation has already been
found. The working principle of the algorithm is illustrated in several
possible scenarios in Fig. 7(a). The metric of Eq. (10) allows our numerical
routine to find continuous solutions embedding isolated vortex singularities,
as those displayed in the LC’s pattern in Fig. 7(b). As expected, the
complexity of these modulations increases with the complexity of the simulated
process (cf. Fig. 3(a)).

###  .3 Reconstruction of the probability distribution

To reconstruct the QW probability distribution from the experimental image,
the coordinates of the lattice site $\ket{m_{x},m_{y}}=\ket{0,0}$ have to be
determined. To do so, we set the birefringence parameter of the three LCMSs to
${\delta_{1}=\delta_{2}=\delta_{3}=2\pi}$. In this configuration, the
metasurfaces act as the identity operator at each point $(x,y)$, and only the
central input mode is transmitted. Starting from the coordinates of the
corresponding spot in the focal plane of $L_{3}$, we build an array of equally
spaced square regions on the image, each associated with an output mode. Then,
we set ${\delta_{1}=\delta_{3}=\pi/2}$ and ${\delta_{2}=\pi}$, so that the
plates implement the desired complex polarization transformation which
simulates the target dynamics $\mathcal{U}(x,y)$. By integrating light
intensity within each square, which provides a good estimate of the amount of
input light coupled to each output mode, and normalizing it to the total
intensity, we eventually reconstruct the experimental probability distribution
$P_{\text{exp}}(m_{x},m_{y})$. This procedure is depicted in Fig. 6(b). The
agreement between theory and experimental results is quantified in terms of
the similarity estimator $S$ (cf. Eq. (8)).

## References

  * Bogaerts  _et al._ [2020] Bogaerts, W., Pérez, D., Capmany, J., Miller, D. A. B., Poon, J., Englund, D., Morichetti, F. and Melloni, A.,  _Programmable photonic circuits_ , Nature 586, 207 (2020).
  * Nikkhah  _et al._ [2024] Nikkhah, V., Pirmoradi, A., Ashtiani, F., Edwards, B., Aflatouni, F. and Engheta, N.,  _Inverse-designed low-index-contrast structures on a silicon photonics platform for vector–matrix multiplication_ , Nat. Photonics (2024).
  * Kirsch  _et al._ [2021] Kirsch, M. S., Zhang, Y., Kremer, M., Maczewsky, L. J., Ivanov, S. K., Kartashov, Y. V., Torner, L., Bauer, D., Szameit, A. and Heinrich, M.,  _Nonlinear second-order photonic topological insulators_ , Nat. Phys. 17, 995 (2021).
  * Hoch  _et al._ [2022] Hoch, F. _et al._ ,  _Reconfigurable continuously-coupled 3D photonic circuit for Boson Sampling experiments_ , npj Quantum Inf. 8, 55 (2022).
  * O’Brien [2007] O’Brien, J. L.,  _Optical Quantum Computing_ , Science 318, 1567 (2007).
  * McMahon [2023] McMahon, P. L.,  _The physics of optical computing_ , Nat. Rev. Phys. 5, 717 (2023).
  * Wetzstein  _et al._ [2020] Wetzstein, G., Ozcan, A., Gigan, S., Fan, S., Englund, D., Soljačić, M., Denz, C., Miller, D. A. B. and Psaltis, D.,  _Inference in artificial intelligence with deep optics and photonics_ , Nature 588, 39 (2020).
  * Raghu and Haldane [2008] Raghu, S. and Haldane, F. D. M.,  _Analogs of quantum-Hall-effect edge states in photonic crystals_ , Phys. Rev. A 78, 033834 (2008).
  * Aspuru-Guzik and Walther [2012] Aspuru-Guzik, A. and Walther, P.,  _Photonic quantum simulators_ , Nat. Phys. 8, 285 (2012).
  * Lustig  _et al._ [2019] Lustig, E., Weimann, S., Plotnik, Y., Lumer, Y., Bandres, M. A., Szameit, A. and Segev, M.,  _Photonic topological insulator in synthetic dimensions_ , Nature 567, 356 (2019).
  * Hiekkamäki and Fickler [2021] Hiekkamäki, M. and Fickler, R.,  _High-Dimensional Two-Photon Interference Effects in Spatial Modes_ , Phys. Rev. Lett. 126, 123601 (2021).
  * Lib and Bromberg [2022] Lib, O. and Bromberg, Y.,  _Quantum light in complex media and its applications_ , Nat. Phys. 18, 986 (2022).
  * Courme  _et al._ [2023] Courme, B., Cameron, P., Faccio, D., Gigan, S. and Defienne, H.,  _Manipulation and Certification of High-Dimensional Entanglement through a Scattering Medium_ , PRX Quantum 4, 010308 (2023).
  * Makowski  _et al._ [2024] Makowski, A., Dabrowski, M., Antolovic, I. M., Bruschini, C., Defienne, H., Charbon, E., Lapkiewicz, R. and Gigan, S.,  _Large reconfigurable quantum circuits with SPAD arrays and multimode fibers_ , Optica 11, 340 (2024).
  * Goel  _et al._ [2024] Goel, S., Leedumrongwatthanakun, S., Valencia, N. H., McCutcheon, W., Tavakoli, A., Conti, C., Pinkse, P. W. H. and Malik, M.,  _Inverse design of high-dimensional quantum optical circuits in a complex medium_ , Nat. Phys. 20, 232 (2024).
  * Zhong  _et al._ [2020] Zhong, H.-S. _et al._ ,  _Quantum computational advantage using photons_ , Science 370, 1460 (2020).
  * Arrazola  _et al._ [2021] Arrazola, J. M. _et al._ ,  _Quantum circuits with many photons on a programmable nanophotonic chip_ , Nature 591, 54 (2021).
  * Wang  _et al._ [2020] Wang, J., Sciarrino, F., Laing, A. and Thompson, M. G.,  _Integrated photonic quantum technologies_ , Nat. Photonics 14, 273 (2020).
  * Flamini  _et al._ [2018] Flamini, F., Spagnolo, N. and Sciarrino, F.,  _Photonic quantum information processing: a review_ , Rep. Prog. Phys. 82, 016001 (2018).
  * Slussarenko and Pryde [2019] Slussarenko, S. and Pryde, G. J.,  _Photonic quantum information processing: A concise review_ , Appl. Phys. Rev. 6, 041303 (2019).
  * [21] Bouchard, F., Fenwick, K., Bonsma-Fisher, K., England, D., Bustard, P. J., Heshami, K. and Sussman, B.,  _Programmable Photonic Quantum Circuits with Ultrafast Time-bin Encoding_ , arXiv:2404.17657 .
  * Bouchard  _et al._ [2022] Bouchard, F., England, D., Bustard, P. J., Heshami, K. and Sussman, B.,  _Quantum Communication with Ultrafast Time-Bin Qubits_ , PRX Quantum 3, 010332 (2022).
  * Matthès  _et al._ [2019] Matthès, M. W., del Hougne, P., de Rosny, J., Lerosey, G. and Popoff, S. M.,  _Optical complex media as universal reconfigurable linear operators_ , Optica 6, 465 (2019).
  * Cristiani  _et al._ [2022] Cristiani, I. _et al._ ,  _Roadmap on multimode photonics_ , J. Opt. 24, 083001 (2022).
  * Piccardo  _et al._ [2021] Piccardo, M. _et al._ ,  _Roadmap on multimode light shaping_ , J. Opt. 24, 013001 (2021).
  * Kupianskyi  _et al._ [2023] Kupianskyi, H., Horsley, S. A. R. and Phillips, D. B.,  _High-dimensional spatial mode sorting and optical circuit design using multi-plane light conversion_ , APL Photonics 8, 026101 (2023).
  * Rubano  _et al._ [2019] Rubano, A., Cardano, F., Piccirillo, B. and Marrucci, L.,  _Q-plate technology: a progress review [Invited]_ , J. Opt. Soc. Am. B 36, D70 (2019).
  * D’Errico  _et al._ [2020a] D’Errico, A., Cardano, F., Maffei, M., Dauphin, A., Barboza, R., Esposito, C., Piccirillo, B., Lewenstein, M., Massignan, P. and Marrucci, L.,  _Two-dimensional topological quantum walks in the momentum space of structured light_ , Optica 7, 108 (2020a).
  * D’Errico  _et al._ [2020b] D’Errico, A., Di Colandrea, F., Barboza, R., Dauphin, A., Lewenstein, M., Massignan, P., Marrucci, L. and Cardano, F.,  _Bulk detection of time-dependent topological transitions in quenched chiral models_ , Phys. Rev. Res. 2, 023119 (2020b).
  * Di Colandrea  _et al._ [2023] Di Colandrea, F., Babazadeh, A., Dauphin, A., Massignan, P., Marrucci, L. and Cardano, F.,  _Ultra-long quantum walks via spin–orbit photonics_ , Optica 10, 324 (2023).
  * Piccirillo  _et al._ [2010] Piccirillo, B., D’Ambrosio, V., Slussarenko, S., Marrucci, L. and Santamato, E.,  _Photon spin-to-orbital angular momentum conversion via an electrically tunable q-plate_ , Appl. Phys. Lett. 97, 241104 (2010).
  * D’Errico  _et al._ [2021] D’Errico, A., Barboza, R., Tudor, R., Dauphin, A., Massignan, P., Marrucci, L. and Cardano, F.,  _Bloch–Landau–Zener dynamics induced by a synthetic field in a photonic quantum walk_ , APL Photonics 6, 020802 (2021).
  * Esposito  _et al._ [2022] Esposito, C., Barros, M. R., Durán Hernández, A., Carvacho, G., Di Colandrea, F., Barboza, R., Cardano, F., Spagnolo, N., Marrucci, L. and Sciarrino, F.,  _Quantum walks of two correlated photons in a 2D synthetic lattice_ , npj Quantum Inf. 8, 34 (2022).
  * Simon and Mukunda [1990] Simon, R. and Mukunda, N.,  _Minimal three-component SU(2) gadget for polarization optics_ , Phys. Lett. A 143, 165 (1990).
  * Sit  _et al._ [2017] Sit, A., Giner, L., Karimi, E. and Lundeen, J. S.,  _General lossless spatial polarization transformations_ , J. Opt. 19, 094003 (2017).
  * Tang  _et al._ [2018] Tang, H. _et al._ ,  _Experimental two-dimensional quantum walk on a photonic chip_ , Sci. Adv. 4, eaat3174 (2018).
  * Yu and Capasso [2014] Yu, N. and Capasso, F.,  _Flat optics with designer metasurfaces_ , Nat. Mater. 13, 139 (2014).
  * Morizur  _et al._ [2010] Morizur, J.-F., Nicholls, L., Jian, P., Armstrong, S., Treps, N., Hage, B., Hsu, M., Bowen, W., Janousek, J. and Bachor, H.-A.,  _Programmable unitary spatial mode manipulation_ , J. Opt. Soc. Am. A 27, 2524 (2010).
  * Fontaine  _et al._ [2019] Fontaine, N. K., Ryf, R., Chen, H., Neilson, D. T., Kim, K. and Carpenter, J.,  _Laguerre-Gaussian mode sorter_ , Nat. Commun. 10, 1865 (2019).
  * Nomerotski  _et al._ [2023] Nomerotski, A., Chekhlov, M., Dolzhenko, D., Glazenborg, R., Farella, B., Keach, M., Mahon, R., Orlov, D. and Svihra, P.,  _Intensified Tpx3Cam, a fast data-driven optical camera with nanosecond timing resolution for single photon detection in quantum applications_ , J. Instrum. 18, C01023 (2023).
  * Zia  _et al._ [2023] Zia, D., Dehghan, N., D’Errico, A., Sciarrino, F. and Karimi, E.,  _Interferometric imaging of amplitude and phase of spatial biphoton states_ , Nat. Photonics 17, 1009 (2023).
  * Oripov  _et al._ [2023] Oripov, B. G., Rampini, D. S., Allmaras, J., Shaw, M. D., Nam, S. W., Korzh, B. and McCaughan, A. N.,  _A superconducting nanowire single-photon camera with 400,000 pixels_ , Nature 622, 730 (2023).

Supplementary Material for:
Large-scale spin-orbit photonic circuits in two dimensions

## Supplementary Data

We provide experimental results for different input coin-polarization states.
Figures S1-S2-S3 show the probability distributions obtained for a $\ket{H}$,
$\ket{V}$, and $\ket{L}$ polarization input state (cf. Fig. 3), respectively.
Figure S4 shows the QW ballistic spreading for the same polarization inputs
(cf. Fig. 4(d)).

Figure S1: 2D QWs via spin-orbit photonics. (a) Experimental images obtained
for a $\ket{H}$-polarized input state, from which the walker probability
distribution $P_{\text{exp}}(m_{x},m_{y})$ is extracted (b), and compared with
the theoretical prediction $P_{\text{th}}(m_{x},m_{y})$ (c). For each
realization, we report the value of the similarity, computed as the average of
four independent measurements. The rows refer to $3$, $5$, $10$, and $20$ time
steps ($t$), respectively. Figure S2: 2D QWs via spin-orbit photonics. (a)
Experimental images obtained for a $\ket{V}$-polarized input state, from which
the walker probability distribution $P_{\text{exp}}(m_{x},m_{y})$ is extracted
(b), and compared with the theoretical prediction $P_{\text{th}}(m_{x},m_{y})$
(c). For each realization, we report the value of the similarity, computed as
the average of four independent measurements. The rows refer to $3$, $5$,
$10$, and $20$ time steps ($t$), respectively. Figure S3: 2D QWs via spin-
orbit photonics. (a) Experimental images obtained for a $\ket{L}$-polarized
input state, from which the walker probability distribution
$P_{\text{exp}}(m_{x},m_{y})$ is extracted (b), and compared with the
theoretical prediction $P_{\text{th}}(m_{x},m_{y})$ (c). For each realization,
we report the value of the similarity, computed as the average of four
independent measurements. The rows refer to $3$, $5$, $10$, and $20$ time
steps ($t$), respectively. Figure S4: Variance of 2D QW distributions.
Variance of the output distribution along $x$ and $y$, $\sigma_{x}^{2}$ and
$\sigma_{y}^{2}$, for different input states: (a) $\ket{H}$, (b) $\ket{V}$,
(c) $\ket{L}$. The experimental points correctly reproduce the expected
ballistic behavior.