# The Deep Space Quantum Link: Prospective Fundamental Physics Experiments
using Long-Baseline Quantum Optics

Makan Mohageg  E-mail<EMAIL_ADDRESS>Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, California, U.S.A.  Luca
Mazzarella  Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, California, U.S.A.  Dmitry V. Strekalov  Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, California, U.S.A.  Nan Yu  Jet
Propulsion Laboratory, California Institute of Technology, Pasadena,
California, U.S.A.  Aileen Zhai  Jet Propulsion Laboratory, California
Institute of Technology, Pasadena, California, U.S.A.  Spencer Johnson  Dept.
of Physics, University of Illinois at Urbana-Champaign; Illinois Quantum
Information Science & Technology Center, Urbana, Illinois, U.S.A.  Charis
Anastopoulos  Department of Physics, University of Patras, Patras, Greece
Jason Gallicchio  Department of Physics, Harvey Mudd College, Claremont,
California, U.S.A.  Bei Lok Hu  Maryland Center for Fundamental Physics and
Joint Quantum Institute, University of Maryland, College Park, Maryland,
U.S.A.  Thomas Jennewein  Institute for Quantum Computing and Dep. of Physics
and Astronomy, University of Waterloo, Waterloo, Canada  Shih-Yuin Lin
Department of Physics, National Changhua University of Education, Changhua,
Taiwan  Alexander Ling  Centre for Quantum Technologies and Department of
Physics, National University of Singapore  Christoph Marquardt  Max Planck
Institute for the Science of Light, Erlangen, Germany  Matthias Meister
Institute of Quantum Technologies, German Aerospace Center (DLR), Ulm, Germany
Albert Roura  Institute of Quantum Technologies, German Aerospace Center
(DLR), Ulm, Germany  Lisa Wörner  Institute of Quantum Technologies, German
Aerospace Center (DLR), Ulm, Germany  Wolfgang P. Schleich  Institute of
Quantum Technologies, German Aerospace Center (DLR), Ulm, Germany  Institut
für Quantenphysik and Center for Integrated Quantum Science and Technology
(IQST), Universität Ulm, Ulm, Germany  Hagler Institute for Advanced Study at
Texas A$\&$M University, Texas A$\&$M AgriLife Research, Institute for Quantum
Science and Engineering (IQSE), and Department of Physics and Astronomy, Texas
A$\&$M University, College Station, Texas, U.S.A  Raymond Newell  Los Alamos
National Laboratory, Los Alamos, New Mexico, U.S.A.  Christian Schubert
Institute for Satellite Geodesy and Inertial Sensing, German Aerospace Center
(DLR), Hanover, Germany  Hanover, Germany Leibniz University Hannover,
Institute for Quantum Optics, Hanover, Germany  Giuseppe Vallone  Dipartimento
di Ingegneria dell’Informazione, Università degli Studi di Padova, Padova,
Italy  Padua Quantum Technologies Research Center, Università degli Studi di
Padova, Padova, Italy  Dipartimento di Fisica e Astronomia, Università di
Padova, Padova, Italy  Paolo Villoresi  Dipartimento di Ingegneria
dell’Informazione, Università degli Studi di Padova, Padova, Italy  Padua
Quantum Technologies Research Center, Università degli Studi di Padova,
Padova, Italy  Paul Kwiat<EMAIL_ADDRESS>Dept. of Physics, University of
Illinois at Urbana-Champaign; Illinois Quantum Information Science &
Technology Center, Urbana, Illinois, U.S.A.

###### Abstract

The National Aeronautics and Space Administration’s Deep Space Quantum Link
mission concept enables a unique set of science experiments by establishing
robust quantum optical links across extremely long baselines. Potential
mission configurations include establishing a quantum link between the Lunar
Gateway moon-orbiting space station and nodes on or near the Earth. In this
publication, we summarize the principal experimental goals of the Deep Space
Quantum Link mission. These include long-range teleportation, tests of
gravitational coupling to quantum states, and advanced tests of quantum
nonlocality.

###### Contents

  1. I Introduction: The Case for Deep Space Quantum Optics
  2. II Deep Space Quantum Link Science Experiments
    1. II.1 GR Effects on Light and Tests of the Equivalence Principle
      1. II.1.1 Polarization Rotation of Photons in General Relativity
      2. II.1.2 COW Test with Classical Light, Superposition States, Entangled and Hyper-entangled Photons
      3. II.1.3 Flight Mission Design for Optical COW Tests
      4. II.1.4 Gravitational Redshift in a HOM Interferometer with Frequency-Entangled Photons
      5. II.1.5 Flight Mission Design for Optical COW Tests using HOM Interference
      6. II.1.6 Gravitational Dephasing, Decorrelation, and Decoherence
      7. II.1.7 GR Effects: Summary
    2. II.2 Long-baseline Bell tests
      1. II.2.1 Verification of Long-baseline Quantum Entanglement
      2. II.2.2 Current Status of State-of-the-art Bell Tests
      3. II.2.3 Bell Tests Between Frames with Large Relative Velocities
      4. II.2.4 Human-decision Bell Tests
      5. II.2.5 Flight Mission Design for Bell Tests
      6. II.2.6 Bell Tests: Summary
    3. II.3 Long-Baseline Quantum Teleportation
      1. II.3.1 The Quantum Teleportation Protocol
      2. II.3.2 State-of-the-Art Long-Range Teleportation
      3. II.3.3 Teleportation in the Earth-Moon System
      4. II.3.4 Flight Quantum Memory
      5. II.3.5 Teleportation Mission Design
      6. II.3.6 Quantum Teleportation: Summary
    4. II.4 Potential Applications of Squeezed Light
  3. III The DSQL Mission
    1. III.1 A Sequence of Missions
    2. III.2 DSQL Technology Challenges
    3. III.3 Other Applications of DSQL
  4. IV Conclusion
  5. V Abbreviations
  6. VI Acknowledgments

##  I Introduction: The Case for Deep Space Quantum Optics

Space-based quantum optical links support future networking applications for
quantum sensing, quantum communications, and quantum information science
Bedington  _et al._ (2017); Sidhu  _et al._ (2021); Kaltenbaek  _et al._
(2021); Belenchia  _et al._ (2021). In addition, such links enable new
scientific experiments impossible to reach in terrestrial experiments Rideout
_et al._ (2012). The Deep Space Quantum Link (DSQL) is a spacecraft mission
concept that aims to use extremely long-baseline quantum optical links to test
fundamental quantum physics in novel special and general relativistic regimes
Mohageg  _et al._ (2018); Mazzarella  _et al._ (2021, 2020a). DSQL is
currently in the pre-project developmental phase, with expected mission
integration planned to begin in the late 2020’s. One or more nodes of a DSQL
network could deploy on deep space platforms, such as the Lunar Gateway (LG)
NASA (2021) or Orion modules.

A key challenge of contemporary physics is the reconciliation of gravity and
quantum mechanics. Quantum Field Theory in Curved Spacetime (QFTCST),
established in the 1970’s, is the most reliable theory combining two well-
established theories: quantum field theory for matter and general relativity
for spacetime dynamics. It predicts effects like Hawking radiation from black
holes Hawking (1974), and, with extension to semi-classical gravity, provides
the theoretical framework for inflationary cosmology which foretells a
spatially-flat universe Guth (1981). However, most tests of QFTCST, either in
the laboratory setting of analog gravity, or set in strong-field astrophysical
processes, are indirect tests. DSQL aims to conduct a series of direct tests
of QFTCST in the weak-field regime, accessible through deployment of long-
baseline quantum optical links in the Earth-Moon system.

The essence of the planned experiments with DSQL is to transmit photons
between inertial reference frames and across gravitational gradients to
implement long-baseline teleportation, test Bell’s inequality, conduct
photonic tests of the Weak Equivalence Principle, and validate relativistic
predictions through single- and two-photon interference. In this context, the
fundamental question is how to describe the evolution of the photonic state
between creation and measurement. The simple definition of a trajectory which
makes sense in classical physics is not a suitable notion for the description
of the motion of quantum particles. In quantum physics, the fundamental
concepts are time-evolving wave functions, or more generally, consistent
histories. Quantum systems can be prepared in highly delocalized states such
as entangled states. When discussing particle propagation in curved spacetime,
it is necessary to express particles in terms of quantum fields. This is
because QFTCST is currently the only theory that provides a consistent and
general way of coupling quantum matter to background gravitational fields.
Furthermore, the quantum field description is essential for describing higher-
order coherences of the electromagnetic field, and for the formulation of a
photodetection theory.

The weak equivalence principle states that all objects “fall” with the same
acceleration regardless of their mass and composition. As embodied in general
relativity, it says that all neutral objects, massive and massless, follow
geodesic trajectories of the metric, dependent only on initial conditions of
position and velocity, and not on mass or composition. The extension of the
equivalence principle established in classical mechanics to quantum physics is
an important challenge for both theory and experiment Anastopoulos and Hu
(2018); Roura (2020). Trajectories emerge for quantum histories as an
approximation that is valid only for specific quantum states (quasi-
localized), under specific conditions (sufficiently decohered) and in specific
experimental setups with suitable measurement protocols. Therefore, it is
necessary to phrase the equivalence principle in terms of quantum mechanical
notions, namely, preparations of the quantum state and statistics of
measurement outcomes. For photons, the equivalence principle must be expressed
in terms of photodetection probabilities.

Any Colella-Overhauser-Werner (COW) Colella  _et al._ (1975) type test of the
equivalence principle that touches upon quantum properties of photons or
massive particles will have the significance of being the first direct test of
QFTCST. This benchmark experiment is proposed to be conducted with photons
using the DSQL. The first photonic COW test planned for DSQL will use weak
coherent pulses. Single-photon superposition state tests will follow.
Subsequently, increasingly complex tests using entangled and hyper-entangled
states will be conducted. This sequence of experiments conclude with measuring
the gravitationally induced phase shift of frequency-entangled photons using
intrinsically nonclassical interference in a Hong-Ou-Mandel (HOM)
interferometer.

Furthermore, the DSQL could test for new physics of gravitational origin
manifested as non-unitary channels in the time evolution of photons and/or
matter waves. Such channels have been proposed in different contexts,
including Ghirardi-Rimini-Weber-type of collapse models Bassi and Ghirardi
(2003), Diosi-Penrose gravity-induced collapse model Diosi (1987); Penrose
(1996), and the gravity-induced decoherence mechanisms Anastopoulos and Hu
(2013); Blencowe (2013). These processes typically result in a loss of
visibility in interference experiments. The predicted effects are too weak in
Earth-based or space-station experiments, but deep-space experiments with long
unbalanced interferometers could provide noticeable constraints that could be
used to test the viability of alternative quantum theories.

Long-baseline links enabled by deep-space platforms create a high-latency
quantum communication network, sufficient to incorporate human decision-making
outside of the light cone of synthesis events. DSQL could use this to perform
a long-range Bell test with and without human involvement. Such an experiment
would test for local hidden variables across long distances, and address the
“free will” Bell test loophole.

The DSQL could perform tests of quantum optical teleportation and entanglement
swapping between inertial frames. The inertial frames are defined by the
relative velocity of the nodes in the communication network, and their
positions with respect to the central mass. Quantum states depend on the
choices of reference frame and gauge condition, and these are not physical
entities; only the measurement outcomes are physical. Consider the scenario
where an entangled-photon source is centered between two receivers, roughly
equidistant on either side, arranged to travel either towards each other or
away from each other. In the rest-frame of the source, the subsequent two
detection events are simultaneous and therefore space-like separated. These
scenarios are interesting because it is possible to construct models of
physics where the detection event at one measurement apparatus seemingly
instantaneously prepares the measurement outcome of the other sub-system, as
initially discussed by Aharonov and Albert in 1981 Aharonov and Albert (1981)
and by Moffat in 1997 Moffat (1997). However, models of this type would be
falsified if quantum correlations satisfy the standard Bell tests.

Beyond their intrinsic scientific interest, demonstrations of long-baseline
Bell tests and photonic teleportation are key technological achievements with
direct relevance to the future deployment of global-scale quantum networks and
sensor systems. The large relative velocities and network latency due to long
baselines could degrade network performance if left uncorrected. Effects of
gravity would also directly impact timing synchronization for high-rate
quantum communications. The proposed DSQL tests of long-baseline quantum
teleportation between different inertial frames, long-baseline Bell tests, and
tests of gravitational phase shifts will thus inform future network
architectures.

Accomplishing these experiments requires access to different mission
configurations. Some of the experiments can be accomplished with a single
spacecraft and ground station, while others require multiple spacecraft, or a
pair of ground stations. The orbital configurations optimized against the
experimental goals include LEO, MEO, Lunar orbits, and exotic orbits about the
Earth-Moon Lagrange points. The flight- and ground-system technology required
to achieve the experimental goals are also of direct relevance to terrestrial
quantum networking: high-rate, high-fidelity, entangled photon pair sources;
high count-rate, low-jitter, and low dark count-rate single-photon counters;
narrow-band optical filters; temporal synchronization and time transfer for
the quantum channels; and, in some cases, quantum memory development.

In the following sections we highlight potential science experiments that
could be conducted by the DSQL mission, and the relevant technology
development required. We also present a link model to estimate the efficacy of
deep space optical links under different experiment configurations. The
article concludes with a survey of graduated space- and ground-based
opportunities to advance the science objectives of the DSQL mission.

##  II Deep Space Quantum Link Science Experiments

This section describes the experiments proposed for the DSQL mission, which
among other outcomes, would enable testing theories predicting novel coupling
mechanisms between a photonic quantum state and gravity, with the goal of
bounding these predictions. The experiments are categorized by the similarity
of their scientific objectives. These are: GR effects on light and tests of
the equivalence principle, long-baseline Bell tests, long-baseline quantum
teleportation, and applications of squeezed light. Individual experiments are
achievable using a diverse set of different mission architectures. These
architectures could involve multiple ground and/or space network nodes, all
with different instrumentation packages. Subsequent sections summarize these
implementation options.

###  II.1 GR Effects on Light and Tests of the Equivalence Principle

Light propagating across a weak gravity field is subject to phase delays
caused by the characteristics of the local spacetime Schleich and Scully
(1982). The DSQL mission will characterize these delays in the quantum optical
regime, in order to test the underlying predictions from general relativity.
Another interpretation is that the proposed experiments will test quantum
optical interference phenomena in the regime where gravitational effects are
resolvable. These tests embody the notion of testing predictions of the weak-
field regime of QFTCST. Essential background on Einstein’s equivalence
principle (EEP), and calculations of the magnitude of the predicted general
relativistic effects, are summarized prior to description of the specific DSQL
experiments.

Einstein’s equivalence principle states that the outcome of any local
experiment in a freely falling frame will be the same as in Minkowski
spacetime, i.e., in the absence of gravitational fields, provided that all
relevant length scales are much smaller than the characteristic curvature
radius of spacetime. This guarantees that gravitational tidal effects will not
affect the outcome of the local experiment. The equivalence principle, which
applies to situations where self-gravitation effects are negligible, comprises
three different (but intertwined) aspects Will (2014):

  * •

universality of free fall (UFF),

  * •

local Lorentz invariance (LLI),

  * •

local position invariance, also referred to as universality of gravitational
redshift (UGR).

These have implications on the propagation of electromagnetic wave packets in
curved spacetime, as well as on the comparison of clocks and frequency
references at different locations. In particular, since the relevant modes of
the electromagnetic field will have a transverse size far smaller than the
spacetime curvature radius, their propagation is well described by the eikonal
approximation, with light rays corresponding to null geodesics.

While this approach has been thoroughly tested for classical electromagnetic
waves, DSQL would offer a unique opportunity to test it for various quantum
states of light, including true single-photon states, quantum superposition
states, and entangled and hyperentangled states.

For optical frequencies, quantum mechanics and general relativity predict the
same effect on single photons as on classical light, i.e., the effect is on
the _mode_ of the field, independent of the precise quantum mechanical
excitation of that mode. The proposed DSQL experiments test different aspects
of these predictions, using long-range quantum optical channels between
inertial frames.

The superposition of quantum states of a single photon was experimentally
tested along space channels by observing the single-photon interference at a
ground station due to the coherent superposition of two temporal modes Vallone
_et al._ (2016). The measurements were carried out using an unbalanced
interferometer to create the two temporal modes, and a matching one at the
receiver to recombine them, thereby recovering the fringe visibility. The
space channel was realized by exploiting retroreflectors mounted on several
spacecraft, as originally exploited for the first single-photon exchange in
space Villoresi  _et al._ (2008); Dequal  _et al._ (2016) and recently
extended to 20,000 km for MEO orbits Calderaro  _et al._ (2018). In contrast
to the experiments conducted to date, which have classical analogs, the tests
proposed for the DSQL mission have the advantage of controlled and verifiable
quantum statistics and entanglement, leading to measurements that are
intrinsically nonclassical.

In addition to verifying that frequency, superposition, and entanglement do
not affect the photon propagation, DSQL could measure independent relativistic
effects in controlled experiments. Experiments could be designed to
independently examine the following effects:

  * •

“Moving clocks run slow” from special relativity

  * •

“Clocks located down a gravitational well run slow” from general relativity

  * •

Doppler effect from differences in path length between successive clock ticks

  * •

Experimental offsets due to drifting equipment

These effects are important to the operation of global navigation satellite
systems and were calculated in the design-phase of GPS Alley (1983). The
effects are large enough to matter, but small enough to be treated as first-
and second-order corrections. For example, a clock near the surface of Earth
will measure an extra 10 ns/day for every kilometer of altitude Burns  _et
al._ (2017). More generally, when we compare the rate of a clock near the
rotating surface of Earth to one on an orbiting satellite, the fractional
change is given by simple dimensionless correction factors involving the
fraction of the speed of light $v/c$ and the Schwarzschild radius of the
gravitational potential well (in this case for the Earth), given by
$R_{\textrm{Schwarzschild}}=2G_{N}M_{\textrm{Earth}}/c^{2}\approx
1\,\textrm{cm}$. The ratio of the clock-tick interval on the satellite to one
moving with the surface of Earth is Taylor and Wheeler (2000)

$\frac{dt_{\textrm{satellite}}}{dt_{\textrm{Earth}}}\approx
1+\underbrace{\frac{1}{2}\left(\frac{R_{\textrm{Schwarzschild}}}{r_{\textrm{Earth}}}+\frac{v^{2}_{\textrm{Earth}}}{c^{2}}\right)}_{\epsilon_{\textrm{observatory}}}-\underbrace{\frac{1}{2}\left(\frac{R_{\textrm{Schwarzschild}}}{r_{\textrm{satellite}}}+\frac{v^{2}_{\textrm{satellite}}}{c^{2}}\right)}_{\epsilon_{\textrm{satellite}}}.$
(1)

The first correction is constant at $7\times 10^{-10}$, with $10^{-13}$
variation for different elevations.

(This is for an observatory at the equator, but for observatories elsewhere,
the reduction in velocity is compensated exactly by accounting for the Earth
geoid.) The second correction depends on the satellite orbit. For a satellite
in a circular orbit, a good approximation to this order is to balance
centripetal force against Newton’s gravitational force, in which case

$\frac{m\,v^{2}_{\textrm{satellite}}}{r_{\textrm{satellite}}}=\frac{G_{N}M_{\textrm{Earth}}\,m}{r^{2}_{\textrm{satellite}}}\qquad\textrm{(circular
orbit)}.$ (2)

Writing this in terms of $R_{\textrm{Schwarzschild}}$, we recover the
somewhat-well-known result for GPS that the time dilation from special
relativity is half of that from general relativity’s gravitational effect. The
overall satellite correction is then simply

$\epsilon_{\textrm{satellite}}=\frac{3}{4}\frac{R_{\textrm{Schwarzschild}}}{r_{\textrm{satellite}}}\qquad\textrm{(circular
orbit)}.$ (3)

and is a small effect given that $R_{\textrm{Schwarzschild}}\approx
1\,\mathrm{cm}$. This term is $10^{-9}$ for a LEO orbit like ISS and falls to
$1.6\times 10^{-10}$ for a GEO orbit. The observatory and satellite terms
completely cancel at $r_{\textrm{satellite}}=10^{7}\,\textrm{m}$, which is in
the inner Van Allen Belt.

For a spacecraft in an elliptic orbit with semi-major axis $a$, the relation
between velocity $v$ and distance $r$ is

$v^{2}=GM\left(\frac{2}{r}-\frac{1}{a}\right)\qquad\textrm{(elliptic orbit)},$
(4)

or in terms of dimensionless ratios and $R_{\textrm{Schwarzschild}}$,

$\frac{v^{2}_{\textrm{satellite}}}{c^{2}}=R_{\textrm{Schwarzschild}}\left(\frac{1}{r_{\textrm{satellite}}}-\frac{1}{2a}\right)\qquad\textrm{(elliptic
orbit)}.$ (5)

The overall satellite correction for a clock in an elliptic orbit (with semi-
major axis $a$), including both special and general relativistic effects is
then

$\epsilon_{\textrm{satellite}}=R_{\textrm{Schwarzschild}}\left(\frac{1}{r_{\textrm{satellite}}}-\frac{1}{4a}\right)\qquad,$
(6)

which reduces to the circular-orbit case when $a=r_{\textrm{satellite}}$, but
varies throughout an elliptic orbit as the satellite approaches and recedes
from Earth. Thus, a highly elliptical orbit breaks the degeneracy of constant
offsets due to experimental drift. Even the sign of the total relativistic
correction may change throughout the orbit. An example for the “typical
Molniya orbit” from Wikipedia111https://en.wikipedia.org/wiki/Molniya_orbit is
shown in Figure 1. If the atmosphere is deemed to be a problem, the effect can
be enhanced by comparing two Earth-orbiting satellites, one near apogee and
one near perigee. Note that for the Lunar Gateway’s near-rectilinear halo
orbit222https://www.esa.int/ESA_Multimedia/Images/2019/07/Orbiting_lunar_Gateway,
the values of $\epsilon_{\textrm{satellite}}$ are more than 50 times smaller
due to the 80 times smaller mass of the moon, while the proposed low lunar
orbit of the Lunar Gateway results in $\epsilon_{\textrm{satellite}}$ about
twice as small.

This order-of-magnitude analysis shows that an Earth-Moon link would not be an
ideal starting point for measuring gravitationally induced phase shifts in
quantum light. The two-body gravitational field of the Earth-Moon system
reaches an inflection point near the first Lagrange point. Future experiments
may leverage Earth-Moon links to test for modulation, or even full
cancellation of the phase shift of a photon propagating across this region
Mohageg  _et al._ (2018). In contrast, other quantum optical experiments, such
as the tests of Bells inequality and tests of quantum teleportation, described
in detail below, are enhanced through use of an Earth-Moon baseline.

Figure 1: Total relativistic time dilation (from both gravitational and
velocity effects) along a highly elliptical satellite orbit (blue) and at a
fixed observatory on the surface of Earth (green). These effects subtract
(Equation 1) to get the relative rate of clocks in the two locations. Note
that the sign of the relative rate can be positive or negative, depending on
the satellite altitude.

These calculations capture the difference in rates that the clocks run on the
satellite as compared to on Earth. As we will show, these “clocks” can be the
time bins of pulses or the optical-frequency oscillations of the light itself.
The gravitational blueshift (redshift) experienced by the photons as they fall
toward (climb away from) Earth is already included in these expressions and
should not be double counted. The same applies to length contraction in any
fiber delay line or interferometer—Einstein’s light-clock thought experiments
shows that these effects are already included in the slow-down of the clock
ticks that an external observer sees.

In addition to these relativistic effects, there is a large Doppler shift
caused by successive pulses or successive crests of waves being sent and
received from different places and therefore taking different amounts of time
to arrive at the observer. In practice, this will be the dominant effect, but
can be separated from the intrinsic time dilation by its different orbit
dependence, i.e., the sign of the effect depends on whether the satellite is
approaching or moving away from the ground terminal. This shift is first order
in the relative velocities $v/c$:

$\left|\frac{\delta t}{t}\right|=\left|\frac{\delta
f}{f}\right|\approx\frac{v}{c}\qquad\textrm{(Doppler)},$ (7)

so any series expansion must be kept to second order to match the size of the
leading-order $v^{2}_{\textrm{satellite}}/c^{2}$ terms above from special
relativity. Note that using the formula for the relativistic Doppler effect
double-counts the source’s time dilation. To avoid double-counting, it is best
to use global coordinates like the Schwarzschild coordinates that asymptote to
Minkowski spacetime infinitely far away:

  * •

Fix an infinitesimal time difference between two pulse transmissions.

  * •

Calculate the global time difference of reception, which will be longer or
shorter by an amount on the order of $v/c$, depending on the orbit and
orientation. This is $10^{-5}$ for LEO satellites.

  * •

The curved path of light through the Schwarzschild metric can be computed
numerically, but its effect is $10^{-15}$, which is small compared to the
special and general relativistic time dilation.

  * •

Each of these time differences can then be translated from global time to the
time experienced by clocks on the satellite and on Earth. This $\approx
v^{2}/c^{2}$ effect is captured by Equation 1 and is of order $10^{-10}$ for
LEO satellites.

####  II.1.1 Polarization Rotation of Photons in General Relativity

Polarization rotation also occurs when light traverses the warped space around
a rotating body. This is the “frame dragging” or “Lense-Thirring” effect. An
introduction to the effect can be found in Schleich and Scully’s Les Houches
lecture Schleich and Scully (1982). Recent calculations provide numbers
relevant for satellite experiments. Brodutch and Terno Brodutch and Terno
(2011) quotes a rotation of 55 milliarcseconds ($3\times 10^{-7}$ rad) when
sending a photon from a satellite out to infinity, but this calculation is
“gauge dependent” and therefore unmeasurable—the fixed reference frame of
stars the satellite is measuring with respect to is also frame dragged.

Brodutch et al. Brodutch  _et al._ (2015) consider a closed loop
interferometer, where the polarization is compared at the same point (as
differential geometry requires) to make comparisons unambiguous. For an
interferometer 100 km by 10 km they find a polarization rotation of only 4
pico-arcseconds ($2\times 10^{-17}$ rad), concluding correctly that the
“minuscule scale of the effect puts it beyond the current experimental
reach.”333For any photon-counting experiment, the statistical uncertainty of
photon polarization angle measurement scales inversely as the square root of
the number of photons detected: $\Delta\theta=\tfrac{1}{2\sqrt{N}}$ radians
for large $N$. Completely disregarding all systematic effects of trying to
establish a common reference frame, reaching the required sensitivity requires
at least 4 million detection events for the observable “out to infinity” case,
but $10^{33}$ (a megawatt of 1550 nm photons continuously for 30 years) for
the observable “compare at the same place” setup.

####  II.1.2 COW Test with Classical Light, Superposition States, Entangled
and Hyper-entangled Photons

Quantum optics experiments enable a unique window to explore the interplay
between quantum mechanics and relativity, both general and special. The
Colella, Overhauser, and Werner (COW) experiment used a neutron interferometer
to test the influence of the gravitational field on a quantum wave function
Colella  _et al._ (1975). Since then many experiments using matter-wave
interferometry with atoms and molecules have been performed. However, all such
experiments to date can be interpreted using a Newtonian gravity framework. In
contrast, tests with massless particles – photons – would require a general
relativistic description. There have been several proposals to detect general
relativistic effects on single photons using Mach-Zehnder interferometers
(MZI) whose arms are located at different altitudes, i.e., at different
gravitational potentials. In this section, an overview of the background of
photonic COW tests is presented, followed by a detailed discussion of specific
experimental concepts.

Both in the original COW experiment and in light-pulse atom interferometers
Kasevich and Chu (1991); Kleinert  _et al._ (2015), the matter-wave packets
are diffracted by periodic gratings (a crystal in neutron interferometers and
one or more optical lattices in the atomic case) but freely falling the rest
of the time. This implies that the proper-time difference between the
interferometer arms is insensitive to gravitational time dilation effects in a
uniform field, as can be argued by considering a freely falling frame Roura
(2020). On the other hand, every time a matter-wave packet is diffracted by a
grating, it acquires a phase that depends on the central position of the wave
packet with respect to the grating, and the total phase shift between the two
interfering wave packets depends on the relative acceleration between the wave
packets and the diffraction gratings. This can be exploited in high-precision
gravimetry measurements Peters  _et al._ (2001) and tests of the universality
of free fall (UFF) Asenbaum  _et al._ (2020).

In contrast, atom interferometers where the wave packets in the two arms are
held at two different constant heights through matter-wave guides can be
sensitive to gravitational time dilation in uniform fields Roura (2020). These
kinds of interferometers are, in fact, closer analogs of the optical
interferometers considered below, where photons in the two interferometer arms
propagate along optical-fiber delay lines located at two different heights in
a gravitational field.

Indeed, Hilweg et al. have proposed a ground experiment that uses an actively
switched, triple-arm MZI, with long fiber loops to extend the interaction
time, and thus the size of the effect Hilweg  _et al._ (2017). For a vertical
separation of 3 m and 100-km fiber delays, their model predicts that an effect
might be observed after 2-4 days of integration time. Obviously, maintaining
stability over that length of time could be quite difficult, particularly with
long pieces of optical fiber that are subject to thermal expansion.
Additionally, to minimize effects of dispersion, narrow-bandwidth photons
(<100 MHz) are needed, which can also be challenging.

Figure 2:  Simplified scheme of the optical COW experiment in space. A time-
bin superposition is generated by injecting a single photon wavepacket into a
unbalanced MZIs. The photon is sent towards a satellite, where an identical
MZI is located. Interference detected at the satellite reveals the
gravitationally induced phase shift.

To increase the magnitude of the gravitational effect, greater variation of
the gravitational potential can be employed Zych  _et al._ (2012). The idea
proposed in Rideout  _et al._ (2012) is to use two identical unbalanced MZIs,
one located on ground and the other on a satellite (see Fig. 2). Single-photon
wavepackets enter the ground MZI and at the output a coherent superposition of
two wavepackets $|t_{0}\rangle$ and $|t_{1}\rangle$, known as time-bin
encoding Brendel  _et al._ (1999), is generated. The single photons are then
sent toward the spacecraft MZI and at the exit of the second MZI an
interference effect can be observed in the “middle” time bin (corresponding to
photons that took the short path in one unbalanced interferometer and the long
path in the other).

The general formalism of the relation of the different optical paths involved
in COW tests in space by means of optical interferometry requires a careful
treatment of the frequency transformation in each path as well as the relation
of time with distance. According to a general analysis by Terno et al. Terno
_et al._ (2020), the crucial asset to obtain a closed form is the phase
difference due to the difference in the emission times. When the two MZIs are
properly calibrated, a phase shift due to the gravitational redshift will be
observed. In this case, if the satellite is not geostationary, the main
challenge is represented by the necessity to compensate for the first-order
Doppler effect. To solve the above issue, an improvement of the above scheme
was recently proposed to measure the Doppler shift introduced by the relative
motion of the satellite to ground, and to remove it from the final result
Terno  _et al._ (2018); Magnani  _et al._ (2019): in addition to MZIs on the
spacecraft and the ground station, satellite retroreflectors located in space
are used to send some portion of the upcoming light back to the ground. The
latter, detected on ground, can be used to measure the first-order Doppler
effect, since gravitational effects compensate in the two-way propagation. The
Doppler shift was assessed in previous experiments Vallone  _et al._ (2016)
and exploited as the modulator of the time-bin qubit phase as a function of
the instantaneous velocity of the satellite. This modulation, though passive,
may also be seen as a resource when ascertaining the visibility of the
interference phenomena.

The photon temporal superposition state along the space channel may be kept in
a single polarization by exploiting suitable corner cube technology for the
retroreflectors, as demonstrated for space quantum communications Vallone  _et
al._ (2015a), thus allowing for a large parameter space for the observables
under test; the latter use was pivotal in the test in space of the Wheeler
“delayed choice Gedanken-experiment”, addressing the well-known wave-particle
duality of quantum physics Vedovato  _et al._ (2018); Agnesi  _et al._ (2019).
We note that the scheme proposed in Terno  _et al._ (2018); Magnani  _et al._
(2019) exploits classical light in order to test the Einstein Equivalence
Principle in the optical domain. However, extending the scheme by using
single-photon wavepackets will allow the measurement of a gravitationally
induced phase shift on a quantum state. Finally, the temporal resolution for
the discrimination of the interference is a sensible parameter for the
experimental design. Present limits in the case of a link to a MEO satellite
are of order of a quarter of nanosecond Agnesi  _et al._ (2019).

The gravitational phase shift measured by the experiment depicted in Fig. 2 is
connected with a small shift, caused by the gravitational field, of the time
difference between the modes encoding the time bins. This shift is of the
order of $10^{-10}$ times the temporal separation between the two time bins,
which implies that the two delay lines (on ground and in space) need to be
matched at that level. For example, 100-m delay lines would need to be matched
with a precision better than 10 nm. One way of achieving this precision is to
independently calibrate the length of each delay line with a local frequency
reference, such as an atomic clock, and convert to distance using the
universal speed of light. The experiment can then be interpreted as a test of
UGR where the two frequency references at different heights are compared
through quantum states of light.

If time-bin entangled photons Brendel  _et al._ (1999) are used, the
gravitational shift also affects the relative phase of the entangled state:
for instance, by generating the entangled pair
$|\psi\rangle=(|t_{0}\rangle_{A}|t_{0}\rangle_{B}+|t_{1}\rangle_{A}|t_{1}\rangle_{B})/\sqrt{2}$
on the ground, and sending photon $A$ to the ground MZI and photon $B$ to the
satellite MZI, the entangled state will be transformed (before the MZIs) into
$|\psi^{\prime}\rangle=(|t_{0}\rangle_{A}|t_{0}\rangle_{B}+e^{i(\delta\Phi_{g}+\delta\Phi_{D})}|t_{1}\rangle_{A}|t_{1}\rangle_{B})/\sqrt{2}$
where $\delta\Phi_{g}$ is the gravitationally induced phase shift and
$\delta\Phi_{D}=k\ell v/c$, for a wave number $k$ and delay length $\ell$, is
the Doppler correction from the fact that the two time bins are emitted and
absorbed at slightly different places; here $v$ is the projection of the
relative velocities along the optical link, which varies from approximately
$-v_{spacecraft}$ to $+v_{spacecraft}$ as the spacecraft flies overhead. For a
10-km delay line and satellite velocity of 10 km/s, $\ell v_{spacecraft}/c$ =
0.33 m, corresponding to $\delta\Phi_{D}=1.3\cdot 10^{6}\,$rad, for a laser
wavelength of 1500 nm; as discussed above, any attempt to precisely determine
$\delta\Phi_{g}$ will need to compensate for this much larger value of
$\delta\Phi_{D}$. For example, one could send a classical beacon along (also
in a superposition of $t_{0}$ and $t_{1}$), and use that to stabilize the
remote MZIs by using a fiber stretcher or a piezo-electrically controlled
optical “trombone”, as was done in Chapman  _et al._ (2019); comparing the
error signal to the expected one would allow one to look for deviations.
Alternatively, as discussed above, corner cubes could be used to directly
measure the satellite’s relative motion, and a local feedback system (with a
stabilized laser) could be used to set the unbalanced MZI path length.

More advanced schemes can also be conceived with time-bin entanglement: for
instance, if the entangled source is located on a satellite, one photon can be
directed towards a ground station and the other photon towards a different
satellite, enhancing the gravitational effect and at the same time allowing
Bell tests between frames with large relative velocities (see Section II.2.3).

Finally, one can also conceive experiments that utilize hyper-entanglement –
photons that are simultaneously entangled in multiple degrees of freedom Kwiat
(1997). As discussed above, special and general relativity result in
predictable time dilation that would affect time-bin entanglement in a
measurable way, but would have very little effect on polarization
entanglement, with only unmeasurably small frame-dragging effects. Extending
the prototypes in Chapman  _et al._ (2019), one could repeat the proposed
optical COW experiments with a source of photons entangled both in time-bin
and in polarization, again with one photon measured on the ground and the
other on the space platform. One can thus directly compare the effect on the
entanglement in the two degrees of freedom, one of which is sensitive to the
relativistic effects and one of which is not. As with the previous
suggestions, if precise enough measurements can be made, the Doppler and
relativistic effects can be distinguished from each other and from constant
offsets by putting the satellite in an elliptical orbit as discussed in
Section II.1.

####  II.1.3 Flight Mission Design for Optical COW Tests

For the quantum photonic COW tests that involve classical light, the
measurement requirements follow the procedure of Dequal  _et al._ (2016). In
Appendix Appendix A: DSQL System Design, the requirements derived from the
parametric model to achieve a statistical confidence level are evaluated using
a quantum-channel model. Here we use these to estimate the high-level system
performance requirements to achieve the DSQL science objectives.

The proposed optical COW experiments are carried out by using quantum optical
interferometry, for example, by using an MZI with a single photon input into
one port (Figure 3). In this example, the photon is in a path superposition.
The paths are characterized by different values of the gravitational
potential; the time dilation between these two paths causes a relative phase
shift along the two arms. A number of specific implementations of this
concept, describing tests using coherent superposition states, single photons,
and different forms of entangled and hyperentangled photon pairs are described
in the previous section.

Figure 3: A simplified MZI for quantum optical COW tests. The source node and
receiver node are at different heights $h$ measured along the direction of the
gravity vector $\vec{g}$. Each node has a long optical delay line of length
$l$. Note that this is equivalent to the arrangement with two unbalanced MZIs
shown in Fig. 2, which allows photons emitted at different times to propagate
along essentially the same spatial mode (modulo whatever lateral shift has
occurred due to the motion of the source in the time between the early and
late emission times).

In a uniform gravitational field, this phase shift is linked to the
interferometer dimensions and gravity via the formula:

$\phi_{GR}=\omega_{0}\tau_{GR}\sim(1+\alpha)\frac{2\pi}{\lambda}\frac{gh\ell}{c^{2}},$
(8)

where $\omega_{0}$ and $\lambda$ are respectively the central frequency and
vacuum wavelength of the photon emitted by the source in its reference frame,
$\tau_{GR}$ is the time delay between the two interferometric paths, $g$ is
the acceleration due to gravity, $h$ is the orbital altitude difference
between source and receiver, and $\ell$ is the horizontal length of optical
delay 444More precisely, $n(\lambda)\ell$, if the delay has some non-unity
refractive index, as would definitely be the case for an optical fiber as
considered here. in both the source and receiver interferometers. The
parameter $\alpha$, which is equal to zero for general relativity,
parametrizes violations of UGR.

The primary goal of this set of experiments is to validate the predicted
gravitational phase shift in quantum light; the expected magnitude of the
phase shift for the spacecraft implementation of this experiment is on the
order of a few to tens of radians. For example, $\phi_{GR}\sim 1\,\,$rad for
$\lambda=1550\,$nm, $h=400\,\,$km, and $\ell=6\,\,$km. The other, equally
important, objective of this test is to bound the magnitude of parameter
$\alpha$, which should be zero according to general relativity, in different
regimes of quantum light. Both $\phi$ and $\alpha$ are measured using quantum
interferometry. There is an associated integration time associated with each
interferometer measurement, where $N$ signal measurement events are captured.
Qualitatively, the instrument and mission design requirements should enable a
sufficiently high rate of signal events, relative to noise events, and a net
interferometer stability that yields a phase error smaller than $\phi$. Note
that there is a tradeoff, due to exponential Beers-law decay in the fiber-
optic delay $\ell$, and diffractive propagation loss associated with the free-
space portion $h$ of the interferometer (assuming the send and receive
telescopes are sufficiently small that the received flux falls as $1/R^{2}$;
see Appendix Appendix A: DSQL System Design for details); increasing $h$ and
$\ell$ both increase the magnitude of $\phi_{GR}$ but decrease the number of
photons with which to measure this.

The development of a quantitative model to express these requirements is
described in Appendix Appendix C: Obtaining the Parametric Model of the COW
Tests. Our approach treats $N$, the total signal flux collected over some
integration period, and the likelihood $p$ that a given measurement is a
legitimate signal, as the principle analytical parameters characterizing the
link; the key mission instrumentation requirements may then be derived from
$N$ and $p$. As shown in Appendix Appendix A: DSQL System Design, $N$ scales
directly with the photon production rate and transmission efficiency, and also
the integration time. The integration time itself is a strong function of
orbital configuration, requiring line-of-sight between source and receiver
nodes. The number of measurements $N$, depends on the number of delivered
photons and is a function of the transmitter $D_{T}$ and receiver aperture
$D_{R}$; in what follows we assume $D_{T}=1\,$ m and $D_{R}=0.3\,$ m. Results
of this analysis help constrain the design of a flight mission to measure
$\phi$ and bound $\alpha$; representative results are shown in Figure 4.

Figure 4:  Plot of the predicted error on $\alpha$ after a satellite passage,
as a function of the satellite altitude in kilometers, assuming circular
orbits, and 1.0-m and 0.3-m transmit and receive telescope apertures. The
inset shows the trend close to the optimal altitude.

We see that there is an optimal altitude, located at about $1,200\,$km,
maximizes the number of received photons, leading to the predicted error on
$\Delta\alpha=3\times 10^{-4}$; such a configuration is a reasonable starting
point to design a full flight mission to conduct the quantum optical COW test
to validate the predicted general relativistic phase shift on various quantum
states of light, and bound the parameter $\alpha$ in the quantum optical
regime. Similar analyses considering more realistic orbits, spacecraft-to-
spacecraft links, and entangled and hyper-entangled photon pairs, are the
subject of a future research article under development.

The tests of the Einstein equivalence principle described in this paper all
involve long baseline optical interferometry. Since many of them use a fiber
optic delay line, the system wavelength should be within the low fiber-loss
optical C, L, or O-bands. The overall link efficiency follows the description
from Appendix Appendix A: DSQL System Design, using the single-channel
expression (Equation (41)) for single photons, and the double channel
expression (Equation 43) for entangled and hyperentangled photons.

Measuring the gravitationally induced phase shift also requires that other
sources of phase shift are accounted for, either through direct measurement or
usage of mitigation strategies. For example, as discussed in Vallone  _et al._
(2015b, a, 2016), active measurement of the spacecraft position and velocity
can compensate error terms associated with these factors. Active stabilization
of the fiber coil length, relative to a local stabilized laser system on both
the ground system and flight system is also required. This laser reference
should propagate through the fiber delay lines in the opposite direction of
the signal photons to facilitate filtering. In realistic orbit configurations,
the time-of-flight between the ground station and receiver is constantly
changing. There is residual error introduced due to the change in time-of-
flight along the long arm of the interferometer over the transit time through
the optical fiber. In this sense, there are some fixed noise sources beyond
the standard quantum limit that will bound the measurement precision. A
detailed description of these noise factors in the context of a space-mission
scenario is the subject of a future publication.

####  II.1.4 Gravitational Redshift in a HOM Interferometer with Frequency-
Entangled Photons

In this subsection we turn our attention to an experiment using an
interference effect that does not have a classical analog, namely the two-
photon Hong-Ou-Mandel (HOM) interference Hong  _et al._ (1987). As shown in
Fig. 5, a photon-pair source directs (indistinguishable) photons onto a 50-50
beamsplitter. There is then a quantum mechanical interference effect, causing
a complete cancellation of the two processes leading to a coincidence between
detectors $b$ and $c$: both photons being transmitted, or both being
reflected.555This effect is sometimes loosely described as “photons
interfering on the beamsplitter”, which is not entirely correct. It is
possible to construct an HOM interferometer with unbalanced arms such that the
photon wavepackets never overlap on the beamsplitter or anywhere else Pittman
_et al._ (1996); Strekalov  _et al._ (1998), or, as we will see here, where
the photons have different colors, and yet the two-photon interference
persists; all that matters is the indistinguishability of the underlying
processes that could give rise to coincident detections.

Figure 5: A balanced Hong-Ou-Mandel interferometer.

To describe a balanced Hong-Ou-Mandel interferometer, consider a two-photon
state $|1\rangle_{l}|1\rangle_{u}$, with one photon in each mode (see Figure
5). These two photons are now incident on a beamsplitter:

$\displaystyle|1\rangle_{l}|1\rangle_{u}\rightarrow\bigg{(}\frac{|1\rangle_{b}+i|1\rangle_{c}}{\sqrt{2}}\bigg{)}\bigg{(}\frac{|1\rangle_{c}+i|1\rangle_{b}}{\sqrt{2}}\bigg{)}$
$\displaystyle=\frac{1}{2}\bigg{(}i\sqrt{2}(|2\rangle_{b}+|2\rangle_{c})+|1\rangle_{b}|1\rangle_{c}-|1\rangle_{c}|1\rangle_{b}\bigg{)}$
$\displaystyle=\frac{i}{\sqrt{2}}(|2\rangle_{b}+|2\rangle_{c}),$ (9)

and photon bunching is seen to occur—both photons go either into Detector b or
into Detector c, and the probability of a coincident detection at Detectors
$b$ and $c$ is therefore

$P_{b,c}=0.$ (10)

The above assumes the two photons are indistinguishable, as otherwise
destructive interference between the terms $|1\rangle_{b}|1\rangle_{c}$ and
$|1\rangle_{c}|1\rangle_{b}$ would not occur. However, for a photon with non-
zero bandwidth, temporal delays between the photons will introduce
distingushability. In this case, the probability of coincidences becomes

$P_{b,c}(\tau)=\frac{1}{2}[1-e^{-2\sigma^{2}\tau^{2}}],$ (11)

where $\tau$ is the relative temporal delay, and $\sigma$ is the half-
bandwidth of the photons. In the limit of zero bandwidth, this reduces to (10)
for any relative delay. Note that the HOM dip does not depend on the relative
phase of the incident photons, only on their relative arrival time. Thus, HOM
interferometry has lower resolution when compared with a Mach-Zehnder
interferometer, but is robust against certain types of group-velocity
dispersion. Specifically, if there is dispersion in one arm of a standard MZI,
the visibility will be negatively affected, whereas the HOM interference
effect is known to be immune to group velocity dispersion (more precisely, to
the odd orders of this) Steinberg  _et al._ (1992). Another key difference
between a standard MZI and a HOM interferometer is that the former suffers
reduced visibility if there is a relative loss in either of the arms, while
the latter does not (in the absence of noise), as such loss equally affects
both of the underlying interfering physical processes.

We now consider a modified HOM interferometer, in which the input state is
entangled in frequency Chen  _et al._ (2019):

$\frac{1}{\sqrt{2}}(|\omega_{1}\rangle_{l}|\omega_{2}\rangle_{u}+|\omega_{2}\rangle_{l}|\omega_{1}\rangle_{u}).$
(12)

Adding some temporal delay $\tau$ to the upper path adds an energy-dependent
phase $\omega_{i}\tau$, producing the state

$\displaystyle\frac{1}{\sqrt{2}}(e^{i\omega_{2}\tau}|\omega_{1}\rangle_{l}|\omega_{2}\rangle_{u}+e^{i\omega_{1}\tau}|\omega_{2}\rangle_{l}|\omega_{1}\rangle_{u})$
$\displaystyle=\frac{e^{i\omega_{2}\tau}}{\sqrt{2}}(|\omega_{1}\rangle_{l}|\omega_{2}\rangle_{u}+e^{i(\omega_{1}-\omega_{2})\tau}|\omega_{2}\rangle_{l}|\omega_{1}\rangle_{u}).$
(13)

Impinging this state upon a beamsplitter as before, the final state at
Detectors $b$ and $c$ is

$\displaystyle\frac{1}{\sqrt{2}}(|\omega_{1}\rangle_{l}|\omega_{2}\rangle_{u}+e^{i(\omega_{1}-\omega_{2})\tau}|\omega_{2}\rangle_{l}|\omega_{1}\rangle_{u})$
$\displaystyle\rightarrow\frac{1}{2\sqrt{2}}\bigg{[}i(1+e^{i(\omega_{1}-\omega_{2})\tau})(|\omega_{1},\omega_{2}\rangle_{b}+|\omega_{1},\omega_{2}\rangle_{c})+$
$\displaystyle\qquad+(1-e^{i(\omega_{1}-\omega_{2})\tau})(|\omega_{1}\rangle_{b}|\omega_{2}\rangle_{c}-|\omega_{2}\rangle_{b}|\omega_{1}\rangle_{c})\bigg{]},$

yielding a coincidence probability

$P(c,b)=2\left|\frac{1}{\sqrt{8}}(1-e^{i(\omega_{1}-\omega_{2})\tau})\right|^{2}=\frac{1}{2}\big{[}1-\cos((\omega_{1}-\omega_{2})\tau)\big{]}.$
(14)

Accounting for photon bandwidth (and assuming identical bandwidths for
simplicity), we have

$\displaystyle P_{b,c}(\tau)$
$\displaystyle=\frac{1}{2}\big{[}1-\cos((\omega_{1}-\omega_{2})\tau)e^{-2\sigma^{2}\tau^{2}}\big{]}.$
(15)

This interferometer therefore combines the sensitivity of Mach-Zehnder
interferometry with the dispersion cancellation and loss-resilience of
degenerate HOM interferometry; as such it can be quite useful for precision
measurements of phase differences and temporal delays. For example, a
frequency-entangled HOM interferometer with wavelengths 800 nm and 1590 nm
should be able to resolve temporal differences of a few attoseconds with only
tens of thousands of detected photons Lualdi  _et al._ (2021).

These techniques may be useful in probing the intersection of quantum
mechanics and general relativity, since the HOM effect is truly nonclassical
Brady and Haldar (2020). Suppose that the two paths of the interferometer are
held at different gravitational potentials. This will create a frequency- and
path-dependent phase difference which can then potentially be resolved by the
interferometer, allowing us to study the effects of a curved spacetime on an
entangled quantum system. An explicit implementation consisting of a ground
station (G), a spacecraft (S) and an optical link is shown in Figure 6.

Figure 6: Frequency-entangled HOM interferometer sensitive to the
gravitational redshift and consisting of a ground station and a spacecraft.
a.) Simplified schematic. Photons experience different gravitational
potentials, depending on which path they take. b.) Overlapping-path
architecture, with a single uplink channel for both photons; polarization
entanglement ensures that only interfering processes are present, i.e., due to
the polarizing beamsplitters, each photon takes one long and one short path.

A pair of frequency-entangled photons is generated in the ground station and
sent through the optical link to the spacecraft, where they are detected. One
interferometer arm involves a delay line in the ground station, whereas the
other arm involves an analogous delay line in the spacecraft, where the two
arms are then recombined on a beamsplitter with single-photon detectors at the
two exit ports. Whereas delay lines with equal physical length $\ell$ would
lead to a balanced interferometer for a traditional experiment on Earth, in
this proposed setup relativistic effects and the changing distance between the
ground station and the spacecraft give rise to the following time shift
between the two interferometer arms, obtained up to order $1/c^{3}$ within a
post-Newtonian expansion in powers of $v/c$ and $U/c^{2}$:

$\tau=\frac{\ell}{c}\left(\left(\frac{1+(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{S}})(t_{\mathrm{r}})/c}{1+(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{G}})(t_{\mathrm{e}})/c}\right)\left(\frac{dt/d\tau_{\mathrm{S}}}{dt/d\tau_{\mathrm{G}}}\right)-1\right),$
(16)

where $(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{G}})(t_{\mathrm{e}})$ and
$(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{S}})(t_{\mathrm{r}})$ are,
respectively, the velocity of the ground station along the direction of the
optical link at the time of emission $t_{\mathrm{e}}$ and similarly for the
spacecraft at the time of reception $t_{\mathrm{r}}$. The factor containing
these velocities corresponds to the Doppler effect associated with the motion
of the two stations (ground and spacecraft)666For sufficiently long delay
lines or quantum memories, the changes of
$(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{G}})$ and
$(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{S}})$ over the time $\ell/c$ would
also need to be accounted for.. The ratio between $(dt/d\tau_{\mathrm{S}})$
and $(dt/d\tau_{\mathrm{G}})$, which respectively account for the special
relativistic time dilation and the gravitational redshift between the two
stations, is given by

$\left(\frac{dt/d\tau_{\mathrm{S}}}{dt/d\tau_{\mathrm{G}}}\right)\approx
1+\left(\frac{1}{2}\frac{\mathbf{v}_{\mathrm{S}}^{2}-\mathbf{v}_{\mathrm{G}}^{2}}{c^{2}}-\frac{U(\mathbf{x}_{\mathrm{S}})-U(\mathbf{x}_{\mathrm{G}})}{c^{2}}\right),$
(17)

where $U(\mathbf{x})$ is the gravitational potential and we neglect higher-
order terms in the post-Newtonian expansion Misner  _et al._ (1973), as they
are suppressed by higher powers of $v/c$. This result, which is also
applicable to more general situations such as the Moon-Earth system, reduces
to Eq. (1) for a Schwarzschild metric.

The probability $P_{b,c}(\tau)$ of two-photon detection in either of the two
ports is therefore determined by Eq. (15), with

$\tau=\frac{\Delta
l}{c}+\frac{l}{c}\left(\left(\frac{1+(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{S}})(t_{\mathrm{r}})/c}{1+(\hat{\mathbf{n}}\cdot\mathbf{v}_{\mathrm{G}})(t_{\mathrm{e}})/c}\right)-1\right)+\frac{l}{c}\left(\frac{1}{2}\frac{\mathbf{v}_{\mathrm{S}}^{2}-\mathbf{v}_{\mathrm{G}}^{2}}{c^{2}}-\frac{U(\mathbf{x}_{\mathrm{S}})-U(\mathbf{x}_{\mathrm{G}})}{c^{2}}\right),$
(18)

where a possible difference $\Delta\ell$ between the proper lengths of the two
delay lines (ideally $\Delta\ell=0$) has been included and we have assumed
that $\Delta\ell\ll\ell$. The third term on the right-hand side of Eq. (18)
corresponds to the relativistic effects that we are interested in and will
typically be of order $10^{-10}$. In contrast, the “classical” Doppler effect
encoded by the second term can be of order $10^{-5}$. Accurately tracking the
trajectory of the spacecraft by means of satellite laser ranging is thus
necessary so that the comparatively large contribution of the Doppler effect
can be suppressed below the $10^{-10}$ level through postcorrection, as
discussed in Section II.1.2. The length of the two delay lines also needs to
be stabilized below the $10^{-10}$ level; moreover, one should guarantee that
they are equal (i.e., $\Delta\ell=0$) at that level, which can be achieved by
simultaneously calibrating (and stabilizing) them with identical frequency
references on ground and in the spacecraft. Note that using an elliptical
orbit would enable one to distinguish the two different relativistic
contributions, namely special relativistic time dilation and gravitational
redshift, and would also help to separate the small signal of interest from
noise sources and systematic effects, as explained below.

A substantial simplification to this complex stabilization and calibration
procedure can be achieved by employing classical light at an intermediate
frequency between $\omega_{1}$ and $\omega_{2}$ for that purpose, which can be
combined with an active compensation method involving a tunable delay, e.g.,
consisting of a movable right-angle prism with a piezo-actuated translation
stage, or a piezoelectrically controlled fiber stretcher. It should be noted
that since the classical light acting as a reference is equally affected by
the relativistic effects, a HOM interferometer stabilized in this manner will
not be able to independently measure such effects. Nevertheless, it could
still be regarded as the first experimental confirmation that quantum states
of light – frequency-entangled photons – experience the same gravitational
redshift as classical light, and moreover that purely quantum mechanical
interference is similarly affected.

While the implementation displayed in Figure 6a is conceptually simpler, a
scheme involving a single uplink channel, depicted in Figure 6b, is preferable
in practice. With this goal in mind, it is useful to consider a source of co-
propagating frequency-nondegenerate photons in the polarization-entangled
state:

$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}\Big{(}|\omega_{1},H\rangle\,|\omega_{2},V\rangle+|\omega_{1},V\rangle\,|\omega_{2},H\rangle\Big{)}.$
(19)

Such a state has been recently demonstrated in the laboratory Chapman  _et
al._ (2019). By using polarizing beamsplitters (PBS) and a half-wave plate
(HWP), as shown in Figure 6b, one can ensure that H-polarized photons will
propagate along the short path ($S$) in the ground station and the delay line
in the spacecraft ($U^{\prime}$), while V-polarized photons will propagate
along the delay line ($U$) in the ground station and the short path in the
spacecraft ($S^{\prime}$). The resulting state (after the satellite PBS) is
then

$|\Psi\rangle=\frac{1}{\sqrt{2}}\Big{(}e^{i\omega_{2}\tau}|\omega_{1},H\rangle_{BD}\,|\omega_{2},V\rangle_{AC}-e^{i\omega_{1}\tau}|\omega_{2},H\rangle_{BD}\,|\omega_{1},V\rangle_{AC}\Big{)}.$
(20)

Finally, by inserting an additional HWP in the delay line of the spacecraft,
the polarization state in that arm is rotated,
$|V\rangle_{SU^{\prime}}\to|H\rangle_{SU^{\prime}}$, so that the polarizations
are no longer correlated with the interferometer arms and the quantum states
from the two different arms can interfere when recombined at the final
beamsplitter. Indeed, the polarization state of the two photons,
$|H\rangle_{SU^{\prime}}\,|H\rangle_{US^{\prime}}$, can then be factored out
and one is left with the desired state, cf. Eq. (13).

We conclude this subsection with an estimate of the actual sensitivity of the
proposed HOM interferometry experiment to relativistic effects. As explained
in the discussion after Eq. (1), the net contribution of these effects is of
order $-3\times 10^{-10}$ for a circular LEO, but just a smaller fraction – of
order $4\times 10^{-11}$ – corresponds to the gravitational redshift. In
contrast, for a GEO the relativistic effects are dominated by the
gravitational redshift, of order $6\times 10^{-10}$ in that case. As with the
optical COW experiments discussed above, for a highly elliptical orbit, such
as that considered in Figure 1, these effects are modulated by the orbital
period and range from $-5\times 10^{-10}$ at the perigee, where special
relativistic time dilation dominates, to $6\times 10^{-10}$ at the apogee,
where the main contribution comes from the gravitational redshift. Thus,
orbital modulation can be very useful to extract the small signal and separate
it from noise sources and systematic effects.

Similarly to GEOs, for the Lunar Gateway the effect would be of order $7\times
10^{-10}$ and dominated by the redshift associated with Earth’s gravitational
field, because the Moon’s mass is 80 times smaller than Earth’s, so that
contributions from the lunar gravitational field and time dilation due to the
orbital velocity would be much smaller, also implying that there are no
significant orbital modulation effects. Hence, if we consider the case in
which the platform in Figure 6 is the Gateway Spacecraft, the time shift due
to relativistic effects (i.e., excluding the classical Doppler contribution)
is given by $\tau_{\mathrm{rel}}=2.3\times 10^{-15}\,\mathrm{s}\
(\ell/1\,\mathrm{km})$ and leads to the following interferometer phase shift:

$\delta\varphi=(\omega_{1}-\omega_{2})\,\tau_{\mathrm{rel}}=0.2\,\mathrm{rad}\,\cdot\frac{\Delta\lambda}{100\,\mathrm{nm}}\frac{1600\,\mathrm{nm}}{\lambda_{2}}\frac{1500\,\mathrm{nm}}{\lambda_{1}}\frac{l}{1\,\mathrm{km}}.$

Quantitatively comparable results hold for GEOs, and also at the apogee of a
highly elliptical orbit. In contrast, for LEOs the result for the total
relativistic effect is reduced by about a half and is dominated by the special
relativistic contribution, whereas the gravitational redshift is nearly 10
times smaller. Nevertheless, because the transmission rate for pairs of
entangled photons scales inversely with the fourth power of the optical link
baseline (see Appendix Appendix A: DSQL System Design), it should be possible
to resolve this smaller effect too, potentially with even milder requirements
on the telescope size. Similar conclusions apply at the perigee of the highly
elliptical orbit.

####  II.1.5 Flight Mission Design for Optical COW Tests using HOM
Interference

In this section we present a brief summary of the mission design trade-space;
a detailed, rigorous summary of the underlying mathematics is the subject of a
future publication. Tests of gravitational effects on HOM interference are
governed by similar processes as described in Section II.1.3; the key
difference is that a pair of photons must be transmitted, which reduces the
overall link efficiency, per Appendix Appendix A: DSQL System Design.
Furthermore, as described in Section II.1.4, the photons comprising the pair
may be non-degenerate in frequency. All these factors couple with available
spacecraft trajectorys to result in a range of possible mission
configurations.

The system diagram is shown in Fig. 6. The net timing delay $\tau$ between the
upper and lower path of the interferometer is

$\tau=\tau_{GR}+\frac{\Delta\ell}{c}+\tau_{c},$ (21)

where $\Delta\ell$ is the geometric length mismatch between the two paths, due
to error terms in the engineering and control of the interferometer,
$\tau_{c}$ is any control signal applied to the interferometer, and
$\tau_{GR}$ is the relativistic shift we seek to measure:

$\tau_{GR}=\frac{\ell}{c}\left(\frac{\Delta v^{2}}{2c^{2}}-\frac{\Delta
U}{c^{2}}\right).$ (22)

Here $\Delta v^{2}$ is the difference in the squares of the velocities of the
two interferometer nodes, $\Delta U$ is their difference in gravitational
potential energy, $c$ is the speed of light, and $\ell$ is the interferometer
path length depicted in Fig.6, equivalent to the arm length of the HOM
interferometer777As before, the effective optical path length should include
any non-unity refractive index..

Assume now that with probability $p$ the interferometer contains the input
photons and that with probability $(1-p)$ the interferometer is injected with
uncorrelated and distinguishable photons, or the detectors have background or
noise counts, leading to noise events that can cause an “accidental”
coincidence count with probability $1/2$. The flux of noise photons
$N_{noise}$ can be linked to system parameters such as receiver aperture,
spectral filtering, and detector dark counts; see Equation
$(\ref{eq:NoiseEq})$. The parameter $p$ can be interpreted as the experiment
quality factor, see Subsection Appendix D: Obtaining the Parametric Model for
Bell Tests:

$p=(1-N_{noise}\Delta t_{R})^{2}F,$ (23)

where $(N_{noise}\Delta t_{R})$ is the probability of recording a count due to
noise falling within the detector timing window $\Delta t_{R}$ (assumed to be
$10^{-9}\,s$), and $F$ is the source fidelity (assumed to be 0.95 in our
simulations). Here for simplicity we assume that all the system parameters in
$N_{noise}$ are fixed, apart from the spectral filtering bandwidth $\sigma$
that we set to match the signal photon bandwidth; therefore, $p=p(\sigma)$.
The coincidence count probability thus becomes:

$P_{c}=\frac{p(\sigma)}{2}(1-\cos(\Delta\omega\tau)e^{-2\sigma^{2}\tau^{2}})+\frac{(1-p(\sigma))}{2}.$
(24)

A more complete analysis should also take into account error sources such as
path length mismatch and attitude determination error; these and other sources
of imperfection will be explored in detail elsewhere.

The error $\Delta\tau$ in a measurement of $\tau$ is computed using the
correlated counts $\hat{M}=\hat{M_{A}}\otimes\hat{M_{B}}$, for detectors $A$
and $B$ in Figure 6:

$\displaystyle\Delta\tau(\tau,\sigma,\Delta\omega)\equiv\frac{\Delta
M(\tau)}{\left|\frac{\partial\left<\hat{M}\right>}{\partial{\tau}}\right|}=\frac{\sqrt{\left<\hat{M^{2}}\right>-\left<\hat{M}\right>^{2}}}{\left|\frac{\partial\left<\hat{M}\right>}{\partial{\tau}}\right|}$
$\displaystyle=\frac{\sqrt{1-p(\sigma)^{2}e^{-4\sigma^{2}\tau^{2}}\cos^{2}(\Delta\omega\tau)}}{p(\sigma)e^{-2\sigma^{2}\tau^{2}}\left|4\tau\sigma^{2}\cos(\Delta\omega\tau)+\Delta\omega\sin(\Delta\omega\tau)\right|,}$
(25)

where $\Delta\omega\equiv\omega_{1}-\omega_{2}$ is the frequency difference of
the photons, here assumed to have (vacuum) wavelengths $\lambda_{1}=780\,$nm
and $\lambda_{2}=1550\,$nm, corresponding to $\Delta\omega=4\cdot 10^{15}\,$
Hz. In order to avoid excessive broadening of the pulse due to propagation in
the atmosphere and to simplify telescope optics, we assume a maximum bandwidth
$\sigma\leq 4.7\cdot 10^{13}\,$Hz ($\delta\lambda_{1}\leq 100\,$nm), i.e., the
individual spectral components are still relatively narrow for the non-
degenerate HOM implementation.

Given an experiment with a certain quality factor, a natural question to ask
is which choice of the overall time delay $\tau$ minimizes the timing error.
More formally, we want to solve the following optimization problem:

$\Delta\tau_{opt}(\sigma)=\min_{\tau}\Delta\tau(\tau,\sigma,\Delta\omega).$
(26)

The result of the optimization is shown in Figures 7(a) and 7(b), which shows
the expected result that the timing error is minimized for the largest
bandwidth $\sigma$ (in the degenerate source case $\omega_{1}=\omega_{2}$),
but is essentially constant for $\Delta\omega\gg\sigma$. The minimum timing
error for the non-degenerate case is at least an order of magnitude smaller
than for the degenerate case in the considered photon bandwidth interval; even
when we let $\sigma\approx\Delta\omega$, we still see a $\sim 40\%$
improvement for the non-degenerate case.

(a)

(b)

Figure 7: Contour plot of $\Delta\alpha$ versus the photon bandwidth $\sigma$
and satellite altitude for a single satellite passage: a) degenerate case
(both photons at 780 nm, yielding only the HOM dip), b) non-degenerate case
(assuming photons at 780 nm and 1550 nm, yielding a HOM dip with beat-note
oscillations).

Assuming that the main source of error is given by the general relativistic
time delay measurement, we can propagate it to obtain:

$\Delta\alpha=\frac{\partial\alpha}{\partial\tau}\Delta\tau=\frac{\Delta\tau}{-\frac{l}{c}\frac{\Delta
U}{c^{2}}\sqrt{N_{c}},}$ (27)

where $N_{c}$ is the number of coincidence counts used for the measurement, a
function of the optics aperture and of the photon wavelength (which effects
diffraction and thereby photon loss; see Equation 38).

(a)

(b)

Figure 8: Contour plot of $\Delta\alpha$ versus the photon bandwidth $\sigma$
and satellite altitude for a single satellite passage: a) degenerate case
(both photons at 780 nm, yielding only the HOM dip), b) non-degenerate case
(assuming photons at 780 nm and 1550 nm, yielding a HOM dip with beat-note
oscillations).

Figure 8(a) and 8(b) shows the contour plots for the minimized error on
$\alpha$ in a satellite passage, assuming a ground (satellite) aperture of 1 m
(0.3 m). For the degenerate case, the error decreases as the photon bandwidth
increases, reaching a minimum of $\Delta\alpha=0.01$ for an altitude around
$1,300\,$km and $\sigma=4\cdot 10^{13}\,$Hz. The non-degenerate case also
shows a slight reduction in $\Delta\alpha$ as the photon bandwidth $\sigma$
increases but the effect is much smaller than in the degenerate case, as
expected since the timing resolution is dominated by $\Delta\omega\gg\sigma$.
Here the peak performance of $\Delta\alpha=0.001$ is reached at an altitude
close to 1,000 km 888The optimal altitude depends on the optics apertures; for
example, for 0.5-m transmitter and receiver apertures, the optimal altitude
would be located lower, at about 500 km, to compensate for the extra
diffractive loss of the smaller apertures.. Figure 9 shows the benefit of
using photons with different wavelengths – the error on $\alpha$ for the non-
degenerate case is always at least one order of magnitude smaller than the
error for the degenerate case.

Figure 9: Contour plot of the ratio $\alpha^{non-degen}/\alpha^{degen}$
between the error on $\alpha$ of the non-degenerate and the degenerate case,
versus photon bandwidth and satellite altitude.

The display of an optimal altitude minimizing the error on $\alpha$ was also
present for the single-photon interference case (Section II.1.3), in that case
at $1,200\,$km for the same 1-m (0.3-m) telescope apertures. The estimation of
$\alpha$ using single-photon interference seems to be advantageous
($\Delta\alpha=3\cdot 10^{-4}$, i.e., three times smaller than the two-photon
case); the lower optimal altitude and higher value of $\Delta\alpha$ in the
two-photon case arise from the additional signal loss when both photons have
to be transmitted successfully (if one had sufficiently large telescopes that
diffractive losses could be ignored, this disadvantage would largely
disappear). Regardless, the HOM-based experiments bring the desirable feature
of using purely nonclassical interference.

####  II.1.6 Gravitational Dephasing, Decorrelation, and Decoherence

The preceding sections describe a set of experiments to test the
gravitationally induced phase shift on quantum photonic modes. In this
Section, the predictions of QFTCST on the gravitational decoherence of quantum
states are reviewed. The estimated magnitude of these effects suggest that
meaningful tests of this sort are beyond the scope of the proposed DSQL
experiments. However, success of DSQL and related missions will improve the
prospect for considering such tests of gravitational decoherence in the
future.

Gravitational decoherence first appeared as an attempt to explain the very
different behaviors between the micro and the macro worlds, i.e., we seem not
to observe quantum superpositions in the latter. The simplest way is to assume
the existence of a length scale which demarcates the micro from the macro —
superpositions can then only persist below this scale; above it, the states
naturally decohere in some basis. If the underlying theory attributes the
decoherence to new physics (e.g., continuous collapse models such as the
Ghirardi-Rimini-Weber-Pearle (GRW-P) theories Bassi and Ghirardi (2003) or
Diosi-Penrose (DP) model Diosi (1987); Penrose (1996)), then decoherence may
happen in the position basis, and it can explain the micro-macro separation in
quantum theory (a.k.a. the quantum measurement problem). In contrast,
gravitational decoherence that originates from known physics—i.e., general
relativity (GR) and quantum field theory, as in the Anastopoulos-Blencowe-Hu
(ABH) theory Anastopoulos and Hu (2013); Blencowe (2013), occurs in the energy
basis.

Here, we shall focus only on genuine gravitational decoherence 999The term
“gravitational decoherence” is sometimes used in the literature when the
authors actually refer to mere gravitational dephasing. A key difference is
time reversibility: decoherence is an irreversible process, while dephasing
can often be reverted. These processes follow from the entanglement of the
quantum system with additional degrees of freedom that can be taken as an
environment. When the environment is traced over, the original quantum system
evolves into a mixed state. As a result, phases are randomized. However, if
the environment is small, in the sense that it has a small Poincaré recurrence
time, the phases will be restored after some time. In this case, we have
dephasing and not decoherence. In decoherence, the phase information is lost
forever., which involves new physics arising from either A) new phenomena
deduced from established GR theoretical foundations (e.g., the ABH master
equation) or B) modifications of either a) quantum mechanics (e.g., the DP or
GRW-P theories), which we refer to collectively as alternative quantum
theories (AQT); or b) the structure of space or time, known as intrinsic,
fundamental, or quantum gravity decoherence.

In what follows we shall briefly highlight the salient features of these
alternatives, and estimate the magnitude of their effects, as they are
directly relevant to DSQL experiments. Models of gravitational decoherence
typically involve one or more free parameters. Requirements of theoretical
consistency and past experiments on gravitational quantum physics have already
excluded regions of the parameter space—see, for example, Refs. Helou  _et
al._ (2017); Donadi  _et al._ (2020). Deep space experiments can improve such
constraints by many orders of magnitude, even if some regions of the parameter
space may be beyond current measurement capabilities.

In the ABH model Anastopoulos and Hu (2013); Blencowe (2013), decoherence
arises from fluctuations of gravitational waves (classical perturbations) or
gravitons (quantized linear perturbations); the source of these fluctuations
may be cosmological Blencowe (2013) (stochastic gravitons produced in the
early universe near the big bang or from inflation), astrophysical Reynaud
_et al._ (2004), or structural when GR is viewed as an emergent theory (e.g.,
Hu (2009)). The corresponding master equation depends on the noise temperature
$\Theta$, which coincides with the graviton temperature if the origin of
perturbations is cosmological, but is unconstrained if gravity is emergent; in
the latter case $\Theta$ is determined by the deeper layers in the structure
of spacetime at the Planck scale.

The ABH master equation is:

$\frac{\partial\hat{\rho}}{\partial
t}=-i[\hat{H},\hat{\rho}]-\frac{\tau}{16m^{2}}(\delta^{ij}\delta^{kl}+\delta^{ik}\delta^{jl})[\hat{p}_{i}\hat{p}_{j},[\hat{p}_{k}\hat{p}_{l},\hat{\rho}]],$
(28)

where $\tau$ is a constant of dimension time and
$\hat{H}=\frac{\hat{p}^{2}}{2m}$. A similar master equation can be derived for
photons Lagouvardos and Anastopoulos (2021). In the ABH model,
$\tau=\frac{32\pi}{9}\tau_{P}(\Theta/T_{P})$, where $T_{P}=1.4\times 10^{32}K$
is the Planck temperature and $\tau_{P}=5.4\times 10^{-44}s$ is the Planck
time. If $\Theta$ is regarded as a noise temperature, it need not be related
to the Planck length, and $\Theta>>T_{P}$ is perfectly acceptable. For motion
in one dimension, the ABH master equation simplifies to

$\displaystyle\frac{\partial\hat{\rho}}{\partial
t}=-i[\hat{H},\hat{\rho}]-\frac{\tau}{2}[\hat{H},[\hat{H},\hat{\rho}]],$ (29)

Eq. (29) also appears in models by Milburn Milburn (1991), Adler Adler (2004),
Diosi Diosi (2005) and Breuer et al. Breuer  _et al._ (2009) from different
physical considerations. In these models, $\tau$ is also a free parameter, but
the natural candidate is the Planck-time $\tau_{P}$.

Diosi Diosi (1987) postulates a collapse term with noise correlator
proportional to gravitational potential, which leads to a master equation of
the form:

$\frac{\partial\hat{\rho}}{\partial t}=-i[\hat{H},\hat{\rho}]-\frac{G}{2}\int
d{\boldsymbol{r}}d{\boldsymbol{r}^{\prime}}\frac{[\hat{f}({\boldsymbol{r}}),[\hat{f}({\boldsymbol{r}}^{\prime}),\hat{\rho}]]}{|{\boldsymbol{r}}-{\boldsymbol{r}}^{\prime}|},$
(30)

where $\hat{f}({\boldsymbol{r}})$ is the mass density operator. Penrose’s
Penrose (1996, 2014) idea is not model specific, but leads to similar
predictions for decoherence time. A key point in the DP model is that
predictions do not involve any free parameters (at least in the experimentally
relevant regime). The decoherence rate is typically of the order of
$\frac{1}{\Delta E}$ where $\Delta E$ is the gravitational self-energy
difference associated to a macroscopic superposition of mass densities.

Other decoherence models that lead to position decoherence have similar
properties to DP, but more free parameters, some of which have no intuitive
physical interpretation. This includes, for example, continuous collapse
models like GRW-P and the Power-Percival Power and Percival (2000) decoherence
model based on fluctuations of the conformal factor. In any event, most AQTs
have to tolerate a small degree of energy-conservation violation. Experimental
tests for such violations lead to significant constraints in the free
parameters of some models Donadi  _et al._ (2020).

Another distinct class of models is based on the Newton-Schrödinger equation
(NSE) Bahrami  _et al._ (2014). One postulates a non-linear equation for the
single-particle wave function,

$\displaystyle i\frac{\partial\psi}{\partial
t}=-\frac{1}{2m}\nabla^{2}\psi+V_{N}[\psi]\psi$ (31)

where $V_{N}(\bf r)$ is the (normalized) gravitational (Newtonian) potential
given by

$\displaystyle V_{N}({\bf r},t)=-G\int d{\bf r^{\prime}}\frac{|\psi({\bf
r^{\prime}},t)|^{2}}{|{\bf r}-{\bf r^{\prime}}|}.$ (32)

Note that the NSE for a single particle is not derivable from GR and Quantum
Theory Anastopoulos and Hu (2014).

There are several advantages of carrying out tests of these theories in space
experiments. They include the high quality of microgravity ($\sim 10^{-9}$g),
very long free-fall times ($>10^{4}$ s), and the combination of low pressure
($\sim 10^{-13}$Pa ) and low temperature ($\sim 10$ K) with full optical
access. Here we outline necessary experimental parameters using optomechanical
systems, atom interferometry, atomic spatial wavefunction spreading, and
photon decoherence.

Optomechanical experiments. Consider a body brought into a superposition of a
zero momentum and a finite momentum state, corresponding to an energy
difference $\Delta E$. For the ABH model, the decoherence rate for the center
of mass is then

$\displaystyle\Gamma_{ABH}=\frac{(\Delta E)^{2}\tau}{\hbar^{2}},$ (33)

where $\tau$ is the free parameter in the master equation (28). A value for
$\Gamma_{ABH}$ of the order of $10^{-3}s$ may be observable in optomechanical
systems, as it is competitive with current environment-induced-decoherence
timescales. Hence, to exclude values of $\tau>\tau_{P}$, we must prepare a
quantum state with $\Delta E\sim 10^{-14}$J.

In the Diosi-Penrose model, the decoherence rate for a sphere of mass $M$ of
radius $R$ in a quantum superposition of states with different center of mass
position (though the predicted decoherence rate is largely independent of the
details of the prepared state) is of the order of

$\displaystyle\Gamma_{DP}=\frac{GM^{2}}{\hbar\sqrt{R^{2}+\ell^{2}}},$ (34)

where $\ell$ is a cut-off length, originally postulated to be of the order of
the size of the nucleus, but recently constrained to $\ell>0.5\cdot 10^{-10}m$
Donadi  _et al._ (2020). Alternative models postulate $\ell$ up to a scale of
$10^{-7}$m. For an optomechanical nanosphere with $M\sim 10^{10}$ amu and
$R\sim 100$ nm, $\Gamma_{DP}\sim 10^{-3}s^{-1}$, a value that is in principle
measurable in optomechanical experiments.

Matter wave interferometry. The ABH model (but not the 1-d master equation
(29)) leads to loss of phase coherence of the order of
$(\Delta\Phi)^{2}=m^{2}v^{3}\tau L/\hbar^{2}$, where $L$ is the propagation
distance inside the interferometer101010While the exact derivation of
$(\Delta\Phi)^{2}$ requires a dynamical analysis, its magnitude is of the
order of $\Gamma_{ABH}t_{int}$, where $t_{int}=L/v$ is the average time of the
particle in the interferometer.. Setting an upper limit of $L=100$ km, and
$v=10^{4}$ m/s, decoherence due to cosmological gravitons requires particles
with masses of the order of $10^{16}$amu. If $\Theta$ is a free parameter,
experiments with particles at $10^{10}$amu will test up to $\Theta\sim
10^{-5}T_{P}$. For comparison, the heaviest molecules used to date in quantum
mechanical interference experiments are oligoporphyrines with mass of “only”
$2.6\cdot 10^{4}\,$amu Fein  _et al._ (2019).

The Diosi-Penrose model and other models that lead to decoherence in the
position basis can also be tested by near-field Kaltenbaek  _et al._ (2016)
and far-field Kaltenbaek  _et al._ (2014) matter-wave interferometry. A rough
estimation for the loss of phase coherence is
$(\Delta\Phi)^{2}\simeq\Gamma_{DP}L/v=\frac{Gm^{2}L}{\hbar Rv}$, where $R$ is
the radius of the particles. In contrast to the ABH model, this loss of
coherence is enhanced at low velocities. Assuming $L=100$ km, $v=10$ m/s, and
$R=100$ nm, an experiment would require a mass $M\sim 10^{9}-10^{10}$ amu to
observe decoherence according to the DP model.

Wave-packet spread. The intrinsic spreading of a matter wave-packet in free
space is a hallmark of Schrödinger evolution. ABH-type models predict
negligible deviations in the wave-packet spread from that of unitary
evolution. The DP model and all other models that involve decoherence in the
position basis predict a wave packet spread of the form

$\displaystyle(\Delta x)^{2}(t)=(\Delta
x)_{S}^{2}(t)+\frac{\Lambda}{2m^{2}}t^{3},$ (35)

where $(\Delta x)_{S}^{2}(t)$ is the usual Schrödinger spreading, and
$\Lambda$ depends on the model. The changes from free Schrödinger evolution
become significant at later times. An exact estimation of this effect depends
on properties of the initially prepared state, and is rather involved. The
MAQRO proposal Kaltenbaek  _et al._ (2016) estimates that for a free-
propagation time equal to $100$ s (accessible in their setup) it is possible
to constrain GRW-type models, some models of quantum gravity decoherence, but
not decoherence of the D-P type.

In contrast, the Newton-Schrödinger Equation predicts a retraction of the
wave-packet spread for masses around $10^{10}$ amu Giulini and Grossardt
(2011). An osmium nanosphere of radius $R\simeq 100$ nm would require a couple
of hours of free propagation in order to observe significant deviation from
Schrödinger spreading Grossard (2017). This effect provides the only realistic
prospect of directly testing the NSE, and it requires a space environment.

Decoherence of photons. Only the ABH model has been generalized for photons
Lagouvardos and Anastopoulos (2021). For interferometer experiments with arm
length $L$, the model predicts loss of visibility of order
$(\Delta\Phi)^{2}=\frac{8G\Theta E^{2}L}{\hbar^{2}c^{6}}$. For $L=10^{5}$km,
$\Theta\sim T_{P}$ and photon energies $E$ of the order of 1eV, this implies a
loss of coherence of the order of $\Delta\Phi=10^{-8}$. In principle, this
would be discernible with EM-field coherent states with mean photon number
$\bar{N}>10^{16}$, though it would be very challenging to suppress all other
systematic errors to this degree.

####  II.1.7 GR Effects: Summary

The untested prediction that propagation across a gravitational potential
induces a phase shift on a single photon, a photon superposition state, and
(hyper)entangled photon pairs was reviewed. A set of experiments involving
interferometers distributed between spacecraft and ground nodes designed to
test this prediction was outlined. The order of magnitude of the phase shift
on the photon states caused by gravity was determined to be compatible with
experimental capabilities. Preliminary mission systems analysis suggests an
optimal regime for a spacecraft mission to achieve these objectives. As such,
tests of the equivalence principle using photons are plausible using a future
DSQL mission. In contrast, the magnitude of gravitationally induced
decoherence, based on QFTCST models, is likely too small to measure without
significant breakthroughs in multiple instrumentation capabilities.

###  II.2 Long-baseline Bell tests

The long baseline of DSQL could enable tests of Bell’s inequality Bell (1985)
up to the lunar orbital radius and between inertial frames with large relative
velocity, well beyond what is possible on earth or earth orbit. As described
in the sections below, conducting Bell tests at extremely long baselines
between inertial frames opens experimental possibilities and addresses
fundamental questions around quantum theory in the regime of general
relativity. Such tests also serve as an important validation benchmark for the
implementation of future quantum technologies.

####  II.2.1 Verification of Long-baseline Quantum Entanglement

A future global-scale quantum network shall be capable of maintaining the
fidelity of distributed photonic states in various degrees of freedom, and
interact with quantum memory devices, effectively establishing a quantum
internet Boone  _et al._ (2015); Khatri  _et al._ (2021); Gündoğan  _et al._
(2021). This network could be useful for fundamental tests of quantum physics,
distributed quantum computing or distributed quantum sensors (e.g., Gottesman
_et al._ (2012)). Based on current technologies, relatively high-rate
entanglement distribution and quantum communication across baselines more than
a few hundred kilometers are only possible using spacecraft links Bedington
_et al._ (2017). Furthermore, all quantum network applications rely on the
validity of quantum mechanics, and a complete understanding of long-baseline
quantum link behavior, which the DSQL tests could provide.

Entangled quantum systems that are shielded from the environment exhibit
correlations that are expected to persist no matter how far apart the systems
travel. An example is polarization-entanglement carried by pairs of photons
traveling separately along different optical fibers. While the local topology
of the individual fibers will induce a specific rotation on the polarization
of each photon – changing the specific form of the entangled state but not the
amount of entanglement present – this transformation can be reversed so that
analysis devices at the respective destinations can still measure correlations
satisfying the Bell test criterion for demonstrating nonlocality. This
reversal is possible in principle, although sometimes it can be quite
difficult in practice; for example, if the photon pairs have a large
bandwidth, polarization mode dispersion – wavelength-dependent polarization
transformations – within the fibers can be difficult to correct, resulting in
an effective depolarization.

In contrast to propagation through optical fiber, photons propagating across
the vacuum of deep space will encounter very few effects known from
conventional physics111Unconventional physical theories predict some
polarization rotation and other rotations through proposed coupling mechanisms
as photons propagate across changing gravitational potential Ralph and Pienaar
(2014); Siddarth Koduru  _et al._ (2018); Bruschi  _et al._ (2014). These
predicted effects were not observed by the first battery of on-ground Fink
_et al._ (2017) and Micius spacecraft experiments Yin  _et al._ (2017a, b); Xu
_et al._ (2019) attempting to measuring them. In the Micius experiments,
entanglement was distributed between a ground station and the spacecraft, and
also between two ground-based observatories approximately 1,100 km apart. In
these experiments, the entanglement fidelity was 0.907$\pm$ 0.007, only
sufficient to determine the Clauser, Horne, Shimony, and Holt (CHSH) parameter
to within 2.4 $\sigma$. The unconventional theories listed above could thus be
much better bounded with a test resulting in higher statistical significance
on the CHSH parameter, achievable using brighter, higher fidelity sources.
that could change the entanglement correlations in the various degrees-of-
freedom. Kinematic effects may cause small shifts in the polarization state,
but these require the detectors (or observers) to be accelerating (e.g., the
non-inertial frame experienced when orbiting a massive body).

The evidence and the prevailing theories to date suggest that polarization
entanglement correlations should persist in most scenarios involving a deep
space communications link. Given the lack of coupling between light and the
environment, we also expect that entanglement in other degrees of freedom
(such as spatial mode, time-energy, time-bin, or even simultaneous
hyperentanglement across these degrees of freedom) should also be preserved
over long distance propagation.

Consequentially, it is expected that a robust quantum communications link
utilizing entanglement correlations should not face fundamental obstacles to
realization based on known, conventional physics. DSQL will help validate the
assumption that deep space holds no further surprises in the form of new
physics that might invalidate our assumptions of the characteristics of a
space-based quantum communications link. In the following sections, we provide
a checklist of fundamental experiments that could be performed in order to
gain confidence that engineering an entanglement-based quantum communications
system is a worthwhile endeavor, while enhancing the distance limits of
fundamental tests of quantum physics. At the same time, the implementation of
these experiments will build up significant know-how and capability that will
aid future quantum network engineering efforts.

In order for the DSQL to characterize and validate ultra-long range Bell
tests, the methodologies to use photon correlation measurements and counting
statistics as described in Appendix Appendix D: Obtaining the Parametric Model
for Bell Tests would apply. A violation of Bell’s inequality by at least five
standard deviations would be considered a viable test. The required number of
successfully detected photon pair counts needed depends on the correlation
visibility, but typically around $1,000$ detected pairs should allow the test
to be conclusive. Furthermore, a detailed signal-noise analysis that involves
quantum optical models of the photon pair source, channel properties, noise
sources and photon detectors would be used to further compare the results with
the known (conventional) physics, similar to the study in Holloway  _et al._
(2013).

####  II.2.2 Current Status of State-of-the-art Bell Tests

The most sophisticated “loophole-free” Bell tests that have been done up to
this point with entangled photons Giustina  _et al._ (2015); Shalm  _et al._
(2015) close the detector-efficiency loophole, locality loophole, and versions
of the freedom-of-choice loophole. These experiments use local sources of
quantum randomness, where each random bit comes into being at a point in
spacetime that is space-like separated from the measurement on the other side.
In Shalm  _et al._ (2015), three different sources of random bits were XORed
together based on the optical phase of a gain-switched laser, sampling the
amplitude of an optical pulse at the single-photon level, and a predetermined
pseudorandom source comprised of popular movies and digits of $\pi$. Any
local-realist explanation for the observed Bell-violating correlations would
be required to predict the outcomes of all of these processes well in advance
of the beginning of each trial.

The assumption that the random bit comes into being when the phase or
amplitude is measured is critically important in Bell tests. If the bit on
either side is determined (or even influenced) in some way by something in its
past that the other side’s measurement also has access to, the Bell violation
can be explained with a local theory. To address this, Gallicchio  _et al._
(2014); Handsteiner  _et al._ (2017); Rauch  _et al._ (2018) used the
unpredictable color of incoming astronomical photons from opposite sides of
the sky. The latest of these Rauch  _et al._ (2018) used two quasar photons
that were emitted when the universe was a half and a tenth as old as today.
This forces any local explanation that takes advantage of the freedom-of-
choice loophole to have access to the past light cones of these quasar
emissions. This access needs to be of sufficient fidelity for one side’s
measurement to predict the other side’s next photon color. These experiments
did not simultaneously close the detector efficiency loophole. Also, there is
no device-independent way to verify that the clicks registered by the quasar
photons were really mostly determined by the distant cosmological past rather
than a local conspiratorial random number generator that was, perhaps,
matching the average rate of incoming cosmological red and blue photons, but
coordinating the specific order with the other side of the experiment.

Recently the BIG Bell Test BIG Bell Test Collaboration (2018) created a web-
browser game where people around the world were rewarded for acting as
unpredictably as possible. For a 24-hour period, their inputs were used to
choose the measurement bases for 13 simultaneous Bell-type tests around the
world. This was a heroic effort to close a loophole in previous experiments,
where something in each experiment’s past could have influenced the settings.
However, given the constraints of Earth-bound participants, their choices were
electronically recorded and used in such a way that (purely in terms of past
light cones) the sources of entanglement and all measurements had access to
the choices in advance, i.e., a substantial loophole remains.

As mentioned above, the measurement of entanglement over long distances is
expected to follow “conventional physics”, no matter what distances are
traversed, or whether the measurement apparatus (the observers) are at rest
relative to each other. Measurement devices in relative motion can lead to a
reference-frame-dependent event sequence, where the expectation of
entanglement preservation becomes less obvious. This is especially striking if
the reference frames are physical, if the wavefunction is an element of
reality as the Pusey-Barrett-Rudolph theorem (PBR theorem) favors Pusey  _et
al._ (2012), or if wavefunction collapse is a physical phenomenon caused by
interaction with a measurement device in different reference frames. These
alternative viewpoints are fundamentally different from general relativity, in
which both super-observers and preferred reference frames are impossibilities.
Only physical collapse is a non-standard viewpoint. In that sense, conducting
experiments along these lines would test the predictions of QFTCST against
these alternative “strawman” theories.

####  II.2.3 Bell Tests Between Frames with Large Relative Velocities

Consider a Bell test scenario involving three inertial frames. The entangled
photon source is in the center while the two receivers, roughly equidistant on
either side, are arranged to travel either towards each other or away from
each other. In the rest-frame of the source, the two detection events are
simultaneous and therefore space-like separated. In the case where the
detectors are traveling away from each other, each detector would “consider”
itself to be the first to receive the incoming photon and to generate a signal
in its own reference frame. When the detectors are moving towards each other,
the opposite case occurs and each detector would “consider” itself to have
received the photon after its distant counterpart. In special relativity, a
reference frame for a local observer is operationally determined as the radar
coordinates, which respect Lorentz symmetry D‘Inverno (1992); Lin and Hu
(2021). With similar operations, the above “considerations” by each detector
can only emerge after the detector receives an ideal radar signal, emitted
earlier by itself, echoed back from the measurement event by the other
detector. This is long after both events of receiving photons at the two
detectors have occurred in all the reference frames of the detector and the
source. Note, however, that such radar signals could have been used prior to
the measurements described, so that reference frames are predicted before the
experiment.

Suarez and Scarani A.Suarez and Scarani (1997) refer to the above as “before-
before” and “after-after” scenarios, respectively; this nomenclature assumes
that, while each observer may not know whether it measures before or after its
distant counterpart at the moment of its local measurement, as each observer
completes their measurement, the collapse of the wave-function propagates
instantly in their respective reference frame. But, since the two observers
are in relative motion, it appears that both reference frames yield
contradictory wave-function collapses 222In the theoretical description of a
quantum system, a reference frame to coordinatize physical events (spacetime
points) is chosen by a specific observer, and wavefunctions to relate the
outcomes of physical measurement events are specific to each observer Aharonov
and Albert (1981). In fact, the description of quantum states requires an
explicit reference frame choice by an observer, because in the canonical
quantization scheme a time-foliation (1+3) scheme needs to be specified before
a Hamiltonian can be written down Anastopoulos and Hu (2013); Lin (2012).
Making a statement that two wavefunctions supported by different time-slices
in two reference frames for two different local observers have contradictory
collapses does not make sense, because it presumes that there exists a super-
observer who can see the co-existence of two time-slices to support those
wavefunctions.. Suarez and Scarani identified the polarization analysis-
determining beamsplitter in each measurement station as the necessary device
that must be moving, and proposed to place them on rotary mounts that spin to
achieve relative speeds of 100 m/s.

Experimental studies with such dynamics, as first demonstrated by Zbinden et
al. Zbinden  _et al._ (2001), utilized rotating absorbers as “detectors”. In
this experiment, part of the entangled photon signal was absorbed on rotators
which either moved towards each other, or away from each other. The rest of
the signal was routed to conventional, stationary detectors; the results
showed no statistically significant deviation from a standard test using only
conventional detectors at rest.333In these experiments the signal was routed
by fibers, and the detectors only separated over 10km, so it was very
difficult to get the space-like separation needed for the test. Typically, the
experiment would begin with slight asymmetrical distances, and then the
experiment would run over several hours, so that the diurnal effect on the
fiber would sweep through the circumstance of simultaneous measurements at
some point. Furthermore, the absorbers used as “detectors” gave no active
signal (such as a heralding flash upon successful completion of an absorption
event) that could directly contribute to the statistics; instead, they served
“null-result” measurements – lack of an absorption event projected the photon
wavepacket into the arm with the photon detector. Another experimental test
was reported by Stefanov et al. in 2002 Stefanov  _et al._ (2002) using time-
bin entangled photons separated over 10 km via an optical fiber, and “moving”
beamsplitters implemented with acousto-optical modulators. Each incoming
photon effectively saw a beamsplitter moving at about 2,500 m/s; as in the
previous experiment the actual single-photon detectors themselves were not
moving 444This is potentially problematic in an experiment whose underlying
hypothesis relies explicitly on a measurement-induced wavefunction “collapse”.
Specifically, the unitary transformations of the beam splitter could be undone
by other (local) optical elements. The measurement basis settings and the
results are not truly fixed until the actual detection of the photons, which
involves the amplification of a detection signal to the macroscopic scale,
involving millions of electrons. Therefore, any convincing test of these
simultaneity arguments should have the same collapse-inducing detectors
themselves in relative motion. . Again, the experiment found that the
entanglement correlations were perfectly preserved without need to consider
the time-sequencing of events, in agreement with standard quantum mechanics
formalism. An experimental scenario where the detection events by the two
moving observers of entangled photons are in each other’s respective future,
or past, has yet to be tested – though it should be noted that Scarani et al.
Scarani  _et al._ (2014) have shown that the multi-simultaneity model could
lead to superluminal communications, in violation of relativity. It is
understood that satellites are the best approach to obtain the required
relative speeds and distances for such a measurement, and these tests could be
considered for the DSQL platform Rideout  _et al._ (2012).

In the previous tests the actual photon detection systems were stationary,
leaving an open question if a moving measurement device is required to
represent a moving “observer", since the beam splitter operation, even while
moving, remains coherent (and reversible). Ultimately the entire observer
system including the detection process must be in the motion Rideout  _et al._
(2012); Scarani  _et al._ (2014). Here we consider space-based experiments,
where satellites can be distant enough from each other and moving fast enough
away from each other such that each detector’s measurement can be considered
complete in its own reference frame before the other detector even begins its
measurement. Similarly, the satellites can move toward each other fast enough
such that in each local detector reference frame the distant detector’s
measurement is completed before the local detector begins its measurement.

Creating high relative velocity between two moving platforms near enough to
each other to maintain high link efficiency over a long enough integration
time is the key requirement for such tests of relativistic simultaneity. The
constraints on timing are stricter in these before-before or after-after
experiments than in a typical Bell test, where locality dictates only that the
two measurements be space-like separated. This space-like separation means
that there exists some frame where one measurement is first and some other
frame where the other is first. In the before-before experiment, these frames
cannot merely exist, but must include the actual rest frame of each detector
(see Figure 10(b)). For satellite speeds much slower than the speed of light,
the measurements must be within $\Delta t<vD/c^{2}$ of each other, where $v$
is the relative velocity of the detectors and $D$ is the instantaneous
separation between the measurements. For reasonable LEO parameters, this is
around 10 m of light travel time, and translates into an accuracy requirement
of approximately $10^{-5}$ in the position of the receivers. Note that it is
not sufficient to merely know the orbits to within 10 m and the detection
times to within 30 ns—the orbits and timing must be controlled to this
accuracy for sufficient duration so that a statistically significant number of
entangled pairs arrive while this condition holds. Furthermore, in a LEO
constellation, these scenarios only exist for brief periods of time and drive
strict orbit determination and station-keeping requirements on the flight
platforms. To satisfy the conditions of this scenario, at least two platforms
should be on satellites in space.

With receivers on two independent spacecraft, the link budgets are likely to
improve—in Figure 10(a), three polar-orbiting satellites would meet regularly
over the poles to perform the experiment repeatedly, potentially both while
approaching and receding. Some of the challenges in this scenario, however,
are the slew rates as the platforms converge, avoiding collisions, and the
additional challenge of operating in high-radiation zones. An alternative
might be to place the source on the equator and to beam photons to two
counter-propagating receiver satellites in an equatorial orbit. To mitigate
uplink losses, the source would have to be placed at a relatively high
altitude, perhaps even on a balloon.

For comparison, consider the scenario where an entangled pair source transmits
one photon to an orbiting platform, and the other to a ground receiver (see
Figure 11). The asymmetrical free-space path lengths necessitate a delay line
or buffer at the terrestrial receiver that can change quickly by the
equivalent of several hundred kilometers, resulting in several technical
challenges. Because the detectors are not moving symmetrically, there is only
a 1 ms (10 m/$v$) time window in each orbit where the alignment can be such
that each detector measures first (or second) in its own reference frame. A
huge possible improvement to such an asymmetric scheme would be utilization of
a quantum memory to achieve variable read-out times that correspond to the
time-of-flight for the ground-to-space link. If the moving observer were
located on a LEO platform, this quantum memory would need to store the ground
photon for around 1-3 ms, and release it at the correct moment, within about 1
ns, to ensure the relativistic separation of the two measurements of the
entangled photons. The low efficiencies of both the quantum uplink and the
quantum memory would certainly present challenging low count rates. It is
nevertheless encouraging that a first proof-of-concept test of such an
asymmetric setup may be possible with a single receiver, such as the Canadian
QEYSSAt mission Jennewein  _et al._ (2014).

(a) After-After Polar Orbits

(b) After-After Spacetime Diagram

Figure 10: (a). One source and two “After-After” satellites in polar orbit
over Antarctica. An alternative would be to site the source on the ground for
double-uplink transmission of entangled photons to the orbiters. (b) Spacetime
diagram in Alice’s rest frame. Alice’s measurement happens at $(t=0,x=0)$. The
tight constraint on Bob’s allowed measurement window $\Delta t$ is also shown.
Figure 11: The asymmetric Bell test using only one space-based observer, and a
ground-based observer with suitable delay (either a fixed path or quantum
memory). A ground-based source (e.g., located at the Canary Islands) could
transmit one of the entangled photons to another terrestrial receiver, and the
other to a receiver located in orbit. The relatively short ranges on Earth
require that there be a substantial delay ($\sim$ 1 to 3 ms) at the
terrestrial receiver to achieve nearly similar optical path lengths. The
144-km free-space link would only be sufficient (ca. 0.5-ms delay) for very
low altitude satellites. Longer delays could be implemented in fiber-optics,
but would entail about -100dB of loss or greater. Ideally, a low-loss quantum
memory with finely adjustable readout times would be used on the ground.

####  II.2.4 Human-decision Bell Tests

Quantum experiments that directly involve human participants are both
scientifically interesting and socially relevant, igniting the public interest
in fundamental science. In a space-based test of nonlocality, astronaut
participants can meaningfully address the free-will loophole of Bell tests in
a new regime. The motivation behind this body of testing is largely
philosophical in nature, dealing with the epistemological underpinnings of
quantum theory. In fact, John Bell himself suggested letting humans choose the
basis in a test of his famous inequality Bell  _et al._ (1985): “It has been
assumed that the settings of instruments are in some sense free variables—say
at the whim of experimenters ” Bell (2004) and “Roughly speaking it is
supposed that an experimenter is quite free to choose among the various
possibilities offered by his equipment” Bell (2004). Furthermore, Leggett
points out Leggett (2009) that any exploitation of a loophole that relies on a
complicit role of the process that chooses the random settings (it is either
not random, or it is somehow influenced by the entangled photons) can maybe
best be settled by having two human observers operate the measurement devices.
Lucien Hardy, in “Proposal to use Humans to switch settings in a Bell
experiment,” Hardy (2017) sums up this experiment well: “The radical
possibility we wish to investigate is that when humans are used to decide the
settings (rather than various types of random number generators) we might then
expect to see a violation of Quantum Theory in agreement with the relevant
Bell inequality. Such a result, while very unlikely, would be tremendously
significant for our understanding of the world.” In this Section we explore
both the feasibility and desirability of doing such an experiment given that
all previous Bell tests have vindicated the quantum prediction.

Allowing humans to choose the settings while simultaneously closing the
locality loophole requires that the experiment be large enough that no
information about the measurement choice on one side can be accessible to the
measurement on the other side. Libet famously used EEG to measure the
Readiness potential 0.3 s before a person consciously decided to move and 0.5
s before they pressed a button Libet  _et al._ (1993). Perhaps specialized
training can improve the reaction times, but it takes somewhat less than 1 s
second for a person to be presented with an choice that they did not know
about in advance, make a decision using what they perceive to be their free
will, and reliably register this choice in a way that can quickly
(electronically) change the analysis basis of a polarizer. This timing makes
the 1.2-1.4-s light-travel time between the Earth and the Moon the right
scale. The space-time diagrams shown in Figure 12 lead to the conclusion that
with a source half-way in between experimenters on the Earth and Moon,
participants would have the full 1.2-1.4 s to carry out each round of the
experiment. Similar timings are relevant for the source at an Earth-Moon
stable Lagrange point. Although the link losses over such distances are
daunting, laboratory analogs are being used to prepare for quantum
communication experiments over such high-loss channels Cao  _et al._ (2018).

Another scenario is an asymmetric configuration of quantum entanglement, as
shown in Fig. 13. This involves creating an entangled pair between Earth and
Moon, sending one photon to Earth, and storing the other photon until a
classical signal arrives from the Moon with the astronaut’s basis choice. The
random choices implemented by the human random choices are still separated by
the Earth-Moon distance, but this configuration has the benefit of requiring
only one long-distance link at the cost of half-second quantum storage. The
figure shows a source halfway between Earth and Moon, but the optical link can
be shortened at the cost of longer storage time. A smaller version of this
scheme was implemented with a fiber delay on the Canary islands in 2010
Scheidl  _et al._ (2010). It was used to study the Freedom-of-Choice Loophole,
albeit with measurement settings randomly chosen based on the behavior of
photons from an LED on a beamsplitter; no deviations from the predictions of
standard quantum mechanics were found. Other similar arrangements are
conceivable.

(a) Earth and Source

(b) Source Only

(c) Moon and Source

Figure 12: Spacetime Diagrams for a Bell test where the source is halfway
between Earth and Moon. In each round, humans on each side have 1.3 seconds to
be presented with a choice, make a decision, and register their decision with
something like a button, whose activation turns that decision into a polarizer
setting. Figure 13: Space-time diagram for a Bell test involving human
observers. In this asymmetric configuration, while the space-like separation
of random choice is made by humans separated by the Earth-Moon distance, only
one of the entangled photons is sent over a long distance link to Earth (blue
arrow); the other entangled half is stored in a quantum memory at the source
(red arrow), awaiting the random choice classical signal transmitted from the
Moon (black double-line arrow). When the source is still located on the direct
line to the moon, but not necessarily halfway, each participant still has a
time to make their choice that is equal to twice their light travel time to
the source. Figure 14: Space-time diagrams for two versions of a Bell test
involving human choices made by astronauts on Earth and Moon. The source is
shown at an arbitrary position between Earth and Moon. There is a limited time
window for humans at each location to be presented with options and make a
choice, indicated by the black arrows. These windows have maximum duration
$t_{E}$ and $t_{M}$. Each time window starts after the last event that could
influence the other side’s measurement and ends when the choice needs to be
used or transmitted. These windows are shortened by hundreds of nanoseconds
(too small to see at this scale) due to fiber delays, bounces within
telescopes, and transmission through atmosphere. Dual Transmissions (Left): If
the source is midway between Earth and Moon, in each round, humans on each
side have $\sim$1.3 seconds to be presented with a choice, make a decision,
register their decision with something like a button, and have that decision
converted into a polarizer setting. When the entangled source is farther from
Earth, people there have more time $t_{E}$ to make their basis choice at the
expense of time on the moon $t_{M}$. The best situation is when the entangled
source is halfway between Earth and Moon, when astronauts at each location
have equal time to make a valid choice; that time equals the Earth-Moon light
travel time, $\sim$1.3 seconds. Single Transmission and Storage (Right): In
this asymmetric configuration, while the space-like separation of random
choice is made by humans separated by the Earth-Moon distance, only one of the
entangled photons is sent over a long distance link to Earth; the other
entangled half is stored in a quantum memory at the source, awaiting the
random choice classical signal transmitted from the Moon. By the geometry of
the causal influences, the maximum time window $t_{M}$ in which the astronauts
on the moon need to make their basis choice is equal to the storage time
$t_{S}$. A longer storage time comes at the expense of Earth decision time
$t_{E}$. The best situation is to divide the decision time equally,
($t_{E}=t_{M}=t_{S}$). In this case, the source near the moon, transmits one
entangled photon all the way to Earth, and stores the other entangled photon
for the full 1.3 seconds.

And finally, the architecture of the human-decision Bell tests, shown in
Figure 15, does not require deployment of quantum optical hardware in deep
space, but ensures that there is no chance the emitted entangled photons nor
the distant measurement could have been influenced by the local measurement
choice. The key requirement is that the decision-events of the astronaut
participants and emission-events of entangled photons comply with specific
timing requirements.

Both astronauts are queried at time $t_{question}$, synchronized using GPS, at
a time well before $t_{entangled}$ when the source is activate to (possibly)
emit a photon pair. One set astronaut is here presumed to be located on the
Moon, while the other is space-like separated from the Earth, with at least
the same distance as the Moon, but in the opposite direction (the third
Lagrange point (L3) of the Earth-Moon system provides such a benchmark, but in
principle the second set of astronauts can be elsewhere in the solar system).
As described above, each astronaut has some reaction interval $t_{choice}$
during which they can consider and respond to the prompt. There is no reason
to expect that both astronauts would submit their answers at precisely the
same time, so their decisions are cached locally up to time
$t_{transmit}(>t_{question}+t_{choice})$, at which point their decisions are
transmitted classically to unmanned spacecraft with measurement stations
“Alice” and “Bob”, and received at time $t_{basis}$. Completing the Bell test
requires entangled photon reception and measurement at the Alice and Bob
stations – at $t_{Bell}$ – before any possible signal emitted at
$t_{question}$ from the other side’s basis-choice astronaut station could
reach the analysis satellites. In order to ensure this, the time-of-flight
from the entangled photon source to either Alice or Bob (labeled in Figure 15
as $\Delta t$) must exceed $t_{transmit}-t_{question}$. 555For simplicity we
have assumed a symmetric arrangement, but clearly the source might be closer
to Alice or Bob; in this case $\Delta t$ is taken to correspond to the smaller
of the two distances. Assuming a human reaction time is $t_{choice}=0.25\,$s,
the cache time $t_{transmit}$ should be greater than this; here we assume 0.4
s. The corresponding distance between source and Alice (Bob) is about 1/3.25
the Earth-Moon distance.

Because the associated link efficiency of the whole experiment (see Appendix
Appendix A: DSQL System Design) goes as $1/R^{4}$ ($R$ being the source-to-
measurement station separation), this is $\sim$100 times more efficient than
the Earth-Moon human Bell test configuration (Fig. 14a), and 7 times more
efficient than the “midway” Earth-Moon Bell test variant (Fig. 14). This
margin may be traded for making the question and answer interface of the
astronauts more relaxed by, e.g., querying them once every two seconds instead
of once every 0.4 s. Even then, the rate improvement is 22 times higher than
the Earth-Moon test, and 1.5 times higher than the midway test. Furthermore,
the rate could be further enhanced by using multiple astronauts at each
station, e.g., a typical capsule crew of three could supply measurement
decisions 3 times as often.

Figure 15: Alternative scheme for human-decision Bell test. Artemis astronauts
on the Moon, and astronauts at the Earth-Moon L3 point or beyond, are queried
at time $t_{question}$. Both astronauts make a decision within the human
reaction time $t_{choice}$. This answer is locally cached until time
$t_{transmit}$, when a classical signal is sent to unmanned measurement
spacecraft “Alice” and “Bob”. A light source, located on or near to Earth,
emits entangled photons toward Alice and Bob before intersection of the world
line of the source and light cones of the query events, i.e., with no chance
that the emitted photons could have been influenced by the measurement basis
choices of the astronauts.

Quantum mechanics itself makes no distinction between basis choices determined
by classical randomness, quantum randomness, or human choices—all of these
should violate Bell’s Inequality equally. If this experiment is performed with
all of these sources of randomness and the Bell violation in each case is
indistinguishable, we would confirm that quantum mechanics holds, but we would
conclude nothing about free will. If we get the unexpected result, which is
that the runs using human choices do not match the quantum prediction, what
would we conclude? First, we would be forced to concede that quantum mechanics
is incorrect. Any local-realist explanation for all of the observations must
include a mechanism for predicting or influencing all non-human random number
generators used in previous experiments that confirmed quantum predictions in
Bell tests. Second, we would be able to say that human choices are at least
sufficiently complicated so as not to be predicted or influenced, though
exactly how one relates this back to concepts of free will would be up for
debate.

In the closing paragraph of Brans (1988), Brans writes “It is sometimes said
that quantum theory saves free will. In the context of this paper, this might
be reversed, so that free will saves quantum theory, at least in the sense of
eliminating hidden variable alternatives. In other words, if there are any
truly “free” events in the experiment, then there can be no classical
determinism and hence no classical hidden variables.” We wish only to caveat
this inspirational quote by noting that for an experiment to rule out a local
hidden variable theory, a single ‘free’ event would not make a significant
difference. Instead, a sufficient majority of the basis choices for each
measurement must be ‘free’ in the sense that they are not able to be predicted
or influenced by anything happening on the other side of the experiment.

####  II.2.5 Flight Mission Design for Bell Tests

All of the proposed Bell tests are characterized by the statistical
significance of the measured violation of Bell’s inequality. This is described
mathematically and conceptually in Appendix Appendix D: Obtaining the
Parametric Model for Bell Tests. Increasing the number of successful, high-
fidelity photon pairs simultaneously detected by Alice and Bob will improve
the statistical significance of the test. Practically, this experiment could
be realized through use of a very broadband entangled photon pair source
undergoing dense-wavelength-division-multiplexing, thereby creating many
simultaneous channels, each approaching the saturation capacity of the
detectors. Leveraging this source architecture requires exceptionally low
timing jitter. Reducing the probability of a noise event also improves the
Bell test statistics. Thus, number of succesful photon pairs measured during a
measurement campaign, $N$, and purity factor, $p$, are the parameters used to
parametrically describe the Bell test mission design. Equation 56, reproduced
here for convenience, relates the statistical significance of violation of
Bell’s inequality,$\sigma$, to parameters $N$ and $p$: 666The purity factor
$p$ used here is not to be confused with the “$p$-value” used in the previous
loophole-free Bell tests Hensen  _et al._ (2015); Giustina  _et al._ (2015) to
characterize the significance of the violation; loosely speaking, the
$p$-value is the likelihood the observed results are compatible with the null
hypothesis, i.e., that a local hidden variable model could explain the
results. We acknowledge that a more thorough mission design should use this
more sophistocated metric in the analysis, but do not believe the conclusions
will be substantially different than our simpler analysis using $\sigma$.

$\textrm{\\# of $\sigma$
violation}=\left<n\right>=\sqrt{N}\frac{p-\tfrac{1}{\sqrt{2}}}{\sqrt{2-p^{2}}}.$
(36)

The parameters $N$ and $p$ can be expanded into instrument performance
parameters, as discussed in Appendix Appendix A: DSQL System Design. Figure 16
represents the $\sigma$ violation significance achievable with a source of
clock rate $f_{clock}$, pair production probability $p(1)$, photon pair
fidelity $F$, total link efficiency (the product of efficiencies to Alice and
Bob) $\eta_{2e}$, for total integration time $T$, with a receiver with
temporal resolution $\Delta t_{R}$, and total background noise flux of
$N_{noise}$.

Figure 16: Parametric representation of general Bell test. The indicated